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[ [ "Multimodal Feature Fusion and Knowledge-Driven Learning via Experts\n Consult for Thyroid Nodule Classification" ], [ "Abstract Computer-aided diagnosis (CAD) is becoming a prominent approach to assist clinicians spanning across multiple fields.", "These automated systems take advantage of various computer vision (CV) procedures, as well as artificial intelligence (AI) techniques, to formulate a diagnosis of a given image, e.g., computed tomography and ultrasound.", "Advances in both areas (CV and AI) are enabling ever increasing performances of CAD systems, which can ultimately avoid performing invasive procedures such as fine-needle aspiration.", "In this study, a novel end-to-end knowledge-driven classification framework is presented.", "The system focuses on multimodal data generated by thyroid ultrasonography, and acts as a CAD system by providing a thyroid nodule classification into the benign and malignant categories.", "Specifically, the proposed system leverages cues provided by an ensemble of experts to guide the learning phase of a densely connected convolutional network (DenseNet).", "The ensemble is composed by various networks pretrained on ImageNet, including AlexNet, ResNet, VGG, and others.", "The previously computed multimodal feature parameters are used to create ultrasonography domain experts via transfer learning, decreasing, moreover, the number of samples required for training.", "To validate the proposed method, extensive experiments were performed, providing detailed performances for both the experts ensemble and the knowledge-driven DenseNet.", "As demonstrated by the results, the proposed system achieves relevant performances in terms of qualitative metrics for the thyroid nodule classification task, thus resulting in a great asset when formulating a diagnosis." ], [ "Introduction", "Thyroid nodules, described by an abnormal growth of the gland tissue, are a common disease affecting the thyroid gland [12].", "Ultrasonography is the most used modality to both detect and diagnose nodules.", "This method is safe, convenient, non-invasive, and has a better capability of distinguishing benign nodules from malignant ones, with respect to other techniques such as computed tomography (CT) and magnetic resonance imaging (MRI), facilitating early diagnosis and treatment choice [17].", "In order to take full advantage of ultrasound images (US), computer-aided diagnosis (CAD) is rapidly evolving, resulting in systems able to provide less subjective interpretations and, consequently, more precise diagnoses.", "A CAD system is generally developed following established phases including image preprocessing (e.g., noise removal, image reconstruction, etc.", "), region-of-interest (ROI) extraction, segmentation, and classification.", "Historically, many of the available works focus on the first three steps, while in the latest years the emphasis is being shifted towards thyroid nodule classification due to the evolution of machine learning approaches.", "A key aspect of all phases lies in the nodule representation, where techniques such as local patterns (e.g., LDP and LBP [25]), or wavelet transform (e.g., DWT [18]), can provide a detailed description of the gland itself.", "By applying these techniques, as well as a plethora of other computer vision approaches, it is ultimately becoming possible to detect and segment the thyroid inside a US image [3], [27], as well as classify nodules [13], thus making CAD a great asset for clinicians during their diagnoses.", "In the latest years, many interesting works using various machine learning approaches concerning the nodule classification task, were presented.", "In [22], a comparison between Bayesian techniques, Support Vector Machines, and Neural Networks, shows promising results for the latter, close to classic radiological methods.", "By using the EM algorithm to train a CNN, a network able to grasp correlations in an image through convolutions, the authors of [21] can further improve performances of an automatic system trained in a semi-supervised way.", "While convolutional networks are certainly powerful, representing the nodules in a meaningful way can give an edge on the diagnosis accuracy, especially when few samples are available.", "In [11], HOG and LBP features are succesfully combined with other high-level features, in order to compensate the lack of thyroid images.", "Another approach to solve the issue of small datasets, common problem for medical researches, is data augmentation.", "For this procedure, traditional methods include image cropping, rotation, and scaling, although even neural networks, such as GANs, can be used to increase the dataset size.", "Indeed, in [28] data augmentation is performed through CNNs, in order to achieve improved performances with respect to standard US.", "A different approach to handle small-sized datasets, is transfer learning.", "Through this method, a network trained on a different task is reused to reach convergence faster and more easily on the new domain.", "The authors of [15] exploit this technique in combination with a feature fusion procedure, where US images of thyroid glands are fused together with their respective elasticity maps, ultimately exceeding state-of-the-art performances.", "While much was done concerning the classification through neural networks and transfer learning, other approaches are also emerging based on either ensemble learning or domain knowledge transfer.", "In [9], by using an ensemble, the authors are able to exceed radiologists performances for thyroid nodule classification.", "Other works successfully applying and, thus, encouraging the use of the ensemble technique are [5] and [24].", "The former is able to perform white matter fiber clustering, while the latter can classify lung nodules from CT scans with high accuracy.", "Concerning the domain knowledge transfer, in [1], from which we have taken inspiration for this study, the authors effectively drive the learning using prior knowledge of a different network.", "In [10], the authors apply a similar knowledge-driven rationale to diagnose breast cancer by integrating domain knowledge during the training phase, ultimately showing that this approach can be used to formulate a medical diagnosis.", "In this study, due to the relatively small dataset at our disposal, we applied both data augmentation and feature fusion in order to fully exploit the available images.", "Moreover, we decided to explore knowledge-driven approaches using an ensemble of experts obtained via transfer learning.", "The ensemble guides a DenseNet during its training by providing consults, and both components finally collaborate to act as a CAD system.", "Experimental results suggest that, while transfer learning is a powerful technique to handle smaller datasets, by leveraging prior knowledge through the experts ensemble consult, it is possible to drive the learning phase, thus training a good performing classifier for thyroid nodule computer-aided diagnosis.", "Figure: Knowledge-driven learning (KDL) via experts consult (EC) framework architecture.", "The feature fusion between US, LBP and DWT images is given as input to the EC module and to the KDL-EC DenseNet.", "Their outputs are then concatenated and elaborated by dense layers to obtain a diagnosis.", "The EC module, through an ensemble of deep neural networks (DNN), can drive the KDL-EC unit learning." ], [ "Method", "The proposed knowledge-driven learning via experts consult framework, shown in Figure REF , can be divided into three components: a data augmentation and feature fusion phase, where detail-rich nodule images are generated; an expert consult (EC) module based on the ensemble stacking technique, where pre-trained deep neural networks are fine-tuned; and a knowledge-driven learning (KDL) unit, where cues are given to a convolutional network during its training.", "The first augmentation and fusion component is a mandatory choice for the proposed framework, due to the used dataset being relatively small.", "Data augmentation is performed computing LBP and DWT images for each US thyroid nodule.", "Feature fusion is then obtained by stacking the nodule US with its corresponding LBP and DWT images, along their channels axis.", "Since all images are grayscale, this procedure results in an object representing a nodule with shape $(w*h*3)$ , where $w$ and $h$ correspond to width and height of the image; while each channel is a different representation of the nodule.", "LBP and DWT representations were chosen since they provide further information about both inner and outer properties of the nodule itself, as shown in Figure REF .", "Finally, the feature fusion object is used as input for the other two components.", "Figure: Data augmented nodule example.", "In (a), the raw US image, while (b) and (c) show the same nodule analysed via LBP and DWT, respectively.In the second component, an ensemble stacking module is fine-tuned in order to later help training another network.", "The ensemble is composed by $n$ pre-trained deep neural networks, defined experts in this work, taking as input the described feature fusion object, which is normalized via a convolutional layer to handle the different representations.", "Each expert is first fine-tuned on the thyroid dataset and then used to build the ensemble, so that all members can operate simultaneously to formulate a diagnosis.", "To obtain a stacking ensemble, predictions of all $n$ experts are concatenated and re-elaborated through 3 dense layers, so that the opinion (i.e., prediction) of each expert is taken into account for the final diagnosis.", "During this re-elaboration phase, all the experts are left untouched.", "As experts, several good performing networks pre-trained on ImageNet [4] were used, namely: AlexNet [8], DenseNet [7], GoogleNet [20], ResNet [6], ResNeXt [23], and VGG [19].", "These pre-trained networks allow to apply the transfer learning technique, where previous knowledge is transferred and used on a new domain.", "Notice that while the chosen models are trained on non-medical images, some common characteristics, such as object contours, are still present in the new domain and can be effectively used on the new task, as already shown by the authors of [15].", "Through transfer learning, it is possible to reduce both time and number of samples required to fully train a network, thus making this technique ideal for the proposed ensemble which is based on hard-to-come-by medical images.", "Finally, once the ensemble stacking module is trained on the thyroid images, its diagnoses are used to provide a medical consult and drive the learning of the last module of the proposed framework.", "The third and last framework component, is the knowledge-driven learning (KDL) unit.", "In this module, a convolutional neural network is given cues to help it achieve better performances on its classification task.", "The convolutional network of choice for this study is a DenseNet that, due to dense connections between convolutions, is able to forward propagate relevant information, thus obtaining interesting results on diverse tasks analysing both medical [16] and non-medical images [2], [26].", "Similarly to the ensemble module, the KDL-EC unit receives as input a feature fusion object, normalized via a convolutional layer to handle the different representations, to fine-tune the ImageNet pre-trained DenseNet.", "The output is then concatenated to the ensemble stacking prediction and re-elaborated with three dense layers, ultimately obtaining a diagnosis prediction through a softmax function.", "Notice that the expert consult is effectively guiding the learning of the DenseNet because the loss function is computed after the re-elaboration, and back-propagated all the way through the DenseNet itself while the experts ensemble remains left untouched." ], [ "Experimental Results", "In this section, the private dataset and the hardware configuration used to test the proposed framework are first introduced, the experimental results for all the mentioned components are then presented." ], [ "Dataset and Hardware Configuration", "All the experimental results shown in this section are carried out on a private dataset of thyroid nodules, provided by the hospital Policlinico Umberto I of Rome.", "The dataset, collected from 230 distinct patients, is composed by 678 unmarked grayscale ultrasound images generated directly from the DICOM format, and cropped to a size of $440\\times 440$ so that the thyroid gland is retained.", "Moreover, each image has a TI-RADS classification associated, utilised to split the available samples into the benign and malignant categories.", "All images with a score $\\le 2$ are labelled as the former, while the remaining samples (i.e., with a score $\\ge 3$ ) are associated to the latter, thus defining a binary classification task.", "Examples of benign and malignant nodules, are shown in Figure REF .", "After this nodule-label association, the dataset was split into two sets, $D1$ and $D2$ , with non overlapping patients.", "$D1$ contains 452 samples, divided into 360 benign and 92 malignant cases while $D2$ comprises 226 images, partitioned into 180 benign and 46 malignant cases.", "This subdivision enables us to obtain unbiased results for both the EC and KLD-EC modules during their experimentation, since they are trained on different datasets.", "Concerning the hardware configuration, all tests are performed on the Google Cloud Platform (GCP), leveraging the pytorch framework and using a Virtual Machine with the following specifications: 4-Core Intel i7 2.60GHz CPU with 16GB of RAM, and a Tesla P100 GPU.", "Figure: Images from the utilised thyroid dataset.", "Benign examples are shown in (a) and (b), while malignant nodules are represented in (c) and (d)." ], [ "Results", "In order to fully assess the proposed KDL-EC framework, a preliminary grid-search is employed to evaluate various ImageNet pre-trained networks.", "The networks of choice are: AlexNet, DenseNet, GoogleNet, ResNet, ResNeXt, and VGG.", "Each network is available with pre-trained weights in the torchvision library of the pytorch framework.", "For each model, the best ImageNet performing version is selected, following the scores reported in [14].", "All networks were trained for 1000 epochs, using a learning rate of 0.001, a batch size of 32, and $[0.2, 1]$ as class weights to handle the discrepancy in the number of samples between the benign and malignant classes, respectively.", "Relevant results on the $D1$ dataset, comparing 10 cross-validation performances with non overlapping patients on raw US images, augmented (i.e., US, LBP, and DWT) and feature fusion datasets, are summarized in Table REF .", "As shown, using either the augmented or feature fusion datasets, results in consistently improved performances.", "The rationale behind this behaviour can be attributed to the extra information both the LBP and DWT can provide, in the augmented dataset, and the ability to directly correlate visual cues among the various representations, in the feature fusion dataset.", "Moreover, as also shown by the authors of [15], by maintaining previously computed weights via frozen layers, it is possible to achieve conspicuous performances boosts through the transfer learning and network fine-tuning techniques.", "Indeed, as shown in Table REF , the best results for all networks are obtained by using both the feature fusion dataset and by freezing either 25% or 50% layers of the corresponding ImageNet pre-trained network.", "Table: Average 10-cross validation models accuracies of baseline pre-trained models and various percentages of frozen layers.", "In each configuration block, accuracies on raw US, augemented (i.e., US, LBP and DWT) and feature fusion datasets (i.e., left, center, and right column of each block, respectively), are shown.Concerning the experts consult unit performances, the best performing networks are chosen to create an ensemble for the EC module according to the preliminary grid-search results.", "The EC members are selected in decreasing performances order.", "All tests are conducted on the feature fusion dataset $D1$ , using an EC module of size 3, 5, and 7, employing a non-overlapping 10 cross-validation approach.", "In the EC-7, two different implementations of a DenseNet (i.e., DenseNet169 and DenseNet201 from [14]) are used to build the ensemble.", "This decision is taken due to the DenseNet obtaining the best performance on the feature fusion dataset.", "In Table REF , common evaluation metrics for a baseline network and the three EC models, are summarized.", "As shown, increasing the number of experts, allows the module to obtain better overall performances, even though the single networks cannot perform as well as the EC module.", "The rationale behind this behaviour can be attributed to the ensemble stacking technique, where the outputs of the single networks (i.e., the experts) are re-elaborated in order to produce a better representation of the input and, consequently, a more accurate output.", "Notice that specificity scores are slightly lower than sensitivity ones, due to the dataset being skewed toward the benign class, even though class weights are utilised to represent the difference in the number of samples.", "Table: Average 10-cross validation performances for the ensemble consult module (EC) at different sizes.", "The baseline scores refer to the best performing network: a feature fusion fine-tuned DenseNet.", "Scores are computed on dataset D1D1.In relation to the KDL-EC unit performances, all experiments are carried out on dataset $D2$ , in order to obtain unbiased results.", "Similarly to the EC module, a non-overlapping 10 cross-validation approach is used to evaluate this component.", "Moreover, the KDL-EC unit is tested using three different experts consults composed by 3, 5, and 7 members.", "The ensemble networks are the same used for the EC module and remain frozen during the training phase of the KDL-EC unit.", "The base network of choice for the KDL-EC is a DenseNet, selected due to obtaining the best scores in the preliminary results.", "The DenseNet is trained for 1000 epochs, a learning rate of 0.001, a batch size of 32, and uses class weights set to $[0.2, 1]$ for benign and malignant samples, respectively.", "In Table REF , common evaluation metrics for a baseline network and the three KDL-EC models, are compared.", "As shown, adding cues based on previous knowledge during the learning phase (i.e., experts consult output), can drastically increase the performance of the DenseNet.", "Even more interesting, is the increase in the specificity score, which is related to the malignant samples.", "In this case, even though the dataset is skewed toward the benign class, the network is still able to increase its performance by leveraging the EC model output.", "Table: Average 10-cross validation performances for the knowledge-driven learning via experts consult module (KDL-EC) at different sizes.", "The baseline scores refer to the best performing EC module.", "Scores are computed on dataset D2D2.Finally, in Table REF , a comparison with other relevant works, is presented.", "Although each method is tested on a different dataset, the reported results still allow to assess the performances of the proposed framework.", "As shown, the KDL-EC framework, thanks to its feature fusion and knowledge-driven learning approach, is able to achieve significant performances.", "Table: State-of-the-art methods performances comparison." ], [ "Conclusion", "In this paper, a knowledge-driven learning via experts consult framework for thyroid nodule classification is presented.", "As shown, by leveraging previous knowledge obtained by an ensemble of experts (i.e., a consult), it is possible to guide a new network during its training phase, and ultimately obtain improved results with respect to both the base network as well as the ensemble itself.", "As future work, more images are going to be collected and possibly released, so that a common ground for other works can be established.", "Moreover, further experiments on the proposed knowledge-driven approach utilising different types of input (e.g., elasticity maps), as well as a complementary module handling the TI-RADS classification in an automatic fashion, will also be considered." ] ]

[ [ "Parameter Estimation for Subgrid-Scale Models Using Markov Chain Monte\n Carlo Approximate Bayesian Computation" ], [ "Abstract We use approximate Bayesian computation (ABC) combined with an \"improved\" Markov chain Monte Carlo (IMCMC) method to estimate posterior distributions of model parameters in subgrid-scale (SGS) closures for large eddy simulations (LES) of turbulent flows.", "The ABC-IMCMC approach avoids the need to directly compute a likelihood function during the parameter estimation, enabling a substantial speed-up and greater flexibility as compared to full Bayesian approaches.", "The method also naturally provides uncertainties in parameter estimates, avoiding the artificial certainty implied by many optimization methods for determining model parameters.", "In this study, we outline details of the present ABC-IMCMC approach, including the use of an adaptive proposal and a calibration step to accelerate the parameter estimation process.", "We demonstrate the approach by estimating parameters in two nonlinear SGS closures using reference data from direct numerical simulations of homogeneous isotropic turbulence.", "We show that the resulting parameter values give excellent agreement with reference probability density functions of the SGS stress and kinetic energy production rate in a priori tests, while also providing stable solutions in forward LES (i.e., a posteriori tests) for homogeneous isotropic turbulence.", "The ABC-IMCMC method is thus shown to be an effective and efficient approach for estimating unknown parameters, including their uncertainties, in SGS closure models for LES of turbulent flows." ], [ "Introduction", "Large eddy simulations (LES) have the potential to provide a nearly ideal blend of physical accuracy and computational cost for the simulation of turbulent flows, but the predictive power of such simulations depends on the accuracy of closure models for unresolved subgrid scale (SGS) fluxes.", "Attempts to develop such models from physics principles alone have thus far failed to yield a universally accurate SGS model.", "Consequently, the overwhelming majority of LES continues to be performed using classical [1] or dynamic [2] Smagorinsky models, or with artificial kinetic energy dissipation produced by low-order numerical schemes (i.e., implicit LES).", "Although such simple or purely numerical models are robust and provide stable LES solutions, they typically perform poorly in flows with complex physics (e.g., combustion) and in situations where the fundamental principles underlying LES break down (e.g., in regions near solid boundaries where there is no longer a clear separation between kinetic energy input and dissipation scales).", "At the same time, attempts to develop more sophisticated models are typically plagued by the presence of many unknown model parameters, which can be difficult to simultaneously calibrate across different flows.", "Traditionally, model parameters have been determined using either optimization techniques or direct inversion of model equations given some reference (e.g., experimental or higher-fidelity simulation) data.", "For example, successful attempts have been made to find the optimal Smagorinsky constant using an optimization technique [3], Kriging-based response surfaces [4], and neural networks [5], and direct inversion techniques have been used to estimate model parameters for both Reynolds averaged Navier Stokes (RANS) simulations [6] and for LES [7].", "However, Oberkampf, Trucano, and co-authors [8], [9], [10], [11] have advocated caution when using optimization approaches, noting that model parameter estimates should also include a quantification of uncertainty.", "This is especially true given the uncertain nature of essentially all reference data (even if only due to statistical non-convergence), as well as the approximate nature of SGS models (even for the most sophisticated models).", "Direct inversion approaches can also become challenging (although not impossible) for complex model forms or when the model itself consists of partial differential equations.", "Statistical methods, such as Bayesian approaches, provide an alternative path to model parameter calibration, giving a posterior probability distribution of unknown parameters.", "For example, [12], [13], [14], [15], [16], [17], and [18] have all applied Bayesian methods to Reynolds stress models for RANS simulations in order to estimate model parameters and uncertainties.", "Recently, [19] used a Bayesian approach to estimate a joint distribution for LES SGS model parameters.", "A benefit of the Bayesian statistical approach is that the posterior probability density also naturally provides uncertainties associated with each estimated parameter, in contrast to other inversion techniques that provide only single value estimates for unknown parameters.", "However, solving the full Bayesian problem requires knowledge of the likelihood function, which can be either difficult or costly to compute, or both.", "In many cases, this likelihood function is approximated using a Gaussian formulation, which is not valid in general for all model forms and all flows.", "In this paper, we outline the use of approximate Bayesian computation (ABC) and an “improved” Markov chain Monte Carlo (IMCMC) method to determine unknown model parameters and their uncertainties.", "Most significantly, the ABC method approximates the posterior distribution of parameters without using a likelihood function.", "The ABC method was introduced and first widely applied in population genetics [20], [21], [22] and molecular genetics [23].", "It has subsequently been used in other scientific areas such as astrophysics [24], [25], chemistry [26], epidemiology [27], [28] and ecology [29].", "More detailed recent reviews of the ABC approach are provided by [30], [31], [32], [33] and, most recently, [34].", "The ABC method has also previously been used in engineering contexts for the estimation of kinetic rate coefficients for chemical reaction mechanisms [35] and for the estimation of boundary conditions in complex thermo-fluid flows [36], [37], [38].", "Here, we are the first to take advantage of both ABC and IMCMC for discovering model parameter values and uncertainties in multi-parameter SGS closures.", "In the following, the inverse problem solved by the ABC-IMCMC approach is outlined in Section and details of the ABC-IMCMC approach, including the use of an adaptive proposal and a calibration step, are described in Sections and .", "Results are presented in Section , and conclusions and directions for future work are provided at the end." ], [ "Subgrid-Scale Modeling as an Inverse Problem", "Coarse-graining of the Navier-Stokes equations using a low-pass filter, denoted $\\widetilde{(\\cdot )}$ , at scale $\\Delta $ gives the LES equations for an incompressible flow, which are written as [39] $\\frac{\\partial \\widetilde{u}_i}{\\partial x_i} &= 0\\,,\\\\\\frac{\\partial \\widetilde{u}_i}{\\partial t} + \\widetilde{u}_j \\frac{\\partial \\widetilde{u}_i}{\\partial x_j} &= - \\frac{\\partial \\widetilde{p}}{\\partial x_i} + \\nu \\frac{\\partial ^2 \\widetilde{u}_i}{\\partial x_j \\partial x_j} - \\frac{\\partial \\tau _{ij}}{\\partial x_j}\\,,$ where $\\widetilde{u}_i$ is the resolved-scale velocity, $\\widetilde{p}$ is the resolved-scale pressure normalized by density, $\\nu $ is the kinematic viscosity, and $\\tau _{ij}$ is the unclosed SGS stress tensor given by $\\tau _{ij} = \\widetilde{u_i u_j} - \\widetilde{u}_i \\widetilde{u}_j\\,.$ The LES scale $\\Delta $ is often termed the grid scale since this is the finest scale represented when using the grid discretization as an implicit LES filter.", "More broadly, $\\Delta $ is the filter cut-off scale, whether that filter is applied explicitly or implicitly.", "The SGS stress $\\tau _{ij}$ prevents closure and, in order to solve Eq.", "(REF ), an appropriate relation for $\\tau _{ij}$ must be found in terms of resolved-scale quantities only.", "Closure of Eq.", "(REF ) can be achieved by modeling the deviatoric part of the stress tensor $\\sigma _{ij} = \\tau _{ij} - \\tau _{kk}(\\delta _{ij}/3)$ , which, it is assumed, can be approximated by an unknown, high dimensional, non-parametric functional $\\mathcal {F}_{ij}$ that takes as its arguments only quantities that can be expressed in terms of the resolved-scale strain rate, $\\widetilde{S}_{ij}$ , and rotation rate, $\\widetilde{R}_{ij}$ , tensors [40]; namely $\\sigma _{ij}(\\textbf {x},t)\\approx \\mathcal {F}_{ij}\\left[ \\widetilde{S}_{ij}(\\textbf {x}+\\textbf {x}^{\\prime },t-t^{\\prime }),\\widetilde{R}_{ij}(\\textbf {x}+\\textbf {x}^{\\prime },t-t^{\\prime }) \\right]\\,,$ for all $\\textbf {x}^{\\prime }$ and $t^{\\prime }\\ge 0$ , where $\\widetilde{S}_{ij}$ and $\\widetilde{R}_{ij}$ are given by $\\widetilde{S}_{ij} = \\frac{1}{2} \\left( \\frac{\\partial \\widetilde{u}_i }{\\partial x_j} + \\frac{\\partial \\widetilde{u}_j}{ \\partial x_i} \\right)\\,,\\quad \\widetilde{R}_{ij} = \\frac{1}{2} \\left( \\frac{\\partial \\widetilde{u}_i }{\\partial x_j} - \\frac{\\partial \\widetilde{u}_j}{ \\partial x_i} \\right)\\,.$ It should be noted that the closure relation in Eq.", "(REF ) allows SGS stresses at location $\\textbf {x}$ and time $t$ to depend on $\\widetilde{S}_{ij}$ and $\\widetilde{R}_{ij}$ , as well as their products, at any point in the flow and at any prior time.", "As a demonstration of the ABC approach for determining SGS model parameters, here we will use a single-point, single-time nonlinear model introduced by [40], which is given as $\\mathcal {F}_{ij}(\\mathbf {c}) = \\sum _{n=1}^{4} c_{n} \\widetilde{G}_{ij}^{(n)}\\,,$ where $c_{n}$ are non-dimensional coefficients (i.e., the unknown model “parameters”) that can depend on invariants of $\\widetilde{S}_{ij}$ and $\\widetilde{R}_{ij}$ , and $\\widetilde{G}_{ij}^{(n)}$ are tensor bases formed from products of $\\widetilde{S}_{ij}$ and $\\widetilde{R}_{ij}$ up to second order, $& \\widetilde{G}_{ij}^{(1)} = \\Delta ^2 |\\widetilde{S}| \\widetilde{S}_{ij}\\,, \\\\& \\widetilde{G}_{ij}^{(2)} = \\Delta ^2\\left(\\widetilde{S}_{ik}\\widetilde{R}_{kj} - \\widetilde{R}_{ik}\\widetilde{S}_{kj}\\right)\\,, \\\\& \\widetilde{G}_{ij}^{(3)} = \\Delta ^2\\left(\\widetilde{S}_{ik}\\widetilde{S}_{kj} - \\frac{1}{3}\\delta _{ij} \\widetilde{S}_{kl}\\widetilde{S}_{kl}\\right) \\,, \\\\& \\widetilde{G}_{ij}^{(4)} = \\Delta ^2\\left(\\widetilde{R}_{ik}\\widetilde{R}_{kj} - \\frac{1}{3}\\delta _{ij} \\widetilde{R}_{kl}\\widetilde{R}_{kl}\\right)\\,,$ where $|\\widetilde{S}| \\equiv \\left( 2\\widetilde{S}_{ij} \\widetilde{S}_{ij}\\right)^{1/2}$ .", "It should be noted that the original formulation by [40] extends to fifth order, but here the model is truncated for simplicity in order to demonstrate the ABC-IMCMC method.", "The first tensor basis corresponds to the classical Smagorinsky model, whereas the remaining bases represent second-order nonlinear corrections to this classical model.", "As outlined by [41], the non-parametric functional $\\mathcal {F}_{ij}$ in Eq.", "(REF ) can also be written as a Volterra series [42] or any other appropriate high-dimensional non-parametric functional.", "The exact non-parametric representation of $\\mathcal {F}_{ij}$ can thus vary and is not fundamental to the present demonstration of the ABC-IMCMC approach for model parameter estimation.", "Using data from experiments or direct numerical simulations (DNS), $\\sigma _{ij}(\\mathbf {x}, t)$ can be calculated and denoted as reference data $\\mathcal {D}_{ij}$ .", "Thus, we obtain the inverse problem $\\mathcal {F}_{ij}(\\mathbf {c}) = \\mathcal {D}_{ij}\\,,$ where the model parameters of $\\mathcal {F}_{ij}$ (i.e., $\\mathbf {c}$ ) must be determined through an appropriate inversion technique given the data $\\mathcal {D}_{ij}$ .", "Using direct inversion, [7] have demonstrated a nonlinear dynamic SGS model similar to that used here.", "Alternatively, the production rate field $\\mathcal {P}=\\sigma _{ij} \\widetilde{S}_{ij}$ can instead be used as the reference data $\\mathcal {D}_\\mathcal {P}$ , giving the inverse problem $\\mathcal {F}_{ij}(\\mathbf {c})\\widetilde{S}_{ij} = \\mathcal {D}_\\mathcal {P}\\,.$ Although optimization techniques (e.g., as in [41]) can be used to solve these inverse problems, the inversion process can be memory intensive and no uncertainty measures for the values of the unknown parameters in $\\mathbf {c}$ are naturally provided.", "Moreover, the formulation of the inverse problem can become difficult for complex forms of $\\mathcal {F}_{ij}$ , including those involving non-algebraic relations (e.g., for models incorporating transport equations of turbulence quantities).", "In the following, we will demonstrate the use of the ABC-IMCMC approach to solve the inverse problems above, and we will compare estimates of the parameters $\\mathbf {c}$ using reference data based on the stresses $\\sigma _{ij}$ alone, the production $\\mathcal {P}$ alone, and a combination of the stresses and production." ], [ "Approximate Bayesian computation", "Statistical inference is a powerful tool for solving inverse problems and Bayes' theorem, in particular, provides a posteriori probability densities of model parameters $\\mathbf {c}\\in \\mathbf {C}$ given reference data $\\mathcal {D}$ as $P(\\mathbf {c}|\\mathcal {D}) = \\dfrac{L(\\mathcal {D}\\,|\\, \\mathbf {c})\\pi (\\mathbf {c})}{\\int _{\\mathbf {C}}L(\\mathcal {D}\\,|\\, \\mathbf {c})\\pi (\\mathbf {c})d\\mathbf {c}}\\,,$ where $L(\\mathcal {D}\\,|\\, \\mathbf {c})$ is the likelihood function and $\\pi (\\mathbf {c})$ is the prior distribution of model parameters.", "The posterior, $P(\\mathbf {c}\\,|\\,\\mathcal {D})$ , provides the probability distribution of model parameters that satisfy the inverse problems outlined in the previous section, for given reference data.", "A benefit of the Bayesian statistical approach is that $P(\\mathbf {c}\\,|\\,\\mathcal {D})$ also naturally provides uncertainties associated with each estimated parameter, in contrast to other inversion techniques that provide only single-point estimates for unknown parameters.", "The practical application of Bayes' theorem is often complicated by the fact that explicit analytical expressions for the likelihood function $L(\\mathcal {D}\\,|\\, \\mathbf {c})$ are rarely available.", "However, when the model parameter space $\\mathbf {C}$ is finite and of low dimension, we can obtain the posterior density without an explicit likelihood function and without approximation using Algorithm REF , which was introduced by [43].", "This algorithm samples model parameters $\\mathbf {c}$ from a prior distribution $\\pi (\\mathbf {c})$ and compares model outcomes (i.e., model data) $\\mathcal {D}^{\\prime }$ with reference data $\\mathcal {D}$ , which may come from experiments or a higher fidelity model.", "The algorithm accepts parameters only if the modeled and reference data are exactly the same, and repeats until a total of $N$ model parameters have been accepted, from which the posterior joint probability density function (pdf) is then computed.", "[H] Bayesian rejection sampling algorithm [43] [1] $i =1$ to $N$ Sample $\\mathbf {c}_i$ from prior distribution $\\pi (\\mathbf {c})$ Calculate $\\mathcal {D}^{\\prime } = \\mathcal {F}(\\mathbf {c}_i)$ from model $\\mathcal {D}^{\\prime } = \\mathcal {D}$ Accept $\\mathbf {c}_i$ Using all accepted $\\mathbf {c}_i$ calculate posterior joint pdf For the determination of SGS model parameters, however, the parameter space $\\mathbf {C}$ is continuous and the model is imperfect, making Algorithm REF impossible to use in the form outlined above since it is highly unlikely that $\\mathcal {D}^{\\prime }$ and $\\mathcal {D}$ will ever be exactly the same.", "As an alternative, the acceptance criterion in Algorithm REF can be relaxed to $d(\\mathcal {D}^{\\prime }, \\mathcal {D}) \\le \\varepsilon \\,,$ where $d$ is a distance function measuring the discrepancy between $\\mathcal {D}^{\\prime }$ and $\\mathcal {D}$ .", "That is, a sampled parameter value $\\mathbf {c}$ is accepted as part of the posterior distribution if the corresponding distance $d$ is within a specified tolerance $\\varepsilon $ .", "The distance function may be a Kullback-Leibler divergence, Hellinger distance, or simply a mean-square error.", "The central idea of ABC is that, if the distance between modeled and reference data, $d(\\mathcal {D}^{\\prime }, \\mathcal {D})$ , is sufficiently small, then the parameter $\\mathbf {c}$ is assumed to have been sampled from the posterior $P(\\mathbf {c}\\,|\\,\\mathcal {D})$ .", "In order to reduce the dimensionality of the data and, hence, the computational expense of ABC, it is common to replace the full reference data $\\mathcal {D}$ with summary statistics $\\mathcal {S}(\\mathcal {D})$ , such as the mean, standard deviation, or pdf of $\\mathcal {D}$ .", "The choice of summary statistic depends heavily on the problem and requires domain knowledge.", "However, a central assumption of ABC is that, if we use an appropriate summary statistic that depends on the choice of $\\mathbf {c}$ , the posteriors based on $\\mathcal {S}$ and $\\mathcal {D}$ will be equivalent, such that $P(\\mathbf {c}\\,|\\,\\mathcal {S}) = P(\\mathbf {c}\\,|\\,\\mathcal {D})$ .", "Returning to the acceptance criterion in Eq.", "(REF ) in the context of summary statistics, the distance between model and reference data can be replaced by a corresponding statistical distance for the modeled and reference summary statistics, denoted $\\mathcal {S}^{\\prime }$ and $\\mathcal {S}$ , respectively.", "By applying the statistical distance function, $d(\\mathcal {S}^{\\prime }, \\mathcal {S})$ , introducing the acceptance threshold $\\varepsilon $ , and using a summary statistic instead of the full data, we finally obtain the ABC rejection sampling method given by Algorithm REF .", "[H] ABC rejection sampling algorithm [1] Calculate reference summary statistic $\\mathcal {S}$ from $\\mathcal {D}$ Sample $N$ sets of parameters $\\mathbf {c}_i$ from prior distribution $\\pi (\\mathbf {c})$ each $\\mathbf {c}_i$ Calculate $\\mathcal {D}^{\\prime } = \\mathcal {F}(\\mathbf {c}_i)$ from model Calculate model summary statistic $\\mathcal {S}^{\\prime }$ from $\\mathcal {D}^{\\prime }$ Calculate statistical distance $d(\\mathcal {S}^{\\prime }, \\mathcal {S})$ $d(\\mathcal {S}^{\\prime }, \\mathcal {S})\\le \\varepsilon $ Accept $\\mathbf {c}_i$ Using all accepted $\\mathbf {c}_i$ calculate posterior joint pdf Although ABC is based on Bayes' theorem, it should be noted that, instead of determining the true posterior, ABC provides an approximation to the posterior distribution using a distance function and summary statistics [44].", "For appropriate summary statistics and as $\\varepsilon \\rightarrow 0$ , the central assumption of ABC is that $P(\\mathbf {c}\\,|\\,\\mathcal {D}) = \\lim _{\\varepsilon \\rightarrow 0} P\\left[\\mathbf {c}\\,|\\, d(\\mathcal {S}^{\\prime }, \\mathcal {S}) \\le \\varepsilon \\right]\\,.$ However, in practice, too small of an $\\varepsilon $ is computationally impractical because it leads to too many rejections of sampled parameters $\\mathbf {c}_i$ and an insufficiently converged posterior distribution.", "Conversely, relaxing the acceptance criterion (i.e., using a larger $\\varepsilon $ ) and not selecting appropriate summary statistics can lead to bias in the final posterior distribution.", "The primary advantages of the ABC approach are the low cost relative to full Bayesian methods and the flexibility in parameter estimation for complex models.", "In particular, ABC does not require a likelihood function and enables parameter estimation for simulation-based models that can contain un-observable random quantities.", "A single set of parameters can be selected as the maximum a posteriori probability (MAP) estimate, a mean value, or another characteristic statistic of the posterior distribution.", "Uncertainty can be quantified using confidence intervals or some other measure of the width of the posterior, including Monte Carlo sampling of parameter values from the posterior." ], [ "ABC with Markov chain Monte Carlo sampling", "Although Algorithm REF is straightforward to implement, it can be computationally expensive and inefficient.", "The number of accepted parameters that form the posterior distribution are only a small fraction of the total number of sampled parameters, and the region (in parameter space) of accepted parameters rapidly shrinks as the number of parameters in the model increases.", "Thus, most of the sampled parameters and evaluations of summary statistics do not contribute to the posterior.", "However, this problem can be mitigated, and the sampling technique can be significantly improved, by using Markov chain Monte Carlo (MCMC) methods.", "The MCMC without likelihood method (or ABC-MCMC method) introduced by [21], is based on the Metropolis-Hastings algorithm.", "For an accepted parameter $\\mathbf {c}_i$ , the Metropolis-Hastings algorithm samples the next candidate parameter using the proposal $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })$ .", "If $d(\\mathcal {S}^{\\prime }, \\mathcal {S})\\le \\varepsilon $ , then the proposed parameters are accepted with probability $h = \\min \\left[1, \\frac{\\pi (\\mathbf {c}^{\\prime })q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })}{\\pi (\\mathbf {c_i})q(\\mathbf {c^{\\prime }}\\rightarrow \\mathbf {c}_i)}\\right]\\,.$ Using the detailed balance condition, [21] demonstrated that this method yields a stationary posterior distribution, $P\\left[\\mathbf {c}\\,|\\, d(\\mathcal {S}^{\\prime }, \\mathcal {S}) \\le \\varepsilon \\right]$ .", "An outline of the ABC-MCMC method is provided in Algorithm REF .", "[H] ABC-MCMC algorithm [1] Calculate reference summary statistic $\\mathcal {S}$ from $\\mathcal {D}$ Define proposal covariance matrix $C$ Start with initial accepted parameters $\\mathbf {c}_0$ $i:=0$ $i < N$ Sample $\\mathbf {c}^{\\prime }$ from proposal $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })=q(\\mathbf {c^{\\prime }}\\,|\\,\\mathbf {c}_i, C)$ Calculate $\\mathcal {D}^{\\prime } = \\mathcal {F}(\\mathbf {c}^{\\prime })$ from model Calculate model summary statistic $\\mathcal {S}^{\\prime }$ from $\\mathcal {D}^{\\prime }$ Calculate statistical distance $d(\\mathcal {S}^{\\prime }, \\mathcal {S})$ $d(\\mathcal {S}^{\\prime }, \\mathcal {S})\\le \\varepsilon $ Accept $\\mathbf {c}^{\\prime }$ with probability $h = \\min \\left[1, \\frac{\\pi (\\mathbf {c}^{\\prime })q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })}{\\pi (\\mathbf {c}_i)q(\\mathbf {c}^{\\prime }\\rightarrow \\mathbf {c}_i)}\\right]$ Accepted Increment $i$ Set $\\mathbf {c}_i = \\mathbf {c}^{\\prime }$ Using all accepted $\\mathbf {c}_i$ calculate posterior joint pdf In the present demonstration of ABC-MCMC for SGS model parameter estimation, we choose the proposal distribution to be Gaussian with the current parameter $\\mathbf {c}_i$ as the mean value and the covariance matrix $C$ as the width and orientation (in parameter space); i.e., $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime }) = q(\\mathbf {c^{\\prime }}\\,|\\,\\mathbf {c}_i, C)$ .", "For a Gaussian proposal, $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime }) = q(\\mathbf {c^{\\prime }}\\rightarrow \\mathbf {c}_i)$ and $h$ depends only on the prior.", "If the prior is uniform, then $\\pi (\\mathbf {c}_i) = \\pi (\\mathbf {c}^{\\prime })$ , and the Gaussian proposal leads to $h = 1$ for any $\\mathbf {c}^{\\prime }$ .", "As a result, the algorithm reduces to a rejection method with correlated outputs [21].", "Here, we update the covariance $C$ as the ABC-MCMC rejection sampling algorithm proceeds, leading to an adaptive proposal." ], [ "Adaptive proposal", "The choice of proposal distribution $q$ has a significant impact on the rate of convergence of the Metropolis-Hastings algorithm.", "For many problems, a variable proposal with an adaptive size and orientation (in parameter space) provides faster convergence, and we thus implement an adaptive proposal in the ABC-MCMC approach.", "Here we follow the adaptive proposal procedure outlined by [45], where the covariance $C_i$ after the $i$ th accepted parameters $\\mathbf {c}_i$ in the Gaussian proposal $q(\\mathbf {c}^{\\prime }\\,|\\,\\mathbf {c}_i,C_i)$ is updated during the process using all previous steps of the chain as $C_i ={\\left\\lbrace \\begin{array}{ll}s_n\\mathrm {var}I, & \\mbox{if } i < k\\\\s_n\\mathrm {cov}(\\mathbf {c}_0, \\dots , \\mathbf {c}_i), & \\mbox{if } i\\ge k\\end{array}\\right.", "},$ where $k>0$ is the length of the initial period without adaptation and $s_n$ is a constant that depends on the dimensionality of the parameter space, $n$ , as $s_n = (2.4)^2/n$ .", "This algorithm does not require substantial additional computational cost, since the covariance $\\mathrm {cov}_{i}\\equiv \\mathrm {cov}(\\mathbf {c}_0, \\dots , \\mathbf {c}_i)$ can be calculated using the recursive formula $\\mathrm {cov}_{i} = \\left(\\frac{i-1}{i}\\right)\\mathrm {cov}_{i-1} + \\left(\\frac{1}{i+1}\\right)\\left(\\mu _{i-1} - \\mathbf {c}_i \\right)\\left(\\mu _{i-1} - \\mathbf {c}_i \\right)^T\\,,$ where $\\mu _i = [1/(i+1)]\\sum _{k=0}^i \\mathbf {c}_k$ is the average of all previous accepted parameter values, which also satisfies the recursive formula $\\mu _i = \\mu _{i-1} - \\left(\\frac{i}{i + 1}\\right) \\left(\\mu _{i-1} - \\mathbf {c}_i \\right)\\,.$ Although the adaptive Metropolis algorithm is no longer Markovian, [45] have shown that it has the correct ergodic properties and the accuracy of the algorithm is close to the accuracy of the standard Metropolis-Hastings algorithm given a properly chosen proposal." ], [ "Calibration step", "The main advantage of the ABC-MCMC method is the high rate of acceptance, since we start from an accepted parameter and stay in the acceptance region.", "However, for parameter spaces with large dimensions $n$ , the acceptance region becomes quite small relative to the total size of the parameter space.", "As a result, determining the initial accepted parameters $\\mathbf {c}_0$ can itself require many iterations.", "The other primary challenge in implementing ABC-MCMC is the same as for ABC rejection approaches more generally; namely, the fixed acceptance threshold $\\varepsilon $ must be defined a priori, and indeed before the entire Markov chain is run.", "The choice of $\\varepsilon $ is important, since too large a tolerance results in a chain that is dominated by the prior.", "On the other hand, too small a tolerance leads to a very small acceptance rate and also increases the initialization cost.", "[H] ABC-IMCMC algorithm with an initial calibration step and an adaptive proposal [1] Calculate reference summary statistic $\\mathcal {S}$ from $\\mathcal {D}$ Calibration step$N_\\mathrm {c}$ , $r$ Sample $N_\\mathrm {c}$ parameters $\\mathbf {c}_i$ from prior distribution $\\pi (\\mathbf {c})$ each $\\mathbf {c}_i$ Calculate $\\mathcal {D}^{\\prime } = \\mathcal {F}(\\mathbf {c}_i)$ from model Calculate model summary statistic $\\mathcal {S}^{\\prime }$ from $\\mathcal {D}^{\\prime }$ Calculate statistical distance $d_i(\\mathcal {S}^{\\prime }, \\mathcal {S})$ Using all $d_i(\\mathcal {S}^{\\prime }, \\mathcal {S})$ calculate distribution $P(d)$ Define tolerance $\\varepsilon $ such that $P(d\\le \\varepsilon )=r$ Randomly choose $\\mathbf {c}_0$ from $\\mathbf {c}_i$ parameters with $d\\le \\varepsilon $ Adjust prior based on variance of parameters with $d\\le \\varepsilon $ Calculate covariance $C_0$ from parameters with $d\\le \\varepsilon $ MCMC without likelihood$\\mathbf {c}_0$ , $\\varepsilon $ , $C_0$ , $k$ , $N$ Start with accepted parameters $\\mathbf {c}_0$ and covariance $C_0$ $i:=0$ $i < N$ Sample $\\mathbf {c}^{\\prime }$ from proposal $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })=q(\\mathbf {c^{\\prime }}\\,|\\,\\mathbf {c}_i, C_i)$ Calculate $\\mathcal {D}^{\\prime } = \\mathcal {F}(\\mathbf {c}^{\\prime })$ from model Calculate model summary statistic $\\mathcal {S}^{\\prime }$ from $\\mathcal {D}^{\\prime }$ Calculate statistical distance $d(\\mathcal {S}^{\\prime }, \\mathcal {S})$ $d(\\mathcal {S}^{\\prime }, \\mathcal {S})\\le \\varepsilon $ Accept $\\mathbf {c}^{\\prime }$ with probability $h = \\min \\left[1, \\frac{\\pi (\\mathbf {c}^{\\prime })q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })}{\\pi (\\mathbf {c}_i)q(\\mathbf {c}^{\\prime }\\rightarrow \\mathbf {c}_i)}\\right]$ Accepted Increment $i$ Set $\\mathbf {c}_i = \\mathbf {c}^{\\prime }$ Update proposal covariance $C_i$ as $C_i ={\\left\\lbrace \\begin{array}{ll}s_n\\mathrm {var}I, & \\mbox{if } i < k\\\\s_n\\mathrm {cov}(\\mathbf {c}_0, \\dots , \\mathbf {c}_i), & \\mbox{if } i\\ge k\\end{array}\\right.", "}$ Using all accepted $\\mathbf {c}_i$ calculate posterior joint pdf Proposal parameters such as the initial variance of the proposal $q$ must also be defined a priori.", "The fraction of accepted parameters in ABC-MCMC depends on the variance in the proposal, such that a smaller variance leads to a larger number of accepted parameters, but more iterations are required to obtain a statistically converged posterior distribution.", "To address these issues, [22] suggested an “improved” version of the MCMC algorithm, denoted IMCMC, that has an initial calibration step.", "In this step, a series of $N_\\mathrm {c}$ simulations are performed, where the parameters are uniformly sampled from their prior $\\pi (\\mathbf {c})$ to obtain an approximate probability distribution, $P(d)$ , of the distances $d(\\mathcal {S}^\\prime ,\\mathcal {S})$ over the entire parameter space.", "Using this calibration step, we can define a threshold distance $\\varepsilon $ such that $P(d\\le \\varepsilon )=r$ , where $r$ is a desired ratio of accepted simulation parameters.", "We can then use any simulation for which $d \\le \\varepsilon $ as a starting point for the Markov chain.", "These $N_\\mathrm {c}$ calibration simulations are also used to adjust the transition kernel $q(\\mathbf {c}_i\\rightarrow \\mathbf {c}^{\\prime })$ .", "In our case, the initial $q$ is a Gaussian distribution with standard deviation equal to the standard deviation of the retained parameters from the calibration step.", "The resulting Algorithm REF is based on IMCMC with the calibration step from [22], and is the complete form of ABC-IMCMC used here to solve the SGS model parameter estimation inverse problem.", "We demonstrate the use of ABC-IMCMC to determine the parameters $\\mathbf {c}$ in Eq.", "(REF ) for three- and four-term versions of the nonlinear SGS model (corresponding to parameter space dimensions $n=3$ and 4, respectively).", "For both of these models, we perform ABC-IMCMC parameter estimation for distance functions based only on summary statistics of $\\sigma _{ij}$ , based only on summary statistics of $\\mathcal {P}$ , and based on a combination of summary statistics of $\\sigma _{ij}$ and $\\mathcal {P}$ .", "As such, we will estimate three sets of parameters $\\mathbf {c}$ for both the three- and four-parameter models, for a total of six different tests of ABC-IMCMC.", "The corresponding joint posterior distributions obtained in these tests are denoted $P^{(n)}_{\\mathcal {D}}(\\mathbf {c})$ , where $n$ denotes the number of parameters in the model (either 3 or 4), and $\\mathcal {D}$ denotes the underlying reference data used in the ABC-IMCMC parameter estimation (i.e., $\\sigma $ for SGS stress tensor reference data, $\\mathcal {P}$ for SGS production reference data, or $\\sigma \\mathcal {P}$ for combined SGS stress and production reference data).", "In the following, we describe the specific setup of the ABC-IMCMC procedure (summarized in general form in Algorithm REF ) used for these tests, including the choices of reference data, summary statistics, distance functions, and priors.", "It is emphasized, however, that the ABC-IMCMC procedure outlined in the previous section is completely general and can be applied to different model forms using different configurations (e.g., different reference data, summary statistics, distance functions, and priors).", "As such, the current ABC-IMCMC configuration for SGS model parameter estimation should be taken as demonstrative of the power and limitations of the approach, rather than as a comprehensive description of the ideal formulation of ABC-IMCMC, either in terms of computational efficiency or accuracy.", "In particular, through the use of different types of reference data, we seek to demonstrate the impact of the choice of reference data on the estimated parameter distributions.", "Similarly, through the use of two different model forms, we show the inherent limitations imposed by the physical accuracy of the model; namely, an imperfect model cannot be made to perfectly agree with reference data simply by calibrating a finite number of model parameters." ], [ "Reference and model data", "The reference data are obtained from the Johns Hopkins Turbulence Database [46] by sampling every fourth point in a triply periodic $1024^3$ cube of DNS data for homogeneous isotropic turbulence (HIT) at $Re_\\lambda = 433$ .", "The resulting $256^3$ cube of sub-sampled data is then filtered at wavenumber $k_{\\Delta } = 30$ using a spectrally sharp filter to obtain fields of the resolved-scale velocity vector $\\widetilde{u}_i$ and the second-order velocity product tensor $\\widetilde{u_i u_j}$ .", "From these fields, we obtain the reference data for the present tests, given as $\\mathcal {D}_{ij}=\\sigma _{ij}$ for tests using the deviatoric SGS stress tensor as reference data and $\\mathcal {D}_\\mathcal {P}=\\sigma _{ij} \\widetilde{S}_{ij}$ for tests using the SGS production as reference data.", "Two-dimensional fields of reference data $\\sigma _{11}$ , $\\sigma _{12}$ , $\\sigma _{13}$ , and $\\mathcal {P}$ from the DNS are shown in Figure REF .", "The corresponding model data are obtained as $\\mathcal {D}^{\\prime }_{ij}=\\mathcal {F}_{ij}(\\mathbf {c})$ and $\\mathcal {D}^{\\prime }_\\mathcal {P}=\\mathcal {F}_{ij}(\\mathbf {c})\\widetilde{S}_{ij}$ , where $\\mathcal {F}_{ij}(\\mathbf {c})$ represents the model approximation for $\\sigma _{ij}$ for a given choice of $\\mathbf {c}$ from either the three ($n=3$ ) or four ($n=4$ ) parameter SGS stress closures given by Eq.", "(REF ).", "The resolved scale strain and rotation rate tensors, $\\widetilde{S}_{ij}$ and $\\widetilde{R}_{ij}$ , respectively, required for the calculation of $\\mathcal {F}_{ij}(\\mathbf {c})$ are obtained from the DNS data." ], [ "Summary statistics", "All summary statistics in the present tests are based on pdfs of the reference and model data.", "For the deviatoric stresses, the summary statistics (i.e., pdfs) are denoted $\\mathcal {S}_{ij}$ and $\\mathcal {S}^{\\prime }_{ij}$ for the reference ($\\mathcal {D}_{ij}$ ) and model ($\\mathcal {D}^{\\prime }_{ij}$ ) data, respectively, where there are separate pdfs for each $(i,j)$ component of the stress tensor.", "We also consider pdfs of the production $\\mathcal {P}$ , denoted $\\mathcal {S}_\\mathcal {P}$ and $\\mathcal {S}^{\\prime }_\\mathcal {P}$ for $\\mathcal {D}_\\mathcal {P}$ and $\\mathcal {D}^{\\prime }_\\mathcal {P}$ , respectively.", "The flexibility of ABC-IMCMC also allows us to combine summary statistics, and in the following we will present parameter estimation results using both the stress and production summary statistics simultaneously.", "The reference data summary statistics for $\\mathcal {S}_{11}$ , $\\mathcal {S}_{12}$ , $\\mathcal {S}_{13}$ , and $\\mathcal {S}_\\mathcal {P}$ are shown in Figure REF .", "It should be noted that, since we do not expect either of the SGS closures examined here to exactly represent the flow physics, a degree of uncertainty must be present in the model summary statistics $\\mathcal {S}^{\\prime }_{ij}$ and $\\mathcal {S}^{\\prime }_\\mathcal {P}$ to avoid over-fitting.", "To introduce this additional uncertainty, we randomly choose $10^5$ data points out of the filtered $256^3$ cube of DNS reference data for each ABC-IMCMC iteration to calculate $\\mathcal {S}^{\\prime }_{ij}$ and $\\mathcal {S}^{\\prime }_\\mathcal {P}$ .", "This approach has the dual benefit of making our model stochastic while also reducing the total number of computations in the ABC-IMCMC procedure.", "Figure: Summary statistics from the HIT DNS reference data used in the present ABC-IMCMC demonstration, showing pdfs of (a) 𝒟 11 =σ 11 \\mathcal {D}_{11}=\\sigma _{11}, (b) 𝒟 12 =σ 12 \\mathcal {D}_{12}=\\sigma _{12}, (c) 𝒟 13 =σ 13 \\mathcal {D}_{13}=\\sigma _{13}, and (d) 𝒟 𝒫 =σ ij S ˜ ij \\mathcal {D}_\\mathcal {P} = \\sigma _{ij} \\widetilde{S}_{ij}.", "These summary statistics are denoted 𝒮 11 \\mathcal {S}_{11}, 𝒮 12 \\mathcal {S}_{12}, 𝒮 13 \\mathcal {S}_{13}, and 𝒮 𝒫 \\mathcal {S}_\\mathcal {P}, respectively." ], [ "Statistical distances and acceptance criteria", "The statistical distances between the modeled and reference summary statistics can be calculated many different ways, and here we use the Mean Square Error (MSE) of the logarithms of the pdfs.", "We use the logarithms of the pdfs in the present tests to emphasize the importance of the pdf “tails”, which correspond to lower probabilities of extreme values, as shown in Figure REF .", "For the SGS stress summary statistics, we calculate the distance, denoted $d_\\sigma (\\mathcal {S}_{ij}^{\\prime }, \\mathcal {S}_{ij})$ , as $d_\\sigma (\\mathcal {S}_{ij}^{\\prime }, \\mathcal {S}_{ij}) = \\sum _{i,j\\,|\\, j\\ge i}\\overline{\\left(\\ln \\mathcal {S}_{ij}^{\\prime } - \\ln \\mathcal {S}_{ij}\\right)^2}\\,,$ where the summation is over all $i$ and $j$ such that $j\\ge i$ , giving six independent terms in the summation; this summation approach is necessary since the stress tensors are symmetric and the terms with $i\\ne j$ should not be double-counted in the combined distance metric.", "The average $\\overline{(\\cdot )}$ is performed over all bins in the pdfs.", "Similarly, for the SGS production summary statistics, we calculate the distance $d_\\mathcal {P}(\\mathcal {S}_\\mathcal {P}^{\\prime }, \\mathcal {S}_\\mathcal {P})$ as $d_\\mathcal {P}(\\mathcal {S}_\\mathcal {P}^{\\prime }, \\mathcal {S}_\\mathcal {P}) = \\overline{\\left(\\ln \\mathcal {S}_\\mathcal {P}^{\\prime } - \\ln \\mathcal {S}_\\mathcal {P}\\right)^2}\\,.$ For both the stress- and production-based distance functions, the acceptance criteria are simply $d_{\\sigma }\\le \\varepsilon $ and $d_\\mathcal {P}\\le \\varepsilon $ , respectively.", "The specific values of $\\varepsilon $ are determined during an initial calibration step, as outlined in the next section.", "Finally, when using a combined stress and production acceptance criterion, we define a new distance function $d_{\\sigma \\mathcal {P}}$ as $d_{\\sigma \\mathcal {P}}=\\alpha d_\\sigma + \\beta d_\\mathcal {P}\\,,$ where $\\alpha $ and $\\beta $ are weighting coefficients that can be changed based on the desired importance of different summary statistic components.", "For the present ABC-IMCMC demonstration, we use $\\alpha =1$ and $\\beta =3$ , although different values can be used to emphasize greater model correspondence to either the $\\sigma _{ij}$ or $\\mathcal {P}$ pdfs.", "Once again, the acceptance criterion is $d_{\\sigma \\mathcal {P}}\\le \\varepsilon $ , where $\\varepsilon $ is determined during the initial calibration step." ], [ "Calibration step and priors", "As described in Section , we perform an initial calibration step to choose different appropriate values of $\\varepsilon $ for each of the three types of distance function considered here.", "We also use this step to determine the initial choices of parameter $\\mathbf {c}_0$ and proposal covariance $C_0$ .", "For all choices of distance function, we used $N_\\mathrm {c}=10^3$ and $10^4$ samples of $\\mathbf {c}$ , taken from uniform priors, in the calibration step for the three- and four-parameter models, respectively.", "For the calibration based on $d_\\sigma $ , the uniform priors were bounded by $c_1 \\in [-0.3, 0.0]$ , $c_2\\in [-0.5, 0.5]$ , $c_3\\in [-0.2, 0.2]$ , and $c_4\\in [-0.2, 0.2]$ , where the $c_4$ dimension is not used for the three-parameter model tests.", "For the calibrations based on $d_\\mathcal {P}$ and $d_{\\sigma \\mathcal {P}}$ , the priors were bounded by $c_1 \\in [-0.3, 0.0]$ , $c_2\\in [-0.5, 0.5]$ , $c_3\\in [-0.5, 0.2]$ and $c_4\\in [-0.2, 0.5]$ , where again the $c_4$ dimension was only used for the four-parameter model tests.", "Figure: Cumulative distribution functions (cdfs) of the distances (a) d σ d_\\sigma , (b) d 𝒫 d_\\mathcal {P}, and (c) d σ𝒫 d_{\\sigma \\mathcal {P}} for the three- (n=3n=3) and four- (n=4n=4) parameter SGS models from Eq.", "() (denoted by blue and red lines, respectively).", "Vertical dashed lines show values of ε 𝒟 (n) \\varepsilon _\\mathcal {D}^{(n)} for each distance function and model type (blue and red dashed lines for the three- and four-parameter models, respectively).", "Values of ε 𝒟 (n) \\varepsilon _\\mathcal {D}^{(n)} were chosen based on an acceptance ratio of r=10%r=10\\% for the three-parameter model and r=3%r=3\\% for the four-parameter model.From the calibration steps, $\\varepsilon $ was chosen for each distance to correspond to an acceptance percentage of $r=10\\%$ for the three-parameter model and $r=3\\%$ for the four-parameter model.", "The cumulative distribution functions for each of the calibration steps are shown in Figure REF , with the values of $\\varepsilon $ resulting from these tests indicated by dashed lines.", "There are six resulting values of $\\varepsilon $ , denoted $\\varepsilon _\\mathcal {D}^{(n)}$ , corresponding to each of the six posterior distributions $P_\\mathcal {D}^{(n)}(\\mathbf {c})$ that will be computed in the present demonstration of ABC-IMCMC." ], [ "Results", "Here we apply ABC-IMCMC to estimate posterior distributions of unknown parameters for the three- and four-parameter versions of the SGS stress model given by Eq.", "(REF ).", "The posteriors are obtained separately for each model using three different types of model and reference data: (i) SGS stresses $\\sigma _{ij}$ , giving the posterior $P^{(n)}_{\\sigma }(\\mathbf {c})$ , where $n$ is the number of parameters in the model; (ii) SGS production $\\mathcal {P}$ , giving $P^{(n)}_{\\mathcal {P}}(\\mathbf {c})$ ; and (iii) a combination of $\\sigma _{ij}$ and $\\mathcal {P}$ , giving $P^{(n)}_{\\sigma \\mathcal {P}}(\\mathbf {c})$ .", "The detailed configuration for each of these tests is described in the previous section.", "In the following, we first present the posterior distributions resulting from ABC-IMCMC for both three- and four-parameter models and all three types of reference data.", "From the posteriors, we obtain the MAP values of the model parameters, which are then used in a priori tests to show that true summary statistics can be recovered by the models using the estimated parameters.", "We also present results from a posteriori tests showing that models using the estimated parameters can be stably integrated in forward LES runs.", "Uncertainties in the estimated parameters are naturally given by the posterior distributions resulting from the ABC-IMCMC approach, and we estimate the impact of these uncertainties in the a posteriori tests by performing the LES for many different parameter samples from the posteriors, and then weighting the results by the posterior value for that sample." ], [ "Model parameter posterior distributions", "Figures REF and REF show posterior distributions obtained from ABC-IMCMC for, respectively, three- and four-parameter SGS closure models.", "After the calibration step described in the previous section, these posteriors were generated by running parallel Markov chains with a total of roughly $N = 10^5$ and $10^7$ accepted parameters for the three- and four-parameter models, respectively.", "There were 6 parallel chains with acceptance rates between 48% and 62% for the three-parameter model and 24 parallel chains with acceptance rates between 31% and 52% for the four-parameter model.", "In order to visualize the resulting three- and four-dimensional posteriors, in Figures REF and REF we show one-dimensional marginal pdfs for each parameter on the diagonal subplots for each of the three types of statistical distances.", "These marginal distributions are denoted here $M^{(n)}_\\mathcal {D}(c_i)$ , where $c_i$ is the parameter value in the pdf after marginalization (i.e., summation) over all other values of $c_j$ , with $i\\ne j$ .", "In both Figures REF and REF , two-dimensional slices (in parameter space) from the posterior joint pdfs are shown at the MAP values of all other parameters, with $P^{(3)}_{\\sigma }(c_i,c_j\\,|\\, c_k^\\mathrm {MAP})$ and $P^{(4)}_{\\sigma }(c_i,c_j\\,|\\, c_k^\\mathrm {MAP},c_l^\\mathrm {MAP})$ on the over-diagonal of each figure and $P^{(3)}_{\\mathcal {P}}(c_i,c_j\\,|\\, c_k^\\mathrm {MAP})$ and $P^{(4)}_{\\mathcal {P}}(c_i,c_j\\,|\\, c_k^\\mathrm {MAP},c_l^\\mathrm {MAP})$ on the under-diagonal.", "Here, $i$ , $j$ , $k$ , and $l$ each take on values between 1 and 4 (or only 3 for the three-parameter model), with none of the indices equal.", "All posterior pdfs are calculated using kernel density estimation with a Gaussian kernel and bandwidth defined by Scott's Rule [47], [48].", "A summary of the MAP parameter values for each test are provided in Table REF .", "Figure: Posterior distributions of accepted parameters c 1 c_1, c 2 c_2, and c 3 c_3 for the three-term SGS model from Eq. ().", "The diagonal subplots show marginal distributions M σ (3) (c i )M^{(3)}_\\mathcal {\\sigma }(c_i) (blue lines), M 𝒫 (3) (c i )M^{(3)}_\\mathcal {\\mathcal {P}}(c_i) (red lines), and M σ (3) 𝒫(c i )M^{(3)}_\\mathcal {\\sigma \\mathcal {P}}(c_i) (green lines) for (a) i=1i=1, (e) i=2i=2, and (i) i=3i=3.", "The over-diagonal subplots (b,c,f) show P σ (3) (c i ,c j |c k MAP )P^{(3)}_{\\sigma }(c_i,c_j\\,|\\, c_k^\\mathrm {MAP}) and the under-diagonal subplots (d,g,h) show P 𝒫 (3) (c i ,c j |c k MAP )P^{(3)}_{\\mathcal {P}}(c_i,c_j\\,|\\, c_k^\\mathrm {MAP}).", "Dashed lines in each subplot show the MAP parameter estimates c i MAP c_i^\\mathrm {MAP} in Table for stress (red dashed lines), production (blue dashed lines), and combined stress and production (green dashed lines) data.Figure: Posterior distributions of accepted parameters c 1 c_1, c 2 c_2, c 3 c_3, and c 4 c_4 for the four-term SGS model from Eq. ().", "The diagonal subplots show marginal distributions M σ (4) (c i )M^{(4)}_\\mathcal {\\sigma }(c_i) (blue lines), M 𝒫 (4) (c i )M^{(4)}_\\mathcal {\\mathcal {P}}(c_i) (red lines), and M σ (4) 𝒫(c i )M^{(4)}_\\mathcal {\\sigma \\mathcal {P}}(c_i) (green lines) for (a) i=1i=1, (f) i=2i=2, (k) i=3i=3, and (p) i=4i=4.", "The over-diagonal subplots (b,c,d,g,h,l) show P σ (4) (c i ,c j |c k MAP ,c l MAP )P^{(4)}_{\\sigma }(c_i,c_j\\,|\\, c_k^\\mathrm {MAP},c_l^\\mathrm {MAP}) and the under-diagonal subplots (e,i,j,m,n,o) show P 𝒫 (4) (c i ,c j |c k MAP ,c l MAP )P^{(4)}_{\\mathcal {P}}(c_i,c_j\\,|\\, c_k^\\mathrm {MAP},c_l^\\mathrm {MAP}).", "Dashed lines in each subplot show the MAP parameter estimates c i MAP c_i^\\mathrm {MAP} in Table for stress (red dashed lines), production (blue dashed lines), and combined stress and production (green dashed lines) data.For the three-parameter model, Figure REF and Table REF show that ABC-IMCMC provides a MAP value of $c_1^\\mathrm {MAP}= -0.069$ when using SGS stress data, and $c_1^\\mathrm {MAP}=-0.047$ when using SGS production data.", "The 1D marginal posteriors $M^{(3)}_\\mathcal {D}(c_2)$ are bimodal for all types of data, with MAP values of $c_2^\\mathrm {MAP}=0.070$ for stress data and $c_2^\\mathrm {MAP}=-0.036$ for production data.", "The bimodality of $M^{(3)}_\\mathcal {D}(c_2)$ indicates that the sign of $c_2^\\mathrm {MAP}$ is of little importance compared to the magnitude.", "The 2D joint pdfs $P^{(3)}_{\\mathcal {D}}(c_2,c_j\\,|\\, c_k^\\mathrm {MAP})$ are also symmetric with respect to $c_2$ for all types of data $\\mathcal {D}$ .", "The greatest dependence on the type of data for the three-parameter model is observed for $c_3$ , where $c_3^\\mathrm {MAP}$ in Table REF is close to zero for stress data and $c_3^\\mathrm {MAP}=-0.29$ for production data.", "The former result indicates that the $c_3$ term is of negligible importance compared to the other two terms when attempting to predict the stress pdfs.", "Moreover, the joint pdf of $P^{(3)}_{\\mathcal {P}}(c_1,c_3 \\,|\\, c_2^\\mathrm {MAP})$ in Figure REF (g) shows that these two parameters are correlated when using production data, but there is no similarly strong correlation when using stress data [i.e., in Figure REF (c)] .", "When using both the stress and production data, Figure REF and Table REF show that the 1D marginal pdfs and associated MAP values are generally a combination of the stress and production data results, with MAP values of $c_1^\\mathrm {MAP}=-0.040$ , $c_2^\\mathrm {MAP}=0.27$ , and $c_3^\\mathrm {MAP}=-0.16$ .", "In general, however, the combined results are most similar to results obtained using production data alone.", "Table: MAP parameter estimates for the three (n=3n=3) and four (n=4n=4) parameter SGS models in Eq.", "() from posterior joint probability density functions calculated using ABC-IMCMC for different choices of data.Figure: Modeled and reference summary statistics (i.e., pdfs) for σ 11 \\sigma _{11} (first column), σ 12 \\sigma _{12} (second column), σ 13 \\sigma _{13} (third column), and 𝒫\\mathcal {P} (last column).", "Reference results (solid black lines) are obtained from DNS of HIT and modeled results are obtained using the MAP parameter estimates given in Table for (a) P σ (n) (𝐜)P^{(n)}_{\\sigma }(\\mathbf {c}), (b) P 𝒫 (n) (𝐜)P^{(n)}_{\\mathcal {P}}(\\mathbf {c}), and (c) P σ𝒫 (n) (𝐜)P^{(n)}_{\\sigma \\mathcal {P}}(\\mathbf {c}).", "Results are shown for both three (n=3n=3; blue dashed lines) and four (n=4n=4; red dash-dot lines) models given by Eq.", "().For the four-parameter model, Figure REF shows that $M^{(4)}_\\mathcal {D}(c_1)$ is largely similar to the corresponding marginal distribution for the three-parameter model, with MAP values of roughly $c_1^\\mathrm {MAP}=-0.04$ for each type of data (see Table REF ).", "All two-dimensional joint pdfs remain symmetric with respect to $c_2$ , as in the three-parameter case, and there are large differences in $c_3^\\mathrm {MAP}$ and $c_4^\\mathrm {MAP}$ depending on whether stress or production data, or a combination of these data types, are used in the ABC-IMCMC approach.", "Once more, the combined results are generally most similar to the results for the production data alone, although there are large differences in $c_4^\\mathrm {MAP}$ for the production only and combined data.", "Finally, in the four-parameter model, $c_3$ and $c_4$ are strongly correlated, as shown in Figure REF (l) for $P^{(4)}_{\\sigma }(c_3,c_4\\,|\\, c_1^\\mathrm {MAP},c_2^\\mathrm {MAP})$ .", "This strong correlation is a good example of when the proposal kernel with adaptive covariance (see Section REF ) can improve the acceptance rate of the ABC-MCMC algorithm.", "It should be noted that the values of $c_1^\\mathrm {MAP}$ summarized in Table REF can be connected to the classical Smagorinsky coefficient as $C_\\mathrm {S}=(-c_1^\\mathrm {MAP}/2)^{1/2}$ .", "The resulting values of $C_\\mathrm {S}$ then range from $C_\\mathrm {S}=0.19$ for the three-parameter model using stress data to $C_\\mathrm {S}=0.13$ for the four-parameter model using combined stress and production data.", "These values are in generally close agreement with the classical range of $C_\\mathrm {S}$ of between 0.1 and 0.2 [39]." ], [ "The primary outcome of any particular ABC-IMCMC test is an estimate of the posterior joint pdf, such as those shown in Figures REF  and REF .", "From these posterior distributions, estimates of the most probable parameter values, as well as their uncertainties, can be obtained.", "Here we use the maxima of the estimated posteriors (i.e., the MAP values, summarized in Table REF ), as the most probable parameter values, and compute the resulting SGS stress and production summary statistics (i.e., pdfs) from the model in Eq.", "(REF ).", "Figure REF shows these modeled summary statistics, denoted $\\mathcal {S}^{\\prime }_{ij}$ and $\\mathcal {S}^{\\prime }_\\mathcal {P}$ , respectively, for both $P^{(3)}_{\\mathcal {D}}(\\mathbf {c})$ and $P^{(4)}_{\\mathcal {D}}(\\mathbf {c})$ , along with the corresponding reference statistics $\\mathcal {S}_{ij}$ and $\\mathcal {S}_\\mathcal {P}$ .", "In general, across all types of reference data $\\mathcal {D}$ , the performance of the four-parameter model is improved compared to the three-parameter model, as indicated by the improved agreement between the modeled and reference summary statistics.", "This demonstrates that the fourth basis function, $\\widetilde{G}^{(4)}_{ij}$ , is important for improving the accuracy of the second-order SGS stress model given by Eq.", "(REF ).", "For ABC-IMCMC using SGS stress data, Figure REF (a) shows that, despite good agreement between the modeled [using the MAP parameter estimates from $P^{(3)}_{\\sigma }(\\mathbf {c})$ and $P^{(4)}_{\\sigma }(\\mathbf {c})$ ] and reference stress pdfs, neither the three- or four-parameter models correctly reproduces the positive values of the production (i.e., energy backscatter) observed in the reference data.", "This limitation is caused by using only SGS stresses during ABC-IMCMC, without explicitly taking into account the correct modeling of the SGS production process.", "By contrast, Figure REF (b) shows that, when using MAP parameters from $P^{(3)}_{\\mathcal {P}}(\\mathbf {c})$ and $P^{(4)}_{\\mathcal {P}}(\\mathbf {c})$ , both models are able to match the SGS production refernce pdf reasonably well, but the stress pdfs are in much worse agreement, particularly for the diagonal elements of the tensor.", "This thus shows that using only SGS production reference data during ABC-IMCMC is not sufficient to correctly reproduce the stresses using either the three- or four-parameter models.", "Finally, an intermediate model where the MAP parameter estimates are taken from $P^{(3)}_{\\sigma \\mathcal {P}}(\\mathbf {c})$ and $P^{(4)}_{\\sigma \\mathcal {P}}(\\mathbf {c})$ is shown in Figure REF (c).", "In this case, the model and reference results are in reasonably close agreement for both the SGS stress and production pdfs, with generally better agreement for the production pdfs.", "Other combined models could be obtained from ABC-IMCMC by using different weightings of the stress and production components of the combined distance function in Eq.", "(REF ), but the present weighting does provide reasonable agreement between the modeled and reference statistics for the stresses and production simultaneously.", "It should be noted that the present results generally indicate that both three- and four-term second order models have difficulties in simultaneously capturing stresses and production.", "This difficulty has been observed previously for other models [39], and ABC-IMCMC effectively provides a user-definable blend of accuracy for these given models.", "Obtaining even better simultaneous agreement for both the stress and production statistics requires a more sophisticated model, most likely with a greater number of degrees of freedom." ], [ "Ultimately, the SGS model parameter values estimated using ABC-IMCMC are intended for use in forward LES runs of both idealized and practically relevant flows.", "Here we show that the present MAP parameter estimates (summarized in Table REF ) result in two new stand-alone SGS models based on Eq.", "(REF ) that permit stable solutions for HIT in forward LES runs.", "These a posteriori tests were performed using spectralLES, a pseudospectral LES solver for model testing and development written in pure Python.", "The solver is based on the open-source, pure Python code, spectralDNS [49].", "The LES were initialized using a random initial velocity field with a prescribed isotropic energy spectrum, and turbulence was sustained using spectrally-truncated linear forcing of wavenumber shells $k=2$ and $k=3$ .", "The resulting simulation thus produced statistically stationary HIT.", "Simulations were performed using the same domain, energy injection rate, viscosity, and LES filter scale as the DNS reference data used in the a priori tests described in the previous section, with a $64^3$ uniform grid.", "For comparison, we also ran a simulation using the static Smagorinsky model with the standard Smagorinsky coefficient $C_\\mathrm {S} = 0.1$ .", "Figure REF shows the resulting kinetic energy spectra from the LES using both three- and four-parameter models with parameters obtained from the MAP estimates in Table REF .", "Parameters determined using stress data result in spectra similar to the spectrum obtained using the Smagorinsky model.", "By contrast, spectra from LES with parameters based on the SGS production data tend to have larger magnitudes at higher wavenumbers.", "This is caused by the fact that, when using production reference data during ABC-IMCMC, parameters are chosen such that the modeled and reference production match at the LES scale.", "Figure: Kinetic energy spectra resulting from forward LES of HIT using the MAP values in Table for the three- and four-parameters SGS models from Eq. ().", "Each line is labeled by the number of parameters in the model (either n=3n=3 or 4), and by the data used during the ABC-IMCMC procedure (either σ ij \\sigma _{ij}, 𝒫\\mathcal {P}, or their combination).", "An LES spectrum obtained using the Smagorinsky model (solid blue line) and a k -5/3 k^{-5/3} slope line are also shown.Figure: Kinetic energy spectra (magenta lines) resulting from forward LES of HIT using parameter values sampled from P σ (n) (𝐜)P^{(n)}_{\\sigma }(\\mathbf {c}) (left column), P 𝒫 (n) (𝐜)P^{(n)}_{\\mathcal {P}}(\\mathbf {c}) (middle column), and P σ𝒫 (n) (𝐜)P^{(n)}_{\\sigma \\mathcal {P}}(\\mathbf {c}) (right column) for the (a) three- and (b) four-parameter SGS models from Eq. ().", "The intensity of the magenta lines indicates the posterior probability of each sampled parameter set.", "Modeled spectra obtained using MAP parameter estimates (solid red lines), an LES spectrum obtained using the Smagorinsky model (solid blue lines), the reference data spectrum from DNS of HIT (solid black lines), and a k -5/3 k^{-5/3} slope line are also shown.The correspondence between the modeled and reference spectra when using production data during ABC-IMCMC parameter estimation is shown more clearly in Figure REF .", "In this figure, we explore the propagation of uncertainty resulting from the various estimated posteriors by sampling 100 parameter sets from each of the six posterior pdfs using a Monte Carlo acceptance-rejection method.", "We then performed LES of HIT for each parameter set, resulting in the range of spectra shown in Figure REF for each posterior.", "Figure REF shows that there is substantial spread in the spectra for high wavenumbers when using parameters sampled from $P^{(3)}_{\\sigma }(\\mathbf {c})$ and $P^{(4)}_{\\sigma }(\\mathbf {c})$ , and that the spectra are frequently overly dissipative at small scales.", "This yields better agreement of the modeled spectra with the purely dissipative Smagorinsky model than with the reference DNS results.", "By contrast, the spectra obtained by sampling from $P^{(3)}_{\\mathcal {P}}(\\mathbf {c})$ and $P^{(4)}_{\\mathcal {P}}(\\mathbf {c})$ are generally close to the DNS spectrum at small scales, with less variability at high wavenumbers.", "However, compared to spectra for the three-parameter model, the four-parameter model spectra are generally relatively close to the DNS spectrum over all wavenumbers.", "Finally, as with many of the other results presented herein, the spectra obtained by sampling the combined posteriors $P^{(3)}_{\\sigma \\mathcal {P}}(\\mathbf {c})$ and $P^{(4)}_{\\sigma \\mathcal {P}}(\\mathbf {c})$ are a blend of the spectra obtained from posteriors based on either the stresses or production alone." ], [ "Conclusions", "In this study, we used ABC and IMCMC to estimate posterior distributions of unknown parameter values in three- and four-parameter SGS closure models for LES of turbulent flows.", "After outlining the general ABC-IMCMC approach, including sample algorithms, we used the method to estimate posterior distributions of model parameters based on pdfs (i.e., summary statistics) of SGS stress, SGS production, and a combination of SGS stresses and production.", "The reference data were obtained from DNS of HIT [46].", "Through a priori tests, we showed that the estimated parameter values can be used to accurately reproduce stress and production pdfs, although simultaneously matching all pdfs was found to be difficult due to limitations of the model form.", "Using a posteriori tests, we further showed that the models obtained using ABC-IMCMC can be stably integrated in forward LES runs using the spectraLES code.", "It is emphasized that the present tests using nonlinear SGS models are a demonstration of the ABC-IMCMC approach for model parameter estimation, and that more complicated models, other types of reference data, and different choices of summary statistics and distance functions can be equally applied.", "The model evaluations that are part of the ABC-IMCMC approach can also be obtained from forward runs of an LES model, and the approach can be extended to the estimation of RANS model parameters, as described in [50].", "In order to ensure even greater accuracy when comparing modeled and reference results, more sophisticated models involving a greater number of unknown parameters are likely to yield improved simultaneous agreement for both stresses and the production.", "For example, [41] used a very high-dimensional form of the SGS stress model to show that both the pdfs and the pointwise fields of the stresses and production can be reproduced.", "It is possible that lower dimensional models than those examined by [41], but with more than the four unknown parameters in the models examined here, may also retain some of the same accuracy.", "It is also possible that the greater computational efficiency enabled by ABC-IMCMC may allow the use of this approach with autonomic closure [41]; an earlier effort focused on combining ABC with autonomic closure did not make use of the IMCMC method outlined here, and instead relied on the much less efficient ABC rejection sampling approach in Algorithm REF [51].", "Finally, we note that the calibration step, adaptive proposal, and MCMC procedure have all been included in order to accelerate the ABC process and reduce the requirement for computational resources during the model parameter estimation.", "However, further improvements to the ABC-IMCMC approach are also possible, including the use of linear regression to correct biases introduced due to the use of nonzero values of the tolerance $\\varepsilon $ .", "These and other techniques to further accelerate ABC-IMCMC and reduce the computational cost are important directions for future research." ], [ "Acknowledgements", "OAD and PEH acknowledge financial support from NASA award NNX15AU24A-03.", "PEH was also supported, in part, by AFOSR award FA9550-17-1-0144.", "Helpful discussions with Profs.", "Werner J.A.", "Dahm, Ian Grooms, Will Kleiber, and Greg Rieker, as well as with Drs.", "Jason Christopher and Scott Murman, are gratefully acknowledged." ] ]

[ [ "Considering light-matter interactions in the Friedmann equations" ], [ "Abstract Recent observations indicate that the Universe is not transparent but partially opaque due to absorption of light by ambient cosmic dust.", "This implies that the Friedmann equations valid for the transparent universe must be modified for the opaque universe.", "The paper studies a scenario when the opacity steeply rises with redshift.", "In this case, the light-matter interactions become important, because cosmic opacity produces radiation pressure that counterbalances gravitational forces.", "The presented theoretical model assumes the Universe expanding according to the standard FLRW metric but with the scale factor $a(t)$ depending on both types of forces: gravity as well as radiation pressure.", "The modified Friedmann equations predicts a cyclic expansion/contraction evolution of the Universe within a limited range of scale factors with no initial singularity.", "The model avoids dark energy and removes some other tensions of the standard cosmological model.", "The paper demonstrates that considering light-matter interactions in cosmic dynamics is crucial and can lead to new cosmological models essentially different from the standard $\\Lambda$CDM model.", "This emphasizes necessity of new observations and studies of cosmic opacity and cosmic dust at high redshifts for more realistic modelling of the evolution of the Universe." ], [ "Introduction", "Dust is an important component of the interstellar and intergalactic medium, which interacts with the stellar radiation.", "Dust grains absorb and scatter the starlight and reemit the absorbed energy at infrared, far-infrared and microwave wavelengths , , , , , .", "Since galaxies contain interstellar dust, they lose their transparency and become opaque , , , , .", "Similarly, the Universe is not transparent but partially opaque due to ambient cosmic dust.", "The cosmic opacity is very low in the local Universe , , but it might steeply increase with redshift , , .", "The fact that the Universe is not transparent but partially opaque might have fundamental cosmological consequences, because the commonly accepted cosmological model was developed for the transparent universe.", "Neglecting cosmic opacity produced by intergalactic dust may lead to distorting the observed evolution of the luminosity density and the global stellar mass density with redshift .", "Non-zero cosmic opacity may invalidate the interpretation of the Type Ia supernova (SNe Ia) dimming as a result of dark energy and the accelerating expansion of the Universe , , , , .", "Intergalactic dust can partly or fully produce the cosmic microwave background (CMB) , , .", "For example, showed that thermal radiation of dust is capable to explain the spectrum, intensity and temperature of the CMB including the CMB temperature/polarization anisotropies.", "If cosmic opacity and light-matter interactions are considered, the Friedmann equations must be modified and the radiation pressure caused by absorption of photons by dust grains must be incorporated.", "Based on numerical modeling and observations of basic cosmological parameters, I show that the modified Friedmann equations avoid the initial singularity and lead to a cyclic model of the Universe with expansion/contraction epochs within a limited range of scale factors.", "The standard Friedmann equations read ${\\left({\\frac{\\dot{a}}{a}}\\right)}^2 = \\frac{8\\pi G}{3} \\rho - \\frac{k c^2}{a^2} \\,,$ $\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\left(\\rho + \\frac{3 p}{c^2} \\right)\\,,$ where $a = R/R_0 = \\left(1+z\\right)^{-1}$ is the relative scale factor, $G$ is the gravitational constant, $\\rho $ is the mass density, $k/a^2$ is the spatial curvature of the universe, $p$ is the pressure, and $c$ is the speed of light.", "Considering the mass density $\\rho $ as a sum of matter and radiation contributions and including the vacuum contribution, we get $\\frac{8\\pi G}{3} \\rho = H^2_0 \\left[{\\Omega _m a^{-3} + \\Omega _r a^{-4} + \\Omega _\\Lambda }\\right] \\,.$ Eq.", "(1) is then rewritten as $H^2\\left(a\\right) = H^2_0 \\left[{\\Omega _m a^{-3} + \\Omega _r a^{-4} + \\Omega _\\Lambda + \\Omega _k a^{-2}}\\right] \\,,$ with the condition $\\Omega _m + \\Omega _r + \\Omega _\\Lambda + \\Omega _k = 1 \\,,$ where $H(a) = \\dot{a}/a$ is the Hubble parameter, $H_0$ is the Hubble constant, and $\\Omega _m$ , $\\Omega _r$ , $\\Omega _\\Lambda $ and $\\Omega _k$ are the normalized matter, radiation, vacuum and curvature terms.", "Assuming $\\Omega _r = 0$ and $\\Omega _k = 0$ in Eq.", "(4), we get the $\\Lambda $ CDM model $H^2\\left(a\\right) = H^2_0 \\left[{\\Omega _m a^{-3} + \\Omega _\\Lambda }\\right] \\,,$ which describes a flat, matter-dominated universe.", "The universe is transparent, because any interaction of radiation with matter is neglected.", "The vacuum term $\\Omega _\\Lambda $ is called dark energy and it is responsible for the accelerating expansion of the Universe.", "The dark energy is introduced into Eqs (3-5) to fit the $\\Lambda $ CDM model with observations of the Type Ia supernova dimming." ], [ "Friedmann equations for the opaque universe", "The basic drawback of the $\\Lambda $ CDM model is its assumption of transparency of the Universe and neglect of the universe opacity caused by interaction of light with intergalactic dust.", "Absorption of light by cosmic dust produces radiation pressure acting against the gravity, but this pressure is ignored in the $\\Lambda $ CDM model.", "Let us assume a space filled by light and cosmic dust formed by uniformly distributed spherical dust grains.", "The dust grains absorb photons and reemit them in the form of thermal radiation.", "The total force produced by absorption of photons, which acts on dust in a unit volume of the Universe, is $M_D \\ddot{R} = S_D p_D \\,,$ where $M_D$ and $S_D$ are the mass and surface of all dust grains in the spherical volume of radius $R$ , and $p_D$ is the radiation pressure caused by dust absorption of the extragalactic background light (EBL) present in the cosmic space $p_D = \\frac{\\lambda }{c} I^{\\mathrm {EBL}} \\,,$ where $\\lambda $ is the bolometric cosmic opacity (defined as attenuation per unit raypath), and $I^{\\mathrm {EBL}}$ is the bolometric intensity of the EBL, which depends on redshift as $I^{\\mathrm {EBL}} = I^{\\mathrm {EBL}}_0 \\left(1+z\\right)^4 \\,,$ where subscript '0' means the quantity at $z = 0$ .", "Since the production and absorption of photons should be in balance, the EBL intensity $I_0^{\\mathrm {EBL}}$ is related to the luminosity density $j$ at $z = 0$ as $\\lambda _0 I^{\\mathrm {EBL}}_0 = \\frac{j_0}{4 \\pi } \\,.$ If the comoving number density of dust grains is constant, the opacity $\\lambda $ in Eq.", "(8) is redshift independent, $\\lambda = \\lambda _0$ (the proper attenuation coefficient per unit ray path increases with $z$ , but the proper length of a ray decreases with $z$ ).", "Hence, the pressure $p_D$ in Eq.", "(8) reads $p_D = \\frac{j_0}{4 \\pi c} \\left(1+z\\right)^4 \\,.$ Inserting Eq.", "(11) into Eq.", "(7) and substituting $R$ by the relative scale factor $a = R/R_0$ , we obtain $\\ddot{a} = \\frac{S_D}{M_D} \\frac{j_0}{4 \\pi c} \\frac{1}{a^4} \\,,$ where $R_0 = 1$ .", "Integrating Eq.", "(12) in time ${\\left({\\frac{\\dot{a}}{a}}\\right)}^2 = \\frac{S_D}{M_D} \\frac{j_0}{6 \\pi c} \\frac{1}{a^5} \\,,$ and including absorption terms defined in Eqs (12-13) into Eqs (1-2), we get a new form of the Friedmann equations valid for a model of the opaque universe ${\\left({\\frac{\\dot{a}}{a}}\\right)}^2 = \\frac{8\\pi G}{3} \\rho - \\frac{S_D}{M_D} \\frac{j_0}{6 \\pi c} \\frac{1}{a^5} - \\frac{k c^2}{a^2} \\,,$ $\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\left(\\rho + \\frac{3 p_D}{c^2} \\right)+ \\frac{S_D}{M_D} \\frac{j_0}{4 \\pi c} \\frac{1}{a^5}\\,,$ which read for dust formed by spherical grains as ${\\left({\\frac{\\dot{a}}{a}}\\right)}^2 = \\frac{8\\pi G}{3} \\rho - \\frac{1}{2 \\pi c} \\frac{j_0}{\\rho _D R_D} \\frac{1}{a^5} - \\frac{k c^2}{a^2} \\,,$ $\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\rho + \\frac{3}{4 \\pi c} \\frac{j_0}{\\rho _D R_D} \\frac{1}{a^5}\\,,$ where $R_D$ and $\\rho _D$ are the radius and the specific density of dust grains.", "In Eq.", "(17), we omit gravity forces produced by pressure $p_D$ , because they are negligible with respect to the other terms.", "Consequently, the Hubble parameter reads $H^2\\left(a\\right) = H^2_0 \\left[{\\Omega _m a^{-3} + \\Omega _r a^{-4}+ \\Omega _a a^{-5} + \\Omega _k a^{-2}}\\right] \\,,$ which simplifies for a matter-dominated opaque universe ($\\Omega _r = 0$ ) as $H^2\\left(a\\right) = H^2_0 \\left[{\\Omega _m a^{-3} + \\Omega _a a^{-5} + \\Omega _k a^{-2}}\\right] \\,,$ with the condition $\\Omega _m + \\Omega _a + \\Omega _k = 1 \\,,$ where $\\Omega _m$ , $\\Omega _a$ and $\\Omega _k$ are the normalized gravity, absorption and curvature terms, respectively, $\\Omega _m = \\frac{1}{H^2_0} \\left({ \\frac{8}{3} \\pi G \\rho _0}\\right) \\,,$ $\\Omega _a = -\\frac{1}{H^2_0} \\left({\\frac{1}{2 \\pi c} \\frac{j_0}{\\rho _D R_D}}\\right) \\,,$ $\\Omega _k = -\\frac{k c^2}{H^2_0} \\,.$ The minus sign in Eq.", "(22) means that the radiation pressure due to absorption acts against the gravity.", "The dark energy is missing in Eqs (18-20), because the Type Ia supernova dimming can successfully be explained by cosmic opacity, as discussed in .", "Eq.", "(19) shows that the increase of the absorption term $\\Omega _a$ with redshift is enormously high.", "The reasons for such a steep rise of $\\Omega _a$ with $z$ are, however, straightforward.", "The steep rise combines the three following effects: (1) the increase of photon density with $(1+z)^3$ due to the space contraction, (2) the increase of absorption of photons with $(1+z)$ due to the shorter distance between dust grains, and (3) the increase of rate of absorbed photons by dust grains with $(1+z)$ due to time dilation." ], [ "Distance-redshift relation", "The scale factor $a$ of the Universe with the zero expansion rate is defined by the zero Hubble parameter in Eq.", "(19), which yields a cubic equation in $a$ $\\Omega _k a^3 + \\Omega _m a^2 + \\Omega _a = 0 \\,.$ Taking into account that $\\Omega _m > 0$ and $\\Omega _a < 0$ , Eq.", "(24) has two distinct real positive roots for $\\left({\\frac{\\Omega _m}{3}}\\right)^2 > \\left({\\frac{\\Omega _k}{2}}\\right)^2 \\left|\\Omega _a\\right| \\,\\,\\,\\mathrm {and} \\,\\,\\,\\Omega _k < 0 \\,.$ Negative $\\Omega _a$ and $\\Omega _k$ imply that $\\Omega _m > 1 \\,\\,\\,\\mathrm {and} \\,\\,\\,\\rho _0 > \\rho _c = \\frac{8 \\pi G}{3 H^2_0} \\,.$ Under these conditions, Eq.", "(19) describes a universe with a cyclic expansion/contraction history and the two real positive roots $a_\\mathrm {min}$ and $a_\\mathrm {max}$ define the minimum and maximum scale factors of the Universe.", "For $\\Omega _a \\ll 1$ , the scale factors $a_\\mathrm {min}$ and $a_\\mathrm {max}$ read approximately $a_\\mathrm {min} \\cong \\sqrt{\\left|{\\frac{\\Omega _a}{\\Omega _m}}\\right|} \\,\\,\\, \\mathrm {and} \\,\\,\\,a_\\mathrm {max} \\cong \\left|{\\frac{\\Omega _m}{\\Omega _k}}\\right| \\,,$ and the maximum redshift is $z_\\mathrm {max} = \\frac{1}{a_\\mathrm {min}} - 1 \\,.$ The scale factors $a$ of the Universe with the maximum expansion/contraction rates are defined by $\\frac{d}{da} H^2 \\left(a\\right) = 0 \\,,$ which yields a cubic equation in $a$ $2 \\Omega _k a^3 + 3 \\Omega _m a^2 + 5 \\Omega _a = 0 \\,.$ Taking into account Eq.", "(17) and Eqs (21-23), the deceleration of the expansion reads $\\ddot{a} = -\\frac{1}{2} H^2_0 \\left[\\Omega _m a^{-2} + 3 \\Omega _a a^{-4}\\right] \\,.$ Hence, the zero deceleration is for the scale factor $a = \\sqrt{\\left|{\\frac{3\\Omega _a}{\\Omega _m}}\\right|} \\,.$ Finally, the comoving distance as a function of redshift is expressed from Eq.", "(19) as follows $\\begin{split}dr = \\frac{c}{H_0} \\frac{dz}{\\sqrt{\\Omega _m \\left(1+z\\right)^3 +\\Omega _a \\left(1+z\\right)^5 + \\Omega _k \\left(1+z\\right)^2}} \\,.\\end{split}$" ] ]

[ [ "Cats climb entails mammals move: preserving hyponymy in compositional\n distributional semantics" ], [ "Abstract To give vector-based representations of meaning more structure, one approach is to use positive semidefinite (psd) matrices.", "These allow us to model similarity of words as well as the hyponymy or is-a relationship.", "Psd matrices can be learnt relatively easily in a given vector space $M\\otimes M^*$, but to compose words to form phrases and sentences, we need representations in larger spaces.", "In this paper, we introduce a generic way of composing the psd matrices corresponding to words.", "We propose that psd matrices for verbs, adjectives, and other functional words be lifted to completely positive (CP) maps that match their grammatical type.", "This lifting is carried out by our composition rule called Compression, Compr.", "In contrast to previous composition rules like Fuzz and Phaser (a.k.a.", "KMult and BMult), Compr preserves hyponymy.", "Mathematically, Compr is itself a CP map, and is therefore linear and generally non-commutative.", "We give a number of proposals for the structure of Compr, based on spiders, cups and caps, and generate a range of composition rules.", "We test these rules on a small sentence entailment dataset, and see some improvements over the performance of Fuzz and Phaser." ], [ "Introduction", "Vector-based representations of words, with similarity measured by the inner product of the normalised word vectors, have been extremely successful in a number of applications.", "However, as well as similarity, there are a number of other important relations between words or concepts, one of these being hyponymy or the is-a relation.", "Examples of this are that cat is a hyponym of mammal, but we can also apply this to verbs, and say that sprint is a hyponym of run.", "Within standard vector-based semantics based on co-occurrence statistics, there is no standard way of representing hyponymy between word vectors.", "There have been a number of alternative approaches to building word vectors that can represent these relationships, but most of these operate at the single word level.", "Of course, words can be composed to form phrases and sentences, and we use a variant of the categorical compositional distributional (DisCoCat) approach introduced in [6].", "This approach uses a category-theoretic stance.", "It models syntax in one category, call it the grammar category, and semantics in another, call it the meaning category.", "A functor from the grammar category to the meaning category is defined, so that the grammatical reductions on the syntactic side can be translated into morphisms on the meaning side.", "The standard instantiation models meaning within the category of vector spaces and linear transformations, so that nouns are represented as vectors, and functional words such as verbs and adjectives are represented as multilinear maps, or alternatively, matrices and tensors.", "Within DisCoCat, choices can be made about the meaning categoryChoices can also be made for the grammar category, but we do not discuss that in this work.. One choice is to use the category $\\mathbf {CPM}(\\mathbf {FHilb})$ of Hilbert spaces and completely positive maps between them.", "In this category, words are represented as positive semidefinite (psd) matrices.", "Psd matrices have a natural partial order called the Löwner order, and this order is used to model hyponymy.", "This approach was developed in [25], [1], [14], and the use of psd matrices to represent words has also been used in [3], [4].", "One of the drawbacks of this approach is that learning psd matrices from text is difficult, in particular the larger matrices that are required for functional words.", "Therefore, in [14], [4], composition rules for psd matrices have been explored.", "In [4] these composition rules are called Fuzz and Phaser, in [14] they are KMult and BMult, respectively.", "For this paper we stick with the guitar pedal terminology.", "One of the drawbacks of these composition rules is that they do not preserve hyponymy.", "That is, given two pairs of words in a hyponym-hypernym relationship, the combination of the two hyponyms is not necessarily a hyponym of the combination of the two hypernyms: $\\textit {noun}_1 \\leqslant \\textit {noun}_2 \\text{ and } \\textit {verb}_1 \\leqslant \\textit {verb}_2 \\text{ does not imply } \\textit {noun}_1 \\ast \\textit {verb}_1 \\leqslant \\textit {noun}_2\\ast \\textit {verb}_2$ where the nouns and verbs are psd matrices, $\\leqslant $ is the Löwner ordering, and $\\ast $ is one of Fuzz or Phaser.", "The goal of this paper is to define a composition rule which is (i) positivity preserving, and (ii) hyponymy preserving.", "In addition we will require it to be bilinear.", "If possible, it should also be non-commutative.", "Our composition rule is called Compression, Compr, and it is in fact an infinite set of rules; namely all completely positive maps from $\\mathcal {M}_m$ to $\\mathcal {M}_m\\otimes \\mathcal {M}_m$ , where $\\mathcal {M}_m$ denotes the set of real matrices of size $m\\times m$ .", "As a special case, we recover Mult.", "We use the following notation.", "$A^*$ denotes complex conjugate transpose, and $A_*$ means complex conjugate.", "$\\mathrm {PSD}_m$ denotes the set of positive semidefinite (psd) matrices of size $m\\times m$ over the real numbers, and a psd element is denoted by $\\geqslant 0$ .", "We use the term functional words for words such as verbs and adjectives that take arguments.", "Nouns are not functional words." ], [ "Representing words as positive semidefinite matrices", "We assume that the reader is familiar with the categorical compositional distributional model of meaning introduced in [6], Frobenius algebras as used in [23], and the $\\mathbf {CPM}$ construction [26] – we have summarised the most important ingredients in Appendix .", "We now jump right in to the representation of words as psd matrices and possible composition rules.", "Positive semidefinite matrices are represented in $\\mathbf {CPM}(\\mathbf {FHilb})$ as morphisms $\\mathbb {R}\\rightarrow M\\otimes M^*$ , where $M$ is some finite-dimensional Hilbert space and $M^*$ is its dual.", "The functor $\\mathsf {S}: \\mathbf {Preg} \\rightarrow \\mathbf {CPM}(\\mathbf {FHilb})$ sends nouns and sentences to psd matrices, and adjectives, verbs, and other functional words to completely positive maps, or equivalently psd matrices in a larger space.", "We represent words as psd matrices in the following way.", "In the vector-based model of meaning, a word $w$ is represented by a column vector, $\\mathinner {|{w}\\rangle } \\in \\mathbb {R}^m$ (for some $m$ ).", "To pass to psd matrices, a subset of words $S$ will be mapped to rank 1 matrices, i.e.", "$\\mathinner {|{w}\\rangle }\\mapsto \\mathinner {|{w}\\rangle }\\mathinner {\\langle {w}|}$ .", "The words in $S$ are the hyponyms.", "The other words, which are hypernyms of the words in $S$ , will be represented as mixtures of hyponyms: $\\rho =\\sum _{w\\in W \\subset S} \\mathinner {|{w}\\rangle }\\mathinner {\\langle {w}|}.$ Within a compositional model of meaning, we view nouns as psd matrices in $\\mathcal {M}_m$ , and sentences as psd matrices in $\\mathcal {M}_s$ (for some $m$ and $s$ ).", "An intransitive verb has type $n^r s$ in the pregroup grammar, and is mapped by $\\mathsf {S}$ to a psd element in $\\mathcal {M}_m \\otimes \\mathcal {M}_s$ .", "Equivalently, an intransitive verb is a completely positive (CP) map $\\mathcal {M}_m \\rightarrow \\mathcal {M}_s$ .", "The method for building psd matrices summarised in equation (REF ) maps words of all grammatical types to psd matrices in $\\mathcal {M}_m$ .", "This is the correct type for nouns, but wrong for other grammatical types.", "Taking the example of intransitive verbs, we need to find a mechanism to lift an intransitive verb as a psd element in $\\mathcal {M}_m$ to a CP map $\\mathcal {M}_m \\rightarrow \\mathcal {M}_s$ .", "There have been various approaches to implemnting this type lifting, which we now summarise.", "Suppose $n$ is a psd matrix for a noun, and $v$ a psd matrix for a verb.", "Proposals in [14], [3], [4] include the following – note in particular that Fuzz and Phaser defined in [4] coincide with KMult and BMult defined in [14]: $\\textsf {Mult}(n,v) = n\\odot v$ where $\\odot $ is the Hadamard product, i.e.", "$(n\\odot v)_{i,j} = n_{ij} v_{ij}$ .", "$\\textsf {Fuzz}(n,v) = \\textsf {KMult} (n,v)= \\sum _ip_i P_i nP_i $ where $v=\\sum _i p_i P_i$ is the spectral decomposition of $v$ .", "That is, $\\textsf {Fuzz}(n,v) = \\sum _{i} \\sqrt{p_i} P_i n P_i \\sqrt{p_i}$ $\\textsf {Phaser}(n,v) = \\textsf {BMult}(n,v)= v^{1/2} nv^{1/2}$ .", "Let $v=\\sum _i p_i P_i$ be the spectral decomposition of $v$ .", "Then $\\textsf {Phaser}(n,v) = \\sum _{i,j} \\sqrt{p_i} P_i n P_j \\sqrt{p_j}$ Some benefits and drawbacks of these operations are as follows.", "Mult is a straightforward use of Frobenius algebra in the category $\\mathbf {CPM}(\\mathbf {FHilb})$ .", "It is linear, completely positive and preserves hyponymy.", "However, linguistically it is unsatisfactory because it is commutative, and so will map `Howard likes Jimmy' to the same psd matrix as `Jimmy likes Howard' – which do not have the same meaning and so should not have the same matrix representation.", "On the other hand, both Phaser and Fuzz are non-commutative, however they are not linear and do not preserve hyponymy.", "In the next section we outline the properties we want from a composition method, and propose a general framework that will allow us to generate a number of suggestions." ], [ "In search of more guitar pedals: Compression", "For the rest of the paper, we assume that both nouns and verbs are represented by psd matrices of the same size $m$ , that is, we let $n \\in \\mathrm {PSD}_m$ and $v\\in \\mathrm {PSD}_m$ .", "We are looking for a composition rule for these psd matrices.", "We call the desired operation $\\textsf {Compr}$ (for reasons we shall later see), and want it to be a map $\\textsf {Compr}: \\mathcal {M}_m \\times \\mathcal {M}_m \\rightarrow \\mathcal {M}_m.$ The minimal two properties required from this map are the following: Positivity preserving If $n,v$ are psd, then $\\textsf {Compr}( n,v)$ is psd: $\\textsf {Compr}: \\mathrm {PSD}_m \\times \\mathrm {PSD}_m \\rightarrow \\mathrm {PSD}_m$ Hyponymy preserving If hyponymy is represented by the Löwner order $\\leqslant $ ,If $\\rho $ , $\\sigma $ are psd, then $\\rho \\leqslant \\sigma $ iff $\\sigma -\\rho \\geqslant 0$ , i.e.", "if $\\sigma - \\rho $ is itself psd.", "$n_1\\leqslant n_2 , \\quad v_1\\leqslant v_2 \\Rightarrow \\textsf {Compr}(n_1,v_1)\\leqslant \\textsf {Compr}(n_2,v_2)$ Although these two properties are the most important ones, we now consider another property: Bilinearity Compr is linear in each of its arguments, namely for $\\alpha \\in \\mathbb {R}$ : $\\textsf {Compr}(\\alpha n,v) = \\alpha \\textsf {Compr}(n,v)$ $\\textsf {Compr}(n,\\alpha v) = \\alpha \\textsf {Compr}(n,v)$ $\\textsf {Compr}(n+n^{\\prime },v) = \\textsf {Compr}(n,v)+ \\textsf {Compr}(n^{\\prime },v)$ $\\textsf {Compr}(n,v+v^{\\prime }) = \\textsf {Compr}(n,v)+ \\textsf {Compr}(n,v^{\\prime })$ Assumption (iii) has two advantages.", "The first one is that if the map is positivity preserving on the Cartesian product [(i)] and bilinear [(iii)], then it is hyponymy preserving [(ii)]: Lemma 1 Assumptions (i) and (iii) imply (ii).", "Assume that $n_2\\geqslant n_1$ and $v_2\\geqslant 0$ .", "Using (i) we have that $\\textsf {Compr}(n_2-n_1, v_2) \\geqslant 0$ , and using (iii) that $\\textsf {Compr}(n_2, v_2) \\geqslant \\textsf {Compr}(n_1, v_2).$ Now assume that $v_2\\geqslant v_1$ and $n_1\\geqslant 0$ .", "Following the same argument we obtain that $\\textsf {Compr}(n_1, v_2)\\geqslant \\textsf {Compr}(n_1, v_1).", "$ By transitivity of being psd, we obtain that $\\textsf {Compr}(n_2, v_2)\\geqslant \\textsf {Compr}(n_1, v_1) $ , which is condition (ii).", "The second advantage of bilinearity is that it allows to reformulate Compr in a convenient way.", "Since Compr is linear in both components, we construct the following linear map: $\\mathcal {M}_m&\\rightarrow \\textrm {Lin}(\\mathcal {M}_m,\\mathcal {M}_m) \\\\ v &\\mapsto \\left(\\textsf {Compr}(\\cdot , v)\\colon n\\mapsto \\textsf {Compr}(n, v)\\right)$ Note that linearity of Compr in the noun component is necessary for the image of this map to be $\\textrm {Lin}(\\mathcal {M}_m,\\mathcal {M}_m)$ , whereas linearity in the verb component is necessary for this map itself to be linear.", "By slight abuse of notation we denote this new map also by Compr: $\\textsf {Compr} \\colon \\mathcal {M}_m\\rightarrow \\textrm {Lin}(\\mathcal {M}_m,\\mathcal {M}_m).$ Now, assumption (i) applied to this new map means that psd matrices are mapped to positivity preserving maps, $\\textsf {Compr}: \\mathrm {PSD}_m\\rightarrow \\textrm {PP}(\\mathcal {M}_m,\\mathcal {M}_m) ,$ where $\\textrm {PP}(\\mathcal {M}_m,\\mathcal {M}_m)$ is the set of positivity preserving linear maps from $\\mathcal {M}_m$ to $\\mathcal {M}_m$ , i.e.", "those that map psd matrices to psd matrices.", "To make things more tractable, one can use the isomorphism $\\textrm {Lin}(\\mathcal {M}_m,\\mathcal {M}_m)&\\rightarrow \\mathcal {M}_m\\otimes \\mathcal {M}_m\\\\\\varphi &\\mapsto \\sum _{i,j} \\varphi (|e_i\\rangle \\langle e_j|)\\otimes |e_i\\rangle \\langle e_j|,$ where $\\lbrace \\mathinner {|{e_i}\\rangle }\\rbrace $ is an orthonormal basis of $\\mathbb {R}^m$ .", "Using this isomorphism, $\\textrm {PP}(\\mathcal {M}_m,\\mathcal {M}_m)$ corresponds to the set of block positive matrices $\\textrm {BP}(\\mathcal {M}_m \\otimes \\mathcal {M}_m)$ on the tensor product space.A matrix $\\rho \\in \\mathcal {M}_m\\otimes \\mathcal {M}_m$ is block positive if $(\\langle v|\\otimes \\langle w|) \\rho (|v\\rangle \\otimes |w\\rangle ) \\ge 0$ for all vectors $|v\\rangle $ , $|w\\rangle $ .", "Summarizing, we are trying to construct a linear map $\\textsf {Compr} \\colon \\mathcal {M}_m\\rightarrow \\mathcal {M}_m\\otimes \\mathcal {M}_m$ that maps psd matrices to block positive matrices.", "So far, this is just a reformulation of conditions (i) and (iii).", "To make thinks more tractable, we now further strengthen the conditions on the map.", "Namely, we require Compr to map psd matrices in $\\mathcal {M}_m$ to psd matrices in $\\mathcal {M}_m\\otimes \\mathcal {M}_m\\cong \\mathcal {M}_{m^2}$ , i.e.", "to be positivity preserving itself.", "Using the isomorphism above, this means that $\\textsf {Compr}$ maps psd matrices to completely positive (CP) maps from $\\mathcal {M}_m$ to $\\mathcal {M}_m$ (see table REF ): $\\textsf {Compr}: \\mathrm {PSD}_m\\rightarrow \\textrm {CP}(\\mathcal {M}_m,\\mathcal {M}_m) .$ Table: Correspondence between linear maps ℳ m →ℳ m \\mathcal {M}_m \\rightarrow \\mathcal {M}_m and elements in ℳ m ⊗ℳ m \\mathcal {M}_m\\otimes \\mathcal {M}_m, known as the Choi-Jamiołkowski isomorphism.And since we are still not running out of steam, we require Compr not only to be positivity preserving, but also to be completely positive itself.", "In total, we are trying to construct a completely positive map $\\nonumber \\textsf {Compr}\\colon \\mathcal {M}_m &\\rightarrow \\mathcal {M}_m \\otimes \\mathcal {M}_m\\cong \\mathcal {M}_{m^2} \\\\v&\\mapsto \\sum _l K_l v K_l^*$ where we have used the well known fact that every completely positive map admits a Kraus decomposition, for certain Kraus operators $K_l\\in \\mathbb {R}^m\\otimes \\mathcal {M}_m\\cong \\mathcal {M}_{m^2,m}$ .", "Recall that ${K}^*$ denotes the complex conjugate transpose of $K$ .", "In summary, we are asking for a stronger condition than just (i) and (iii).", "On the other hand, the weaker forms of maps mentioned above do not admit a closed description, whereas completely positive maps do.", "By Stinespring's Dilation Theorem, all completely positive map can be expressed as a $*$ -representation followed by a compression, hence the name Compr.", "This is precisely what gives rise to the Kraus decomposition of the map.", "This also fits well with the electric guitar pedal notation from [4], as can be seen in figure REF .", "Figure: Not only Fuzz and Phaser, but also Compression is an important guitar pedal.", "The operation Compr takes as input an element of ℳ m ×ℳ m \\mathcal {M}_m \\times \\mathcal {M}_m (denoted n,vn, v) representing the meaning of a noun and the meaning of a verb, and it outputs 𝖢𝗈𝗆𝗉𝗋(n,v)\\textsf {Compr}(n,v), representing the meaning of the sentence nvn \\: v.Note that in general such Compr is non-commutative, i.e.", "when translated back to the initial setup of $\\textsf {Compr}: \\mathcal {M}_m \\times \\mathcal {M}_m \\rightarrow \\mathcal {M}_m $ we will generally have $\\textsf {Compr}( n,v) \\ne \\textsf {Compr}( v ,n) $ .", "This is a good property, as it reflects the position of the words in a sentence has both syntactic and semantic roles, e.g.", "`woman bites dog' versus `dog bites woman'.", "Note also that Fuzz and Phaser (or KMult and BMult) are not linear in the verb component, i.e.", "they do not fulfill (iii), and are thus not special cases of Compr.", "However, Mult is a special case of Compr, as we shall see." ], [ "Building nouns and verbs in $\\mathbf {CPM}(\\mathbf {FHilb})$", "In order to build psd matrices to represent words, we can use pretrained word embeddings such as word2vec [18] or GloVe [21], together with information about hyponymy relations.", "The word embedding methods produce vectors for each word, all represented in one vector space $W$ .", "Information about hyponymy relations can be found in WordNet [19] or in a less supervised manner by extracting hyponym-hypernym pairs using Hearst patterns [10], [22].", "Given a word $w$ , we can gather a set of hyponyms $\\lbrace h_i\\rbrace _i$ from WordNet, Hearst patterns, or some other source.", "We then take the vectors for the $h_i$ from pretrained word embeddings, and form the matrix $\\rho ( w ) = \\sum _i \\mathinner {|{h_i}\\rangle }\\mathinner {\\langle {h_i}|} \\in W \\otimes W^*.$ where $W^*$ is the dual of the column vector space $W$ .", "$\\rho ( w )$ is then normalised.", "In this work, we normalise using the infinity norm, that is, we divide by the maximum eigenvalue.", "This has been shown to have nice properties [28].", "This approach to building word representations puts all word representations in the shared space $W \\otimes W^*$ .", "If we are working in the category $\\mathbf {CPM}(\\mathbf {FHilb})$ , this is the right kind of representation for nouns, but not for functional words.", "To transform a psd matrix $\\rho ( verb ) \\in \\mathrm {PSD}_m$ to a psd matrix in $\\mathcal {M}_{m^2}$ , we use the composition rule Compr proposed in section : $\\textsf {Compr}\\colon \\mathcal {M}_m&\\rightarrow \\mathcal {M}_m\\otimes \\mathcal {M}_m\\cong \\mathcal {M}_{m^2} \\\\v&\\mapsto \\textsf {Compr}(\\cdot ,v)$" ], [ "Characterising $\\mathsf {Compr}$ diagrammatically", "We now characterise $\\mathsf {Compr}$ , $\\mathsf {Compr}(\\cdot ,v)$ , and $\\mathsf {Compr}(n,v)$ in the diagrammatic calculus for $\\mathbf {FHilb}$ .", "This will allow us to generate simple examples of $\\mathsf {Compr}$ in a systematic manner.", "Equation (REF ) states: $\\mathsf {Compr}(\\cdot ,v) = \\sum _l K_l v K_l^*$ Diagrammatically, this gives us: $\\mathsf {Compr} = {\\begin{array}{c}\\end{array}}, \\qquad \\mathsf {Compr}(\\cdot ,v) = {\\begin{array}{c}\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.25, 1.5) {};\\node [style=none] (1) at (1, 1.5) {};\\node [style=none] (2) at (-0.25, 1) {};\\node [style=none] (3) at (0.75, 1) {};\\node [style=none] (4) at (0, 2) {};\\node [style=none] (5) at (0, 1.5) {};\\node [style=none] (6) at (0.5, 1) {};\\node [style=none] (7) at (0.5, 0.5) {};\\node [style=none] (8) at (0.25, 1.25) {K^*};\\node [style=none] (9) at (0.5, 2) {};\\node [style=none] (10) at (0.5, 1.5) {};\\node [style=none] (11) at (-0.25, -1) {};\\node [style=none] (12) at (1, -1) {};\\node [style=none] (13) at (-0.25, -0.5) {};\\node [style=none] (14) at (0.75, -0.5) {};\\node [style=none] (15) at (0, -1.5) {};\\node [style=none] (16) at (0, -1) {};\\node [style=none] (17) at (0.5, -0.5) {};\\node [style=none] (18) at (0.5, 0) {};\\node [style=none] (19) at (0.25, -0.75) {K};\\node [style=none] (20) at (0.5, -1.5) {};\\node [style=none] (21) at (0.5, -1) {};\\node [style=none] (22) at (0.25, 0.5) {};\\node [style=none] (23) at (0.75, 0.5) {};\\node [style=none] (24) at (0.75, 0) {};\\node [style=none] (25) at (0.25, 0) {};\\node [style=none] (26) at (0.5, 0.25) {v};\\node [style=none] (27) at (0, 1) {};\\node [style=none] (28) at (0, -0.5) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (2.center);(2.center) to (3.center);(3.center) to (1.center);(1.center) to (0.center);(4.center) to (5.center);(7.center) to (6.center);(9.center) to (10.center);(11.center) to (13.center);(13.center) to (14.center);(14.center) to (12.center);(12.center) to (11.center);(15.center) to (16.center);(18.center) to (17.center);(20.center) to (21.center);(22.center) to (25.center);(25.center) to (24.center);(24.center) to (23.center);(23.center) to (22.center);(27.center) to (28.center);\\end{pgfonlayer}\\end{tikzpicture}\\end{array}}$ The application of $\\mathsf {Compr}(\\cdot ,v)$ to $n$ is then: $\\mathsf {Compr}(n,v) = {\\begin{array}{c}\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.25, 1.5) {};\\node [style=none] (1) at (1, 1.5) {};\\node [style=none] (2) at (-0.25, 1) {};\\node [style=none] (3) at (0.75, 1) {};\\node [style=none] (4) at (0, 2) {};\\node [style=none] (5) at (0, 1.5) {};\\node [style=none] (6) at (0.5, 1) {};\\node [style=none] (7) at (0.5, 0.5) {};\\node [style=none] (8) at (0.25, 1.25) {K^*};\\node [style=none] (9) at (0.5, 2) {};\\node [style=none] (10) at (0.5, 1.5) {};\\node [style=none] (11) at (-0.25, -1) {};\\node [style=none] (12) at (1, -1) {};\\node [style=none] (13) at (-0.25, -0.5) {};\\node [style=none] (14) at (0.75, -0.5) {};\\node [style=none] (15) at (0, -1.5) {};\\node [style=none] (16) at (0, -1) {};\\node [style=none] (17) at (0.5, -0.5) {};\\node [style=none] (18) at (0.5, 0) {};\\node [style=none] (19) at (0.25, -0.75) {K};\\node [style=none] (20) at (0.5, -1.5) {};\\node [style=none] (21) at (0.5, -1) {};\\node [style=none] (22) at (0.25, 0.5) {};\\node [style=none] (23) at (0.75, 0.5) {};\\node [style=none] (24) at (0.75, 0) {};\\node [style=none] (25) at (0.25, 0) {};\\node [style=none] (26) at (0.5, 0.25) {v};\\node [style=none] (27) at (0, 1) {};\\node [style=none] (28) at (0, -0.5) {};\\node [style=none] (29) at (-1, 2) {};\\node [style=none] (30) at (-1, 0.5) {};\\node [style=none] (31) at (-1, -1.5) {};\\node [style=none] (32) at (-1, 0) {};\\node [style=none] (33) at (-1.25, 0.5) {};\\node [style=none] (34) at (-0.75, 0.5) {};\\node [style=none] (35) at (-0.75, 0) {};\\node [style=none] (36) at (-1.25, 0) {};\\node [style=none] (37) at (-1, 0.25) {n};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (2.center);(2.center) to (3.center);(3.center) to (1.center);(1.center) to (0.center);(4.center) to (5.center);(7.center) to (6.center);(9.center) to (10.center);(11.center) to (13.center);(13.center) to (14.center);(14.center) to (12.center);(12.center) to (11.center);(15.center) to (16.center);(18.center) to (17.center);(20.center) to (21.center);(22.center) to (25.center);(25.center) to (24.center);(24.center) to (23.center);(23.center) to (22.center);(27.center) to (28.center);(30.center) to (29.center);(32.center) to (31.center);(33.center) to (36.center);(36.center) to (35.center);(35.center) to (34.center);(34.center) to (33.center);[bend left=90, looseness=1.25] (29.center) to (4.center);[bend right=90] (31.center) to (15.center);\\end{pgfonlayer}\\end{tikzpicture}\\end{array}}$ Note that this style corresponds to the usual DisCoCat diagram style via some reshaping, explained in equation (REF ) of the appendix.", "Given that we have representations of $v$ and of $n$ , what should the $K_l$ look like?", "In full generality, parameters of the $K_l$ could be perhaps inferred using regression techniques, in a similar approach to that suggested in [15], inspired by [27], or via methods like those in [2], [8].", "However, we can also consider purely “structural” morphisms, generated from cups, caps, swaps, and spiders (for details, see table REF in the appendix).", "In the following, we give a number of options to specify $K$ .", "We divide up the internal structure of $K$ by specifying the number of spiders inside $K$ ." ], [ "3 spiders", "The instances with 3 spiders are subsumed by the instances with 2 spiders, since to have 3 spiders we would need two spiders with one leg and one spider with two legs.", "A spider with two legs is either a cup, cap, or the identity morphism, hence these have been included in the 2 spider instances." ], [ "4 spiders", "This gives us a whole range of possible instantiations of $\\mathsf {Compr}$ .", "Some of these options are more interesting than others.", "Options that give us a multiple of the identity matrix or a multiple of $\\sum _{ij} \\mathinner {|{e_i}\\rangle }\\mathinner {\\langle {e_j}|} $ for orthonormal basis $\\lbrace \\mathinner {|{e_i}\\rangle }\\rbrace _i$ are less interesting since this means that all phrase representations will be mapped to the same psd matrix, differing only by a scalar.", "This means that although hyponymy information may be preserved, information about similarity will be lost.", "In the following section we test out a number of options on some phrase entailment datasets.", "We test equations: (REF ), (), (REF ), (REF ) (), (REF ), (REF ), and (REF ), the last of which was already shown to work well in [14]." ], [ "Demonstration", "To test these composition methods, we follow the setup in [14], [16].", "We firstly build psd matrices using GloVe vectors.", "For this small scale demonstration we use GloVe vectors of dimension 50.", "We use a set of datasets that contain pairs of short phrases, for which the first either does or does not entail the second.", "In addition, we use a graded form of the Löwner ordering to measure hyponymy, since in general the crisp Löwner ordering will not be obtained between two psd matrices $A$ and $B$ .", "This graded form is measured as follows.", "Given two psd matrices $A$ and $B$ , if $A \\leqslant B$ then $ A + D = B $ where $D$ is itself a psd matrix.", "If this does not hold, we can add an error term $E$ so that $A + D = B + E.$ In the worst case, we can set $E = A$ , so that $D = B$ , and in fact we will always have that $E \\leqslant A$ .", "A graded measure of hyponymy is obtained by comparing the size of $E$ and $A$ .", "We set $k_E = 1 - \\frac{||E||}{||A||},$ where $||\\cdot ||$ denotes the Euclidean norm, $||A|| =\\sqrt{\\textrm {tr}(A^*A)}$ .", "The crisp Löwner order is recovered in the case that $E = 0$ , so that $k_E = 1$ .", "A second measure of graded hyponymy is obtained as follows: $k_{BA} = \\frac{\\sum _i \\lambda _i}{\\sum _i |\\lambda _i|}$ where $\\lambda _i$ is the $i$ th eigenvalue of $B - A$ and $| \\cdot |$ indicates absolute value.", "This measures the proportions of positive and negative eigenvalues in the expression $B-A$ .", "If all eigenvalues are negative, $k_{BA} = -1$ , and if all are positive, $k_{BA} = 1$ .", "This measure is balanced in the sense that $k_{BA} = - k_{AB}$ ." ], [ "Datasets", "The datasets were originally collected for [11].", "They consist of ordered pairs of short phrases in which the first entails the second, and also the same pair in the opposite order, so that the first phrase does not entail the second.", "The datasets were gathered using WordNet as source.", "The datasets contain intransitive sentences, of the form $\\textit {subject verb}$ , verb phrases, of the form $\\textit {verb object}$ and transitive sentences, of the form $\\textit {subject verb object}$ .", "For example: summer finish, season end, true season end, summer finish, false The datasets have a binary classification, so we measure performance using area under receiver operating characteristic (ROC) curve.", "If we imagine that our graded measure is converted to a binary measure by giving a threshold, area under ROC curve measures performance at all cutoff thresholds.", "A value of 1 means that the graded values are in fact a completely correct binary classification, a value of 0.5 means that the graded values are randomly correlated with the correct classification, and a value of 0 means that the graded values are binary values that are classified in exactly the wrong way (a value of 1 is mapped to 0 and 0 to 1)." ], [ "Models", "We test the following models, for $n, v \\in \\mathcal {M}_m$ .", "We denote by $\\mathrm {diag}(A)$ the matrix obtained by setting all off-diagonal elements of $A$ to 0.", "In order to retain the property that the maximum eigenvalue is less than or equal to 1, we divide by the dimension $m$ or by $m^2$ where necessary.", "Traced noun: $\\textsf {Compr}(n, v) = \\frac{\\textrm {tr}(n)}{m} v$ Traced verb: $\\textsf {Compr}(n, v) = \\frac{\\textrm {tr}(v)}{m} n$ Diag: $\\textsf {Compr}(n, v) = \\mathrm {diag}(n)\\mathrm {diag}(v)$ Summed noun: $\\textsf {Compr}(n, v) = \\frac{v}{m^2} \\sum _{ij} n_{ij}$ Summed verb: $\\textsf {Compr}(n, v) = \\frac{n}{m^2} \\sum _{ij} v_{ij}$ Diag verb: $\\textsf {Compr}(n, v) = \\frac{\\mathrm {diag}(v)}{m^2} \\sum _{ij} n_{ij}$ Diag noun: $\\textsf {Compr}(n, v) = \\frac{\\mathrm {diag}(n)}{m^2} \\sum _{ij} v_{ij}$ Mult: $\\textsf {Compr}(n, v) = \\sum _{ij} v_{ij}n_{ij} \\mathinner {|{e_i}\\rangle }\\mathinner {\\langle {e_j}|}$ Above, we have specified models for sentences of the form $\\textit {subj verb}$ .", "For verb phrases, we treat the verb as $v$ and the object as $n$ , so the models differ based on the grammatical type of the word, rather than its position in the argument list.", "For sentence of the form $\\textit {subject verb object}$ , we first combine the verb and the object, according to their grammatical type, and then treating this verb phrase as an intransitive verb, combine the subject and verb phrase, again according to grammatical type.", "So, for example, iterating the composition Traced Verb on psd matrices $s$ , $v$ , $o$ for subject, verb, and object, we obtain: $\\textsf {Compr}(s, \\textsf {Compr}(o, v)) = \\textsf {Compr}(s, \\frac{\\textrm {tr}(v)}{m} o) = \\frac{\\textrm {tr}(v)\\textrm {tr}(o)}{m^2} s$ We also test two combined models: Traced addition: $\\textsf {Compr}(n, v) = \\frac{\\textrm {tr}(n)}{2m} v + \\frac{\\textrm {tr}(v)}{2m} n$ Summed addition: $\\textsf {Compr}(n, v) = \\frac{v}{m^2} \\sum _{ij} n_{ij} + \\frac{n}{m^2} \\sum _{ij} v_{ij}$ We compare with a verb-only baseline and with Fuzz and Phaser.", "These last two are tested in two directions: Verb only: $\\textsf {Verb only}(n, v) = v$ Fuzz: $\\textsf {Fuzz}(n, v) = \\sum _{i} \\sqrt{p_i} P_i n P_i \\sqrt{p_i}$ where $\\sum _i p_i P_i$ is the spectral decomposition of $v$ Fuzz switched: $\\textsf {Fuzz-s}(n, v) = \\sum _{i} \\sqrt{q_i} Q_i v Q_i \\sqrt{q_i}$ where $\\sum _i q_i Q_i$ is the spectral decomposition of $n$ Phaser: $\\textsf {Phaser}(n, v) = \\sqrt{v} n \\sqrt{v}$ Phaser switched: $\\textsf {Phaser-s}(n, v) = \\sqrt{n} v \\sqrt{n}$ To test for significance of our results, we bootstrap the data with 100 repetitions [7] and compare between models using a two sample t-test.", "We apply the Bonferroni correction to compensate for multiple comparisons." ], [ "Results", "Results are presented in table REF .", "A key point is that as in previous work, the $k_{BA}$ measure performs better than the $k_E$ measure.", "Furthermore, across all datasets, the models Traced verb and Summed verb perform much more highly than simply taking the verb on its own, indicating that information about at least the size of the noun is crucial.", "Across both measures, performance is highest on the SVO dataset and lowest on the VO dataset.", "This may be due to the construction of the datasets, or it may be due to these composition methods working well on longer phrases.", "Within the results using the $k_E$ measure, the model Diag verb is strong across all datasets.", "This is surprising, as taking the diagonal of the verb would seem to result in information loss.", "Within the results using the $k_E$ measure, the new models largely outperform Phaser, and on the VO dataset, outperform Fuzz too.", "Within the results using the $k_{BA}$ measure, the picture is less clear.", "Diag, Mult, and Traced addition are all fairly strong, but there is no outright best model.", "Perhaps looking at some other combination possibilities would be useful.", "Table: Area under ROC curve for k E k_E and k BA k_{BA} graded hyponymy measures.", "Figures are mean values of 100 samples taken from each dataset with replacement.", "- * -^* indicates significantly better than both variants of Fuzz, p<0.01p < 0.01, - + -^+ indicates significantly better than both variants of Phaser, p<0.01p < 0.01." ], [ "Discussion", "We have presented a general composition rule called $\\textsf {Compr}$ for converting a psd matrix for a functional word such as a verb or an adjective into a CP map that matches the grammatical type of the word.", "$\\textsf {Compr}$ preserves hyponymy, in contrast to previous approaches like $\\textsf {Fuzz}$ and $\\textsf {Phaser}$ .", "While in full generality we would want to learn the parameters of $\\textsf {Compr}$ from a text corpus, as a first step we have defined the structure of $\\textsf {Compr}$ using just cups, caps, and spiders.", "Results on the text datasets are promising, although there is no completely clear advantage over $\\textsf {Fuzz}$ or $\\textsf {Phaser}$ .", "The approach we have taken, namely that of defining a map that converts representations of functional words to a higher-order type, has also been seen in vector-based models of meaning.", "In [12], [9], word vectors and also matrices are converted using Frobenius algebras, of which the composition Mult is a direct analogue.", "Furthermore, in [20], and recapitulated in [15], a bilinear map $C: N \\otimes N \\rightarrow N$ that gives the composition of two vectors is proposed.", "Under this approach, we would have $\\rho ( \\textit {subj verb} ) = C(\\rho ( \\textit {subj} ) \\otimes \\rho ( \\textit {verb} ))\\text{ and }\\rho ( \\textit {subj verb obj} ) = C(\\rho ( \\textit {subj} ) \\otimes C(\\rho ( \\textit {verb} ) \\otimes \\rho ( \\textit {obj} )))$ Our approach is an analogue to this one within the realm of psd matrices and CP maps.", "There are a number of strands to this work to be continued.", "We would like to learn $\\textsf {Compr}$ directly from text, rather than specifying the structure by hand.", "More freedom in the parameters of $\\textsf {Compr}$ means that we could define it as a CP map $\\textsf {Compr}\\colon \\mathcal {M}_m\\rightarrow \\mathcal {M}_m\\otimes \\mathcal {M}_s.$ where we are then matching the grammatical types more exactly.", "In addition, the psd matrices we use are built using human curated resources – ideally these would be learnt in a less supervised manner directly from text corpora.", "At present, we have given two possible graded measures of hyponymy – more research into these measures is needed, including how they interact with the composition methods we have specified.", "Work is currently ongoing to develop a model of negation within this framework [16]." ], [ "Categorical compositional distributional semantics", "In this appendix we give a brief introduction to the categorical compositional approach to distributional semantics – for details see [6], [12]." ], [ "Compositional distributional semantics", "We start by reviewing some of the category theory used in categorical compositional models of meaning.", "Definition 1 A monoidal category is a tuple $(\\mathbf {C}, \\otimes , I)$ where $\\mathbf {C}$ is a category, meaning that: $\\mathbf {C}$ has a collection of objects $A, B, ...$ and each ordered pair of objects $(A, B)$ has a collection of morphisms $f: A \\rightarrow B$ For each triple of objects $(A,B, C)$ and morphisms $f: A \\rightarrow B$ , $g: B \\rightarrow C$ there is a sequential composite $g \\circ f: A \\rightarrow C$ that is associative, i.e.", "$h \\circ (g\\circ f) = (h \\circ g) \\circ f$ for each object $A$ there is an identity morphism $1_A: A \\rightarrow A$ such that for $f: A \\rightarrow B$ $f \\circ 1_A = f \\quad and \\quad 1_B \\circ f = f$ for each ordered pair of objects $(A, B)$ , there is a composite object $A \\otimes B$ , and we moreover require that: $A \\otimes (B \\otimes C) \\cong (A\\otimes B) \\otimes C$ where $\\cong $ means `is isomorphic to'.", "there is a unit object $I$ , which satisfies $I \\otimes A \\cong A \\cong A \\otimes I$ for each ordered pair of morphisms $f: A \\rightarrow B$ , $g: B \\rightarrow C$ there is a parallel composite $f \\otimes g : A \\otimes B \\rightarrow C \\otimes D$ which satisfies: $(g_1 \\otimes g_2) \\circ (f_1 \\otimes f_2) = (g_1 \\circ f_1) \\otimes (g_2 \\circ f_2)$ For a precise statement and discussion of the above definition, we direct the reader to [17], and [5] for a more gentle introduction.", "Monoidal categories can be given a graphical calculus as in figure REF .", "Figure: Monoidal graphical calculus.By convention the wire for the monoidal unit is omitted.", "As will be seen in section REF , we will require that the grammar category and the meaning category are both a particular kind of monoidal category, called compact closed.", "Definition 2 A monoidal category $(C,\\otimes , I)$ is compact closed if for each object $A\\in C$ there are objects $A^l,A^r\\in C$ (the left and right duals of $A$ ) and morphisms $\\eta _A^l : I\\rightarrow A\\otimes A^l, \\quad \\eta _A^r : I\\rightarrow A^r\\otimes A, \\quad \\epsilon _A^l : A^l\\otimes A \\rightarrow I, \\quad \\epsilon _A^r : A\\otimes A^r\\rightarrow I$ satisfying the snake equations $&(1_A \\otimes \\epsilon ^l_A )\\circ (\\eta ^l_A \\otimes 1_A) = 1_A &\\:& (\\epsilon ^r_A \\otimes 1_A) \\circ (1_A \\otimes \\eta ^r_A ) = 1_A\\\\&(\\epsilon ^l_A \\otimes 1_{A^l})\\circ (1_{A^l} \\otimes \\eta ^l_A)= 1_{A^l} &\\:& (1_{A^r} \\otimes \\epsilon ^r_A ) \\circ (\\eta ^r_A \\otimes 1_{A^r} ) = 1_{A^r}$ The $\\epsilon $ and $\\eta $ maps are called cups and caps respectively, and can also be depicted in the graphical calculus as in figure REF .", "The snake equations are depicted graphically as in figure REF .", "Figure: Compact structure graphically.Figure: The snake equations.Finally, we introduce the notion of a Frobenius algebra over a real finite-dimensional Hilbert space.", "For a mathematically rigorous presentation see [24].", "A real Hilbert space with a fixed orthonormal basis $\\lbrace \\mathinner {|{v_i}\\rangle }\\rbrace _i$ has a Frobenius algebra given by: $\\Delta : V \\rightarrow V \\otimes V ::\\mathinner {|{v_i}\\rangle } \\mapsto \\mathinner {|{v_i}\\rangle } \\otimes \\mathinner {|{v_i}\\rangle } \\quad \\iota : V \\rightarrow \\mathbb {R} :: \\mathinner {|{v_i}\\rangle } \\mapsto 1$ $\\mu : V \\otimes V \\rightarrow V :: \\mathinner {|{v_i}\\rangle } \\otimes \\mathinner {|{v_j}\\rangle }\\mapsto \\delta _{ij} \\mathinner {|{v_i}\\rangle } \\quad \\zeta : \\mathbb {R} \\rightarrow V :: 1 \\mapsto \\sum _i \\mathinner {|{v_i}\\rangle }$ This algebra is commutative, so for the swap map $\\sigma : X \\otimes Y \\rightarrow Y\\otimes X$ , we have $\\sigma \\circ \\Delta = \\Delta $ and $\\mu \\circ \\sigma = \\mu $ .", "It is also special so that $\\mu \\circ \\Delta = 1$ .", "Essentially, the $\\mu $ morphism amounts to taking the diagonal of a matrix, and $\\Delta $ to embedding a vector within a diagonal matrix.", "Diagrammatically, these are represented as follows: $\\Delta ={\\begin{array}{c}\\begin{tikzpicture}[scale=0.3]\\begin{pgfonlayer}{nodelayer}\\node [style=small circ] (0) at (0, 0) {};\\node [style=none] (1) at (1, -1) {};\\node [style=none] (2) at (0, 1.5) {V};\\node [style=none] (3) at (-1, -1.5) {V};\\node [style=none] (4) at (-1, -1) {};\\node [style=none] (5) at (0, 1) {};\\node [style=none] (6) at (1, -1.5) {V};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[bend right=45, looseness=1.25] (0) to (4.center);[bend left=45, looseness=1.25] (0) to (1.center);(5.center) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}},\\qquad \\iota ={\\begin{array}{c}\\begin{tikzpicture}[scale=0.5]\\begin{pgfonlayer}{nodelayer}\\node [style=small circ] (0) at (0, -0.5) {};\\node [style=none] (1) at (0, 1) {V};\\node [style=none] (2) at (0, 0.5) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(2.center) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}}, \\qquad \\mu ={\\begin{array}{c}\\begin{tikzpicture}[scale=0.3]\\begin{pgfonlayer}{nodelayer}\\node [style=small circ] (0) at (0, 0) {};\\node [style=none] (1) at (1, 1) {};\\node [style=none] (2) at (0, -1.5) {V};\\node [style=none] (3) at (-1, 1.5) {V};\\node [style=none] (4) at (-1, 1) {};\\node [style=none] (5) at (0, -1) {};\\node [style=none] (6) at (1, 1.5) {V};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[bend left=45, looseness=1.25] (0) to (4.center);[bend right=45, looseness=1.25] (0) to (1.center);(5.center) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}},\\qquad \\zeta ={\\begin{array}{c}\\begin{tikzpicture}[scale=0.5]\\begin{pgfonlayer}{nodelayer}\\node [style=small circ] (0) at (0, 0.5) {};\\node [style=none] (1) at (0, -1) {V};\\node [style=none] (2) at (0, -0.5) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(2.center) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}}$ Frobenius structures can have any number of wires in and out, and moreover, fuse together, so that the only thing that matters about these morphisms are how many wires in and out they have." ], [ "Meaning: from vectors to positive semidefinite matrices", "We wish to model words as positive semidefinite matrices.", "For this to work within the categorical compositional approach, we require that psd matrices have a home within a compact closed category.", "We can provide this home by using the $\\mathbf {CPM}$ construction [26], applied to $\\mathbf {FHilb}$ .", "Throughout this section $\\mathcal {C}$ denotes an arbitrary $\\dag $ -compact closed category.", "Definition 3 (Completely positive morphism [26]) A $\\mathcal {C}$ -morphism $\\varphi : A^* \\otimes A \\rightarrow B^* \\otimes B$ is said to be completely positive if there exists $C \\in \\mathsf {Ob}(\\mathcal {C})$ and $k \\in \\mathcal {C}(C\\otimes A, B)$ , such that $\\varphi $ can be written in the form: $(k_* \\otimes k) \\circ (1_{A^*} \\otimes \\eta _C \\otimes 1_A)$ where in $\\mathbf {FHilb}$ , $k_*$ is the complex conjugate of $k$ .", "Identity morphisms are completely positive, and completely positive morphisms are closed under composition in $\\mathcal {C}$ , leading to the following: Definition 4 If $\\mathcal {C}$ is a $\\dag $ -compact closed category then $\\mathbf {CPM}(\\mathcal {C})$ is a category with the same objects as $\\mathcal {C}$ and its morphisms are the completely positive morphisms.", "Note that morphisms are defined for objects of the form $A^*\\otimes A$ , where $A$ is an object in $\\mathcal {C}$ .", "The $\\dagger $ -compact structure required for interpreting language in our setting lifts to $\\mathbf {CPM}(\\mathcal {C})$ : Theorem 2 ([26]) $\\mathbf {CPM}(\\mathcal {C})$ is also a $\\dagger $ -compact closed category.", "There is a functor: $\\mathsf {E}: \\mathcal {C} &\\rightarrow \\mathbf {CPM}(\\mathcal {C})\\\\k &\\mapsto k_* \\otimes k$ where $k_*$ denotes the complex conjugate of $k$ .", "This functor preserves the $\\dagger $ -compact closed structure, and is faithful “up to a global phase”.", "It follows that applying the $\\mathbf {CPM}$ construction to $\\mathbf {FHilb}$ , we obtain a $\\dagger $ -compact closed category.", "The objects of the category are finite dimensional Hilbert spaces, and the morphisms are completely positive maps.", "The compact closed structure is summarised in table REF .", "Objects of $\\mathbf {CPM}(\\mathbf {FHilb})$ can be given a Frobenius algebra, summarised in table REF .", "Table: Table of diagrams in 𝐂𝐏𝐌(𝒞)\\mathbf {CPM}(\\mathcal {C}) and 𝒞\\mathcal {C}Table: Table of diagrams for Frobenius algebras in 𝐂𝐏𝐌(𝒞)\\mathbf {CPM}(\\mathcal {C}) and 𝒞\\mathcal {C}" ], [ "Pregroup grammar", "We use Lambek's pregroup grammar [13].", "A pregroup $(P, \\le , \\cdot , 1, (-)^l, (-)^r)$ is a partially ordered monoid $(P, \\le , \\cdot , 1)$ where each element $p\\in P$ has a left adjoint $p^l$ and a right adjoint $p^r$ , such that the following inequalities hold: $p^l\\cdot p \\le 1 \\le p\\cdot p^l \\quad \\text{ and } \\quad p\\cdot p^r \\le 1 \\le p^r \\cdot p$ Intuitively, we think of the elements of a pregroup as linguistic types.", "The monoidal structure allows us to form composite types, and the partial order encodes type reduction.", "The important right and left adjoints then enable the introduction of types requiring further elements on either their left or right respectively.", "We understand a pregroup as a compact closed category in the following way.", "The objects of the category are the elements of the set $P$ .", "The tensor and unit are the monoid multiplication and unit $\\cdot $ , 1, and cup and caps are the morphisms witnessed by the inequalities in (REF ).", "The pregroup grammar $\\mathsf {Preg}_{\\mathcal {B}}$ over an alphabet $\\mathcal {B}$ is freely constructed from the atomic types in $\\mathcal {B}$ .", "Here we use an alphabet $\\mathcal {B} = \\lbrace n, s\\rbrace $ , where we use the type $s$ to denote a declarative sentence and $n$ to denote a noun.", "A transitive verb can then be denoted $n^r s n^l$ .", "If a string of words and their types reduces to the type $s$ , the sentence is judged grammatical.", "The sentence $\\textit {dogs chase cars}$ is typed $n~(n^r s n^l)~ n$ , and can be reduced to $s$ as follows: $n~(n^r s n^l)~ n \\le 1\\cdot s n^l n \\le 1 \\cdot s \\cdot 1 \\le s$ This symbolic reduction can also be expressed graphically, as shown in figure REF .", "Figure: A transitive sentence in the graphical calculus." ], [ "Mapping from grammar to meaning", "We now describe a functor from the grammar category $\\mathsf {Preg}_{\\lbrace n, s\\rbrace }$ to $\\mathbf {CPM}(\\mathbf {FHilb})$ that tells us how to compose word representations to form phrases and sentences.", "The reductions of the pregroup grammar may be mapped into $\\mathbf {CPM}(\\mathbf {FHilb})$ using a strong monoidal functor $\\mathsf {S}$ : $\\mathsf {S}: \\mathbf {Preg} \\rightarrow \\mathbf {CPM}(\\mathbf {FHilb})$ Strong monoidal functors automatically preserve the compact closed structure.", "For our example $\\mathsf {Preg}_{\\lbrace n,s\\rbrace }$ , we must map the noun and sentence types to appropriate finite dimensional vector spaces: $\\mathsf {S}(n) = N^* \\otimes N \\qquad \\mathsf {S}(s) = S^* \\otimes S$ where $N$ , $S$ are finite dimensional Hilbert spaces, i.e.", "objects of $\\mathbf {FHilb}$ .", "Composite types are then constructed functorially using the corresponding structure in $\\mathbf {FHilb}$ .", "Each morphism $\\alpha $ in the pregroup is mapped to a completely positive map interpreting sentences of that grammatical type.", "Since the only basic morphisms in the pregroup are identity, cups, and caps, $\\alpha $ consists of tensor products and compositions of these.", "Then, given psd matrices for words $\\rho ( w_i )$ with pregroup types $p_i$ , and a type reduction in the pregroup grammar $\\alpha : p_1, p_2, ... p_n \\rightarrow s$ , the meaning of the sentence $w_1 w_2 ... w_n$ is given by: $\\rho ( w_1 w_2 ... w_n ) = \\mathsf {S}(\\alpha )(\\rho ( w_1 ) \\otimes \\rho ( w_2 ) \\otimes ... \\otimes \\rho ( w_n ))$ Example 1 Let the space $N$ be a real Hilbert space with basis vectors given by $\\lbrace \\mathinner {|{n_i}\\rangle }\\rbrace _i$ , and suppose we have $\\rho ( \\textit {cars} ) = \\sum _{ij} c_{ij}\\mathinner {|{n_i}\\rangle }\\mathinner {\\langle {n_j}|}, \\quad \\rho ( \\textit {dogs} ) = \\sum _{kl} d_{kl}\\mathinner {|{n_k}\\rangle }\\mathinner {\\langle {n_l}|}$ Let $S$ have basis $\\lbrace \\mathinner {|{s_i}\\rangle }\\rbrace _i$ .", "The verb $\\rho ( \\textit {chase} )$ is given by: $\\rho ( \\textit {chase} ) = \\sum _{pqrtuv} C_{pqrtuv} \\mathinner {|{n_p}\\rangle }\\mathinner {\\langle {n_t}|} \\otimes \\mathinner {|{s_q}\\rangle }\\mathinner {\\langle {s_u}|} \\otimes \\mathinner {|{n_r}\\rangle }\\mathinner {\\langle {n_v}|}$ The meaning of the composite sentence is $(\\epsilon _{N^* \\otimes N} \\otimes 1_{S^*\\otimes S} \\otimes \\epsilon _{N^* \\otimes N})$ applied to $(\\rho ( \\textit {dogs} ) \\otimes \\rho ( \\textit {chase} ) \\otimes \\rho ( \\textit {cars} ))$ as shown below in (REF ), with interpretation in $\\mathbf {FHilb}$ shown in (REF ).", "$\\mathbf {CPM}(\\mathbf {FHilb}): {\\begin{array}{c}\\begin{tikzpicture}[scale=0.8]\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-2, 2) {dogs};\\node [style=none] (1) at (0, 2) {chase};\\node [style=none] (2) at (2, 2) {cars};\\node [style=blank] (3) at (0, 0.5) {S};\\node [style=blank] (4) at (2, 0.5) {N};\\node [style=none] (5) at (-2.5, 1) {};\\node [style=none] (6) at (-1.5, 1) {};\\node [style=none] (7) at (-2, 1.5) {};\\node [style=none] (8) at (-0.75, 1) {};\\node [style=none] (9) at (0.75, 1) {};\\node [style=none] (10) at (0, 1.5) {};\\node [style=blank] (11) at (-0.5, 0.5) {N};\\node [style=blank] (12) at (0.5, 0.5) {N};\\node [style=none] (13) at (1.5, 1) {};\\node [style=none] (14) at (2.5, 1) {};\\node [style=none] (15) at (2, 1.5) {};\\node [style=none] (16) at (0, -0.5) {};\\node [style=blank] (17) at (-2, 0.5) {N};\\node [style=none] (18) at (-2, 1) {};\\node [style=none] (19) at (0, 1) {};\\node [style=none] (20) at (2, 1) {};\\node [style=none] (21) at (0.5, 1) {};\\node [style=none] (22) at (-0.5, 1) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[ultra thick] (7.center) to (5.center);[ultra thick] (5.center) to (6.center);[ultra thick] (6.center) to (7.center);[ultra thick] (8.center) to (10.center);[ultra thick] (8.center) to (9.center);[ultra thick] (9.center) to (10.center);[ultra thick] (15.center) to (13.center);[ultra thick] (13.center) to (14.center);[ultra thick] (14.center) to (15.center);[ultra thick, bend right=90, looseness=1.25] (17) to (11);[ultra thick] (3) to (16.center);[ultra thick, bend left=90, looseness=1.25] (4) to (12);[ultra thick] (11) to (22.center);[ultra thick] (3) to (19.center);[ultra thick] (12) to (21.center);[ultra thick] (4) to (20.center);[ultra thick] (17) to (18.center);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}}$ $\\mathbf {FHilb}: {\\begin{array}{c}\\begin{tikzpicture}[scale=0.8, text height = 1.5 ex]\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0.25, -0.75) {};\\node [style=none] (1) at (0.75, 0) {};\\node [style=none] (2) at (1.25, -0.75) {};\\node [style=none] (3) at (3.25, -0.75) {};\\node [style=none] (4) at (3.75, 0) {};\\node [style=none] (5) at (3, 0) {};\\node [style=none] (6) at (3, 0.5) {};\\node [style=none] (7) at (3.5, 0.5) {};\\node [style=none] (8) at (3.25, 0.5) {};\\node [style=none] (9) at (2.75, 0.5) {};\\node [style=none] (10) at (2, 0) {};\\node [style=none] (11) at (2.25, 0.5) {};\\node [style=none] (12) at (2.5, 0.5) {};\\node [style=none] (13) at (2.5, -0.75) {};\\node [style=none] (14) at (2.75, 0) {};\\node [style=none] (15) at (1.75, 0) {};\\node [style=none] (16) at (1.5, 0.5) {};\\node [style=none] (17) at (0, 0.5) {};\\node [style=none] (18) at (0, 0) {};\\node [style=blank] (19) at (0.25, -0.5) {N^*};\\node [style=blank] (20) at (0.75, -0.5) {S};\\node [style=blank] (21) at (1.25, -0.5) {N^*};\\node [style=blank] (22) at (2.5, -0.5) {N^*};\\node [style=blank] (23) at (3.25, -0.5) {N};\\node [style=none] (24) at (0.25, 0.5) {};\\node [style=none] (25) at (0.75, 0.5) {};\\node [style=none] (26) at (1.25, 0.5) {};\\node [style=none] (27) at (0.75, -2) {};\\node [style=none] (28) at (-1, 0) {};\\node [style=none] (29) at (-3, 0.5) {};\\node [style=none] (30) at (-1.5, 0.5) {};\\node [style=blank] (31) at (-0.5, -0.5) {N};\\node [style=blank] (32) at (-2.75, -0.5) {N};\\node [style=none] (33) at (-1.75, 0.5) {};\\node [style=none] (34) at (-0.5, -0.75) {};\\node [style=none] (35) at (-3.25, 0) {};\\node [style=none] (36) at (-2, 0) {};\\node [style=blank] (37) at (-3.5, -0.5) {N^*};\\node [style=none] (38) at (-2.75, -0.75) {};\\node [style=none] (39) at (-2.25, 0) {};\\node [style=none] (40) at (-4, 0) {};\\node [style=none] (41) at (-3.5, 0.5) {};\\node [style=none] (42) at (-0.25, 0) {};\\node [style=none] (43) at (-2.5, 0.5) {};\\node [style=none] (44) at (-1, -2) {};\\node [style=none] (45) at (-3.25, 0.5) {};\\node [style=blank] (46) at (-1.5, -0.5) {N};\\node [style=none] (47) at (-1, 0.5) {};\\node [style=none] (48) at (-0.5, 0.5) {};\\node [style=none] (49) at (-1.5, -0.75) {};\\node [style=none] (50) at (-2.75, 0.5) {};\\node [style=none] (51) at (-3, 0) {};\\node [style=none] (52) at (-3.75, 0.5) {};\\node [style=none] (53) at (-0.25, 0.5) {};\\node [style=blank] (54) at (-1, -0.5) {S^*};\\node [style=none] (55) at (-3.5, -0.75) {};\\node [style=none] (56) at (-3.5, 0) {};\\node [style=none] (57) at (-2.75, 0) {};\\node [style=none] (58) at (-1.5, 0) {};\\node [style=none] (59) at (-0.5, 0) {};\\node [style=none] (60) at (0.25, 0) {};\\node [style=none] (61) at (1.25, 0) {};\\node [style=none] (62) at (2.5, 0) {};\\node [style=none] (63) at (3.25, 0) {};\\node [style=none] (64) at (-3, 1.5) {dogs};\\node [style=none] (65) at (0, 1.5) {chase};\\node [style=none] (66) at (2.75, 1.5) {cars};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(52.center) to (40.center);(40.center) to (35.center);(35.center) to (45.center);(45.center) to (52.center);(29.center) to (51.center);(51.center) to (39.center);(39.center) to (43.center);(43.center) to (29.center);(33.center) to (36.center);(36.center) to (42.center);(42.center) to (53.center);(53.center) to (33.center);(17.center) to (18.center);(18.center) to (15.center);(15.center) to (16.center);(16.center) to (17.center);(54) to (44.center);(20) to (27.center);(11.center) to (10.center);(9.center) to (14.center);(10.center) to (14.center);(9.center) to (11.center);(6.center) to (5.center);(5.center) to (4.center);(4.center) to (7.center);(7.center) to (6.center);(28.center) to (54);(1.center) to (20);[bend left=90] (41.center) to (50.center);[bend left=90] (48.center) to (24.center);[bend left=90, looseness=0.75] (47.center) to (25.center);[bend left=90, looseness=0.75] (30.center) to (26.center);[bend right=90] (8.center) to (12.center);(56.center) to (37);(57.center) to (32);(58.center) to (46);(59.center) to (31);(60.center) to (19);(61.center) to (21);(62.center) to (22);(63.center) to (23);[bend left=90, looseness=0.75] (3.center) to (2.center);[bend left=90, looseness=0.50] (13.center) to (49.center);[bend right=90, looseness=0.50] (55.center) to (34.center);[bend right=90, looseness=0.75] (38.center) to (0.center);\\end{pgfonlayer}\\end{tikzpicture} \\end{array}} \\cong {\\begin{array}{c}\\begin{tikzpicture}[scale=0.8, text height=1.5 ex]\\begin{pgfonlayer}{nodelayer}\\node [style=none] (1) at (0, 0) {};\\node [style=none] (4) at (2.25, 0.25) {};\\node [style=none] (5) at (1.25, 0.25) {};\\node [style=none] (6) at (1.25, 0.75) {};\\node [style=none] (7) at (2.25, 0.75) {};\\node [style=none] (8) at (1.75, 0.75) {};\\node [style=none] (15) at (0.75, 0) {};\\node [style=none] (16) at (0.75, 1) {};\\node [style=none] (17) at (-0.75, 1) {};\\node [style=none] (18) at (-0.75, 0) {};\\node [style=blank] (19) at (-0.5, -0.5) {N^*};\\node [style=blank] (20) at (0, -0.5) {S};\\node [style=blank] (21) at (0.5, -0.5) {N^*};\\node [style=blank] (23) at (1.75, -0.5) {N};\\node [style=none] (24) at (-0.5, 1) {};\\node [style=none] (25) at (0, 1) {};\\node [style=none] (26) at (0.5, 1) {};\\node [style=none] (27) at (0, -1.5) {};\\node [style=none] (29) at (-2.25, 0.75) {};\\node [style=blank] (32) at (-1.75, -0.5) {N};\\node [style=none] (39) at (-1.25, 0.25) {};\\node [style=none] (43) at (-1.25, 0.75) {};\\node [style=none] (49) at (-1.75, -0.75) {};\\node [style=none] (50) at (-1.75, 0.75) {};\\node [style=none] (51) at (-2.25, 0.25) {};\\node [style=none] (57) at (-1.75, 0.25) {};\\node [style=none] (60) at (-0.5, 0) {};\\node [style=none] (61) at (0.5, 0) {};\\node [style=none] (63) at (1.75, 0.25) {};\\node [style=none] (64) at (-1.75, 0.5) {dogs};\\node [style=none] (65) at (0, 0.5) {chase};\\node [style=none] (66) at (1.75, 0.5) {cars};\\node [style=none] (67) at (0, 2) {};\\node [style=none] (68) at (0.5, 1.25) {};\\node [style=none] (69) at (1.75, 1.25) {};\\node [style=none] (70) at (-0.5, 1.25) {};\\node [style=none] (71) at (-1.75, 1.25) {};\\node [style=none] (72) at (-0.5, -0.75) {};\\node [style=none] (73) at (0.5, -0.75) {};\\node [style=none] (74) at (1.75, -0.75) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(29.center) to (51.center);(51.center) to (39.center);(39.center) to (43.center);(43.center) to (29.center);(17.center) to (18.center);(18.center) to (15.center);(15.center) to (16.center);(16.center) to (17.center);(20) to (27.center);(6.center) to (5.center);(5.center) to (4.center);(4.center) to (7.center);(7.center) to (6.center);(1.center) to (20);(57.center) to (32);(60.center) to (19);(61.center) to (21);(63.center) to (23);[bend left=90, looseness=1.25] (71.center) to (70.center);(67.center) to (25.center);[bend left=90, looseness=1.25] (68.center) to (69.center);(68.center) to (26.center);(70.center) to (24.center);(69.center) to (66.north);[bend right=90, looseness=1.25] (49.center) to (72.center);[bend right=90, looseness=1.25] (73.center) to (74.center);(71.center) to (50.center);\\end{pgfonlayer}\\end{tikzpicture}\\end{array}}$" ] ]

[ [ "Causal Impact of Masks, Policies, Behavior on Early Covid-19 Pandemic in\n the U.S" ], [ "Abstract This paper evaluates the dynamic impact of various policies adopted by US states on the growth rates of confirmed Covid-19 cases and deaths as well as social distancing behavior measured by Google Mobility Reports, where we take into consideration people's voluntarily behavioral response to new information of transmission risks.", "Our analysis finds that both policies and information on transmission risks are important determinants of Covid-19 cases and deaths and shows that a change in policies explains a large fraction of observed changes in social distancing behavior.", "Our counterfactual experiments suggest that nationally mandating face masks for employees on April 1st could have reduced the growth rate of cases and deaths by more than 10 percentage points in late April, and could have led to as much as 17 to 55 percent less deaths nationally by the end of May, which roughly translates into 17 to 55 thousand saved lives.", "Our estimates imply that removing non-essential business closures (while maintaining school closures, restrictions on movie theaters and restaurants) could have led to -20 to 60 percent more cases and deaths by the end of May.", "We also find that, without stay-at-home orders, cases would have been larger by 25 to 170 percent, which implies that 0.5 to 3.4 million more Americans could have been infected if stay-at-home orders had not been implemented.", "Finally, not having implemented any policies could have led to at least a 7 fold increase with an uninformative upper bound in cases (and deaths) by the end of May in the US, with considerable uncertainty over the effects of school closures, which had little cross-sectional variation." ], [ "Introduction", "Accumulating evidence suggests that various policies in the US have reduced social interactions and slowed down the growth of Covid-19 infections.See [23], [39], [61], [2], and [82].", "An important outstanding issue, however, is how much of the observed slow down in the spread is attributable to the effect of policies as opposed to a voluntarily change in people's behavior out of fear of being infected.", "This question is critical for evaluating the effectiveness of restrictive policies in the US relative to an alternative policy of just providing recommendations and information such as the one adopted by Sweden.", "More generally, understanding people's dynamic behavioral response to policies and information is indispensable for properly evaluating the effect of policies on the spread of Covid-19.", "This paper quantitatively assesses the impact of various policies adopted by US states on the spread of Covid-19, such as non-essential business closure and mandatory face masks, paying particular attention to how people adjust their behavior in response to policies as well as new information on cases and deaths.", "We present a conceptual framework that spells out the causal structure on how the Covid-19 spread is dynamically determined by policies and human behavior.", "Our approach explicitly recognizes that policies not only directly affect the spread of Covid-19 (e.g., mask requirement) but also indirectly affect its spread by changing people's behavior (e.g., stay-at-home order).", "It also recognizes that people react to new information on Covid-19 cases and deaths and voluntarily adjust their behavior (e.g., voluntary social distancing and hand washing) even without any policy in place.", "Our casual model provides a framework to quantitatively decompose the growth of Covid-19 cases and deaths into three components: (1) direct policy effect, (2) policy effect through behavior, and (3) direct behavior effect in response to new information.", "Guided by the causal model, our empirical analysis examines how the weekly growth rates of confirmed Covid-19 cases and deaths are determined by (the lags of) policies and behavior using US state-level data.", "To examine how policies and information affect people's behavior, we also regress social distancing measures on policy and information variables.", "Our regression specification for case and death growths is explicitly guided by a SIR model although our causal approach does not hinge on the validity of a SIR model.", "As policy variables, we consider mandatory face masks for employees in public businesses, stay-at-home orders (or shelter-in-place orders), closure of K-12 schools, closure of restaurants except take out, closure of movie theaters, and closure of non-essential businesses.", "Our behavior variables are four mobility measures that capture the intensity of visits to “transit,” “grocery,” “retail,” and “workplaces” from Google Mobility Reports.", "We take the lagged growth rate of cases and deaths and the log of lagged cases and deaths at both the state-level and the national-level as our measures of information on infection risks that affects people's behavior.", "We also consider the growth rate of tests, month dummies, and state-level characteristics (e.g., population size and total area) as confounders that have to be controlled for in order to identify the causal relationship between policy/behavior and the growth rate of cases and deaths.", "Our key findings from regression analysis are as follows.", "We find that both policies and information on past cases and deaths are important determinants of people's social distancing behavior, where policy effects explain more than $50\\%$ of the observed decline in the four behavior variables.The behavior accounts for the other half.", "This is in line with theoretical study by [27] that investigates the role of private behavior and negative external effects for individual decisions over policy compliance as well as information acquisition during pandemics.", "Our estimates suggest that there are both large policy effects and large behavioral effects on the growth of cases and deaths.", "Except for mandatory masks, the effect of policies on cases and deaths is indirectly materialized through their impact on behavior; the effect of mandatory mask policy is direct without affecting behavior.", "Using the estimated model, we evaluate the dynamic impact of the following three counterfactual policies on Covid-19 cases and deaths: (1) mandating face masks, (2) allowing all businesses to open, and (3) not implementing a stay-at-home order.", "The counterfactual experiments show a large impact of those policies on the number of cases and deaths.", "They also highlight the importance of voluntary behavioral response to infection risks when evaluating the dynamic policy effects.", "Our estimates imply that nationally implementing mandatory face masks for employees in public businesses on March 14th would have reduced the growth rate of cases and that of deaths by approximately 10 percentage points in late April.", "As shown in Figure REF , this leads to reductions of 21% and 34% in cumulative reported cases and deaths, respectively, by the end of May with 90 percent confidence intervals of $[9,32]$ % and $[19,47]$ %, which roughly implies that 34 thousand lives could have been saved.", "This finding is significant: given this potentially large benefit of reducing the spread of Covid-19, mandating masks is an attractive policy instrument especially because it involves relatively little economic disruption.", "These estimates contribute to the ongoing efforts towards designing approaches to minimize risks from reopening [72].", "Figure REF illustrates how never closing any businesses (no movie theater closure, no non-essential business closure, and no closure of restaurants except take-out) could have affected cases and deaths.", "We estimate that business shutdowns have roughly the same impact on growth rates as mask mandates, albeit with more uncertainty.", "The point estimates indicate that keeping all businesses open could have increased cumulative cases and deaths by $40\\%$ at the end of May (with 90 percent confidence intervals of $[17,78]$ % for cases and $[1,97]$ % for deaths).", "Figure REF shows that stay-at-home orders had effects of similar magnitude as business closures.", "No stay-at-home orders could have led to 37% more cases by the start of June with a 90 percent confidence interval given by 6% to 63%.", "The estimated effect of no stay-at-home orders on deaths is a slightly smaller with a 90 percent confidence interval of $-7$ % to 50%.", "We also conducted sensitivity analysis with respect to changes to our regression specification, sample selection, methodology, and assumptions about delays between policy changes and changes in recorded cases.", "In particular, we examined whether certain effect sizes can be ruled out by various more flexible models or by using alternative timing assumptions that define forward growth rates.", "The impact of mask mandates is more robustly and more precisely estimated than that of business closure policies or stay-at-home orders, and an undesirable effect of increasing the weekly death growth by 5 percentage points is ruled out by all of the models we consider.This null hypothesis can be generated by looking at the meta-evidence from RCTs on the efficacy of masks in preventing other respiratory cold-like deceases.", "Falsely rejecting this null is costly in terms of potential loss of life, and so it is a reasonable null choice for the mask policy from decision-theoretic point of view.", "This is largely due to the greater variation in the timing of mask mandates across states.", "The findings of shelter-in-place and business closures policies are considerably less robust.", "For example, for stay-at-home mandates, models with alternative timing and richer specification for information set suggested smaller effects.", "Albeit after application of machine learning tools to reduce dimensionality, the range of effects $[0,0.15]$ could not be ruled out.", "A similar wide range of effects could not be ruled out for business closures.", "We also examine the impact of school closures.", "Unfortunately, there is very little variation across states in the timing of school closures.", "Across robustness specifications, we obtain point estimates of the effect of school closures as low as 0 and as high as -0.6.", "In particular, we find that the results are sensitive to whether the number of past national cases/deaths is included in a specification or not.", "This highlights the uncertainty regarding the impact of some policies versus private behavioral responses to information.", "Figure: Relative cumulative effect on confirmed cases and fatalitiesof nationally mandating masks for employees on March14th in the USFigure: Relative cumulative effect of no business closure policies on cases and fatalities in the USFigure: Relative cumulative effect of not implementing stay-at-home order on cases and fatalities in the USA growing number of other papers have examined the link between non-pharmaceutical interventions and Covid-19 cases.We refer the reader to [10] for a comprehensive review of a larger body of work researching Covid-19; here we focus on few quintessential comparisons on our work with other works that we are aware of.", "[39] estimate the effect of policies on the growth rate of cases using data from the United States, China, Iran, Italy, France, and South Korea.", "In the United States, they find that the combined effect of all policies they consider on the growth rate is $-0.347$ $(0.061)$ .", "[23] use US county level data to analyze the effect of interventions on case growth rates.", "They find that the combination of policies they study reduced growth rates by 9.1 percentage points 16-20 days after implementation, out of which 5.9 percentage points are attributable to shelter in place orders.", "Both [39] and [23] adopt a reduced-form approach to estimate the total policy effect on case growth without using any social distancing behavior measures.Using a synthetic control approach, [20] finds that the cases would have been lower by 75 percent had Sweden adopted stricter lockdown policies.", "Existing evidence for the impact of social distancing policies on behavior in the US is mixed.", "[2] employ a difference-in-differences methodology to find that statewide stay-at-home orders have strong causal impacts on reducing social interactions.", "In contrast, using data from Google Mobility Reports, [51] find that the increase in social distancing is largely voluntary and driven by information.Specifically, they find that of the 60 percentage point drop in workplace intensity, 40 percentage points can be explained by changes in information as proxied by case numbers, while roughly 8 percentage points can be explained by policy changes.", "Another study by [30] also found little evidence that stay-at-home mandates induced distancing by using mobility measures from PlaceIQ and SafeGraph.", "Using data from SafeGraph, [6] shows that there has been substantial voluntary social distancing but also provide evidence that mandatory measures such as stay-at-home orders have been effective at reducing the frequency of visits outside of one's home.", "[61] use county-level observations of reported infections and deaths in conjunction with mobility data from SafeGraph to conduct simulation of implementing all policies 1-2 weeks earlier and found that it would have resulted in reducing the number of cases and deaths by more than half.", "However, their study does not explicitly analyze how policies are related to the effective reproduction numbers.", "Epidemiologists use model simulations to predict how cases and deaths evolve for the purpose of policy recommendation.", "As reviewed by [10], there exists substantial uncertainty about the values of key epidimiological parameters [8], [71].", "Simulations are often done under strong assumptions about the impact of social distancing policies without connecting to the relevant data [25].", "Furthermore, simulated models do not take into account that people may limit their contact with other people in response to higher transmission risks.See [9] and [71] for the implications of the SIR model for Covid-19 in the US.", "[26] estimate a SIRD model in which time-varying reproduction numbers depend on the daily deaths to capture feedback from daily deaths to future behavior and infections.", "When such a voluntary behavioral response is ignored, simulations would necessarily exhibit exponential spread and over-predict cases and deaths.", "In contrast, as cases and deaths rise, a voluntary behavioral response may possibly reduce the effective reproduction number below 1, potentially preventing exponential spread.", "Our counterfactual experiments illustrate the importance of this voluntary behavioral change.", "Whether wearing masks in public place should be mandatory or not has been one of the most contested policy issues with health authorities of different countries providing contradictory recommendations.", "Reviewing evidence, [28] recognize that there is no randomized controlled trial evidence for the effectiveness of face masks, but they state “indirect evidence exists to support the argument for the public wearing masks in the Covid-19 pandemic.", "\"The virus remains viable in the air for several hours, for which surgical masks may be effective.", "Also, a substantial fraction of individual who are infected become infectious before showing symptom onset.", "[38] also review available medical evidence and conclude that “mask wearing reduces the transmissibility per contact by reducing transmission of infected droplets in both laboratory and clinical contexts.” The laboratory findings in [37] suggest that the nasal cavity may be the initial site of infection followed by aspiration to the lung, supporting the argument “for the widespread use of masks to prevent aersol, large droplet, and/or mechanical exposure to the nasal passages.” [33] examined temporal patterns of viral shedding in COVID-19 patients and found the highest viral load at the time of symptom onset; this suggests that a significant portion of transmission may have occurred before symptom onset and that universal face masks may be an effective control measure to reduce transmission.", "[46] find evidence that viral loads in asymptomatic patients are similar to those in symptomatic patients.", "Aerosol transmission of viruses may occur through aerosols particles released during breathing and speaking by asymptomatic infected individuals; masks reduce such airborne transmission [63].", "[7] provide visual evidence of speech-generated droplet as well as the effectiveness of cloth masks to reduce the emission of droplets.", "[21] conduct a meta-analysis of observational studies on transmission of the viruses that cause COVID-19 and related diseases and find the effectiveness of mask use for reducing transmission.", "[57] provide a meta-analysis of randomized controlled trials of non-surgical face masks in preventing viral respiratory infections in non-hospital and non-household settings, finding that face masks decreased infections across all five studies they reviewed.Whether wearing masks creates a false sense of security and leads to decrease in social distancing is also a hotly debated topic.", "A randomized field experiment in Berlin, Germany, conducted by [68] finds that wearing masks actually increases social distancing, providing no evidence that mandatory masks leads to decrease in social distancing.", "Given the lack of experimental evidence on the effect of masks in the context of COVID-19, conducting observational studies is useful and important.", "To the best of our knowledge, our paper is the first empirical study that shows the effectiveness of mask mandates on reducing the spread of Covid-19 by analyzing the US state-level data.", "This finding corroborates and is complementary to the medical observational evidence in [38].", "Analyzing mitigation measures in New York, Wuhan, and Italy, [86] conclude that mandatory face coverings substantially reduced infections.", "[1] find that the growth rates of cases and of deaths in countries with pre-existing norms that sick people should wear masks are lower by 8 to 10% than those rates in countries with no pre-existing mask norms.", "[55] find that country's COVID-19 death rates are negatively associated with mask wearing rates.", "The Institute for Health Metrics and Evaluation at the University of Washington assesses that, if 95% of the people in the US were to start wearing masks from early August of 2020, 66,000 lives would be saved by December 2020 [41], which is largely consistent with our results.", "Our finding is also independently corroborated by a completely different causal methodology based on synthetic control using German data in [54].Our study was first released in ArXiv on May 28, 2020 whereas [54] was released at SSRN on June 8, 2020.", "Our empirical results contribute to informing the economic-epidemiological models that combine economic models with variants of SIR models to evaluate the efficiency of various economic policies aimed at the gradual “reopening\" of various sectors of economy.", "[4] analyzes the effect of policies reducing interpersonal contacts such as school closures or the closure of public transportation networks on the spread of influenza, gastroenteritis, and chickenpox using high frequency data from France.", "For example, the estimated effects of masks, stay-home mandates, and various other policies on behavior, and of behavior on infection can serve as useful inputs and validation checks in the calibrated macro, sectoral, and micro models (see, e.g., [5], [11], [26], [3], [44], [52] and references therein).", "Furthermore, the causal framework developed in this paper could be applicable, with appropriate extensions, to the impact of policies on economic outcomes replacing health outcomes (see, e.g., [19], [22]).", "Finally, our causal model is framed using the language of structural equations models and causal diagrams of econometrics ([83], [31], [77], [81], [58]) and genetics [84],The father and son, P. Wright (economist) and S. Wright (geneticist) collaborated to develop structural equation models and causal path diagrams; P. Wright's key work represented supply-demand system as a directed acyclical graph and established its identification using exclusion restrictions on instrumental variables.", "We view our work as following this classical tradition.", "with natural unfolding potential/structural outcomes representation [66], [76], [56], [42].", "The work on causal graphs has been modernized and developed by [58], [29], [59], [60] and many others (e.g., [60], [79], [65], [62], [12], [35]), with applications in computer science, genetics, epidemiology, and econometrics (see, e.g., [34], [40], [79] for applications in econometrics).", "The particular causal diagram we use has several “mediation\" components, where variables affect outcomes directly and indirectly through other variables called mediators; these structures go back at least to [84]; see, e.g., [13], [36], [65] for modern treatments." ], [ "The Causal Model and Its Structural Equation Form", "We introduce our approach through the Wright-style causal diagram shown in Figure REF .", "The diagram describes how policies, behavior, and information interact together: The forward health outcome, $Y_{i,t+\\ell }$ , is determined last, after all other variables have been determined; The adopted policies, $P_{it}$ , affect health outcome $Y_{i,t+\\ell }$ either directly, or indirectly by altering human behavior $B_{it}$ ; Information variables, $I_{it}$ , such as lagged values of outcomes can affect human behavior and policies, as well as outcomes; The confounding factors $W_{it}$ , which vary across states and time, affect all other variables.", "The index $i$ denotes observational unit, the state, and $t$ and $t+\\ell $ denotes the time, where $\\ell $ is a positive integer that represents the time lag between infection and case confirmation or death.", "Figure: S. & P. Wright type causal path diagram for our model.Our main outcomes of interest are the growth rates in Covid-19 cases and deaths, behavioral variables include proportion of time spent in transit, shopping, and workplaces, policy variables include mask mandates, stay-at-home orders, and school and business closures, and the information variables include lagged values of outcome.", "We provide a detailed description of these variables and their timing in the next section.", "The causal structure allows for the effect of the policy to be either direct or indirect – through behavior or through dynamics; all of these effects are not mutually exclusive.", "The structure also allows for changes in behavior to be brought by change in policies and information.", "These are all realistic properties that we expect from the contextual knowledge of the problem.", "Policies such as closures of schools, non-essential business, and restaurants alter and constrain behavior in strong ways.", "In contrast, policies such as mandating employees to wear masks can potentially affect the Covid-19 transmission directly.", "The information variables, such as recent growth in the number of cases, can cause people to spend more time at home, regardless of adopted state policies; these changes in behavior in turn affect the transmission of Covid-19.", "Importantly, policies can have the informational content as well, guiding behavior rather than constraining it.", "The causal ordering induced by this directed acyclical graph is determined by the following timing sequence: (1) information and confounders get determined at $t$ , (2) policies are set in place, given information and confounders at $t$ ; (3) behavior is realized, given policies, information, and confounders at $t$ ; (4) outcomes get realized at $t+\\ell $ given policies, behavior, information, and confounders.", "The model also allows for direct dynamic effects of information variables on the outcome through autoregressive structures that capture persistence in growth patterns.", "As highlighted below, realized outcomes may become new information for future periods, inducing dynamics over multiple periods.", "Our quantitative model for causal structure in Figure REF is given by the following econometric structural (or potential) outcomes model: $ \\begin{aligned}& Y_{i,t+\\ell } (b,p,\\iota ) &:=& {\\color {ForestGreen}\\alpha ^{\\prime } b} + {\\color {blue}\\pi ^{\\prime }p} +{\\color {magenta}\\mu ^{\\prime }\\iota } + {\\color {gray}\\delta _Y ^{\\prime }W_{it}} + \\varepsilon ^y_{it}, \\\\& B_{it} (p,\\iota ) &:= & {\\color {blue}\\beta ^{\\prime }p } + {\\color {magenta}\\gamma ^{\\prime }\\iota } + {\\color {gray}\\delta _B ^{\\prime }W_{it} } + \\varepsilon ^b_{it},\\end{aligned}\\qquad \\mathrm {(SO)}$ which is a collection of functional relations with stochastic shocks, decomposed into observable part $\\delta ^{\\prime } W$ and unobservable part $\\varepsilon $ .", "The terms $\\varepsilon ^y_{it}$ and $\\varepsilon ^b_{it} $ are the centered stochastic shocks that obey the orthogonality restrictions posed below.", "The policies can be modeled via a linear form as well, $ P_{it} (\\iota ) := {\\color {magenta}\\eta ^{\\prime }\\iota } + {\\color {gray}\\delta _P^{\\prime } W_{it}} + \\varepsilon ^p_{it}, \\qquad \\mathrm {(P)}$ although linearity is not critical.Under some additional independence conditions, this can be replaced by an arbitrary non-additive function $P_{it}(\\iota ) = p (\\iota , W_{it}, \\varepsilon ^p_{it})$ , such that the unconfoundedness condition stated in the next footnote holds.", "The exogeneity restrictions on the stochastic shocks are as follows: $ \\begin{aligned}& \\varepsilon ^y_{it} & \\perp & \\quad (\\varepsilon ^b_{it}, \\varepsilon ^p_{it}, {\\color {gray}W_{it}}, {\\color {magenta}I_{it}}), \\\\& \\varepsilon ^b_{it} & \\perp & \\quad (\\varepsilon ^p_{it}, {\\color {gray}W_{it}}, {\\color {magenta}I_{it}}), \\\\& \\varepsilon ^p_{it} & \\perp & \\quad ({\\color {gray}W_{it}}, {\\color {magenta}I_{it}}),\\end{aligned}\\qquad \\mathrm {(E)}$ where we say that $V \\perp U$ if ${\\mathrm {E}}VU = 0$ .An alternative useful starting point is to impose the Rosenbaum-Rubin type unconfoundedness condition: $Y_{i,t+\\ell } (\\cdot ,\\cdot ,\\cdot ) \\protect \\mathchoice{\\protect \\mathrel {\\displaystyle \\perp }\\copy 0\\hspace{0.0pt}\\hspace{2.22214pt}\\box 0}{}{}{}$ $\\textstyle \\perp $$\\textstyle \\perp $$\\scriptstyle \\perp $$\\scriptstyle \\perp $$\\scriptscriptstyle \\perp $$\\scriptscriptstyle \\perp $ (Pit, Bit, Iit) Wit, Bit (,) $\\displaystyle \\perp $$\\displaystyle \\perp $$\\textstyle \\perp $$\\textstyle \\perp $$\\scriptstyle \\perp $$\\scriptstyle \\perp $$\\scriptscriptstyle \\perp $$\\scriptscriptstyle \\perp $ (Pit, Iit) Wit, Pit () $\\displaystyle \\perp $$\\displaystyle \\perp $$\\textstyle \\perp $$\\textstyle \\perp $$\\scriptstyle \\perp $$\\scriptstyle \\perp $$\\scriptscriptstyle \\perp $$\\scriptscriptstyle \\perp $ Iit Wit, $which imply, with treating stochastic errors as independent additive components, the orthogonal conditions stated above.The same unconfoundedness restrictions can be formulated using formal causal DAGs, and also imply orthogonality restrictions stated above, once stochastic errors are modeled as independent additive components.$ This is a standard way of representing restrictions on errors in structural equation modeling in econometrics.The structural equations of this form are connected to triangular structural equation models, appearing in microeconometrics and macroeconometrics (SVARs), going back to the work of [73].", "The observed variables are generated by setting $\\iota = I_{it}$ and propagating the system from the last equation to the first: $ \\begin{aligned}& {\\color {red}Y_{i,t+\\ell }} & := & Y_{i,t+\\ell } ( {\\color {ForestGreen}B_{it} } ,{\\color {blue}P_{it}}, {\\color {magenta}I_{it}}), \\\\& {\\color {ForestGreen}B_{it} } & := & B_{it}({\\color {blue}P_{it} } ,{\\color {magenta}I_{it}}), \\\\& {\\color {blue}P_{it} }& := & P_{it}({\\color {magenta}I_{it}}).", "\\end{aligned}\\qquad \\mathrm {(O)}$ The specification of the model above grasps one-period responses.", "The dynamics over multiple periods will be induced by the evolution of information variables, which include time, lagged and integrated values of outcome:Our empirical analysis also considers a specification in which information variables include lagged national cases/deaths as well as lagged behavior variables.", "${\\color {magenta}I_{it} } := I_t( {\\color {red}Y_{it}}, { \\color {magenta}I_{i, t-\\ell } }) := \\Big (g(t) , {\\color {red}Y_{it}}, \\sum _{m=0}^{ \\lfloor t/\\ell \\rfloor }{\\color {red}Y_{i,t - \\ell m}} \\Big )^{\\prime } \\qquad \\mathrm {(I)}$ for each $t \\in \\lbrace 0,1,...,T\\rbrace $ , where $g$ is deterministic function of time, e.g., month indicators, assuming that the log of new cases at time $t \\le 0$ is zero, for notational convenience.", "The general formula for $I_{i, t-1}$ is $S_{i, t, \\ell } + \\sum _{m=1}^{ \\lfloor t/\\ell \\rfloor } Y_{i,t - \\ell m}$ , where $ S_{i, t, \\ell }$ is the initial condition, the log of new cases at time $- t \\ \\mathrm { mod } \\ \\ell $ .", "In this structure, people respond to both global information, captured by a function of time such as month dummies, and local information sources, captured by the local growth rate and the total number of cases.", "The local information also captures the persistence of the growth rate process.", "We model the reaction of people's behavior via the term ${\\color {magenta}\\gamma ^{\\prime }I_{t}}$ in the behavior equation.", "The lagged values of behavior variable may be also included in the information set, but we postpone this discussion after the main empirical results are presented.", "With any structure of this form, realized outcomes may become new information for future periods, inducing a dynamical system over multiple periods.", "We show the resulting dynamical system in a diagram of Figure REF .", "Specification of this system is useful for studying delayed effects of policies and behaviors and in considering the counterfactual policy analysis.", "Figure: Diagram for Information Dynamics in SEMNext we combine the above parts together with an appropriate initializations to give a formal definition of the model we use.", "Structural Equations Model (SEM).", "Let $i \\in \\lbrace 1,..., N\\rbrace $ denote the observational unit, $t$ be the time periods, and $\\ell $ be the time delay.", "(1) For each $i$ and $t \\le -\\ell $ , the confounder, information, behavior, and policy variables $W_{it}, I_{it}, B_{it}, P_{it}$ are determined outside of the model, and the outcome variable $Y_{i, t+\\ell }$ is determined by factors outside of the model for $t \\le 0$ .", "(2) For each $i$ and $t \\ge -\\ell $ , confounders $W_{it}$ are determined by factors outside of the model, and information variables $I_{it}$ are determined by (I); policy variables $P_{it}$ are determined by setting $\\iota = I_{it}$ in (P) with a realized stochastic shock $\\varepsilon ^p_{it}$ that obeys the exogeneity condition (E); behavior variables $B_{it}$ are determined by setting $\\iota = I_{it}$ and $p= P_{it}$ in (SO) with a shock $\\varepsilon ^b_{it}$ that obeys (E); finally, the outcome $Y_{i, t + \\ell }$ is realized by setting $\\iota = I_{it}$ , $p= P_{it}$ , and $b = B_{it}$ in (SO) with a shock $\\varepsilon ^y_{it}$ that obeys (E)." ], [ "Main Testable Implication, Identification, Parameter Estimation", "The system above together with orthogonality restrictions (REF ) implies the following collection of projection equations for realized variables: $& {\\color {red}Y_{i,t+\\ell }}= {\\color {ForestGreen}\\alpha ^{\\prime } B_{it}} + {\\color {blue}\\pi ^{\\prime }P_{it}} + {\\color {magenta}\\mu ^{\\prime }I_{it}} + {\\color {gray}\\delta _Y ^{\\prime }W_{it}} + \\varepsilon ^y_{it},& & \\varepsilon ^y_{it} \\perp {\\color {ForestGreen}B_{it}}, {\\color {blue}P_{it}}, {\\color {magenta}I_{it}}, {\\color {gray}W_{it}} \\\\& {\\color {ForestGreen}B_{it}}= {\\color {blue}\\beta ^{\\prime } P_{it}} + {\\color {magenta}\\gamma ^{\\prime }I_{it}} + {\\color {gray}\\delta _B^{\\prime } W_{it}} + \\varepsilon ^b_{it},& & \\varepsilon ^b_{it} \\perp {\\color {blue}P_{it}}, {\\color {magenta}I_{it}}, {\\color {gray}W_{it}} \\\\& {\\color {blue}P_{it}}= {\\color {magenta}\\eta ^{\\prime }I_{it}} + {\\color {gray}\\delta _P^{\\prime } W_{it}} + \\varepsilon ^p_{it}, & & \\varepsilon ^p_{it} \\perp {\\color {magenta}I_{it}}, {\\color {gray}W_{it}} \\\\& {\\color {red}Y_{i,t+\\ell }}= ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime } }+{\\color {blue}\\pi ^{\\prime }} ) {\\color {blue}P_{it}} + ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime }} + {\\color {magenta}\\mu ^{\\prime }}){\\color {magenta}I_{it} }+ {\\color {gray}\\bar{\\delta } ^{\\prime }}{\\color {gray}W_{it}} + {\\bar{\\varepsilon }}_{it}, && {\\bar{\\varepsilon }}_{it} \\perp {\\color {blue}P_{it}}, {\\color {magenta}I_{it}}, {\\color {gray}W_{it}}.", "$ Therefore the projection equation: ${\\color {red}Y_{i,t+\\ell }}= \\mathsf {a}^{\\prime }{\\color {blue}P_{it}} + \\mathsf {b}^{\\prime } {\\color {magenta}I_{it} }+ {\\color {gray}\\tilde{\\delta } ^{\\prime }}{\\color {gray}W_{it}} + {\\bar{\\varepsilon }}_{it}, \\quad {\\bar{\\varepsilon }}_{it} \\perp {\\color {blue}P_{it}}, {\\color {magenta}I_{it}}, {\\color {gray}W_{it}}.", "\\qquad \\mathrm {(Y\\sim PI)}$ should obey: $ \\mathsf {a}^{\\prime } = ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime } }+{\\color {blue}\\pi ^{\\prime }} ) \\text{ and }\\mathsf {b}^{\\prime } = ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime }} + {\\color {magenta}\\mu ^{\\prime }}).\\qquad \\mathrm {(TR)}$ Without any exclusion restrictions, this equality is just a decomposition of total effects into direct and indirect components and is not a testable restriction.", "However, in our case we rely on the SIR model with testing to motivate the presence of change in testing rate as a confounder in the outcome equations but not in the behavior equation (therefore, a component of $\\delta _B$ and $\\delta _P$ is set to 0), implying that (TR) does not necessarily hold and is testable.", "Furthermore, we estimate () on the data set that has many more observations than the data set used to estimate the outcome equations, implying that (TR) is again testable.", "Later we shall also try to utilize the contextual knowledge that mask mandates only affect the outcome directly and not by changing mobility (i.e., $\\beta =0$ for mask policy), implying again that (TR) is testable.", "If not rejected by the data, (TR) can be used to sharpen the estimate of the causal effect of mask policies on the outcomes.", "Validation of the model by (TR) allows us to check exclusion restrictions brought by contextual knowledge and check stability of the model by using different data subsets.", "However, passing the (TR) does not guarantee that the model is necessarily valid for recovering causal effects.", "The only fundamental way to truly validate a causal model for observational data is through a controlled experiment, which is impossible to carry out in our setting.", "The parameters of the SEM are identified by the projection equation set above, provided the latter are identified by sufficient joint variation of these variables across states and time.", "We can develop this point formally as follows.", "Consider the previous system of equations, after partialling out the confounders: $\\begin{aligned}& {\\color {red}{ \\tilde{Y}_{i,t+\\ell }}} & =& {\\color {ForestGreen}{\\alpha ^{\\prime } \\tilde{B}_{it}}} +{\\color {blue}{\\pi ^{\\prime }\\tilde{P}_{it}}} +{\\color {magenta}\\mu ^{\\prime }\\tilde{I}_{it} } + \\varepsilon ^y_{it},&\\varepsilon ^y_{it} &\\perp {\\tilde{B}_{it}}, {\\tilde{P}_{it}}, { \\tilde{I}_{it}}, \\\\& {\\color {ForestGreen}{ \\tilde{B}_{it}}} & = &{\\color {blue}{ \\beta ^{\\prime } \\tilde{P}_{it}} } + {\\color {magenta}{ \\gamma ^{\\prime } \\tilde{I}_{it}} } + \\varepsilon ^b_{it},&\\quad \\varepsilon ^b_{it} &\\perp { \\tilde{P}_{it}}, { \\tilde{I}_{it}}, \\\\& {\\color {blue}{ \\tilde{P}_{it}} } &= & {\\color {magenta}{\\eta ^{\\prime } \\tilde{I}_{it}} } + \\varepsilon ^p_{it}, &\\quad \\varepsilon ^p_{it} &\\perp {\\tilde{I}_{it}} \\\\& {\\color {red}{ \\tilde{Y}_{i,t+\\ell }}} & =& \\mathsf {a} ^{\\prime } {\\color {blue}{\\tilde{P}_{it}}} + \\mathsf {b}^{\\prime } {\\color {magenta}\\tilde{I}_{it} } + \\bar{\\varepsilon }^y_{it},&\\bar{\\varepsilon }^y_{it} &\\perp {\\tilde{P}_{it}}, { \\tilde{I}_{it}}, \\\\\\end{aligned}$ where $ \\tilde{V}_{it} = V_{it} - {\\color {gray}W_{it}^{\\prime }} {\\mathrm {E}}[{\\color {gray}W_{it}W_{it}^{\\prime }}]^{-} {\\mathrm {E}}[{\\color {gray}W_{it}} V_{it}]$ denotes the residual after removing the orthogonal projection of $V_{it}$ on ${\\color {gray}W_{it}}$ .", "The residualization is a linear operator, implying that (REF ) follows immediately from the above.", "The parameters of (REF ) are identified as projection coefficients in these equations, provided that residualized vectors appearing in each of the equations have non-singular variance, that is ${\\mathrm {Var}}( {\\color {blue}\\tilde{P}_{it}^{\\prime }} ,{\\color {ForestGreen}\\tilde{B}_{it}^{\\prime }},{\\color {magenta}\\tilde{I}_{it}^{\\prime }})>0,\\ {\\mathrm {Var}}({\\color {blue}\\tilde{P}_{it}^{\\prime }}, {\\color {magenta}\\tilde{I}_{it}^{\\prime }})> 0 , \\ \\text{ and } {\\mathrm {Var}}( {\\color {magenta}\\tilde{I}_{it}^{\\prime }}) >0.$ Our main estimation method is the standard correlated random effects estimator, where the random effects are parameterized as functions of observable characteristic, ${\\color {gray}W_{it}}$ , which include both state-level and time random effects.", "The state-level random effects are modeled as a function of state level characteristics, and the time random effects are modeled by including month dummies and their interactions with state level characteristics (in the sensitivity analysis, we also add weekly dummies).", "The stochastic shocks $\\lbrace \\varepsilon _{it}\\rbrace _{t=1}^T$ are treated as independent across states $i$ and can be arbitrarily dependent across time $t$ within a state.", "Another modeling approach is the fixed effects panel data model, where ${\\color {gray}W_{it}}$ includes latent (unobserved) state level confonders ${\\color {gray}W_i}$ and and time level effects ${\\color {gray}W_t}$ , which must be estimated from the data.", "This approach is much more demanding of the data and relies on long time and cross-sectional histories to estimate ${\\color {gray}W_i}$ and ${\\color {gray}W_t}$ , resulting in amplification of uncertainty.", "In addition, when histories are relatively short, large biases emerge and they need to be removed using debiasing methods, see e.g., [16] for overview.", "We present the results on debiased fixed effect estimation with weekly dummies as parts of our sensitivity analysis.", "Our sensitivity analysis also considers a debiased machine learning approach using Random Forest in which observed confounders enter the model nonlinearly.", "With exclusion restrictions there are multiple approaches to estimation, for example, via generalized method of moments.", "We shall take a more pragmatic approach where we estimate the parameters of equations separately and then compute $\\frac{1}{2} \\hat{\\mathsf {a}}^{\\prime } + \\frac{1}{2} ( {\\color {ForestGreen}\\hat{\\alpha }^{\\prime }} {\\color {blue}\\hat{\\beta }^{\\prime } }+{\\color {blue}\\hat{\\pi }^{\\prime }} ) \\text{ and }\\frac{1}{2} \\hat{\\mathsf {b}}^{\\prime } + \\frac{1}{2} ( {\\color {ForestGreen}\\hat{\\alpha }^{\\prime }} {\\color {magenta}\\hat{\\gamma }^{\\prime }} + {\\color {magenta}\\hat{\\mu }^{\\prime }}),$ as the estimator of the total policy effect.", "Under standard regularity conditions, these estimators concentrate around their population analogues $\\frac{1}{2} {\\mathsf {a}}^{\\prime } + \\frac{1}{2} ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime } }+{\\color {blue}\\pi ^{\\prime }} ) \\text{ and }\\frac{1}{2} {\\mathsf {b}}^{\\prime } + \\frac{1}{2} ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime }} + {\\color {magenta}\\mu ^{\\prime }}),$ with approximate deviations controlled by the normal laws, with standard deviations that can be approximated by the bootstrap resampling of observational units $i$ .", "Under correct specification the target quantities reduce to $ {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime } }+{\\color {blue}\\pi ^{\\prime }} \\text{ and } {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime }} + {\\color {magenta}\\mu ^{\\prime }},$ respectively.This construction is not as efficient as generalized method of moments but has a nicer interpretation under possible misspecification of the model: we are combining predictions from two models, one motivated via the causal path, reflecting contextual knowledge, and another from a “reduced form\" model not exploiting the path.", "The combined estimator can improve on precision of either estimator." ], [ "Counterfactual Policy Analysis", "We also consider simple counterfactual exercises, where we examine the effects of setting a sequence of counterfactual policies for each state: $ \\lbrace {\\color {blue}P^\\star _{it}} \\rbrace _{t=1}^T, \\quad i=1, \\ldots N.\\qquad \\mathrm {(CF-P)}$ We assume that the SEM remains invariant, except for the policy equation.It is possible to consider counterfactual exercises in which policy responds to information through the policy equation if we are interested in endogenous policy responses to information.", "Counterfactual experiments with endogenous government policy would be important, for example, to understand the issues related to the lagged response of government policies to higher infection rates due to incomplete information.", "The assumption of invariance captures the idea that counterfactual policy interventions would not change the structural functions within the period of the study.", "The assumption is strong but is necessary to conduct counterfactual experiments, e.g.", "[69] and [73].", "To make the assumption more plausible we limited our study to the early pandemic period.Furthermore, we conducted several stability checks, for example, checking if the coefficients on mask policies remain stable (reported in the previous version) and also looking at more recent data during reopening, beyond early pandemic, to examine the stability of the model.", "Given the policies, we generate the counterfactual outcomes, behavior, and information by propagating the dynamic equations: $ \\begin{aligned}& {\\color {red}Y^\\star _{i,t+\\ell }} & := & Y_{i,t+\\ell } ( {\\color {ForestGreen}B^\\star _{it} } , {\\color {blue}P^\\star _{it}}, {\\color {magenta}I^\\star _{it}}), \\\\& {\\color {ForestGreen}B^\\star _{it} } & := & B_{it}({\\color {blue}P^\\star _{it} } ,{\\color {magenta}I^\\star _{it}}), \\\\& {\\color {magenta}I^\\star _{it} }& := & I_t ( {\\color {red}Y^\\star _{it}}, \\color {magenta}I^\\star _{i,t-\\ell }), \\end{aligned}\\qquad \\mathrm {(CF-SEM)}$ with the same initialization as the factual system up to $t \\le 0$ .", "In stating this counterfactual system of equations, we assume that structural outcome equations (SO) and information equations (I) remain invariant and so do the stochastic shocks, decomposed into observable and unobservable parts.", "Formally, we record this assumption and above discussion as follows.", "Counterfactual Structural Equations Model (CF-SEM).", "Let $i \\in \\lbrace 1,..., N\\rbrace $ be the observational unit, $t$ be time periods, and $\\ell $ be the time delay.", "(1) For each $i$ and $t \\le 0$ , the confounder, information, behavior, policy, and outcome variables are determined as previously stated in SEM: $W^\\star _{it}= W_{it}$ , $I^\\star _{it}= I_{it}$ , $B^\\star _{it} = B_{it}$ , $P^\\star _{it}=P_{it}$ , $Y^\\star _{it} =Y_{it}$ .", "(2) For each $i$ and $t \\ge 0$ , confounders $W^\\star _{it} =W_{it}$ are determined as in SEM, and information variables $I^\\star _{it}$ are determined by (I); policy variables $P^\\star _{it}$ are set in (REF ); behavior variables $B^\\star _{it}$ are determined by setting $\\iota = I^\\star _{it}$ and $p= P^\\star _{it}$ in (SO) with the same stochastic shock $\\varepsilon ^b_{it}$ in (SO); the counterfactual outcome $Y^\\star _{i, t + \\ell }$ is realized by setting $\\iota = I^\\star _{it}$ , $p= P^\\star _{it}$ , and $b = B^\\star _{it}$ in (SO) with the same stochastic shock $\\varepsilon ^y_{it}$ in (SO).", "Figures REF and REF present the causal path diagram for CF-SEM as well as the dynamics of counterfactual information in CF-SEM.", "Figure: Causal path diagram for CF-SEM.Figure: A Diagram for Counterfactual Information Dynamics in CF-SEMThe counterfactual outcome $Y^\\star _{i,t+\\ell }$ and factual outcome $Y_{i,t+\\ell }$ are given by: ${\\color {red}Y^\\star _{i,t+\\ell }} = ( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime }} + {\\color {blue}\\pi ^{\\prime }}){\\color {blue}P^\\star _{it}} + ({\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime } + \\mu ^{\\prime }}){\\color {magenta}I^\\star _{it} }+ {\\color {gray}\\bar{\\delta }^{\\prime }W_{it}} + \\bar{\\varepsilon }_{it}^y, $ $\\ {\\color {red}Y_{i,t+\\ell }} \\ =( {\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {blue}\\beta ^{\\prime }} + {\\color {blue}\\pi ^{\\prime }}){\\color {blue}P_{it}} + ({\\color {ForestGreen}\\alpha ^{\\prime }} {\\color {magenta}\\gamma ^{\\prime } + \\mu ^{\\prime }}){\\color {magenta}I_{it} }+ {\\color {gray}\\bar{\\delta }^{\\prime }W_{it}} + \\bar{\\varepsilon }_{it}^y.$ In generating these predictions, we explore the assumption of invariance stated above.", "We can write the counterfactual contrast into the sum of three components: ${{\\color {red}Y^\\star _{i,t+\\ell } - Y_{i,t+\\ell } }}_{\\text{CF Change}} & ={{\\color {ForestGreen}\\alpha }^{\\prime } {\\color {blue}\\beta }^{\\prime } {\\color {blue}\\left(P^\\star _{it} - P_{it} \\right)}}_{\\text{CF Policy Effect via Behavior}} + {{\\color {blue}\\pi ^{\\prime } \\left( P^\\star _{it} - P_{it} \\right)}_{\\text{CF Direct Effect}} } \\nonumber \\\\&\\qquad + {{{\\color {ForestGreen}\\alpha }^{\\prime } {\\color {magenta}{\\gamma }^{\\prime } \\left( I^\\star _{it} - I_{it} \\right)}+ {\\color {magenta}{\\mu }^{\\prime } \\left( I^\\star _{it} - I_{it} \\right)}}_{\\text{ CF Dynamic Effect}}} \\nonumber \\\\& =: \\mathrm {PEB}^\\star _{it} + \\mathrm {PED}^\\star _{it} + \\mathrm {DynE}^\\star _{it}, $ which describe the immediate indirect effect of the policy via behavior, the direct effect of the policy, and the dynamic effect of the policy.", "By recursive substitutions the dynamic effect can be further decomposed into a weighted sum of delayed policy effects via behavior and a weighted sum of delayed policy effects via direct impact.", "All counterfactual quantities and contrasts can be computed from the expressions given above.", "For examples, given $\\Delta C_{i0}>0$ , new confirmed cases are linked to growth rates via relation (taking $t$ divisible by $\\ell $ for simplicity): $\\frac{\\Delta C^{\\star }_{it}}{\\Delta C_{i0}} = \\exp \\left( \\sum _{m=1}^{t/\\ell } Y^\\star _{i,m\\ell } \\right) \\text{ and }\\frac{ \\Delta C_{it}}{\\Delta C_{i0}} = \\exp \\left( \\sum _{m=1}^{t/\\ell } Y_{i, m \\ell }\\right).$ The cumulative cases can be constructed by summing over the new cases.", "Various contrasts are then calculated from these quantities.", "For example, the relative contrast of counterfactual new confirmed cases to the factual confirmed cases is given by: $\\Delta C^{\\star }_{it}/\\Delta C_{it} = \\exp \\left( \\sum _{m=1}^{t/\\ell } ( Y^\\star _{i,m\\ell } - Y_{i,m \\ell }) \\right).$ We refer to the appendix for further details.", "Similar calculations apply for fatalities.", "Note that our analysis is conditional on the factual history and structural stochastic shocks.For unconditional counterfactual, we need to make assumptions about the evolution of stochastic shocks appearing in $Y_{it}$ .", "See, e.g, previous versions of our paper in ArXiv, where unconditional counterfactuals were calculated by assuming stochastic shocks are i.i.d.", "and resampling them from the empirical distribution.", "The differences between conditional and unconditional contrasts were small in our empirical analysis.", "The estimated counterfactuals are smooth functionals of the underlying parameter estimates.", "Therefore, we construct the confidence intervals for counterfactual quantities and contrasts by bootstrapping the parameter estimates.", "We refer to the appendix for further details." ], [ "Outcome and Key Confounders via SIRD model", "We next provide details of our key measurement equations, defining the outcomes and key confounders.", "We motivate the structural outcome equations via the fundamental epidemiological model for the spread of infectious decease called the Susceptible-Infected-Recovered-Dead (SIRD) model with testing.", "Letting $C_{it}$ denote the cumulative number of confirmed cases in state $i$ at time $t$ , our outcome $ {\\color {red}Y_{it}} =\\Delta \\log (\\Delta C_{it}):= \\log ( \\Delta C_{it} ) -\\log ( \\Delta C_{i,t-7})$ approximates the weekly growth rate in new cases from $t-7$ to $t$ .", "Here $\\Delta $ denotes the differencing operator over 7 days from $t$ to $t-7$ , so that $\\Delta C_{it}:=C_{it}-C_{i,t-7}$ is the number of new confirmed cases in the past 7 days.", "We chose this metric as this is the key metric for policy makers deciding when to relax Covid mitigation policies.", "The U.S. government's guidelines for state reopening recommend that states display a “downward trajectory of documented cases within a 14-day period” [80].", "A negative value of $Y_{it}$ is an indication of meeting this criteria for reopening.", "By focusing on weekly cases rather than daily cases, we smooth idiosyncratic daily fluctuations as well as periodic fluctuations associated with days of the week.", "Our measurement equation for estimating equations (REF ) and () will take the form: ${\\color {red}\\Delta \\log (\\Delta C_{it})} = X_{i,t-14} ^{\\prime } \\theta -\\gamma + \\delta _T \\Delta \\log (T_{it}) + \\epsilon _{it}, $ where $i$ is state, $t$ is day, $C_{it}$ is cumulative confirmed cases, $T_{it}$ is the number of tests over 7 days, $\\Delta $ is a 7-days differencing operator, and $\\epsilon _{it}$ is an unobserved error term.", "$X_{i,t-14}$ collects other behavioral, policy, and confounding variables, depending on whether we estimate (REF ) or (), where the lag of 14 days captures the time lag between infection and confirmed case (see the Appendix REF ).", "Here $\\Delta \\log (T_{it} ):= \\log (T_{it}) - \\log (T_{i,t-7}) $ is the key confounding variable, derived from considering the SIRD model below.", "We are treating the change in testing rate as exogenous.To check sensitivity to this assumption we performed robustness checks, where we used the further lag of $\\Delta \\log (T_{it} )$ as a proxy for exogenous change in the testing rate, and we also used that as an instrument for $\\Delta \\log (T_{it} )$ ; this did not affect the results on policy effects, although the instrument was not sufficiently strong.", "We describe other confounders in the empirical section.", "Our main measurement equation (REF ) is motivated by a variant of SIRD model, where we add confirmed cases and infection detection via testing.", "Let $S$ , ${\\mathcal {I}}$ , ${R}$ , and $D$ denote the number of susceptible, infected, recovered, and deceased individuals in a given state.", "Each of these variables are a function of time.", "We model them as evolving as $\\dot{S}(t) & = -\\frac{S(t)}{N} \\beta (t) {\\mathcal {I}}(t) \\\\\\dot{{\\mathcal {I}}}(t) & = \\frac{S(t)}{N} \\beta (t) {\\mathcal {I}}(t) - \\gamma {\\mathcal {I}}(t) \\\\\\dot{{R}}(t) & = (1-\\kappa ) \\gamma {\\mathcal {I}}(t) \\\\ \\dot{D}(t) & = \\kappa \\gamma {\\mathcal {I}}(t) $ where $N$ is the population, $\\beta (t)$ is the rate of infection spread, $\\gamma $ is the rate of recovery or death, and $\\kappa $ is the probability of death conditional on infection.", "Confirmed cases, $C(t)$ , evolve as $\\dot{C}(t) = \\tau (t) {\\mathcal {I}}(t), $ where $\\tau (t)$ is the rate that infections are detected.", "Our goal is to examine how the rate of infection $\\beta (t)$ varies with observed policies and measures of social distancing behavior.", "A key challenge is that we only observed $C(t)$ and $D(t)$ , but not ${\\mathcal {I}}(t)$ .", "The unobserved ${\\mathcal {I}}(t)$ can be eliminated by differentiating (REF ) and using () as $\\frac{\\ddot{C}(t)}{\\dot{C}(t)}& =\\frac{S(t)}{N} \\beta (t) -\\gamma + \\frac{\\dot{\\tau }(t)}{\\tau (t)}.", "$ We consider a discrete-time analogue of equation (REF ) to motivate our empirical specification by relating the detection rate $\\tau (t)$ to the number of tests $T_{it}$ while specifying $\\frac{S(t)}{N}\\beta (t)$ as a linear function of variables $X_{i,t-14}$ .", "This results in ${\\Delta \\log (\\Delta C_{it})}_{\\frac{\\ddot{C}(t)}{\\dot{C}(t)}}={X_{i,t-14}^{\\prime } \\theta + \\epsilon _{it}}_{\\frac{S(t)}{N}\\beta (t) -\\gamma }+& {\\delta _T \\Delta \\log (T)_{it}}_{\\frac{\\dot{\\tau }(t)}{\\tau (t)} } \\nonumber $ which is equation (REF ), where $X_{i,t-14}$ captures a vector of variables related to $\\beta (t)$ .", "Structural Interpretation.", "Early in the pandemic, when the number of susceptibles is approximately the same as the entire population, i.e.", "$S_{it}/N_{it} \\approx 1$ , the component $X_{i,t-14}^{\\prime } \\theta $ is the projection of infection rate $ \\beta _i(t)$ on $X_{i,t-14}$ (policy, behavioral, information, and confounders other than testing rate), provided the stochastic component $\\epsilon _{it}$ is orthogonal to $X_{i,t-14}$ and the testing variables: $\\beta _i(t)S_{it}/N_{it} - \\gamma = X_{i,t-14}^{\\prime } \\theta + \\epsilon _{it}, \\quad \\epsilon _{it} \\perp X_{i,t-14}.$ The specification for growth rate in deaths as the outcome is motivated by SIRD as follows.", "By differentiating () and (REF ) with respect to $t$ and using (REF ), we obtain $\\frac{\\ddot{D}(t) }{\\dot{D}(t)}& = \\frac{\\ddot{C}(t) }{\\dot{C}(t)} - \\frac{\\dot{\\tau }(t) }{ \\tau (t)} = \\frac{S(t)}{N}\\beta (t) - \\gamma .$ Our measurement equation for the growth rate of deaths is based on equation (REF ) but accounts for a 21 day lag between infection and death as ${\\color {red}\\Delta \\log (\\Delta D_{it})} = X_{i,t-21}^{\\prime } \\theta + \\epsilon _{it}, $ where $ \\Delta \\log (\\Delta D_{it}):= \\log ( \\Delta D_{it} ) -\\log ( \\Delta D_{i,t-7})$ approximates the weekly growth rate in deaths from $t-7$ to $t$ in state $i$ ." ], [ "Data", "Our baseline measures for daily Covid-19 cases and deaths are from The New York Times (NYT).", "When there are missing values in NYT, we use reported cases and deaths from JHU CSSE, and then the Covid Tracking Project.", "The number of tests for each state is from Covid Tracking Project.", "As shown in the lower right panel of Figure REF in the appendix, there was a rapid increase in testing in the second half of March and then the number of tests increased very slowly in each state in April.", "We use the database on US state policies created by [64].", "In our analysis, we focus on 6 policies: stay-at-home, closed nonessential businesses, closed K-12 schools, closed restaurants except takeout, closed movie theaters, and face mask mandates for employees in public facing businesses.", "We believe that the first four of these policies are the most widespread and important.", "Closed movie theaters is included because it captures common bans on gatherings of more than a handful of people.", "We also include mandatory face mask use by employees because its effectiveness on slowing down Covid-19 spread is a controversial policy issue [38], [28], [86].", "Table REF provides summary statistics, where $N$ is the number of states that have ever implemented the policy.", "We also obtain information on state-level covariates mostly from [64], which include population size, total area, unemployment rate, poverty rate, a percentage of people who are subject to illness, and state governor's party affiliations.", "These confounders are motivated by [78] who find that case growth is associated with residential density and per capita income.", "Table: State PoliciesWe obtain social distancing behavior measures from“Google COVID-19 Community Mobility Reports” [50].", "The dataset provides six measures of “mobility trends” that report a percentage change in visits and length of stay at different places relative to a baseline computed by their median values of the same day of the week from January 3 to February 6, 2020.", "Our analysis focuses on the following four measures: “Grocery & pharmacy,\" “Transit stations,” “Retail & recreation,” and “Workplaces.”The other two measures are “Residential” and “Parks.” We drop “Residential” because it is highly correlated with “Workplaces” and “Retail & recreation” at correlation coefficients of -0.98 and -0.97, respectively.", "We also drop “Parks” because it does not have clear implication on the spread of Covid-19.", "Figure REF shows the evolution of “Transit stations” and “Workplaces,” where thin lines are the value in each state and date while thicker colored lines are quantiles conditional on date.", "The figures illustrate a sharp decline in people's movements starting from mid-March as well as differences in their evolutions across states.", "They also reveal periodic fluctuations associated with days of the week, which motivates our use of weekly measures.", "Figure: The Evolution of Google Mobility Measures: Transit stations and WorkplacesIn our empirical analysis, we use weekly measures for cases, deaths, and tests by summing up their daily measures from day $t$ to $t-6$ .", "We focus on weekly cases and deaths because daily new cases and deaths are affected by the timing of reporting and testing and are quite volatile as shown in the upper right panel of Figure REF in the appendix.", "Aggregating to weekly new cases/deaths/tests smooths out idiosyncratic daily noises as well as periodic fluctuations associated with days of the week.", "We also construct weekly policy and behavior variables by taking 7 day moving averages from day $t-14$ to $t-21$ for case growth, where the delay reflects the time lag between infection and case confirmation.", "The four weekly behavior variables are referred to “Transit Intensity,” “Workplace Intensity,” “Retail Intensity,” and “Grocery Intensity.” Consequently, our empirical analysis uses 7 day moving averages of all variables recorded at daily frequencies.", "Our sample period is from March 7, 2020 to June 3, 2020.", "Table: Correlations among Policies and BehaviorTable REF reports that weekly policy and behavior variables are highly correlated with each other, except for the“masks for employees” policy.", "High correlations may cause multicolinearity problems and could limit our ability to separately identify the effect of each policy or behavior variable on case growth.", "For this reason, we define the “business closure policies” variable by the average of closed movie theaters, closed restaurants, and closed non-essential businesses variables and consider a specification that includes business closure policies in place of these three policy variables separately.", "Figure REF shows the portion of states that have each policy in place at each date.", "For most policies, there is considerable variation across states in the time in which the policies are active.", "The one exception is K-12 school closures.", "About 80% of states closed schools within a day or two of March 15th, and all states closed schools by early April.", "This makes the effect of school closings difficult to separate from aggregate time series variation.", "Figure: Portion of states with eachpolicy" ], [ "The Effect of Policies and Information on Behavior", "We first examine how policies and information affect social distancing behaviors by estimating a version of (): ${\\color {ForestGreen}B_{it}^j}& = {\\color {blue}(\\beta ^j)^{\\prime } P_{it}} + {\\color {magenta}(\\gamma ^j)^{\\prime } I_{it}} +{\\color {gray}(\\delta _B^j)^{\\prime } W_{it}} + \\varepsilon _{it}^{bj}, $ where ${\\color {ForestGreen}B_{it}^j}$ represents behavior variable $j$ in state $i$ at time $t$ .", "${\\color {blue}P_{it}}$ collects the Covid related policies in state $i$ at $t$ .", "Confounders, ${\\color {gray}W_{it}}$ , include state-level covariates, month indicators, and their interactions.", "${\\color {magenta}I_{it}}$ is a set of information variables that affect people's behaviors at $t$ .", "As information, we include each state's growth of cases (in panel A) or deaths (in panel B), and log cases or deaths.", "Additionally, in columns (5)-(8) of Table REF , we include national growth and log of cases or deaths.", "Table REF reports the estimates with standard errors clustered at the state level.", "Across different specifications, our results imply that policies have large effects on behavior.", "Comparing columns (1)-(4) with columns (5)-(8), the magnitude of policy effects are sensitive to whether national cases or deaths are included as information.", "The coefficient on school closures is particularly sensitive to the inclusion of national information variables.", "As shown in Figure REF , there is little variation across states in the timing of school closures.", "Consequently, it is difficult to separate the effect of school closures from a behavioral response to the national trend in cases and deaths.", "The other policy coefficients are less sensitive to the inclusion of national case/death variables.", "After school closures, business closure policies have the next largest effect followed by stay-at-home orders.", "The effect of masks for employees is small.Similar to our finding, [45] find no evidence that introduction of compulsory face mask policies affect community mobility in Germany.", "The row “$\\sum _j \\mathrm {Policy}_j$ ” reports the sum of the estimated effect of all policies, which is substantial and can account for a large fraction of the observed declines in behavior variables.", "For example, in Figure REF , transit intensity for a median state was approximately -50% at its lowest point in early April.", "The estimated policy coefficients in columns imply that imposing all policies would lead to roughly 75% (in column 4) or roughly 35% (in column 8) of the observed decline.", "The large impact of policies on transit intensity suggests that the policies may have reduced the Covid-19 infection by reducing people's use of public transportation.Analyzing the New York City's subway ridership, [32] finds a strong link between public transit and spread of infection.", "In Table REF (B), estimated coefficients of deaths and death growth are generally negative.", "This suggests that the higher number of deaths reduces social interactions measured by Google Mobility Reports perhaps because people are increasingly aware of prevalence of Covid-19 [51].", "The coefficients on log cases and case growth in Table REF (A) are more mixed.Rewrite a regression specification after omitting other variables as $B_{it} = \\gamma _1 \\Delta \\log \\Delta C_{it} + \\gamma _2 \\log \\Delta C_{it} = (\\gamma _1 +\\gamma _2) \\log \\Delta C_{it} - \\gamma _1 \\log \\Delta C_{i,t-7}$ .", "In columns (1)-(4) of Table REF (A), the estimated values of both $(\\gamma _1 +\\gamma _2)$ and $-\\gamma _1$ are negative except for grocery.", "This suggests that a higher level of confirmed cases reduces people's mobility in workplaces, retails, and transit.", "For grocery, the positive estimated coefficient of $(\\gamma _1 +\\gamma _2)$ may reflect stock-piling behavior in early pandemic periods.", "In columns (5)-(8) of both panels, we see that national case/death variables have large, negative coefficients.", "This suggests that behavior responded to national conditions although it is also likely that national case/death variables capture unobserved aggregate time effects beyond information which are not fully controlled by month dummies (e.g., latent policy variables and time-varying confounders that are common across states).", "Figure: Case and death growth conditional on mask mandates" ], [ "The Direct Effect of Policies and Behavior on Case\nand Death Growth", "We now analyze how behavior and policies together influence case and death growth rates.", "We begin with some simple graphical evidence of the effect of policies on case and death growth.", "Figure REF shows average case and death growth conditional on date and whether masks are mandatory for employees.We take 14 and 21 day lags of mask policies for case and death growths, respectively, to identify the states with a mask mandate because policies affect cases and deaths with time lags.", "See our discussion in the Appendix REF .", "The left panel of the figure shows that states with a mask mandate consistently have 0-0.2 lower case growth than states without.", "The right panel also illustrates that states with a mask mandate tend to have lower average death growth than states without a mask mandate.", "Similar plots are shown for other policies in Figures REF and REF in the appendix.", "The figures for stay-at-home orders and closure of nonessential businesses are qualitatively similar to that for masks.", "States with these two policies appear to have about 0.1 percentage point lower case growth than states without.", "The effects of school closures, movie theater closures, and restaurant closures are not clearly visible in these figures.", "These figures are merely suggestive; the patterns observed in them may be driven by confounders.", "We more formally analyze the effect of policies by estimating regressions.", "We first look at the direct effect of policies on case and death growth conditional on behavior by estimating equation (REF ): ${\\color {red}Y_{i,t+\\ell }}& = {\\color {ForestGreen}\\alpha ^{\\prime } B_{it}} + {\\color {blue}\\pi ^{\\prime }P_{it}} + {\\color {magenta}\\mu ^{\\prime }I_{it}} + {\\color {gray}\\delta _Y ^{\\prime }W_{it}} + \\varepsilon ^y_{it},$ where the outcome variable, $\\color {red}Y_{i,t+\\ell }$ , is either case growth or death growth.", "For case growth as the outcome, we choose a lag length of $\\ell =14$ days for behavior, policy, and information variables to reflect the delay between infection and confirmation of case.As we review in the Appendix REF , a lag length of 14 days between exposure and case reporting, as well as a lag length of 21 days between exposure and deaths, is broadly consistent with currently available evidence.", "Section provides a sensitivity analysis under different timing assumptions.", "$\\color {ForestGreen}B_{it}=(B_{it}^1,...,B_{it}^4)^{\\prime }$ is a vector of four behavior variables in state $i$ .", "$\\color {blue}P_{it}$ includes the Covid-related policies in state $i$ that directly affect the spread of Covid-19 after controlling for behavior variables (e.g., masks for employees).", "We include information variables, $\\color {magenta}I_{it}$ , that include the past cases and case growths because the past cases may be correlated with (latent) government policies or people's behaviors that are not fully captured by our observed policy and behavior variables.", "We also consider a specification that includes the past cases and case growth at the national level as additional information variables.", "$\\color {gray}W_{it}$ is a set of confounders that includes month dummies, state-level covariates, and the interaction terms between month dummies and state-level covariates.Month dummies also represent the latent information that is not fully captured by the past cases and growths.", "For case growth, $\\color {gray}W_{it}$ also includes the test rate growth $\\Delta \\log (T)_{it}$ to capture the effect of changing test rates on confirmed cases.", "Equation (REF ) corresponds to (REF ) derived from the SIR model.", "For death growth as the outcome, we take a lag length of $\\ell =21$ days.", "The information variables $\\color {magenta}I_{it}$ include past deaths and death growth rates; $\\color {gray}W_{it}$ is the same as that of the case growth equation except that the growth rate of test rates is excluded from $\\color {gray}W_{it}$ as implied by equation (REF ).", "Table REF shows the results of estimating (REF ) for case and death growth rates.", "Column (1) represents our baseline specification while column (2) replaces business closure policies with closed movie theaters, closed restaurants, and closed non-essential businesses.", "Columns (3) and (4) include past cases/deaths and growth rates at national level as additional regressors.", "The estimates indicate that mandatory face masks for employees reduce the growth rate of infections and deaths by 9-15 percent, while holding behavior constant.", "This suggests that requiring masks for employees in public-facing businesses may be an effective preventive measure.Note that we are not evaluating the effect of universal mask-wearing for the public but that of mask-wearing for employees.", "The effect of universal mask-wearing for the public could be larger if people comply with such a policy measure.", "[75] considered a model in which mask wearing reduces the reproduction number by a factor $(1-e \\cdot pm)^2$ , where $e$ is the efficacy of trapping viral particles inside the mask and $pm$ is the percentage of mask-wearing population.", "Given an estimate of $R_0=2.4$ , [38] argue that 50% mask usage and a 50% mask efficacy level would reduce the reproduction number from 2.4 to 1.35, an order of magnitude impact.", "The estimated effect of masks on death growth is larger than the effect on case growth, but this difference between the two estimated effects is not statistically significant.", "Closing schools has a large and statistically significant coefficient in the death growth regressions.", "As discussed above, however, there is little cross-state variation in the timing of school closures, making estimates of its effect less reliable.", "Neither the effect of stay-at-home orders nor that of business closure policies is estimated significantly different from zero, suggesting that these lockdown policies may have little direct effect on case or death growth when behavior is held constant.", "The row “$\\sum _k w_k \\mathrm {Behavior}_k$ ” reports the sum of estimated coefficients weighted by the average of the behavioral variables from April 1st-10th.", "The estimates of $-0.80$ and $-0.84$ for “$\\sum _k w_k \\mathrm {Behavior}_k$ ” in column (1) imply that a reduction in mobility measures relative to the baseline in January and February have induced a decrease in case and death growth rates by 80 and 84 percent, respectively, suggesting an importance of social distancing for reducing the spread of Covid-19.", "When including national cases and deaths in information, as shown in columns (3) and (4), the estimated aggregate impact of behavior is substantially smaller but remains large and statistically significant.", "A useful practical implication of these results is that Google Mobility Reports and similar data might be useful as a leading indicator of potential case or death growth.", "This should be done with caution, however, because other changes in the environment might alter the relationship between behavior and infections.", "Preventative measures, including mandatory face masks, and changes in habit that are not captured in our data might alter the future relationship between Google Mobility Reports and case/death growth.", "The negative coefficients of the log of past cases or deaths in Table REF is consistent with a hypothesis that higher reported cases and deaths change people's behavior to reduce transmission risks.", "Such behavioral changes in response to new information are partly captured by Google mobility measures, but the negative estimated coefficient of past cases or deaths imply that other latent behavioral changes that are not fully captured by Google mobility measures (e.g., frequent hand-washing, wearing masks, and keeping 6ft/2m distancing) are also important for reducing future cases and deaths.", "If policies are enacted and behavior changes, then future cases/deaths and information will change, which will induce further behavior changes.", "However, since the model includes lags of cases/deaths as well as their growth rates, computing a long-run effect is not completely straightforward.", "We investigate dynamic effects that incorporate feedback through information in section .", "Table: The Total Effect of Policies on Case and Death Growth (PI→YPI {\\rightarrow }Y)Table: Direct and Indirect Policy Effectswithout national case/death variablesTable: Direct and Indirect Policy Effectswithout national case/death variables, restrictions imposedTable: Direct and Indirect Policy Effects with national case/death variables" ], [ "The Total Effect of Policies on Case Growth", "In this section, we focus our analysis on policy effects when we hold information constant.", "The estimated effect of policy on behavior in Table REF and those of policies and behavior on case/death growth in Table REF can be combined to calculate the total effect of policy as well as its decomposition into direct and indirect effects.", "The first three columns of Table REF show the direct (holding behavior constant) and indirect (through behavior changes) effects of policy under a specification that excludes national information variables.", "These are computed from the specification with national cases or deaths included as information (columns (1)-(4) of Table REF and column (1) of Table REF ).", "The estimates imply that all policies combined would reduce the growth rate of cases and deaths by 0.69 and 0.90, respectively, out of which more than one-half to two-third is attributable to the indirect effect through their impact on behavior.", "The estimate also indicates that the effect of mandatory masks for employees is mostly direct.", "We can also examine the total effect of policies and information on case or death growth, by estimating ().", "The coefficients on policy in this regression combine both the direct and indirect effects.", "Table REF shows the full set of coefficient estimates for ().", "The results are broadly consistent with what we found above.", "As in Table REF , the effect of school closures is sensitive to the inclusion of national information variables.", "Also as above, mask mandates have a significant negative effect on growth rates.", "Similarly to Table REF , the first three columns of Table REF report the estimated direct and indirect effects of policy but impose that masks for employees only affect cases/deaths directly without affecting behavior and that business closure policies only affect cases/deaths indirectly through their effects on behavior.These are computed from estimating the specification in columns (1)-(4) of Table REF and column (1) of Table REF but imposing that the coefficient of masks for employees is zero in (REF ) and that the coefficient of business closure policies is zero in ().", "The estimated total effect of masks for employees in the third column of Table REF is higher than that of Table REF .", "Similarly, the total effect of business closure policies is estimated to be larger in Table REF than in Table REF .", "Column “Difference” in Tables REF and REF show the difference between the estimate of () in column “PI${\\rightarrow }$ Y Coefficient” and the implied estimate from (REF )-() in column “Total.” Differences are generally small except for the coefficient of closed K-12 schools and the sum of all policies in Table REF .", "The large differences in school closures may be due to the difficulty in identifying the effect of school closures given a lack of cross-sectional variation.", "Imposing the coefficients of masks for employees and business closure policies to be zero in (REF ) and (), respectively, does not increase differences between “Total\" and “PI${\\rightarrow }$ Y Coefficient” as reported in the last column of Table REF .", "Tables REF and REF present the results for a specification that includes national information variables, where the estimates on masks for employees are similar to those in Tables REF and REF .", "Column “Difference” of Table REF indicates that the restrictions of zero coefficients for masks for employees in (REF ) and for business closure policies in () are not statistically rejected.", "Column “Average” of Tables REF and REF reports the average of “Total” and “PI${\\rightarrow }$ Y Coefficient” columns.", "The average is an appealing and simple way to combine the two estimates of the total effect: one relying on the causal structure and another inferred from a direct estimation of equation (PI ${\\rightarrow }$ Y).", "We shall be using the average estimate in generating the counterfactuals in the next section.", "Turning to the results, the estimates of Tables REF and REF imply that all policies combined would reduce $\\Delta \\log \\Delta D$ by 0.93 and 0.39, respectively.", "For comparison, the median of $\\Delta \\log \\Delta D_{it}$ reached its peak in mid-March of about 1.3 (see Figure REF in the appendix).", "Since then it has declined to near 0.", "Therefore, -0.93 and -0.39 imply that policy changes can account for roughly one-third to two-third of the observed decrease in death growth.", "The remainder of the decline is likely due to changes in behavior from information." ], [ "Specifications, Timings, and Flexible Controls via Machine Learning", "In this section, we provide sensitivity analysis by estimating () with alternative specifications and methods.", "Figure REF shows the 90% confidence intervals of coefficients of (A) masks for employees, (B) closed K-12 school, (C) stay-at-home, and (D) the average variable of closed movie theaters, closed restaurants, and closed non-essential businesses for the following specifications and estimation methods: (1) Baseline specification in columns (1) and (3) of Table REF .", "(2) Exclude the state of New York from the sample because it may be viewed as an outlier in the early pandemic period.", "(3) Add the percentage of people who wear masks to protect themselves in March and April as a confounder for unobserved personal risk-aversion and initial attitude toward mask wearing.The survey is conducted online by YouGov and is based on the interviews of 89,347 US adults aged 18 and over between March 26-April 29, 2020.", "The survey question is “Which, if any, of the following measures have you taken in the past 2 weeks to protect yourself from the Coronavirus (COVID-19)?”.", "(4) Add the log of Trump's vote share in the 2016 presidential election as a confounder for unobserved private behavioral response.", "(5) Add past behavior variables as information used to set policies.", "Under this specification, our causal interpretation is valid when policy variables are sufficiently random conditional on past behavior variables.", "(6) Include all additional controls in (3)-(5) with the sample that excludes New York as in (2).", "(7) Add weekly dummies to the baseline specification.", "(8) Baseline specification estimated by instrumenting $\\Delta \\log T_{it}$ with one week lagged logarithm value of the number of tests per 1000 people.", "(9) Estimated by Double Machine Learning (DML) [18] with Lasso to reduce dimensionality while including all additional controls in (3)-(5).", "(10) Estimated by DML with Random Forest to reduce dimensionality and capture some nonlinearities while including all additional controls in (3)-(5).", "In Figure REF , “red”, “green”, “blue”, and “purple\" indicate the regression models for case growth without national information variables, case growth with national information variables, death growth without national information variables, and death growth with national information variables, respectively.", "The left panel of “(i) baseline timing” assume that the times from exposure to case confirmation and death reporting are 14 and 21 days, respectively, while they are 7 and 24 days, respectively, in the right panel of “(ii) alternative timing.”This alternative timing assumption is motivated by the lower bound estimate of median times from exposure to case confirmation or death reporting in Table 2 of https://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scenarios.html, which are based on data received by CDC through June 29, 2020.", "We thank C. Jessica E. Metcalf for recommending us to do a sensitivity analysis on the timing assumption while suggesting this reference to us.", "Panel (A) of Figure REF illustrates that the estimated coefficients of mask mandates are negative and significant in most specifications, methods, and timing assumptions, confirming the importance of mask policy on reducing case and death growths.", "In Panel (B) of Figure REF , many estimates of closures of K-12 schools suggest that the effect of school closures is large.", "The visual evidence on growth rates for states with and without school closures in Figure REF also suggests that there may be a potentially large effect, though the history is very short.", "This evidence is consistent with the emerging evidence of prevalence of Covid-19 among children [47], [74].", "[24] find that although children's transmission and susceptibility rates are half that of ages 20-30, children's contact rates are much higher.", "This type of evidence, as well as, evidence that children carry viral loads similar to older people ([43]), led Germany to make the early decision of closing schools.", "Our estimates of school closures substantially vary across specifications, however.", "In particular, the estimated effects of school closures on case/death growth become notably smaller once national cases/deaths or weekly dummies are controlled for.", "In the US state-level data, there is little variation across states in the timing of school closures.", "Consequently, its estimate is particularly sensitive to an inclusion of some aggregate variables such as national cases or weekly dummies.", "Given this sensitivity, there still exists a lot of uncertainty as to the magnitude of the effect of school closures.", "Any analyses of re-opening plans need to be aware of this uncertainty.", "An important research question is how to resolve this uncertainty using additional data sources.", "Panel (C) of Figure REF indicates that the estimated coefficients of stay-at-home orders are generally negative and often significant, providing evidence that stay-at-home orders reduce the spread of COVID-19, although the estimates are sometimes sensitive to timing assumptions.", "Panel (D) shows that the estimated coefficients of business closure policies substantially vary across specifications, providing a mixed evidence for the effect of closures of businesses on case/death growth.", "Figure: Estimated Coefficients for Policy Variables: Sensitivity AnalysisFigure: Estimated Coefficients for Policy Variables: Sensitivity Analysis (cont.", ")Figure: Case and death growth conditional on school closures" ], [ "Fixed Effects Specification", "Table REF presents the results of estimating () with state fixed effects and weekly dummies.", "Because fixed effects estimator could be substantially biased when the time dimension is relatively short, we report not only the standard fixed effects estimator in columns (1) and (3) but also the debiased fixed effects estimator [16] in columns (2) and (4), where the state-level clustered bootstrap standard errors are reported in all columns.", "The estimated coefficients of masks for employees largely confirm our finding that mandatory mask policies reduce the case and death growths.", "The estimated coefficients of stay-at-home orders and business closures are negative in columns (1) and (2) but their magnitudes as well as statistical significance are somewhat sensitive to whether bias corrections are applied or not.", "The results in Table REF should be interpreted with caution.", "The fixed effects approach may not be preferred to random effects approach here because the former relies on long time and cross-sectional histories but, in our data, the effective time-dimension is short and the number of states is not large.", "Furthermore, the fixed effects approach could suffer more from measurement errors, which could be a concern for our behavior and policy variables.", "Table: Fixed Effects Specification: the Total Effect of Policies on Case and Death Growth (PI→YPI {\\rightarrow }Y)" ], [ "Empirical Evaluation of Counterfactual Policies", "We now turn our focus to dynamic feedback effects.", "Policy and behavior changes that reduce case and death growth today can lead to a more optimistic, riskier behavior in the future, attenuating longer run effects.", "We perform the main counterfactual experiments using the average of two estimated coefficients as reported in column “Average” of Table REF under a specification that excludes the number of past national cases and deaths from information variables and constrains masks to have only direct effects and business policies only indirect effects.Results (not shown) using the unconstrained estimates in Table REF are very similar.", "Details of the counterfactual computations are in appendix REF ." ], [ "Business Mask Mandate", "We first consider the impact of a nationwide mask mandate for employees beginning on March 14th.", "As discussed earlier, we find that mask mandates reduce case and death growth even when holding behavior constant.", "In other words, mask mandates may reduce infections with relatively little economic disruption.", "This makes mask mandates a particularly attractive policy instrument.", "In this section we examine what would have happened to the number of cases if all states had imposed a mask mandate on March 14th.We feel this is a plausible counterfactual policy.", "Many states began implement business restrictions and school closures in mid March.", "In a paper made publicly available on April 1st, [1] argued for mask usage based on comparisons between countries with and without pre-existing norms of widespread mask usage.", "For illustrative purpose, we begin by focusing on Washington State.", "The left column of Figure REF shows the change in growth rate from mandating masks on March 14th.", "The shaded region is a 90% pointwise confidence interval.", "As shown, mandating masks on March 14th lowers the growth of cases or deaths 14 or 21 days later by 0.07 or 0.14.", "This effect then gradually declines due to information feedback.", "Mandatory masks reduce past cases or deaths, which leads to less cautious behavior, attenuating the impact of the policy.", "The reversal of the decrease in growth in late May is due to our comparison of a mask mandate on March 14th with Washington's actual mask mandate in early May.", "By late May, the counterfactual mask effect has decayed through information feedback, and we are comparing it to the undecayed impact of Washington's actual, later mask mandate.", "The middle column of Figure REF shows how the changes in case and death growth translate into changes in daily cases and deaths.", "The estimates imply that mandating masks on March 14th would have led to about 70 fewer cases and 5 fewer deaths per day throughout April and May.", "Cumulatively, this implies 19% $[9,29]$ % fewer cases and 26% $[14,37]$ % fewer deaths in Washington by the start of June.", "Figure: Effect of nationally mandating masks for employees on March14th in the USThe results for other states are similar to those for Washington.", "Figure REF shows the national effect of mask mandate on March 14th.", "The top panel shows the effect on cases and the bottom panel shows the effect on deaths.", "The left column shows the change in growth rates.", "Dots are the average change in growth rate in each state (i.e.", "the dots are identical to the solid line in the left panel of Figure REF , except for each state instead of just Washington).", "The solid blue line is the national average change in growth rate.", "The shaded region is a 90% pointwise confidence band for the national average change in growth rate.", "The middle column shows the changes in cases and deaths per day.", "Dots are the expected change in each state.", "The thick line is the national total change in daily cases or deaths.", "The shaded region is a pointwise 90% confidence band for the national total change.", "The estimates that a mask mandate in mid March would have decreased cases and deaths by about 6000 and 600 respectively throughout April and May.", "The right column of Figure REF shows how these daily changes compare to the total cumulative cases and deaths.", "The right column shows the national relative change in cumulative cases or deaths.", "The point estimates indicate that mandating masks on March 14th could have led to 21% fewer cumulative cases and 34% fewer deaths by the end of May with their 90 percent intervals given by $[9,32]$ % and $[19,47]$ %, respectively.", "The result roughly translates into 19 to 47 thousand saved lives." ], [ "Business Closure Policies", "Business closures are particularly controversial.", "We now examine a counterfactual where there are no business closure policies (no restaurant, movie theater, or non-essential business closure).", "Figure REF shows the national effect of never restricting businesses on cases and deaths.", "The point estimates imply that, without business closures, cases and deaths would be about 40% higher at the end of May.", "The confidence intervals are wide.", "A 90 percent confidence interval for the increase in cases at the end of May is $[17,78]$ %.", "The confidence interval for deaths is even wider, $[1, 97]$ %.", "Figure: Effect of having no business closure policies in the US" ], [ "Stay-at-home orders", "We next examine a counterfactual where stay-at-home orders had been never issued.", "Figure REF shows the national effect of no stay-at-home orders.", "On average, without stay-at-home orders, case growth rate would have been nearly 0.075 higher in late April.", "This translates to 37% $[6,63]$ % more cases per week by the start of June.", "The estimated effect for deaths is a bit smaller, with no increase included in a 90 percent confidence interval, $[-7,50]$ %.", "Figure: Effect of having no stay-at-home orders in the US" ], [ "Conclusion", "This paper assesses the effects of policies on the spread of Covid-19 in the US using state-level data on cases, tests, policies, and social distancing behavior measures from Google Mobility Reports.", "Our findings are summarized as follows.", "First, our empirical analysis robustly indicates that mandating face masks has reduced the spread of Covid-19 without affecting people's social distancing behavior measured by Google Mobility Reports.", "Our counterfactual experiment suggests that if all states had adopted mandatory face mask policies on March 14th of 2020, then the cumulative number of deaths by the end of May would have been smaller by as much as 19 to 47%, which roughly translates into 19 to 47 thousand saved lives.", "Second, our baseline counterfactual analysis suggests that keeping all businesses open would have led to 17 to 78% more cases while not implementing stay-at-home orders would have increased cases by 6 to 63% by the end of May although we should interpret these numbers with some caution given that the estimated effect of business closures and stay-at-home orders vary across specifications in our sensitivity analysis.", "Third, we find considerable uncertainty over the impact of school closures on case or death growth because it is difficult to identify the effect of school closures from the US state-level data due to the lack of variation in the timing of school closures across states.", "Fourth, our analysis shows that people voluntarily reduce their visits to workplace, retail stores, grocery stores, and people limit their use of public transit when they receive information on a higher number of new cases and deaths.", "This suggests that individuals make decisions to voluntarily limit their contact with others in response to greater transmission risks, leading to an important feedback mechanism that affects future cases and deaths.", "Model simulations that ignore this voluntary behavioral response to information on transmission risks would over-predict the future number of cases and deaths.", "Beyond these findings, our paper presents a useful conceptual framework to investigate the relative roles of policies and information on determining the spread of Covid-19 through their impact on people's behavior.", "Our causal model allows us to explicitly define counterfactual scenarios to properly evaluate the effect of alternative policies on the spread of Covid-19.", "More broadly, our causal framework can be useful for quantitatively analyzing not only health outcomes but also various economic outcomes [14], [19]." ], [ "Measuring $\\Delta C$ and {{formula:03aa5b32-1db2-485c-9d44-6a74eeb7865f}}", "We have three data sets with information on daily cumulative confirmed cases in each state.", "As shown in Table REF , these cumulative case numbers are very highly correlated.", "However, Table REF shows that the numbers are different more often than not.", "Table: Correlation of cumulative casesTable: Portion of cumulative cases that are equal between data setsThe upper left panel of Figure REF shows the evolution of new cases in the NYT data, where daily changes in cumulative cases exhibit some excessive volatility.", "This is likely due to delays and bunching in testing and reporting of results.", "Table REF shows the variance of log new cases in each data set, as well as their correlations.", "As shown, the correlations are approximately $0.9$ .", "The NYT new case numbers have the lowest variance.This comparison is somewhat sensitive to how you handle negative and zero cases when taking logs.", "Here, we replaced $\\log (0)$ with $-1$ .", "In our main results, we work with weekly new cases, which are very rarely zero.", "In our subsequent results, we will primarily use the case numbers from The New York Times.", "Figure: Daily Cases, Weekly Cases, Weekly Deaths from NYT Data, and Weekly Tests from JHUTable: Correlation and variance of log daily new casesFor most of our results, we focus on new cases in a week instead of in a day.", "We do this for two reasons as discussed in the main text.", "First, a decline in new cases over two weeks has become a key metric for decision makers.", "Secondly, aggregating to weekly new cases smooths out the noise associated with the timing of reporting and testing.", "Table REF reports the correlation and variance of weekly log new cases across the three data sets.", "The upper right panel of Figure REF shows the evolution of weekly new cases in each state over time from the NYT data.", "The upper panel of Figure REF shows the evolution of the weekly growth rate of new cases in each state over time.", "Table: Correlation and variance of log weekly new cases" ], [ "Deaths", "Table REF reports the correlation and variance of weekly deaths in the three data sets.", "The lower left panel of Figure REF shows the evolution of weekly deaths in each state from the NYT data.", "As with cases, we use death data from The New York Times in our main results.", "The lower panel of Figure REF shows the evolution of the weekly growth rate of new deaths in each state over time.", "Table: Correlation and variance of weekly deathsFigure: Case and death growth" ], [ "Tests", "Our test data comes from The Covid Tracking Project.", "The lower right panel of Figure REF shows the evolution of the weekly number of tests over time." ], [ "Social Distancing Measures", "In measuring social distancing, we focus on Google Mobility Reports.", "This data has international coverage and is publicly available.", "Figure REF shows the evolution of the four Google Mobility Reports variables that we use in our analysis.", "Figure: Evolution of Google Mobility Reports" ], [ "Policy Variables", "We use the database on US state policies created by [64].", "As discussed in the main text, our analysis focuses on six policies.", "For stay-at-home orders, closed nonessential businesses, closed K-12 schools, closed restaurants except takeout, and closed movie theaters, we double-checked any state for which [64] does not record a date.", "We filled in a few missing dates.", "Our modified data is available here.", "Our modifications fill in 1 value for school closures, 2 for stay-at-home orders, 3 for movie theater closure, and 4 for non-essential business closures.", "Table REF displays all 25 dated policy variables in [64]'s database with our modifications described above.", "Figures REF -REF show the average case and death growth conditional on date and whether each of six policies is implemented or not.", "Table: State Policies" ], [ "Timing", "There is a delay between infection and when a person is tested and appears in our case data.", "[53] maintain a list of estimates of the duration of various stages of Covid-19 infections.", "The incubation period, the time from infection to symptom onset, is widely believed to be 5 days.", "For example, using data from Wuhan, [48] estimate a mean incubation period of 5.2 days.", "[70] reviews the literature and concludes the mean incubation period is 3-9 days.", "Estimates of the time between symptom onset and case reporting or death are less common.", "Using Italian data, [15] estimate an average of 7.3 days between symptom onset and reporting.", "[85] find an average of 7.4 days using Chinese data from December to early February, but they find this period declined from 8.9 days in January to 5.4 days in the first week of February.", "Both of these papers on time from symptom onset to reporting have large confidence intervals covering approximately 1 to 20 days.", "Studying publicly available data on infected persons diagnosed outside of Wuhan, [49] estimate an average of 15 days from onset to death.", "Similarly, using publicly available reports of 140 confirmed Covid-19 cases in China, mostly outside Hubei Province, [67] estimate the time from onset to death to be 16.1 days.", "Based on the above, we expect a delay of roughly two weeks between changes in behavior or policies, and changes in reported cases while a corresponding delay of roughly three weeks for deaths." ], [ "Direct and Indirect Policy Effects with national case/death variables", "Tables REF and REF present the estimates of direct and indirect effects of policies for the specification with past national case/death variables.", "The effects of school closures and the sum of policies are estimated substantially smaller in Table REF when national case/death variables are included than in Table REF .", "This sensitivity reflects the difficulty in identifying the aggregate time effect—which is largely captured by national cases/deaths—given little cross-sectional variation in the timing of school closures across states.", "On the other hand, the estimated effects of policies other than school closures are similar between Table REF and Table REF ; the effect of other policies are well-identified from cross-sectional variations." ], [ "Double Machine Learning", "To estimate the coefficient of the $j$ -th policy variable, $P_{it}^j$ , using double machine learning ([18]), we consider a version of () as follows: $Y_{i,t+\\ell } & = P_{it}^j \\theta _0 + \\phi ^{\\prime } X_{it} + g_0(W_{it}) + \\xi _{it}, && E[\\xi _{it}|P_{it}^j,X_{it},W_{it}]=0,\\\\P_{it}^j &= \\psi ^{\\prime } X_{it} +m_0(W_{it}) +V_{it},&& E[V_{it}|X_{it},W_{it}]=0,$ where $X_{it}=((P_{it}^{-j})^{\\prime },B_{it}^{\\prime },I_{it}^{\\prime })^{\\prime }$ collects policy variables except for the $j$ -th policy variables $P_{it}^{-j}$ , behavior variables $B_{it}$ , and information variables $I_{it}$ .", "The confounding factors $W_{it}$ affect the policy variable $P_{it}^j$ via the function $m_0(W)$ and the outcome variable $Y_{i,t+\\ell }$ via the function $g_0(W)$ .", "We apply Lasso or Random Forests to estimate $g_0(W)$ for dimension reductions or for capturing non-linearity while the coefficients of $X_{it}$ are estimated under linearity without imposing any dimension reductions.", "We omit the details of estimation procedure here.", "Please see the discussion in Example 1.1 of [18] for reference." ], [ "Debiased Fixed Effects Estimator", "We apply cross-over Jackknife bias correction as discussed in [17] in details.", "Here, we briefly describe our debiased fixed effects estimator.", "Given our panel data with $N=51$ states and $T=75$ days for case growth equation (), consider two subpanels as follows: ${\\bf S}_1 = \\lbrace (i,t) : i \\le \\lceil N/2 \\rceil , t \\le \\lceil T/2 \\rceil \\rbrace \\cup \\lbrace (i,t) : i \\ge \\lfloor N/2 + 1 \\rfloor , t \\ge \\lfloor T/2 + 1\\rfloor \\rbrace $ and ${\\bf S}_2 = \\lbrace (i,t) : i \\le \\lceil N/2 \\rceil , t \\ge \\lfloor T/2 + 1\\rfloor \\rbrace \\cup \\lbrace (i,t) : i \\ge \\lfloor N/2 + 1 \\rfloor , t \\le \\lceil T/2 \\rceil \\rbrace $ where $\\lceil .", "\\rceil $ and $\\lfloor .", "\\rfloor $ are the ceiling and floor functions.", "Each of these two subpanels includes observations for all cross-sectional units and time periods.", "We form the debiased fixed effects estimator as $\\widehat{\\beta }_{\\rm BC} = 2 \\widehat{\\beta }- \\widetilde{\\beta }_{{\\bf S}_1\\cup {\\bf S}_2},$ where $\\widehat{\\beta }$ is the standard fixed effects estimator while $\\widetilde{\\beta }_{{\\bf S}_1\\cup {\\bf S}_2}$ denotes the fixed effects estimator using the data set ${\\bf S}_1\\cup {\\bf S}_2$ but treats the states in ${\\bf S}_1$ differently from those in ${\\bf S}_2$ to form the fixed effects estimator; namely, we include approximately twice more state fixed effects to compute $\\widetilde{\\beta }_{{\\bf S}_1\\cup {\\bf S}_2}$ .Alternatively, we may form the cross-over jackknife corrected estimator as $\\widehat{\\beta }_{\\rm CBC} = 2 \\widehat{\\beta }- (\\widetilde{\\beta }_{{\\bf S_1}}+\\widetilde{\\beta }_{{\\bf S_2}})/2$ , where $\\widetilde{\\beta }_{{\\bf S}_j}$ denotes the fixed effect estimator using the subpanel ${\\bf S}_j$ for $j=1,2$ .", "In our empirical analysis, these two cross-over jackknife bias corrected estimators give similar result.", "We obtain bootstrap standard errors by using multipler bootstrap with state-level clustering." ], [ "Details for Computing Counterfactuals ", "We compute counterfactuals from the “total effect\" version of the model, with behavior concentrated out.", "$Y_{i,t+\\ell }= \\mathsf {a}^{\\prime }P_{it} + \\mathsf {b}^{\\prime } I_{it} + {\\tilde{\\delta }}^{\\prime } W_{it} + {\\bar{\\varepsilon }}_{it}$ We consider a counterfactual change of $P_{it}$ to $P_{it}^\\star $ , while $W_{it}$ and $\\bar{\\varepsilon }_{it}$ are held constant.", "In response to the policy change, $Y_{i,t+\\ell }$ and the part of $I_{it}$ that contains $Y_{it}$ , change to $Y_{i,t+\\ell }^\\star $ and $I_{it}^\\star $ .", "To be specific, let $C_{it}$ denote cumulative cases in state $i$ on day $t$ .", "Our outcome is: $Y_{i,t+\\ell } \\equiv \\log \\left(C_{i, t+\\ell } - C_{i, t + \\ell - 7} \\right) - \\log \\left(C_{i, t+\\ell -7} - C_{i, t + \\ell - 14} \\right) \\equiv \\Delta \\log \\Delta C_{i,t+\\ell }$ where $\\Delta $ is a 7 day difference operator.", "Writing the model in terms of $\\Delta \\log \\Delta C$ we have: $\\Delta \\log \\Delta C_{i,t+\\ell } = \\mathsf {a}^{\\prime } P_{it} + \\mathsf {b}_D\\Delta \\log \\Delta C_{i,t} + \\mathsf {b}_L \\log \\Delta C_{i,t} +{\\tilde{\\delta }}^{\\prime } W_{it} + {\\bar{\\varepsilon }}_{it}$ To simplify computation, we rewrite this model in terms of $\\log \\Delta C$ : $\\log \\Delta C_{i,t+\\ell } = \\mathsf {a}^{\\prime } P_{it} + \\log \\Delta C_{i,t+\\ell -7} + (\\mathsf {b}_D + \\mathsf {b}_L) \\log \\Delta C_{i,t} -\\mathsf {b}_D \\log \\Delta C_{i,t-7} + {\\tilde{\\delta }}^{\\prime } W_{it} + {\\bar{\\varepsilon }}_{it}$ This equation is used to iteratively compute $\\log \\Delta C_{i,t+\\ell }^\\star $ conditional on initial $\\lbrace \\log \\Delta C_{i,s}\\rbrace _{s=-\\ell -7}^{0}$ , and the entire sequence of $\\lbrace W_{i,t}, P_{i,t}^\\star , \\bar{\\varepsilon }_{it} \\rbrace _{t=0}^T$ .", "Weekly cases instead of log weekly cases are given simply by $ \\Delta C_{i,t+\\ell }^\\star = \\exp (\\log \\Delta C_{i,t+\\ell }^\\star )$ Cumulative cases can be recursively computed as: $C_{i,t} = C_{i,t-7} + \\Delta C_{i,t} = C_{i,t-7} + \\exp (\\log \\Delta C_{i,t})$ given initial $\\lbrace C_{i,s}\\rbrace _{s=-7}^0$ .", "Note that $\\log \\Delta C_{i,t+\\ell }^\\star $ depends linearly on $\\bar{\\varepsilon }$ , so the residuals (and our decision to condition on versus integrate them out) do not matter when considering linear constasts of $\\log \\Delta C_{it}$ or $\\Delta \\log \\Delta C_{it}$ , or when considering relative contrasts of $\\Delta C_{it}$ ." ], [ "Inference", "Let $S(\\theta , \\mathbf {\\varepsilon })$ denote some counterfactual quantity or contrast of interest, where $\\theta = (\\mathsf {a},\\mathsf {b}, \\tilde{\\delta })$ are the parameters, and $\\mathbf {\\varepsilon }$ is the vector of residuals.", "Examples of $S$ that we compute include: Contrasts of growth rates: $S(\\theta , \\mathbf {\\varepsilon }) =\\Delta \\log \\Delta C_{it}^\\star - \\Delta \\log \\Delta C_{it}$ Relative contrasts of weekly cases: $S(\\theta , \\mathbf {\\varepsilon }) =\\Delta C_{it}^\\star /\\Delta C_{it}$ Relative contrasts of cumulative cases: $S(\\theta , \\mathbf {\\varepsilon }) =\\frac{C_{it}^\\star - C_{it}}{C_{it}}$ The first two examples do not actually depend on $\\mathbf {\\varepsilon }$ , but the third one does.", "Inference is by simulation.", "Let $\\hat{\\theta }$ denote our point estimates and $\\hat{\\mathbf {\\varepsilon }}$ the associated residuals.", "We draw $\\tilde{\\theta }_j$ from the asymptotic distribution of $\\hat{\\theta }$ .", "Let $\\tilde{\\varepsilon }_j$ denote the residuals associated with $\\tilde{\\theta }_j$ .", "We then compute $ \\tilde{s}_j = S(\\tilde{\\theta }_j, \\tilde{\\varepsilon }_j) $ for $j=1,..., 200$ and plots the mean across $j$ as a point estimate and quantiles across $j$ for confidence intervals." ] ]

[ [ "Data Analysis Recipes: Products of multivariate Gaussians in Bayesian\n inferences" ], [ "Abstract A product of two Gaussians (or normal distributions) is another Gaussian.", "That's a valuable and useful fact!", "Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly.", "That is, a linear, Gaussian, Bayesian model.", "This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian.", "The means and variance tensors of the refactored Gaussians are straightforward to obtain as closed-form expressions; here we deliver these expressions, with discussion.", "The closed-form expressions can be used to speed up and improve the precision of inferences that contain linear parameters with Gaussian priors.", "We connect these methods to inferences that arise frequently in physics and astronomy.", "If all you want is the answer, the question is posed and answered at the beginning of Section 3.", "We show two toy examples, in the form of worked exercises, in Section 4.", "The solutions, discussion, and exercises in this Note are aimed at someone who is already familiar with the basic ideas of Bayesian inference and probability." ], [ "Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences", "David W. Hogg11The authors would like to thank Will Farr (Stony Brook), Dan Foreman-Mackey (Flatiron), Rodrigo Luger (Flatiron), Hans-Walter Rix (MPIA), and Sam Roweis (deceased), for help with all these concepts.", "This project was developed in part at AstroHackWeek 2016, which was hosted by the Moore–Sloan Data Science Environment.", "This research was supported by the National Science Foundation and National Aeronautics and Space Administration.", "The source text, example IPython notebooks, and data files used below are available via Zenodo archive [6].", "Center for Cosmology and Particle Physics, Dept.", "Physics, New York University Max-Planck-Institut für Astronomie, Heidelberg Flatiron Institute, a division of the Simons Foundation Adrian M. Price-Whelan Flatiron Institute, a division of the Simons Foundation Boris Leistedt Department of Physics, Imperial College, London Center for Cosmology and Particle Physics, Dept.", "Physics, New York University" ], [ "Abstract:", "A product of two Gaussians—or normal distributions—is another Gaussian.", "That's a valuable and useful fact!", "Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly.", "That is, a linear, Gaussian, Bayesian model.", "This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian.", "The means and variance tensors of the refactored Gaussians are straightforward to obtain as closed-form expressions; here we deliver these expressions, with discussion.", "The closed-form expressions can be used to speed up and improve the precision of inferences that contain linear parameters with Gaussian priors.", "We connect these methods to inferences that arise frequently in physics and astronomy.", "If all you want is the answer, the question is posed and answered at the beginning of Section .", "We show two toy examples, in the form of worked exercises, in Section .", "The solutions, discussion, and exercises in this Note are aimed at someone who is already familiar with the basic ideas of Bayesian inference and probability." ], [ "Inferences with linear parameters", "It is common in physics, astronomy, engineering, machine learning, and many other fields that likelihood functions (probabilities of data given parameters) are chosen to be Gaussian (or normal22In this Note, we obey physics and astronomy conventions and refer to the normal pdf as the Gaussian pdf.", "We realize that it is irresponsible to name things after people when those same things also have generic, descriptive names.", "On the other hand, “normal” isn't the finest name either.", "Perhaps Gaussian pdfs should be called “central” since they are produced by the central limit theorem.", "Anyway, we will continue with the name “Gaussian” despite our own reservations.", "We apologize to our reader.", "): One reason is that a likelihood function is basically a noise model, and it is often case that the noise is treated as Gaussian.", "This assumption for the likelihood function is accurate when the noise model has benefitted from the central limit theorem.", "This is true, for example, when the noise is thermal, or when the noise is shot noise and the numbers (numbers of detected photons or other particles) are large.", "Another reason that the likelihood function is often treated as Gaussian is that Gaussians are generally tractable: Many computations we like to perform on Gaussians, like integrals and derivatives and optimizations, have closed-form solutions.", "Even when we don't use the closed-form solutions, there are many contexts in which Gaussians lead to convex optimizations, providing guarantees to resulting inferences.", "It is also common in physics and astronomy that models for data include parameters such that the expectation value for the data (in, say, a set of repeated experiments) is linearly proportional to some subset of the parameters.", "This is true, for example, when we fit a histogram of Large Hadron Collider events affected by the Higgs boson,33See, for example, [1], and [2], and references therein.", "where the expected number of counts in each energy bin is proportional to a linear combination of the amplitudes of various backgrounds and some coupling to the Higgs.", "Another linear-parameter context, for example, arises when we fit for the radial-velocity variation of a star in response to a faint, orbiting companion.44See, for an example in our own work, [10], [11].", "That project and those papers would have been impossible without the speed-ups provided by the expressions derived in this Note.", "Indeed, the writing of this Note was motivated by the work presented in those papers.", "(Okay full disclosure: It was motivated in part by a mistake made by DWH in one of those papers!)", "In this problem, the expectation of the radial-velocity measurements depends linearly on the binary system velocity and some combination of masses and system inclination (with respect to the line of sight).", "In both of these cases, there are both linear parameters (like the amplitudes) and nonlinear parameters (like the mass of the Higgs, or the orbital period of the binary-star system).", "In what follows, we will spend our energies on the linear parameters, though our work on them is in service of learning the nonlinear parameters too, of course.", "In Bayesian inference contexts, the “models” to which we are referring are expressions for likelihood functions and prior pdfs; these are the things that will be Gaussians here.", "Bayes' theorem is often written as a ratio of probability density functions (pdfs in what follows), but it can also be written as a pdf factorization:55For a tutorial on probability factorizations, see [4].", "$p(y,\\theta \\,|\\,\\textrm {\\ding {114}}) = p(y\\,|\\,\\theta ,\\textrm {\\ding {114}})\\,p(\\theta \\,|\\,\\textrm {\\ding {114}}) = p(\\theta \\,|\\,y,\\textrm {\\ding {114}})\\,p(y\\,|\\,\\textrm {\\ding {114}})$ where $p(y,\\theta \\,|\\,\\textrm {\\ding {114}})$ is the joint probability of data $y$ and parameters $\\theta $ given your model assumptions and hyper parameters (symbolized jointly as $\\textrm {\\ding {114}}$ ),66In this Note, we typeset different mathematical objects according to their mathematical or transformation properties.", "We typeset vectors (which are column vectors) as $a, b, \\theta $ , we typeset variance tensors (which in this case are square, non-negative semi-definite matrices) as $\\mathbf {C}, \\mathbf {\\Lambda }$ , we typeset other matrices (which will in general be non-square) as ${M}, {U}$ , and we typeset blobs or unstructured collections of information as $\\textrm {\\ding {114}}, \\textrm {\\ding {80}}$ .", "Related to this typography is an implicit terminology: We distinguish variance tensors from matrices.", "This distinction is somewhat arbitrary, but the strong constraints on the variance tensors (non-negative, real eigenvalues) make them special beasts, with special geometric properties, like that they can be used as metrics in their respective vector spaces.", "$p(y\\,|\\,\\theta ,\\textrm {\\ding {114}})$ is the likelihood, or probability of data $y$ given parameters (and assumptions), $p(\\theta \\,|\\,\\textrm {\\ding {114}})$ is the prior pdf for the parameters $\\theta $ , $p(\\theta \\,|\\,y,\\textrm {\\ding {114}})$ is the posterior pdf for the parameters $\\theta $ given the data, and $p(y\\,|\\,\\textrm {\\ding {114}})$ is the pdf for the data, marginalizing out all of the linear parameters (hereafter, we refer to this as the marginalized likelihood77In the case that the problem has no parameters other than the linear parameters $\\theta $ , this term, $p(y\\,|\\,\\textrm {\\ding {114}})$ , is sometimes called the Bayesian evidence or the fully marginalized likelihood.).", "If the likelihood is Gaussian, and the expectation of the data depends linearly on the parameters, and if we choose the prior pdf to also be Gaussian, then all the other pdfs (the joint, the posterior, and the marginalized likelihood) all become Gaussian too.", "The main point of this Note is that the means and variances of these five Gaussians are all related by simple, closed-form expressions, given below.", "One consequence of this math is that if you have a Gaussian likelihood function, and if you have a subset of parameters that are linearly related to the expectation of the data, then you can obtain both the posterior pdf $p(\\theta \\,|\\,y,\\textrm {\\ding {114}})$ and the marginalized likelihood $p(y\\,|\\,\\textrm {\\ding {114}})$ with closed-form transformations of the means and variances of the likelihood and prior pdf.", "A currently popular data-analysis context in which Gaussian likelihoods are multiplied by Gaussian priors is Gaussian processes (GPs), which is a kind of non-parametric fitting in which a kernel function sets the flexibility of a data-driven model.", "A full discussion of GPs is beyond the scope of this Note, but excellent discussions abound.88We like the free book by [12].", "The math below can be applied in many GP contexts.", "Indeed, most linear model fits of the kind we describe below can be translated into the language of GPs, because any noise process that delivers both a prior pdf and a likelihood with Gaussian form is technically identical to a (probably non-stationary) GP.", "We leave that translation as an exercise to the ambitious reader.99If you want a cheat sheet, we come close to performing this translation in [8]." ], [ "Marginalization by refactorization", "Imagine that we are doing an inference using data $y$ (which is a $N$ -dimensional vector, say).", "We are trying to learn linear parameters $\\theta $ (a $K$ -dimensional vector) and also nonlinear parameters $\\textrm {\\ding {80}}$ (an arbitrary vector, list, or blob).1010Here, $\\textrm {\\ding {80}}$ represents the nonlinear parameters and assumptions or hyper parameters.", "That is, it contains everything on which the linear model is conditioned, including not just nonlinear parameters but also investigator choices.", "Note our subjectivism here!", "Whether we are Bayesian or frequentist, the inference is based on a likelihood function, or probability for the data given parameters $\\mbox{\\small likelihood:} ~~p(y\\,|\\,\\theta ,\\textrm {\\ding {80}}) ~~.$ Now let's imagine that the parameters $\\theta $ are either nuisance parameters, or else easily marginalized, so we want to marginalize them out.", "This will leave us with a lower-dimensional marginalized likelihood function $\\mbox{\\small marginalized likelihood:} ~~p(y\\,|\\,\\textrm {\\ding {80}}) ~~.$ That's good, but the marginalization comes at a cost: We have to become Bayesian, and we have to choose a prior $\\mbox{\\small prior on nuisance parameters:} ~~p(\\theta \\,|\\,\\textrm {\\ding {80}}) ~~.$ This is the basis for the claim1111A claim that perhaps hasn't been made clearly yet, but will eventually be by at least one of these authors.", "that Bayesian inference requires a likelihood function, and priors on the nuisance parameters.", "It does not require a prior on everything, contrary to some statements in the literature.1212It is very common for papers or projects with Bayesian approaches to claim that the goal of Bayesian inference is to create posterior pdfs.", "That isn't correct.", "Different Bayesian inferences have different objectives.", "The fundamental point of Bayesian inference is that consistently held beliefs obey the rules of probability.", "That, in turn, says that if you want to communicate to others things useful to the updating of their beliefs, you want to communicate about your likelihood.", "Your posterior pdf isn't all that useful to them!", "We have said “$p(\\theta \\,|\\,\\textrm {\\ding {80}})$ ” because this prior pdf may depend on the nonlinear parameters $\\textrm {\\ding {80}}$ , but it certainly doesn't have to.", "Armed with the likelihood and prior—if you want it—you can construct the posterior pdf for the linear parameters $\\mbox{\\small posterior for nuisance parameters:} ~~p(\\theta \\,|\\,y,\\textrm {\\ding {80}}) ~~.$ To perform a marginalization of the likelihood, we have two choices.", "We can either do an integral: $p(y\\,|\\,\\textrm {\\ding {80}}) = \\int p(y\\,|\\,\\theta ,\\textrm {\\ding {80}})\\,p(\\theta \\,|\\,\\textrm {\\ding {80}})\\,\\mathrm {d}\\theta ~~,$ where the integral is implicitly over the entire domain of the linear parameters $\\theta $ (or the entire support of the prior).", "Or we can re-factorize the expression using Bayes' theorem: $p(y\\,|\\,\\theta ,\\textrm {\\ding {80}})\\,p(\\theta \\,|\\,\\textrm {\\ding {80}})= p(\\theta \\,|\\,y,\\textrm {\\ding {80}})\\,p(y\\,|\\,\\textrm {\\ding {80}})~~.$ That is, in certain magical circumstances it is possible to do this re-factorization without explicitly doing any integral.", "When this is true, the marginalization is sometimes far easier than the relevant integral.", "The point of this Note is that this magical circumstance arises when the two probability distributions—the likelihood and the prior—are both Gaussian in form, and when the model is linear over the parameters we would like to marginalize over.", "In detail we will assume the likelihood $p(y\\,|\\,\\theta ,\\textrm {\\ding {80}})$ is a Gaussian in $y$ , the prior $p(\\theta \\,|\\,\\textrm {\\ding {80}})$ is a Gaussian in $\\theta $ , the mean of the likelihood Gaussian depends linearly on the linear parameters $\\theta $ , and the linear parameters $\\theta $ don't enter the likelihood anywhere other than in the mean.", "In equations, this becomes: $p(y\\,|\\,\\theta ,\\textrm {\\ding {80}}) = \\mathcal {N}\\!\\,(y\\,|\\,{M}\\cdot \\theta ,\\mathbf {C})$ $p(\\theta \\,|\\,\\textrm {\\ding {80}}) = \\mathcal {N}\\!\\,(\\theta \\,|\\,\\mu ,\\mathbf {\\Lambda })$ $\\mathcal {N}\\!\\,(x\\,|\\,m,\\mathbf {V}) \\equiv \\frac{1}{||2\\pi \\,\\mathbf {V}||^{1/2}}\\,\\exp \\left(-\\frac{1}{2}\\,[x-m]\\mathbf {V}^{-1}\\cdot [x- m]\\right)~~,$ where $\\mathcal {N}\\!\\,(x\\,|\\,m,\\mathbf {V})$ is the multivariate Gaussian pdf1313Check out what we did with the “$2\\pi $ ” in the determinant in equation (REF ): We wrote an expression for the multivariate Gaussian that never makes any reference to the dimension $d$ of the $x$ -space.", "Most expressions in the literature have a pesky $d/2$ in them, which is ugly and implies some need for code or equations to know the dimension explicitly, even though all the terms (determinant, inner product) are coordinate-free scalar forms.", "If you use the expression as we have written it here, you never have to explicitly access the dimensions.", "for a vector $x$ given a mean vector $m$ and a variance tensor $\\mathbf {V}$ , ${M}$ is a $N\\times K$ rectangular design matrix (which depends, in general, on the nonlinear parameters $\\textrm {\\ding {80}}$ ), $\\mathbf {C}$ is a $N\\times N$ covariance matrix of uncertainties for the data (diagonal if the data dimensions are independent).", "That is, the likelihood is a Gaussian with a mean that depends linearly on the parameters $\\theta $ , and $\\mu $ and $\\mathbf {\\Lambda }$ are the $K$ -vector mean and $K\\times K$ variance tensor for the Gaussian prior.", "Figure: NO_CAPTION 1: The one-dimensional case: If the prior pdf and the likelihood are both Gaussian in a single parameter, their product (and hence the posterior pdf) is also Gaussian, with a narrower width (smaller variance) than either the prior pdf or the likelihood.In this incredibly restrictive—but also surprisingly common—situation, the re-factored pdfs $p(\\theta \\,|\\,y,\\textrm {\\ding {80}})$ (the posterior for the linear parameters, conditioned on the nonlinear parameters in $\\textrm {\\ding {80}}$ ) and $p(y\\,|\\,\\textrm {\\ding {80}})$ (the marginalized likelihood, similarly conditioned) will also both be Gaussian.", "We will solve this problem for general multivariate Gaussians in spaces of different dimensionality (and units) but the one-dimensional case is illustrated in  .", "Obtaining the specific form for the general Gaussian product is the object of this Note." ], [ "Products of two Gaussians", "On the internets, there are many documents, slide decks, and videos that explain products of Gaussians in terms of other Gaussians.1414Two good examples are [13], and [9].", "The closest—that we know of—to a discussion with the generality of what is shown here is perhaps our own previous contribution [8].", "The vast majority of these consider either the univariate case (where the data $y$ and the parameter $\\theta $ are both simple scalars, which is not useful for our science cases), or the same-dimension case (where the data $y$ and the parameter vector $\\theta $ are the same length, which never occurs in our applications).", "Here we solve this problem in the general case:1515We solve this general case, but we are not claiming priority in any sense: This mathematics has been understood for many many decades or even centuries.", "This Note is a pedagogical contribution, not a research contribution.", "The inputs are multivariate (vectors) and the two Gaussians we are multiplying live in spaces of different dimensions.", "That is, we solve the following problem:" ], [ "Problem:", "Find $K$ -vector $a$ , $K\\times K$ variance tensor $\\mathbf {A}$ , $N$ -vector $b$ , and $N\\times N$ variance tensor $\\mathbf {B}$ such that $\\mathcal {N}\\!\\,(y\\,|\\,{M}\\cdot \\theta ,\\mathbf {C})\\,\\mathcal {N}\\!\\,(\\theta \\,|\\,\\mu ,\\mathbf {\\Lambda })= \\mathcal {N}\\!\\,(\\theta \\,|\\,a,\\mathbf {A})\\,\\mathcal {N}\\!\\,(y\\,|\\,b,\\mathbf {B}) ~~,$ and such that $a$ , $\\mathbf {A}$ , $b$ , and $\\mathbf {B}$ don't depend on $\\theta $ at all.", "Note that $y$ is a $N$ -vector, ${M}$ is a $N\\times K$ matrix, $\\theta $ is a $K$ -vector, $\\mathbf {C}$ is a $N\\times N$ non-negative semi-definite variance tensor, $\\mu $ is a $K$ -vector, and $\\mathbf {\\Lambda }$ is a $K\\times K$ non-negative semi-definite variance tensor." ], [ "Proof:", "The two sides of equation (REF ) are identical if two things hold.", "The first thing is that the determinant products must be equal: $||\\mathbf {C}||\\,||\\mathbf {\\Lambda }|| = ||\\mathbf {A}||\\,||\\mathbf {B}||~~,$ because the determinants are involved in the normalizations of the functions.", "This equality of determinant products follows straightforwardly from the matrix determinant lemma1616See, for example, [16], and [3].", "$||\\mathbf {Q}+ {U}\\cdot {V}| = ||\\mathbf {I}+ {V}\\mathbf {Q}^{-1}\\cdot {U}||\\,||\\mathbf {Q}||~~,$ where ${U}$ and ${V}$ can be rectangular, and $\\mathbf {I}$ is the correct-sized identity matrix.", "This identity implies that $||\\mathbf {A}^{-1}|| = ||\\mathbf {I}+ {M}\\mathbf {C}^{-1}\\cdot {M}\\cdot \\mathbf {\\Lambda }||\\,||\\mathbf {\\Lambda }^{-1}||$ $||\\mathbf {B}|| = ||\\mathbf {I}+ {M}\\mathbf {C}^{-1}\\cdot {M}\\cdot \\mathbf {\\Lambda }||\\,||\\mathbf {C}||~~,$ where we had to apply the identity twice to get the $||\\mathbf {A}^{-1}||$ expression.", "We can ratio these as follows to prove this first thing: $||\\mathbf {A}||\\,||\\mathbf {B}||= \\frac{||\\mathbf {B}||}{||\\mathbf {A}^{-1}||}= \\frac{||\\mathbf {C}||}{||\\mathbf {\\Lambda }^{-1}||}= ||\\mathbf {C}||\\,||\\mathbf {\\Lambda }||~~.$ The second thing required for the proof is that the quadratic scalar form $[y-{M}\\cdot \\theta ]\\mathbf {C}^{-1}\\cdot [y-{M}\\cdot \\theta ]+ [\\theta -\\mu ]\\mathbf {\\Lambda }^{-1}\\cdot [\\theta -\\mu ]$ must equal the quadratic scalar form $[\\theta -a]\\mathbf {A}^{-1}\\cdot [\\theta -a]+ [y-b]\\mathbf {B}^{-1}\\cdot [y-b]~~,$ because these quadratic scalar forms appear in the exponents in the functions.", "This equality follows from straightforward expansion of all the quadratic forms, plus some use of the matrix inversion lemma1717This useful lemma is also called the Woodbury matrix identity.", "See also [17], and [3].", "$[\\mathbf {Q}+ {U}\\cdot \\mathbf {S}\\cdot {V}^{-1}= \\mathbf {Q}^{-1}- \\mathbf {Q}^{-1}\\cdot {U}\\cdot [\\mathbf {S}^{-1}+ {V}\\mathbf {Q}^{-1}\\cdot {U}]^{-1}\\cdot {V}\\mathbf {Q}^{-1}~~,$ which gives an expression for the inverse $\\mathbf {B}^{-1}$ of the marginalized likelihood variance: $\\mathbf {B}^{-1}= \\mathbf {C}^{-1}- \\mathbf {C}^{-1}\\cdot {M}\\cdot [\\mathbf {\\Lambda }^{-1}+ {M}\\mathbf {C}^{-1}\\cdot {M}]^{-1}\\cdot {M}\\mathbf {C}^{-1}~~.$ After that it's just a lot of grinding through matrix expressions.1818We leave this grinding to the avid reader.", "For guidance, it might help to realize that there are terms that contain $\\theta \\theta $ , $\\theta y$ , $yy$ , $\\theta \\mu $ , and $\\mu \\mu $ .", "If you expand out each of these five kinds of terms, each of the five should lead to an independent-ish equality." ], [ "Solution notes:", "In principle we found this factorization by expanding the quadratic in (REF ) and then completing the square.", "Of course we didn't really; we used arguments (which physicists love) called detailed balance: We required that the terms that look like $\\theta \\mathbf {Q}\\cdot \\theta $ were equal between the LHS (REF ) and the RHS (REF ), and then all the terms that look like $\\mu \\mathbf {S}\\cdot \\mu $ , and so on.", "It turns out you don't have to consider them all to get the right solution.", "There is an alternative derivation or proof involving the canonical form for the multivariate Gaussian.", "This form is $\\mathcal {N}\\!\\,(x\\,|\\,m,\\mathbf {V}) = \\exp \\left(-\\frac{1}{2}x^T \\cdot \\mathbf {H}\\cdot x+ \\eta ^T \\cdot x- \\frac{1}{2}\\,\\xi \\right)$ $\\mathbf {H}\\equiv \\mathbf {V}^{-1}~~;~~\\eta \\equiv \\mathbf {V}^{-1}\\cdot m~~;~~\\xi \\equiv \\ln ||2\\pi \\,\\mathbf {V}||+\\eta ^T\\cdot \\mathbf {V}\\cdot \\eta ~~.$ In the canonical form, many products and other manipulations become simpler, so it is worth trying this route if you get stuck when manipulating Gaussian expressions.", "Because the matrix ${M}$ is not square, it has no inverse.", "And because this is a physics problem, ${M}$ has units (which are the units of $\\mathrm {d}y/\\mathrm {d}\\theta $ ).", "It's beautiful in the solution that ${M}$ and ${M} appear only where theunits make sense.They make sense because the units of $ C-1$ are inverse data-squared (where $ y$is the data vector) and the units of $$ are parameters-squared and the unitsof $ M$ are data over parameters.And they are all different sizes.$ If you remember the Bayesian context around equation (REF ) and the Bayesian discussion thereafter, the Gaussian $\\mathcal {N}\\!\\,(\\theta \\,|\\,a,\\mathbf {A})$ is the posterior pdf for the linear parameters $\\theta $ , and the Gaussian $\\mathcal {N}\\!\\,(y\\,|\\,b,\\mathbf {B})$ is the marginalized likelihood, marginalizing out the linear parameters $\\theta $ .", "This marginalization is usually thought of as being an integral, like the one given in equation (REF ).", "How are these linear-algebra expressions in any sense “doing this integral”?", "The answer is: That integral is a correlation of two Gaussians,1919Astronomers like to say that it is the “convolution” of two Gaussians, but it is really the correlation of two Gaussians.", "The differences between convolution and correlation are minimal, though, and we aren't sticklers.", "and the correlation of two Gaussians delivers a new Gaussian with a shifted mean that is wider than either of the original two Gaussians.", "This factorization does, indeed, deliver the correct marginalization integral.", "Continuing along these lines, various parts of the solution are highly interpretable in terms of the objects of Bayesian inference.", "For example, because the term $p(\\theta \\,|\\,a,\\mathbf {A})$ is the conditional posterior pdf2020We say “conditional” here because it is conditioned on nonlinear parameters $\\textrm {\\ding {80}}$ .", "The vector $a$ and variance tensor $\\mathbf {A}$ will depend on the nonlinear parameters $\\textrm {\\ding {80}}$ through the design matrix ${M}$ .", "for the linear parameters $\\theta $ , the vector $a$ is the maximum a posteriori (or MAP) value for the parameter vector $\\theta $ .", "It is found by inverse-variance-weighted combinations of the data and the prior.", "In some projects, posterior pdfs or MAP parameter values are the goal (although we don't think they often should be2121Although—in subjective Bayesian inference—the posterior pdf is the valid statement of your belief, it is not so useful to your colleagues, who start with different beliefs from yours.", "See Note.).", "The variance tensor $\\mathbf {A}$ is the posterior variance in the parameter space.", "It is strictly smaller (in eigenvalues or determinant) than either the prior variance $\\mathbf {\\Lambda }$ or the parameter-space data-noise variance $[{M}\\mathbf {C}^{-1}\\cdot {M}]^{-1}$ .", "The vector $b$ is the prior-optimal (maximum a priori) value for the data $y$ .", "It is the most probable data vector (prior to seeing any data), and also the prior expectation for the data, under the prior pdf.", "The variance tensor $\\mathbf {B}$ is the prior variance expanded out to the data space, and including the noise variance in the data.", "It is strictly larger than both the data noise variance $\\mathbf {C}$ and the data-space prior variance ${M}\\cdot \\mathbf {\\Lambda }\\cdot {M}.$" ], [ "Implementation notes:", "The solution gives an expression for the variance tensor $\\mathbf {B}$ , but note that when you actually evaluate the pdfs you probably need to have either the inverse of $\\mathbf {B}$ , or else an operator that computes the product of the inverse and vectors, as in $\\mathbf {B}^{-1}\\cdot y$ and the same for $b$ .", "To get the inverse of the tensor $\\mathbf {B}$ stably, you might want to use the matrix inversion lemma (REF ) given above.", "This is often useful because you often know or are given the data inverse variance tensor $\\mathbf {C}^{-1}$ for the noise, and the prior variance inverse $\\mathbf {\\Lambda }^{-1}$ , and the lemma manipulates these into the answer without any heavy linear algebra.", "The lemma saves you the most time and precision when the parameter size $K$ is much smaller than the data size $N$ (or vice versa); that is, when ${M}$ is “very non-square”.", "We also give the general advice that one should avoid taking an explicit numerical inverse (unless you know the inverse exactly in closed form, as you do for, say, diagonal tensors).", "In your code, it is typically stabler to use a solve() function instead of the inv() function.", "The reason is that the code operation inv(B) returns the best possible inverse to machine precision (if you are lucky), but what you really want instead is the best possible product of that inverse times a vector.", "So, in general, solve(B,y) will deliver more precise results than the mathematically equivalent dot(inv(B),y).", "The expressions in equation (REF ) do not require that the variance tensors $\\mathbf {C}$ , $\\mathbf {\\Lambda }$ , $\\mathbf {A}$ , $\\mathbf {B}$ be positive definite; they only require that they be non-negative semi-definite.", "That means that they can have zero eigenvalues.", "As can their inverses $\\mathbf {C}^{-1}$ , $\\mathbf {\\Lambda }^{-1}$ , $\\mathbf {A}^{-1}$ , $\\mathbf {B}^{-1}$ .", "If either of these might happen in your problem—like if your prior freezes the parameters to a subspace of the $\\theta $ -space, which would lead to a zero eigenvalue in $\\mathbf {\\Lambda }$ , or if a data point is unmeasured or missing, which would lead to a zero eigenvalue in $\\mathbf {C}^{-1}$ —you might have to think about how you implement the linear algebra operations to be zero-safe.2222A completely zero-safe implementation is somewhat challenging, but one comment to make is that if, say, $\\mathbf {A}$ contains a zero eigenvalue, then there is a direction in the parameter space (the $\\theta $ space) in which the variance vanishes.", "This means that all valid parameter combinations lie on a linear subspace of the full $K$ -dimensional space.", "All parameter combinations that wander off the subspace get strictly zero probability or negative infinities in the log.", "If your inference is valid, it will probably be the case that the vectors at which you want to evaluate always lie in the non-zero subspace.", "It makes sense, then, in this case, to work in a representation in which it is easy to enforce or ensure that.", "This usually involves some kind of coordinate transformation or rotation or projection.", "Doing this correctly is beyond the scope of this Note." ], [ "Simplification: single multiplicative scaling", "One interesting case is when $K=1$ , so the design matrix in fact reduces to a model vector $m$ , multiplied by a scalar $\\theta $ , and $\\mu $ and $\\Lambda $ are now scalars as well: $\\mathcal {N}\\!\\,(y\\,|\\,\\theta \\,m,\\mathbf {C})\\,\\mathcal {N}\\!\\,(\\theta \\,|\\,\\mu ,\\Lambda ) = \\mathcal {N}\\!\\,(\\theta \\,|\\,a,A)\\,\\mathcal {N}\\!\\,(y\\,|\\,b,\\mathbf {B})~~.$ This can arise if one wants to multiplicatively scale a model to the data.", "$a$ would then correspond to the maximum a posteriori value of the multiplicative scaling to fit $y$ with $m$ .", "In this case, the previous equations are simplified and no longer involve many matrix operations.", "It's a nice exercise to simplify the solution above for this scalar case." ], [ "Special case: wide prior", "Another interesting case that often arises in inferences is the use of an improper (infinitely wide) prior on the parameters $\\theta $ .", "Rather than ignoring the prior pdf on the left-hand side of equation (REF ), which is technically incorrect, the correct posterior pdf can be obtained by taking the limit $\\mathbf {\\Lambda }^{-1}\\rightarrow 0$ in the fiducial results derived above.", "It is perhaps worth noting that in the improper-prior case, the posterior can still be fine, but the marginalized likelihood will make no sense (it will technically vanish)." ], [ "Generalization: product of many Gaussians", "A case that arises in some applications is that the likelihood is made of multiple Gaussian terms, each of which is a different linear combination of the linear parameters $\\theta $ .", "That is, there are $J$ data vectors $y_j$ , each of which has size or length $N_j$ , and each of which has an expectation set linearly by the parameters $\\theta $ but through a different design matrix ${M}_j$ .", "Provided that the different data vectors $y_j$ are independently “observed” (that is, they have independent noise draws with noise variance tensors $\\mathbf {C}_j$ ), the total likelihood is just the product of the individual-data-vector likelihoods.", "An example of this case arises in astronomy, for example, when considering radial velocity measurements of a star taken with different instruments that may have systematic offsets between their velocity zero-points.", "In principle we could work around this problem—reduce it to the previously solved problem—by forming a large vector $y$ which is the concatenation of all the individual data vectors $y_j$ , and a large design matrix ${M}$ which is the concatenation of all the individual design matrices ${M}_j$ , and a large total covariance matrix $\\mathbf {C}$ which is a block diagonal matrix containing the noise variance tensors $\\mathbf {C}_j$ on the diagonal blocks.", "We could then apply the result of the single-data-vector problem above.", "However, this can result in significant unnecessary computation, and it is hard to write the answer in a simple form.", "Instead we can take advantage of the separability of the likelihoods, and write the following generalized problem statement: Find $K$ -vector $a$ , $K\\times K$ variance tensor $\\mathbf {A}$ , $J$ vectors $b_j$ (each of which is a different length $N_j$ ), and $J$ variance tensors $\\mathbf {B}_j$ (each of which is a different size $N_j\\times N_j$ ) such that $\\mathcal {N}\\!\\,(\\theta \\,|\\,\\mu ,\\mathbf {\\Lambda })\\,\\prod _{j=1}^J\\mathcal {N}\\!\\,(y_j\\,|\\,{M}_j\\cdot \\theta ,\\mathbf {C}_j)\\,= \\mathcal {N}\\!\\,(\\theta \\,|\\,a,\\mathbf {A})\\,\\prod _{j=1}^J\\mathcal {N}\\!\\,(y_j\\,|\\,b_j,\\mathbf {B}_j) ~~,$ and such that $a$ , $\\mathbf {A}$ , all the $b_j$ , and all the $\\mathbf {B}_j$ don't depend on $\\theta $ at all.", "Note that $\\theta $ is a $K$ -vector, $\\mu $ is a $K$ -vector, $\\mathbf {\\Lambda }$ is a $K\\times K$ non-negative semi-definite variance tensor, each $y_j$ is an $N_j$ -vector, each ${M}_j$ is a $N_j\\times K$ matrix, and each $\\mathbf {C}_j$ is a $N_j\\times N_j$ non-negative semi-definite variance tensor.", "One way to solve this problem is to write all Gaussians in their canonical form, then separate the elements that depend on $\\theta $ and on the individual $y_j$ .", "The result can be written as an iteration over data vectors $y_j$ : $\\mbox{\\small initialize:}~~\\mathbf {A}_0^{-1}= \\mathbf {\\Lambda }^{-1}~~;~~a_0 = \\mu ~~;~~x_0 = \\mathbf {\\Lambda }^{-1}\\cdot \\mu $ $\\mbox{\\small iterate:}~~\\mathbf {B}_j &= \\mathbf {C}_j + {M}_j \\cdot \\mathbf {A}_{j-1} \\cdot {M}_jb_j &= {M}_j \\cdot a_{j-1} \\\\\\mathbf {A}_j^{-1}&= \\mathbf {A}_{j-1}^{-1}+ {M}_j\\mathbf {C}_j^{-1}\\cdot {M}_j \\\\x_j &= x_{j-1} + {M}_j\\mathbf {C}_j^{-1}\\cdot y_j \\\\a_j &= \\mathbf {A}_j \\cdot x_j$ $\\mbox{\\small finish:}~~\\mathbf {A}&= \\mathbf {A}_J \\\\a&= a_J~~.$ The solution is an iteration because you can think of adding each new data set $y_j$ sequentially, with the prior for set $j$ being the posterior from set $j-1$ .", "The way this solution is written is unpleasant, because the specific values you get for the vectors $b_j$ and tensors $\\mathbf {B}_j$ depend on the order in which you insert the data.", "But—and very importantly for the rules of Bayesian inference—the posterior mean $a$ and variance $\\mathbf {A}$ do not depend on the order!2323It is literally part of the fundamental justification of Bayesian inference that the knowledge you eventually have (your final beliefs) does not depend on the order in which you observed the data.", "This is one of the axioms or inputs to the theorems that underlie the consistency of Bayesian reasoning.", "There is an illuminating discussion of all this in Chapter 1 of [7]." ], [ "Worked Examples", "When working with a probabilistic model that meets the strong requirements imposed above (Gaussians everywhere; expectations linear in parameters), the identities described in this Note have practical uses: (1) To simplify the posterior pdf of your model (which makes generating samples or computing integrals far simpler), and (2) to reduce the dimensionality of your model (by enabling closed-form marginalizations over linear parameters).", "Reducing the dimensionality of your parameter-space will in general improve convergence of Markov Chain Monte Carlo2424We have also written a tutorial on MCMC in this series [5].", "(MCMC) sampling methods, or enable alternate sampling methods (for example, rejection sampling) that may be intractable when the parameter dimensionality is large: These two benefits also typically make inference procedures (like sampling) far faster.", "Here, we demonstrate the utility of the identities shown above with two worked exercises." ], [ "Exercise 1: A fully linear model:", "We observe a set of data $(x_i, y_i)$ (indexed by $i$ ) with known, Gaussian uncertainties in $y$ , $\\sigma _y$ , and no uncertainty in $x$ .", "The parametric model we will use for these data is a quadratic polynomial, $f(x \\,;\\, \\alpha , \\beta , \\gamma ) = \\alpha \\,x^2 + \\beta \\,x + \\gamma $ and we assume we have Gaussian prior pdfs on all of the $K=3$ linear parameters $(\\alpha , \\beta , \\gamma )$ , $p(\\alpha ) &= \\mathcal {N}\\!\\,(\\alpha \\,|\\,\\mu _\\alpha , \\sigma _\\alpha )\\\\p(\\beta ) &= \\mathcal {N}\\!\\,(\\beta \\,|\\,\\mu _\\beta , \\sigma _\\beta )\\\\p(\\gamma ) &= \\mathcal {N}\\!\\,(\\gamma \\,|\\,\\mu _\\gamma , \\sigma _\\gamma )~~.$ While this example may seem overly simple or contrived, quadratic models are occasionally useful in astronomy and physics, for example, when centroiding a peak,2525Fitting second-order polynomials has been shown to be great for centroiding peaks in astronomy contexts.", "See, for example, [15], and [14].", "and polynomial models are often used to capture smooth trends in data.", "Table: NO_CAPTION  Figure: NO_CAPTION 2: Top: Data generated with “true” parameters $(\\alpha , \\beta , \\gamma ) = (3.21, 2.44, 14.82)$ .", "Bottom: The solution to Exercise 1.", "The data points (black markers) show the data from the table.", "The MAP parameter values were found using equation (REF ).", "All data files and solution notebooks are available via Zenodo [6].The data table shown in  REF contains $N=4$ data points, $(x_i, y_i, \\sigma _{y_i})$ , generated using this quadratic model.", "Assuming values for the prior means, $\\mu $ , and prior variance tensor, $\\mathbf {\\Lambda }$ , $\\mu = (\\mu _\\alpha , \\mu _\\beta , \\mu _\\gamma ) = (1, 3, 9)\\\\\\mathbf {\\Lambda }&=\\begin{pmatrix}5^2 & 0 & 0\\\\0 & 2^2 & 0\\\\0 & 0 & 8^2\\end{pmatrix}$ compute the MAP parameter values $a (\\alpha _{\\rm MAP}, \\beta _{\\rm MAP},\\gamma _{\\rm MAP})$ .", "Plot the data (with error bars) and over-plot the model evaluated at the MAP parameter values.", "Generate 4096 posterior samples of the linear parameters.", "Over-plot a shaded region showing the 68 percent credible region for the model, estimated using these samples." ], [ "Solution:", "Given the assumptions and prior parameter values above, the design matrix, ${M}$ , is ${M}&= \\begin{pmatrix}0.36 & -0.6 & 1\\\\4.0 & 2.0 & 1\\\\7.29 & 2.7 & 1\\\\12.96 & 3.6 & 1\\end{pmatrix} ~~.$ By plugging in to equation (REF ), we find MAP parameter values for the linear parameters $a(\\alpha _{\\rm MAP}, \\beta _{\\rm MAP}, \\gamma _{\\rm MAP}) =(3.61, 1.98, 14.26) ~~.$  REF shows the data (black points), the model computed with the MAP parameter values (blue line), and the 68-percent credible region (shaded blue region) estimated using posterior samples generated from $\\mathcal {N}\\!\\,(\\theta \\,|\\,a,\\mathbf {A})$ .", "The companion IPython notebook (Exercise1.ipynb) contains the full solution." ], [ "Exercise 2: A model with a nonlinear parameter:", "We observe a set of data $(x_i, y_i)$ (indexed by $i$ ) with known, Gaussian uncertainties in $y$ , $\\sigma _{y_i}$ , and no uncertainty in $x$ .", "The parametric model we will use for these data is a generalized sinusoid with a constant offset, $f(x \\,;\\, \\alpha , \\beta , \\gamma , \\omega ) =\\alpha \\,\\cos (\\omega \\, x) + \\beta \\,\\sin (\\omega \\, x) + \\gamma $ and we again assume we have Gaussian prior pdfs on all of the linear parameters $(\\alpha , \\beta , \\gamma )$ .", "Models like this (a periodic model with both linear and nonlinear parameters) are common in astronomy, especially in the context of asteroseismology, light curve analysis, and radial velocity variations from massive companions (binary star systems or exoplanets).", "For this setup, we can no longer analytically express the posterior pdf because of the nonlinear parameter $\\omega $ , but we can compute the marginal likelihood (marginalizing over the linear parameters) conditioned on the frequency $\\omega $ .", "Table: NO_CAPTION  Figure: NO_CAPTIONFigure: NO_CAPTION 3: Top: Data generated with “true” parameters $(\\alpha , \\beta , \\gamma , \\omega ) = (3.21, 2.44, 13.6, 1.27)$ .", "Middle and Bottom: The solution to Exercise 2.", "All data files and solution notebooks are available via Zenodo [6].The table shown in  REF contains $N=4$ data points, $(x_i, y_i, \\sigma _{y_i})$ , generated with this sinusoid model.", "Assuming values for the prior means, $\\mu $ , and prior variance tensor, $\\mathbf {\\Lambda }$ , $\\mu = (\\mu _\\alpha , \\mu _\\beta , \\mu _\\gamma ) = (0, 0, 0)\\\\\\mathbf {\\Lambda }&=\\begin{pmatrix}5^2 & 0 & 0\\\\0 & 5^2 & 0\\\\0 & 0 & 10^2\\end{pmatrix}$ write a function to compute the vectors and matrices we need for the linear parameters (the design matrix and components $a, \\mathbf {A}, b, \\mathbf {B}$ of the factorization) at a given value of the frequency $\\omega $ .", "Assuming a prior on $\\omega $ that is uniform in $\\ln \\omega $ over the domain $(0.1, 100)$ , $p(\\omega ) \\propto \\frac{1}{\\omega }$ evaluate the log-marginal likelihood $\\ln p(y\\,|\\,\\omega )$ and add to the log-frequency prior $\\ln p(\\omega )$ over a grid of 16,384 frequencies $\\omega $ between (0.1, 100).", "Plot both the marginal likelihood (not log!)", "$p(y\\,|\\,\\omega )$ and the posterior pdf $p(y\\,|\\,\\omega )\\,p(\\omega )$ as a function of this frequency grid.", "Generate 512 posterior samples2626The posterior pdf over $\\omega $ will be extremely multimodal.", "Don't fire up standard MCMC!", "Try using rejection sampling instead: Generate a dense prior sampling in the nonlinear parameter $\\omega $ , evaluate the marginalized likelihood at each sample in $\\omega $ , and use this to reject prior samples.", "in the full set of parameters $(\\alpha ,\\beta ,\\gamma ,\\omega )$ .", "Make a scatter plot showing a 2D projection of these samples in $(\\alpha , \\ln \\omega )$ .", "Plot the data, and over-plot 64 models (equation REF ) computed using a fair subset of these posterior samples.", "The companion IPython notebook (Exercise2.ipynb) contains the full solution." ] ]

[ [ "Unlucky Explorer: A Complete non-Overlapping Map Exploration" ], [ "Abstract Nowadays, the field of Artificial Intelligence in Computer Games (AI in Games) is going to be more alluring since computer games challenge many aspects of AI with a wide range of problems, particularly general problems.", "One of these kinds of problems is Exploration, which states that an unknown environment must be explored by one or several agents.", "In this work, we have first introduced the Maze Dash puzzle as an exploration problem where the agent must find a Hamiltonian Path visiting all the cells.", "Then, we have investigated to find suitable methods by a focus on Monte-Carlo Tree Search (MCTS) and SAT to solve this puzzle quickly and accurately.", "An optimization has been applied to the proposed MCTS algorithm to obtain a promising result.", "Also, since the prefabricated test cases of this puzzle are not large enough to assay the proposed method, we have proposed and employed a technique to generate solvable test cases to evaluate the approaches.", "Eventually, the MCTS-based method has been assessed by the auto-generated test cases and compared with our implemented SAT approach that is considered a good rival.", "Our comparison indicates that the MCTS-based approach is an up-and-coming method that could cope with the test cases with small and medium sizes with faster run-time compared to SAT.", "However, for certain discussed reasons, including the features of the problem, tree search organization, and also the approach of MCTS in the Simulation step, MCTS takes more time to execute in Large size scenarios.", "Consequently, we have found the bottleneck for the MCTS-based method in significant test cases that could be improved in two real-world problems." ], [ "Introduction and Background", "Graph Traversal is an important and famous problem in computer science with many applications in memory and storage systems [1], network flow [2], as well as computer games [3].", "Advances in different aspects such as insuring the security [4], efficient path fining [5], and Graph Exploration in the robotics are also addressed by the researchers [6].", "As the conventional approaches to solving this problem and other variations like Tree Traversal, Depth-first search (DFS), and Breadth-first search (BFS) are known to be effective in general.", "On the other hand, random-based approaches such as Monte-Carlo Tree Search (MCTS) are demonstrated to be efficient in many search-based problems and games as well as [7].", "The fundamental problem of many simple computer games lays on solving specific computer or mathematical puzzles.", "The solution methodology used in many of these games is very relevant to fundamental approaches.", "For instance, Flow-Free is a variant of a known mathematical puzzle named Numberlink, and interestingly, the problem could be addressed as a Multi-Agent Path Finding [8].", "In this context, Icosian Game [9] as an old mathematical game invented by W.R Hamilton could be considered as a modified version of a Graph Traversal problem.", "The objective in Icosian is finding a Hamiltonian cycle along the edges of a dodecahedron, visiting all the vertexes of the graph by ending at the same point as the starting vertex.", "The Hamiltonian Path problem as an NP-Complete problem [10] has its own applications in various fields [11] with many solution methods [12].", "In this article, we investigate the foundation of the Maze Dash game.", "We demonstrate that the constraints involved in solving the game, inevitably minimize the number of Turning Movement in the grid exploration procedure.", "Satisfying this particular condition applied in this game could be interesting in terms of real-world robot exploration since an extra cost is often associated with the Turning Movement in smart explorer vehicles [13].", "Hence, finding an efficient and effective solution to the focused Maze Dash game could lead to faster Grid Traversal approaches where realistic restrictions in the robots are considered.", "Furthermore, We mathematically define the underlying primary problem of the game as a particular case of a Hamiltonian Path problem.", "Then, by studying the problem's specifications, we tackle the problem with different possible approaches, including the MCTS, as one of the promising methods.", "We examine the unique characteristics of the involved tree search in detail and study the exclusive attributes of Hamiltonian Path in this problem.", "Maze Dash is a puzzle game with a single agent and a 2-Dimensional grid map.", "The map might have some obstacles or blocking cells.", "The agent moves in the map and marks the cells after visiting them by changing their color and can not return to the marked cells.", "So, each cell must be visited just once.", "Eventually, the puzzle aims to visit all the cells or to explore the whole of the map.", "The essential rule in this puzzle is that if the agent chooses to go to the one the quad directions, it will continue to move until it reaches an obstacle or wall.", "As shown in Figure REF , the agent starts to move from the initial cell (S) and decides to go down to reach the wall or the border of the grid.", "Then, it keeps moving to explore all of the possible cells, finishing the traversal at the last cell (E).", "Figure: Maze Dash game solving process" ], [ "Monte-Carlo Tree Search", "MCTS is a best-first search algorithm with four main steps which are Selection, Expansion, Simulation, and Backpropagation.", "This algorithm uses Monte-Carlo methods to sample steps and create the search tree indeterminately to solve problems in their particular domain [7].", "Like other tree-based approaches, the algorithm requires considering an initial state as its root to construct the search tree.", "In the context of our problem, the initial state is the state that the agent craves to move from its current cell.", "The first step of the MCTS is the selection that the algorithm chooses the best node, which is a leaf at the moment based on the Tree Policy.", "Then, at the expansion point, all non-terminal children of the selected node, if exist, will be expanded.", "In the next step, simulation, MCTS strides in the search tree aimlessly based on a policy until it reaches a leaf.", "The obtained result will be evaluated and measured how much is this result is analogous to the desired result, and how many of the rules and conditions of the problem are satisfied.", "Finally, in the backpropagation, the results are propagated back through the tree, and all related node values are updated.", "After that, the next rounds will be iterated to find the suitable solution." ], [ "Problem Definition", "The exact definition of the problem as a modification of a Hamiltonian Path in a 2-D grid is described below.", "The the set of $O=\\lbrace o_1,o_2,...,o_m\\rbrace $ is demonstrated as the obstacle set which determines the coordination of the obstacle cells in the grid.", "By considering an $N\\times N$ grid, the function $\\pi $ identifies the movement path in the grid: $ \\begin{split}&\\pi : \\mathcal {N} \\rightarrow \\mathcal {N}\\times \\mathcal {N}\\end{split}$ The input of $\\pi $ function rises incrementally to represents the movement path in the grid.", "The output represents the coordinates with the constrain of moving a single cell at each step to ensure the consistency of the solution path: $ \\begin{split}&\\pi (0)=S\\\\&Direction=\\lbrace (1,0),(0,1),(-1,0),(0,-1)\\rbrace \\\\&\\forall (N^2 - |O|) > i > 0 :\\\\&\\pi (i+1)-\\pi (i) \\in Direction, \\pi (i+1) \\notin O\\\\&\\forall i,j: \\pi (i) \\ne \\pi (j)\\\\\\end{split}$ In Equation REF , $S$ is the initial coordinate of the beginning cell.", "The Direction set $D$ is the set of all possible movements that can be used in this context.", "As for the constraints regarding the minimum turning movement restrictions in the game, the $\\pi $ function falls into either one of the two Straight Movement, Turning Movement conditions as defined in Equation REF , respectively: $ \\begin{split}&\\forall (N^2 - |O|) > i > 1 : one of Three: \\\\&Straight:\\pi (i) + (\\pi (i)- \\pi (i-1)) = \\pi (i+1)\\\\&Turning:\\pi (i) + (\\pi (i)- \\pi (i-1))\\in O \\\\&Turning:\\exists j < i:\\pi (i) + (\\pi (i)- \\pi (i-1)) = \\pi (j) \\\\\\end{split}$ Note that the turning is occurred either by a blocking obstacle or a previously occupied cell by the earlier path.", "Hence, the $i^{th}$ step in a cell must be as the same previous direction, or a turning movement happens.", "Eventually, the solution of the game comprise of the adequate assignment for the $\\pi $ function satisfying all the constraints presented in Equations REF and REF .", "The accurate mathematical definition discussed, as the foundation of the problem, could be fed to a Satisfiability solver (SAT Solver, e.g., Z3[14]) as a simple solution approach by converting the conditions into boolean constraints.", "In this respect, the boolean assignments of $t,s : \\mathcal {N} \\rightarrow \\lbrace 0,1\\rbrace $ , should be defined to determined whatever a cell falls into the Turning Movement or the Straight Movement sets.", "Then, by creating the same constraints in the function $\\pi $ for each cell of the grid, the Satisfiability check could be performed over the described assignments of $\\lbrace t,s,\\pi \\rbrace $ .", "Figure: Procedure of Backtracking in solving the puzzle" ], [ "Backtracking", "As a naive approach to solve the problem, one could apply the Backtracking method to find the correct solution.", "The backtracking process is very similar to a DFS method.", "Considering the conditions of the game, a single branch of the possible solution is pursued until all the cells are traversed or a deadlock occurs.", "In the case of a failure, the movement path backtraces itself to the previous state and changes the branch by choosing another possible path.", "A simple backtrack demonstration is shown in Figure REF .", "The algorithm has to eliminate its current path to correct it since one of the cells is not visited." ], [ "MCTS", "As mentioned in the background section, one of the particular and pronounced features of the game is that the agent must explore the grid with the minimum number of turns.", "More precisely, the agent only could change its direction when it reaches the end of the current path.", "This constraint intensely affects the tree search of the puzzle.", "As shown in Figure REF , each non-terminal node of the tree search has only one or two children.", "Therefore, the Branch Factor of the tree is equal to or smaller than two making the tree significantly long in-depth.", "One of the approaches that could solve the problem is MCTS.", "As explained earlier, MCTS runs its four steps iteratively to construct the tree search and find the solution.", "Figure: Construction of the tree search of the puzzleTo solve the problem more efficiently, we have disregarded the nodes with a certain failure at their final stats to prune the tree.", "More precisely, the Expansion function was amended to block a terminal state with an inadequate value of the desired final state.", "Then, the Selection function blocks a node if all of its children were blocked before, by assigning 0 as the value of the node.", "Therefore, gradually wrong paths could be blocked efficiently.", "Nevertheless, even by employing these modifications to the algorithm, the execution time increases immensely due to the computational overhead caused by huge depth in the tree.", "Another optimization has been employed by applying the Fast Rollout Function, where instead of constructing a new state for each Child Node, the state of the parent is updated each time.", "This optimization reduces the memory consumption of $\\mathcal {O}(N^2)$ to $\\mathcal {O}(1)$ for each node [15].", "These details will be discussed in the result section.", "Note that, another approach to the problem could be a BFS method where all of the branches are searched at the time without traversing deep into the final state.", "Not surprisingly, this method burdens considerable memory usage to store the branch states.", "This memory overuse could be predicted due to the tremendous depth of the tree search.", "Consequently, a BFS approach would not be a suitable solution compared to other possible methods.", "As the test cases of the Maze Dash game are not large enough to assess the methods correctly, we decided to utilize a method to generate large random test cases.", "First, an empty grid with the desired size is assumed.", "Then an agent starts to move through the grid randomly.", "Whenever the agent turns or changes its direction, an obstacle is placed at the next cell of the current cell.", "It means that there was a hypothetical obstacle in the path, so the agent decided to change the direction.", "For avoiding creating a unique path for solving the test case, we defined a variable as the number of obstacles.", "After generating the test cases, all of them have been tested by the Backtracking approach to make sure that they are solvable.", "All of the compiled files were executed on a machine running Ubuntu 16.04 equipped with two Intel XEON E5 2697V3 CPUs clocked at 2.6 GHz and 128 GB of DDR3 RAM.", "Table: evaluation of different algorithms for solving Maze Dash GameAs shown in Table REF , we have implemented and compared possible approaches to solve the defined problem.", "Each test case is executed by each method, 50 times, and the average values are presented.", "It is worth mentioning that the Backtracking algorithm is used to indicate our worst-case scenario, not to be a good rival.", "The MCTS approach performs well in small and medium-size test cases but could not cope with large ones.", "The results of the MCTS method could be discussed.", "First, most of the execution run-time of the algorithm is spent in the simulation step.", "Assume that $ I$ defines the number of iterations of the simulation in each MCTS traversal.", "In each simulation, the algorithm would traverse the tree down to $N^{2}$ depth.", "After selecting each node and adding one depth to the MCTS tree, the algorithm would be repeated.", "Thus, the search would be performed for another $N^{2}$ times.", "Ultimately, simulations process enlarges and will have an immense cost, calculated as follows: $ Simulations Cost = I \\times N^{4}\\\\$ This order is a huge cost for the problem since the agent could not determine or predict a complete solution until it tests all possible movements.", "For instance, the Evaluation Function could return 0.98 as a value of a final state, meaning that only 2 cells are not visited among 100.", "We know that this state is not the accurate answer but the algorithm would recognize it as a promising state in the previous steps.", "So, the algorithm would never prune these kinds of states.", "Furthermore, owing to the non-overlapping feature of this modification of the exploration, after each movement, the agent would be faced by new obstacles.", "Moreover, previously visited cells are considered as dynamic obstacles and walls, which makes the algorithm unable to be optimized by any pre-process methods.", "Furthermore, as explained, the branch factor of the problem is equal or smaller than 2, constructing a tree with a huge depth without many branches.", "This kind of tree produces an arduous circumstance for MCTS in its simulation step." ], [ "Discussion and Conclusion", "In this article, we first introduced the Maze Dash puzzle as a modification of exploration problem or, more precisely, a non-overlapping exploration that could be solved by Hamiltonian Path.", "This means that the primary purpose of this problem is to visit all of the cells in 2-D grid.", "We have investigated promising approaches to find a proper solution.", "Although four methods implemented to compare with each other, the main focus was on MCTS and SAT.", "As expected, our implemented SAT could solve the auto-generated test cases accurately.", "Nevertheless, by reducing the number of obstacles in the grid, the execution time increased exponentially.", "In small and medium-size test cases, MCTS could outperform SAT.", "In the large test cases, as explored, the simulation time increased uncontrollably since the algorithm recognizes the failure early in the simulation.", "As our observation, due to the non-overlapping feature of the problem, the agent considers its previous path as dynamic walls or obstacles.", "Therefore, the algorithm could not be optimized by pre-processing methods, but the SAT performs more beneficial in these test cases.", "Investigating the introduced intricacy, the authors of this article came to realize two more practical and pronounced problems that should be considered.", "The first one is reasonably analogous to the current problem but differs in the non-overlapping constraint.", "It means that the aim of the problem is that an agent must explore all of the environment fast and accurately.", "However, the agent's previous path would not be defined as a new obstacle.", "Thus, the agent prefers not to use the visited cells but is not forced to do this.", "The second problem is the exploration of a grid to find a goal with minimum numbers of turns, by assuming that turning movement has an additional cost, since the agent must reduce its velocity, stop, and then start to move again [16], [13].", "By this constraint, the agent prefers to choose a path with lesser turnings.", "In our future studies, we would concentrate on these two problems, which would be useful in the real world applications, such as 2-D Robotic soccer." ] ]

[ [ "Stability estimates for an inverse Steklov problem in a class of hollow\n spheres" ], [ "Abstract In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric.", "Precisely, we aim at studying the continuous dependence of the warping function dening the warped product with respect to the Steklov spectrum.", "We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary.", "Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem.", "As a last result, we prove a log-type stability estimate for the corresponding Calder{\\'o}n problem." ], [ "Framework", "For $n\\ge 2$ , let us consider a class of $n$ -dimensional manifolds $M=[0,1]\\times \\mathbb {S}^{n-1}$ equipped with a warped product metric $g=f(x)(dx^2+g_{\\mathbb {S}})$ where $g_\\mathbb {S}$ denotes the usual metric on $\\mathbb {S}^{n-1}$ induced by the euclidean metric on $\\mathbb {R}^n$ and $f$ is a smooth and positive function on $[0,1]$ .", "Let $\\displaystyle \\psi $ belong to $ H^{1/2}(\\partial M)$ and $\\omega $ be in $\\mathbb {R}$ .", "$\\quad $ The Dirichlet problem is the following elliptic equation with boundary condition $\\left\\lbrace \\begin{aligned}& -\\Delta _g u=\\omega u \\:\\:{\\rm {in}}\\:\\:M\\\\& u=\\psi \\:\\:{\\rm {on}}\\:\\:\\partial M, \\end{aligned}\\right.$ where, in a local coordinate system $(x_i)_{i=1,...,n}$ , and setting $|g|=\\det (g_{ij})$ and $(g^{ij})=(g_{ij})^{-1}$ , the Laplacian operator $-\\Delta _g$ has the expression $-\\Delta _g=-\\sum _{1\\le i,j\\le n}\\frac{1}{\\sqrt{|g|}}\\partial _i\\big (\\sqrt{|g|}g^{ij}\\partial _j\\big ).$ $\\quad $ If $\\omega $ does not belong to the Dirichlet spectrum of $-\\Delta _g$ , the equation (REF ) has a unique solution in $H^1(M)$ (see [11], [14]), so we can define the so-called Dirichlet-to-Neumann operator (\"DN map\") $\\Lambda _g(\\omega )$ as : $\\begin{array}{ccccc}\\Lambda _g(\\omega ) & : & H^{1/2}(\\partial M) & \\rightarrow & H^{-1/2}(\\partial M) \\\\& & \\displaystyle \\psi & \\mapsto & \\displaystyle \\frac{\\partial u}{\\partial \\nu } \\big |_{\\partial M}\\\\\\end{array}$ where $\\nu $ is the unit outer normal vector on $\\partial M$ .", "The previous definition has to be understood in the weak sense by: $\\forall (\\psi ,\\phi )\\in H^{1/2}(\\partial M)^2\\: :\\: \\langle \\Lambda _g(\\omega )\\psi ,\\phi \\rangle =\\int _{M}\\langle du,dv\\rangle _g,\\ \\mathrm {dVol}_g+\\omega \\int _{M}uv\\,\\mathrm {dVol}_g.$ where $v$ is any element of $H^1(M)$ such that $v|_{\\partial M}=\\phi $ , $\\langle .,.\\rangle $ is the standard $L^2$ duality pairing between $H^{1/2}(\\partial M)$ and its dual, and $\\mathrm {dVol}_g$ is the volume form induced by $g$ on $M$ .", "$\\quad $ The DN map $\\Lambda _g(\\omega )$ is a self-adjoint pseudodifferential operator of order one on $L^2(\\partial M)$ .", "Then, it has a real and discrete spectrum accumulating at infinity.", "We shall denote the Steklov eigenvalues counted with multiplicity by $\\sigma (\\Lambda _g(\\omega ))=\\lbrace 0=\\lambda _0<\\lambda _1\\le \\lambda _2\\le ...\\le \\lambda _k\\rightarrow +\\infty \\rbrace ,$ usually called the Steklov spectrum (see [8], p.2 or [7] for details).", "$\\quad $ The inverse Steklov problem adresses the question whether the knowledge of the Steklov spectrum is enough to recover the metric $g$ .", "Precisely: If $\\:\\displaystyle \\sigma \\big (\\Lambda _g(\\omega )\\big )=\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ , is it true that $ g=\\tilde{g}$ ?", "It is known that the answer is negative because of some gauge invariances in the Steklov problem.", "These gauge invariances are (see [8]): 1) Invariance under pullback of the metric by the diffeomorphisms of $M$ : $\\forall \\psi \\in \\mathrm {Diff}(M),\\quad \\Lambda _{\\psi ^*g}(\\lambda )=\\varphi ^*\\circ \\Lambda _g(\\lambda )\\circ \\varphi ^{*-1}.$ where $\\varphi :=\\psi |_{\\partial M}$ and where $\\varphi ^* : C^\\infty (\\partial M)\\rightarrow C^\\infty (\\partial M)$ is the application defined by $\\varphi ^*h:=h\\circ \\varphi $ .", "2) In dimension $n=2$ and for $\\omega =0$ , there is one additional gauge invariance.", "Indeed, thanks to the conformal invariance of the Laplacian, for every smooth function $c>0$ , we have $\\Delta _{cg}=\\frac{1}{c}\\Delta _g.$ Consequently, the solutions of the Dirichlet problem (REF ) associated to the metrics $g$ and $cg$ are the same when $\\omega =0$ .", "Moreover, if we assume that $c\\equiv 1$ on the boundary, the unit outer normal vectors on $\\partial M$ are also the same for both metrics.", "Therefore, $\\Lambda _{cg}(0)=\\Lambda _{g}(0).$ In our particular model, we have shown in [6] that the only gauge invariance is given by the involution $\\eta :x\\mapsto 1-x$ when $n=2$ and also when $n\\ge 3$ under some technical estimates on the conformal factor $f$ on the boundary.", "Precisely, we proved: Theorem 1.1 Let $M=[0,1]\\times \\mathbb {S}^{n-1}$ be a smooth Riemannian manifold equipped with the metrics $ g=f(x)(dx^2+g_\\mathbb {S}),$ $\\tilde{g}=\\tilde{f}(x)(dx^2+g_\\mathbb {S})$ , and let $\\omega $ be a frequency not belonging to the Dirichlet spectrum of $-\\Delta _g$ and $-\\Delta _{\\tilde{g}}$ on $M$ .", "Then, For $n=2$ and $\\omega \\ne 0$ , $\\big (\\sigma (\\Lambda _g(\\omega )) = \\sigma (\\Lambda _{\\tilde{g}}(\\omega ))\\big ) \\Leftrightarrow \\big (f=\\tilde{f}\\quad {\\rm {or}}\\quad f=\\tilde{f}\\circ \\eta \\big )$ where $\\eta (x)=1-x$ for all $x \\in [0,1]$ .", "For $n\\ge 3$ , and if moreover $f,\\tilde{f}\\in \\mathcal {C}_b := \\bigg \\lbrace f\\in C^\\infty ([0,1]),\\: \\bigg |\\frac{f^{\\prime }(k)}{f(k)}\\bigg |\\le \\frac{1}{n-2},\\:k=0 \\ {\\rm {and}} \\ 1 \\bigg \\rbrace ,$ $\\big (\\sigma (\\Lambda _g(\\omega )) = \\sigma (\\Lambda _{\\tilde{g}}(\\omega ))\\big ) \\Leftrightarrow \\big (f=\\tilde{f}\\quad {\\rm {or}}\\quad f=\\tilde{f}\\circ \\eta \\big )$ Remark 1 In Theorem REF , there is no need to assume that $\\omega \\ne 0$ when $n\\ge 3$ whereas this condition is necessary in dimension 2, due to the gauge invariance by a conformal factor.", "In this paper, we will show two additional results on the Steklov inverse problem, that follow and precise the question of uniqueness.", "Namely, we will prove some local uniqueness and stability results.", "Before stating our results, recall that the boundary $\\partial M$ of $M$ has two connected components.", "If we denote $-\\Delta _{g_\\mathbb {S}}$ the Laplace-Beltrami operator on $(\\mathbb {S}^{n-1},g_\\mathbb {S})$ and $\\sigma (-\\Delta _{g_\\mathbb {S}}):=\\lbrace 0=\\mu _0<\\mu _1\\le \\mu _2\\le ...\\le \\mu _m\\le ...\\rightarrow +\\infty \\rbrace $ the sequence of the eigenvalues of $-\\Delta _{g_\\mathbb {S}}$ , counted with multiplicity, one can show that the spectrum of $\\Lambda _g(\\omega )$ is made of two sets of eigenvalues $\\lbrace \\lambda ^-(\\mu _m)\\rbrace $ and $\\lbrace \\lambda ^+(\\mu _m)\\rbrace $ whose asymptotics are given later in Lemma REF ." ], [ "Closeness of two Steklov spectra", "Let us define first what is the closeness between two spectra $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ and $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ .", "Definition 1.2 Let $(\\varepsilon _m)_m$ a sequence of positive numbers.", "We say that $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ is close to $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ up to the sequence $(\\varepsilon _m)_m$ if, for every $\\lambda ^{\\pm }(\\mu _m)\\in \\sigma \\big (\\Lambda _g(\\omega )\\big )$ : there is $\\tilde{\\lambda }^\\pm $ in $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ such that $|\\lambda ^{\\pm }(\\mu _m)-\\tilde{\\lambda }^\\pm |\\le \\varepsilon _m$ .", "$\\mathrm {Card}\\big \\lbrace \\lambda ^\\pm \\in \\sigma \\big (\\Lambda _{g}(\\omega )\\big ),\\:\\: |\\lambda ^\\pm (\\mu _m)-\\lambda ^\\pm |\\le \\varepsilon _m\\big \\rbrace =\\mathrm {Card}\\big \\lbrace \\tilde{\\lambda }^\\pm \\in \\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big ),\\:\\: |\\lambda ^\\pm (\\mu _m)-\\tilde{\\lambda }^\\pm |\\le \\varepsilon _m\\big \\rbrace $ .", "where $\\mathrm {Card} \\:A$ is the cardinal of the set $A$ .", "We denote it $\\sigma \\big (\\Lambda _{g}(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}}\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ .", "Remark 2 The second point of Definition REF amounts to taking into account the multiplicity of the eigenvalues.", "Definition 1.3 We say that $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ and $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ are close up to $(\\varepsilon _m)_m$ if $\\sigma \\big (\\Lambda _{g}(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}}\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )\\quad $ and $\\quad \\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}}\\sigma \\big (\\Lambda _{g}(\\omega )\\big )$ .", "We denote it $\\sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\asymp }\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ .", "Constant sequence : if $(\\varepsilon _m)$ is a constant sequence such that, for all $m$ , $\\varepsilon _m=\\varepsilon $ , we just denote $\\sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{\\varepsilon }{\\asymp }\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big ).$ Definition 1.4 If $A$ and $\\tilde{A}$ are any finite subset of $\\mathbb {R}$ , we will denote $A\\underset{\\varepsilon }{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}} \\tilde{A}$ if, for every $a\\in A$ There is $\\tilde{a}\\in \\tilde{A}$ such that $|a-\\tilde{a}|\\le \\varepsilon $ , $\\mathrm {Card}\\big \\lbrace \\lambda \\in A,\\:\\: |\\lambda -a|\\le \\varepsilon \\big \\rbrace =\\mathrm {Card}\\big \\lbrace \\tilde{\\lambda }\\in \\tilde{A},\\:\\: |a-\\tilde{\\lambda }|\\le \\varepsilon \\big \\rbrace $ .", "We denote $A\\underset{\\varepsilon }{\\asymp }\\tilde{A}\\:$ if $\\: A\\underset{\\varepsilon }{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}} \\tilde{A}$ and $\\tilde{A}\\underset{\\varepsilon }{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}} A$ .", "$\\quad $ This work is based on ideas developped by Daudé, Kamran and Nicoleau in [5].", "However, due to the specific structure of our model that possesses a disconnected boundary (contrary to the model studied in [5]), some new difficulties arise.", "$\\quad $ Local uniqueness.", "We would like to answer the following question : if the data of the Steklov spectrum is known up to some exponentially decreasing sequence, is it possible to recover the conformal factor $f$ in the neigbourhood of the boundary (or one of its component) up to a natural gauge invariance ?", "The main difficulty that appears here is due to the presence of two sets of eigenvalues, in each spectrum $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ and $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ , instead of one as in [5].", "With the previous definitions of closeness, it is not clear that we can get, for example, this kind of implication : $\\big ($ $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ and $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ close $\\big )$ $\\Rightarrow $ $\\big ($ $\\lambda ^-(\\mu _m)$ and $\\tilde{\\lambda }^-(\\mu _m)$ close for all $m\\in \\mathbb {N}$ $\\big )$ $\\hspace{204.85974pt}$ or $\\big ($ $\\lambda ^-(\\mu _m)$ and $\\tilde{\\lambda }^+(\\mu _m)$ close for all $m\\in \\mathbb {N}$ $\\big )$ .", "In order to overcome this problem, we will need to do some hypotheses on the warping function $f$ to introduce a kind of asymmetry on the metric on each component.", "In that way, the previous implication will be true by replacing $\\mathbb {N}$ with an infinite subset $\\mathcal {L}\\subset \\mathbb {N}$ that satisfies, for $m$ large enough, $\\mathcal {L}\\cap \\lbrace m,m+1\\rbrace \\ne \\emptyset $ .", "In other word, the frequency of integers belonging to $\\mathcal {L}$ will be greater than $1/2$ .", "$\\quad $ Stability.", "As regards the problem of stability, if the Steklov eigenvalues are known up to a positive, fixed and small $\\varepsilon $ , is it possible to find an approximation of the conformal factor $f$ depending on $\\varepsilon $ ?", "Thanks to Theorem REF , we know that there is no uniqueness in the problem of recovering $f$ from $\\sigma \\big (\\Lambda _g(\\omega )\\big )$ .", "This seems to be a serious obstruction to establish any result of stability in a general framework.", "Indeed, the uniqueness problem solved in Theorem REF is quite rigid (as well as the local uniqueness result) and is based on analyticity results that can no longer be used here.", "On the contrary, the condition $\\sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{\\varepsilon }{\\asymp }\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ seems much less restrictive than an equality, and the non-uniqueness makes this new problem quite difficult to tackle.", "From Theorem REF , we see that the only way to get a strict uniqueness result is to assume that $f$ is symmetric with respect to $1/2$ .", "This natural - albeit restrictive - condition will be made on $f$ in Section 4 devoted to the stability result." ], [ "The main results", "Definition 1.5 The class of functions $\\mathcal {D}_b$ is defined by $\\mathcal {D}_b=\\lbrace h\\in C^\\infty ([0,1])\\:\\:\\big |\\:\\:\\:\\exists k\\in \\mathbb {N},\\:\\: h^{(k)}(0)\\ne (-1)^kh^{(k)}(1)\\rbrace $ .", "Definition 1.6 The potential associated to the conformal factor $f$ is the function $q_f$ defined on $[0,1]$ by $\\displaystyle q_f=\\frac{(f^{\\frac{n-2}{4}})^{\\prime \\prime }}{f^{\\frac{n-2}{4}}}-\\omega f$ .", "The potential $q_f$ naturally appears when we solve the problem (REF ) by separating the variables in order to get an infinite system of ODE.", "We have at last to precise the following notation that will appear in the statement of Theorem REF .", "$\\quad $ Notation: Let $x_0$ be in $\\mathbb {R}$ and $g$ be a real function such that $\\lim \\limits _{x\\rightarrow x_0}g(x)=0$ .", "We say that $f(x)\\underset{x_0}{=}\\tilde{O}\\big (g(x)\\big )$ if $\\displaystyle \\forall \\varepsilon >0\\,\\:\\: \\lim \\limits _{x\\rightarrow x_0}\\frac{|f(x)|}{|g(x)|^{1-\\varepsilon }}=0$ .", "Here is our local uniqueness result.", "Theorem 1.7 Let $M=[0,1]\\times \\mathbb {S}^{n-1}$ be a smooth Riemannian manifold equipped with the metrics $ g=f(x)(dx^2+g_\\mathbb {S}),$ $\\tilde{g}=\\tilde{f}(x)(dx^2+g_\\mathbb {S})$ , and let $\\omega $ be a frequency not belonging to the Dirichlet spectrum of $-\\Delta _g$ or $-\\Delta _{\\tilde{g}}$ on $M$ .", "Let $a\\in ]0,1[$ and $\\mathcal {E}$ be the set of all the positive sequences $(\\varepsilon _m)_m$ satisfying $\\varepsilon _m= \\tilde{O}\\big (e^{-2a\\sqrt{\\mu _m}}\\big ),$ $\\quad $ In order to simplify the statements of the results, let us denote the propositions: $(P_1)$ : $f=\\tilde{f}\\:\\:\\mathrm {on}\\:\\: [0,a]$ $(P_2)$ : $f=\\tilde{f}\\circ \\eta \\:\\:\\mathrm {on}\\:\\: [0,a]$ $(P_3)$ : $f=\\tilde{f}\\:\\:\\mathrm {on}\\:\\: [1-a,1]$ $(P_4)$ : $f=\\tilde{f}\\circ \\eta \\:\\:\\mathrm {on}\\:\\: [1-a,1]$ where $\\eta (x)=1-x$ for all $x \\in [0,1]$ .", "$\\quad $ Assume that $f$ and $\\tilde{f}$ belong to $\\displaystyle C^\\infty ([0,1])\\cap \\mathcal {C}_b$ where $\\mathcal {C}_b$ is defined as $\\displaystyle \\mathcal {C}_b=\\bigg \\lbrace \\bullet \\:\\: \\bigg |\\frac{f^{\\prime }(k)}{f(k)}\\bigg |\\le \\frac{1}{n-2},\\:k\\in \\lbrace 0,1\\rbrace ,\\qquad \\bullet \\:\\: q_f\\in \\mathcal {D}_b\\bigg \\rbrace $ .", "Then : $\\quad $ $\\bullet $ For $n=2$ and $\\omega \\ne 0$ or $n\\ge 3$ : $\\bigg (\\exists \\:(\\varepsilon _m)\\in \\mathcal {E},\\:\\:\\sigma (\\Lambda _g(\\omega )) \\underset{(\\varepsilon _m)}{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega ))\\bigg ) \\Rightarrow (P_1)\\:\\:\\mathrm {or}\\:\\: (P_2)\\:\\:\\mathrm {or}\\:\\: (P_3)\\:\\:\\mathrm {or}\\:\\: (P_4),$ $\\quad $ Remark 3 When $n=2$ , the condition $\\displaystyle \\bigg |\\frac{f^{\\prime }(k)}{f(k)}\\bigg |\\le \\frac{1}{n-2}$ is always satisfied.", "Remark 4 The converse is not true if $f(0)\\ne f(1)$ .", "If one of the $(P_i)$ is satisfied, it cannot imply more than the closeness of one of the subsequence $\\big (\\lambda ^-(\\mu _m)\\big )$ or $\\big (\\lambda ^+(\\mu _m)\\big )$ with $\\big (\\tilde{\\lambda }^-(\\mu _m)\\big )$ or $\\big (\\tilde{\\lambda }^+(\\mu _m)\\big )$ .", "Special case : when $f(0)=f(1)$ , we have the following equivalence : $\\begin{aligned}\\bigg (\\exists \\:(\\varepsilon _m)\\in \\mathcal {E},\\:\\:\\sigma (\\Lambda _g(\\omega )) \\underset{(\\varepsilon _m)}{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega ))\\bigg ) \\Leftrightarrow \\bigg ((P_1)\\:\\:\\mathrm {and}\\:\\: (P_3)&\\bigg )\\:\\:\\mathrm {or}\\:\\: \\bigg ((P_2)\\:\\:\\mathrm {and}\\:\\: (P_4)\\bigg ).\\end{aligned}$ $\\quad $ $\\quad $ Let us also give our stability result.", "It requires to assume that, for some $A>0$ , the unknown conformal factor belongs to the set of $A$ -admissible functions that we define now.", "Definition 1.8 Let $A>0$ .", "The set of $A$ -admissible functions is defined as : $\\displaystyle \\mathcal {C}(A)=\\bigg \\lbrace \\bullet \\:f\\in C^2([0,1])\\quad \\:\\:\\quad \\bullet \\:\\forall k\\in [\\![0,2]\\!", "],\\:\\: \\Vert f^{(k)}\\Vert _\\infty +\\bigg \\Vert \\frac{1}{f}\\bigg \\Vert _\\infty \\le A\\bigg \\rbrace $ $\\quad $ Our stability result for the Steklov problem is the following: Theorem 1.9 Let $M=[0,1]\\times \\mathbb {S}^{n-1}$ be a smooth Riemanniann manifold equipped with the metrics $ g=f(x)(dx^2+g_\\mathbb {S})$ , $\\tilde{g}=\\tilde{f}(x)(dx^2+g_\\mathbb {S})$ , with $f$ , $\\tilde{f}$ positive on $[0,1]$ and symmetric with respect to $x=1/2$ .", "$\\quad $ Let $A>0$ be fixed and $\\omega $ be a frequency not belonging to the Dirichlet spectrum of $-\\Delta _g$ and $-\\Delta _{\\tilde{g}}$ on $M$ .", "Then, for $n\\ge 2$ , for a sufficiently small $\\varepsilon >0$ and under the assumption $f,\\tilde{f}\\in \\mathcal {C}(A)$ we have the implication : $\\sigma (\\Lambda _g(\\omega )) \\underset{\\varepsilon }{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega )\\big ) \\Rightarrow \\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _2\\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}$ $\\quad $ where $C_A$ is a constant that only depends on $A$ .", "As a by-product, we get two corollaries : Corollary 1.10 Using the same notations and assumptions as in Theorem REF , for all $0\\le s\\le 2$ , we have $\\sigma (\\Lambda _g(\\omega )) \\underset{\\varepsilon }{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega )\\big ) \\Rightarrow \\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _{H^s(0,1)}\\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )^\\theta }$ where $\\theta = (2-s)/2$ and $C_A$ is a constant that only depends on $A$ .", "In particular, from the Sobolev embedding, one gets $\\sigma (\\Lambda _g(\\omega )) \\underset{\\varepsilon }{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega )\\big ) \\Rightarrow \\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _{\\infty }\\le C_A\\:\\sqrt{\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}}$ Corollary 1.11 Using the same notations and assumptions as in Theorem REF , if moreover $\\omega =0$ and $n\\ge 3$ , one has $\\sigma (\\Lambda _g(\\omega )) \\underset{\\varepsilon }{\\asymp } \\sigma (\\Lambda _{\\tilde{g}}(\\omega )\\big ) \\Rightarrow \\big \\Vert f-\\tilde{f}\\big \\Vert _\\infty \\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}$ where $C_A$ is a constant that only depends on $A$ .", "$\\quad $ The stability in the inverse Calderón problem somehow precedes the inverse Steklov problem, so we say few words about it.", "Let $\\mathcal {B}(H^{1/2}(\\partial M))$ be the set of bounded operators from $H^{1/2}(\\partial M)$ to $H^{1/2}(\\partial M)$ equipped with the norm $\\Vert F\\Vert _*=\\sup _{\\psi \\in H^{1/2}(\\partial M)\\backslash \\lbrace 0\\rbrace }\\frac{\\Vert F\\psi \\Vert _{H^{1/2}}}{\\Vert \\psi \\Vert _{H^{1/2}}}.$ In Lemma REF (Section 5) we show the equivalence $\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\in \\mathcal {B}(H^{1/2}(\\partial M))\\Leftrightarrow \\left\\lbrace \\begin{aligned}&f(0)=\\tilde{f}(0)\\\\&f(1)=\\tilde{f}(1)\\end{aligned}\\right.$ and prove the following stability result for the Calderón problem.", "We draw the reader's attention to the fact that the symmetry hypothesis no longer occurs here since the strict uniqueness is true (see [4]).", "However, it is replaced by a technical assumption on the mean of the difference of the potentials.", "Theorem 1.12 Let $M=[0,1]\\times \\mathbb {S}^{n-1}$ be a smooth Riemanniann manifold equipped with the metrics $ g=f(x)(dx^2+g_\\mathbb {S})$ , $\\tilde{g}=\\tilde{f}(x)(dx^2+g_\\mathbb {S})$ , with $f$ and $\\tilde{f}$ positive on $[0,1]$ .", "$\\quad $ Let $A>0$ be fixed and $\\omega $ be a frequency not belonging to the Dirichlet spectrum of $-\\Delta _g$ and $-\\Delta _{\\tilde{g}}$ on $M$ .", "Let $n\\ge 2$ , $\\varepsilon >0$ and assume that $f(0)=\\tilde{f}(0)\\:$ and $\\:f(1)=\\tilde{f}(1),$ $f,\\tilde{f}\\in \\mathcal {C}(A)$ , $\\displaystyle \\bigg |\\int _{0}^{1}\\big (q_f-\\tilde{q}_f\\big )\\bigg |+\\Vert \\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\Vert _*\\le \\varepsilon $ .", "Then : $\\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _2\\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )},$ $\\quad $ where $C_A$ is a constant that only depends on $A$ .", "$\\quad $ Corollary 1.13 Using the same notations and assumptions as in Theorem REF , for all $0\\le s\\le 2$ , we obtain also $\\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _{H^s(0,1)}\\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )^\\theta }$ where $\\theta = (2-s)/2$ and $C_A$ is a constant that only depends on $A$ .", "In particular, from the Sobolev embedding, one gets $\\big \\Vert q_f-\\tilde{q}_f\\big \\Vert _{\\infty }\\le C_A\\:\\sqrt{\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}}$ Corollary 1.14 Using the same notations and assumptions as in Theorem REF , if moreover $\\omega =0$ and $n\\ge 3$ , one has the stronger conclusion: $\\big \\Vert f-\\tilde{f}\\big \\Vert _\\infty \\le C_A\\:\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}$ where $C_A$ is a constant that only depends on $A$ .", "$\\quad $" ], [ "Asymptotics of the Steklov spectrum", "The proof of both theorems is based on the separation of variables that leads to reformulating the Dirichlet problem in terms of boundary value problems for ordinary differential equations.", "All the details can be found in [6], [4] but we outline the main points for the sake of completeness." ], [ "From PDE to ODE using separation of variables", "The equation $\\left\\lbrace \\begin{aligned}& -\\Delta _g u=\\omega u \\:\\:{\\rm {in}}\\:\\:M\\\\& u=\\psi \\:\\:{\\rm {on}}\\:\\:\\partial M \\end{aligned}\\right.$ can be reduced to a countable system of Sturm Liouville boundary value problems on $[0,1]$ .", "Indeed, the boundary $\\partial M$ of the manifold $M$ has two distinct connected components $\\Gamma _0=\\lbrace 0\\rbrace \\times \\mathbb {S}^{n-1}$ and $\\Gamma _1=\\lbrace 1\\rbrace \\times \\mathbb {S}^{n-1}$ , so we can decompose $H^{1}(\\partial M)$ as the direct sum : $H^{1/2}(\\partial M)=H^{1/2}(\\Gamma _0) \\bigoplus H^{1/2}(\\Gamma _1)$ .", "Each element $\\psi $ of $H^{1/2}(\\partial M)$ can be written as $\\displaystyle \\psi =\\begin{pmatrix}\\psi ^0\\\\\\psi ^1\\end{pmatrix},\\quad \\quad $ $\\psi ^0\\in H^{1/2}(\\Gamma _0)\\:$ and $\\:\\psi ^1\\in H^{1/2}(\\Gamma _1)$ .", "Using separation of variables, one can write the solution of (REF ) as $\\begin{aligned}&u(x,y)=\\sum _{m=0}^{+\\infty }u_m(x)Y_m(y),\\\\\\end{aligned}$ and $\\psi ^0$ and $\\psi ^1$ as $\\psi ^0=\\sum _{m\\in \\mathbb {N}}\\psi _m^0 Y_m,\\quad \\quad \\psi ^1=\\sum _{m\\in \\mathbb {N}}\\psi _m^1 Y_m,$ where $(Y_m)$ represents an orthonormal basis of eigenfunctions of $-\\Delta _\\mathbb {S}$ associated to the sequence of its eigenvalues counted with multiplicity $\\sigma \\big (\\Delta _{\\mathbb {S}}\\big )=\\lbrace 0=\\mu _0\\le \\mu _1...\\le \\mu _m \\rightarrow +\\infty \\rbrace .$ By setting $\\displaystyle v(x,y)=f^{\\frac{n-2}{4}}u(x,y)=\\sum _{m=0}^{+\\infty }v_m(x)Y_m(y)$ , $\\quad x\\in [0,1]$ , $\\:\\:y\\in \\mathbb {S}^{n-1}$ , one can show the equivalence (cf [6]) : $u\\:\\:\\mathrm {solves}\\:\\:(\\ref {Schr_2})\\Leftrightarrow \\forall m\\in \\mathbb {N},\\:\\:\\left\\lbrace \\begin{aligned}& -v^{^{\\prime \\prime }}_m(x)+q_f(x)v_m(x)=-\\mu _mv_m(x),\\quad \\forall x\\in ]0,1[\\\\& v_m(0)=f^{\\frac{n-2}{4}}(0)\\psi _m^0,\\:\\:v_m(1)=f^{\\frac{n-2}{4}}(1)\\psi _m^1, \\end{aligned}\\right.$ with $\\displaystyle \\:\\:q_f=\\frac{(f^{\\frac{n-2}{4}})^{\\prime \\prime }}{f^{\\frac{n-2}{4}}}-\\omega f$ (the dependence in $f$ will be omitted in the following and we will just write $q$ instead of $q_f$ ).", "$\\quad $ We thus are brought back to a countable system of 1D Schrödinger equations whose potential does not depend on $m\\in \\mathbb {N}$ .", "Thanks to the Weyl-Titchmarsh theory, we are able to give a nice representation of the DN map that involves the Weyl-Titchmarsh functions of (REF ) evaluated at the sequence $(\\mu _m)$ ." ], [ "Diagonalization of the DN map", "From the equation on $[0,1]$ $-u^{\\prime \\prime }+qu=-zu,\\quad z\\in \\mathbb {C}.$ one can define two fundamental systems of solutions of (REF ), $\\lbrace c_0,s_0\\rbrace $ and $\\lbrace c_1,s_1\\rbrace $ , whose initial Cauchy conditions satisfy $\\left\\lbrace \\begin{aligned}& c_0(0,z)=1,\\:c_0^{\\prime }(0,z)=0\\\\& c_1(1,z)=1,\\:c_1^{\\prime }(1,z)=0 \\end{aligned}\\right.\\quad {\\rm {and}}\\quad \\left\\lbrace \\begin{aligned}& s_0(0,z)=0,\\:s_0^{\\prime }(0,z)=1\\\\& s_1(1,z)=0,\\:s_1^{\\prime }(1,z)=1.", "\\end{aligned}\\right.$ We shall add the subscript $\\quad \\tilde{}\\quad $ to all the quantities referring to $\\tilde{q}$ .", "The characteristic function $\\Delta (z)$ associated to the equation (REF ) is defined by the Wronskian $\\Delta (z)=W(s_0,s_1):=s_0s_1^{\\prime }-s_0^{\\prime }s_1.$ Furthermore, there are two (uniqueLy defined) Weyl-solutions $\\psi $ and $\\phi $ of (REF ) having the form : $\\psi (x)=c_0(x)+M(z)s_0(x),\\quad \\phi (x)=c_1(x)-N(z)s_1(x)$ with Dirichlet boundary conditions at $x=1$ and $x=0$ respectively.", "The meromorphic functions $M$ and $N$ are called the Weyl-Titchmarsh functions.", "Denoting $D(z):=W(c_0,s_1),\\quad \\quad E(z):=W(c_1,s_0)$ an easy calculation leads to $M(z)=-\\frac{D(z)}{\\Delta (z)},\\quad \\quad N(z)=-\\frac{E(z)}{\\Delta (z)}.$ Remark 5 The function $N$ has the same role as $M$ for the potential $q(1-x)$ , i.e : $N(z,q)=M(z,q\\circ \\eta )$ where, for all $x\\in [0,1]$ , $\\eta (x)=1-x$ .", "Those meromorphic functions naturally appear in the expression of the DN map $\\Lambda _g(\\omega )$ in a specific basis of $H^{1/2}(\\Gamma _0)\\oplus H^{1/2}(\\Gamma _1)$ .", "More precisely, in the basis ${B}=\\big (\\lbrace e_m^1,e_m^2\\rbrace \\big )_{m\\ge 0}$ where $e_m^1$ and $e_m^2$ are defined as : $e_m^1=(Y_m,0)\\quad \\quad e_m^2=(0,Y_m)$ one can prove the that the operator $\\Lambda _g(\\omega )$ can be bloc-diagonalized : $[\\Lambda _g]_{{B}}=\\begin{pmatrix}&\\Lambda _g^1(\\omega )&&0&&0&\\cdots &\\\\&0&&\\Lambda _g^2(\\omega )&&0&\\cdots &\\\\&0&&0&&\\Lambda _g^3(\\omega )&\\cdots &\\\\&\\vdots &&\\vdots &&\\vdots &\\ddots \\end{pmatrix},$ with, for every $m\\in \\mathbb {N}$ and setting $h=f^{n-2}$ (cf [4]): $\\Lambda _g^m(\\omega )=\\begin{pmatrix}-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{1}{4\\sqrt{f(0)}}\\frac{h^{\\prime }(0)}{h(0)}&-\\frac{1}{\\sqrt{f(0)}}\\frac{h^{1/4}(1)}{h^{1/4}(0)}\\frac{1}{\\Delta (\\mu _m)}\\\\-\\frac{1}{\\sqrt{f(1)}}\\frac{h^{1/4}(0)}{h^{1/4}(1)}\\frac{1}{\\Delta (\\mu _m)}&-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-\\frac{1}{4\\sqrt{f(1)}}\\frac{h^{\\prime }(1)}{h(1)}\\end{pmatrix}.$" ], [ "Asymptotics of the eigenvalues", "It is then possible, with this representation of $\\Lambda _g(\\omega )$ , to get the following precise asymptotics of the eigenvalues $\\lambda ^\\pm (\\mu _m)$ of $\\Lambda _g(\\omega )$ : Lemma 2.1 When $q$ belongs to $\\mathcal {D}_b$ , $\\Lambda _g^m(\\omega )$ has two eigenvalues $\\lambda ^-(\\mu _m)$ and $\\lambda ^+(\\mu _m)$ whose asymptotics are given by : $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\mu _m)=-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-\\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}+\\tilde{O}\\bigg (e^{-2\\sqrt{\\mu _m}}\\bigg ) \\\\&\\lambda ^+(\\mu _m)=-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{(\\ln h)^{\\prime }(0)}{4\\sqrt{f(0)}}+\\tilde{O}\\bigg (e^{-2\\sqrt{\\mu _m}}\\bigg ).\\end{aligned}\\right.$ The characteristic polynomial $\\chi _m(X)$ of $\\Lambda _g^m(\\omega )$ is $\\chi _m(X)=X^2-{\\rm {Tr}}(\\Lambda _g^m(\\omega ))X+\\det (\\Lambda _g^m(\\omega )).$ To simplify the notations, we set $\\displaystyle C_0=\\frac{\\ln (h)^{\\prime }(0)}{4\\sqrt{f(0)}},\\quad $ $\\quad \\displaystyle C_1=\\frac{\\ln (h)^{\\prime }(1)}{4\\sqrt{f(1)}}$ .", "Thanks to the matrix representation of $\\Lambda _g(\\omega )$ , we see that ${\\rm {Tr}}(\\Lambda _g^m(\\omega ))$ and $\\det (\\Lambda _g^m(\\omega ))$ satisfy the equalities: $\\left\\lbrace \\begin{aligned}&{\\rm {Tr}}(\\Lambda _g^m(\\omega ))=-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+C_0-C_1.", "\\\\&\\det (\\Lambda _g^m(\\omega ))=\\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0\\bigg )\\bigg (-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-C_1\\bigg )+O(\\mu _m e^{-2\\sqrt{\\mu _m}}),\\qquad m\\rightarrow +\\infty \\end{aligned}\\right.$ The asymptotics of the discriminant $\\delta _m$ of $\\chi _m(X)$ is thus : $\\begin{aligned}\\delta _m &= \\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-C_1\\bigg )^2-4\\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0\\bigg )\\bigg (-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-C_1\\bigg )\\\\&\\hspace{312.9803pt}+O(\\mu _m e^{-2\\sqrt{\\mu _m}}).\\\\&=\\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+C_1\\bigg )^2+O(\\mu _m e^{-2\\sqrt{\\mu _m}}).\\end{aligned}$ Now, let us recall the result obtained by Simon in [13] : Theorem 2.2 $M(z^2)$ has the following asymptotic expansion : $\\forall B\\in \\mathbb {N},\\: \\:-M(z^2)\\underset{z \\rightarrow \\infty }{=}z+\\sum _{j=0}^{B}\\frac{\\beta _j(0)}{z^{j+1}}+o\\bigg (\\frac{1}{z^{B+1}}\\bigg )$ where, for every $x\\in [0,1]$ , $\\beta _j(x)$ is defined by : $\\left\\lbrace \\begin{aligned}&\\beta _0(x)=\\frac{1}{2}q(x)\\\\&\\beta _{j+1}(x)=\\frac{1}{2}\\beta ^{\\prime }_j(x)+\\frac{1}{2}\\sum _{l=0}^{j}\\beta _l(x)\\beta _{j-l}(x).\\end{aligned}\\right.$ From Remark REF , by symmetry, one has immediately: Corollary 2.3 $N(z^2)$ has the following asymptotic expansion : $\\forall B\\in \\mathbb {N},\\: \\:-N(z^2)\\underset{_{\\begin{array}{c}z \\rightarrow \\infty \\end{array}}}{=}z+\\sum _{j=0}^{B}\\frac{\\gamma _j(0)}{z^{j+1}}+o\\bigg (\\frac{1}{z^{B+1}}\\bigg )$ where, for all $x\\in [0,1]$ , $\\gamma _j(x)$ is defined by : $\\left\\lbrace \\begin{aligned}&\\gamma _0(x)=\\frac{1}{2}q(1-x)\\\\&\\gamma _{j+1}(x)=\\frac{1}{2}\\gamma ^{\\prime }_j(x)+\\frac{1}{2}\\sum _{l=0}^{j}\\gamma _l(x)\\gamma _{j-l}(x).\\end{aligned}\\right.$ If $f(0)\\ne f(1)$ , we deduce from Theorem REF and Corollary REF : $-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}=\\underbrace{\\bigg (\\frac{1}{\\sqrt{f(0)}}-\\frac{1}{\\sqrt{f(1)}}\\bigg )}_{\\ne 0}\\sqrt{\\mu _m}+O\\bigg (\\frac{1}{\\sqrt{\\mu _m}}\\bigg ).$ If $f(0)= f(1)$ , we will need the elementary general following lemma: Lemma 2.4 We have the equivalence : $q^{(k)}(0)=(-1)^kq^{(k)}(1),\\:\\:\\forall k\\in \\mathbb {N}\\quad $ $\\Leftrightarrow $ $\\quad \\beta _k(0)=\\gamma _k(0),\\:\\:\\forall k\\in \\mathbb {N}$ .", "$\\quad $ $\\quad $ Let us prove by induction that, for every $j\\in \\mathbb {N}$ , there exists $P_j\\in \\mathbb {R}[X_1,...,X_j]$ such that $\\left\\lbrace \\begin{aligned}&\\displaystyle \\beta _j(x)=\\frac{1}{2^{j+1}} q^{(j)}(x)+P_j(q,q^{\\prime },...,q^{(j-1)})(x)\\\\&\\displaystyle \\gamma _j(x)=\\frac{1}{2^{j+1}} (q\\circ \\eta ) ^{(j)}(x)+P_j(q\\circ \\eta ,(q\\circ \\eta )^{\\prime },...,(q\\circ \\eta )^{(j-1)})(x)\\end{aligned}\\right.$ where $\\eta (x)=1-x$ .", "$\\quad $ $\\displaystyle \\beta _0(x)=\\frac{1}{2}q(x)$ and $\\displaystyle \\gamma _0(x)=\\frac{1}{2}q(1-x)$ , so the result holds with $P_0(X)=0$ .", "Let $j\\in \\mathbb {N}$ and assume that $\\left\\lbrace \\begin{aligned}&\\displaystyle \\beta _k(x)=\\frac{1}{2^{k+1}} q^{(k)}(x)+P_k(q,q^{\\prime },...,q^{(k-1)})(x)\\\\&\\displaystyle \\gamma _k(x)=\\frac{1}{2^{k+1}} (q\\circ \\eta ) ^{(k)}(x)+P_k(q\\circ \\eta ,(q\\circ \\eta )^{\\prime },...,(q\\circ \\eta )^{(k-1)})(x),\\end{aligned}\\right.$ for every $0\\le k\\le j$ .", "Then : $\\begin{aligned}\\beta _{j+1}(x)&=\\frac{1}{2}\\beta ^{\\prime }_j(x)+\\frac{1}{2}\\sum _{l=0}^{j}\\beta _l(x)\\beta _{j-l}(x)\\\\&= \\frac{1}{2^{j+2}}q^{(j+1)}(x)+P_{j+1}(q,q^{\\prime },...,q^{(j)})(x),\\end{aligned}$ where we have set $\\displaystyle P_{j+1}(q,q^{\\prime },...,q^{(j)})= \\frac{1}{2}\\big [P_{j}(q,q^{\\prime },...,q^{(j-1)})\\big ]^{\\prime }+\\frac{1}{2}\\sum _{l=0}^{j}\\beta _l(x)\\beta _{j-l}(x)$ .", "In the same way, one also has $\\begin{aligned}\\gamma _{j+1}(x)&=\\frac{1}{2}\\gamma ^{\\prime }_j(x)+\\frac{1}{2}\\sum _{l=0}^{j}\\gamma _l(x)\\gamma _{j-l}(x)\\\\&= \\frac{1}{2^{j+2}}(q\\circ \\eta )^{(j+1)}(x)+P_{j+1}(q\\circ \\eta ,(q\\circ \\eta )^{\\prime },...,(q\\circ \\eta )^{(j)})(x).\\end{aligned}$ Hence, we get the result by induction.", "We are now able to prove the equivalence.", "$\\quad $ $(\\Rightarrow )$ If $q^{(j)}(0)=(-1)^jq^{(j)}(1)$ for every $j\\in \\mathbb {N}$ then one has, for every $k\\in \\mathbb {N}$ and every $P\\in \\mathbb {R}[X_1,...,X_k]$ : $P(q,q^{\\prime },...,q^{(k-1)})(0)=P(q\\circ \\eta ,(q\\circ \\eta )^{\\prime },...,(q\\circ \\eta )^{(k-1)})(0),$ so, thanks to (REF ): $\\beta _j(0)=\\gamma _j(0)$ for every $j\\in \\mathbb {N}$ .", "$(\\Leftarrow )$ Conversely, assume that there is $j\\in \\mathbb {N}$ such that $q^{(j)}(0)\\ne (-1)^jq^{(j)}(1)$ and set $k=\\min \\lbrace j\\in \\mathbb {N},\\: q^{(j)}(0)\\ne (-1)^jq^{(j)}(1) \\rbrace $ .", "As previously, for every $P\\in \\mathbb {R}[X_1,...,X_{k}]$ : $P(q,q^{\\prime },...,q^{(k-1)})(0)=P(q\\circ \\eta ,(q\\circ \\eta )^{\\prime },...,(q\\circ \\eta )^{(k-1)})(0)$ .", "Hence : $\\begin{aligned}\\beta _{k}(0)\\ne \\gamma _{k}(0) &\\Leftrightarrow \\frac{1}{2^{k+1}} q^{(k)}(0)+P_k(q,...,q^{(k-1)})(0) \\ne \\frac{1}{2^{k+1}} (q\\circ \\eta )^{(k)}(0)\\\\&\\hspace{199.16928pt}+P_k\\big ((q\\circ \\eta ),...,(q\\circ \\eta )^{(k-1)}\\big )(0)\\\\&\\Leftrightarrow \\frac{1}{2^{k+1}} q^{(k)}(0)\\ne \\frac{1}{2^{k+1}} (q\\circ \\eta )^{(k)}(0)\\\\&\\Leftrightarrow q^{(k)}(0)\\ne (-1)^kq^{(k)}(1),\\end{aligned}$ and that is true by definition of $k$ .", "As we have assumed that $q$ belongs to $\\mathcal {D}_b$ , by setting $k=\\min \\lbrace j\\in \\mathbb {N},\\: q^{(j)}(0)\\ne (-1)^jq^{(j)}(1) \\rbrace $ , we get, thanks to (REF ), (REF ) and Lemma REF : $-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}=\\underbrace{\\bigg (\\frac{\\beta _k(0)-\\gamma _k(0)}{\\sqrt{f(0)}}\\bigg )}_{\\ne 0}\\frac{1}{(\\sqrt{\\mu _m})^{k+1}}+O\\bigg (\\frac{1}{(\\sqrt{\\mu _m})^{k+2}}\\bigg ).$ In both cases, there is $A\\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ and $k\\in \\mathbb {Z}$ such that $-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}=A(\\sqrt{\\mu _m})^k + o\\big ((\\sqrt{\\mu _m})^k\\big )$ $\\quad $ Thus, recalling that $\\begin{aligned}\\delta &=\\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+C_1\\bigg )^2+O(\\mu _m e^{-2\\sqrt{\\mu _m}}),\\end{aligned}$ we obtain, as $A$ is not 0: $\\begin{aligned}\\sqrt{\\delta }&=\\bigg [\\bigg (-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0+\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+C_1\\bigg )^2+O(\\mu _m e^{-2\\sqrt{\\mu _m}})\\bigg ]^{\\frac{1}{2}}\\\\&=\\bigg [\\bigg (A(\\sqrt{\\mu _m})^k + C_0+C_1+ o\\big ((\\sqrt{\\mu _m})^k\\big )\\bigg )^2+O(\\mu _m e^{-2\\sqrt{\\mu _m}})\\bigg ]^{\\frac{1}{2}}\\\\&= \\bigg |A(\\sqrt{\\mu _m})^k + C_0+C_1+ o\\big ((\\sqrt{\\mu _m})^k\\big )\\bigg |\\bigg [1+O\\bigg (\\big (\\sqrt{\\mu _m}\\big )^{-2k+2}e^{-2\\sqrt{\\mu _m}}\\bigg )\\bigg ]^{\\frac{1}{2}} \\\\&=\\bigg |\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0+C_1\\bigg |\\bigg [1+O\\bigg (\\big (\\sqrt{\\mu _m}\\big )^{-2k+2}e^{-2\\sqrt{\\mu _m}}\\bigg )\\bigg ]\\\\&=\\bigg |\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0+C_1\\bigg |+\\tilde{O}\\big (e^{-2\\sqrt{\\mu _m}}\\big ).\\end{aligned}$ Therefore, the two eigenvalues $\\lambda ^\\pm (\\mu _m)$ of $\\Lambda _g^m(\\omega )$ satisfy the asymptotics equalities $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\mu _m)=-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-\\frac{\\ln (h)^{\\prime }(1)}{4\\sqrt{f(1)}}+\\tilde{O}\\big (e^{-2\\sqrt{\\mu _m}}\\big ) \\\\&\\lambda ^+(\\mu _m)=-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+\\frac{\\ln (h)^{\\prime }(0)}{4\\sqrt{f(0)}}+\\tilde{O}\\big (e^{-2\\sqrt{\\mu _m}}\\big ).\\end{aligned}\\right.$ $\\quad $ In fact, the eigenvalues $\\mu _m$ being counted with multiplicity, the asymptotics of Lemma REF won't be sufficiently precise for our purpose.", "Indeed, by Theorem REF and its Corollary, the Weyl-Titchmarsh functions always satisfy $\\left\\lbrace \\begin{aligned}&-N(z^2)=z+O\\bigg (\\frac{1}{z}\\bigg )\\\\&-M(z^2)=z+O\\bigg (\\frac{1}{z}\\bigg ).\\end{aligned}\\right.$ So, using the Weyl law, one can prove immediately that $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\mu _m)=\\frac{\\sqrt{\\mu _m}}{\\sqrt{f(1)}} -\\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}+ O\\bigg (\\frac{1}{\\sqrt{\\mu _m}}\\bigg ) = C_1m^{\\frac{1}{n-1}}+O(1)\\\\&\\lambda ^+(\\mu _m)=\\frac{\\sqrt{\\mu _m}}{\\sqrt{f(0)}} +\\frac{(\\ln h)^{\\prime }(0)}{4\\sqrt{f(0)}}+O\\bigg (\\frac{1}{\\sqrt{\\mu _m}}\\bigg ) = C_0m^{\\frac{1}{n-1}}+O(1)\\end{aligned}\\right.$ with $C_0,C_1>0$ .", "In order to have a much more precise asymptotic expansion in $m$ , let us introduce the set $\\Sigma \\big (\\Lambda _g(\\omega )\\big )=\\lbrace \\lambda ^\\pm (\\kappa _m),\\: m\\in \\mathbb {N}\\rbrace $ where the $\\kappa _m$ are the eigenvalues of $-\\Delta _{\\mathbb {S}}$ counted without multiplicity.", "We have an explicit formula for $\\kappa _m$ (cf [12]) given by $\\kappa _m=m(m+n-2)$ .", "From now on, we will always use the asymptotics of Lemma REF by replacing $\\mu _m$ by $\\kappa _m$ .", "Of course, one can also define the closeness between $\\Sigma \\big (\\Lambda _g(\\omega )\\big )$ and $\\Sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ up to a sequence $(\\varepsilon _m)$ by replacing $\\mu _m$ by $\\kappa _m$ in Definitions REF and REF ." ], [ "A local uniqueness result", "Now, let us give the proof of Theorem REF .", "Let $(\\varepsilon _m)$ be a sequence such that $\\displaystyle \\varepsilon _m=\\tilde{O}(e^{-2a\\sqrt{\\mu _m}})\\:$ and $\\:\\displaystyle \\sigma (\\Lambda _{g}(\\omega ))\\underset{(\\varepsilon _m)}{\\asymp }\\sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ .", "Then, there is a subsequence of $(\\varepsilon _m)$ , that we will still denote $(\\varepsilon _m)$ which satisfies the estimate $\\varepsilon _m=\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})$ and the relation $\\Sigma (\\Lambda _{g}(\\omega ))\\underset{(\\varepsilon _m)}{\\asymp }\\Sigma (\\Lambda _{\\tilde{g}}(\\omega )).$ Lemma 3.1 Under the hypothesis $\\Sigma (\\Lambda _{g}(\\omega ))\\underset{(\\varepsilon _m)}{\\asymp }\\Sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ , we have the alternative : $\\left\\lbrace \\begin{aligned}&f(0)=\\tilde{f}(0) \\\\&f(1)=\\tilde{f}(1)\\end{aligned}\\right.\\:\\:$ or $\\:\\:\\left\\lbrace \\begin{aligned}&f(0)=\\tilde{f}(1) \\\\&f(1)=\\tilde{f}(0).\\end{aligned}\\right.$ $\\quad $ $\\quad $ $\\quad $ We first show the equality $\\sqrt{f(0)}+\\sqrt{f(1)}=\\sqrt{\\tilde{f}(0)}+\\sqrt{\\tilde{f}(1)}$ As $\\displaystyle \\sqrt{\\kappa _m}=m+\\frac{n-2}{2}+O\\bigg (\\frac{1}{m}\\bigg )$ , we get from Lemma REF the following asymptotics $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=\\frac{m}{\\sqrt{f(1)}} +\\frac{n-2}{2\\sqrt{f(1)}} -\\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}+ O\\bigg (\\frac{1}{m}\\bigg ) \\\\&\\lambda ^+(\\kappa _m)=\\frac{m}{\\sqrt{f(0)}}+\\frac{n-2}{2\\sqrt{f(0)}} +\\frac{(\\ln h)^{\\prime }(0)}{4\\sqrt{f(0)}}+O\\bigg (\\frac{1}{m}\\bigg )\\end{aligned}\\right.$ Let $L>0$ .", "The sequences $(\\lambda ^{\\pm }(\\kappa _m))$ are asymptotically in arithmetic progression.", "Combined with the relation $\\Sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\asymp }\\Sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big ),$ this leads to the equality (when $L\\rightarrow +\\infty $ ) $\\begin{aligned}\\mathrm {Card}&\\big \\lbrace m\\in \\mathbb {N},\\: \\lambda ^-(\\kappa _m)\\le L\\big \\rbrace \\quad +\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: \\lambda ^+(\\kappa _m)\\le L\\big \\rbrace \\\\&\\quad =\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: \\tilde{\\lambda }^-(\\kappa _m)\\le L\\big \\rbrace \\quad +\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: \\tilde{\\lambda }^+(\\kappa _m)\\le L\\big \\rbrace + O(1).\\end{aligned}$ $\\quad $ Thanks to the asymptotics (REF ), we deduce that : $\\begin{aligned}\\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: m&\\le \\sqrt{f(1)}L\\big \\rbrace \\quad +\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: m\\le \\sqrt{f(0)}L\\big \\rbrace \\\\&\\quad =\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: m\\le \\sqrt{\\tilde{f}(1)}L\\big \\rbrace \\quad +\\quad \\mathrm {Card}\\big \\lbrace m\\in \\mathbb {N},\\: m\\le \\sqrt{\\tilde{f}(0)}L\\big \\rbrace + O(1),\\end{aligned}$ and then that : $\\sqrt{f(1)}L+\\sqrt{f(0)}L=\\sqrt{\\tilde{f}(1)}L+\\sqrt{\\tilde{f}(0)}L+O(1),\\qquad L\\rightarrow +\\infty .$ As $L$ is any positive number, this proves (REF ).", "$\\quad $ $\\bullet $ Now, we have to show : $f(0)\\in \\lbrace \\tilde{f}(0),\\tilde{f}(1)\\rbrace $ .", "$\\quad $ Assume that it is not true, for example $\\displaystyle f(0)<\\min \\lbrace \\tilde{f(0)},\\tilde{f}(1)\\rbrace .$ Then (REF ) implies $f(1)>\\max \\lbrace \\tilde{f}(0),\\tilde{f}(1)\\rbrace .$ $\\quad $ Let $m$ be in $\\mathbb {N}$ .", "There is $\\ell \\in \\mathbb {N}$ such that $\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _\\ell )= O(\\varepsilon _m),$ with $|O(\\varepsilon _m)|\\le \\varepsilon _m$ .", "From the assumption $\\sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\asymp }\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ , each of the eigenvalues $\\lambda ^-(\\kappa _{m-1})$ , $\\lambda ^-(\\kappa _{m+1})$ and $\\lambda ^-(\\kappa _{m+2})$ is also close to an element of $\\sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ .", "If $m$ is large enough, the situation is necessary the following: [scale=6.7] [domain=0:4] [->] (0,0) – (2,0) node[right] $\\big (\\lambda ^-(\\kappa _m)\\big )$ ; [black,line width=1.2pt](0.5,-0.02)–(0.5,0.02)node[above]$\\lambda ^-(\\kappa _{m-1})$ ; [black,line width=1.2pt](0.8,-0.02)–(0.8,0.02)node[above]$\\lambda ^-(\\kappa _m)$ ; [black,line width=1.2pt](1.1,-0.02)–(1.1,0.02)node[above]$\\lambda ^-(\\kappa _{m+1})$ ; [black,line width=1.2pt](1.4,-0.02)–(1.4,0.02)node[above]$\\lambda ^-(\\kappa _{m+2})$ ; [->] (0,-0.3) – (2,-0.3) node[right] $\\big (\\tilde{\\lambda }^-(\\kappa _\\ell )\\big )$ ; [black!40,line width=1.2pt](0.19,-0.32)–(0.19,-0.28)node[above]$\\tilde{\\lambda }^-(\\kappa _{\\ell -1})$ ; [black,line width=1.2pt](0.79,-0.32)–(0.79,-0.28)node[above]$\\tilde{\\lambda }^-(\\kappa _\\ell )$ ; [black,line width=1.2pt](1.39,-0.32)–(1.39,-0.28)node[above]$\\tilde{\\lambda }^-(\\kappa _{\\ell +1})$ ; [->] (0,-0.6) – (2,-0.6) node[right] $\\big (\\tilde{\\lambda }^+(\\kappa _p)\\big )$ ; [black,line width=1.2pt](0.52,-0.62)–(0.52,-0.58)node[above]$\\tilde{\\lambda }^+(\\kappa _{p-1})$ ; [black,line width=1.2pt](1.12,-0.62)–(1.12,-0.58)node[above]$\\tilde{\\lambda }^+(\\kappa _{p})$ ; [black!40,line width=1.2pt](1.72,-0.62)–(1.72,-0.58)node[above]$\\tilde{\\lambda }^+(\\kappa _{p+1})$ ; [dashed,color=violet] (0.74,0.02) – (0.74,-0.18) ; [dashed,color=violet] (0.84,0.02) – (0.84,-0.18) ; [dashed,color=violet] (0.74,-0.28) – (0.74,-0.36) ; [dashed,color=violet] (0.84,-0.28) – (0.84,-0.36) ; [<->,samples=100,color=black!40,line width=1pt](0.74,-0.39)–(0.84,-0.39)node[below]$\\varepsilon _m \\qquad $ ; [dashed,color=violet] (1.05,0.02) – (1.05,-0.48) ; [dashed,color=violet] (1.15,0.02) – (1.15,-0.48) ; [dashed,color=violet] (1.05,-0.58) – (1.05,-0.7) ; [dashed,color=violet] (1.15,-0.58) – (1.15,-0.7) ; [<->,samples=100,color=black!40,line width=1pt] (1.05,-0.74)–(1.15,-0.74)node[below]$\\varepsilon _{m+1} \\qquad $ ; Indeed, since $\\displaystyle f(1)>\\tilde{f}(1)$ , for $m$ large enough, from (REF ) and (REF ), we have $\\begin{aligned}\\lambda ^-(\\kappa _{m+1})&=\\lambda ^-(\\kappa _{m}) +\\frac{1}{\\sqrt{f(1)}}+o(1)\\\\&=\\tilde{\\lambda }^-(\\kappa _\\ell )+O(\\varepsilon _m)+\\frac{1}{\\sqrt{f(1)}}+o(1)\\\\&=\\tilde{\\lambda }^-(\\kappa _{\\ell +1})+\\underbrace{\\frac{1}{\\sqrt{f(1)}}-\\frac{1}{\\sqrt{\\tilde{f}(1)}}}_{<0}+O(\\varepsilon _m)+o(1)\\end{aligned}$ Let us chose $m$ large enough such that $\\left\\lbrace \\begin{aligned}&\\frac{1}{\\sqrt{f(1)}}-\\frac{1}{\\sqrt{\\tilde{f}(1)}}+O(\\varepsilon _m)+o(1) <\\varepsilon _{m+1}\\\\&O(\\varepsilon _m)+\\frac{1}{\\sqrt{f(1)}} +o(1)>\\varepsilon _{m+1}\\end{aligned}\\right.$ Then: $\\tilde{\\lambda }^-(\\kappa _{\\ell })+\\varepsilon _{m+1}<\\lambda ^-(\\kappa _{m+1})<\\tilde{\\lambda }^-(\\kappa _{\\ell +1})-\\varepsilon _{m+1}$ so, as $\\big (\\tilde{\\lambda }^-(\\kappa _\\ell )\\big )$ is a strictly increasing sequence, for $m$ large enough $\\lambda ^-(\\kappa _{m+1})$ is not $\\varepsilon _{m+1}$ -close to any element of $\\big (\\tilde{\\lambda }^-(\\kappa _\\ell )\\big )$ : there is thus $p\\in \\mathbb {N}$ such that $\\lambda ^-(\\kappa _{m+1})-\\tilde{\\lambda }^+(\\kappa _p)=O(\\varepsilon _{m+1}),$ with $|O(\\varepsilon _{m+1})|\\le \\varepsilon _{m+1}$ .", "$\\quad $ For the same reasons, we get also $\\displaystyle \\tilde{\\lambda }^-(\\kappa _{\\ell -1})+\\varepsilon _{m-1}<\\lambda ^-(\\kappa _{m-1})<\\tilde{\\lambda }^-(\\kappa _{\\ell })-\\varepsilon _{m-1}$ and one deduces that (the previous picture helps to visualize it) $\\lambda ^-(\\kappa _{m-1})-\\tilde{\\lambda }^+(\\kappa _{p-1})=O(\\varepsilon _{m-1}),$ with $|O(\\varepsilon _{m-1})|\\le \\varepsilon _{m-1}$ .", "Since we have $f(1)>\\tilde{f}(0)$ , we get also $\\tilde{\\lambda }^+(\\kappa _{p})+\\varepsilon _{m+2}<\\lambda ^-(\\kappa _{m+2})<\\tilde{\\lambda }^+(\\kappa _{p+1})-\\varepsilon _{m+2}.$ Consequently $\\lambda ^-(\\kappa _{m+2})-\\tilde{\\lambda }^-(\\kappa _{\\ell +1})=O(\\varepsilon _{m+2})$ with $|O(\\varepsilon _{m+2})|\\le \\varepsilon _{m+2}$ .", "$\\quad $ Then, by (REF ), (REF ), (REF ) and (REF ), we have for $m$ large enough : $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _{m+1})-\\lambda ^-(\\kappa _{m-1})=\\tilde{\\lambda }^+(\\kappa _{p})-\\tilde{\\lambda }^+(\\kappa _{p-1})+O(\\varepsilon _{m+1}) - O(\\varepsilon _{m-1})\\\\&\\lambda ^-(\\kappa _{m+2})-\\lambda ^-(\\kappa _{m})=\\tilde{\\lambda }^-(\\kappa _{\\ell +1})-\\tilde{\\lambda }^-(\\kappa _{\\ell })+O(\\varepsilon _{m+2}) - O(\\varepsilon _{m}).\\\\\\end{aligned}\\right.$ In particular: $\\left\\lbrace \\begin{aligned}&\\frac{2}{\\sqrt{f(1)}}=\\frac{1}{\\sqrt{\\tilde{f}(0)}}+o(1)\\\\&\\frac{2}{\\sqrt{f(1)}}=\\frac{1}{\\sqrt{\\tilde{f}(1)}}+o(1),\\\\\\end{aligned}\\right.$ and so, taking $m\\rightarrow +\\infty $ , one deduces $2\\sqrt{\\tilde{f}(0)}=\\sqrt{f(1)}\\quad \\mathrm {and}\\quad 2\\sqrt{\\tilde{f}(1)}=\\sqrt{f(1)}.$ As $\\sqrt{f(0)}+\\sqrt{f(1)}=\\sqrt{\\tilde{f}(0)}+\\sqrt{\\tilde{f}(1)}$ , we get $\\begin{aligned}2\\sqrt{f(0)}= 2\\bigg (\\sqrt{\\tilde{f}(0)}+\\sqrt{\\tilde{f}(1)} -\\sqrt{f(1)}\\bigg ) &=\\bigg (2\\sqrt{\\tilde{f}(0)}-\\sqrt{f(1)}\\bigg )+\\bigg (2\\sqrt{\\tilde{f}(1)}-\\sqrt{f(1)}\\bigg )\\\\&=0.\\end{aligned}$ and we get a contradiction as $f(0)>0$ .", "$\\quad $ Hence $f(0)\\in \\lbrace \\tilde{f}(0),\\tilde{f}(1)\\rbrace $ .", "The equality (REF ) gives the conclusion.", "Assume from now that $f(0)=\\tilde{f}(0)$ and $f(1)=\\tilde{f}(1)$ .", "The other case is obtained by substituting the roles of $\\tilde{\\lambda }^-(\\kappa _m)$ and $\\tilde{\\lambda }^+(\\kappa _m)$ ." ], [ "The case $\\bf f(0)\\ne f(1)$", "Without loss of generality, we assume that $f(0)<f(1)$ .", "The following lemma focuses on the sequence $\\big (\\lambda ^-(\\kappa _p)\\big )$ since this is the sequence that grows slower.", "If we had treated the case $f(0)>f(1)$ , the sequence considered in this section would have been $\\big (\\lambda ^+(\\kappa _p)\\big )$ .", "Lemma 3.2 Assume that $f(0)<f(1)$ .", "There is an infinite subset $\\mathcal {L}$ of $\\mathbb {N}$ such that $\\:\\:\\lambda ^-(\\kappa _p)-\\tilde{\\lambda }^-(\\kappa _p)=\\tilde{O}(e^{-2a\\kappa _p}),\\quad p\\in \\mathcal {L}$ .", "For all $m$ in $\\mathbb {N}$ large enough, $\\lbrace m,m+1\\rbrace \\cap \\mathcal {L}\\ne \\emptyset $ .", "Let us denote $U$ the subset of $\\lbrace \\lambda ^-(\\kappa _m),\\:\\: m\\in \\mathbb {N}\\rbrace $ such that : $U\\underset{(\\varepsilon _m)}{\\mathrel {\\begin{array}{c}\\textstyle \\subset \\\\[-0.2ex]\\textstyle \\sim \\end{array}}}\\:\\lbrace \\tilde{\\lambda }^+(\\kappa _m),\\:\\: m\\in \\mathbb {N}\\rbrace $ Case 1 : $U$ is finite.", "Then there is $m_0\\in \\mathbb {N}$ such that : $\\forall m\\ge m_0,\\quad \\exists p\\in \\mathbb {N},\\:\\: |\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _p)|\\le \\varepsilon _m$ This can be written as : $\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _p)= O(\\varepsilon _m)$ with $|O(\\varepsilon _m)|\\le \\varepsilon _m$ .", "By replacing the eigenvalues by their asymptotics (REF ) in the previous equality, one finds, as $f(1)=\\tilde{f}(1)$ : $\\frac{m}{\\sqrt{f(1)}}+\\frac{n-2}{2\\sqrt{f(1)}}-\\frac{\\ln (h)^{\\prime }(1)}{4\\sqrt{f(1)}}=\\frac{p}{\\sqrt{f(1)}}+\\frac{n-2}{2\\sqrt{f(1)}}-\\frac{\\ln (\\tilde{h})^{\\prime }(1)}{4\\sqrt{f(1)}}+O(\\varepsilon _m)$ If $n=2$ then $h=f^{n-2}$ is a constant.", "One has $\\displaystyle \\frac{m}{\\sqrt{f(1)}}=\\frac{p}{\\sqrt{f(1)}}+O(\\varepsilon _m)$ .", "Hence, as $m$ and $p$ are integers, if $m$ is large enough, we have $m=p$ .", "If $n\\ge 3$ , then $\\displaystyle m-p=\\frac{\\ln (h)^{\\prime }(1)}{4}-\\frac{\\ln (\\tilde{h})^{\\prime }(1)}{4}+O(\\varepsilon _m)$ .", "By hypothesis : $\\bigg |\\frac{\\ln (h)^{\\prime }(1)}{4}-\\frac{\\ln (\\tilde{h})^{\\prime }(1)}{4}\\bigg |=\\frac{n-2}{4}\\bigg |\\frac{f^{\\prime }(1)}{f(1)}-\\frac{\\tilde{f}^{\\prime }(1)}{f(1)}\\bigg |\\le \\frac{n-2}{4}\\times \\frac{2}{n-2}= \\frac{1}{2}.$ Hence, $m=p$ for $m,p$ greater than some integer $m_0$ .", "$\\quad $ The set $\\mathcal {L}=\\lbrace m\\in \\mathbb {N},\\:\\:m\\ge m_0 \\rbrace $ satisfies the properties of Lemma REF .", "$\\quad $ Case 2 : $U$ is infinite.", "Then, there exists $\\varphi ,\\psi :\\mathbb {N}\\rightarrow \\mathbb {N}$ two strictly increasing functions such that : $\\lambda ^-(\\kappa _{\\psi (m)})-\\tilde{\\lambda }^+(\\kappa _{\\varphi (m)})=O(\\varepsilon _{\\psi (m)}),$ with $|O(\\varepsilon _{\\psi (m)})|\\le \\varepsilon _{\\psi (m)}$ .", "Remark 6 $\\varphi $ and $\\psi $ are built in such a way that an integer $m\\in \\mathbb {N}$ that is not in the range of $\\psi $ (respectively not in the range of $\\varphi $ ) satisfies $|\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _n)|<\\varepsilon _m$ for some $n\\in \\mathbb {N}$ (respectively $|\\lambda ^+(\\kappa _n)-\\tilde{\\lambda }^+(\\kappa _m)|<\\varepsilon _n$ for some $n\\in \\mathbb {N}$ ).", "$\\quad $ By replacing $\\lambda ^+(\\kappa _{\\varphi (m)})$ and $\\tilde{\\lambda }^-(\\kappa _{\\psi (m)})$ with their asymptotics in the equality (REF ), one has : $\\frac{\\varphi (m)}{\\sqrt{f(0)}}=\\frac{\\psi (m)}{\\sqrt{f(1)}}+C+O(\\varepsilon _{\\psi (m)})+O\\bigg (\\frac{1}{\\varphi (m)}\\bigg )$ $\\quad $ where $\\displaystyle C=-\\frac{\\ln (h)^{\\prime }(1)}{4\\sqrt{f(1)}}-\\frac{\\ln (h)^{\\prime }(0)}{4\\sqrt{f(0)}}+\\frac{n-2}{2\\sqrt{f(1)}}-\\frac{n-2}{2\\sqrt{f(0)}}$ .", "Lemma 3.3 There is an integer $m_0\\in \\mathbb {N}$ such that ($m\\ge m_0\\Rightarrow \\psi (m+1)\\ge \\psi (m)+ 2\\big )$ .", "Set $\\displaystyle B=\\frac{\\sqrt{f(1)}}{\\sqrt{f(0)}}>1\\:$ and $\\: C^{\\prime }=-\\sqrt{f(1)}C$ .", "From (REF ), one gets : $\\psi (m)=B\\varphi (m)+C^{\\prime }+o(1).$ Assume $\\psi (m+1)=\\psi (m)+1$ .", "Then : $\\begin{aligned}\\psi (m)+1=\\psi (m+1)&=B\\varphi (m+1)+C^{\\prime }+o(1)\\\\&\\ge B(\\varphi (m)+1)+C^{\\prime }+o(1)\\\\&=B\\varphi (m)+C^{\\prime }+B+o(1)\\\\&=\\psi (m)+B+o(1).\\end{aligned}$ Thus, we find : $1\\ge B+o(1)$ which is false if $m\\ge m_0$ for some $m_0\\in \\mathbb {N}$ .", "Consequently, the range of $\\psi $ does not contain two consecutive integers.", "Let us set : $\\mathcal {L}=\\lbrace m\\in \\mathbb {N}\\:\\big |\\: m\\ge m_0,\\:\\: m\\notin \\mathrm {range}(\\psi )\\rbrace .$ Then $\\mathcal {L}$ satisfies the condition $\\mathcal {L}\\cap \\lbrace m,m+1\\rbrace \\ne \\emptyset $ for any $m\\ge m_0$ .", "Moreover, for any $m\\in \\mathcal {L}$ , there is $\\ell \\in \\mathbb {N}$ such that $|\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _\\ell )|\\le \\varepsilon _m$ .", "One deduces, as previously, $m=\\ell $ .", "Remark 7 Lemma REF and asymptotics (REF ) imply in particular $\\displaystyle \\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}=\\frac{(\\ln \\tilde{h})^{\\prime }(1)}{4\\sqrt{\\tilde{f}(1)}}$ .", "Now, let us recall an asymptotic integral representation of the Weyl-Titschmarsh function $N(z^2)$ obtained by Simon in [13] (Theorem 3.1) : Theorem 3.4 For every $0<a<1$ , there is $A\\in L^1([0,a])$ such that $N(z^2)=-z-\\int _{0}^{a}A(x)e^{-2xz}dx + \\tilde{O}(e^{-2az}),\\qquad z\\rightarrow +\\infty .$ $\\quad $ From the asymptotic of $\\lambda ^-(\\kappa _m)$ obtained in Lemma REF , we have $\\lambda ^-(\\kappa _m)=-\\frac{N(\\kappa _m)}{\\sqrt{f(1)}}-\\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}+\\tilde{O}\\bigg (e^{-2\\sqrt{\\kappa _m}}\\bigg ).", "$ $\\quad $ Hence $\\begin{aligned}\\lambda ^-(\\kappa _m)-&\\tilde{\\lambda }^-(\\kappa _m)=\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}}) \\\\&\\Rightarrow -\\frac{N(\\kappa _m)}{\\sqrt{f(1)}}-\\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}=-\\frac{\\tilde{N}(\\kappa _m)}{\\sqrt{\\tilde{f}(1)}}-\\frac{(\\ln \\tilde{h})^{\\prime }(1)}{4\\sqrt{\\tilde{f}(1)}}+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})+\\tilde{O}\\bigg (e^{-2\\sqrt{\\kappa _m}}\\bigg )\\end{aligned}$ The equalities $f(1)=\\tilde{f}(1)\\:$ and $\\:\\displaystyle \\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}=\\frac{(\\ln \\tilde{h})^{\\prime }(1)}{4\\sqrt{\\tilde{f}(1)}}\\:$ (see Remark REF ) imply $\\begin{aligned}\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m)=\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})&\\Rightarrow N(\\kappa _m)=\\tilde{N}(\\kappa _m)+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\Rightarrow \\int _{0}^{a}A(x)e^{-2x\\sqrt{\\kappa _m}}dx=\\int _{0}^{a}\\tilde{A}(x)e^{-2x\\sqrt{\\kappa _m}}dx + \\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\Rightarrow \\int _{0}^{a}(A(x)-\\tilde{A}(x))e^{-2x\\sqrt{\\kappa _m}}dx = \\tilde{O}(e^{-2a\\sqrt{\\kappa _m}}).\\end{aligned}$ Let $\\varepsilon >0$ and set $\\displaystyle F(z)=e^{2a(1-\\varepsilon )z}\\int _{0}^{a}(A(x)-\\tilde{A}(x))e^{-2xz}dx$ .", "The function $F$ is entire and satisfies $\\forall z\\in \\mathbb {C},\\quad \\mathrm {Re}(z)> 0 \\Rightarrow |F(z)|\\le \\Vert A-\\tilde{A}\\Vert _{1}e^{2a(1-\\varepsilon )\\mathrm {Re}(z)}$ Let $m$ be an integer large enough.", "From Lemma REF , we can find an integer $p$ in $\\lbrace 2m,2m+1\\rbrace $ such that $|\\lambda ^-(\\kappa _p)-\\tilde{\\lambda }^-(\\kappa _p)|\\le \\varepsilon _p$ .", "We can thus build a sequence $(u_m)$ by setting, for each $m$ large enough, $\\displaystyle u_m=\\frac{\\sqrt{\\kappa _p}}{2}$ .", "This sequence satisfies $u_m-m= O(1).$ We set at last $G(z)=F(2z)$ .", "Then $|G(z)|\\le \\Vert A-\\tilde{A}\\Vert _{1}e^{4a(1-\\varepsilon )\\mathrm {Re}(z)}$ and moreover : $G(u_m) = o(1)$ Consequently (cf [2], Theorem 10.5.1, p.191), $G$ is bounded on $\\mathbb {R}_+$ , and so is $F$ : $\\forall u\\in \\mathbb {R}_+,\\quad \\int _{0}^{a}(A(x)-\\tilde{A}(x))e^{-2xu}dx=O\\big (e^{-2a(1-\\varepsilon )u}\\big )$ As this estimate is true for all $\\varepsilon >0$ , we have : $\\forall u\\in \\mathbb {R}_+,\\quad \\int _{0}^{a}(A(x)-\\tilde{A}(x))e^{-2xu}dx=\\tilde{O}\\big (e^{-2au}\\big ),$ hence ([13], Lemma A.2.1) $A=\\tilde{A}$ on $[0,a]$ .", "One deduces : $\\forall t\\in \\mathbb {R},\\quad N(t^2)-\\tilde{N}(t^2)=\\tilde{O}(e^{-2at})$ From Remark REF , $N$ (resp.", "$\\tilde{N}$ ) has the same role as $M$ (resp.", "$\\tilde{M}$ ) for the potential $x\\mapsto q(1-x)$ (resp.", "$x\\mapsto \\tilde{q}(1-x)$ ).", "Now, it follows, from [13], Theorem A.1.1, that we get $q(1-x)=\\tilde{q}(1-x)$ for all $x\\in [0,a]$ , i.e $\\frac{(f^{n-2})^{\\prime \\prime }(x)}{f^{n-2}(x)}-\\omega f(x) = \\frac{(\\tilde{f}^{n-2})^{\\prime \\prime }(x)}{\\tilde{f}^{n-2}(x)}-\\omega \\tilde{f}(x):=r(x),\\quad \\forall x\\in [1-a,1].$ The functions $f$ and $\\tilde{f}$ solve on $[1-a,1]$ the same ODE $(y^{n-2})^{\\prime \\prime }(x)-\\lambda y^{n-1}(x) = r(x) y^{n-2}(x)$ Moreover $f(1)=\\tilde{f}(1)$ and, thanks to the equality $\\displaystyle \\frac{(\\ln h)^{\\prime }(1)}{4\\sqrt{f(1)}}=\\frac{(\\ln \\tilde{h})^{\\prime }(1)}{4\\sqrt{\\tilde{f}(1)}}$ , we also have $f^{\\prime }(1)=\\tilde{f}^{\\prime }(1)$ .", "Hence, the Cauchy-Lipschitz Theorem entails that $f=\\tilde{f}$ on $[1-a,1]$ .", "$\\quad $ Remark 8 If we had assumed that $f(0)>f(1)$ , we would have worked with $\\big (\\lambda ^+(\\kappa _p)\\big )$ and $\\big (\\tilde{\\lambda }^+(\\kappa _p)\\big )$ , and found that $f=\\tilde{f}$ on $[0,a]$ ." ], [ "The case ${\\bf f(0)= f(1)}$", "Assume, without loss of generality, that $f(0)=f(1)=1$ .", "From Lemma REF , the eigenvalues $\\lambda ^\\pm (\\kappa _m)$ satisfy the asymptotics : $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=-N(\\kappa _m)-\\frac{(\\ln h)^{\\prime }(1)}{4}+\\tilde{O}\\bigg (e^{-2\\sqrt{\\kappa _m}}\\bigg ) \\\\&\\lambda ^+(\\kappa _m)=-M(\\kappa _m)+\\frac{(\\ln h)^{\\prime }(0)}{4}++\\tilde{O}\\bigg (e^{-2\\sqrt{\\kappa _m}}\\bigg ).\\end{aligned}\\right.$ Let us denote $\\displaystyle C_0=\\frac{(\\ln h)^{\\prime }(0)}{4}$ , $\\displaystyle C_1=\\frac{(\\ln h)^{\\prime }(1)}{4}$ , $\\displaystyle \\tilde{C}_0=\\frac{(\\ln \\tilde{h})^{\\prime }(0)}{4}$ and $\\displaystyle \\tilde{C}_1=\\frac{(\\ln \\tilde{h})^{\\prime }(1)}{4}$ .", "Using the asymptotics of $M$ and $N$ given in Theorem REF and Corollary REF , and the explicit expression of $\\kappa _m=m(m+n-2)$ , we get $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=m+\\frac{n-2}{2}-C_1+O\\bigg (\\frac{1}{m}\\bigg ) \\\\&\\lambda ^+(\\kappa _m)=m+\\frac{n-2}{2}+C_0+O\\bigg (\\frac{1}{m}\\bigg ).\\end{aligned}\\right.\\qquad \\quad m\\rightarrow +\\infty .$ Let us set also $V_m=\\displaystyle \\bigg \\lbrace \\lambda ^-(\\kappa _m)-\\frac{n-2}{2}\\:,\\:\\lambda ^+(\\kappa _m)-\\frac{n-2}{2}\\bigg \\rbrace \\:\\:$ and $\\:\\:\\tilde{V}_m=\\displaystyle \\bigg \\lbrace \\tilde{\\lambda }^-(\\kappa _m)-\\frac{n-2}{2}\\:,\\:\\tilde{\\lambda }^+(\\kappa _m)-\\frac{n-2}{2}\\bigg \\rbrace $ .", "As $f$ and $\\tilde{f}$ belong to $\\mathcal {C}_b$ , one has $|C_i|\\le \\frac{1}{4},\\quad |\\tilde{C}_i|\\le \\frac{1}{4} ,\\quad i\\in \\lbrace 0,1\\rbrace .$ Hence, thanks to (REF ) and (REF ), we get for $m$ large enough: $V_m,\\tilde{V}_m\\subset \\bigg [m-\\frac{1}{3},m+\\frac{1}{3}\\bigg ]$ Of course, the assumption $\\Sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\asymp }\\Sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ implies $\\displaystyle -\\frac{n-2}{2}+\\Sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{(\\varepsilon _m)}{\\asymp }-\\frac{n-2}{2}+\\Sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ From (REF ) and (REF ), for each $m$ large enough, we have the alternative $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\\\&\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\end{aligned}\\right.\\quad \\mathrm {or}\\quad \\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\\\&\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\end{aligned}\\right.$ There is thus an infinite set $\\mathcal {S}\\subset \\mathbb {N}$ such that either $\\begin{aligned}\\forall m\\in \\mathcal {S},\\:\\:\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\\\&\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\end{aligned}\\right.&\\qquad \\mathrm {or}\\\\ \\forall m\\in \\mathcal {S},&\\:\\:\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\\\&\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m) =O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big ).\\end{aligned}\\right.\\end{aligned}$ Assume, for example, that the former is true.", "Then we have, using (REF ) : $C_1=\\tilde{C}_1\\quad \\mathrm {and}\\quad C_0=\\tilde{C}_0.$ Case 1 : $C_0\\ne -C_1$ .", "Let us denote $\\displaystyle \\delta =\\frac{|C_0+C_1|}{3}\\in \\big ]0,\\frac{1}{4}\\big [$ .", "$\\quad $ For $m$ large enough, we have, thanks to (REF ): $\\displaystyle \\lambda ^-(\\kappa _m)-\\frac{n-2}{2}$ and $\\displaystyle \\tilde{\\lambda }^-(\\kappa _m)-\\frac{n-2}{2}$ both belong to the interval $-C_1+[m-\\delta ,m+\\delta ]:=I_{m,1}$ $\\displaystyle \\lambda ^+(\\kappa _m)-\\frac{n-2}{2}$ and $\\displaystyle \\tilde{\\lambda }^+(\\kappa _m)-\\frac{n-2}{2}$ both belong to the interval $C_0+[m-\\delta ,m+\\delta ]:=I_{m,0}$ .", "Moreover, as $C_0\\ne -C_1$ , we have $d(I_{m,1} , I_{m,0})\\ge \\delta $ for all $m$ large enough, where $\\displaystyle d(I,J)=\\underset{x\\in I,y\\in J}{\\inf }|x-y|$ .", "We can therefore associate eigenvalues as follows : $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m) = \\tilde{\\lambda }^-(\\kappa _m) +O(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\lambda ^+(\\kappa _m) = \\tilde{\\lambda }^+(\\kappa _m) +O(e^{-2a\\sqrt{\\kappa _m}})\\end{aligned}\\right.\\qquad \\qquad m\\rightarrow +\\infty .$ One shows, as in Section 3.1, that $\\left\\lbrace \\begin{aligned}&N(t^2)-\\tilde{N}(t^2)=\\tilde{O}(e^{-2at})\\\\&M(t^2)-\\tilde{M}(t^2)=\\tilde{O}(e^{-2at})\\end{aligned}\\right.\\qquad t\\rightarrow +\\infty $ and then, that $f(x)=\\tilde{f}(x)\\quad \\forall x\\in [1-a,1]\\quad \\mathrm {and}\\quad f(x)=\\tilde{f}(x)\\quad \\forall x\\in [0,a].$ $\\quad $ Case 2 : $C_0= -C_1$ .", "$\\quad $ By hypothesis : $f,\\tilde{f}$ belong to $\\mathcal {C}_b$ so $q$ belongs to $\\mathcal {D}_b$ .", "Thanks to Lemma REF , there is $j_0\\in \\mathbb {N}$ such that $\\beta _j(0)\\ne \\gamma _j(0)$ .", "Let us set $j_0=\\min \\lbrace j\\ge 2,\\: \\beta _{j}(0)\\ne \\gamma _{j}(0) \\rbrace $ .", "The asymptotics given by Theorem REF and Corollary (REF ) imply $\\displaystyle \\lambda ^-(\\kappa _m)-\\lambda ^+(\\kappa _m)= \\frac{\\gamma _{j_0}-\\beta _{j_0}}{m^{j_0}}+O\\bigg (\\frac{1}{m^{j_0+1}}\\bigg )$ .", "$\\quad $ Because of the relation $\\Sigma (\\Lambda _g(\\omega )) \\underset{(\\varepsilon _m)}{\\asymp } \\Sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ , one can show that, for the same $j_0$ : $\\displaystyle \\tilde{\\lambda }^-(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m)= \\frac{\\tilde{\\gamma }_{j_0}-\\tilde{\\beta }_{j_0}}{m^{j_0}}+O\\bigg (\\frac{1}{m^{j_0+1}}\\bigg )$ $\\quad $ It is then possible to order the eigenvalues $\\lambda ^-(\\kappa _m)$ and $\\lambda ^+(\\kappa _m)$ (also $\\tilde{\\lambda }^-$ and $\\tilde{\\lambda }^+$ ), and this order depends on the sign of $\\gamma _{j_0}-\\beta _{j_0}$ (resp.", "$\\tilde{\\gamma }_{j_0}-\\tilde{\\beta }_{j_0}$ ).", "If $\\gamma _{j_0}-\\beta _{j_0}$ and $\\tilde{\\gamma }_{j_0}-\\tilde{\\beta }_{j_0}$ have the same sign, we claim that $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=\\tilde{\\lambda }^-(\\kappa _m)+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\lambda ^+(\\kappa _m)=\\tilde{\\lambda }^+(\\kappa _m)+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}}).\\end{aligned} \\right.$ $\\quad $ Indeed, if not, from (REF ), there is an infinite subset $\\mathcal {F}\\subset \\mathbb {N}$ such that : $\\displaystyle \\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=\\tilde{\\lambda }^+(\\kappa _m)+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\lambda ^+(\\kappa _m)=\\tilde{\\lambda }^-(\\kappa _m)+\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}}).\\end{aligned} \\right.,\\qquad \\quad m\\rightarrow +\\infty ,\\:\\: m\\in \\mathcal {F}$ .", "Then $\\lambda ^-(\\kappa _m)-\\lambda ^+(\\kappa _m)=\\tilde{\\lambda }^+(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m)+O(e^{-2a\\sqrt{\\kappa _m}})$ , and letting $m$ go to infinity : $\\gamma _{j_0}-\\beta _{j_0}= \\tilde{\\beta }_{j_0}-\\tilde{\\gamma }_{j_0}$ and we have a contradiction.", "Using (REF ) and the same method as in Section 3.1, we find $\\displaystyle \\forall t\\in \\mathbb {R}_+,\\quad M(t^2)-\\tilde{M}(t^2)=\\tilde{O}(e^{-2at})\\quad $ and $\\quad \\displaystyle N(t^2)-\\tilde{N}(t^2)=\\tilde{O}(e^{-2at})$ and at last $f=\\tilde{f}$ on $[0,a]\\quad $ and $\\quad f=\\tilde{f}$ on $[1-a,1]$ .", "$\\quad $ If $\\gamma _{j_0}-\\beta _{j_0}$ and $\\tilde{\\gamma }_{j_0}-\\tilde{\\beta }_{j_0}$ have opposite sign, then : $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=\\tilde{\\lambda }^+(\\kappa _m)+O(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\lambda ^+(\\kappa _m)=\\tilde{\\lambda }^-(\\kappa _m)+O(e^{-2a\\sqrt{\\kappa _m}}).\\end{aligned} \\right.$ In this case, one can prove that $f=\\tilde{f}\\circ \\eta $ on $[0,a]\\quad $ and $\\quad f=\\tilde{f}\\circ \\eta $ on $[1-a,1]$ ." ], [ "Special case", "When $f(0)=f(1)$ , the direct implication has already been established.", "Now, we prove the converse in this case.", "Let $a\\in ]0,1[$ and assume that, for example: $\\begin{aligned}f=\\tilde{f}&\\:\\:\\mathrm {on}\\:\\: [0,a] \\quad {\\rm {and}}\\quad f=\\tilde{f}\\:\\:\\mathrm {on}\\:\\: [1-a,1]\\end{aligned}$ $\\quad $ In that case, $q=\\tilde{q}$ on $[0,a]$ and $q\\circ \\eta =\\tilde{q}\\circ \\eta $ on $[0,a]$ .", "But thanks to Theorem 3.1 in [13], the potential $q$ determines the function $A$ that appears in the representation (REF ).", "Hence, $\\left\\lbrace \\begin{aligned}&M(z^2)-\\tilde{M}(z^2)=\\tilde{O}(e^{-2az})\\\\&N(z^2)-\\tilde{N}(z^2)=\\tilde{O}(e^{-2az})\\end{aligned}\\right.$ The hypothesis $(P_1)$ implies in particular that $f(0)=\\tilde{f}(0)$ and $f^{\\prime }(0)=\\tilde{f}^{\\prime }(0)$ .", "From $(P_3)$ we have also $f(1)=\\tilde{f}(1)$ and $f^{\\prime }(1)=\\tilde{f}^{\\prime }(1)$ .", "Using the asymptotics given by Lemma REF , one deduces immediately that $\\left\\lbrace \\begin{aligned}&\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _m)=\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\\\&\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _m)=\\tilde{O}(e^{-2a\\sqrt{\\kappa _m}})\\end{aligned}\\right.\\qquad m\\rightarrow +\\infty $ which concludes the proof.", "Remark 9 We emphasize that if $f(0)=f(1)$ and $\\displaystyle \\frac{1}{2}\\le a<1$ , then we have a global uniqueness result.", "$\\quad $ Corollary 3.5 If $f$ and $\\tilde{f}$ are analytic functions on $[0,1]$ the previous local uniqueness result becomes a global uniqueness result without the additional constraint that $q,\\tilde{q}\\in \\mathcal {D}_b$ .", "In proving Theorem REF , we needed the hypothesis $q,\\tilde{q}\\in \\mathcal {D}_b$ only in the case where $f(0)=f(1)$ and $C_0=-C_1$ .", "In all other cases, without this hypothesis, one of the properties $(P_1)$ , $(P_2)$ , $(P_3)$ or $(P_4)$ was obtained and, as $f$ and $\\tilde{f}$ are assumed to be analytic, the corresponding equalities extend over $[0,1]$ by analytic continuation.", "Then, only this latter case remains to be dealt with.", "Let us assume, without loss of generality, that $f(0)=1$ .", "Subcase 1 : $q,\\tilde{q}\\in \\mathcal {D}_b$ and the situation has already been studied.", "Subcase 2 : $q$ or $\\tilde{q}$ does not belong to $\\mathcal {D}_b$ .", "Assume, for example, that $q\\notin \\mathcal {D}_b$ .", "As $f$ is analytic, so is $q$ .", "The function $\\varphi :[0,1]\\mapsto \\mathbb {R},\\:\\: x\\mapsto q(x)-q(1-x)$ is also analytic and, as $q$ is not in $\\mathcal {D}_b$ , satisfies : $\\forall k\\in \\mathbb {N},\\quad \\varphi ^{(k)}(0)=0.$ By analytic continuation, $\\varphi $ vanishes everywhere on $[0,1]$ , i.e $q=q\\circ \\eta $ on $[0,1]$ .", "This is equivalent to $M=N$ .", "The eigenvalues $\\lambda ^\\pm (\\kappa _m)$ then satisfy the asymptotics $\\left\\lbrace \\begin{aligned}&\\lambda ^-(\\kappa _m)=M(\\kappa _m)+C_0+\\tilde{O}\\bigg (e^{-2a\\sqrt{\\kappa _m}}\\bigg ) \\\\&\\lambda ^+(\\kappa _m)=M(\\kappa _m)+C_0+\\tilde{O}\\bigg (e^{-2a\\sqrt{\\kappa _m}}\\bigg )\\end{aligned}\\right.$ Using the assumption $\\sigma (\\Lambda _{g}(\\omega ))\\underset{(\\varepsilon _m)}{\\asymp }\\sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ and the same arguments as above, we prove that $C_0=\\tilde{C_0}=-\\tilde{C}_1$ and $\\left\\lbrace \\begin{aligned}&M(\\kappa _m) = \\tilde{M}(\\kappa _m)+ O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big )\\\\&M(\\kappa _m) = \\tilde{N}(\\kappa _m)+ O\\big (e^{-2a\\sqrt{\\kappa _m}}\\big ) \\end{aligned}\\right.\\qquad m\\rightarrow +\\infty $ One can show, as in the proof of Theorem REF , that : $M(t^2)-\\tilde{M}(t^2) = O(e^{-2at})\\quad \\mathrm {and}\\quad M(t^2)-\\tilde{N}(t^2) = O(e^{-2at}).$ Hence $f=\\tilde{f}\\:\\: \\mathrm {on}\\:\\: [0,a]$ and $f=\\tilde{f}\\circ \\eta \\:\\: \\mathrm {on}\\:\\: [0,a]$ .", "By analytic continuation : $f=\\tilde{f}=\\tilde{f}\\circ \\eta \\:\\: \\mathrm {sur}\\:\\: [0,1].$ The next section is devoted to the proof of Theorem REF ." ], [ "Discrete estimates on Weyl-Titchmarsh functions", "Preliminary remarks: Until the end of the paper, we will denote by $C_A$ any constant depending only on $A$ , even within the same calculation.", "In this section, each factor $f$ and $\\tilde{f}$ is supposed so be symmetric.", "This simplifies many formula.", "However, in order to generalize our arguments as much as possible, we will use this property of symmetry only when it seems necessary and write the formulas in their generic forms.", "For example, we will distinguish $M$ from $N$ whereas those two functions are equal.", "$\\quad $ The goal of this subsection is to prove the following result which will be useful in Subsection 4.2.", "Proposition 4.1 Let $\\varepsilon >0$ small enough.", "Assume that $\\sigma (\\Lambda _g(\\omega ))\\underset{\\varepsilon }{\\asymp }\\sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ .", "There is $C_A>0$ and $m_0\\in \\mathbb {N}$ (independant of $\\varepsilon $ ) such that, for all $\\displaystyle m\\ge m_0$ and by setting $y_m=\\sqrt{\\kappa _m}$ : $\\bigg |\\bigg (M(\\kappa _m)N(\\kappa _m)-\\frac{1}{\\Delta ^2(\\kappa _m)}\\bigg )-\\bigg (\\tilde{M}(\\kappa _m)\\tilde{N}(\\kappa _m)-\\frac{1}{\\tilde{\\Delta }^2(\\kappa _m)}\\bigg )\\bigg | \\le C_A \\varepsilon \\times y_m$ We first need the following result.", "Lemma 4.2 Under the hypothesis $\\sigma (\\Lambda _{g}(\\omega ))\\underset{\\varepsilon }{\\asymp }\\sigma (\\Lambda _{\\tilde{g}}(\\omega ))$ , and under the hypothesis that $f$ and $\\tilde{f}$ are symmetric, we have : $f(0)=\\tilde{f}(0).$ Using the same argument as in the proof of Lemma REF , one proves the equality $\\sqrt{f(1)} + \\sqrt{f(0)} = \\sqrt{\\tilde{f}(1)}+\\sqrt{\\tilde{f}(0)}.$ As $f$ and $\\tilde{f}$ are symmetric with respect to $1/2$ , we have $f(0)=f(1)$ and $\\tilde{f}(0)=\\tilde{f}(1)$ .", "Hence $2f(0)=2\\tilde{f}(0)$ .", "This proves Lemma REF .", "Lemma 4.3 For $m$ large enough, we have : $\\lbrace \\lambda ^-(\\kappa _m),\\lambda ^+(\\kappa _m) \\rbrace \\underset{\\varepsilon }{\\asymp } \\lbrace \\tilde{\\lambda }^-(\\kappa _m),\\tilde{\\lambda }^+(\\kappa _m) \\rbrace $ For every $m\\in \\mathbb {N}$ , there $p$ such that : $|\\lambda ^{\\pm }(\\kappa _m)- \\tilde{\\lambda }^{\\pm }(\\kappa _{p}|\\le \\varepsilon $ Let us denote $C_1=\\frac{1}{4\\sqrt{f(1)}}\\frac{h^{\\prime }(1)}{h(1)}\\quad \\mathrm {et}\\quad C_0=\\frac{1}{4\\sqrt{f(0)}}\\frac{h^{\\prime }(0)}{h(0)}.$ Since $f$ and $\\tilde{f}$ are supposed symmetric, we have $C_0=-C_1\\quad \\mathrm {and}\\quad \\tilde{C}_0=-\\tilde{C}_1$ $\\quad $ Thus, setting $C=C_0-\\tilde{C}_0$ one has, from Lemma REF : $\\sqrt{f(0)}\\bigg (\\lambda ^{\\pm }(\\kappa _m)- \\tilde{\\lambda }^{\\pm }(\\kappa _{p})\\bigg ) = \\big (m-p\\big ) + C + o(1).$ Let $k=\\lfloor C \\rfloor $ .", "Then $m-p+k= \\sqrt{f(0)}\\bigg (\\lambda ^{\\pm }(\\kappa _m)- \\tilde{\\lambda }^{\\pm }(\\kappa _{p})\\bigg ) + \\underbrace{k- C}_{\\in ]-1,0[}+o(1)$ and, as $m-p+k$ is an integer, using (REF ), this leads, for $m$ large enough and $\\varepsilon $ small enough, to $p=m+k.$ Hence, for $m$ large enough, (REF ) is equivalent to $\\left\\lbrace \\begin{aligned}&|\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _{m+k})|\\le \\varepsilon \\\\&|\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _{m+k})|\\le \\varepsilon \\end{aligned}\\right.\\qquad \\mathrm {or}\\qquad \\left\\lbrace \\begin{aligned}&|\\lambda ^-(\\kappa _m)-\\tilde{\\lambda }^+(\\kappa _{m+k})|\\le \\varepsilon \\\\&|\\lambda ^+(\\kappa _m)-\\tilde{\\lambda }^-(\\kappa _{m+k})|\\le \\varepsilon \\end{aligned}\\right.$ The relation $\\Sigma \\big (\\Lambda _g(\\omega )\\big )\\underset{\\varepsilon }{\\asymp }\\Sigma \\big (\\Lambda _{\\tilde{g}}(\\omega )\\big )$ implies $2m = 2(m+k)$ and then $k=0$ .", "$\\quad $ This means that, for $m$ greater than some $m_0$ (that does not depend on $\\varepsilon $ ) one has $\\lbrace \\lambda ^-(\\kappa _m),\\lambda ^+(\\kappa _m) \\rbrace \\underset{\\varepsilon }{\\asymp } \\lbrace \\tilde{\\lambda }^-(\\kappa _m),\\tilde{\\lambda }^+(\\kappa _m) \\rbrace $ Of course, Lemma REF is still true by replacing $\\kappa _m$ by $\\mu _m$ .", "For $m$ large enough, we have : $\\lbrace \\lambda ^-(\\mu _m),\\lambda ^+(\\mu _m) \\rbrace \\underset{\\varepsilon }{\\asymp } \\lbrace \\tilde{\\lambda }^-(\\mu _m),\\tilde{\\lambda }^+(\\mu _m) \\rbrace $ Recall that $\\Lambda _g^m(\\omega )=\\begin{pmatrix}-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0&-\\frac{1}{\\sqrt{f(0)}}\\frac{h^{1/4}(1)}{h^{1/4}(0)}\\frac{1}{\\Delta (\\mu _m)}\\\\-\\frac{1}{\\sqrt{f(1)}}\\frac{h^{1/4}(0)}{h^{1/4}(1)}\\frac{1}{\\Delta (\\mu _m)}&-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-C_1\\\\\\end{pmatrix}$ Hence $\\begin{aligned}\\mathrm {Tr}\\big (\\Lambda _g^m(\\omega )\\big )- \\mathrm {Tr}&\\big (\\Lambda _{\\tilde{g}}^m(\\omega )\\big )\\\\ &= \\bigg [-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0 -\\frac{N(\\mu _m)}{\\sqrt{f(1)}}-C_1\\bigg ] - \\bigg [-\\frac{\\tilde{M}(\\mu _m)}{\\sqrt{\\tilde{f}(0)}}+\\tilde{C}_0 -\\frac{\\tilde{N}(\\mu _m)}{\\sqrt{\\tilde{f}(1)}}-\\tilde{C}_1\\bigg ]\\\\&= -\\frac{1}{\\sqrt{f(0)}}\\bigg (M(\\mu _m)-\\tilde{M}(\\mu _m)\\bigg ) -\\frac{1}{\\sqrt{f(1)}}\\bigg (N(\\mu _m)-\\tilde{N}(\\mu _m)\\bigg )\\\\&\\hspace{170.71652pt}+ (\\tilde{C}_0-C_0) + (C_1-\\tilde{C}_1)\\\\\\end{aligned}$ Thanks to (REF ), with $k=0$ and $m-p=0$ , we have $|C|=|\\tilde{C}_0-C_0|=|C_1-\\tilde{C}_1| \\le C_A\\varepsilon .$ Hence, combining (REF ), (REF ) and (REF ), we get : $\\begin{aligned}\\bigg |\\frac{1}{\\sqrt{f(0)}}\\bigg (M(\\mu _m)-\\tilde{M}(\\mu _m)\\bigg ) +\\frac{1}{\\sqrt{f(1)}}\\bigg (N(\\mu _m)-\\tilde{N}(\\mu _m)\\bigg )\\bigg | &\\le \\underbrace{\\big |\\mathrm {Tr}\\big (\\Lambda _g^m(\\omega )\\big )- \\mathrm {Tr}\\big (\\Lambda _{\\tilde{g}}^m(\\omega )\\big )\\big |}_{\\le 2\\varepsilon } + C_A\\varepsilon \\\\&\\le C_A\\varepsilon .\\end{aligned}$ As $f$ and $\\tilde{f}$ are symmetric with respect to $1/2$ , this leads to $\\bigg |M(\\mu _m)-\\tilde{M}(\\mu _m)\\bigg |\\le C_A \\varepsilon $ We also have an estimate on the determinant.", "From (REF ), assume for example that $\\big |\\lambda ^+(\\mu _m)-\\tilde{\\lambda }^+(\\mu _m)\\big |\\le \\varepsilon \\quad \\mathrm {and}\\quad \\big |\\tilde{\\lambda }^-(\\mu _m)-\\lambda ^-(\\mu _m)\\big |\\le \\varepsilon .$ Then : $\\begin{aligned}\\bigg |\\det (\\Lambda _g^m(\\omega ))-\\det (\\Lambda _{\\tilde{g}}^m(\\omega )) \\bigg |&= \\bigg |\\lambda ^-(\\mu _m)\\lambda ^+(\\mu _m)-\\tilde{\\lambda }^-(\\mu _m)\\tilde{\\lambda }^+(\\mu _m)\\bigg |\\\\&\\le \\big |\\lambda ^-(\\mu _m)\\big |\\big |\\lambda ^+(\\mu _m)-\\tilde{\\lambda }^+(\\mu _m)\\big |+\\big |\\tilde{\\lambda }^-(\\mu _m)-\\lambda ^-(\\mu _m)\\big |\\big |\\tilde{\\lambda }^+(\\mu _m)\\big |\\\\&\\le C_A \\varepsilon \\times \\sqrt{\\mu _m}\\end{aligned}$ We write : $\\det (\\Lambda _g^m(\\lambda ))-\\det (\\Lambda _{\\tilde{g}}^m(\\lambda ))= \\mathrm {I}(\\mu _m)+\\mathrm {II}(\\mu _m)+\\mathrm {III}(\\mu _m)+\\mathrm {IV}$ with $\\mathrm {I}(\\mu _m)=\\frac{1}{\\sqrt{f(0)f(1)}}\\bigg [\\bigg (M(\\mu _m)N(\\mu _m)-\\frac{1}{\\Delta ^2(\\mu _m)}\\bigg )-\\bigg (\\tilde{M}(\\mu _m)\\tilde{N}(\\mu _m)-\\frac{1}{\\tilde{\\Delta }^2(\\mu _m)}\\bigg )\\bigg ],$ $\\mathrm {II}(\\mu _m)=\\frac{1}{\\sqrt{f(0)}}\\bigg [C_1(M(\\mu _m)-\\tilde{M}(\\mu _m))+(C_1-\\tilde{C_1})\\tilde{M}(\\mu _m)\\bigg ]$ $\\mathrm {III}(\\mu _m)=\\frac{1}{\\sqrt{f(1)}}\\bigg [\\tilde{C}_0(\\tilde{N}(\\mu _m)-N(\\mu _m))+(\\tilde{C}_0-C_0)N(\\mu _m)\\bigg ]$ and $\\mathrm {IV} = (\\tilde{C}_0-C_0)\\tilde{C}_1 + C_0(\\tilde{C}_1-C_1)$ We have: $\\begin{aligned}|\\mathrm {II}(\\mu _m)| &\\le \\frac{1}{\\sqrt{f(0)}}|C_1||M(\\mu _m)-\\tilde{M}(\\mu _m)| + \\frac{1}{\\sqrt{f(0)}}|C_1-\\tilde{C}_1||\\tilde{M}(\\mu _m)|\\\\&\\le C_A \\varepsilon + C_A \\varepsilon \\sqrt{\\mu _m}\\\\&\\le C_A \\varepsilon \\sqrt{\\mu _m}.\\end{aligned}$ Similarly: $|\\mathrm {II}(\\mu _m)|\\le C_A \\varepsilon \\sqrt{\\mu _m}\\quad \\mathrm {and}\\quad |\\mathrm {IV}|\\le C_A\\varepsilon .$ Finally : $\\begin{aligned}|I(\\mu _m)|&\\le \\big |\\det (\\Lambda _g^m(\\lambda ))-\\det (\\Lambda _{\\tilde{g}}^m(\\lambda )) \\big | + C_A\\varepsilon \\sqrt{\\mu _m}\\\\&\\le C_A\\varepsilon \\sqrt{\\mu _m}\\end{aligned}$ As this is true for $\\mu _m$ , with $m\\ge m_0$ with $m_0$ not depending on $\\varepsilon $ , this is also true for $\\kappa _m$ with $m\\ge m_0$ (with $m_0$ different from the other one but still independent of $\\varepsilon $ ).", "Hence, by setting $y_m=\\sqrt{\\kappa _m}$ , we have proved that there exists $C_A>0$ such that, for $m\\ge m_0$ : $\\bigg |\\bigg (M(\\kappa _m)N(\\kappa _m)-\\frac{1}{\\Delta ^2(\\kappa _m)}\\bigg )-\\bigg (\\tilde{M}(\\kappa _m)\\tilde{N}(\\kappa _m)-\\frac{1}{\\tilde{\\Delta }^2(\\kappa _m)}\\bigg )\\bigg | \\le C_A \\varepsilon \\times y_m$" ], [ "An integral estimate", "In all this section, we will use the estimate of Proposition REF in order to show that $\\Vert q-\\tilde{q}\\Vert _{L^2(0,1)}\\le C_{A} \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.$ where $q$ is the potential defined in (REF ).", "$\\quad $ $\\quad $ Let us go back to the Sturm-Liouville equation $-u^{\\prime \\prime }+qu=-zu,\\quad z\\in \\mathbb {C}$ and to the fundamental system of solutions $\\lbrace c_0,s_0\\rbrace $ and $\\lbrace c_1,s_1\\rbrace $ given by (REF ).", "We define $\\psi $ and $\\phi $ as the two unique solutions of (REF ) that can be written as $\\psi (x,z)=c_0(x,z)+M(z)s_0(x,z),\\quad \\phi (x,z)=c_1(x,z)-N(z)s_1(x,z),$ with Dirichlet boundary conditions at $x=1$ and $x=0$ respectively.", "$\\quad $ Proposition 4.4 We have the following relations $\\begin{aligned}&s_0(1,z)=\\Delta (z)\\\\&s_0^{\\prime }(1,z)=-N(z)\\Delta (z)\\\\&c_0(1,z)=-M(z)\\Delta (z) \\\\&c_0^{\\prime }(1,z)=M(z)N(z)\\Delta (z)-\\frac{1}{\\Delta (z)}\\end{aligned}\\qquad \\quad \\mathrm {and}\\qquad \\quad \\begin{aligned}&s_1(0,z)=-\\Delta (z)\\\\&s_1^{\\prime }(0,z)=-M(z)\\Delta (z)\\\\&c_1(0,z)=-N(z)\\Delta (z)\\\\&c_1^{\\prime }(0,z)=\\frac{1}{\\Delta (z)} - N(z)M(z)\\Delta (z).\\end{aligned}$ First of all, the equalities $s_0(1,z)=\\Delta (z)\\:$ and $\\: s_1(0,z)=-\\Delta (z)$ come from the relation (REF ).", "The set of solutions of (REF ) that satisfy $u(0,z)=0$ is a one dimensional vector space.", "Therefore, there exists $A(z)\\in \\mathbb {C}$ such that : $\\forall x\\in [0,1],\\quad s_0(x,z)=A(z)\\phi (x,z)$ The conditions on $c_1$ and $s_1$ at $x=1$ lead to the equality $A(z)=s_0(1,z)=\\Delta (z)$ .", "We get also, by differentiating $(\\ref {s1_2})$ : $s_0^{\\prime }(1,z)= A(z)\\big (c_1^{\\prime }(1,z)-N(z)s_1^{\\prime }(1,z)\\big ) =-N(z)A(z)=-\\Delta (z)N(z).$ Analogously, there is $B(z)\\in \\mathbb {C}$ such that $s_1(x,z)=B(z)\\psi (x,z)$ and we show $B(z)=-\\Delta (z)$ .", "Hence $s_1^{\\prime }(1,z)=-\\Delta (z)\\psi ^{\\prime }(1,z)$ and so $\\displaystyle \\psi ^{\\prime }(1,z)=-\\frac{1}{\\Delta (z)}$ .", "By differentiating $(\\ref {s1_2})$ and taking $x=1$ , we get: $c_0^{\\prime }(1,z)+M(z)s_0^{\\prime }(1,z) = -\\frac{1}{\\Delta (z)}$ and then $c_0^{\\prime }(1,z) = M(z)N(z)\\Delta (z)-\\frac{1}{\\Delta (z)}.$ $\\quad $ This proves the equalities on $c_0$ and $s_0$ .", "We proceed similarly to establish those on $c_1$ and $s_1$ .", "Thanks to those relations, we are now able to prove the following lemma: Lemma 4.5 Denote $\\mathcal {P}$ the poles of $N$ .", "For any $z\\in \\mathbb {C}\\backslash \\mathcal {P}$ we have the equality : $\\begin{aligned}\\tilde{\\Delta }(z)\\Delta (z)\\bigg (M(z)N(z)-\\frac{1}{\\Delta (z)^2}\\bigg )-M(z)\\tilde{N}(z)\\Delta (z)&\\tilde{\\Delta }(z)+1\\\\&=\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )c_0(x,z)\\tilde{s}_0(x,z)dx\\end{aligned}$ Let us define $\\displaystyle \\theta :x\\mapsto c_0(x,z)\\tilde{s}_0^{\\prime }(x,z)-s_0^{\\prime }(x,z)\\tilde{c}_0(x,z)$ .", "Then : $\\begin{aligned}\\theta ^{\\prime }(x)&=c_0(x,z)\\tilde{s}_0^{\\prime \\prime }(x,z)+c_0^{\\prime }(x,z)\\tilde{s}_0^{\\prime }(x,z)-c_0^{\\prime }(x,z)\\tilde{s}_0^{\\prime }(x,z)-c^{\\prime \\prime }_0(x,z)\\tilde{s}_0(x,z)\\\\&=c_0(x,z)(\\tilde{q}(x)\\tilde{s}_0(x,z)+z\\tilde{s}_0(x,z))-(q(x)c_0(x,z)+zc_0(x,z))\\tilde{s}_0(x,z)\\\\&=(\\tilde{q}(x)-q(x)\\big )c_0(x,z)\\tilde{s}_0(x,z)\\end{aligned}$ Hence, by integrating between 0 and 1 : $\\theta (1)-\\theta (0)=\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )c_0(x,z)\\tilde{s}_0(x,z)dx.$ By replacing $c_0^{\\prime }(1,z)$ , $\\tilde{s}_0(1,z)$ , $c_0(1,z)$ and $\\tilde{s}_0^{\\prime }(1,z)$ by the expressions given in Proposition REF , we get the relation of Lemma REF .", "By inverting the roles of $q$ and $\\tilde{q}$ , we get $\\begin{aligned}\\Delta (z)\\tilde{\\Delta }(z)\\bigg (\\tilde{M}(z)\\tilde{N}(z)-\\frac{1}{\\tilde{\\Delta }(z)^2}\\bigg )-\\tilde{M}(z)N(z)&\\Delta (z)\\tilde{\\Delta }(z)+1\\\\&=\\int _{0}^{1}\\big (\\tilde{q}(x)-q(x)\\big )\\tilde{c}_0(x,z)s_0(x,z)dx\\end{aligned}$ At last, from Remark REF , if we replace $q(x)$ and $\\tilde{q}(x)$ by $q(1-x)$ and $\\tilde{q}(1-x)$ , then, the roles of $M$ and $N$ are inverted.", "Moreover, we remark that $c_1(1-x)$ and $-s_1(1-x)$ play the roles of $c_0(x)$ and $s_0(x)$ but for the potential $q(1-x)$ , i.e.", "denoting $\\eta (x)=1-x$ : $c_0(x,z,,q\\circ \\eta )=c_1(1-x,z,q)\\quad \\mathrm {and}\\quad s_0(x,z,,q\\circ \\eta )=-s_1(1-x,z,q)$ Hence : $\\begin{aligned}\\Delta (z)\\tilde{\\Delta }(z)\\bigg (\\tilde{M}(z)\\tilde{N}(z)-\\frac{1}{\\tilde{\\Delta }(z)^2}\\bigg )-&\\tilde{N}(z)M(z)\\Delta (z)\\tilde{\\Delta }(z)+1\\\\&=-\\int _{0}^{1}\\big (\\tilde{q}(1-x)-q(1-x)\\big )\\tilde{c}_1(1-x,z)s_1(1-x,z)dx\\\\\\end{aligned}$ As $q$ is symmetric, we have $c_1(1-x)=c_0(x)$ and $s_1(1-x)=-s_0(x)$ .", "The previous equality can be written $\\begin{aligned}\\Delta (z)\\tilde{\\Delta }(z)\\bigg (\\tilde{M}(z)\\tilde{N}(z)-\\frac{1}{\\tilde{\\Delta }(z)^2}\\bigg )-\\tilde{N}(z)M(z)\\Delta (z)&\\tilde{\\Delta }(z)+1\\\\&=\\int _{0}^{1}\\big (\\tilde{q}(x)-q(x)\\big ) \\tilde{c}_0(x,z)s_0(x,z)dx\\\\\\end{aligned}$ Hence, by substracting the relation of Lemma REF from equality (REF ), we get $\\begin{aligned}\\Delta (x)\\tilde{\\Delta }(z)\\bigg [\\bigg (M(z)N(z)-\\frac{1}{\\Delta (z)^2}\\bigg )-\\bigg (\\tilde{M}(z)\\tilde{N}(z)-\\frac{1}{\\tilde{\\Delta }(z)^2}\\bigg )\\bigg ]=&\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )c_0(x,z)\\tilde{s}_0(x,z)dx\\\\&+\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )\\tilde{c}_0(x,z)s_0(x,z)dx\\end{aligned}$ Using Proposition REF , we have proved: Proposition 4.6 There is $m_0\\in \\mathbb {N}$ such that, for $m\\ge m_0$ : $\\bigg |\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )\\big [c_0(x,\\kappa _m)\\tilde{s}_0(x,\\kappa _m)+\\tilde{c}_0(x,\\kappa _m)s_0(x,\\kappa _m)\\big ]dx\\bigg | \\le C_A y_m |\\Delta (\\kappa _m)||\\tilde{\\Delta }(\\kappa _m)|\\varepsilon $ $\\quad $" ], [ "Construction of an inverse integral operator", "From now on, we set $L(x)=q(x)-\\tilde{q}(x)$ .", "We want to express the integrand in the left-hand-side of (REF ) in terms of an operator acting on $L$ .", "Proposition 4.7 There is an operator $B:L^2([0,1]) \\rightarrow L^2([0,1])$ such that : For all $m\\in \\mathbb {N}$ , $\\displaystyle \\int _{0}^{1}\\big [c_0(x,\\kappa _m)\\tilde{s}_0(x,\\kappa _m)+\\tilde{c}_0(x,\\kappa _m)s_0(x,\\kappa _m)\\big ]L(x)dx=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)BL(\\tau )d\\tau .$ The function $\\tau \\mapsto BL(\\tau )$ is $C^1$ on $[0,1]$ and $BL$ and $(BL)^{\\prime }$ are uniformly bounded by a constant $C_A$ .", "Let us extend on $[-1,0]$ $q$ and $\\tilde{q}$ into even functions.", "From [10] (page 9) we have the following integral representations of the functions $c_0$ and $s_0$ : $\\begin{aligned}&s_0(x,-z^2)=\\frac{\\sin (z x)}{z}+\\int _{0}^{x}H(x,t)\\frac{\\sin (z t)}{z}dt\\\\&c_0(x,-z^2)= \\cos (zx) + \\int _{0}^{x}P(x,t)\\cos (z t)dt\\end{aligned}$ where $H(x,t)$ and $P(x,t)$ can be written as $\\begin{aligned}&H(x,t)=K(x,t)-K(x,-t)\\\\&P(x,t)=K(x,t)+K(x,-t)\\end{aligned}$ with $K$ a $C^1$ function on $[-1,1]\\times [-1,1]$ satisfying some good estimates.", "More precisely ([10], p.14), we have: Theorem 4.8 On $[-1,1]\\times [-1,1]$ , $K$ satisfies the estimate $|K(x,t)|\\le \\frac{1}{2}w\\bigg (\\frac{x+t}{2}\\bigg )\\exp \\bigg (\\sigma _1(x)-\\sigma _1\\bigg (\\frac{x+t}{2}\\bigg )-\\sigma _1\\bigg (\\frac{x-t}{2}\\bigg )\\bigg )$ with $\\displaystyle w(u)=\\underset{0\\le \\xi \\le u}{\\max }\\bigg |\\int _{0}^{\\xi }q(y)dy\\bigg |,\\quad $ $\\displaystyle \\quad \\sigma _0(x)=\\int _{0}^{x}|q(t)|dt,\\quad $ $\\displaystyle \\quad \\sigma _1(x)=\\int _{0}^{x}\\sigma _0(t)dt$ .", "$\\quad $ $\\quad $ We thus have the following estimate : Proposition 4.9 There is a constant $C_A>0$ , which only depends on $A$ , such that $\\Vert K\\Vert _\\infty + \\bigg \\Vert \\frac{\\partial K}{\\partial x}\\bigg \\Vert _\\infty + \\bigg \\Vert \\frac{\\partial K}{\\partial t}\\bigg \\Vert _\\infty \\le C_A.$ Since $f\\in C(A)$ , the potential $q$ is bounded by a constant that only depends on $A$ , so are $\\sigma _0$ , $\\sigma _1$ , $w$ and $K$ .", "Denote $J(u,v)=K(u+v,u-v)$ .", "Then $J$ is uniformly bounded by $C_A$ and moreover (cf [10], p. 14 and 16), one has the equalities : $\\left\\lbrace \\begin{aligned}& \\frac{\\partial J(u,v)}{\\partial u}=\\frac{1}{2}q(u)+\\int _{0}^{v}q(u+\\beta )J(u,\\beta )d\\beta \\\\&\\frac{\\partial J(u,v)}{\\partial v}=\\int _{0}^{u}q(v+\\alpha )J(\\alpha ,v)d\\beta \\ \\end{aligned}\\right.$ We deduce that the partial derivative of $J$ are uniformly bounded by $C_A$ .", "Returning to the $(x,t)$ coordinates, the conclusion of Proposition REF follows.", "$\\quad $ For $z=i\\sqrt{\\kappa _m}=:iy_m$ , we have thus : $\\begin{aligned}&s_0(x,\\kappa _m)=\\frac{\\sinh (y_m x)}{y_m}+\\int _{0}^{x}H(x,t)\\frac{\\sinh (y_mt)}{y_m}dt\\\\&c_0(x,\\kappa _m)=\\cosh (y_m x)+\\int _{0}^{x}H(x,t)\\cosh (y_mt)dt\\end{aligned}$ $\\quad $ We will take advantage of this representation to write the estimates of Proposition REF as an integral estimate.", "We have $\\begin{aligned}\\int _{0}^{1}L(x)s_0(x)\\tilde{c}_0(x)dx&=\\int _{0}^{1}L(x)\\bigg [\\frac{\\sinh (y_m x)}{y_m}+\\int _{0}^{x}H(x,t)\\frac{\\sinh (y_m t)}{y_m}dt\\bigg ]\\times \\\\&\\hspace{113.81102pt}\\bigg [\\cosh ( y_m x)+\\int _{0}^{x}\\tilde{P}(x,u)\\cosh (y_m u)du\\bigg ]dx\\\\ &=\\mathrm {I}_0+\\mathrm {II}_0+\\mathrm {III}_0+\\mathrm {IV}_0,\\end{aligned}$ with $\\displaystyle \\mathrm {I}_0=\\int _{0}^{1}L(x)\\frac{\\sinh (y_m x)\\cosh (y_mx)}{y_m}dx$ $\\displaystyle \\mathrm {II}_0=\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\tilde{P}(x,u)\\frac{\\sinh (y_m x)\\cosh (y_m u)}{y_m}du\\bigg ]dx$ $\\displaystyle \\mathrm {III}_0 =\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}H(x,t)\\frac{\\sinh (y_m t)\\cosh (y_m x)}{y_m}dt\\bigg ]dx$ $\\displaystyle \\mathrm {IV}_0 =\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\int _{0}^{x}\\tilde{P}(x,u)H(x,t)\\frac{\\sinh (y_m t)\\cosh (y_m u)}{y_m}du\\:dt\\bigg ]dx$ $\\quad $ Let us compute those four quantities independently.", "$\\displaystyle \\mathrm {I}_0=\\int _{0}^{1}L(x)\\frac{\\sinh (y_m x)\\cosh (y_mx)}{y_m}dx=\\frac{1}{2y_m}\\int _{0}^{1}\\sinh (2xy_m)L(x)dx$ .", "$\\quad $ $\\begin{aligned}\\mathrm {II}_0&=\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\tilde{P}(x,u)\\frac{\\sinh (y_m x)\\cosh (y_m u)}{y_m}du\\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\tilde{P}(x,u)\\frac{\\sinh (y_m (x+u))+\\sinh (y_m (x-u))}{2}du\\bigg ]dx\\\\&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\tilde{P}(x,u)\\sinh (y_m (x+u))du+\\int _{0}^{x}\\tilde{P}(x,u)\\sinh (y_m (x-u))du\\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{\\frac{x}{2}}^{x}\\tilde{P}(x,2\\tau -x)\\sinh (2 \\tau y_m)d\\tau +\\int _{0}^{\\frac{x}{2}}\\tilde{P}(x,x-2\\tau )\\sinh (2\\tau y_m)d\\tau \\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2 \\tau y_m)\\bigg [\\int _{\\tau }^{2\\tau }\\tilde{P}(x,2\\tau -x)L(x)dx+\\int _{2\\tau }^{1}\\tilde{P}(x,x-2\\tau )L(x)dx\\bigg ]d\\tau \\end{aligned}$ $\\quad $ But, for all $(x,\\tau )$ in $\\mathbb {R}^2$ , we have $\\tilde{P}(x,x-2\\tau )=\\tilde{P}(x,2\\tau -x)$ .", "Then $\\begin{aligned}\\mathrm {II}_0&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2 \\tau y_m)\\bigg [\\int _{\\tau }^{1}\\tilde{P}(x,2\\tau -x)L(x)dx\\bigg ]d\\tau \\\\\\end{aligned}$ Let us compute $\\mathrm {III}_0$ : $\\begin{aligned}\\displaystyle \\mathrm {III}_0 &=\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}H(x,t)\\frac{\\sinh (y_m t)\\cosh (y_m x)}{y_m}dt\\bigg ]dx\\\\&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}H(x,t)\\sinh (y_m (t+x))dt+\\int _{0}^{x}H(x,t)\\sinh (y_m (t-x))dt\\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{\\frac{x}{2}}^{x}H(x,2\\tau -x)\\sinh (2\\tau y_m)d\\tau +\\int _{-\\frac{x}{2}}^{0}H(x,2\\tau +x)\\sinh (2\\tau y_m)d\\tau \\bigg ]dx\\\\\\end{aligned}$ By changing $\\tau $ in $-\\tau $ , we get $\\begin{aligned}\\displaystyle \\mathrm {III}_0 &= \\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{\\frac{x}{2}}^{x}H(x,2\\tau -x)\\sinh (2\\tau y_m)dt+\\int _{0}^{\\frac{x}{2}}H(x,-2\\tau +x)\\sinh (-2\\tau y_m)dt\\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{\\frac{x}{2}}^{x}H(x,2\\tau -x)\\sinh (2\\tau y_m)d\\tau -\\int _{0}^{\\frac{x}{2}}H(x,-2\\tau +x)\\sinh (2\\tau y_m)d\\tau \\bigg ]dx\\\\\\end{aligned}$ $\\quad $ As $H$ is odd with respect to the second variable : $\\begin{aligned}\\displaystyle \\mathrm {III}_0&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}H(x,2\\tau -x)\\sinh (2\\tau y_m)d\\tau \\bigg ]dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\bigg [\\int _{\\tau }^{1}H(x,2\\tau -x)L(x)dx\\bigg ]d\\tau \\end{aligned}$ At last : $\\begin{aligned}\\mathrm {IV}_0 &=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\bigg [\\int _{0}^{x}\\int _{0}^{x}\\tilde{P}(x,u)H(x,t)\\big (\\sinh (y_m (t+u))+\\sinh (y_m (t-u))\\big )du\\:dt\\bigg ]dx\\\\&=\\mathrm {IV}_0(1) + \\mathrm {IV}_0(2)\\end{aligned}$ where $\\begin{aligned}\\mathrm {IV}_0(1)&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\int _{0}^{x}\\int _{0}^{x}\\tilde{P}(x,u)H(x,t)\\sinh (y_m (t+u))dudtdx\\\\&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\int _{0}^{1}{\\bf 1}_{[0,x]}(t)\\int _{\\frac{t}{2}}^{\\frac{x+t}{2}}2\\tilde{P}(x,2\\tau -t)H(x,t)\\sinh (2\\tau y_m)d\\tau dtdx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\int _{0}^{1}\\sinh (2\\tau y_m){\\bf 1}_{[0,x]}(\\tau )\\int _{2\\tau -x}^{2\\tau }\\tilde{P}(x,2\\tau -t)H(x,t){\\bf 1}_{[0,x]}(t)dtd\\tau dx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{\\tau }^{1}L(x)\\int _{2\\tau -x}^{2\\tau }\\tilde{P}(x,2\\tau -t)H(x,t){\\bf 1}_{[0,x]}(t)dtdxd\\tau \\\\\\end{aligned}$ and $\\begin{aligned}\\mathrm {IV}_0(2)&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\int _{0}^{x}\\int _{0}^{x}\\tilde{P}(x,u)H(x,t)\\sinh (y_m (t-u))du\\:dtdx\\\\&=\\frac{1}{2y_m}\\int _{0}^{1}L(x)\\int _{0}^{1}{\\bf 1}_{[0,x]}(t)\\int _{\\frac{t-x}{2}}^{\\frac{t}{2}}2\\tilde{P}(x,t-2\\tau )H(x,t)\\sinh (2\\tau y_m)d\\tau \\:dtdx\\\\&=\\frac{1}{y_m}\\int _{0}^{1}L(x)\\int _{-1}^{1}\\sinh (2\\tau y_m){\\bf 1}_{[-x,x]}(2\\tau )\\int _{2\\tau }^{2\\tau +x}\\tilde{P}(x,t-2\\tau )H(x,t){\\bf 1}_{[0,x]}(t)\\:dtd\\tau dx\\\\&=\\frac{1}{y_m}\\int _{-1}^{1}\\sinh (2\\tau y_m)\\int _{2|\\tau |}^{1}L(x)\\int _{2\\tau }^{2\\tau +x}\\tilde{P}(x,t-2\\tau )H(x,t){\\bf 1}_{[0,x]}(t)\\:dtdxd\\tau \\\\&=\\mathrm {IV}_0(2,1)+\\mathrm {IV}_0(2,2)\\end{aligned}$ with $\\begin{aligned}\\mathrm {IV}_0(2,1)&=\\frac{1}{y_m}\\int _{-1}^{0}\\sinh (2\\tau y_m)\\int _{-2\\tau }^{1}L(x)\\int _{2\\tau }^{2\\tau +x}\\tilde{P}(x,t-2\\tau )H(x,t){\\bf 1}_{[0,x]}(t)\\:dtdxd\\tau \\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{2\\tau }^{1}L(x)\\int _{-2\\tau }^{-2\\tau +x}\\tilde{P}(x,t+2\\tau )H(x,t){\\bf 1}_{[0,x]}(t)\\:dtdxd\\tau \\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{2\\tau }^{1}L(x)\\int _{0}^{-2\\tau +x}\\tilde{P}(x,t+2\\tau )H(x,t)\\:dtdxd\\tau \\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{P}(x,t)H(x,t-2\\tau )\\:dtdxd\\tau \\end{aligned}$ and $\\begin{aligned}\\mathrm {IV}_0(2,2)&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{2\\tau +x}\\tilde{P}(x,t-2\\tau )H(x,t){\\bf 1}_{[0,x]}(t)\\:dtdxd\\tau \\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{P}(x,t-2\\tau )H(x,t)\\:dtdxd\\tau \\\\\\end{aligned}$ Finally : $\\int _{0}^{1}s_0(x)\\tilde{c}_0(x)L(x)dx = \\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)QL(\\tau )d\\tau $ with $\\begin{aligned}QL(\\tau ) = \\frac{1}{2}L(\\tau )&+\\int _{\\tau }^{1}\\tilde{P}(x,2\\tau -x)L(x)dx+\\int _{\\tau }^{1}H(x,2\\tau -x)L(x)dx\\\\&+\\int _{\\tau }^{1}L(x)\\int _{2\\tau -x}^{2\\tau }\\tilde{P}(x,2\\tau -t)H(x,t){\\bf 1}_{[0,x]}(t)dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{P}(x,t)H(x,t-2\\tau )\\:dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{P}(x,t-2\\tau )H(x,t)\\:dtdx\\end{aligned}$ $\\quad $ Similarly, inverting the $\\:\\tilde{}\\:$ , we construct as well an operator $R:L^2(0,1)\\rightarrow L^2(0,1)$ such that $\\int _{0}^{1}\\tilde{s}_0(x)c_0(x)L(x)dx = \\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)RL(\\tau )d\\tau $ with $\\begin{aligned}RL(\\tau ) = \\frac{1}{2}L(\\tau )&+\\int _{\\tau }^{1}P(x,2\\tau -x)L(x)dx+\\int _{\\tau }^{1}\\tilde{H}(x,2\\tau -x)L(x)dx\\\\&+\\int _{\\tau }^{1}L(x)\\int _{2\\tau -x}^{2\\tau }P(x,2\\tau -t)\\tilde{H}(x,t){\\bf 1}_{[0,x]}(t)dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}P(x,t)\\tilde{H}(x,t-2\\tau )\\:dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}P(x,t-2\\tau )\\tilde{H}(x,t)\\:dtdx\\end{aligned}$ $\\quad $ Let us denote $B=Q+R$ .", "Then $\\begin{aligned}\\int _{0}^{1}\\big [c_0(x,z)\\tilde{s}_0(x,z)+\\tilde{c}_0(x,z)s_0(x,z)\\big ]L(x)dx&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)(R+Q)L(\\tau )d\\tau \\\\&=\\frac{1}{y_m}\\int _{0}^{1}\\sinh (2\\tau y_m)BL(\\tau )d\\tau .\\end{aligned}$ Now, let us prove the second part of the proposition.", "As the conformal factors $f$ and $\\tilde{f}$ belong to $C(A)$ , and thanks to Proposition REF , we know that $H$ and $\\tilde{H}$ are $C^1$ and uniformly bounded by a constant $C_A$ (and also are their partial derivatives).", "Moreover, it is known that, for a function $g$ that is $C^1$ on $[0,1]$ , for any $a\\in ]0,1[$ the function $G_a$ defined as $\\displaystyle G_a(\\tau )= \\int _{a}^{\\tau }g(\\tau ,x)dx$ is also $C^1$ and its derivative is $G_a^{\\prime }(\\tau )=\\int _{a}^{\\tau }\\frac{\\partial g}{\\partial \\tau }(\\tau ,x)dx + g(\\tau ,\\tau ).$ Hence $BL$ and its derivative are also bounded by some constant $C_A$ .", "$\\quad $ Thus, we have obtained: $\\bigg |\\frac{1}{y_m^2}\\int _{0}^{1}\\sinh (2\\tau y_m)BL(\\tau )d\\tau \\bigg | \\le C_A \\varepsilon \\times \\Delta (\\kappa _m)\\tilde{\\Delta }(\\kappa _m)$ Moreover $\\begin{aligned}y_m^2 e^{-2ym}\\times \\frac{1}{y_m^2}\\int _{0}^{1}\\sinh (2\\tau y_m)BL(\\tau )d\\tau &= \\frac{1}{2}\\bigg [e^{-2ym}\\int _{0}^{1}e^{2\\tau y_m}BL(\\tau )d\\tau + e^{-2ym}\\int _{0}^{1}e^{-2\\tau y_m}BL(\\tau )d\\tau \\bigg ]\\\\&=\\frac{1}{2}\\bigg [\\int _{0}^{1}e^{2(\\tau -1) y_m}BL(\\tau )d\\tau + \\int _{0}^{1}e^{-2(\\tau +1) y_m}BL(\\tau )d\\tau \\bigg ]\\\\&=\\frac{1}{2}\\bigg [\\int _{0}^{1}e^{-2\\tau y_m}BL(1-\\tau )d\\tau + \\int _{1}^{2}e^{-2\\tau y_m}BL(\\tau -1)d\\tau \\bigg ]\\end{aligned}$ and so, by multiplying (REF ) by $y_m^2e^{-2y_m}$ , one gets, for $m\\ge m_0$ : $\\begin{aligned}\\bigg |\\int _{0}^{+\\infty }e^{-2\\tau y_m}\\bigg (BL(1-\\tau ){\\bf 1}_{[0,1]}(\\tau ) + BL(\\tau -1){\\bf 1}_{[1,2]}(\\tau )\\bigg )d\\tau \\bigg | &\\le C_A \\varepsilon \\times \\big [y_m^2 e^{-2ym}\\Delta (\\kappa _m)\\tilde{\\Delta }(\\kappa _m)\\big ]\\\\&\\le C_A \\varepsilon .\\end{aligned}$" ], [ "A Hausdorf moment problem", "Let us set $g(\\tau )=BL(1-\\tau ){\\bf 1}_{[0,1]}(\\tau ) + BL(\\tau -1){\\bf 1}_{[1,2]}(\\tau )$ The change of variable $t=e^{-\\tau }$ leads to the estimates : $\\forall m\\ge m_0,\\quad \\bigg |\\int _{0}^{1}t^{2y_m-1}g(-\\ln (t))dt\\bigg |\\le C_A\\varepsilon .$ We recall that, for all $m\\in \\mathbb {N}$ , we have set $y_m=\\sqrt{\\kappa _m}$ , where $\\kappa _m=m(m+n-2)$ .", "Let us set $\\alpha =2y_{m_0}-1$ and $\\lambda _m:=2y_m-1-\\alpha $ Then, by denoting $\\displaystyle h(t)=t^\\alpha g(-\\ln (t)),$ we get: $\\bigg |\\int _{0}^{1}t^{\\lambda _m}h(t)dt\\bigg |\\le C_A\\varepsilon ,\\quad \\forall m\\in \\mathbb {N}.$ $\\quad $ Thus, we would like now to answer the following question : does the approximate knowledge of the moments of $h$ on the sequence $(\\lambda _m)_{m\\in \\mathbb {N}}$ determine $h$ up to a small error in $L^2$ norm ?", "$\\quad $ Let us fix $m\\in \\mathbb {N}$ (we will precise it later) and consider the finite real sequence : $\\Lambda _m:0= \\lambda _0<\\lambda _1<...<\\lambda _m.$ Definition 4.10 The subspace of the Müntz polynomials of degree $\\lambda _m$ is defined as : $\\mathcal {M}(\\Lambda _m)=\\lbrace P:\\: P(x)=\\sum _{k=0}^{m}a_kx^{\\lambda _k}\\rbrace .$ Definition 4.11 The $L^2$ -error of approximation from $\\mathcal {M}(\\Lambda _m)$ of a function $f\\in L^2([0,1])$ is : $E_2(f,\\Lambda _m)=\\underset{P\\in \\mathcal {M}(\\Lambda _m)}{\\inf }\\Vert f-P\\Vert _2.$ $\\quad $ $E_2(h,\\Lambda _m)$ appears in an estimate of $\\Vert h\\Vert _{2}$ given by Proposition REF .", "Thanks to the Gram-Schmidt process, we define the sequence of Müntz polynomials $\\big (L_p(x)\\big )$ as $L_0\\equiv 1$ and, for $p\\ge 1$ : $L_p(x)=\\sum _{j=0}^{p}C_{pj}x^{\\lambda _j},$ where : $C_{pj}=\\sqrt{2\\lambda _p+1}\\frac{\\prod _{r=0}^{p-1}(\\lambda _j+\\lambda _r+1)}{\\prod _{\\begin{array}{c}r=0,r\\ne j\\end{array}}^{p}(\\lambda _j-\\lambda _r)}.$ $\\quad $ Proposition 4.12 Under the assumption (REF ), we have the following estimate : We have the following estimate : $\\Vert h\\Vert _2^2\\le C_A\\varepsilon ^2\\sum _{k=0}^{m}\\bigg (\\sum _{\\ell =0}^{k}|C_{k\\ell }|\\bigg )^2 +E_2(h,\\Lambda _m)^2.$ Let us denote $\\displaystyle \\pi (h)=\\sum _{k=0}^{m}\\langle L_k,h\\rangle L_k$ the orthogonal projection of $h$ on $\\mathcal {M}(\\Lambda _m)$ .", "$\\begin{aligned}\\Vert h\\Vert ^2_2&=\\Vert \\pi (h)\\Vert ^2+ \\Vert h-\\pi (h)\\Vert ^2_2\\\\&=\\sum _{k=0}^{m}\\langle L_k,h\\rangle ^2+ E_2(\\Lambda _m,h)^2.\\end{aligned}$ As $\\displaystyle |\\langle L_k,h\\rangle | = \\bigg |\\sum _{\\ell =0}^{k}C_{k\\ell }\\underbrace{\\int _{0}^{1}x^{\\lambda _\\ell }h(x)dx}_{\\le C_A\\varepsilon } \\bigg |\\le C_A\\varepsilon \\sum _{\\ell =0}^{k}|C_{k\\ell }|$ , one gets $\\Vert h\\Vert ^2_2 \\le C_A\\varepsilon ^2\\sum _{k=0}^{m}\\bigg (\\sum _{\\ell =0}^{k}|C_{k\\ell }|\\bigg )^2+E_2(\\Lambda _m,h)^2.$ We would like to find $m(\\varepsilon )\\in \\mathbb {N}$ satisfying : $\\lim \\limits _{\\varepsilon \\rightarrow 0}m(\\varepsilon )=+\\infty $ and such that $\\displaystyle \\sum _{k=0}^{m(\\varepsilon )}\\bigg (\\sum _{\\ell =0}^{k}|C_{k\\ell }|\\bigg )^2 \\le \\frac{1}{\\varepsilon }$ , in order to obtain $\\Vert h\\Vert ^2_2\\le C_A\\varepsilon + E_2(\\Lambda _{m(\\varepsilon )},h)$ .", "Lemma 4.13 $\\quad $ For all $m\\in \\mathbb {N}$ , $\\lambda _{m+1}-\\lambda _m\\ge 2$ .", "For all $m\\in \\mathbb {N}$ , $\\displaystyle \\lambda _{m+1}-\\lambda _m= 2+O\\bigg (\\frac{1}{m}\\bigg )$ .", "$\\quad $ $\\quad $ 1.", "Let $m\\in \\mathbb {N}$ and set $a=n-2$ .", "From (REF ) we have the equivalence $\\lambda _{m+1}-\\lambda _m\\ge 2\\Leftrightarrow y_{m+1}-y_m\\ge 1$ , where $y_m=\\sqrt{m^2+am}$ .", "For $m\\in \\mathbb {N}$ , one has : $\\begin{aligned}y_{m+1}-y_m\\ge 1&\\Leftarrow \\sqrt{(m+1)^2+a(m+1)}-\\sqrt{m^2+am}\\ge 1\\\\&\\Leftrightarrow (m+1)^2+a(m+1)-m^2-am\\ge \\sqrt{(m+1)^2+a(m+1)}+\\sqrt{m^2+am}\\\\&\\Leftrightarrow 2m+1+a \\ge m+1 + \\frac{a}{2}-\\frac{a}{8(m+1)}+m+\\frac{a}{2}-\\frac{a}{8m}+o\\bigg (\\frac{1}{m}\\bigg )\\\\&\\Leftrightarrow \\frac{a}{8(m+1)}\\ge -\\frac{a}{8m}+ o\\bigg (\\frac{1}{m}\\bigg ),\\end{aligned}$ and that is true for $m$ large enough.", "We assume, without loss of generality, that it is true for all $m\\ge m_0$ .", "Hence, for all $m\\in \\mathbb {N}$ , $\\lambda _{m+1}-\\lambda _m \\ge 2$ .", "$\\quad $ 2.", "Let $m\\in \\mathbb {N}$ and $u_m=\\sqrt{\\kappa _\\ell }\\:$ for some $\\ell \\in \\mathbb {N}$ .", "Then $\\displaystyle y_{m+1}=\\sqrt{\\kappa _{m+1}}=\\sqrt{\\kappa _m}+1+O\\bigg (\\frac{1}{m}\\bigg )=y_m+1+O\\bigg (\\frac{1}{m}\\bigg ),$ so we have the result.", "Hence, there is $C>0$ such that, for all $m\\in \\mathbb {N}$ , $\\lambda _m\\le 2m+C$ .", "By setting $M_1=\\max (2,2C+1)$ , one gets : $\\prod _{r=0}^{p-1}(\\lambda _j+\\lambda _r+1)\\le \\prod _{r=0}^{p-1}(2j+2r+2C+1)\\le M_1^p \\prod _{r=0}^{p-1}(j+r+1).$ On the other hand, for all $m\\in \\mathbb {N}$ , $\\lambda _{m+1}-\\lambda _m\\ge 2$ .", "Let $m\\in \\mathbb {N}$ and $(r,j)\\in \\mathbb {N}$ such that $0\\le r,j\\le m$ , $r\\ne j$ .", "$\\begin{aligned}|\\lambda _j-\\lambda _r| &= |\\lambda _j-\\lambda _{j-1}|+|\\lambda _{j-1}-\\lambda _{j-2}|+...+|\\lambda _{r+1}-\\lambda _r|\\\\&\\ge 2|j-r|.\\end{aligned}$ Consequently: $|\\prod _{\\begin{array}{c}r=0,r\\ne j\\end{array}}^{p}(\\lambda _j-\\lambda _r)|\\ge 2^p\\bigg |\\prod _{\\begin{array}{c}r=0,r\\ne j\\end{array}}^{p}(j-r)\\bigg |$ It follows that $\\begin{aligned}|C_{pj}|&\\le \\sqrt{4p+2C+1}{\\bigg (\\frac{M_1}{2}\\bigg )^p}\\frac{\\prod _{r=0}^{p-1}|j+r+1|}{\\prod _{\\begin{array}{c}r=0,r\\ne j\\end{array}}^{p}|j-r|}\\\\&=\\sqrt{4p+2C+1}{\\bigg (\\frac{M_1}{2}\\bigg )^p}\\frac{(j+1)...(j+p)}{j(j-1)...2\\times 1\\times 2\\times ... (p-j)}\\\\&=\\sqrt{4p+2C+1}{\\bigg (\\frac{M_1}{2}\\bigg )^p}\\frac{(j+p)!}{(j!)^2(p-j)!", "}\\end{aligned}$ The multinomial formula stipulates that for any real finite sequence $(x_0,...,x_m)$ and any $n\\in \\mathbb {N}$ : $\\bigg (\\sum _{k=0}^{m}x_k\\bigg )^n=\\sum _{k_1+...+k_m=n}\\binom{n}{k_1,k_2,...,k_m}x_1^{k_1}...x_m^{k_m},$ where $\\displaystyle \\binom{n}{k_1,k_2,...,k_m}=\\frac{n!}{k_1!k_2!...k_m!", "}$ .", "$\\quad $ As $j+j+(p-j)=j+p$ , one deduces that : $\\frac{(j+p)!}{(j)!(j)!(p-j)!", "}\\le (1+1+1)^{j+p} = 3^{j+p}$ Hence (see [5] or [1], chapter 4, for similar computations) : $\\begin{aligned}\\varepsilon ^2\\sum _{k=0}^{m}\\bigg (\\sum _{\\ell =0}^{k}|C_{k\\ell }|\\bigg )^2&\\le \\varepsilon ^2\\sum _{k=0}^{m}\\bigg (\\sum _{\\ell =0}^{k}\\sqrt{4k+2C+1}\\bigg (\\frac{M_1}{2}\\bigg )^k3^{k+\\ell }\\bigg )^2\\\\&= \\varepsilon ^2\\sum _{k=0}^{m}\\bigg (\\frac{3M_1}{2}\\bigg )^{2k}(4k+2C+1)\\bigg (\\sum _{\\ell =0}^{k}3^{\\ell }\\bigg )^2\\\\&\\le \\varepsilon ^2(4m+2C+1)\\sum _{k=0}^{m}\\bigg (\\frac{3M_1}{2}\\bigg )^{2k}\\bigg (\\sum _{\\ell =0}^{k}3^{\\ell }\\bigg )^2\\\\&\\le \\varepsilon ^2(4m+2C+1)\\sum _{k=0}^{m}\\bigg (\\frac{3M_1}{2}\\bigg )^{2k}\\frac{3}{2}\\times 3^{2k}\\\\&\\le \\varepsilon ^2\\times \\frac{3}{2} (4m+2C+1)\\sum _{k=0}^{m}\\bigg (\\frac{9M_1}{2}\\bigg )^{2k}\\\\&\\le \\varepsilon ^2\\times \\frac{3}{2} (4m+2C+1)(m+1)\\bigg (\\frac{9M_1}{2}\\bigg )^{2m}\\\\&=\\varepsilon ^2 g(m)^2\\end{aligned}$ where $\\displaystyle g(t):=\\frac{3}{2}(4t+2C+1)(t+1)\\bigg (\\frac{9M_1}{2}\\bigg )^{2t}$ .", "$\\quad $ As $g$ is a strictly increasing function on $\\mathbb {R}_+$ , we can set, for $\\varepsilon $ small enough, $\\displaystyle m(\\varepsilon )=E\\bigg (g^{-1}\\bigg (\\frac{1}{\\sqrt{\\varepsilon }}\\bigg )\\bigg )$ .", "Thanks to this choice, we have $g\\big (m(\\varepsilon )\\big )\\le \\frac{1}{\\sqrt{\\varepsilon }},$ so that $\\varepsilon ^2\\sum _{k=0}^{m(\\varepsilon )}\\bigg (\\sum _{p=0}^{k}|C_{kp}|\\bigg )^2\\le \\varepsilon .$ $\\quad $ Let us now estimate $E_2(\\Lambda _m,h)$ .", "To this end, we recall some definitions.", "Definition 4.14 The index of approximation of $\\Lambda _m$ in $L^2([0,1])$ is : $\\varepsilon _2(\\Lambda _m)=\\underset{y\\ge 0}{\\max }\\bigg |\\frac{B(1+iy)}{1+iy}\\bigg |$ where $B:\\mathbb {C}\\rightarrow \\mathbb {C}$ is the Blaschke product defined as : $B(z):=B(z,\\Lambda _m)=\\prod _{k=0}^{m}\\frac{z-\\lambda _k-\\frac{1}{2}}{z+\\lambda _k+\\frac{1}{2}}$ We will take advantage of a much simpler expression of $\\varepsilon _2\\big (\\Lambda _m\\big )$ , thanks to the following Theorem ([9], p.360): Theorem 4.15 Let $\\Lambda _m:0=\\lambda _0<\\lambda _1<...<\\lambda _m$ be a finite sequence.", "Assume that $\\lambda _{k+1}-\\lambda _k\\ge 2$ for $k\\ge 0$ .", "Then : $\\varepsilon _2\\big (\\Lambda _m\\big )=\\prod _{k=0}^{m}\\frac{\\lambda _k-\\frac{1}{2}}{\\lambda _k+\\frac{3}{2}}$ Definition 4.16 For a function $f\\in L^2([0,1])$ , its $L^2$ -modulus of continuity $w(f,.", "):\\:]0,1[\\rightarrow \\mathbb {R}$ is defined as: $w(f,u)=\\underset{0\\le r\\le u}{\\mathrm {sup}}\\bigg (\\int _{0}^{1-r}|f(x+r)-f(x)|^2dx\\bigg )^{\\frac{1}{2}}.$ The introduction of the two previous concepts is motivated by the following result (cf [9], Theorem 2.7 p.352) : $\\quad $ Theorem 4.17 Let $f\\in L^2([0,1])$ .", "Then there is an universal constant $C>0$ such that : $E_2(\\Lambda _m)\\le C\\omega (f,\\varepsilon \\big (\\Lambda _m)\\big )$ Lemma 4.18 $w(h,u)\\le C_{A}u$ , $\\forall u\\in [0,1/e^2]$ .", "We write $h(x)$ as the sum of two functions with disjoint support : $h=h_{1}+h_{2},$ with : $\\displaystyle h_{1}(t)=t^\\alpha BL(-\\ln (x)-1){\\bf 1}_{[\\frac{1}{e^2},\\frac{1}{e}]}(t)$ , $\\displaystyle h_{2}(t)=t^\\alpha BL(1+\\ln (t)){\\bf 1}_{[\\frac{1}{e},1[}(t)$ .", "Thanks to the second part of Proposition REF , the function $BL$ is bounded by a constant $C_A$ so, for $i\\in [\\![1,2]\\!", "]$ , each of the function $h_{i}$ is bounded by some constant $C_{A}$ depending on $A$ .", "Moreover, $BL$ is $C^1$ on $[\\frac{1}{e^2},\\frac{1}{e}]$ and $[\\frac{1}{e},1]$ , and, for $i\\in [\\![1,2]\\!", "]$ , $h_{i}^{\\prime }$ is bounded by a constant $C_{A}$ .", "Let $x\\in [0,1/e^2]$ , $r\\in [0,x]$ .", "We have : $\\begin{aligned}\\int _{0}^{1-r}|h(t+r)-h(t)|^2dt &=\\int _{\\frac{1}{e^2}}^{\\frac{1}{e}-r^2}|h_{1}(t+r)-h_{1}(t)|^2dt + \\int _{\\frac{1}{e}-r^2}^{\\frac{1}{e}}|h_{2}(x+r)-h_{1}(t)|^2dt\\\\&+\\int _{\\frac{1}{e}}^{1-r}|h_{2}(t+r)-h_{2}(t)|^2dt\\\\&\\le \\bigg (\\frac{1}{e}-\\frac{1}{e^2}-r^2\\bigg )\\Vert h_{1}^{\\prime }\\Vert _\\infty ^2 r^2 + r^2\\bigg (\\Vert h_{2}\\Vert _\\infty +\\Vert h_{1}\\Vert _\\infty \\bigg )^2\\\\&+\\bigg (1-\\frac{1}{e}-r\\bigg )\\Vert h_{2}^{\\prime }\\Vert _\\infty ^2 r^2\\\\&\\le C_{A}r^2.\\end{aligned}$ Taking the square root and the supremum on $r$ on each side, the result is proved.", "Lemma 4.19 $\\varepsilon _2\\big (\\Lambda _m\\big )=O\\bigg (\\frac{1}{m}\\bigg ),\\qquad m\\rightarrow +\\infty .$ $\\quad $ $\\quad $ Using Theorem REF and Lemma REF , the expression of $\\varepsilon _2(\\Lambda _m)$ defined above can be written as $\\varepsilon _2(\\Lambda _m)=\\prod _{k=0}^{m}\\frac{\\lambda _k-\\frac{1}{2}}{\\lambda _k+\\frac{3}{2}}.$ $\\quad $ Recall there exists $C>0$ such that for all $m\\in \\mathbb {N}$ , $\\lambda _m\\le 4m+C$ .", "Consequently, one has : $\\begin{aligned}\\forall m\\in \\mathbb {N},\\quad \\ln \\bigg (\\prod _{k=0}^{m}\\frac{\\lambda _{k}-\\frac{1}{2}}{\\lambda _k+\\frac{3}{2}}\\bigg )&=\\ln \\bigg (\\prod _{k=0}^{m}\\bigg (1-\\frac{2}{\\lambda _k+\\frac{3}{2}}\\bigg )\\bigg )\\\\&=\\sum _{k=0}^{m}\\ln \\bigg (1-\\frac{2}{\\lambda _k+\\frac{3}{2}}\\bigg )\\bigg )\\\\&\\le -2\\sum _{k=0}^{m} \\frac{1}{\\lambda _k+\\frac{3}{2}}\\\\&\\le -2\\sum _{k=0}^{m} \\frac{1}{2k+C+\\frac{3}{2}}.\\end{aligned}$ But $\\displaystyle -2\\sum _{k=0}^{m} \\frac{1}{2k+C+\\frac{3}{2}}\\underset{m\\rightarrow +\\infty }{=}-\\ln (m)+O(1)$ .", "Hence $\\varepsilon _2\\big (\\Lambda _m\\big )=O\\bigg (\\frac{1}{m}\\bigg ).$ Hence, as $\\displaystyle \\varepsilon _2(\\Lambda _{m(\\varepsilon )})\\in [0,1/e^2]$ for $\\varepsilon $ small enough, we get thanks to Lemma REF and Theorem REF : $\\displaystyle E(h,\\Lambda _{m(\\varepsilon )})_2\\le C_{A}\\varepsilon _2(\\Lambda _{m(\\varepsilon )})$ .", "To sum up, we have shown that : $\\Vert h\\Vert _2^2\\le C_{A}\\bigg (\\varepsilon +\\varepsilon _2(\\Lambda _{m(\\varepsilon )})^2\\bigg ).$ Now, we know that $\\displaystyle \\varepsilon _2(\\Lambda _{m(\\varepsilon )})^2\\le \\frac{C_A}{m(\\varepsilon )^2}$ .", "By virtue of the double inequality $\\displaystyle \\frac{1}{\\sqrt{\\varepsilon }}+o(1)\\le g\\big (m(\\varepsilon )\\big )\\le \\frac{1}{\\sqrt{\\varepsilon }}$ one has $\\displaystyle \\frac{1}{2}\\ln \\bigg (\\frac{1}{\\varepsilon }\\bigg )\\underset{\\varepsilon \\rightarrow 0}{\\sim }\\ln \\bigg (g\\big (m(\\varepsilon )\\big )\\bigg )\\underset{\\varepsilon \\rightarrow 0}{\\sim } C_Am(\\varepsilon )$ .", "Hence (for another $C_A>0$ ) : $\\displaystyle \\frac{1}{m(\\varepsilon )}\\le \\frac{C_A}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}$ .", "Consequently : $\\Vert h\\Vert _2^2\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )^2}.$ $\\quad $ Since $h_{1}$ and $h_{2}$ have disjoint support, we have $\\Vert h\\Vert _2^2=\\Vert h_{1}\\Vert _2^2+\\Vert h_{2}\\Vert _2^2.$ In particular $\\Vert h_{2}\\Vert _2^2\\le \\Vert h\\Vert _2^2$ But as $\\Vert h_{2}\\Vert _2^2 = \\int _{\\frac{1}{e}}^{1}t^{2\\alpha }\\bigg |BL\\big (1+\\ln (t)\\big )\\bigg |^2dt$ we get $\\int _{\\frac{1}{e}}^{1}t^{2\\alpha +1}\\bigg | BL\\big (1+\\ln (t)\\big )\\bigg |^2\\frac{dt}{t}\\le \\frac{C_{A,a}}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )^2}$ Hence, as we integrate over $\\displaystyle \\bigg [\\frac{1}{e^{1}},1\\bigg ]$ , the term $t^{2\\alpha +1}$ is minorated by $(1/e)^{(2\\alpha +1)}$ .", "By returning to the $\\tau $ coordinate, we obtain : $\\Vert BL(1-\\tau )\\Vert _{L^2([0,1])}\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )},$ and then $\\Vert BL\\Vert _{L^2([0,1])}\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.$ $\\quad $" ], [ "Invertibility of the $B$ operator", "$\\quad $ Now, we want to prove that $B:L^2(0,1)\\rightarrow L^2(0,1)$ is invertible and that its inverse is bounded with respect to $C_A$ .", "We can write : $B=I+C$ where $\\displaystyle Ch(\\tau )=\\int _{\\tau }^{1}H_1(x,\\tau )h(x)dx$ , with : $\\begin{aligned}H_1(x,\\tau )&= \\tilde{P}(x,2\\tau -x)+H(x,2\\tau -x)+\\int _{2\\tau -x}^{2\\tau }\\tilde{P}(x,2\\tau -t)H(x,t){\\bf 1}_{[0,x]}(t)dt\\\\&+\\int _{2\\tau }^{x}\\tilde{P}(x,t)H(x,t-2\\tau )\\:dt{\\bf 1}_{[2\\tau ,1]}(x)+\\int _{2\\tau }^{x}\\tilde{P}(x,t-2\\tau )H(x,t)\\:dt{\\bf 1}_{[2\\tau ,1]}(x)\\\\&+P(x,2\\tau -x)+\\tilde{H}(x,2\\tau -x)+\\int _{2\\tau -x}^{2\\tau }P(x,2\\tau -t)\\tilde{H}(x,t){\\bf 1}_{[0,x]}(t)dt\\\\&+\\int _{2\\tau }^{x}P(x,t)\\tilde{H}(x,t-2\\tau )\\:dt{\\bf 1}_{[2\\tau ,1]}(x)+\\int _{2\\tau }^{x}P(x,t-2\\tau )\\tilde{H}(x,t)\\:dt{\\bf 1}_{[2\\tau ,1]}(x).\\end{aligned}$ $\\quad $ Lemma 4.20 There is a constant $C_A>0$ such that, for all $h$ in $L^2(0,1)$ : $\\displaystyle \\forall n\\in \\mathbb {N^*},\\:\\:\\forall \\tau \\in [0,1],\\quad |C^nh(\\tau )|\\le C_A \\frac{\\big ((1-\\tau )\\Vert H_1\\Vert _{L^\\infty }\\big )^{n-1}}{(n-1)!", "}\\Vert h\\Vert _{L^2([0,1])}$ $\\quad $ $\\quad $ By induction : $\\quad $ $\\bullet $ From the estimates of Proposition REF , $H$ , $\\tilde{H}$ and $H_1$ are bounded by a constant $C_A$ .", "Using the triangle inequality and the Cauchy-Schwarz inequality, one immediately gets : $\\begin{aligned}|Ch(\\tau )|&\\le C_A\\int _{\\tau }^{1}|h(x)|dx\\le C_A(1-\\tau )\\Vert h\\Vert _{L^2([0,1])}\\le C_A\\Vert h\\Vert _{L^2([0,1])}\\end{aligned}$ $\\bullet $ Assume it is true for some $n\\in \\mathbb {N}^*$ .", "Then : $\\begin{aligned}|C^{n+1}h(\\tau )|=\\bigg |\\int _{\\tau }^{1}H_1(x,t)C^nh(x)dx\\bigg | &\\le \\int _{\\tau }^{1}\\Vert H_1\\Vert _\\infty C_A\\frac{(1-x)^{n-1}\\Vert H_1\\Vert _\\infty ^{n-1}}{(n-1)!", "}\\Vert h\\Vert _{L^2(0,1)}dx\\\\&=C_A\\frac{\\Vert H_1\\Vert _\\infty ^n}{(n-1)!", "}\\Vert h\\Vert _{L^2(0,1)}\\int _{\\tau }^{1}(1-x)^{n-1}dx\\\\&=C_A \\frac{\\big ((1-\\tau )\\Vert H_1\\Vert _{\\infty }\\big )^{n}}{n!", "}\\Vert h\\Vert _{L^2([0,1])}\\end{aligned}$ Thus $\\:\\:\\displaystyle \\Vert C^n\\Vert \\le C_A \\frac{\\big ((1-\\tau )\\Vert H_1\\Vert _{\\infty }\\big )^{n-1}}{(n-1)!", "}\\:\\:$ for all $n\\in \\mathbb {N^*}$ .", "It follows that the serie $\\sum (-1)^nC^n$ is convergent.", "Consequently $B$ is invertible, $\\displaystyle B^{-1}=\\sum _{n=0}^{+\\infty }(-1)^nC^n$ and : $\\Vert B^{-1}\\Vert \\le C_A.$ Hence : $\\Vert q-\\tilde{q}\\Vert _{L^2(0,1)}=\\Vert L\\Vert _{L^2(0,1)}\\le \\Vert B^{-1}\\Vert \\Vert BL\\Vert _{L^2(0,1)} \\le C_{A} \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}$ and the proof of Theorem REF is complete.", "$\\quad $ Let us prove Corollary REF .", "Let $s_1,s_2\\ge 0$ and $\\theta \\in (0,1)$ .", "Using the Gagliardo-Nirenberg inequalities (see [3]), one can write $\\Vert g\\Vert _{H^s(0,1)}\\le \\Vert g\\Vert _{H^{s_1}(0,1)}^\\theta \\Vert g\\Vert _{H^{s_2}(0,1)}^{1-\\theta }$ for every $g\\in H^{s_1}(0,1)\\cap H^{s_2}(0,1)$ and $s=\\theta s_1+(1-\\theta )s_2$ .", "As $f$ and $\\tilde{f}$ belong to $C(A)$ then $q-\\tilde{q}$ belong to $H^2(0,1)$ and $\\Vert q-\\tilde{q}\\Vert _{H^2(0,1)}\\le C_A$ .", "Hence, for $s_1=0$ and $s_2=2$ , we have: $\\begin{aligned}\\Vert q-\\tilde{q}\\Vert _{H^s(0,1)}&\\le \\Vert q-\\tilde{q}\\Vert _{L^2(0,1)}^\\theta \\Vert q-\\tilde{q}\\Vert _{H^{2}(0,1)}^{1-\\theta }\\\\&\\le C_A^{1-\\theta }\\Vert q-\\tilde{q}\\Vert _{L^2(0,1)}^\\theta \\\\&\\le C_A\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )^\\theta }\\end{aligned}$ with $\\displaystyle \\theta =\\frac{2-s}{2}$ .", "Using the Sobolev embedding $H^1(0,1)\\hookrightarrow C^0(0,1)$ with $\\Vert .\\Vert _{\\infty }\\le 2 \\Vert .\\Vert _{H^1(0,1)}$ , one gets (for $s=1$ and $\\theta =1/2$ ): $\\begin{aligned}\\Vert q-\\tilde{q}\\Vert _{\\infty }\\le 2\\Vert q-\\tilde{q}\\Vert _{H^1(0,1)}\\le C_A \\sqrt{\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}}\\end{aligned}$" ], [ "Uniform estimate of the conformal factors", "Now we give the proof of Corollary REF .", "Assume that $n\\ge 3$ , $\\omega =0$ and let us set $F=f^{n-2}$ .", "We can write $\\displaystyle q=\\frac{F^{\\prime \\prime }}{F}$ and then $(\\tilde{F}F^{\\prime }-\\tilde{F}^{\\prime }F)^{\\prime }(t)=\\tilde{F}F(q-\\tilde{q})(t).$ For all $t\\in [0,1]$ , we have : $\\begin{aligned}\\tilde{F}(t)F^{\\prime }(t)-\\tilde{F}^{\\prime }(t)F(t)&=(n-2)\\tilde{f}^{n-2}f^{n-3}(t)f^{\\prime }(t)-(d-2)\\tilde{f}^{n-3}f^{n-2}(t)\\tilde{f}^{\\prime }(t)\\\\&=(n-2)f^{n-3}(t)\\tilde{f}^{n-3}(t)\\bigg (\\tilde{f}(t)f^{\\prime }(t)-f(t)\\tilde{f}^{\\prime }(t)\\bigg )\\end{aligned}$ Assume that for all $t$ in $[0,1]$ , $\\tilde{f}(t)f^{\\prime }(t)-f(t)\\tilde{f}^{\\prime }(t)\\ne 0$ , for example $\\tilde{f}(t)f^{\\prime }(t)>f(t)\\tilde{f}(t)$ .", "Then : $\\frac{f^{\\prime }(t)}{f(t)}>\\frac{\\tilde{f}^{\\prime }(t)}{\\tilde{f}(t)}.$ Then, by integrating between 0 and 1, one gets : $\\ln \\big (f(1)\\big )-\\ln \\big (f(0)\\big )>\\ln \\big (\\tilde{f}(1)\\big )-\\ln \\big (\\tilde{f}(0)\\big ).$ and this is not true as $f(0)=f(1)$ and $\\tilde{f}(0)=\\tilde{f}(1)$ .", "Consequently, there is $x_0\\in [0,1]$ such that $\\big (\\tilde{f}f^{\\prime }-f\\tilde{f}^{\\prime }\\big )(x_0) =0$ .", "Setting $G(x)=(\\tilde{F}F^{\\prime }-\\tilde{F}^{\\prime }F)(x)$ , we have : $\\forall x\\in [0,1],\\quad G(x)=\\int _{x_0}^{x}\\tilde{F}F(q-\\tilde{q})(t)dt.$ From the $L^2$ estimate previously established on $q-\\tilde{q}$ , one has : $\\begin{aligned}\\forall x\\in [0,1],\\quad |G(x)|&\\le \\sqrt{|x-x_0|}C_A\\Vert q-\\tilde{q}\\Vert _2 \\\\&\\le C_A \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}\\end{aligned}$ Hence : $\\bigg |\\bigg (\\frac{F}{\\tilde{F}}\\bigg )^{\\prime }(x)\\bigg |=\\bigg |\\frac{G(x)}{\\tilde{F}(x)^2}\\bigg |\\le C_A \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )},$ and by integrating betwwen 0 and $x$ : $\\begin{aligned}\\bigg |\\frac{F(x)}{\\tilde{F}(x)}-1\\bigg |=\\bigg |\\int _{0}^{x}\\bigg (\\frac{F}{\\tilde{F}}\\bigg )^{\\prime }(t)dt\\bigg |\\le \\int _{0}^{1}\\bigg |\\frac{G(t)}{\\tilde{F}(t)^2}\\bigg |dt \\le C_A \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.\\end{aligned}$ and this last inequality leads to the estimate : $\\forall x\\in [0,1],\\quad |f^{n-2}(x)-\\tilde{f}^{n-2}(x)|\\le C_A \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.$ Setting $k=n-2$ , thanks to the relation $\\displaystyle a^k-b^k=(a-b)\\sum _{j=0}^{k}a^jb^{k-j}$ , we get at last : $\\forall x\\in [0,1],\\quad |f(x)-\\tilde{f}(x)|\\le C_A \\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.$ $\\quad $" ], [ "About the Calderón problem", "Now, we prove Theorem REF .", "For $s\\in \\mathbb {R}$ , $H^{s}(\\partial M)$ can be defined as $H^{s}(\\partial M)=\\bigg \\lbrace \\psi \\in \\mathcal {D}^{\\prime }(\\partial M),\\:\\: \\psi =\\sum _{m\\ge 0}\\begin{pmatrix}\\psi _m^1\\\\ \\psi _m^2\\end{pmatrix}\\otimes Y_m, \\quad \\sum _{m\\ge 0}(1+\\mu _m)^s\\bigg (|\\psi _m^1|^2+|\\psi _m^2|^2\\bigg ) <\\infty \\bigg \\rbrace .$ Recall that we have denoted $\\mathcal {B}(H^{1/2}(\\partial M))$ the set of bounded operators from $H^{1/2}(\\partial M)$ to $H^{1/2}(\\partial M)$ and equipped $\\mathcal {B}(H^{1/2}(\\partial M))$ with the norm $\\Vert F\\Vert _*=\\sup _{\\psi \\in H^{1/2}(\\partial M)\\backslash \\lbrace 0\\rbrace }\\frac{\\Vert F\\psi \\Vert _{H^{1/2}}}{\\Vert \\psi \\Vert _{H^{1/2}}}.$ Lemma 5.1 We have the equivalence : $\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\in \\mathcal {B}(H^{1/2}(\\partial M))\\Leftrightarrow \\left\\lbrace \\begin{aligned}&f(0)=\\tilde{f}(0)\\\\&f(1)=\\tilde{f}(1).\\end{aligned}\\right.$ Let us set $\\displaystyle C_0=\\frac{1}{4\\sqrt{f(0)}}\\frac{h^{\\prime }(0)}{h(0)},\\:$ $\\quad \\displaystyle C_1=\\frac{1}{4\\sqrt{f(1)}}\\frac{h^{\\prime }(1)}{h(1)},\\:$ $\\quad \\displaystyle A_0=\\frac{1}{f(0)}-\\frac{1}{\\tilde{f}(0)}\\quad $ and $\\quad \\displaystyle A_1=\\frac{1}{f(1)}-\\frac{1}{\\tilde{f}(1)}$ .", "For $m\\ge 0$ , one has, using the block diagonal representation of $\\Lambda _g(\\omega )$ and the asymptotics of $M(\\mu _m)$ and $N(\\mu _m)$ given in Theorem REF and Corollary REF : $\\begin{aligned}\\Lambda _g^m(\\omega )-\\Lambda _{\\tilde{g}}^m(\\omega )&=\\begin{pmatrix}\\frac{\\tilde{M}(\\mu _m)}{\\sqrt{\\tilde{f}(0)}}-\\frac{M(\\mu _m)}{\\sqrt{f(0)}}+C_0-\\tilde{C}_0&O\\big (e^{-2\\mu _m}\\big )\\\\O\\big (e^{-2\\mu _m}\\big )&\\frac{\\tilde{N}(\\mu _m)}{\\sqrt{\\tilde{f}(1)}}-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+\\tilde{C}_1-C_1\\end{pmatrix}\\\\&=\\begin{pmatrix}A_0\\sqrt{\\mu _m}+(C_0-\\tilde{C}_0)&0\\\\0&A_1\\sqrt{\\mu _m}+(\\tilde{C}_1-C_1)\\end{pmatrix}+ \\begin{pmatrix}O\\bigg (\\frac{1}{\\sqrt{\\mu _m}}\\bigg )&O\\big (e^{-2\\mu _m}\\big )\\\\O\\big (e^{-2\\mu _m}\\big )&O\\bigg (\\frac{1}{\\sqrt{\\mu _m}}\\bigg )\\end{pmatrix}\\end{aligned}$ Hence, for any $\\big (\\psi _m^1,\\psi _m^2\\big )\\in \\mathbb {R}^2$ : $\\big (\\Lambda _g^m(\\omega )-\\Lambda _{\\tilde{g}}^m(\\omega )\\big )\\begin{pmatrix}\\psi _m^1\\\\ \\psi _m^2\\end{pmatrix}=\\sqrt{\\mu _m}\\begin{pmatrix}A_0\\psi ^1_m\\\\A_1\\psi ^2_m\\end{pmatrix}+\\begin{pmatrix}(C_0-\\tilde{C}_0)\\psi ^1_m\\\\(\\tilde{C}_1-C_1)\\psi ^2_m\\end{pmatrix}+O\\bigg (\\frac{\\psi ^1_m+\\psi ^2_m}{\\sqrt{\\mu _m}}\\bigg )$ For $\\displaystyle \\psi =\\sum _{m\\ge 0}\\begin{pmatrix}\\psi _m^1\\\\ \\psi _m^2\\end{pmatrix}\\otimes Y_m\\in H^{1/2}(\\partial M)$ , one has $\\begin{aligned}\\Vert \\big (\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\big )\\psi \\Vert ^2_{H^{1/2}(\\partial M)}&=\\sum _{m\\ge 0}(1+\\mu _m)^{1/2}\\mu _m\\bigg (A_0^2|\\psi _m^1|^2+A_1^2|\\psi _m^2|^2\\bigg )\\\\&+\\sum _{m\\ge 0}2(1+\\mu _m)^{1/2}\\sqrt{\\mu _m}\\bigg (|A_0(C_0-\\tilde{C}_0)||\\psi _m^1|^2+|A_1(\\tilde{C}_1-C_1)||\\psi _m^2|^2\\bigg )\\hspace{56.9055pt}\\\\&+\\sum _{m\\ge 0}(1+\\mu _m)^{1/2}O\\big (|\\psi _m^1|^2 + |\\psi _m^2|^2 \\big )\\end{aligned}$ Then $\\Vert \\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\Vert _*<\\infty \\Leftrightarrow \\left\\lbrace \\begin{aligned}&A_0=0\\\\&A_1=0\\end{aligned}\\right.\\Leftrightarrow \\left\\lbrace \\begin{aligned}&\\tilde{f}(0)=\\tilde{f}(0)\\\\&\\tilde{f}(1)=\\tilde{f}(1).\\end{aligned}\\right.$ Under the assumptions of Theorem REF , the following estimate holds: Proposition 5.2 Let $\\varepsilon >0$ .", "Assume that $\\Vert \\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\Vert _*\\le \\varepsilon $ .", "There is $C_A>0$ such that : $\\forall m\\in \\mathbb {N},\\quad \\bigg |N(\\kappa _m)-\\tilde{N}(\\kappa _m)\\bigg |\\le C_A\\varepsilon .$ For $m\\in \\mathbb {N}$ , consider $\\displaystyle \\psi _m=\\begin{pmatrix}0\\\\1\\end{pmatrix}\\otimes Y_m \\in H^{1/2}(\\partial M)$ .", "One has : $\\begin{aligned}\\big (\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\big )\\psi _m &= \\big (\\Lambda _g^m(\\omega )-\\Lambda _{\\tilde{g}}^m(\\omega )\\big ) \\begin{pmatrix}0\\\\1\\end{pmatrix}\\otimes Y_m\\\\ &= \\displaystyle \\begin{pmatrix}0&\\frac{1}{\\sqrt{f(0)}}\\frac{h^{1/4}(1)}{h^{1/4}(0)}\\big (\\frac{1}{\\tilde{\\Delta }(\\mu _m)}-\\frac{1}{\\Delta (\\mu _m)}\\big )\\\\0&\\bigg (\\frac{\\tilde{N}(\\mu _m)}{\\sqrt{f(1)}}-\\frac{N(\\mu _m)}{\\sqrt{f(1)}}\\bigg )+(\\tilde{C}_1-C_1)\\end{pmatrix}\\otimes Y_m\\end{aligned}$ Then $\\begin{aligned}\\Vert \\big (\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\big )\\psi _m\\Vert ^2_{H^{1/2}(\\partial M)} = (\\mu _m+1)^{1/2}\\bigg [\\bigg (\\frac{\\tilde{N}(\\mu _m)}{\\sqrt{f(1)}}-&\\frac{N(\\mu _m)}{\\sqrt{f(1)}}+(\\tilde{C}_1-C_1)\\bigg )^2\\\\&+\\frac{1}{f(0)}\\frac{h^{1/2}(1)}{h^{1/2}(0)}\\bigg (\\frac{1}{\\tilde{\\Delta }(\\mu _m)}-\\frac{1}{\\Delta (\\mu _m)}\\bigg )^2\\bigg ].\\end{aligned}$ so, for all $m\\ge 0$ : $\\begin{aligned}(\\mu _m+1)^{1/2}\\bigg |\\frac{1}{\\sqrt{f(1)}}\\big (\\tilde{N}(\\mu _m)-N(\\mu _m)\\big )+(\\tilde{C}_1-C_1)\\bigg |^2 &\\le \\Vert \\big (\\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega \\big )\\psi _m\\Vert ^2_{H^{1/2}(\\partial M)}\\\\&\\le \\Vert \\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\Vert _*^2 \\Vert \\psi _m\\Vert ^2_{H^{1/2}(\\partial M)}\\\\&= \\Vert \\Lambda _g(\\omega )-\\Lambda _{\\tilde{g}}(\\omega )\\Vert ^2_*(\\mu _m+1)^{1/2} \\\\&\\le \\varepsilon ^2(\\mu _m+1)^{1/2}.\\end{aligned} $ Hence $\\bigg |\\frac{1}{\\sqrt{f(1)}}\\big (\\tilde{N}(\\mu _m)-N(\\mu _m)\\big )+(\\tilde{C}_1-C_1)\\bigg |\\le \\varepsilon .$ Using the asymptotic $N(\\mu _m)=-\\mu _m+o(1)$ , we deduce from (REF ) that $|\\tilde{C}_1-C_1|\\le \\varepsilon $ and then that there is $C_A>0$ such that, for all $m\\in \\mathbb {N}$ : $\\bigg |N(\\mu _m)-\\tilde{N}(\\mu _m)\\bigg |\\le C_A\\:\\varepsilon .$ As in Lemma REF , one gets an integral relation between $N(z)-\\tilde{N}(z)$ and $q-\\tilde{q}$ : Lemma 5.3 The following integral relation holds: $\\big (N(z)-\\tilde{N}(z)\\big )\\Delta (z)\\tilde{\\Delta }(z)=\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )s_0(x,z)\\tilde{s}_0(x,z)dx$ Let us define $\\displaystyle \\theta :x\\mapsto s_0(x,z)\\tilde{s_0}^{\\prime }(x,z)-s_0^{\\prime }(x,z)\\tilde{s}_0(x,z)$ .", "Then : $\\begin{aligned}\\theta ^{\\prime }(x)&=\\big (\\tilde{q}(x)-q(x)\\big )s_0(x,z)\\tilde{s}_0(x,z)\\end{aligned}$ By integrating between 0 and 1, one gets: $s_0^{\\prime }(1,z)\\tilde{s}_0(1,z)-s_0(1,z)\\tilde{s}_0^{\\prime }(1,z)=\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )s_0(x,z)\\tilde{s}_0(x,z)dx$ As $s_0^{\\prime }(1,z)=N(z)\\Delta (z)$ and $s_0(1,z)=\\Delta (z)$ , one gets for all $z\\in \\mathbb {C}\\backslash \\mathcal {P}$ : $\\big (N(z)-\\tilde{N}(z)\\big )\\Delta (z)\\tilde{\\Delta }(z)=\\int _{0}^{1}\\big (q(x)-\\tilde{q}(x)\\big )s_0(x,z)\\tilde{s}_0(x,z)dx.$ Just as in Section 4, let us extend on $[-1,0]$ $q$ and $\\tilde{q}$ into even functions and denote $L(x)=q(x)-\\tilde{q}(x)$ .", "We recall that for all $m\\in \\mathbb {N}$ , we have set $y_m=\\sqrt{\\kappa _m}$ .", "$\\quad $ We will take advantage of this representation to write in another way the equalities $\\begin{aligned}\\big (N(\\kappa _m)-\\tilde{N}(\\kappa _m)\\big )\\Delta (\\kappa _m)\\tilde{\\Delta }(\\kappa _m)&=\\int _{0}^{1}(q(x)-\\tilde{q}(x)s_0(x,\\kappa _m)\\tilde{s}_0(x,\\kappa _m)dx.\\\\\\end{aligned}$ Proposition 5.4 There is an operator $D:L^2([0,1]) \\rightarrow L^2([0,1])$ such that : For all $m\\in \\mathbb {N}$ , $\\displaystyle \\big (N(\\kappa _m)-\\tilde{N}(\\kappa _m)\\big )s_0(1,\\kappa _m)\\tilde{s}_0(1,\\kappa _m)=\\frac{1}{y_m^2}\\int _{0}^{1}\\cosh (2\\tau y_m)DL(\\tau )d\\tau -\\frac{1}{y_m^2}\\int _{0}^{1}L(\\tau )d\\tau .$ The function $\\tau \\mapsto DL(\\tau )$ is $C^1$ on $[0,1]$ and $DL$ and $(DL)^{\\prime }$ are uniformly bounded by a constant $C_A$ .", "Using the same calculations as in Proposition REF together with the representation formula for $s_0$ $s_0(x,\\kappa _m)=\\frac{\\sinh (y_m x)}{y_m}+\\int _{0}^{x}H(x,t)\\frac{\\sinh (y_mt)}{y_m}dt$ one can prove that the operator $D$ is given by $\\begin{aligned}DL(\\tau )=L(\\tau )+\\int _{\\tau }^{1}\\tilde{H}(x,2\\tau -x)L(x)dx &+\\int _{\\tau }^{1}H(x,2\\tau -x)L(x)dx\\\\&+\\int _{\\tau }^{1}L(x)\\int _{2\\tau -x}^{2\\tau }\\tilde{H}(x,2\\tau -t)H(x,t){\\bf 1}_{[0,x]}(t)dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{H}(x,t)H(x,t-2\\tau )\\:dtdx\\\\&+\\int _{2\\tau }^{1}L(x)\\int _{2\\tau }^{x}\\tilde{H}(x,t-2\\tau )H(x,t)\\:dtdx\\\\\\end{aligned}$ and so that $DL$ and its derivative are bounded by some constant $C_A>0$ .", "For all $m\\in \\mathbb {N}$ , one has $\\begin{aligned}(N(\\kappa _m)-\\tilde{N}(\\kappa _m))s_0(1,\\kappa _m)\\tilde{s}_0(1,\\kappa _m)&=\\frac{1}{y_m^2}\\int _{0}^{1}\\cosh (2\\tau y_m)DL(\\tau )d\\tau -\\frac{1}{y_m^2}\\int _{0}^{1}L(\\tau )d\\tau \\\\&=\\frac{1}{2y_m^2}\\int _{0}^{1}e^{2\\tau y_m}DL(\\tau )d\\tau +\\frac{1}{2y_m^2}\\int _{0}^{1}e^{-2\\tau y_m}DL(\\tau )d\\tau \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad -\\frac{1}{y_m^2}\\int _{0}^{1}L(\\tau )d\\tau .\\end{aligned}$ Hence, by multiplying both sides by $2y_m^2e^{-2y_m}$ , one has : $\\begin{aligned}2y_m^2e^{-2y_m}\\big (N(\\kappa _m)-\\tilde{N}(\\kappa _m)\\big )s_0(1,\\kappa _m)\\tilde{s}_0(1,\\kappa _m)&=\\int _{0}^{1}e^{2 y_m(\\tau -1)}DL(\\tau )d\\tau +\\int _{0}^{1}e^{-2y_m(\\tau +1)}DL(\\tau )d\\tau \\\\&\\quad \\quad -2e^{-2y_m}\\int _{0}^{1}L(\\tau )d\\tau \\end{aligned}$ The asymptotic $s_0(1,\\kappa _m)\\sim \\frac{e^{y_m}}{y_m},\\qquad m\\rightarrow +\\infty ,$ ensures that $y_m^2e^{-2y_m}s_0(1,\\kappa _m)\\tilde{s}_0(1,\\kappa _m)$ is bounded uniformly in $m$ .", "Moreover, by hypothesis: $\\bigg |\\int _{0}^{1}L(\\tau )d\\tau \\bigg |\\le \\varepsilon $ so $\\bigg |\\int _{0}^{1}e^{2 y_m(\\tau -1)}DL(\\tau )d\\tau +\\int _{0}^{1}e^{-2y_m(\\tau +1)}DL(\\tau )d\\tau \\bigg |\\le C_A\\varepsilon .$ We write : $\\quad $ $\\bullet $ $\\displaystyle \\int _{0}^{1}e^{2y_m(\\tau -1)}DL(\\tau )d\\tau = \\int _{0}^{1}e^{-2\\tau y_m}DL(1-\\tau )d\\tau $ , $\\bullet $ $\\displaystyle \\int _{0}^{1}e^{-2y_m(t+1)}DL(\\tau )d\\tau =\\int _{1}^{2}e^{-2\\tau y_m}DL(\\tau -1)d\\tau $ , $\\quad $ Setting $\\displaystyle RL(\\tau )=DL(1-\\tau ){\\bf 1}_{[0,1]}(\\tau )+DL(\\tau -1){\\bf 1}_{[1,2]}(\\tau )$ one has for all $m\\in \\mathbb {N}$ $\\bigg |\\int _{0}^{+\\infty }e^{-2\\tau y_m}RL(\\tau )d\\tau \\bigg |\\le C_A\\varepsilon $ By the change of variable $\\tau = -\\ln (t)$ , we obtain the moment problem : $\\forall m\\in \\mathbb {N},\\quad \\bigg |\\int _{0}^{1}t^{2 y_m}RL(-\\ln (t))dt\\bigg |\\le C_A\\varepsilon $ Using the same technique as in section 4.4.1, we prove the stability estimate $\\Vert DL\\Vert _{L^2([0,1])}\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.$ But $D$ can be written as (see section 4.4.2) $D=I+C$ with, for all $n\\ge 1$ , $\\displaystyle \\Vert C^n\\Vert \\le \\frac{\\big (C_A(1-\\tau )\\big )^{n-1}}{(n-1)!", "}$ .", "Consequently $B$ is invertible and its inverse is bounded by some constant $C_A>0$ .", "Hence $\\begin{aligned}\\Vert q-\\tilde{q}\\Vert _{2}&\\le \\Vert D^{-1}\\Vert \\Vert DL\\Vert _{L^2([0,1])}\\\\&\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.\\end{aligned}$ At last, if $\\omega =0$ and $n\\ge 3$ , we deduce as previously that $\\begin{aligned}\\Vert f-\\tilde{f}\\Vert _{\\infty }&\\le C_{A}\\frac{1}{\\ln \\big (\\frac{1}{\\varepsilon }\\big )}.\\end{aligned}$ $\\quad $ $\\quad $ Aknowledgements.", "The author would like to deeply thank Thierry Daudé and François Nicoleau for their encouragements, helpful discussions and careful reading." ] ]

[ [ "Human Recognition Using Face in Computed Tomography" ], [ "Abstract With the mushrooming use of computed tomography (CT) images in clinical decision making, management of CT data becomes increasingly difficult.", "From the patient identification perspective, using the standard DICOM tag to track patient information is challenged by issues such as misspelling, lost file, site variation, etc.", "In this paper, we explore the feasibility of leveraging the faces in 3D CT images as biometric features.", "Specifically, we propose an automatic processing pipeline that first detects facial landmarks in 3D for ROI extraction and then generates aligned 2D depth images, which are used for automatic recognition.", "To boost the recognition performance, we employ transfer learning to reduce the data sparsity issue and to introduce a group sampling strategy to increase inter-class discrimination when training the recognition network.", "Our proposed method is capable of capturing underlying identity characteristics in medical images while reducing memory consumption.", "To test its effectiveness, we curate 600 3D CT images of 280 patients from multiple sources for performance evaluation.", "Experimental results demonstrate that our method achieves a 1:56 identification accuracy of 92.53% and a 1:1 verification accuracy of 96.12%, outperforming other competing approaches." ], [ "Introduction", "Computed tomography (CT) has become an essential imaging modality for medical imaging since its introduction into clinical practice in 1973 [1].", "It can produce tomographic images combining many X-ray measurements taken from different angles and thus provide internal body structure and pathological characteristics.", "It has often been used for disease screening of various diseases in head, brain, lung heart, etc.", "The significant use of CT in clinical diagnosis also brings challenges in CT image management.", "The CT images may be lost partially or corrupted due to erroneous operations.", "For example, the CT images are usually stored in the form of Digital Imaging and Communications in Medicine (DICOM), which can contain not only the CT data, but also the users' name and ID.", "When there are input errors with the name or ID information, or these fields get corrupted, the correspondence between the multiple CT scans may no longer be reliable, leading to difficulties when the doctors need to review all the historical CT scans.", "In addition, people may go to more than one hospital in order to get a confirmed diagnosis result.", "Doctors need to confirm whether the CT images obtained in other hospitals belong to the same patient.", "There is an increasing demand for automatically associating different CT scans to the corresponding subjects, so that the 3D CT images are identified, registered, and archived.", "Exploring biometrics in medical images has already attracted some research interest [2], [3].", "Medical biometrics such as electrocardiogram (ECG), electroencephalogram (EEG), or blood pressure signals have been considered to be informative modalities for the next generation of biometric characteristics [2].", "In addition, exploration of biometric features in medical images is able to give assistance to the protection of medical data and provide references for improving biometric security system.", "For example, ECG has been employed in biometric applications [3].", "A biometric authentication algorithm by utilizing ECG is proposed to unlock the mobile devices [4].", "To the best of our knowledge, our work is the first attempt to discover a biometric identifier from 3D CT images.", "The challenge of our work is to capture the most subject discriminative features from 3D CT images.", "CT images are usually stacked by a series of slices, which can be represented by a volume that is a collection of voxels.", "We note that the number of slices, slice spacing, and voxel resolution of different CT images are generally different, which can cause big diversities of CT images.", "Moreover, 3D CT images are susceptible to a number of artifacts, such as patient movement [5], representation method [6] and radiation dose [7].", "To achieve the best performance, we focus on the face area in CT, which occupies the majority of CT images, especially in head and neck CT.", "Furthermore, face recognition has been the prominent and increasingly mature biometric technique for identity authentication [8].", "Thus, we utilize the 3D CT face as our biometrics.", "We investigate existing 3D data processing methods.", "3D volumetric data has endowed a variety of representations and corresponding learning methods.", "For example, 3D convolutional neural networks (CNNs) [9], [10] are proposed to directly deal with volumetric data.", "In [9], V-Net, a special type of CNN, is proposed for the 3D clinical prostate magnetic resonance imaging (MRI) data for specific segmentation tasks.", "However, such methods usually require unifying input data (i.e., a fixed number of voxels and volume size), which may not be suitable in dealing with diverse 3D CT images.", "Furthermore, due to the computational cost of 3D convolution and data sparsity, volume representation is limited by its resolution.", "Vote3D [11] is a method proposed for dealing with sparsity problem.", "The main idea is utilizing a sliding window and a voting mechanism to handle 3D volume data.", "However, the operation is still based on sparse data, which may result in loss of detailed information.", "Again, these 3D CNN-based methods require unifying input, which may not be suitable for our irregular CT images.", "PointNet [12] and PointNet++ [13] are proposed for irregular 3D point cloud data representation learning, and have reported promising performance on classification and segmentation.", "However, such superiority is dispensable to the 3D data that can be regularly arranged.", "There is a size limitation of PointNet based method, i.e., usually smaller than 2048 points.", "Multiview CNNs [14] is proposed to render 3D data into 2D images and then applying 2D CNNs for classification tasks.", "By their design, promising performance has been reported in [14] in shape classification and retrieval tasks.", "Since there are a number of excellent classification networks for 2D image-based classification and identification tasks, we also propose to perform 3D CT based person identification by projecting CT into multi-view 2D renderings.", "Unlike common computer vision tasks like image classification and face recognition, which usually have millions of images, the typical amount of medical images is relatively small.", "To cope with this, one popular solution is to apply transfer learning [15], [16], which has been proved efficient in quantifying the severity of radiographic knee osteoarthritis [15] and improving schizophrenia (SZ) classification performance [16].", "A popular and simple strategy is to fine-tune the pretrained network on the target dataset.", "In such cases, finding a suitable source domain dataset and minimizing the gap between source and target domain is crucial.", "In our tasks, we utilize the ample amount of face depth images obtain with RGB-D sensors to perform transfer learning towards CT based face recognition.", "We summarize the main contributions of this work as follows: To the best of our knowledge, this is the first attempt to explore the biometric characteristic of 3D CT images for human recognition, and we show 3D CT images do contain subject discriminative information, and achieve 92.53% rank-1 identification rate on a dataset with 600 CT images.", "We propose an automatic processing pipeline for 3D CTs, which first detects facial landmarks in 3D CTs for ROI extraction and then generates aligned 2D depth images projected from 3D CTs.", "We employ transfer learning and a group sampling strategy to handle the small data issue, which is found to improve the person recognition performance by a large margin.", "We validate the effectiveness and robustness of our method using datasets from multiple sources.", "The results show our model is able to capture the discriminative characteristics from CT images.", "The remainder of this paper is organized as follows.", "We briefly review related works in Section ii@.", "The proposed 3D CT based recognition method is detailed in Section iii@.", "In Section iv@, we provide the experiments on several public 3D CT datasets.", "Discussions of the method and results are provided in Section v@.", "We finally conclude this work in Section vi@.", "PointNet [12] is proposed for data representation learning from point cloud, which is data format consisting of point coordinates listed irregularly.", "PointNet is a novel convolutional neural network that takes point cloud as input and learns the spatial pattern of each point, and finally aggregates the individual features into a global view.", "The network structure is simple but efficient.", "It has been successfully applied in object classification, partial segmentation, and scene semantic analysis tasks.", "A hierarchical feature learning method PointNet++ [13] is proposed to better capture local structure and improve the ability of managing the variable density of point cloud.", "The main idea is to extract features to a higher dimension in a global pattern from a small scale.", "It introduces a strategy to divide and group the input points into some overlapping local regions by measuring the distance in a metric space and produces higher level features by utilizing PointNet.", "Such a method can achieve good performance in dealing with irregular point cloud data.", "However, such superiority is dispensable to the 3D data that can be regularly arranged.", "Moreover, it has a strong demand for high computational consumption and large memory.", "VoxelNet [17] proposed by Apple Inc offers a novelty for handling point cloud data.", "It encodes point cloud data to a descriptive volumetric representation by introducing a voxel feature encoding layer.", "Then the problem can be translated to the task of operating 3D volume data.", "But it is proved to perform well only in LiDAR dataset and its performance is uncertain for other tasks.", "PointGrid [18] integrates point and grid for dense 3D data.", "It holds a constant number of points in a grid cell and gets a local, geometrical shape representation.", "However, the random screening of points causes a loss of detailed information, thus difficult to recognize the features on a small scale.", "We note that those methods are usually applied in object identification tasks, in which the target objects have macroscopic differences in shapes or colors and do not request for high resolution of source data.", "It may not be suitable for those tasks that require finer granularity such as facial texture." ], [ "2D Rendering Based 3D Modeling", "3D shape models can be naturally encoded by a 3D convolutional neural network.", "The method may be limited by computational complexity and restricted memory.", "In addition, 3D model is usually defined on its surface, which ignores the observed information reflected in the pixels of 2D images [19].", "Su et al.", "prove that applying a 2D CNN on multiple 2D views can achieve a better recognition result [20].", "They build classifiers of 3D shapes based on 2D rendered images, which outperform the classifiers directly based on 3D representations.", "Compared to the general 3D volume-based method, the 2D CNN on multiple 2D images consumes less memory.", "It allows an increased resolution and better expresses the fine-grained pattern.", "Another superiority of using 2D representations is that it offers an additional training data augmentation, since the 3D model of a subject is a single data point, which makes the 3D training dataset short of richness and multiformity.", "By projecting a 3D model into a 2D space, we can easily obtain a series of related 2D images for training, which is beneficial for network training.", "2D images have wide sources and have already organized as massive image databases such as ImageNet [21] which offers a plethora of information conducive to pre-training powerful features.", "The depth information and observed information can also be generated for improved performance [20].", "It has been also proved to achieve high results in recent retrieval benchmarks [22].", "Due to its advantages, it has designed for some classification and detection tasks.", "For example, Hegde et al.", "[23] propose FusionNet for 3D object classification after generating multiple representations.", "Deng et al.", "[22] develop a 3D Shape retrieval method named CM-VGG by clock matching and using convolutional neural networks.", "Kalogerakis et al.", "[24], inspired by the projecting strategy, develop a deep architecture for 3D objects segmentation tasks by combining multi-view FCNs and CRF.", "It learns to adaptively select part-based views to obtain special view-based shape representations.", "2D CNN has achieved a series of breakthroughs in image classification and segmentation tasks.", "It can naturally integrate low/medium/high level features and classifiers in an end-to-end manner.", "The network is flexible due to different ways of stacking layers.", "Among many promising methods, ResNet [25] proposed by He et.al is an efficient and powerful structure which can be easily implemented and modified.", "It is capable of solving the network degradation problem as the depth of the network increases.", "Many improvements based on ResNet have been proposed, such as ResNeXt, DenseNet, MobileNet, and ShuffleNet.", "Adopting the great benefits of Muti-views CNN and mature implementation of ResNet, our framework is capable of offering a remedy for the small samples of source data, which is common in medical imaging field." ], [ "Depth Representation for 3D Object", "Recent developments in depth imaging sensors have induced an effective application of depth cameras which are actively used for a variety of image recognition tasks [26].", "The depth camera is used to produce high quality depth images and is getting more significance in multimedia analysis and man-machine interaction [27].", "The major principle of depth imaging system is extracting a depth target object silhouette and discarding the background, which renders insensitivity to lighting conditions and offers more spatial characteristics.", "Depth camera is proved to be helpful for many tasks by its benefits.", "For example, Papazov et al.", "[28] employ a commodity depth sensor for 3D head pose estimation and facial landmark localization.", "Kamal et al.", "[29] describe a novel method for recognizing human activities from video using depth silhouettes.", "Raghavendra et al.", "[30] present a novel face recognition system by focusing on the multiple depth images rendered by the Light Field Camera.", "Ge et al.", "[31] generate 2D multi-view projections from a single depth image and fuse to produce a robust 3D hand pose estimation.", "They have observed the transformation of 3D model, depth image, and multiple 2D projections, which is practicable and flexible.", "The idea has also been employed in a multi-view video coding system [32].", "Depth representation can also play an important role in assisting detection or surface segmentation tasks combining with RGB images [33], [34].", "Such studies all take advantage of the strong relevancy between the depth representation and 3D model.", "In other words, these indicate one specific subject.", "And the flexibility of such transformation can be utilized for improving the algorithm performance.", "In our tasks, we exploit our medical imaging data and make use of the depth representation of facial part for human recognition.", "In this paper, we propose an effective approach for exploring the biometric features in CT images for face recognition.", "We first normalize different 3D CT images into the same spacing and organize them in a 3D manner.", "Then, we detect 3D facial landmarks from 3D CT volumes by learning a modified face alignment network (denoted as FAN below).", "The detected 3D facial landmarks are then used for cropping the facial region from CTs.", "Next, 3D rendering and 3D-to-2D projection are performed to obtain the face depth images, which will be used for face recognition.", "Finally, we leverage transfer learning to adapt a pretrained face recognition model from face depth images in the RGB-D domain to the computed face depth images from CTs.", "The overview of our approach is illustrated in Fig.", "REF .", "Our method can exploit the uniqueness of CT images into consideration and learn the most discriminative feature for face recognition tasks.", "We adopt transfer learning and a novel training strategy to improve performance.", "We describe the detailed procedure below.", "Figure: An overview of the proposed approach for 3D CT based face recognition." ], [ "CT Normalization and Face Landmark Detection", "While CTs are widely used in medical diagnosis, there are big diversities among different 3D CT images.", "First, the slice spacing of 3D CT images captured by different CT imaging machines are usually different, i.e., the slice spacing for CT images can vary from about $1mm$ to $4mm$ in practice.", "Second, the body areas presented in different CT scans are usually different.", "For example, some CT scans may contain body from the legs to the head, but some may just contain the head.", "Therefore, our first step is to extract the facial regions from these various CT scans, and normalize their slice spacing to the same scale.", "We extract facial regions from CTs by detecting facial landmarks as widely used in face recognition.", "However, while facial landmark detection in 2D and 3D face images has been widely studied [35], [36], facial landmark detection from CTs is a relatively new problem because of the modality difference between CTs and traditional 2D and 3D face images.", "So, we propose a CT facial alignment network (CT-FAN) to localize a set of pre-defined facial landmarks from each CT image.", "Our CT-FAN consists of an input block with 4 convolution layers and an Hourglass (HG) [37] block.", "Following the widely used settings in visible RGB image-based face recognition [38], [39], we also pre-define facial landmarks on each 3D CT face image, including left eye center, right eye center, nose tip, and left mouth corner.", "We have manually annotated these landmarks for all the CT images using 3D Slicer [40].", "We denote the landmarks for each CT image as $P=\\lbrace p_j\\rbrace $ , in which $p_j = [x_j, y_j, z_j]$ denoting a coordinate in 3D space.", "Then, landmark detection from CTs can be formulated as $P = \\psi _{CT-FAN} (X),$ where $X$ denotes an input CT image.", "Considering the ambiguity of landmarks in 3D CTs, instead of directly predicting the coordinate values of landmarks, we represent each landmark with a Gaussian blob (or heatmap) centered at the landmark, i.e., $s_j = G(p_j)$ .", "Then, the goal of our CT-FAN becomes $S = \\psi _{CT-FAN} (X),$ where $\\psi _{CT-FAN}$ can be learned with the conventional Adam solver.", "After $\\psi _{CT-FAN}$ is learned, given a testing CT image $X$ , its predicted extended landmarks can be denoted as $\\hat{S}$ .", "Then, the final facial landmarks $\\hat{P}$ can be computed with an average of the 3D points in $\\hat{S}$ .", "A simple mean square error (MSE) loss is adapted for minimizing the differences.", "$L_{CT-FAN} =\\frac{1}{2n} \\sum {\\Vert P - \\hat{P} \\Vert ^2}$ Then, the facial ROI volume from each CT image $X^{\\prime }$ can be easily computed based on the predicted landmarks and the bounding box.", "Figure: The results of face landmark detection.", "(a)-(c) are nose landmark detection results from a 3D CT image, and (d)-(f) are nose landmark detection results from another 3D CT image." ], [ "3D Rendering and Depth Image Generation", "After obtaining facial ROI volume $X^{\\prime }$ , we perform 3D rendering and 3D-to-2D projection to generate CT depth images which will be used for the final face recognition task.", "Due to the diversity of slice spacing among CTs, scale normalization is crucial.", "However, the conventional scaling method by utilizing interpolation may lead to voxel holes (i.e., no value assigned to voxels).", "Therefore, we employ the visualization toolkit (VTK) [41], [42] tool and construct a 3D rendering model for each CT face image.", "Compared to other visualization methods for natural images, VTK is more suitable for medical image management and better restoring the original 3D contour information of subjects.", "The 3D rendering of a CT face image can be written as: $CX = \\psi _{REN}\\lbrace X^{\\prime } | \\gamma , \\theta \\rbrace ,$ where $\\psi _{REN}$ indicates 3D rendering.", "$\\gamma $ and $\\theta $ indicate iso-surface extraction threshold and surface normal rendering angle for 3D model construction, respectively.", "Specifically, we set $\\theta =60^{\\circ }$ , and $\\gamma $ ranges in [-400, -300].", "Then, we project the rendered 3D CTs to 2D to obtain CT face depth images.", "The key of projection is to find the relationship between 2D and 3D coordinates.", "VTK is capable of simulating the action of taking pictures in reality, like “photographing”, and it can transform the coordinates naturally.", "Assuming $CX$ consists of a set of 3D voxels ${(x_j, y_j, z_j)}$ coordinate.", "Then, we can obtain the depth image $DX$ with a set of 2D pixels $({x_j}^{^{\\prime }},{y_j}^{^{\\prime }})$ as follows: $DX = \\psi _{TS}(CX|cl, co, sc, u_0, v_0, u, v)$ where $\\psi _{TS} ()$ is a transformation function, which is defined based on a number of parameters, like camera location $cl$ , camera orientation $co$ , scaling parameter $sc$ , the coordinates of the image center $(u_0, v_0)$ and the size of the 2D images $(u, v)$ .", "Specifically, we set $co$ as default, which means facing the object.", "We set $u=v=256, u_0=v_0=128$ , and $sc=180^{\\circ }$ .", "The parameter setting is largely related to our task.", "In a common CT scan, there exist different rotation degrees of the human head, e.g., some patients will look up or bow, and others will turn right or left.", "In addition, our dataset has a limited number of images per patient (most individual only has one CT).", "Such situation causes difficulties in acquiring CTs of different head pose.", "Therefore, we utilize different camera locations for data augmentation to simulate different head poses.", "Camera location $cl$ is determined by three types of rotation, i.e., pitch rotation angle $rp$ , roll rotation $rr$ , and yaw rotation $ry$ .", "Specifically, we set $rp\\in [-20, 20]$ , and $rr\\in [-25, 25]$ and $ry=0$ .", "For each 3D CT image, we obtain 90 depth images with different head poses." ], [ "Face Separation", "In order to suppress the influence of non-facial noises (neck or hands) and get more distinguishable CT images, we then adopt face segmentation.", "A fully convolutional encoder-decoder network (CEDN) [43] is employed for face contour extraction.", "We utilize VGG16 [44] as a basic framework and modified it as an encoder.", "The decoder part is a corresponding deconvolution neural network.", "We first train the model on the Helen [45] dataset, which includes 2,000 face images with 68 landmarks, like eyes, nose, mouth, eyebrows, and chin, et al.", "Then the network is transferred to depth images of CTs.", "The separated depth images $SX$ can be obtained as follows: $SX =\\psi _{CEDN}(DX)$ where $\\psi _{CEDN}$ represents the output of CEDN network." ], [ "Image Adjustment", "Due to the limited number of samples, which is common in medical image analysis, training -from-scratch is insufficient to accomplish complex assignments.", "We employ transfer learning and adopt a depth image dataset collected in natural environment as source dataset for better performance.", "To reduce the gap between the source and target domain, we analyze the data distribution of the two datasets.", "The analysis results are shown in Fig.", "REF (a) and Fig.", "REF (b).", "It can be discovered that the pixel values of natural depth images are gathered in $40\\sim 220$ , while the pixel values of the CT depth images are gathered in $80\\sim 140$ .", "Then, we select an appropriate threshold and normalize the depth images.", "The normalization function is described as: $NX = C \\frac{SX - \\theta }{\\max (SX) - \\theta }$ where $C$ is a scaling parameter and $\\theta $ is threshold.", "The adjusted result is shown in Fig.", "REF (c).", "The normalized image $NX$ is used as the input to the network.", "Figure: The procedure and results of image adjustment.", "(a) and (b) are data distribution maps from source domain and target domain, respectively.", "(c) is the normalization results." ], [ "Classification Model and Training Strategy", "We introduce a modified ResNet50 [25] as our backbone for face identification and verification tasks, and utilize triplet loss as a loss [46] function for limited samples.", "To this end, the anchor, positive, and negative samples in each batch are required to select appropriately.", "The loss function for face recognition task is formulated as: $L\\!=\\sum {abs\\begin{pmatrix}\\begin{Vmatrix}X_a\\!-X_p\\end{Vmatrix}_2\\!-\\begin{Vmatrix}X_a\\!-X_n\\end{Vmatrix}_2\\end{pmatrix}\\!+m},$ where $abs()$ represents the absolute value; $X_a$ , $X_p$ and $X_n$ represent anchor, positive sample and negative sample of each batch, respectively, and $m$ is a margin threshold.", "To eliminate the error for determining the three samples of loss calculation, the diversity of each batch is largely required.", "Since our datasets are collected from multiple sources, there is a possible dissimilarity between 3D CTs of different patients.", "We employ a training strategy for 3D CTs, which forces the network to focus on the inter-individual differences of CTs.", "We first divide depth images of the same patient into $G_k$ random subgroups, where $k$ is patient ID (denote as label).", "The size of subgroups $E$ is related to the batch size.", "Then, $L$ labels are randomly selected.", "Finally, each batch consists of one random subgroup from each selected label." ], [ "Dataset and Protocol", "Our CT dataset consists of five open-source head and neck datasets from TCIA Collectionshttps:/www.cancerimagingarchive.net, including Head-Neck-PET-CT, QIN-HEADNECK, HNSCC-3DCT-RT, CPTAC-HNSCC, and Head-Neck Cetuximab.", "The Cancer Imaging Archive (TCIA) is a large archive of medical images.", "Some datasets also offer other image-related information such as patient treatment results, treatment details, genomics and pathology related information, and expert analysis.", "Head-Neck-PET-CT [47] consists of 298 patients from FDG-PET/CT and radiation therapy programs for head and neck (H&N) cancer patients supported by 4 different institutions in Quebec.", "All patients underwent FDG-PET/CT scans between April 2006 and November 2014.", "The same transformation is applied to all images, preserving the time interval between serial scans.", "The patient treatments and image scanning protocols are elaborated in [47].", "QIN-HEADNECK is a set of multiple positron emission tomography/computed tomography (PET/CT).", "18F-FDG scans-before and after therapy-with follow up scans of head and neck cancer patients.", "The data contributes to the research activities of the National Cancer Institute's (NCI's) Quantitative Imaging Network (QIN).", "HNSCC-3DCT-RT consists of 3D high-resolution fan-beam CT scans of 31 head-and-neck squamous cell carcinoma (HNSCC) patients by using a Siemens 16-slice CT scanner with standard clinical protocol.", "CPTAC-HNSCC contains subjects from the National Cancer Institute’s Clinical Proteomic Tumor Analysis Consortium Head-and-Neck cancer (CPTAC-HNSCC) cohort.", "Head-Neck Cetuximab is a subset of RTOG 0522/ACRIN 4500.", "The protocol is randomized chemotherapy and phase III trial of radiation therapy for stage III and IV head and neck cancer.", "In total, 600 3D CT images of 280 patients are collected after integrating and screening the above five datasets.", "We randomly select 224 subjects for training and the remaining 56 subjects for testing.", "We obtain a total of 54,000 CT depth images through data augmentation.", "Since the number of CT depth images of different classes may be different, we divide the training and testing sets with a similar ratio for each class, i.e., split each class by 8:2.", "We perform both face identification and verification experiments using the CT depth images obtained by our approach to verify the effectiveness of leveraging CT depth images for human identification.", "Firstly, we perform face identification and calculate the classification or identification accuracy (ACC) for performance evaluation.", "One sample from each class in test dataset is selected to the gallery, and the remaining CT depth images are used as probes.", "We measure the distance between each probe and all gallery samples.", "And class of the gallery with the smallest distance is regarded as predicted label.", "Our gallery and probe sets contain 54 CT depth images and $700\\sim 800$ CT depth images, respectively.", "Face identification is a simulation of 1 vs. N matching tasks for an unknown 3D CT in reality.", "The illustration is shown in Fig.", "REF .", "In addition, we perform face verification to replicate the 1 vs. 1 face matching scenarios.", "Specifically, given a random pair of CT depth images, it is called genuine pair if the two images are from the same subject; otherwise, the pair is called impostor pair.", "The verification accuracy (VACC) and the area under the ROC curve (AUC) is adopted for evaluation.", "The threshold calculated by ROC is utilized as indicator for positive and negative classification.", "Figure: An example of a CT probe face image and several gallery CT face images after our data processing pipeline.Table: Face identification and verification results of our method and the other baseline methods.", "`Pretrain' indicates if a pretrained model is used or not.", "Mean ACC, Mean VACC and Mean AUC indicates the mean identification accuracy, the mean verification accuracy and the mean area under ROC curve, respectively.", "The standard deviations of the 5-fold test are shown in brackets.Figure: The box plot figures of face identification and verification.", "Four baselines, e.g., VGG, ResNet34, pretrained Pointnet, pretrained PointNet++ and our proposed model are shown.Figure: The ROC curves of face verification, in which (a) highlights the true positive rates (TPR) at false positive rates (FPR) ranging in [0, 0.05], and (b) shows the whole ROC for FPR ranging in [0, 1].Table: Face identification and verification results of our method with different training settings.", "Grouping and proportion are related to different selections of training strategy.", "Grouping setting, Pretrain and Data stages indicate different parameters (E and L), a pretrained model is used or not, and different data processing stages, respectively.Figure: Face identification and verification results on our CT dataset when gradually adding the individual steps of our processing pipeline.Figure: Examples of (a) are the source domain face depth images of one subject from the RGB-D dataset in , and (b-d) are the 2D face depth images generated from 3D CTs for three subjects." ], [ "Implement Details", "We employ transfer learning to mitigate the challenges caused by small sample of our CT dataset.", "We first train our classification model ResNet50 by using depth images from an RGB-D face dataset, which contains 581,366 depth images of different shooting angles from 450 subjects [49], [48].", "Then, the model is fine-tuned to CT depth images.", "We set $E=15$ and $L=18$ for training strategy, and use a batch size of 270 in our experiments.", "The CT depth images are normalized to $256\\times 256$ with a $224\\times 224$ random crop as input.", "The classification model for CT depth is implemented using PyTorch.", "The initial learning rate is set to $1e^{-3}$ .", "The Adam solver is used as the optimizer for network training.", "For the baseline methods of PointNet and PointNet++, we extract point cloud from CT depth images so that they can be input to the baseline methods.", "Specifically, we downsample 4096 points from each depth image as input, and set the batch size to 48.", "For 3D volume-based baseline methods like 3D VGG, 3D ResNet10, 3D ResNet18, and 3D ResNet34, the 3D CTs of different pre-processing methods are fed into 3D network directly.", "We utilize multi-scale grouping (MSG) model for PoinNet++.", "We employ 5-fold cross validation for all baseline methods and our model." ], [ "Results of Our Model and Baselines.", "Firstly, we compare our method with other 3D based methods for 1 vs. 54 identification task, such as PointNet, PointNet++ and 3D ResNet.", "The results are summarized in Table REF .", "It is easy to discover that our method can obtain much better performance than the baseline methods.", "Specifically, our pre-trained method obtains an identification accuracy of 92.53%, more than 52.53% higher than the best of the baseline method, e.g., 3D ResNet34.", "And our un-pretrained obtains an identification accuracy of 87.12%, 50.78% higher than the best of the baseline method, e.g., PointNet.", "The 3D CNNs are observed to get lower accuracy, which may because of the small number of training samples, since each 3D CT only has one data point.", "We also evaluate the effectiveness of data augmentation by random 3D rotation for 3D CNN.", "However, such augmentation proved to be unprofitable, which is affected by 3D deformation.", "In addition, it is difficult to find a large 3D dataset which is similar to 3D CTs for pre-training.", "In addition, the performances of PointNet and PointNet++ are weak.", "Because their limited input representation may cause detailed information loss.", "We also evaluate the pre-trained performances of PointNet and PointNet++ as mentioned in [13].", "The transfer results from ModelNet40 are proved to be helpless, which largely due to the gap between CT depth images and 3D natural point cloud.", "And such point-cloud-based models are also limited in terms of input size.", "Thus, our model is capable of learning discriminatice feature by its design.", "We also draw the box plot and ROC in Fig.", "REF and Fig.", "REF by using the best model of each network, e.g., VGG with cropping and rotation, ResNet34 with cropping and rotation, PointNet with ModelNet40 pretrained and PointNet++ with ModelNet40 pretrained.", "The ROC results suggest that our model leads to the best performance, which shows a robustness feature learning capacity.", "We evaluate the performance of 1 vs. 1 verification task.", "It can be discovered that our method can obtain the best verification performance than other baseline methods.", "The results are shown in Table REF .", "Specifically, our method obtains a verification rate of 96.12%, with 18.41% higher than the best of the baseline method, e.g., ResNet34.", "Also, the AUC (0.9936 by our proposed method) suggests that our method can yield a discriminative performance, indicating powerful identification abilities for 3D CTs.", "The 3D volume-based baseline methods and point-cloud-based methods also proved to less beneficial, compared to our model." ], [ "Results of Different Training Settings.", "To validate the robustness of our training strategy, we also compare our training strategy with other training settings.", "There are two important operations in our proposed method.", "First, we shuffle the samples in subgroup generation, and each subgroup could contain depth images generated from different 3D CTs of the same patient.", "Second, we split the training and testing set according to the number of 3D CTs obtained from the same patient, and we expect to find the superiority and effectiveness of such a partition.", "To this end, we perform experiments without at least one of the above operations, where other parameters are kept the same to those of the original setting.", "And results are shown in rows $9\\sim 14$ of Table REF .", "We discover that the two operations of our training strategy are beneficial and each operation shows some boost.", "And the model with two operations can improve accuracy by 3.35% and 1.56% over un-pretrained and pretrained models, respectively.", "Such results indicate that a training method with a suitable partition for training and testing for 3D CTs of multiple sources is crucial.", "There are two parameters in our training.", "i.e., subgroup size $E$ and selected label number $L$ .", "To validate the robustness of our training strategy, we perform different parameter settings with the same batch size 270.", "The results are shown in rows $3\\sim 8$ of Table REF .", "It can be discovered that the best result is obtained by setting $E=15$ and $L=18$ .", "The triple loss is sensitive to the positive and negative samples.", "Thus, our design for each mini-batch enables the network to focus on the inter-individual differences of 3D CTs as designed.", "In order to evaluate the necessity of our data processing pipeline, we utilize the data generated by each step as network input, including original projected 2D images (Projected), face segmented images (Segmented), and final adjusted images (Final).", "The results are shown in rows $1\\sim 2$ of Table REF and Fig.", "REF .", "It can be seen that the final processed data can achieve the best accuracy and AUC value.", "Each stage can improve the identification performance, e.g.", "0.64% and 4.11% boost, respectively.", "The results demonstrate that each step of our model can be essential and can improve the accuracy accordingly." ], [ "Discussions", "Transfer learning is employed in our tasks.", "However, there exists a gap between natural depth images and CT depth images.", "The samples of natural depth images come from every frame of four videos.", "And the samples are shown in Fig.", "REF (a).", "Each row indicates a video.", "We discover that the images from different video images have high similarity.", "It demonstrates the ROI (face area) of a subject is consistent across the different videos.", "Meanwhile, we analyze our depth CT images, especially the images generated from different 3D CTs with the same patient.", "Different from natural depth images, the images are not consistent.", "For example, the patient may have a tube in his nose when undergoing one CT scans and not have a tube when undergoing the other.", "The samples are shown in Fig REF (b).", "Such condition is unusual but exists in the medical field.", "In addition, the time interval of undergoing multiple CT scans may vary, which can lead to changes of facial shape.", "The patients underwent CT scans are generally diseased.", "And they may become more and more emaciated in case of serious conditions.", "For example, the images in Fig.", "REF (c) come from the same patient, and different rows indicate different 3D CTs.", "It is obvious that the images in the first row are thinner than the second row, which leads to a variation of face.", "The images in Fig.", "REF (d) come from the same patient as well.", "We found that the images in the second row have a bigger cheek, and the patient may take a deep breath when scanning.", "Such conditions may not appear in the laboratory environment.", "However, our datasets contain the above 3D CTs, which may common in other medical images.", "It is difficult for a network to recognize the differences and similarities between an unusual image and usual images.", "For example, the network may not be able to distinguish the CT depth images with a tube during testing, due to such images not appearing in the training dataset.", "Our model has its limitation to handle such 3D CTs.", "Utilizing style transfer learning method to generate more images with different cases may be helpful.", "The style transfer learning methods largely based on generative adversarial networks (GANs).", "However, training such networks from 3D CTs may be limited by the small sample problem.", "And the generated images may not close to the real images.", "Our paper employs model transferring from natural depth images.", "Minimizing the gap between the source domain (natural depth images) and the target domain (CT depth images) is crucial." ], [ "Conclusions", "In this paper, we explore the biometric characteristic of 3D CTs and use them to perform face recognition and verification.", "We propose an automatic processing pipeline for human recognition based on 3D CT.", "The pipeline first detects facial landmarks for ROI extraction and then project 3D CT faces to 2D to obtain depth images.", "To address small training data issue and improve the inter-class separability, we use transfer learning and a group sampling strategy to train our classification network from face depth images obtained from 3D CTs.", "We perform experiments on 3D CTs collected from multiple sources, which shows the proposed method performs better than the state-of-the-art methods like point cloud networks and 3D CNNs.", "We also validate and discuss the effectiveness of the data processing and transfer learning modules.", "Our work is of vital importance to the integrity of medical image big data.", "because medical images are inevitably expanding, to assure correct associations between different scans of the same patient it is necessary to leverage automatic human recognition technologies like the proposed CT based face recognition.", "In addition, the proposed CT based face recognition method can be used to identify unknown CT images when the meta data is lost.", "Our work can improve the efficiency of hospital operations to a certain extent, avoid cumbersome manual inspections, and bring convenience to hospital visits and medical treatment.", "Our future work includes combining the CT images with other modalities of the same individual to perform multi-modality or cross-modality human recognition." ] ]

[ [ "A Plasmoid Model for the Sgr A* Flares Observed with GRAVITY and Chandra" ], [ "Abstract The Galactic Center black hole Sgr A* shows significant variability and flares in the submillimeter, infrared, and X-ray wavelengths.", "Owing to its exquisite resolution in the IR bands, the GRAVITY experiment for the first time spatially resolved the locations of three flares and showed that a bright region moves in ellipse-like trajectories close to but offset from the black hole over the course of each event.", "We present a model for plasmoids that form during reconnection events and orbit in the coronal region around a black hole to explain these observations.", "We utilize general-relativistic radiative transfer calculations that include effects from finite light travel time, plasmoid motion, particle acceleration, and synchrotron cooling and obtain a rich structure in the flare lightcurves.", "This model can naturally account for the observed motion of the bright regions observed by the GRAVITY experiment and the offset between the center of the centroid motion and the position of the black hole.", "It also explains why some flares may be double-peaked while others have only a single peak and uncovers a correlation between the structure in the lightcurve and the location of the flare.", "Finally, we make predictions for future observations of flares from the inner accretion flow of Sgr A* that will provide a test of this model." ], [ "Introduction", "The low luminosity and the broadband spectrum of the supermassive black hole at the center of the Milky Way, Sagittarius A$^*$ (Sgr A$^*$ ), is thought to arise from a high-temperature, low-density, collisionless, and radiatively inefficient accretion flow (see [36] for a review).", "Long-term monitoring has revealed significant multiwavelength variability and flaring behavior from the submillimeter ([24]) to the infrared ([16], [17], [10]) and X-ray wavelengths ([26], [21]).", "The timescales, polarization measurements, and spectra of these observations suggest that the flares most likely originate in the inner accretion flow from compact magnetized structures emitting synchrotron radiation ([13], [12], [27], [1], [28]).", "When interpreted in the context of the radiatively inefficient accretion models, these flares offer unique insight into particle acceleration and heating mechanisms in collisionless plasmas, which are fundamental plasma physics processes that are largely unconstrained by laboratory experiments.", "Numerous theoretical studies have attempted to explain the observed flaring and variability of Sgr A$^*$ , often invoking transient structures or the episodic release of energy in compact regions of accretion flows.", "Some models explored hot spots orbiting in the equatorial plane ([5]) or along the jet of a black hole ([35]).", "Others discussed plasma instabilities that cause buoyant magnetic bubbles to rise in the accretion disk, eventually erupting into the corona and forming a current sheet where reconnection may occur ([23]).", "Indeed, magnetic reconnection has been recognized as potentially playing an important role in the observed variability of Sgr A$^*$ and other low-luminosity accretion flows, leading to localized and episodic energy release ([15], [11], [1], [23], [2]).", "Various magnetohydrodynamic (MHD) simulations have incorporated this effect (e.g., [11], [8], [1]).", "Coupling general-relativistic radiative transfer calculations to general relativistic MHD simulations, these latter studies have found that intermittent magnetized structures (or, \"flux tubes”) that copiously radiate synchrotron emission, coupled to the strong gravitational lensing when one of these structures passes behind the black hole, can cause significant IR and X-ray variability and flaring behavior.", "Recent observations have revealed additional properties of these multiwavelength flares that are not easy to account for in simple models.", "Using the Chandra X-ray Observatory, [21] characterized the time evolution and spectra of very bright X-ray flares, one of which shows a distinct double peak in its lightcurve, with a time separation between the two peaks of about $\\sim 40$  min.", "Observations with the GRAVITY interferometer on the Very Large Telescope measured astrometrically the motions of the centroids of the emission during three IR flares, which were within 100$\\;\\mu $ as of the black hole ([19]; hereafter, \"the GRAVITY paper\").", "One of these flares also showed a distinct double-peaked lightcurve with a timescale similar to that of the Chandra flares.", "The GRAVITY paper (see also [31]) interpreted the astrometric excursions as orbital motions centered around the black hole, although the central positions of the inferred orbits are different among the three flares and offset from the location of the central black hole.", "For a black hole mass of $M=4.1 \\times 10^6 M_\\odot $ at a distance of 8.1kpc (e.g., [20]), 100$\\;\\mu $ as corresponds to 10 $R_{\\rm S}$ and 40 minutes corresponds to the orbital period at 3.5 $R_{\\rm S}$ , where $R_{\\rm S} \\equiv 2GM/c^2$ is the Schwarzschild radius.", "In this paper, we show that emission from hot plasmoids orbiting in the funnel region of a black hole accretion flow can account for several previously unexplained aspects of the flares.", "Plasmoids, which are compact structures of magnetized plasma that are formed from the collapse of a current sheet and contain heated and accelerated particles, are a generic byproduct of reconnection events and are a natural way to explain the presence of hot, compact emitting regions that are offset from the black hole.", "Plasmoids hierarchically merge and may eventually coalesce into an astrophysically large structure (such as “monster” plasmoids in [32], [18]).", "Because these large plasmoids are highly magnetized and contain all of the high-energy electrons accelerated in the reconnection event, they will radiate copiously as the high-energy electrons cool via synchrotron radiation.", "Furthermore, as these regions can occur away from the equatorial plane, when viewed from an inclined angle with respect to the black hole spin axis, the observed center of emission will trace out ellipse-like shapes, with their centers offset from the position of the black hole.", "Finally, because they are prevalent in regions containing low-$\\beta $ plasma (where $\\beta \\equiv kT_e/B^2$ is the ratio of of gas pressure to magnetic pressure), such as in the innermost accretion flow and jet/coronal regions around a black hole, the positions of the centroids will naturally have two preferred directions along an axis: either aligned or anti-aligned with the black hole spin.", "This has the potential to explain the aligned trajectories of the flares as well as the differences observed between them, as we will discuss in the next section.", "Making use of these characteristics, we construct a plasmoid model orbiting in the jet or coronal region of a black hole, with properties informed by microphysical studies of reconnection and incorporating cooling via synchrotron emission.", "We include the physics of finite light-travel time, which we show can have a significant effect on both the observed lightcurves and the centroid motion.", "We show that this model can not only explain the differences in the orientations between different flares but also the connection between the orientation of the flare's trajectory and the structure in its lightcurve.", "In §2, we discuss the characteristics of the flares observed with GRAVITY.", "In §3, we introduce the formalism for the orbital motion and the energetics of plasmoids.", "In §4 and 5, we present trajectories and lightcurves for the two possible orientations of the plasmoid motion with respect to the black hole axis.", "In §6, we explore the correlations between the various properties of the flares expected in the plasmoid model and conclude in §7 with testable predictions and outlook for future observations.", "Figure: Centroid positions from all three flares observed with GRAVITY, centered on the position of Sgr A* (black point).", "The May 27th and July 22nd flares both appear to be similar in their orbital orientation, while the July 28th flare centroids largely point in the opposite direction.", "None of the centers of the projected motion are centered on the position of Sgr A * ^*.", "We depict the direction of orbital motion with a black arrow in the upper left hand corner.Figure: Lightcurves from all three flares observed with GRAVITY.", "The May 27th and July 22nd flares show a single peak, while the July 28th flare has a secondary peak separated from the first by ≈40\\approx 40min." ], [ "Characteristics of the Flares Observed with Gravity", "In 2018, the GRAVITY collaboration presented high spatial-resolution observations during three flares from Sgr A$^*$ and reported that the centroids of the bright spots followed ellipse-like trajectories, with excursions of the order of $\\sim 200 \\; \\mu a s$ ([19]).", "Of the three flares, two occurring on July 22nd and July 28th were significantly brighter than the third occurring on May 27th.", "We show in Figure REF the centroid positions from the GRAVITY data for all three observed flares, but with adjusted coordinates compared to those used in the figures in the GRAVITY paper.", "In the latter, the coordinates are centered on the median centroid position of a given flare, whereas here we centered the coordinate system on the position of Sgr A* (shown as a black dot at $x=0, \\; y=0$ ) in order to be able to compare their relative locations.", "It is evident from this figure that none of the orbits are centered on the position of the black hole.", "Another interesting feature of these centroid tracks is that the trajectory followed by the flare on July 28th (blue dots) appears practically in the polar opposite direction with respect to the black hole (i.e., rotated $180^\\circ $ ) compared to the other two flares, which share a common directionality.", "In their original analysis, the GRAVITY paper invoked a hot spot model orbiting in the equatorial plane around the black hole to explain these observations.", "It attributed this offset between the center of the centroid motion and position of the black hole to the fact that the observations may span only $50-70$ percent of the orbit.", "In the follow-up analysis, [31] also considered non-equatorial orbits.", "Although finding models that better fit the trajectories, the solutions for the different flare orbits presented in the follow-up paper still did not show a common center.", "In addition to the trajectories of the bright spots, GRAVITY also reported on the evolution of the total observed flux over the course of the flares.", "We show the combined lightcurves in Figure REF .", "For the May 27th and July 22nd flares, the lightcurves show relatively little structure, falling off monotonically after the first peak.", "The July 28th flare, on the other hand, shows two distinct peaks in flux that are separated by about $\\sim 40$ minutes, which is comparable to the doubly-peaked lightcurve reported by Chandra during an X-ray flare in [21].", "Overall, the properties of the July 28th flare are significantly different than the other two: it appears to be oriented on the opposite side of the black hole from the others and shows a different evolution in its lightcurve.", "We note, however, that the total observing time was longer for the July 28th flare than for the other two, so it is possible that there may have been a second peak associated with one of the other flares.", "The differences between these flares motivated us to search for a physical model that will naturally explain the offset of the centroid orbits, the difference in orientation between different flares, and the reason why some flares are single-peaked while others are double-peaked." ], [ "The Plasmoid Model", "In this section, we discuss the framework needed to calculate the time-dependent emission from a plasmoid that forms during a reconnection event in the low-$\\beta $ funnel region of a black hole and its appearance to a distant observer.", "The observed properties of the such an event will depend both on small scale processes, such as energy injection and cooling, as well as large-scale processes, such as the motion of the plasmoid and the transport of emitted radiation in the spacetime of the black hole.", "On microphysical scales, reconnection injects a distribution of energetic electrons, the properties of which depend on the plasma conditions.", "Here, we will adopt injection parameters that are appropriate for the low density, low$-\\beta $ , and high magnetization conditions that are appropriate for the funnel region of Sgr A$^*$ ([33], [3]), where magnetization is defined via the parameter $\\sigma =B^2/4\\pi \\rho c^2$ .", "The electrons then cool by emitting synchrotron radiation.", "On larger scales, the plasmoid is not expected to be stationary but to move in the ambient gravitational and magnetic field that is present in the funnel region, on dynamical timescales that are relevant for its position.", "In addition to this motion, radiation emitted from the plasmoid will also be affected by light bending as it travels from the region near the black hole to the observer at infinity.", "Because each of these physical effects play a role in determining the observables in the plasmoid model, we describe in the following subsections our treatment of the trajectory of the plasmoid, the evolution of the particle energies within the plasmoid, and the general relativistic radiative transfer that allow us to calculate lightcurves and observed centroid motions at infinity." ], [ "The Plasmoid Motion", "We treat the plasmoid as a compact magnetized structure that contains energetic particles and moves in the gravitational and magnetic fields near the black hole.", "To this end, we define a simple orbit that can be representative of such a motion, by restricting the trajectory of the plasmoid to be on a conical helix.", "The initial conditions are defined by parameters $r_0, \\; \\phi _0, \\; \\theta _0$ , $v_{r0}$ , and $v_{\\phi 0}$ , where $r$ , $\\theta $ , and $\\phi $ represent the usual spherical polar coordinates, centered on the black hole, and the subscript 0 reflects the initial values for these parameters.", "For simplicity, we use a constant velocity $v_r=v_{r0}$ , such that the radial coordinate simply increases in time as $r(t)=r_0+v_r t$ .", "The plasmoid moves on a surface of constant $\\theta $ , such that $\\theta (t)=\\theta _0$ .", "We then solve for an orbit that conserves the Newtonian angular momentum, i.e., $\\dot{\\phi }(t)=\\dot{\\phi }_0 r_0^2 / r(t)^2.$ Naturally, a plasmoid may move on a more complicated trajectory under gravitational, hydrodynamic, and magnetic forces, which would introduce a larger number of parameters to the model.", "However, our goal here is to identify the simplest physical model that can approximate a set of likely trajectories with the potential to explain the GRAVITY observations.", "Because of that, we choose to limit the current scope to the conical helix trajectories defined above.", "We note that both $v_r>0$ and $v_r<0$ cases are allowed in this setup.", "The case of $v_r>0$ is applicable to a plasmoid forming in the vicinity of an outflowing jet, which pushes the structure along with the outflow; whereas the case of $v_{r}<0$ represents a scenario where hydromagnetic forces are negligible and the plasmoid falls in towards the black hole.", "The $v_r=0$ case restricted to the equatorial plane reduces to the traditional hot spot model.", "We summarize in Table REF the orbital parameters we use for the two models we explore in this paper, which we will elaborate on further in sections and .", "cccccc Summary of orbital parameters.", "Model $r_0$ $\\phi _0$ $\\theta _0$ $v_{r}$ $v_{\\phi }$ Posterior Plasmoid 36 $200^\\circ $$15^\\circ $ 0.01c 0.41c Anterior Plasmoid 50 $0^\\circ $ $165^\\circ $ -0.5c 0.5c Figure: Top: snapshots from three distinct times in a simulation where the plasmoid is in the posterior region of the black hole, ordered chronologically from left to right.", "The background color scale from white to black shows the intensity (logarithmically scaled and normalized) at each pixel in the image.", "The red line shows the motion of the centroid, up to the time of a given snapshot.", "The black cross in the middle shows the location of the black hole.Bottom: Lightcurve from the same simulation; the three colored dots indicate the time along the lightcurve that the snapshots in the top panels correspond to.", "The first peak in the lightcurve appears at t≈4GM/c 3 t \\approx 4 \\; GM/c^3, at the end of the injection phase.", "The second peak at t≈132GM/c 3 t \\approx 132 \\; GM/c^3 is the result of intense gravitational lensing." ], [ "Evolution of the Electron Energy Distribution", "Having specified the orbital motion of the plasmoid, we turn to developing a physically motivated model for the evolution of the electron energy distribution in time throughout the flare.", "The event begins with an injection phase, where reconnection heats the plasma and loads it into the plasmoid.", "The injection phase lasts for a time given by $t_{\\rm {inj}}$ that is set by the reconnection timescale.", "At this point, the electrons in the plasmoid begin to cool via synchrotron radiation, which we refer to as the onset of the cooling phase.", "Even though these phases can in principle overlap, i.e., the cooling can begin before the acceleration phase is over, we choose to treat the two processes as distinct and sequential phases because of the fact that the acceleration timescale is much shorter than the cooling timescale.", "To describe the physics of the injection phase, we define an “injection energy”, $\\gamma _{\\rm {inj}}$ , which we will take as the typical Lorentz factor the electrons are heated to by reconnection.", "In general, the heating and acceleration processes generate a distribution of electron energies that depends on the conditions of the plasma (e.g., [30], [22], [34], [3]).", "GRMHD simulations indicate that the magnetization in the jet or corona region of Sgr A$^*$ is of order $\\sigma =1$ (e.g., [2]) and a correspondingly low plasma-$\\beta $ ($\\beta \\approx 0.1$ ) due to the low density of particles in this region.", "[3] used particle-in-cell simulations to calculate the energization and acceleration of particles in a low-$\\beta $ plasma at $\\sigma =1$ and found that reconnection heats the peak of the electron distribution to approximately $\\gamma =500$ .", "We set this to be equal to $\\gamma _{\\rm {inj}}$ in this studyThe energy distributions resulting from reconnection also typically contain a power-law tail of higher energy electrons, especially at the lowest values of the plasma $\\beta $ .", "We ignore this component here..", "In addition to the typical Lorentz factor of the electron at the end of the reconnection event, we also estimate the typical rate at which electrons gain energy during the receonnection event, which we denote as the injection rate $\\dot{n}$ .", "The canonical reconnection rate is given by $v_{\\rm {rec}}=0.1 c (\\sigma /\\sigma + 1)^{1/2}$ , and the relevant length scale of our problem is the plasmoid size, which we take here to be $L=1 GM/c^2$ , consistent with recent observations that infer the size of the emission region (e.g., [28]) of flaring electrons.", "Using these relations, we can express our injection rate as $\\dot{n}=n_0 v_{\\rm {rec}}/L$ .", "For $\\sigma =1$ , this yields $\\dot{n}= \\frac{0.07 n_0}{GM/c^3},$ where $n_0$ is the background electron density that flows into the reconnection region and gets energized up to $\\gamma _{\\rm {inj}}$ .", "For the purposes of this calculation, we take a thermal distribution with a temperature equal to $\\theta \\approx \\bar{\\gamma }/3$ , i.e., the average Lorentz factor of the electron energy distribution, which is valid in the relativistic limit.", "During the injection phase, the temperature is fixed to $\\theta =\\gamma _{\\rm {inj}}/3$ and the the number density of electrons at a given time, $t$ , is simply given by $n=\\dot{n}t=\\frac{0.07 n_0}{GM/c^3}t.$ After reconnection and the corresponding particle heating ceases at $t=t_{\\rm {inj}}$ , the number density of electrons is fixed, and they cool via synchrotron radiation.", "The synchrotron power emitted by an electron is (e.g., [29]) $P=\\frac{4}{3}\\sigma _T c \\beta ^2 \\gamma ^2 B^{2}/8\\pi .$ In the limit of very relativistic electrons (electrons near $\\gamma _{\\rm {inj}}$ are indeed highly relativistic), this gives $m_e c^2 \\dot{\\gamma }=\\frac{4}{3}\\sigma _T c \\gamma ^2 B^{2}/8\\pi .$ We can then write an expression for the cooling rate, $\\dot{\\gamma }$ , as $\\dot{\\gamma } = 3.2 \\times 10^{-6} \\times \\left(\\frac{B}{50}\\right)^2 \\gamma ^2\\; s^{-1}.$ Solving this equation for the Lorentz factor as a function of time yields $\\gamma (t)=\\frac{1}{1/\\gamma _0 + Ct},$ where $C=3.2 \\times 10^{-6}$ is the coefficient in equation (REF )." ], [ "Computing Observables at Infinity", "We now calculate the lightcurves and trajectories resulting from the emission from the plasmoid as viewed by a distant observer.", "We implement the motion of the plasmoid as well as the expressions for the number density and the Lorentz factor during injection and cooling phases given in Eq.", "(REF ) and Eq.", "(REF ) into a general relativistic radiative transfer simulation.", "To this end, we use GRay ([7]), which we have modified to account for the finite speed of light.", "GRay integrates the radiative transfer equation along geodesics in a black-hole spacetime and includes all of the general-relativistic effects.", "In GRay, we initialize a square grid of $1024 \\times 1024$ \"rays\" over a field of view that we vary depending on the particular problem, set at a distance $1000 GM/c^2$ away from the black hole.", "More specifically, we choose a field of view, centered on the black hole, that is just wider than the plasmoid orbit.", "In this way, we ensure that the entire motion of the plasmoid falls within the field of view while maximizing the resolution given our number of rays.", "We integrate each ray backwards along a null geodesic towards the black hole and numerically integrate the radiative transfer equation along these paths.", "For the results presented in this paper, we perform this calculation for the case of a high-spin Kerr black hole with $a=0.9$ , where $a$ is the dimensionless black hole spin parameter $J/M^2$ .", "We note that the choice of black hole spin here is somewhat arbitrary due to the large uncertainty on measurements of the spin of Sgr A*.", "We did, however, test various values of the black hole spin and found that the salient qualitative features in both the centroid orbits and lightcurves persisted across a wide range of spins.", "The precise quantitative details (e.g., time delay, magnitude of secondary peaks, and centroid motion), however, can vary somewhat depending on the spin and geometry of the setup." ], [ "Plasmoids in the Posterior Region", "Using the setup described in the previous section, we explore the parameter space of plasmoid orbits to identify models that can adequately fit the centroid motion and the lightcurve observed during the July 22nd flare.", "In principle, a number of combinations of parameters within our model can reproduce the general circular shape that is apparent in the July 22nd flare orbit.", "Because we opted to keep the level of complexity of the model and the number of model parameters fairly small, our goal is to identify a class of models that are able to describe the data reasonably well rather than to perform a formal multi-parameter search to find the best-fit orbit.", "We focus, in particular, on reproducing the following set of salient features: the general characteristics of the centroid motion, the number of re-brightening events in the lightcurve, the time between re-brightening events, and the luminosity ratio between peaks, when more than one peak exists.", "As a representative example of a plasmoid orbit that describes the centroid motion of the July 22nd flare reasonably well, we show a model where the plasmoid is in the funnel region on the opposite side of the black hole from the observer, which we refer to as the \"posterior plasmoid\" model.", "We use the following parameters for the plasmoid motion: the polar angles are $\\phi _0=200 ^\\circ $ and $\\theta =15^\\circ $ ; the initial distance from the black hole is $r_0=36 \\; GMc^{-2}$ , and the initial radial and azimuthal components of the velocity are $v_{\\phi }=0.41c$ , and $v_r=0.01c$ .", "The centroid motion from the July 22nd flare exhibits an orbit with a fairly constant radius, which suggests $v_r \\ll v_{\\phi }$ .", "For this reason, we use a value of $v_r \\sim 0$ for this particular model.", "We set the inclination of the observer to $\\theta _{\\rm {obs}}=168^\\circ $ , which is close to the fiducial inclination of $160^{\\circ }$ that was used by the GRAVITY collaboration.", "(Note that this places the observer on the opposite side of the black hole from the plasma, which is oriented at $\\theta =15 ^\\circ $ ; hence the designation of this set up as a \"posterior plasmoid” model.)", "For the local properties of the plasmoid, we set its size equal to $1 \\; \\rm {GM}/\\rm {c}^2$ , its magnetic field strength to 35 G, and the background density, $n_0$ , to $10^6 \\; cm^{-3}$ .", "Figure: Observed centroid positions from July 22nd flare (orange dots) and centroid track from posterior plasmoid model (blue line).", "We see that this model captures the general features from the data, including a mostly circular orbit, offset from the center of the black hole.We show in the top panels of Figure REF the snapshots of the image at three notable times in the simulation.", "In these plots, the color scale corresponds to the intensity of light at a given position in the image plane (where each pixel in the image corresponds to a null geodesic optical path, or, \"ray\").", "The red line depicts the centroid motion up to the time in that snapshot, while the black cross shows the location of the black hole.", "In the bottom panel, we show the lightcurve with a blue line, and mark the times we show with colored dots along the lightcurve.", "In the first of the snapshots (top left panel), the injection phase has just ended and the light from the secondary image has not yet reached the observing plane.", "This is because the path length associated with the secondary image is slightly longer than the path length associated with the primary image, causing a delay in the appearance of the secondary image.", "In the second snapshot (top middle panel), a secondary image forms in the bottom right quadrant, roughly mirrored across the black hole from the direct image of the plasmoid in the upper left quadrant.", "This secondary image is the result of null geodesics that are strongly lensed by the black hole.", "Although the secondary image appears in the second panel, it does not result in a significant flux increase that is discernible in the lightcurve: the flux from the secondary image here is subdominant to the flux from the primary image, but it does result in a slight shift of the centroid inwards.", "In the final snapshot (top right), we see a strong lensing event, where both the primary and secondary images are highly elongated.", "The lensing causes a significant increase in flux, by a factor of approximately 5.", "However, at this point in the simulation, the plasmoid has cooled sufficiently such that, despite the factor of $\\sim 5$ increase from the strong lensing, the luminosity of the second peak is $\\sim 2$ orders of magnitude smaller than the peak associated with the initial rise of the flare.", "We show in Figure REF the full centroid motion from the posterior plasmoid model with a blue line and the centroid data from the July 22nd flare with orange dots.", "We see that the data are reasonably well described by $\\sim 75\\%$ of the full orbit from our model.", "This is similar to the GRAVITY interpretation that the centroid motion associated with the July 22nd flare can be explained by a partial orbit.", "Our model differs, however, because it naturally accounts for the offset between the black hole and the center of the centroid motion.", "In order to see how our model compares to the orbital model that was proposed by the GRAVITY collaboration to interpret the observations (i.e., an orbit in the equatorial plane with a radius of $7 GM/c^2$ , viewed at an inclination of $160^\\circ $ ), we show in Figure REF the X (blue) and Y (red) centroid positions from the posterior plasmoid model described above (solid lines), the model shown in the GRAVITY paper (dashed lines), as well as the observed centroid positions (points).", "With the caveat that neither study employed formal model fitting to estimate the best parameters, we find that the plasmoid model provides a similar description of the data as the orbital model.", "Figure: X (blue) and Y (red) centroid positions from our posterior plasmoid model (solid lines), the model from the GRAVITY paper (dashed lines), and the observed centroid positions (points).", "Both models fit the data equally well." ], [ "Phenomenology of other Plasmoid Orbits", "When the plasmoid orbit is in the posterior region, the interplay of cooling, finite light travel time, and lensing results in a rich variety of possible perceived centroid motions.", "For instance, in Figure REF , we show two models, both with an initial radial location $r_0=40 GM/c^2$ , radial velocity $v_{\\rm r}=0.01 c$ , and observed at an inclination of $i=160^{\\circ }$ .", "We denote the starting positions with filled-in dots.", "The difference between the two models is the direction of the plasmoid orbit: the left panel shows the centroid motion of a plasmoid moving at $v_{\\phi }=0.5c$ and the right panel shows a nearly identical model but with $v_{\\phi }=-0.5c$ and an associated black hole spin flipped (a=-0.9), such that the plasmoid is always corotating with the spacetime.", "Both of these models exhibit a distinct warp in the observed trajectory, displaying a teardrop-like shape.", "The latter is caused when the plasmoid passes closely behind the black hole with respect to the observer's line of sight and is strongly lensed.", "The light that forms the lensed image originates from an earlier time, when the plasmoid was hotter, and hence can pull the position of the centroid substantially away from the primary image, resulting in a teardrop-like shape.", "Interestingly, by changing the direction of travel of the plasmoid, the perceived centroid motion is mirrored.", "Initially, one may expect that reversing the direction of the plasmoid orbit to result in the same shape of the centroid motion, but traveling in the opposite direction.", "Instead, we demonstrate here that the finite light travel time coupled to the cooling of the plasmoid breaks the time symmetry such that reversing the direction of the plasmoid motion results in a differently shaped centroid trajectory (or, in this case, a mirrored trajectory).", "Figure: Two identical posterior plasmiod models, but with the direction of plasmoid motion reversed (left: v φ =0.5cv_{\\phi }=0.5c, right: v φ =-0.5cv_{\\phi }=-0.5c).", "We denote the starting position of the plasmoid with a filled-in dot.", "We see that by reversing the direction of motion of the plasmoid, that the shape of the centroid orbit fundamentally changes." ], [ "Plasmoids in the Anterior Region", "We now explore the features of the model when the plasmoid is on the same side of the black hole as the observer, which we refer to as the \"anterior plasmoid” model.", "This is motivated not only by a desire to explore the full range of phenomena that can occur in the simulations but also by the fact that the trajectory of the July 28 flare occurs in the opposite quadrant with respect to the black hole than do the July 28 and May 27 flares.", "We use the same plasmoid properties (i.e., magnetic field strength, size, and density) for this model, but employ slightly different orbital parameters to highlight some of the most interesting phenomena that can occur in the anterior setup.", "For the orbital parameters, we use a radial distance $r_0=50$ , polar angles $\\phi _0=0^\\circ $ and $\\theta =165^\\circ $ , and azimuthal and radial velocities $v_\\phi =0.5c$ and $v_r=-0.5c$ .", "There are two main differences between this model and the posterior model.", "First, as previously mentioned, is that the plasmoid is on the same side of the black hole as the observer (the observer's inclination is set to $\\theta _{\\rm {obs}}=165^\\circ $ for all models).", "Second, the plasmoid has the opposite sign of radial velocity and is falling in towards the black hole, which as we will see below, can result in strong double peaks in the lightcurve.", "We show in Figure REF an analogous plot to Figure REF , but for the anterior setup.", "We see that the secondary image does not occur until very late times (middle panel, corresponding to $t\\approx 132 GM/c^3$ in this model), when the centroid shifts rapidly and dramatically, as the delayed light from the secondary image hits the observer's plane.", "In order to form the secondary image in the anterior plasmoid model, the light must travel from the plasmoid towards the black hole (away from the observer) a distance $r_0$ , then an additional distance $\\sim 15 GM/c^2$ around the black hole in the vicinity of the photon ring such that the photon direction changes substantially ($\\sim 180^\\circ $ ), and then back again to the observer, another $r_0$ in distance.", "We note here that we are using the term photon ring loosely to describe spherical photon trajectories that exist in the vicinity of Kerr black holes; these trajectories are rings only for the case of non-spinning black holes.", "In either case, the locations of the spherical photon orbits provide a useful mark for the region in the spacetime at which the photon trajectory can change direction by $\\sim 180^\\circ $ .", "Because of this extra distance traveled, the light that the observer sees from the secondary image is delayed with respect to light emitted at the same time that travels directly to the observer by $\\approx 2r_0/c+15 GM/c^3$ .", "We see a strong second peak occur at this time (also see last panel at the top): when light from the delayed lensed image reaches the observer's plane, it shifts the centroid position so rapidly that it appears as a superluminal motion and also results in a strong second peak in the lightcurve.", "The time difference between peaks is comparable to the time between the two peaks in flux observed by the GRAVITY collaboration during the July 28th flare, which is $\\approx 40$ minutes, or 120 $GM/c^3$ ).", "It is noteworthy that the simple set up that explains the particular observed trajectories of the flares with respect to the position of the black hole also naturally produces the differences observed in the lightcurves between the flares.", "Because the double-peaked lightcurves is an unusual characteristic of Sgr A$^*$ flares that are observed both with GRAVITY and Chandra, we turn to a more thorough exploration of this phenomenon in the next section.", "Figure: Top: snapshots from three distinct times in the anterior plasmoid simulation, increasing chronologically from left to right.", "The background color scale from white to black shows the logarithmically scaled and normalized intensity of the image.", "The red line shows the motion of the centroid, up to the time of a given snapshot.", "The black cross in the middle shows the location of the black hole.Bottom: Lightcurve from the simulation, with the three colored dots showing the time along the lightcurve that the snapshots in the top panels correspond to.", "There is a second peak at t=132GM/c 3 t=132 GM/c^3, accompanied by a sudden change in centroid position, similar to that seen in the July 28 flare.", "This is caused by light emitted at the beginning of the simulation, but has traveled around the back of the black hole to reach the observer.", "Because this light is blue shifted due to the motion of the centroid and is emitted at a time before the plasmoid had significantly cooled, it results in a sudden increase in flux.", "The solid angle associated with this secondary image, however, is small, resulting in only a few pixels capturing the extent of the secondary image." ], [ "The Appearance of Double Peaked Lightcurves", "We showed that an anterior plasmoid model with an initial plasmoid distance from the black hole of $r=50\\; GM/c^2$ naturally results in a distinct double-peaked lightcurve, with a time between the two peaks that matches the time between observed double peaked lightcurves in GRAVITY and Chandra observations.", "In this section, we further explore how the properties of these two peaks depend on the orbital parameters of the plasmoid.", "First, we focus on how the infall speed, $v_r$ , influences the relative amplitude of the two peaks.", "In the context of the anterior plasmoid model, we expect a larger infall speed to cause the direct image of the plasmoid to be redshifted and dimmed, while the secondary image is boosted.", "Figure: Lightcurves from a number of anterior plasmoid models holding all parameters fixed while varying the radial velocity v r v_r (defined as pointing inward, towards the black hole).", "Increasing the infall velocity has the effect of redshifting (and hence dimming) the direct image (first peak), while blueshifting (boosting) the light from the secondary image (second peak).", "The secondary image appears at a significantly delayed time at ∼2r 0 \\sim 2 r_0, the light travel time corresponding to the initial distance of the plasmoid.", "The relative height of the two main peaks is set by the infall speed, with the expected general behavior.We show in Figure REF the lightcurves from four anterior plasmoid simulations, with varying values of the radial velocity $v_r$ , while all other model parameters are held constant.", "Each lightcurve is normalized to the maximum flux from the $v_{r}=0$ model.", "We see that all of the models show a distinct second bump that starts at roughly $t=120 \\; GM/c^3$ .", "The amplitude of this second peak, however, depends strongly on $v_r$ , with larger infall speeds causing a brighter second peak.", "Additionally, we see that the amplitude of the first peak also depends on $v_r$ , but with the opposite trend of the second bump.", "This is easy to understand in the context of redshifts and blueshifts of the direct and secondary images.", "The first peak originates from light emitted towards the observer, forming the \"direct” image.", "As the infall speed $v_r$ increases, the plasmoid is moving away from the observer faster, resulting in a redshift and dimming of the light that the observer receives directly from the plasmoid.", "The second peak, however, is dominated by light that is emitted in the direction of the plasmoid motion, orbits around the black hole, and then reaches the observer's plane.", "The solid angle associated with this secondary image will be very small relative to the direct image, which is why the amplitude of the second peak is much smaller when the plasmoid is not moving ($v_r=0$ ).", "However, as the infall speed increases, we see that the amplitude of the second peak rises, and eventually becomes greater than the amplitude of the first peak, as in the $v_r=0.6 \\; c$ case.", "This occurs because the plasmoid is blueshifted along the line of sight that forms the secondary lensed image." ], [ "Decomposing the Effects of Boosting, Lensing, and Gravitational Redshift", "The lightcurves shown in Figure REF reveal a rich structure, beginning with an initial peak and relatively slow cooling, followed by a faster decline with numerous smaller rises and falls in the lightcurve, and ending with a strong second peak.", "We now aim to delineate the physical mechanisms responsible for the dominant features.", "To this end, we run a simplified set of models, where we systematically and artificially exclude various pieces of physics and explore how these choices affect the resulting structures in the lightcurve.", "First, we aim to cleanly isolate the cause of the strong double-peak, without the confounding effects of any additional features in the lightcurve.", "We have compelling reasons to believe that this delayed peak is caused by the delayed lensed image of the plasmoid, because (i) the second peak has the expected $v_r$ dependence, (ii) the time difference between the peaks matches the difference in the light travel time between the direct and secondary images, and (iii) the sudden centroid motion is coincident with the appearance of the second peak.", "However, we further test this interpretation here by removing the effects of the changing position of the plasmoid.", "We show in Figure REF lightcurves from a set of models with a stationary plasmoid, with a varying initial plasmoid distance $r_0$ from the black hole, in order to cause different amounts of time delay.", "Due to the lack of motion, changes in the gravitational redshift and lensing as the plasmoid moves will not occur.", "Additionally, because the azimuthal velocity is set to 0, there is no Doppler boosting or dimming on orbital timescales.", "However, in order to achieve a distinct and large second peak, we artificially incorporate a Doppler boost corresponding to $v_r = -0.5 \\; c$ in the radiative transfer calculation, even though the plasmoid does not move.", "We emphasize that this is not meant to be a physical model, but we use it to cleanly show the effects of cooling and finite light travel time without the confounding effects due to the changing position of the plasmoid.", "We see in Figure REF the general features that we would expect from varying the radial distance of the plasmoid from the black hole.", "In particular, the time between the first and the second peak, $\\Delta t_{}$ , increases by $2\\Delta r_0$ as $r_0$ increases.", "This particular scaling happens simply due to the geometry of the setup: the secondary image is formed by light that travels from the plasmoid towards the black hole, wraps around the black hole, and then comes back towards the plasmoid, ultimately hitting the observer's plane, i.e., it travels the space between the plasmoid and the black hole twice.", "Figure: Lightcurves from a number of anterior plasmoid models with no plasmoid motion.", "By removing additional effects from plasmoid motion, we cleanly isolate the effect of finite light travel time.", "We clearly see the expected behavior of both the timing of the second peak as well as its relative brightness to the primary peak.", "As the plasmoid gets further from the black hole, the time to the secondary peak is delayed, and the secondary peak also becomes dimmer because the solid angle subtended by the secondary image becomes smaller, as fewer optical paths intercept the plasmoid.In order to further understand the simple properties of the double peak without other physical effects, we show in Figure REF lightcurves from a set of simulations with a stationary plasmoid, but varying the radial velocity used in the radiative transfer equation.", "We see that we cleanly recover the distinct double-peaked behavior and underlying trends we found in Figure REF that included all of the physics resulting from plasmoid motion.", "Specifically, we see that when the plasmoid's inward velocity is higher, the first peak corresponding to the direct image is dimmed while the secondary lensed image is boosted.", "By comparing the lightcurves in Figure REF and REF , we can identify the additional effects of plasmoid motion.", "As the plasmoid falls into the gravitational well of the black hole, the gravitational redshift dims the light the observer receives from the plasmoid at a rate that becomes faster than the cooling, which explains the sudden dip in the lightcurves in Figure REF that sets in faster for models with a higher infall velocity.", "Finally, we see a number of smaller peaks and dips in the lightcurve in Figure REF that are not present in Figure REF .", "These occur due to the effects of Doppler boosting/dimming on orbital timescales, and hence are not present in the model shown in Figure REF where the plasmoid has no azimuthal velocity.", "Figure: Lightcurves from a set of anterior plasmoid models with no plasmoid motion.", "Here we vary the radial velocity used in the radiative-transfer portion of the calculation, despite the plasmoid not physically moving throughout the simulation, to isolate the effect of Doppler boosting on the properties of the double peaked behavior in the lightcurve." ], [ "Predictions and Future Outlook", "In this paper, we presented a plasmoid model that explains the offset between the location of Sgr A* and the centroid orbits during IR flares, their orientation relative to each other, and how one preferred direction (the anterior plasmoid model) naturally produces secondary peaks in the lightcurve if the plasmoid falls toward the black hole.", "From this model, we can make a number of predictions for future observations regarding the relationship between the structures in the lightcurve and orientation of the centroid motion during flares.", "First, this model predicts that future observations will continue to reveal centroid orbits with centers that are offset from the position of the black hole.", "In this interpretation, the offset is caused by the physical location of the orbit and is not just a consequence of observing only part of the orbit.", "Second, this model predicts that the centroid positions will orbit along the two sides of a preferred axis, which correspond to the plasmoid being either above or below the black hole and relatively close to the spin axis.", "In other words, we expect that future observations will see centroids that either align with the directionality of the May 27th and July 22nd centroid positions (up and to the right in Figure REF ), or along the opposite orientation corresponding to the July 28th centroids (down and to the left in Figure REF ), but not in between (up and to the left, or down and to the right).", "Third, we predict that double-peaked flares with separations between the two peaks on timescales of $\\sim 40$ minutes should occur frequently for flares with the same orientation as the July 28th flare, but not for flares with the same orientation as the May 27th and July 22nd flares.", "This is not to say that double peaks are impossible in the framework of a posterior plasmoid setup (in the direction of May 27th and July 22nd flares): we already saw that a second peak can form due to the strong gravitational lensing during the plasmoids orbit.", "This second peak, however, will be small unless it occurs shortly after the formation of the plasmoid, before it has significantly cooled.", "Because of that, any double-peaked feature that may arise in a posterior plasmoid model (and hence, we predict, in the same direction as the May 27th and July 22nd flares) will be on much shorter timescales than the observed $\\sim 40$ minute delay between peaks for the July 28th flare.", "Furthermore, within this model, the spectrum during the second peak should look like a blue-shifted version of the spectrum during the first peak." ], [ "Conclusions", "In the tenuous and low-$\\beta $ plasma regions present in the inner accretion flow surrounding Sgr A*, reconnection events leading to short-lived particle acceleration episodes can be commonly expected.", "Earlier work had shown that such events are likely to be associated with flares and impart on these flares characteristics that are unique to the motion of plasmoids close to black holes.", "The GRAVITY events resolved the positions of bright regions for the first time during flaring activity, providing an opportunity to see if plasmoid motions, coupled with the effects of GR in the vicinity of the black hole, provide a natural explanation for the properties that have been observed.", "In this paper, we showed that a plasmoid model in the funnel region of a black hole can reproduce some of the important features of the GRAVITY observations and explains why there may be a preferred axis along which these flares are oriented and why the center of their motion is offset from the black hole.", "Additionally, we show that strong double-peaked flares are a generic consequence of a blob of plasma falling in towards a black hole when it is oriented on the same side of the black hole as the observer.", "In this setup, we show that the second peak in the lightcurve comes from the delayed lensed image and occurs at a time roughly $2r_0 + 15 \\; GM/c^3$ after the first peak, and that the relative strength of this second peak is amplified when the infall velocity is higher due to the relativistic Doppler boosting.", "We also make a number of predictions for future high-resolution observations of IR flares from Sgr A* that will provide a thorough test of this model.", "The centroid positions of the emission during the three GRAVITY flares define an axis, which we identify here with the spin axis of the black hole (or more precisely, in case the black hole is not spinning, with the angular momentum axis of the accretion flow).", "Within our model, this axis is oriented approximately 135 degrees East of North (or equivalently 135+180=315 degrees East of North).", "This orientation is broadly consistent with the inferred angular momentum axis of a cold disk around Sgr A$^*$ recently detected by ALMA at much larger distances than those probed by GRAVITY ($\\simeq 20,000$ Schwarzschild radii; [25]).", "Observations of Sgr A* with the Event Horizon Telescope offer the possibility of inferring the orientation of the angular momentum axis of the inner accretion flow by measuring the Doppler-induced asymmetry in the brightness of the emission surrounding the black hole, as was done for the case of M87 [14].", "Modeling early EHT observations of Sgr A* with semi-analytic [4], [6] and GRMHD simulations of accretion flows [9] resulted in orientations of 156$^{+10}_{-17}$  degrees and 160$^{+15}_{-86}$  degrees, respectively, which are consistent with the orientation we infer here.", "Besides inferring the orientation of the angular momentum axis in Sgr A*, EHT observations may also be able to identify whether significant morphological changes in the inner accretion flow are associated with an increase in emission during flares.", "This will constrain the possible flaring mechanisms, such as gravitational lensing, Doppler boosting of hot spots, and other such processes that will leave a distinct imprint on the image of the inner accretion flow.", "We thank Lorenzo Sironi for useful discussions.", "We gratefully acknowledge support for this work from NSF AST-1715061, Chandra Award No.", "TM6- 17006X, NASA ATP 80NSSC20K0521, and NSF PIRE 1743747." ] ]

[ [ "Light-nuclei production and search for the QCD critical point" ], [ "Abstract We discuss the potential of light-nuclei measurement in heavy-ion collisions at intermediate energies for the search of the hypothetical QCD critical end-point.", "A previous proposal based on neutron density fluctuations has brought appealing experimental evidences of a maximum in a ratio involving tritons, protons and deuterons, ${\\cal O}_{tpd}$.", "However these results are difficult to reconcile with the state-of-the-art statistical thermal model predictions.", "Based on the idea that the QCD critical point can lead to a substantial attraction among nucleons, we propose new ratios involving $^4$He in which the maximum would be more evident.", "We argue that the experimental extraction is feasible by presenting actual measurements at low and high collision energies.", "We also illustrate the possible behavior of these ratios at intermediate energies applying the semiclassical method based on flucton paths using preliminary STAR data for ${\\cal O}_{tpd}$." ], [ "Introduction", "Significant efforts, in both theory and experiment, have been devoted in heavy-ion collision physics to the search of the hypothetical QCD critical point and first-order transition between confined and deconfined matter [1].", "For the former, the bulk of these studies follows a common driving idea [2], [3]: the search of indicative observables showing a nonmonotonous behavior when measured as functions of a control parameter (collision energy, system size, rapidity acceptance...).", "The ultimate reason is the existence of physical quantities which present critical enhancement with the correlation length close to $T_c$  [1], [4].", "Such critical behavior should manifest as a nonmonotonous behavior when approaching and subsequently overpassing the critical end-point.", "An example of such observables have been the higher-order cumulants of the net-baryon distribution [5].", "In particular, the scaled kurtosis of the net-proton distribution as a function of the collision energy has led to some clues that might be related to the critical dynamics [5], [6], [7].", "In Ref.", "[8] we focused on nucleon dynamics and the $NN$ interaction as a contributing source for these quantities.", "The attractive part of the pairwise potential, as being dominated by the long-ranged critical ($\\sigma $ ) mode of QCD [9], is expected to have an important role in these observables.", "Focusing on more global observables we proposed that the increase of the nuclear attraction might lead to an extra production of light nuclei if the critical mode acts substantially among nucleons.", "In a favorable situation, we expect an enhanced production of light nuclei close to $T_c$ with respect to a noncritical scenario [10].", "The statistical thermal model (STM) [11] has been used as a good tool to describe multiplicities of hadronic states $N_i$ at high energies ($i$ denotes the type of hadron).", "In particular, its application to hadrons and light nuclei has been successfully tested at LHC energies [12].", "The application of this model typically ignores the effects of the $NN$ potential (although extensions of it do include interactions between baryons e.g. [13]).", "A sizable modification of the $NN$ potential would bring a small positive correction to the light-nuclei multiplicities $\\delta N_i$ .", "One expects that such modification cannot be observed when looking at total nuclear yield, as $\\delta N_i \\ll N_i$ .", "However one can construct certain nuclear multiplicity ratios in which the common thermodynamic factors (temperature, chemical potentials, volume...) cancel out for thermal abundances.", "Only the presence of nonideal thermodynamic effects (e.g.", "when the interaction potential is comparable to the temperature) makes these ratios non trivially dependent on the collision energy, giving a more sensitive observable to detect such modifications [10].", "STMs implement feed down corrections from higher unstable hadronic states and excited nuclei [14].", "This is a relevant effect which must be included in the calculation of these nuclear ratios.", "It turns out [15] that after the implementation of these feed down effects, some these ratios might remain, after all, rather flat in the relevant collision energy region.", "Therefore, a further attractive $NN$ potential—comparable to the typical freeze-out temperatures—would still introduce corrections producing eventual nonmonotonous ratios in consonance with the location of the critical region [8].", "In the following we will review the recent theoretical advances in this direction, and motivate the study of such ratios by presenting the status of the ratio ${\\cal O}_{tpd}$ involving tritons, protons and deuterons.", "Later, we will introduce different proposals in which the effect of the critical point is expected to be enhanced, and motivate our experimental colleagues to address these ratios by presenting some experimental results measured away from the critical region, and the expectations coming from a novel theoretical approach based on a semiclassical method at finite temperature." ], [ "\"Preclusters\" and the special role of four-nucleon systems", "It is a well-known fact that nuclear forces include two delicate cancellations, between the attractive and repulsive parts of the $NN$ potential, as well as between potential and kinetic energies [9].", "As a result, neutron systems are all unbound, while deuteron and triton have only one shallow state.", "The champion in small binding is perhaps the $pn\\Lambda $ system, with an exact value of the binding energy still in dispute [16], [17].", "However the situation drastically changes for the four-nucleon systems.", "The $^4$ He binding energy of 28 MeV is much larger than that of lighter systems.", "And—as we pointed out in [10]— unlike these, it possesses multiple bound and resonant states.", "Their decay should feed down into production of lighter systems.", "Even if they eventually decay back into four independent nucleons, their correlation in phase space is important, contributing positively to the kurtosis of the proton multiplicity distribution [8].", "In our previous work [10] we introduced the notion of “preclusters”, or statistical spatial correlation of several nucleons at the kinetic freeze-out.", "While matter reaches a good degree of equilibration by that time, and formation of preclusters themselves is subject to equilibrium statistical mechanics, the subsequent transition of preclusters to clusters—light nuclei states or resonances at zero temperature and density—have a completely different temporal and spatial scales and is dynamical.", "For example, for four nucleons (to be mostly discussed) the splitting between states and widths all have the magnitude of few MeV, and thus their separation and decay require times of hundreds of fm$/c$ .", "While preclusters are compact objects, with sizes $1-2$ fm, the clusters and their decay products are much larger in size.", "This distinction helps to understand a well-known paradox of apparently copious production of large-size and fragile clusters.", "Quantum statistical calculations of four nucleons is in practice not trivial, as the system has 9 coordinates.", "In [10] we also developed two theoretical tools able to calculate properties and production rates of preclusters.", "We extended a novel semiclassical method for the density matrix, based on special classical paths called “fluctons” [18], [19] to finite temperatures.", "We now prepare a separate methodical paper [20] on this method, as its applications can be extended well beyond heavy-ion physics.", "Alternatively, we extended the method of hyperspherical harmonics, previously used only for ground states of few-nucleon systems, to calculation of ${\\cal O}(100)$ quantum states, which also allowed to calculate the thermal density matrix.", "Furthermore, in a soon-coming work [21] the most straightforward quantum mechanical method, that of path integral simulations, is used.", "It allows to calculate quantum effects for many-body systems in multidimensional settings, without any approximations.", "In all of the above mentioned works the “preclustering” phenomenon is well documented, both for the standard (unmodified) nuclear forces and those with potential modifications, due to either finite temperature-density effects or a hypothetical critical point.", "In all of them one observes the same basic fact: preclustering is extremely sensitive to forces, especially at large distances.", "Therefore, light nuclei production must be a good observable to study.", "Later we will apply the finite-temperature flucton method described in Ref.", "[10] to obtain the distribution of the relative nucleon distance as a function of the $NN$ potential.", "For the latter, we will make use of the Serot-Walecka potential as given in [8], [10], whose attractive part is dominated—close to the critical point—by the $\\sigma $ -excitation mass.", "This method will be used to compute several nuclear multiplicity ratios.", "Before, we start by reviewing the experimental situation of one of these special ratios." ], [ "Triton-proton-deuteron ratio", "Based on the coalescence model for nuclei production, the authors of Ref.", "[22] considered neutron density fluctuations quantified by $\\Delta n=\\langle (\\delta n)^2\\rangle /\\langle n\\rangle ^2$ , which is sensitive to the QCD critical point.", "This quantity controls the ratio $ {\\cal O}_{tpd} \\equiv \\frac{N_t N_p}{N_d^2} \\simeq 0.29 (1+\\Delta n) \\ , $ which involves tritons, protons and deuterons.", "The coefficient $0.29$ is a combination of numerical factors and spin degeneracies of the involved nuclei.", "In a situation with negligible fluctuations one simply has ${\\cal O}_{tpd}=0.29$ , independent of the collision energy.", "From a different perspective, the pure thermodynamical production of a nucleus $i$ composed by $A$ nucleons is the Boltzmann factor describing the well-known exponential suppression with $A$  [11].", "The equilibrium number of nuclei is $ N_i \\simeq \\frac{g_i {\\cal V}}{(2\\pi )^{3/2}} (m_i T)^{3/2} \\exp \\left(-\\frac{m_i}{T}\\right) \\exp \\left(\\frac{\\sum _{a} Q_i^a \\mu _i^a}{T} \\right)\\ , $ where $g_i$ is the spin degeneracy of nucleus $i$ , ${\\cal V}$ the volume, $m_i \\simeq Am_N$ the mass of the nucleus, $T$ the temperature and $Q_i^a,\\mu _i^a$ the corresponding charges and chemical potentials ($a=\\lbrace B,Q,S \\rbrace )$ .", "By forming multiplicity ratios in which this exponential factor cancels out, one might obtain some interesting results.", "As we emphasized in [10], that should come from different powers of the interaction potential per nucleon pair.", "In particular, for the ratio in (REF ), $ {\\cal O}_{tpd} \\simeq 0.29 \\frac{\\langle e^{-3V(r)/T} \\rangle }{\\langle e^{-V(r)/T}\\rangle ^2 } \\ , $ where the thermal average $\\langle \\rangle $ reflects the nonideal contribution of the internucleon potential $V(r)$ averaged in space.", "In the ideal case (REF ), where the interaction energy is negligible with respect to temperature, one again obtains ${\\cal O}_{tpd}=0.29$ .", "While a rigorous treatment of clusterization in the vicinity of the critical point is not yet developed, it is intuitively clear that the increase of attraction might lead to formation of precluster structures, if the nucleons can feel the presence of the critical region long enough.", "Therefore, the conceptual conclusion in both approaches (REF ) and (REF ) is similar: close to $T_c$ both $\\Delta n$ and $|V(r)/T|$ might become ${\\cal O}(1)$ and the ratio will present a maximum for those collisions passing close to the QCD critical point.", "The authors of Ref.", "[22] presented evidences of such a maximum around $\\sqrt{s_{NN}}=8.8$ GeV in the results of the NA49 experiment [23] (a second peak was also suggested in [24]).", "More recently, preliminary STAR data showed a more prominent peak in ${\\cal O}_{tpd}$ around $\\sqrt{s_{NN}} =27$ GeV [25].", "These results are really intriguing.", "However the main puzzle resides in the comparison of the experimental data with the up-to-date calculations using the STM, and also coalescence studies [26].", "Not only there is a lack of agreement among models, but also between theory and experimental data.", "One of the known problems [25] is the apparent overestimate of tritons by the STM, whereas protons and deuterons seem to be well described [15], [25].", "Figure: 𝒪 tpd {\\cal O}_{tpd} ratio as a function of the center-of-mass collision energy.", "Upper panel: Ratio formed from FOPI multiplicity results at low energies .", "Bottom panel: Same FOPI-based data together with NA49 , STAR  and ALICE , , preliminary results.", "NA49, STAR and ALICE results are based, not in ratios of total multiplicities, but in dN/dydN/dy ratios at midrapidity, and in the later, the triton yield is traded by 3 ^3He.", "The horizontal line indicated the value 0.29, cf.", "Eq ().Focusing on experimental results, we present the ratio ${\\cal O}_{tpd}$ formed from published FOPI data [27] at low energies in the upper panel of Fig REF .", "FOPI Collaboration has collected yields of several nuclei in heavy-ion collisions for 25 combination of colliding heavy-nuclei and collision energies.", "We focus on the most central Au+Au collisions for the beam energies (per nucleon) $E/A=$ 0.12, 0.15, 0.25, 0.4, 0.6, 0.8, 1.0, 1.2, 1.5 GeV (we eventually show them converted to center-of-mass energy).", "From the final products we consider $p, d, t, ^3$ He and $^4$ He, measured after a $4\\pi $ reconstruction [27].", "From the reported total yields we build the needed light-nuclei multiplicity ratio, and assign an error bar by propagating the experimental uncertainties.", "We observe that the ratio ${\\cal O}_{tpd}$ using FOPI results is rather flat, and the data might be already affected by the presence of the multifragmentation regime at low energies.", "In the lower panel, we plot together the FOPI results with NA49 data [23], and the preliminary STAR [25] and ALICE [28], [29], [30] results.", "While FOPI uses $4\\pi $ -reconstructed multiplicities, the other experiments consider $dN/dy$ values at midrapidity, instead of total yields.", "Moreover tritons in ALICE are difficult to disentangle from other hadrons via energy loss only, and additional analyses (with the time-of-flight detector) are needed.", "Yet it is possible to substitute their yields by $^3$ He, which is expected to be similar at these energies based on isospin symmetry We thank Benjamin Dönigus for pointing this out to us [12]..", "It is interesting that FOPI data at its top energies are compatible with the ALICE results.", "However ALICE point is known to be above the simple-minded thermal value for a well-known reason: protons get a significant contribution from feed down of excited baryons [11].", "There is no feasible feed down to $t,d$  [15].", "The opposite situation is at FOPI energies: here temperature is too low to excite baryons, but $\\mu _B$ is close to the nucleon mass and there should be large clustering feed downs, both to $t$ and $d$  [15].", "Therefore, the similar result of FOPI and LHC must be a coincidence.", "It is also curious to observe that while the degree of feed-down should be similar, the STAR point with the smallest error (corresponding to the main energy of RHIC operations, $\\sqrt{s_{NN}} =200$ GeV) is away from the ALICE point, and close to the Boltzmann prediction (without feed-down).", "As we already said, ALICE data was very well described by the STM [12], and the STAR result on triton multiplicity [25] presents a discrepancy with the STM prediction.", "This discrepancy remains for lower energies, and should certainly be addressed in the future.", "Both NA49 and STAR data tell us that some nontrivial dynamics might be happening at intermediate energies, with an important medium modification with respect to the ideal gas case.", "Among them, there is the possible peak signaling the critical point, in accordance with the physics encoded in Eqs.", "(REF ) and (REF )." ], [ "Ratios involving $^4$ He", "Based on the idea that a modified $NN$ potential could lead to preclustering effects close to $T_c$  [8], more interesting ratios can be considered where the net effect is enhanced [10].", "A more prominent ratio would involve larger nuclei thus increasing the number of mutual nucleon pairs.", "In that case one automatically increases the powers of $V/T$ in the exponent of (REF ).", "However the bigger the nuclei, the larger “penalty factor” in the thermal multiplicity [12], as nuclei with increasing mass number $A$ are much more scarce in the final state as can be seen from Eq.", "(REF ).", "The nucleus of $^4$ He ($=\\alpha $ ), with six mutual distances, provides the optimal case.", "This state has been measured in low- and high-energy heavy-ion collisions, so it is a good candidate to form light-nuclei multiplicity ratios.", "The importance of the excited states of $^4$ He in the context of thermal models has been addressed in Ref.", "[10], and put into practice in Refs.", "[31], [15] for the ${\\cal O}_{tpd}$ ratio following the ideas of Refs.", "[14], [32].", "Constructing ratios that cancel most of the factors in Eq.", "(REF ) leads to [10] $ {\\cal O}_{\\alpha p^3{\\rm He}d} \\equiv \\frac{N_{\\alpha } N_p}{N_{^3\\rm {He}} N_d} \\simeq 0.18 \\frac{\\langle e^{-6V(r)/T}\\rangle }{\\langle e^{-3V(r)/T} \\rangle \\langle e^{-V(r)/T} \\rangle } \\ , $ and $ {\\cal O}_{\\alpha tp^3{\\rm He}d} \\equiv \\frac{N_{\\alpha } N_t N_p^2}{N_{^3\\rm {He}} N_d^3} \\simeq 0.05 \\frac{\\langle e^{-6V(r)/T}\\rangle }{\\langle e^{-V(r)/T} \\rangle ^3} \\ , $ where $0.18$ and $0.05$ are the remaining numerical factors after cancellations (the last one corrects the incorrect prefactor quoted in Ref.", "[10] for this ratio).", "Notice that in comparison to ${\\cal O}_{tpd}$ , these ratios are globally enhanced by two and three powers of $V(r)/T$ in the exponential, respectively.", "Figure: 𝒪 αp 3 He d {\\cal O}_{\\alpha p^3{\\rm He}d} ratio as a function of the center-of-mass collision energy.", "Upper panel: Ratio formed from FOPI multiplicity results  at low energies.", "Lower panel: Same FOPI-based data together with ALICE results , , at high energies.", "The ALICE result is formed, not from total multiplicities, but from dN/dydN/dy ratios at midrapidity.", "We provide a theory estimate of such ratio in the intermediate energies (points without error bars) as explained in the text.", "The horizontal line indicated the value 0.18, cf.", "Eq ().In the upper panel of Fig.", "REF we present the first multiplicity ratio ${\\cal O}_{\\alpha p^3{\\rm He} d}$ built from FOPI yield data [27] at low collision energies.", "As opposed to the previous case, here there is a clear decreasing tendency with $\\sqrt{s_{NN}}$ .", "At very low energies might be difficult to make contact with thermal estimates, as the system enters into the multifragmentation region.", "In the lower panel of Fig.", "REF we add the preliminary result at LHC energies, from ALICE experiment [28], [29], [30].", "As for ${\\cal O}_{tdp}$ , the top energy FOPI result and the LHC are compatible between them, but should be a coincidence as explained before.", "Unfortunately, these two sets of experimental data are located at opposed limits in collision energy, away from the expected range corresponding to a possible critical point.", "We know no experimental data that could be analyzed in that range, calling for an experimental effort to explore that interesting region.", "We can offer an estimate of this ratio applying a simple model, and using the NA49 and STAR data for the ratio (REF ) as input.", "For this we will neglect feed down contributions (which can be added along the lines of [15]), and assume that the structure shown by NA49 and STAR in ${\\cal O}_{tpd}$ is entirely due to the effect of a modified potential as described by Eq.", "(REF ).", "We apply the semiclassical thermal flucton method described in [10] to the Serot-Walecka potential [9] between nucleons.", "After a Wick rotation in time, the path integral representation of the density matrix contains the exponential factor of (minus) the Euclidean action of the field configuration, $S_E$ .", "For a pair of nucleons interacting via the potential $V(r)$ , the corresponding Euclidean action reads [10] $S_E [r(\\tau )]=\\int _0^\\beta d\\tau \\left[ \\frac{m_N}{4} \\left( \\frac{dr}{d\\tau } \\right)^2 + V(r) \\right] \\ .", "$ The periodic configuration that minimizes $S_E[r(\\tau )]$ , i.e.", "solution of the classical equation of motion, is the so-called flucton path $r_{{\\rm fluc}}(\\tau )$  [18].", "In the semiclassical approximation we assume that this flucton path dominates the path integral, so we can compute the probability density of the mutual distances as $P(r)= \\exp \\lbrace -S_{E} [r =r_{\\rm fluc} (\\tau )] \\rbrace $ .", "In turn, this probability density allows us to define the spatial average of any observable $A(r)$ by doing $\\langle A \\rangle \\equiv \\frac{4\\pi \\int dr r^2 A(r) [P(r)-1]}{4\\pi \\int dr r^2 [P(r)-1]} \\ , $ where the $-1$ is used to render the probability density integrable [10], and the denominator normalizes it to one.", "In this way, the ratios of Eqs.", "(REF ),(REF ),(REF ) are directly determined by the flucton solution of the Walecka potential with the $\\sigma $ mass and $T$ as parameters.", "The temperature dependence on $\\sqrt{s_{NN}}$ is obtained from the STM as given in Ref. [11].", "Therefore, the method utilizes the experimental light-nuclei multiplicity ratios (REF ) from STAR and NA49 experiments to fix the $\\sigma $ mass (and therefore set the $NN$ potential) for each collision energy.", "Then, the same potential is applied to the flucton path method to generate the new multiplicity ratios (REF ) and (REF ).", "This gives an estimate of the expected behavior in the intermediate energy range.", "Note that the semiclassical method is not appropriate for very low energies where the multifragmentation region and quantum effects start to dominate the nucleon dynamics.", "In the conditions of high baryon densities the validity of the thermal flucton path is doubtful.", "The theoretical points (without error bars) are shown in the lower panel of Fig.", "REF .", "We observe the same peak structure as in Fig.", "REF but the strength of the effect, i.e.", "the difference between the top and bottom theory points, is a factor of 2 larger than in the ratio ${\\cal O}_{tpd}$ .", "Figure: 𝒪 αtp 3 He d {\\cal O}_{\\alpha tp^3{\\rm He}d} ratio as a function of the center-of-mass collision energy.", "Upper panel: Ratio formed from FOPI multiplicity results  at low energies.", "Lower panel: Same FOPI-based data together with ALICE at high energies , , .", "The ALICE result is formed, not from total multiplicities, but from dN/dydN/dy ratios at midrapidity.", "We provide a theory estimate of such ratio in the intermediate energies (points without error bars) as explained in the text.", "The horizontal line indicated the value 0.05, cf.", "Eq ().This effect is enhanced in the multiplicity ratio ${\\cal O}_{\\alpha tp^3 {\\rm He} d}$ , which we show in Fig.", "REF .", "The upper panel shows the ratio built with FOPI data [27], which presents again a decreasing pattern with $\\sqrt{s_{NN}}$ , not seen in ${\\cal O}_{tdp}$ .", "Such a pattern deserves more attention in the context of a possible medium-modified nuclear dynamics.", "In the lower panel we plot ALICE data at $\\sqrt{s_{NN}}=2760$ GeV.", "In this case we again substitute $t$ by $^3$ He.", "Finally, we also plot the theory estimate using the procedure explained before, but applied to the new ratio.", "The structure follow the same qualitative pattern as the previous ratio, namely, the top energy FOPI data and LHC are similar, being the intermediate energy prediction lower in magnitude, but with the familiar peak around $\\sqrt{s_{NN}}=27$ GeV.", "Notice that in this case, the strength of the predicted effect (difference between top and bottom points in the intermediate energy region) is a factor of 5 larger with respect to ${\\cal O}_{tpd}$ , indicating that the signal-to-noise ratio considerably improves in this ratio.", "This is the reason why we encourage the experimental collaborations to look into these new ratios at intermediate energies.", "Finally, we would like to point out that some studies which incorporate partial chemical equilibration, are able to describe light-nuclei production in heavy-ion collisions using kinetic freeze-out temperatures as low as $T_{{\\rm kin}} \\simeq 100$ MeV [33], [34].", "To check the sensitivity to the temperature of our method, we have repeated this calculation for the ratios ${\\cal O}_{\\alpha p ^3 {\\rm He} d}$ and ${\\cal O}_{\\alpha t p^3 {\\rm He}d}$ using a constant $T=100$ MeV for all intermediate collision energies.", "As this temperature is lower than the temperatures we used from Ref.", "[11], one requires less degree of modification of the $V(r)$ (remember that the main effect enters as $|V(r)|/T$ ).", "Therefore, we obtain systematically larger $\\sigma $ masses, and the final ratios become systematically smaller.", "The largest differences with respect to Figs.", "REF ,REF appear around the peak structures, being in any case no greater than 5% (the biggest deviation occurs for ${\\cal O}_{\\alpha t p^3 {\\rm He}d}$ at $\\sqrt{s_{NN}} = 27$ GeV where using $T=157$ MeV or $T=100$ MeV leads to ${\\cal O}_{\\alpha t p^3 {\\rm He}d}=0.70$ and $0.67$ , respectively).", "Therefore the differences between using $T=100$ MeV and the values of the chemical freeze-out curve in Ref.", "[11] are insignificant in the flucton method as applied here." ], [ "Summary", "In this communication we have proposed new light-nuclei multiplicity ratios (REF )(REF ) involving $^4$ He ($=\\alpha $ ), ${\\cal O}_{\\alpha p ^3 {\\rm He} d} \\ , \\qquad {\\cal O}_{\\alpha t p^3 {\\rm He}d} \\,$ which are sensitive to the nonideal production of light-nuclei in heavy-ion collisions.", "Should the QCD critical point enhance the attraction between nucleons to the extend of an increase of light-nuclei formation, then these ratios would present maxima as functions of the collision energy.", "Such peaks would be more compelling than the one considered so far in Eq.", "(REF ), as the variability in ${\\cal O}_{\\alpha p ^3 {\\rm He} d}$ and ${\\cal O}_{\\alpha t p^3 {\\rm He}d}$ is, respectively, a factor of 2 and 5, compared to ${\\cal O}_{tpd}$ .", "We have demonstrated the feasibility of such measurements by presenting experimentally-based multiplicity ratios at low and high energies by FOPI and ALICE experiments, respectively.", "We have also motivated the usefulness of our proposal by computing a theoretical estimate of these new ratios at intermediate energies based on previous NA49 and STAR data for the ratio of Eq.", "(REF ).", "We exploited the semiclassical thermal flucton method applied to the Serot-Walecka potential, as detailed in Ref. [10].", "We hope that experiments running at intermediate energies can address these observables in the future to assess the possible effect of critical dynamics via light-nuclei production.", "We acknowledge Benjamin Dönigus for bringing the FOPI and LHC data on light-nuclei production to our attention, and for useful discussions.", "Work supported by the Office of Science, U.S. Department of Energy under Contract No.", "DE-FG-88ER40388 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Projects No.", "411563442 (Hot Heavy Mesons) and No.", "315477589 - TRR 211 (Strong-interaction matter under extreme conditions)." ] ]

[ [ "Quantum Sampling Algorithms for Near-Term Devices" ], [ "Abstract Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning.", "We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution.", "We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples including the Ising model and sampling from weighted independent sets of two different graphs.", "Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup.", "Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays." ], [ "Parent Hamiltonians", "Our construction of the parent Hamiltonian follows the prescription in reference [23] (see references [24], [25], [26] for related earlier work).", "We first define a Markov chain that samples from the desired Gibbs distribution $p_\\beta (s)$ .", "The Markov chain is specified by a generator matrix $M_\\beta $ , where the probability distribution $q_t(s)$ at time $t$ evolves according to $q_{t+1}(s) = \\sum _{s^{\\prime }} q_t(s^{\\prime }) M_\\beta (s^{\\prime }, s)$ .", "By construction, $p_\\beta (s)$ is a stationary distribution of the Markov chain and therefore constitutes a left eigenvector of $M_\\beta $ with eigenvalue unity.", "We assume in addition that the Markov chain satisfies detailed balance, which can be expressed as $e^{- \\beta H_c(s^{\\prime })} M_\\beta (s^{\\prime }, s) = e^{- \\beta H_c(s)} M_\\beta (s, s^{\\prime })$ .", "This property implies that $H_q(\\beta ) = n \\left( \\mathbb {I} - e^{-\\beta H_c / 2} M_\\beta e^{\\beta H_c/2} \\right)$ is a real, symmetric matrix.", "It can be viewed as a quantum Hamiltonian with the Gibbs state $\\vert \\psi (\\beta ) \\rangle $ being its zero-energy eigenstate.", "The Gibbs state is a ground state because the spectrum of $H_q(\\beta )$ is bounded from below by 0.", "For every eigenvalue $n (1 - \\lambda )$ of $H_q(\\beta )$ , there exists an eigenvalue $\\lambda $ of $M_\\beta $ , where $\\lambda \\le 1$ because $M_\\beta $ is a stochastic matrix.", "If the Markov chain is fully mixing and aperiodic, the Perron–Frobenius theorem [27] guarantees that $\\vert \\psi (\\beta ) \\rangle $ is the unique ground state of $H_q(\\beta )$ .", "The factor of $n$ in equation (REF ) ensures that the spectrum of the parent Hamiltonian is extensive.", "To account for the natural parallelization in adiabatic evolution, we divide the mixing time of the Markov chain by $n$ for a fair comparison, denoting the result by $t_m$ .", "The correspondence between the spectra of $M_\\beta $ and $H_q(\\beta )$ establishes the bound $t_m \\ge 1/\\Delta (\\beta )$ , where $\\Delta (\\beta )$ is the gap between the ground state and first excited state of the parent Hamiltonian [28].", "We now illustrate this procedure by considering a ferromagnetic Ising model composed of $n$ spins in one dimension.", "The classical Hamiltonian is given by $H_c = -\\sum _{i=1}^n \\sigma _i^z \\sigma _{i+1}^z$ with periodic boundary conditions, letting $\\sigma _{n+1}^z = \\sigma _1^z$ and $\\sigma _0^z = \\sigma _n^z$ .", "We choose Glauber dynamics as the Markov chain, in which at each time step, a spin is selected at random and its value is drawn from a thermal distribution with all other spins fixed [29].", "Up to a constant, the corresponding parent Hamiltonian takes the form Hq() = -i=1n [ h() ix .", "+ J1() iz i+1z - .", "J2() i-1z ix i+1z ] , where $4 h(\\beta ) = 1+1/\\cosh (2 \\beta )$ , $2 J_1(\\beta ) = \\tanh (2 \\beta )$ , and $4 J_2(\\beta ) = 1 - 1/\\cosh (2 \\beta )$ (see supplementary information (SI) for details and references [30], [31] for early derivations of this result).", "At infinite temperature ($\\beta = 0$ ), we have $J_1 = J_2 = 0$ and $h = 1/2$ .", "The ground state is a paramagnet aligned along the $x$ -direction, which corresponds to an equal superposition of all classical spin configurations, consistent with the Gibbs distribution at infinite temperature.", "When the temperature is lowered, the parameters move along a segment of a parabola in the two-dimensional parameter space $(J_1/h, J_2/h)$ shown by the red curve (ii) in Fig.", "REF a.", "We highlight that any point along the segment can be viewed as a generalized Rokhsar–Kivelson point [32], [25], where the Hamiltonian is by construction stoquastic and frustration free [26], [33].", "The quantum phase diagram of the parent Hamiltonian for arbitrary values of $h$ , $J_1$ , and $J_2$ is obtained by performing a Jordan–Wigner transformation that maps equation (REF ) onto a free-fermion model [34], [35] (see SI).", "The distinct quantum phases are displayed in Fig.", "REF a.", "The model reduces to the transverse field Ising model on the $J_2/h = 0$ axis, in which a phase transition from a paramagnet to a ferromagnet occurs at $J_1/h = 1$ .", "Along the $J_1/h = 0$ axis, the ground state undergoes a symmetry-protected topological phase transition at $J_2/h = \\pm 1$ from the paramagnet to a cluster-state-like phase [36].", "We note that the tricritical point at $(J_1/h, J_2/h) = (2,1)$ describes the parent Hamiltonian corresponding to zero temperature ($\\beta \\rightarrow \\infty $ ).", "To prepare the Gibbs state $\\vert \\psi (\\beta ) \\rangle $ , one may start from the ground state of $H_q(0)$ before smoothly varying the parameters $(h, J_1, J_2)$ to bring the Hamiltonian into its final form at the desired inverse temperature $\\beta $ .", "States with finite $\\beta $ can be connected to the infinite temperature state by a path that lies fully in the paramagnetic phase.", "Both adiabatic state preparation and the Markov chain are efficient in this case.", "Indeed, it has been shown previously that there exists a general quantum algorithm with run time $\\sim \\log n$ for gapped parent Hamiltonians [22], which is identical to the Markov mixing time $t_m$ for the Ising chain [37].", "Figure: Sampling from the 1D Ising model.", "a, Phase diagram of the parent Hamiltonian corresponding to the Ising chain.", "The black lines indicate the boundaries between paramagnetic (PM), ferromagnetic (FM), and cluster-state-like (CS) phases.", "The curves labeled (i)–(iv) show four different choices of adiabatic paths with (ii) representing the one-parameter family H q (β)H_q(\\beta ).", "b, The time dependence of J 1 /hJ_1/h for a chain of n=100n = 100 spins according to equation () of the Methods section.", "c, Fidelity as a function of the sweep time along these trajectories.", "d, The time t a t_a required to reach a fidelity exceeding 1-10 -3 1-10^{-3} as a function of the number of spins nn.", "The black lines are guides to the eye showing the expected linear, quadratic, and cubic dependencies of t a t_a on nn.Sampling at zero temperature is more challenging with the mixing time of the Markov chain bounded by $t_m \\gtrsim n^2$ (see SI).", "For the quantum algorithm, we consider the four different paths in Fig.", "REF a.", "To evaluate the dynamics quantitatively, we choose the rate of change with the aim of satisfying the adiabatic condition at every point along the path (see Fig.", "REF b and Methods) and numerically integrate the Schrödinger equation with the initial state $\\vert \\psi (0) \\rangle $ to obtain $\\vert \\phi (t_\\mathrm {tot}) \\rangle $ after total evolution time $t_\\mathrm {tot}$ (without loss of generality we set $h = 1$ ).", "Figure REF c shows the resulting fidelity $\\mathcal {F} = | \\langle \\phi (t_\\mathrm {tot}) | \\psi (\\infty ) \\rangle |^2$ for a chain of $n = 100$ spins.", "The total variation distance $d = ||p - q||$ between the desired Gibbs distribution and the prepared distribution $q(s) = |\\langle s | \\phi (t_\\mathrm {tot}) \\rangle |^2$ is bounded by $d \\le \\sqrt{1-\\mathcal {F}}$  [38].", "To determine the dependence on the number of spins, we extract the time $t_a$ at which the fidelity exceeds $1-10^{-3}$ , Fig.", "REF d. We find three different scalings of the time $t_a$ : along path (i), it roughly scales as $t_a \\sim n^3$ , along (ii) as $t_a \\sim n^2$ , while (iii) and (iv) exhibit a scaling close to $t_a \\sim n$ .", "These scalings follow from the nature of the phase transitions.", "The dynamical critical exponent at the tricritical point is $z=2$ , meaning that the gap closes with system size as $\\Delta \\sim 1/n^2$ , which is consistent with the time required along path (ii).", "As shown in the SI, the dynamical critical exponent at all phase transitions away from the tricritical point is $z=1$ and the gap closes as $\\Delta \\sim 1/n$ .", "Therefore, the paramagnetic to ferromagnetic phase transition can be crossed adiabatically in a time proportional to $n$ , only limited by ballistic propagation of domain walls as opposed to diffusive propagation in the Markov chain.", "There is no quadratic slowdown as paths (iii) and (iv) approach the tricritical point, which we attribute to the large overlap of the final state with ground states in the ferromagnetic phase.", "Path (i) performs worse than path (ii) because the gap between the paramagnetic and cluster-state-like phases vanishes exactly for certain parameters even in a finite-sized system (see SI).", "To further support the statement that the speedup is of quantum mechanical origin, we note that the half-chain entanglement entropy of the ground state diverges logarithmically with the number of spins when paths (iii) and (iv) cross from the paramagnetic into the ferromagnetic phase.", "Hence, it is impossible to find a representation for the ground state at the phase transition in the form of equation (REF ) with a local, classical Hamiltonian $H_c$ because any such representation would be a matrix product state with constant bond dimension and bounded entanglement entropy [23].", "While this example illustrates a mechanism for quantum speedup, sampling from large systems is hard only at zero temperature [39].", "Since sampling at zero temperature is equivalent to optimization, there may exist more suitable algorithms to solve the problem.", "In addition, the parent Hamiltonian, equation (REF ), does not have a simple physical realization.", "We address these limitations by considering the weighted independent set problem.", "An independent set of a graph is any subset of vertices in which no two vertices share an edge, see Fig.", "REF a.", "We say vertex $i$ is occupied if it is in the independent set and assign the occupation number $n_i = 1$ .", "All other vertices are unoccupied with $n_i = 0$ .", "In the weighted independent set problem, each vertex is further assigned a weight $w_i$ and we seek to minimize the energy $H_c = - \\sum _i w_i n_i$ subject to the independent set constraint.", "The corresponding Gibbs distribution has been studied extensively in probability theory and computer science [40] as well as in statistical physics [41], [42].", "To construct a quantum algorithm, each vertex is associated with a spin variable $\\sigma _i^z = 2 n_i - 1$ .", "Single spin flips with the Metropolis–Hastings update rule [43] yield the parent Hamiltonian Hq() = i Pi [ Ve,i() ni + .", "[ .", "Vg,i() (1 - ni) - i() ix ], where we only consider the subspace spanned by the independent sets (see SI).", "In equation (REF ), $P_i= \\prod _{j \\in \\mathcal {N}_i} (1 - n_j)$ projects onto states in which the nearest neighbors $\\mathcal {N}_i$ of vertex $i$ are all unoccupied.", "The remaining parameters are given by $V_{e,i}(\\beta ) = e^{- \\beta w_i}$ , $V_{g,i}(\\beta ) = 1$ , and $\\Omega _i(\\beta ) = e^{- \\beta w_i/2}$ .", "The projectors $P_i$ involve up to $d$ -body terms, where $d$ is the degree of the graph.", "Nevertheless, they can be implemented e.g.", "using programmable atom arrays with minimal experimental overhead for certain classes of graphs.", "In the case of so-called unit disk graphs, Fig.", "REF a, these operators are naturally realizable using highly excited Rydberg states of neutral atoms (see Fig.", "REF b and Methods).", "As a simple example of a unit disk graph, we consider a chain of length $n$ and choose equal weights $w_i = 1$ .", "The resulting parent Hamiltonian has been studied both theoretically [44], [45] and experimentally using Rydberg atoms [46].", "Its quantum phases can be characterized by the staggered magnetization $M_k = (1/n) \\sum _{j=1}^n e^{2 \\pi i j / k} \\sigma _j^z$ .", "Figure REF c shows the ground state expectation value of $|M_2| + |M_3|$ for $n=30$ , clearly indicating the presence of three distinct phases.", "For large $\\Omega /V_g$ or large, positive $V_e/V_g$ , assuming $V_g > 0$ throughout, the ground state respects the full translational symmetry of the Hamiltonian and $|M_k|$ vanishes for all integers $k > 1$ .", "When $V_e/V_g$ is sufficiently small, the ground state is $\\mathbb {Z}_2$ ordered with every other site occupied and $|M_2| \\ne 0$ .", "Owing to next-to-nearest neighbor repulsive terms in the Hamiltonian, there further exists a $\\mathbb {Z}_3$ ordered phase, in which $|M_3| \\ne 0$ and the ground state is invariant only under translations by three lattice sites or multiples thereof.", "Figure: Sampling from independent sets of unit disk graphs.", "a, Example of an independent set (red vertices) on a unit disk graph.", "Two vertices are connected if and only if they are separated by a distance less than RR.", "b, Physical realization of the parent Hamiltonian, equation (), using Rydberg blockade as discussed in the Methods section.", "The Hamiltonian parameters are determined by the drive amplitudes ℰ i \\mathcal {E}_i and ℰ i ' \\mathcal {E}_i^{\\prime } as well as their detuning from resonance.", "c, Parameter space and order parameter of the parent Hamiltonian for a chain of length n=30n=30.", "The order parameter |M 2 |+|M 3 ||M_2| + |M_3| distinguishes the disordered phase from the ℤ 2 \\mathbb {Z}_2 and ℤ 3 \\mathbb {Z}_3 ordered phases.", "The red curve (i) indicates the one-parameter family H q (β)H_q(\\beta ), while the blue curve (ii) is an alternative adiabatic path crossing into the ℤ 2 \\mathbb {Z}_2 phase.", "d, Adiabatic state preparation time t a t_a along the two paths in (c).", "Path (i) terminates at β c =2logn\\beta _c = 2 \\log n while (ii) continues to β→∞\\beta \\rightarrow \\infty .", "The sweep rate was chosen according to equation () and the threshold fidelity was set to ℱ=1-10 -3 \\mathcal {F} = 1- 10^{-3}.", "The black lines are guides to the eye showing the scalings t a ∼nt_a \\sim n and t a ∼n 4 t_a \\sim n^4.The one-parameter family $H_q(\\beta )$ is indicated by the red curve (i) in Fig.", "REF c. We note that $\\vert \\psi (0) \\rangle $ is not a product state.", "However, the Hamiltonian $H_q(0)$ can be adiabatically connected to $\\Omega /V_g = 0$ and $V_e/V_g > 3$ , where the ground state is a product state of all sites unoccupied.", "Since such a path may lie fully in the disordered phase, adiabatic state preparation of the zero temperature Gibbs state is efficient.", "Similarly, the Markov chain at infinite temperature is efficient as the parent Hamiltonian is gapped.", "More generally, we show numerically in the SI that the gap is proportional to $e^{- 2\\beta }$ at high temperature and $e^{- \\beta }/n^2$ at low temperature.", "In contrast to the Ising chain, the gap vanishes as $\\beta \\rightarrow \\infty $ even for finite sized systems.", "The physical reason is that defects in the $\\mathbb {Z}_2$ ordering must overcome an energy barrier to propagate.", "Since the Markov chain is not ergodic at zero temperature, it is only possible to sample approximately from the ground state by running the Markov chain at a low but nonzero temperature $\\beta \\gtrsim \\beta _c$ , where $\\beta _c = 2 \\log n$ corresponds to the temperature at which the correlation length is comparable to the system size.", "The gap of the parent Hamiltonian bounds the mixing time by $t_m \\gtrsim e^{2 \\beta _c} \\sim n^4$ .", "As shown in Fig.", "REF d, the adiabatic state preparation time along the one-parameter family $H_q(\\beta )$ appears to follow the same scaling.", "A quantum speedup is obtained by choosing a different path.", "For example, an approximately linear scaling $t_a \\sim n$ , is observed along path (ii) in Fig.", "REF c. In analogy to the Ising chain, we attribute the linear scaling to the dynamical critical exponent $z = 1$ at the phase transition between the disordered and the $\\mathbb {Z}_2$ ordered phases.", "Note that for the independent set problem, the quantum speedup is quartic owing to the more slowly mixing Markov chain.", "We remark that it is possible to improve the performance of the Markov chain by adding simultaneous spin flips on neighboring sites, though the quantum algorithm still retains at least a quadratic speedup in this case similar to the Ising model (see Methods).", "We next consider a graph for which it is hard to sample from independent sets even at nonzero temperature.", "The graph takes the shape of a star with $b$ branches and two vertices per branch (see Fig.", "REF a).", "The weight of the vertex at the center is $b$ , while all other weights are set to 1.", "The classical model exhibits a phase transition at $\\beta _c = \\log \\varphi \\approx 0.48$ , where $\\varphi $ is the golden ratio (see Methods).", "Above the phase transition temperature, the free energy is dominated by the entropic contribution from the $3^b$ states with the center unoccupied.", "Below the transition temperature, it is more favorable to decrease the potential energy by occupying the center at the cost of reducing the entropy as the independent set constraint limits the number of available states to $2^b$ (see Fig.", "REF b).", "The Markov chain on this graph has severe kinetic constraints since changing the central vertex from unoccupied to occupied requires all neighboring vertices to be unoccupied.", "Assuming that each individual branch is in thermal equilibrium, the probability of accepting such a move is given by $p_{0 \\rightarrow 1} = [(1 + e^{\\beta })/(1 + 2 e^{\\beta })]^{b}$ .", "Similarly, the reverse process is energetically suppressed with an acceptance probability $p_{1 \\rightarrow 0} = e^{-b \\beta }$ .", "The central vertex can thus become trapped in the thermodynamically unfavorable configuration, resulting in a mixing time that grows exponentially with $b$ at any finite temperature.", "When starting from a random configuration, the Markov chain will nevertheless sample efficiently at high temperature because the probability of the central vertex being initially occupied is exponentially small.", "By the same argument, the Markov chain almost certainly starts in the wrong configuration in the low temperature phase and convergence to the Gibbs distribution requires a time $t_m \\gtrsim 1/p_{0 \\rightarrow 1}$ .", "Figure: Hard sampling at nonzero temperature.", "a, Sampling from a star graph with two vertices per branch is kinetically constrained because occupying the central vertex requires all adjacent vertices to be unoccupied.", "The mixing of the Markov chain is limited at low temperature by the probability p 0→1 p_{0 \\rightarrow 1} of changing the central vertex from unoccupied to occupied.", "The quantum algorithm achieves a quadratic speedup over the Markov chain by tunneling between such configurations.", "b, Entropy per branch S/bS/b of the Gibbs distribution corresponding to the weighted independent set problem on this star graph.", "The system exhibits a discontinuous phase transition at β c ≈0.48\\beta _c \\approx 0.48 (dashed, vertical line).", "The central vertex is occupied with high probability when β>β c \\beta > \\beta _c and unoccupied otherwise.", "The blue curves were obtained for finite-sized systems, while the black curve indicates the thermodynamic limit (see Methods).The corresponding quantum dynamics are captured by a two-state model formed by $\\vert \\psi _0(\\beta ) \\rangle $ and $\\vert \\psi _1(\\beta ) \\rangle $ , which are Gibbs states with the central vertex fixed to be respectively unoccupied or occupied (see Fig.", "REF a and Methods).", "The tunneling rate between these states, i.e.", "the matrix element $\\langle \\psi _0 \\vert H_q \\vert \\psi _1 \\rangle $ , is given by $J = \\Omega _\\mathrm {cen} \\sqrt{p_{0 \\rightarrow 1}}$ , where $\\Omega _\\mathrm {cen}$ denotes the coefficient $\\Omega _i$ in equation (REF ) associated with the central vertex.", "The time required to adiabatically cross the phase transition is bounded by $t_a \\gtrsim 1/J$ with $J$ evaluated at the phase transition.", "Along the one-parameter family $H_q(\\beta )$ , we have $\\Omega _\\mathrm {cen} = \\sqrt{p_{1 \\rightarrow 0}}$ .", "In addition, at the phase transition, $p_{0 \\rightarrow 1} = p_{1 \\rightarrow 0}$ such that $t_a \\gtrsim 1/p_{0 \\rightarrow 1}$ , yielding the same time complexity as the Markov chain that samples at the phase transition.", "However, the square-root dependence of the tunneling rate on $p_{0\\rightarrow 1}$ immediately suggests that a quadratic speedup may be attainable by crossing the phase transition with $\\Omega _\\mathrm {cen} = 1$ .", "An example of such a path is provided in the SI along with a demonstration of the quadratic speedup by numerically evaluating the adiabatic state preparation time.", "Our approach to quantum sampling algorithms unveils a connection between computational complexity and phase transitions and provides physical insight into the origin of quantum speedup.", "The quantum Hamiltonians appearing in the construction are guaranteed to be local given that the Gibbs distribution belongs to a local, classical Hamiltonian and that the Markov chain updates are local.", "Consequently, time evolution under these quantum Hamiltonians can be implemented using Hamiltonian simulation [47].", "Moreover, a hardware efficient implementation in near-term devices may be possible for certain cases such as the independent set problem.", "While the proposed realization utilizing Rydberg blockade is restricted to unit disk graphs, a wider class of graphs may be accessible using anisotropic interactions [48] and individual addressing with multiple atomic sublevels [49].", "Our approach can be extended along several directions.", "These include quantum algorithms corresponding to Markov chains with cluster updates, which are often effective in practice.", "To address practically relevant sampling problems, our method should be expanded to disordered systems in two or more dimensions, where it may be challenging to directly identify an efficient adiabatic path.", "Extensions to hybrid algorithms that combine quantum evolution with classical optimization may offer a solution to this problem.", "For instance, the energy of the parent Hamiltonian $H_q(\\beta )$ could be minimized using variational quantum algorithms, similar to previous proposals that directly minimize the free energy [50], without requiring complex measurements of the entanglement entropy.", "Apart from testing such algorithms, their realization on near-term quantum devices can open the door to exploration of novel applications in areas ranging from physical science to machine learning.", "The authors thank J. I. Cirac, E. A. Demler, A. Polkovnikov, and P. Zoller for insightful discussions.", "We acknowledge support from the National Science Foundation, the MIT–Harvard Center for Ultracold Atoms, the Department of Energy, and the DARPA ONISQ program.", "DS acknowledges support from the FWO as post-doctoral fellow of the Research Foundation – Flanders.", "HP acknowledges support by the Gordon and Betty Moore Foundation.", "The rate of change of the Hamiltonian parameters are chosen to satisfy the adiabatic condition at every point along a given path Messiah2014,Rezakhani2009.", "For a general set of parameters $\\lambda _\\mu $ , we let $\\sum _{\\mu , \\nu } g_{\\mu \\nu } \\frac{\\mathrm {d}\\lambda _\\mu }{\\mathrm {d}t} \\frac{\\mathrm {d}\\lambda _\\nu }{\\mathrm {d}t}= \\varepsilon ^2,$ where $\\varepsilon $ is a small, dimensionless number and $g_{\\mu \\nu } = \\sum _{k \\ne 0} \\frac{\\langle \\partial _\\mu 0 | k \\rangle \\langle k | \\partial _\\nu 0 \\rangle }{(E_k - E_0)^2}.$ Here the $\\vert k \\rangle $ and $E_k$ label the eigenstates and corresponding eigenenergies of the system with $k=0$ denoting the ground state.", "The notation $\\partial _\\mu $ is a shorthand for $\\mathrm {d}/ \\mathrm {d}\\lambda _\\mu $ .", "Equations (REF ) and (REF ) ensure that the parameters change slowly when the gap is small while simultaneously taking into account the matrix elements $\\langle k | \\partial _\\mu 0 \\rangle $ , which determine the coupling strength of nonadiabatic processes to a particular excited state $\\vert k \\rangle $ .", "The total evolution time can be adjusted by varying $\\varepsilon $ and is given by $t_\\mathrm {tot} = \\frac{1}{\\varepsilon } \\int \\sqrt{\\sum _{\\mu , \\nu } \\mathrm {d}\\lambda _\\mu \\mathrm {d}\\lambda _\\nu \\, g_{\\mu \\nu }},$ where the integral runs along the path of interest.", "We show in the SI that for the cases studied here, a constant fidelity close to unity is reached at a small value of $\\varepsilon $ that is independent of $n$ .", "Hence, the parametric dependence of the adiabatic state preparation time on $n$ only depends on the integral in equation (REF ).", "Indeed, we find that the scalings along the various paths for the 1D Ising model can be analytically established from the singular properties of $g_{\\mu \\nu }$ at the tricritical point.", "A similar numerical analysis is provided in the SI for both of the independent set problems.", "For unit disk graphs, the parent Hamiltonian for the weighted independent set problem, equation (REF ), can be efficiently implemented using highly excited Rydberg states of neutral atoms [49].", "As illustrated in Fig.", "REF b, the ground state $\\vert g_i \\rangle $ of an atom $i$ encodes the unoccupied state of a given vertex $i$ .", "Similarly, the occupied state is encoded in a Rydberg state $\\vert r_i \\rangle $ .", "We implement the first and last term in equation (REF ) by driving a transition from $\\vert g_i \\rangle $ to $\\vert r_i \\rangle $ .", "The value of $V_{e,i}$ is set by the detuning of the drive, whereas $\\Omega _i$ is proportional to the amplitude of the drive, $\\mathcal {E}_i$ .", "The projectors $P_i$ arise due to Rydberg blockade.", "If an atom is excited to the Rydberg state, the strong van der Waals interaction $U_\\mathrm {vdW}$ shifts the Rydberg states of all neighboring atoms out of resonance, effectively turning off the drive and thereby enforcing the independent set constraint.", "The remaining second term in equation (REF ) can be realized using a similar approach, combining the Rydberg blockade with an AC Stark shift induced by an off-resonant drive from the ground state to an auxiliary Rydberg state $\\vert r_i^{\\prime } \\rangle $ .", "The Rydberg interaction contributes additional terms to the Hamiltonian that decay as $1/r^6$ with the distance $r$ between two atoms.", "We have neglected these terms throughout, noting that a strategy to mitigate their role has been proposed in a related context Pichler2018b.", "Interactions between Rydberg atoms can also be used to implement more complicated parent Hamiltonians.", "For instance, Förster resonances between Rydberg states can give rise to simultaneous flips of two neighboring spins Barredo2015.", "Such updates allow defects of two adjacent, unoccupied vertices to diffuse without an energy barrier.", "This results in a gap $\\Delta \\sim 1/n^2$ at low temperature along the one-parameter family $H_q(\\beta )$ without the factor $e^{-\\beta }$ .", "Hence, the corresponding Markov chain samples quadratically faster from the ground state than the Markov chain with only single spin flips.", "The quantum algorithm does not experience a parametric speedup with these updates as the computation time is limited by the time for correlations to spread over the entire system.", "Both the classical and quantum algorithms associated with correlated updates will be discussed in detail elsewhere.", "Finally, we note that even though the star graph is not a unit disk graph, its parent Hamiltonian could potentially be implemented using anisotropic interactions between Rydberg states [48].", "The temperature at which the classical model associated with the weighted independent set problem for the star graph undergoes a phase transition can be computed exactly.", "The partition function is given by $\\mathcal {Z} = \\left(1 + 2 e^{\\beta }\\right)^b + e^{b \\beta } \\left(1 + e^{\\beta }\\right)^b.$ The two terms correspond to the different configurations of the central vertex.", "The probability that the central site is occupied is given by $p_1 = \\frac{1}{\\mathcal {Z}} e^{b \\beta } \\left( 1 + 2 e^\\beta \\right)^b = \\left[ 1 + \\left( \\frac{1 + 2 e^\\beta }{e^{\\beta } + e^{2 \\beta }} \\right)^b \\right]^{-1}.$ In the thermodynamic limit $b \\rightarrow \\infty $ , this turns into the step function $p_1 = \\Theta (\\beta - \\beta _c)$ , where $\\beta _c = \\log \\varphi $ with $\\varphi = (\\sqrt{5} + 1)/2$ being the golden ratio.", "The entropy $S = \\beta (U - F)$ can be computed from the Helmholtz free energy $F = - \\log \\mathcal {Z} / \\beta $ and the total energy $U = - \\partial \\log \\mathcal {Z} / \\partial \\beta $ .", "The star graph has three types of vertices: the vertex at the center and the inner and outer vertices on each branch.", "Restricting our analysis to the subspace that is completely symmetric under permutations of the branches, we introduce the total occupation numbers $n_\\mathrm {in} = \\sum _{i=1}^b n_{\\mathrm {in}, i}$ and $n_\\mathrm {out} = \\sum _{i = 1}^b n_{\\mathrm {out}, i}$ as well as the number of unoccupied branches $n_0$ .", "The symmetric subspace is spanned by the states $\\vert n_\\mathrm {cen}, n_\\mathrm {in}, n_\\mathrm {out}, n_0 \\rangle $ , where $n_\\mathrm {cen} \\in \\lbrace 0, 1\\rbrace $ , while the other occupation numbers are nonnegative integers satisfying $n_\\mathrm {in} + n_\\mathrm {out} + n_0 = b$ .", "If $n_\\mathrm {cen} = 1$ , the independent set constraint further requires $n_\\mathrm {in} = 0$ .", "The state $\\vert n_\\mathrm {cen}, n_\\mathrm {in}, n_\\mathrm {out}, n_0 \\rangle $ is an equal superposition of $b!/ (n_\\mathrm {in}!", "\\, n_\\mathrm {out}!", "\\, n_0!", ")$ independent configurations.", "The permutation symmetry leads to a bosonic algebra.", "We define the bosonic annihilation operators $b_\\mathrm {in}$ , $b_\\mathrm {out}$ , and $b_0$ respectively associated with the occupation numbers $n_\\mathrm {in}$ , $n_\\mathrm {out}$ , and $n_0$ .", "The Hamiltonian can be split into blocks where the central vertex is either occupied or unoccupied as well as an off-diagonal term coupling them.", "Explicitly, $H_q = H_q^{(0)} \\otimes (1 - n_\\mathrm {cen}) + H_q^{(1)} \\otimes n_\\mathrm {cen} + H_q^{(\\mathrm {od})} \\otimes \\sigma _\\mathrm {cen}^x.$ It follows from equation (REF ) that in terms of the bosonic operators Hq(0) = Ve,in binbin + Ve,out boutbout + (Vg,in + Vg,out) b0b0       - in (binb0 + h.c.) - out (boutb0 + h.c.)       + Vg, cen P(nin = 0), Hq(1) = Ve,out boutbout + Vg,out b0b0       - out (boutb0 + h.c.) + Ve,cen, Hq(od) = - cen P(nin = 0), where $P(n_\\mathrm {in} = 0)$ projects onto states with no occupied inner vertices.", "The parameters are labeled in accordance to the definitions in equation (REF ) with the vertex index $i$ replaced by the type of the vertex.", "To diagonalize the Hamiltonian, we treat $P(n_\\mathrm {in} = 0)$ perturbatively.", "We identify the lowest energy modes that diagonalize the quadratic parts of $H_q^{(0)}$ and $H_q^{(1)}$ and associate with them the bosonic annihilation operators $c_0$ and $c_1$ , respectively.", "Both modes have zero energy while the other modes are gapped for any finite value of $\\beta $ .", "We may thus expect the ground state to be well approximated in the subspace spanned by $\\vert \\psi _0 \\rangle = c_0^{\\dagger b} \\vert 0 \\rangle / \\sqrt{b!", "}$ and $\\vert \\psi _1 \\rangle = c_1^{\\dagger b} \\vert 0 \\rangle / \\sqrt{b!", "}$ , where $\\vert 0 \\rangle $ denotes the bosonic vacuum.", "We focus on the situation where all parameters follow the one-parameter family $H_q(\\beta )$ except for $\\Omega _\\mathrm {cen}$ and $V_{e,\\mathrm {cen}}$ , which may be adjusted freely.", "One can show that in this case, $\\vert \\psi _0 \\rangle $ and $\\vert \\psi _1 \\rangle $ correspond to the Gibbs state of the star with the central vertex held fixed.", "To include the effect of the terms involving $P(n_\\mathrm {in} = 0)$ , we perform a Schrieffer–Wolff transformation for the subspace spanned by $\\vert \\psi _0 \\rangle $ and $\\vert \\psi _1 \\rangle $  Bravyi2011.", "We arrive at the effective Hamiltonian $H_\\mathrm {eff} =\\begin{pmatrix}\\varepsilon _0 + \\delta \\varepsilon _0 & -J - \\delta J\\\\-J - \\delta J & V_{e,\\mathrm {cen}} + \\delta \\varepsilon _1\\end{pmatrix},$ where the terms $\\varepsilon _0 = \\langle \\psi _0 \\vert P(n_\\mathrm {in} = 0) \\vert \\psi _0 \\rangle = \\left( \\frac{1 + e^{\\beta }}{1 + 2 e^{\\beta }} \\right)^b, \\\\J = \\Omega _\\mathrm {cen} \\langle \\psi _1 \\vert P(n_\\mathrm {in} = 0) \\vert \\psi _0 \\rangle = \\Omega _\\mathrm {cen}\\left( \\frac{1 + e^{\\beta }}{1 + 2 e^{\\beta }} \\right)^{b/2}$ are obtained by projecting the full Hamiltonian onto the low-energy subspace.", "The corrections from coupling to excited states as given by the Schrieffer–Wolff transformation up to second order are 0 = - 0 n 1En(0) | $\\langle E_n^{(0)} \\vert $ cenx $\\vert \\psi _1 \\rangle $ |2, 1 = - cen2 n | $\\langle E_n^{(0)} \\vert $ cenx $\\vert \\psi _1 \\rangle $ |2En(0) - Ve,cen, J = - 12 cen 0 n ( 1En(0) +1En(0) - Ve,cen )                              | $\\langle E_n^{(0)} \\vert $ cenx $\\vert \\psi _1 \\rangle $ |2.", "Here, we made use of the relation $P(n_\\mathrm {in} = 0) \\vert \\psi _0 \\rangle = \\sqrt{\\varepsilon _0} \\sigma _\\mathrm {cen}^x \\vert \\psi _1 \\rangle $ , which holds along the paths of interest.", "The sums run over all excited states $\\vert E_n^{(0)} \\rangle $ with energy $E_n^{(0)}$ of the quadratic part of $H_q^{(0)}$ (excluding $\\vert \\psi _0 \\rangle $ ).", "We will neglect the term $V_{e,\\mathrm {cen}}$ in the energy denominators of equations (REF ) and (REF ), which is justified when $V_{e,\\mathrm {cen}}$ is small compared to $E_n^{(0)}$ .", "The discussion remains valid even if this is not the case because the second-order corrections from the Schrieffer–Wolff transformation can then be ignored.", "Combining these results, the complete effective Hamiltonian may be written as $H_\\mathrm {eff} =\\begin{pmatrix}(1 - f) \\varepsilon _0 & -( 1 - f) J\\\\-(1- f) J & V_{e,\\mathrm {cen}} - f \\Omega _\\mathrm {cen}^2\\end{pmatrix},$ where $f = \\sum _n \\left| \\langle E_n^{(0)} \\vert \\sigma _\\mathrm {cen}^x \\vert \\psi _1 \\rangle \\right|^2 / E_n^{(0)}$ .", "We find numerically that $f$ decays as an inverse power law in $b$ such that our approximations are well justified in the thermodynamic limit (see SI).", "Along the one-parameter family $H_q(\\beta )$ , we have $V_{e, \\mathrm {cen}} = \\Omega _\\mathrm {cen}^2$ .", "Hence, $H_\\mathrm {eff}$ depends on $f$ only through an overall factor $(1 - f)$ , which tends to 1 in the limit of large $b$ .", "The phase transition of the underlying classical model manifests itself as a first-order quantum phase transition from $\\vert \\psi _0 \\rangle $ to $\\vert \\psi _1 \\rangle $ .", "The transition occurs when the two states are resonant, $\\varepsilon _0 = V_{e,\\mathrm {cen}}$ , which can be solved to give $\\beta _c = \\log \\varphi $ as expected.", "naturemag bibliography" ] ]

[ [ "Primordial magnetic fields during the cosmic dawn in light of EDGES\n 21-cm signal" ], [ "Abstract We study prospects of constraining the primordial magnetic field (PMF) and its evolution during the dark ages and cosmic dawn in light of EDGES 21-cm signal.", "Our analysis has been carried out on a `colder IGM' background which is one of the promising avenues to interpret the EDGES signal.", "We consider the dark matter-baryon interactions for the excess cooling.", "We find that the colder IGM suppresses both the residual free electron fraction and the coupling coefficient between the ionised and neutral components.", "The Compton heating also gets affected in colder IGM background.", "Consequently, the IGM heating rate due to the PMF enhances compared to the standard scenario.", "Thus, a significant fraction of the magnetic energy, for $B_0 \\lesssim 0.5 \\, {\\rm nG}$, gets transferred to the IGM and the magnetic field decays at a much faster rate compared to the simple $(1+z)^2$ scaling during the dark ages and cosmic dawn.", "This low PMF is an unlikely candidate for explaining the rise of the EDGES absorption signal at lower redshift.", "We also see that the PMF and DM-baryon interaction together introduces a plateau-like feature in the redshift evolution of the IGM temperature.", "We find that the upper limit on the PMF depends on the underlying DM-baryon interaction.", "Higher PMF can be allowed when the interaction cross-section is higher and/or the DM particle mass is lower.", "Our study shows that the PMF with $B_0$ up to $\\sim 0.4 \\, {\\rm nG}$, which is ruled out in the standard model, can be allowed if DM-baryon interaction with suitable cross-section and DM mass is considered." ], [ "Introduction", "The global redshifted HI 21-cm signal from the dark ages and cosmic dawn is a promising tool to study the primordial magnetic field [36].", "The primordial magnetic field can heat up the Hydrogen and Helium gas in the inter galactic medium (IGM) by processes such as the ambipolar diffusion (AD) and decaying turbulence (DT) [12], [37], [15], [6].", "This indirectly affects the spin temperature and the globally averaged redshifted HI 21-cm signal [33].", "Furthermore, growth of structures during the cosmic dawn gets accelerated in presence of magnetic field in the IGM.", "As a consequence, the primordial magnetic field can have an important impact on the formations of the first luminous sources [35], [29].", "A substantial amount of theoretical work has been carried out to understand, in detail, the role of the primordial field on the HI 21-cm signal [39], [30], [41], [14], early structure formation during the cosmic dawn and reionization [13], [42], [23].", "The measurements of the global HI 21-cm absorption signal by the EDGES experiments in the redshift range $z \\sim 14$ to 20 [5] have opened up a possibility to constrain the primordial magnetic field and understand its evolution during the cosmic dawn and dark ages.", "In a recent work, [19] has exploited the EDGES data to put an upper limit on the primordial magnetic field.", "The analysis has been carried out on the backdrop of the standard cosmological model and baryonic interactions of the IGM.", "However, the measured EDGES absorption signal is $\\sim 2 -3$ times stronger compared to predictions by the standard model.", "If the measurements are confirmed, one promising way to explain the measured signal is to consider the IGM to be significantly `colder' compared to the IGM kinetic temperature predicted by the standard scenario.", "Thus, one needs to consider a non-standard cooling mechanism such as the DM-baryon interaction in order to make the IGM colder [40], [21].", "This avenue has been widely explored to explain the unusually strong absorption signal found by the EDGES experiment [1], [2], [22], [20].", "Constraints on the primordial magnetic field using the global 21-cm absorption signal in the colder IGM background would, in principle, be different from constraints obtained in the standard scenario.", "Because, colder IGM enhances the Hydrogen recombination rate which, in turn, reduces the residual free electron fraction during the dark ages and cosmic dawn [7].", "In addition, the coupling between the ionized and neutral component which has direct impact on the IGM heating also get suppressed in the 'colder IGM' scenario.", "Moreover, the heating rate due to the Compton process, which depends on the IGM kinetic temperature and the residual free electron fraction, too gets affected when the background IGM temperature is lower.", "Together all these effects enhance the IGM heating rate due to the primordial magnetic field.", "Consequently, small amount of magnetic field would be enough to keep IGM temperature at a certain label.", "On the contrary, significantly more magnetic energy would be transferred to the IGM due to the enhanced heating rate which would affect the redshift evolution of the primordial magnetic field itself and the heating at later reshifts.", "If one considers DM-baryon interaction in order to make the IGM colder, the exact constraints on primordial magnetic field should also depend on the mass of the DM particles and the interaction cross section between the DM particles and baryons.", "Thus, it is important to highlight these aspects in order to understand the role of the primordial magnetic field on the 21-cm absorption signal and put limits on primordial magnetic field.", "Recently, [4] has used the EDGES low band measurements to study constraints on the primordial magnetic field in presence DM-baryonic interaction.", "Various other observations such the CMBR, the Sunyaev-Zel'dovich effect, the star formation, blazar light curve have been exploited to constrain the primordial magnetic field [26], [28], [18], [17], [38].", "Constraining the primordial magnetic field is very important as it can shed light on its origin and evolution.", "In this work, we study the constraints on the primordial magnetic field using the EDGES 21-cm absorption profile on the backdrop of the colder IGM scenario.", "We consider interactions between cold DM particles and baryons [40], [21] which makes the IGM colder as compared to that in the standard predictions.", "In addition, we study the redshift evolution of the primordial magnetic field during dark ages and cosmic dawn.", "The differential brightness temperature in the EDGES absorption profile starts increasing at redshift $z \\sim 16$ which suggests that heating of the IGM started around that redshift.", "Here, we also investigate if the primordial magnetic heating is able to explain this behavior.", "Our analysis also allows us to study the constraints on the mass of the DM particle and the interaction cross section in presence of the primordial magnetic field.", "The structure of the paper is as follows.", "A brief discussion and essential equations regarding the redshifted HI 21-cm signal, dark matter - baryon interaction, heating due to the primordial magnetic field and redshift evolution of IGM temperature in presence of both the primordial magnetic field and DM-baryon interaction are presented in subsections 2.1, 2.2, 2.3 and 2.4 respectively.", "We discuss our results in section 3 and present summary and discussion in section 4.", "Throughout our work we use cosmological parameters $\\Omega _{m0}= 0.3$ , $\\Omega _{b0} = 0.0486$ , $h=0.677$ , $\\Omega _{\\Lambda 0} = 0.7$ consistent with the Plank measurements [25].", "The globally averaged differential brightness temperature corresponding to the redshifted HI 21-cm at redshift $z$ can be written as [3], [9], $\\frac{T_{21}}{\\rm mK} = 27 x_{\\rm HI} \\left(1-\\frac{T_\\gamma }{T_s}\\right) \\left(\\frac{\\Omega _{\\rm b0} h^2}{0.02} \\right) \\left(\\frac{0.15}{\\Omega _{\\rm m0} h^2} \\right)^{0.5} \\left( \\frac{1+z}{10} \\right)^{0.5},$ where $T_\\gamma $ and $x_{HI}$ are the CMBR temperature and neutral Hydrogen fraction respectively.", "The spin temperature $T_{\\rm s}$ which is a measure of population ratio of the ground state Hydrogen atoms in the triplet and singlet states is defined as, $\\frac{n_1}{n_0} = \\frac{g_1}{g_0} \\exp {(-T_\\ast /T_{\\rm s})},$ where $n_0$ and $n_1$ are the number densities of ground state Hydrogen atoms in the singlet and triplet states respectively, and $g_0=1$ and $g_1=3$ are the degeneracies of these states.", "Further, $T_\\ast = h_p \\nu _e/k_B = 0.068 $  K is the characteristic temperature corresponding to the HI 21-cm transition.", "The Ly-$\\alpha $ photons emitted from the very first stars/galaxies help the spin temperature $T_{\\rm s}$ to couple with the IGM kinetic temperature $T_g$ .", "Since we are interested in the Ly-$\\alpha $ saturated part of the EDGES absorption profile, we assume $T_s=T_g$ for the rest of the paper." ], [ "Dark matter-baryon interaction", "Interactions between the cold DM particles and baryons are expected to help the IGM to cool faster than the standard adiabatic cooling and can explain the unusually strong absorption signal found by the EDGES experiments [1].", "We consider Rutherford like velocity dependent interaction cross section which is modelled as $\\sigma = \\sigma _0 (v/c)^{-4}$ .", "The milli-charged dark matter model follows this kind of interaction and is a potential candidate for explaining the EDGES trough [22], [20].", "Here we adopt the DM-baryon interaction model presented in [21].", "The cooling rate of baryon due to such interaction is modelled as, $\\frac{dQ_b}{dt} = \\frac{2 m_b \\rho _{\\chi } \\sigma _0 e^{-r^{2} / 2} (T_{\\chi } - T_g) k_B c^4}{(m_b + m_{\\chi })^2 \\sqrt{2\\pi } u^{3}_{th}} \\nonumber \\\\+ \\frac{\\rho _{\\chi }}{\\rho _m} \\frac{m_{\\chi } m_b}{m_{\\chi } + m_b} V_{\\chi b} \\frac{D(V_{\\chi b})}{c^2}.$ Similarly, the heating rate of the DM, $\\dot{Q_\\chi }$ can be obtained by just replacing $b \\leftrightarrow \\chi $ in the above expression due to symmetry.", "Here, $m_\\chi $ , $m_b$ and $\\rho _\\chi $ , $\\rho _b$ are the masses and energy densities of dark matter and baryon respectively.", "We can see from equation (REF ) that the heating rate is proportional to the temperature difference between two fluids i.e.", "$(T_{\\chi } - T_g)$ .", "The second term in equation (REF ) arises due to the friction between dark matter and baryon fluids as they flow at different velocities.", "Hence both the fluids get heated up depending on their relative velocity $V_{\\chi b}$ and the drag term $D(V_{\\chi b})$ given as, $\\frac{dV_{\\chi b}}{dz} = \\frac{V_{\\chi b}}{1+z} + \\frac{D(V_{\\chi b})}{H(z)(1+z)}$ and $D(V_{\\chi b}) = \\frac{\\rho _m \\sigma _0 c^4}{m_b + m_\\chi } \\frac{1}{V^2_{\\chi b}} F(r).$ The variance of the thermal relative motion of dark matter and baryon fluids $u_{th}^2 = k_B(T_b/m_b + T_\\chi /m_\\chi $ ) and $r =V_{\\chi b}/u_{th}$ .", "The function $F(r)$ is given by $F(r) = erf \\Big ( \\frac{r}{\\sqrt{2}} \\Big ) - \\sqrt{ \\frac{2}{\\pi }} r e^{-r^2/2}.$ We see that $F(r)$ grows with r, $F(0) = 0$ when $r = 0$ and $F(r) \\rightarrow 1$ when $r \\rightarrow \\infty $ .", "This ensures that the heating due to the friction is negligible when the relative velocity $V_{\\chi b}$ is smaller compared to the thermal motion of dark matter and baryon fluid $u_{th}$ .", "However, it can be significant if $V_{\\chi b}$ is higher than $u_{th}$ ." ], [ "IGM heating due to primordial magnetic field", "Magnetic field exerts Lorentz force on the ionized component of the IGM.", "This causes rise in the IGM temperature, $T_g$ .", "There are mainly two processes namely the ambipolar diffusion (AD) and decaying turbulence (DT) by which the magnetic field can heat up the IGM during the cosmic dawn and dark ages.", "We follow the prescription presented in [34] and [6] to calculate the rate of heating due to these two processes.", "The heating rate (in unit of energy per unit time per unit volume) due to the ambipolar diffusion is given by, $\\Gamma _{\\rm AD} = \\frac{(1-x_e)}{\\gamma x_e \\rho _b^2} \\frac{\\Big \\langle \\vert (\\nabla \\times B)\\times B \\vert ^2 \\Big \\rangle }{16\\pi ^2},$ where $x_e = n_e/n_{\\rm H}$ is the residual free electron fraction and $n_{\\rm H} = n_{\\rm HI} + n_{\\rm HII}$ .", "We assume $n_{\\rm HII} = n_e$ as Helium is considered to be fully neutral in the redshift range of our interest.", "Further, $\\rho _b$ is the baryon mass density at redshift $z$ , and the coupling coefficient between the ionized and neutral components is $\\gamma =\\langle \\sigma v \\rangle _{HH^+}/{2m_H}= 1.94 \\times 10^{14} \\, (T_g/{\\rm K})^{0.375} \\, {\\rm cm}^3 {\\rm gm}^{-1} {\\rm s}^{-1}$ .", "The Lorentz force can be approximated as $ \\Big \\langle \\vert (\\nabla \\times B)\\times B \\vert ^2 \\Big \\rangle \\approx 16 \\pi ^2 \\, \\rho _B(z)^2 \\, l_d(z)^{-2} f_L(n_B+3)$ [6], where $\\rho _B(z)=|{\\bf B}|^2/8\\pi $ is the magnetic field energy density at redshift $z$ , $f_L(p)=0.8313[1-1.02 \\times 10^{-2} p]p^{1.105}$ , and $l_d^{-1}= (1+z) \\,k_D$ .", "The damping scale is given by $k_D \\approx 286.91 \\, (B_0/{\\rm nG})^{-1}\\, {\\rm Mpc}^{-1}$ [15].", "We note that the above heating rate is inversely proportional to the coupling coefficient $\\gamma $ and the residual electron fraction $x_e$ .", "Furthermore, both $\\gamma $ and the ionization fraction, $x_e$ gets suppressed when the IGM is colder compared to the standard scenario.", "As a result, the ambipolar heating rate becomes more efficient during the cosmic dawn and dark ages.", "The heating rate due to the decaying turbulence is described by, $\\Gamma _{\\rm DT} = \\frac{3m}{2} \\frac{\\Big [\\ln {\\left(1+\\frac{t_i}{t_d} \\right)}\\Big ]^m}{\\Big [\\ln {\\left(1+\\frac{t_i}{t_d}\\right)} + \\frac{3}{2} \\ln {\\left(\\frac{1+z_i}{1+z}\\right)}\\Big ]^{m+1}} H(z)\\, \\rho _B(z),$ where $m=2 \\, (n_B+3)/(n_B+5)$ , and $n_B$ is the spectral index corresponding to the primordial magnetic field.", "The physical decay time scale ($t_d$ ) for turbulence and the time ($t_i$ ) at which decaying magnetic turbulence becomes dominant are related as $t_i/t_d \\simeq 14.8(B_0/{\\rm nG})^{-1}(k_D/{\\rm Mpc}^{-1})^{-1}$ [6].", "The heating rate due to the decaying turbulence is more efficient at early times as it is proportional to the Hubble rate, $H(z)$ and the primordial magnetic energy density.", "The effect monotonically decreases at lower redshifts and becomes sub-dominant during the cosmic dawn and dark ages.", "It is often assumed that, like the CMBR energy density, the primordial magnetic field and energy density scale with redshift $z$ as $B(z) = B_0 (1+z)^2$ and $\\rho _B(z) \\sim (1+z)^4$ respectively under magnetic flux freezing condition.", "However, the magnetic field energy continuously gets transferred to the IGM through the ambipolar diffusion and decaying turbulence processes.", "For the magnetic field with $B_0 \\gtrsim 1 \\, {\\rm nG}$ , the transfer may be insignificant compared to the total magnetic filed energy and the above scalings holds.", "However, this may not be a valid assumption for lower magnetic field $B_0 \\lesssim 0.1 $ nG.", "Therefore, we self-consistently calculate the redshift evolution of the magnetic field energy using the following equation, $\\frac{d}{dz} \\left( \\frac{\\vert B \\vert ^2}{8 \\pi } \\right) = \\frac{4}{1+z} \\left( \\frac{\\vert B \\vert ^2}{8 \\pi } \\right) + \\frac{1}{H(z)\\,(1+z)} (\\Gamma _{\\rm DT} + \\Gamma _{\\rm AD}).$ The first term in the rhs quantifies the effect due to the adiabatic expansion of universe, and the second term quantifies the loss of the magnetic energy due to the IGM heating described above." ], [ "Temperature evolution", "This section focuses on the evolution of the IGM kinetic temperature $T_g $ from the recombination epoch to the cosmic dawn.", "Considering the effects described in Section REF and REF , the evolution of IGM gas temperature ($T_g$ ) can be written as, $\\frac{dT_g}{dz} = \\frac{2T_g}{1+z} - \\frac{32\\sigma _T\\sigma _{SB} T_0^4}{3m_ec^2H_0\\sqrt{\\Omega _{m0}}}\\left(T_\\gamma -T_g\\right)\\left(1+z\\right)^{3/2} \\frac{x_e}{1+x_e} \\nonumber \\\\- \\frac{2}{3k_B H(z)\\,(1+z) } \\left[ \\dot{Q_b}+ \\frac{\\Gamma }{n_{\\rm tot} } \\right].$ The first two terms on the rhs describe the adiabatic cooling due to expansion of the universe and Compton heating due to interaction between CMBR and free electrons respectively.", "Further, $\\dot{Q_b}$ is the heating/cooling rate per baryon due to interactions between the DM particles and baryons (see eq.", "REF ) and $\\Gamma =\\Gamma _{\\rm AD}+\\Gamma _{\\rm DT}$ is the total rate of heating per unit volume due to the primordial magnetic field described in eqs.", "REF and REF .", "Also, $ k_B$ , $ \\sigma _T$ , $\\sigma _{SB}$ , $m_e$ are the Boltzman constant, Thomson scattering cross-section, Stefan Boltzman constant and the rest mass of an electron respectively.", "Further, $n_{\\rm tot} \\approx n_{\\rm H}(1+f_{He}+x_e)$ denotes the total number density of baryon particles, and $n_{\\rm H}$ is the number density of Hydrogen.", "Taking the Helium mass fraction $Y_P=0.24$ , the fraction of Helium atoms with respect to hydrogen atoms, $f_{\\rm He}$ becomes $0.079$ .", "The evolution of the DM temperature $T_{\\chi }$ can be calculated using, $\\frac{dT_\\chi }{dz} = \\frac{2T_\\chi }{1+z} - \\frac{2}{3k_B} \\frac{\\dot{Q_\\chi }}{H(z)\\, (1+z)}.$ The first and second terms on the rhs quantify the adiabatic cooling and heating rate per dark matter particle due to its interactions with baryons respectively.", "Figure: The upper and middle panels show the IGM kinetic temperature T g T_g and residual free electron fraction x e x_e as a function of redshift in presence of the primordial magnetic field.", "The lower solid (black), dashed-dotted (red) and dashed (blue) lines correspond to the primordial magnetic field with B 0 =0B_0 = 0, 0.050.05 and 0.5 nG 0.5\\, {\\rm nG} respectively.", "The upper solid (black) line shows the CMBR temperature T γ T_{\\gamma }.", "We do not include the DM-baryon interaction here.", "The lower panel shows the normalised magnetic field i.e.", "B(z) B 0 (1+z) 2 \\frac{B(z)}{B_0 (1+z)^2} for the same B 0 B_0 values mentioned above.We note that the residual free electron fraction $x_e$ influences the IGM heating through the Compton heating (eq.", "REF ) and ambipolar diffusion (eq.", "REF ).", "We calculate the residual free electron fraction using the equation [24], $\\frac{dx_e}{dz} = \\frac{C}{H(z)\\,(1+z)} \\left[\\alpha _e\\,x_e^2 n_{\\rm H} -\\beta _e\\,(1-x_e) \\, e^\\frac{-h_p\\nu _\\alpha }{k_BT_g}\\right] \\nonumber \\\\- \\frac{\\gamma _e\\,n_H(1-x_e)x_e}{H(z)\\,(1+z)},$ where $\\alpha _e(T_g)$ , $\\beta _e(T_{\\gamma )}$ and $\\gamma _e(T_g)$ are the recombination, photoionization and collisional ionization coefficients respectively.", "We note that $\\alpha _e$ and $\\gamma _e$ depend on the IGM temperature $T_g$ .", "In contrast, $\\beta _e$ depends on the CMBR temperature [6].", "For the recombination co-efficient we use $\\alpha _e (T_g) = F \\times 10^{-19} (\\frac{a t^b}{1 + c t^d}) \\hspace{2.84544pt} {\\rm m^3 s^{-1} } $ , where $a = 4.309$ , $b = -0.6166$ , $c = 0.6703$ , $d = 0.53$ , $F = 1.14$ (the fudge factor) and $t = \\frac{T_g}{10^4 \\, {\\rm K}} $ .", "Further, $\\beta _e$ is calculated using the relations $\\beta _e (T_{\\gamma }) = \\alpha _e (T_{\\gamma }) \\Big ( \\frac{2 \\pi m_e k_B T_\\gamma }{h^2_p} \\Big )^{3/2} e^{-E_{2s}/k_B T_\\gamma }$ [31], [32].", "The Peebles factor is given by $C = \\frac{1+ K \\Lambda (1-x) n_{H}}{1+K(\\Lambda +\\beta _{e})(1-x) n_{H}}$ , where $\\Lambda =8.3 \\, {\\rm s}^{-1}$ is the rate of transition from (hydrogen ground state) $2s\\rightarrow 1s$ state through decaying two photons.", "Further, $K=\\frac{\\lambda _{\\alpha }^{3}}{8\\pi H(z)}$ , $\\gamma _e(T_g) =0.291 \\times 10^{-7} \\times U ^{0.39} \\frac{\\exp (-U)}{0.232+U} \\, {\\rm cm^3/s} $ [18] with $h_{p} \\nu _{\\alpha } = 10.2 \\, {\\rm eV}$ and $U=\\vert E_{1s}/k_B T_g \\vert $ .", "We see from equations (REF ) and (REF ), that the IGM temperature becomes velocity ($V_{\\chi b}$ ) dependent as soon as the dark matter-baryon interaction is taken into consideration which, in turn, modifies the brightness temperature $T_{21}$ .", "Therefore, the observable global HI 21-cm brightness temperature is calculated by averaging over the velocity $V_{\\chi b}$ as, $\\langle T_{21} (z) \\rangle = \\int d^3 V_{\\chi b} T_{21} (V_{\\chi b}) P(V_{\\chi b}),$ where the initial velocity $V_{\\chi b, 0}$ follows the probability distribution $P(V_{\\chi b, 0}) = \\frac{e^{-3 V^2_{\\chi b, 0}/( 2 V^2_{\\rm rms})}}{(\\frac{2 \\pi }{3} V^2_{\\rm rms})^{3/2}}.$ In order to calculate the velocity averaged IGM temperature $\\langle T_g(z) \\rangle $ and ionization fraction $\\langle x_e \\rangle $ , the same procedure is followed." ], [ "Results and Discussion", "We simultaneously solve equations (REF ), (REF ), (REF ), (REF ), (REF ) and (REF ) to evaluate $T_g$ and $x_e$ for a range possible values of the dark matter particle mass $m_{\\chi }$ , the interaction cross-section $\\sigma _{45}=\\frac{\\sigma _0}{10^{-45} \\, {\\rm m^2}}$ , and the initial magnetic field ($B_0$ ) for a given $V_{\\chi b}$ .", "We then use eq.", "REF to calculate the averaged quantities such as $\\langle T_{21} (z) \\rangle $ , $\\langle T_g(z) \\rangle $ and $\\langle x_e \\rangle $ .", "Note that all values/results quoted below are these average quantities even if we don't mention them explicitly.", "Below we discuss our results on the heating due the primordial magnetic field, impacts of the DM-baryonic interaction in presence/absence of the magnetic field.", "In addition we study, in the context of the colder IGM background, the role of the residual free electron fraction $x_e$ , evolution of the primordial magnetic field and the upper limit on the primordial magnetic field using EDGES absorption profile.", "We set the following initial conditions at redshift $z_i=1010$ : $T_{\\rm gi} = 2.725(1+z_i) \\, {\\rm K};\\, T_{\\chi i}= 0,\\, V_{\\chi b,i} = V_{\\rm rms i} = 29 \\, {\\rm km/s}$ , $B_i=B_0(1+z_i)^2$ and $x_{ei}=0.055$ [31], [32]." ], [ "Impact on heating due to the primordial magnetic field", "The upper panel of Fig.", "REF shows the evolution of IGM temperature $T_g$ in presence of the primordial magnetic field with $B_0=0.05 \\, {\\rm nG}$ and $0.5 \\, {\\rm nG}$ .", "We fix $n_B = -2.9$ throughout our analysis.", "In order to understand the role of the primordial magnetic field alone we do not include the DM-baryon interaction in Fig.", "REF .", "Note that our results for $B_0=3 \\, {\\rm nG}$ and Hydrogen only scenario is very similar to that presented in [6] for the similar scenario.", "We find that the primordial magnetic field makes a noticeable change in the IGM temperature during the cosmic dawn and dark ages ($z \\lesssim 100$ ) for $B_0 \\gtrsim 0.03 \\, {\\rm nG}$ .", "This is because the ambipolar diffusion becomes very active at lower redshifts as it is inversely proportional to the square of the baryon density, $\\rho _b$ .", "It also scales with the IGM temperature as $T^{-0.375}_{g}$ (see eq.", "REF ).", "The Effects due to the decaying turbulence, which scales as $\\Gamma _{DT} \\propto H(z) \\, \\rho _B(z)$ , gets diluted at lower redshifts.", "We find that for $B_0 \\sim 0.1 \\, {\\rm nG}$ , the IGM temperature rises to the CMBR temperature and, consequently, the global differential brightness temperature $T_{21}$ becomes nearly zero.", "Further increase of the primordial magnetic field causes the IGM temperature goes above the CMBR temperature and $T_{21}$ becomes positive.", "This is completely ruled out as the EDGES measured the HI 21-cm signal in absorption i.e., $T_{21}$ is negative.", "This put an upper limit on the primordial magnetic field and we find $B_0 \\lesssim 0.1 \\, {\\rm nG}$ , similar to the upper limit found by [19].", "The middle panel of Fig.", "REF shows the history of residual free electron fraction, $x_e$ .", "We see that $x_e$ increases if we increase the magnetic field $B_0$ .", "This is because of suppression in the Hydrogen recombination rate $\\alpha _e$ due to increase in the IGM temperature $T_g$ .", "The increase is more prominent during the cosmic dawn and dark ages.", "For example, $x_e$ increases by a factor of $\\sim 1.5$ as compared to the standard prediction at redshift $z=17$ if $B_0=0.5 \\, {\\rm nG}$ .", "Conversely, the residual free electron fraction $x_e$ directly influences the magnetic heating and its evolution through the ambipolar diffusion process (eq.", "REF and REF ) which is dominant over the decaying turbulence during the cosmic dawn and dark ages.", "Therefore it is important to highlight the role of $x_e$ in constraining the primordial magnetic field using the global 21-cm signal.", "Moreover, $x_e$ also affects the standard Compton heating (see eq.", "REF ).", "The bottom panel of Fig.", "REF shows the evolution of the primordial magnetic field.", "The normalised primordial magnetic field ($\\frac{B(z)}{B_0(1+z)^2}$ ) has been plotted here to highlight any departure from the simple $B_0(1+z)^2$ scaling.", "We find that the primordial normalised magnetic field maintains a constant value at higher redshifts $z \\gtrsim 100$ , and then decays at lower redshifts during the cosmic dawn and dark ages.", "Because, a considerable fraction of the magnetic field energy is transferred to the IGM for its heating through the ambipolar diffusion process.", "The ambipolar diffusion becomes very active at lower redshifts for reasons explained in subsection REF .", "We also notice that the amount of decay of the magnetic field depends on $B_0$ .", "For example, the normalised primordial magnetic field goes down to $\\sim 0.4$ and $\\sim 0.6$ for $B_0=0.05 \\, {\\rm nG}$ and $0.5 \\, {\\rm nG}$ at redshift $z \\sim 17.2$ .", "This implies that the fractional decay of the magnetic energy is more when the primordial magnetic filed is weaker.", "For higher primordial magnetic field with $B_0 \\gtrsim 1 \\, {\\rm nG}$ , the fractional decay is not significant and it can be safely assumed to scale as $(1+z)^2$ ." ], [ "Effect of dark matter-baryon interaction", "We consider the DM-baryon interaction model that was discussed in Sec.", "REF .", "As mentioned there, the model has two free parameters i.e., the mass of the dark matter particle, $m_{\\chi }$ and the interaction cross-section between the dark matter particles and baryons, $\\sigma _{45}$ .", "Below we briefly discuss the impact of the DM-baryon interaction on the IGM temperature, $T_g$ and residual free electron fraction, $x_e$ .", "We refer readers to [7] for a more elaborate discussion.", "The upper panel of Fig.", "REF shows the evolution of IGM temperature for two sets of dark matter mass $m_\\chi $ and interaction cross-section $\\sigma _{45}$ i.e., $(1\\, {\\rm GeV},\\, 1)$ and $(0.01\\, {\\rm GeV},\\, 50)$ .", "It also plots the IGM temperature as predicted in the standard model.", "As expected, the interaction helps the IGM to cool faster and the IGM temperature becomes lower than the standard scenario during the cosmic dawn.", "Lower the dark matter mass, $m_\\chi $ and/or larger the cross-section $\\sigma _{45}$ , more is the rate of IGM cooling and, consequently, lower is the IGM temperature.", "We note that for higher cross section $\\sigma _{45}$ the IGM temperature gets decoupled from the CMBR temperature early and coupled to the dark matter temperature $T_{\\chi }$ .", "This helps the IGM and the dark matter to reach the thermal equilibrium.", "After that both the IGM and dark matter temperatures scale as $(1+z)^2$ which is seen at redshifts $z \\lesssim 100$ for $m_{\\chi }=0.01\\, {\\rm GeV}$ and $\\sigma _{45}=50$ (the blue-dashed curve in Fig.", "REF ).", "Here we note that, there are mainly two effects arising due to the interaction between the cold DM and baryon.", "First, it helps to cool down the IGM faster (first term of rhs.", "of eq.", "REF ).", "Second, the friction due to the relative velocity between the DM and baryon can heat up both the DM and IGM (second term of rhs.", "of eq.", "REF )).", "We find that the friction heating dominates over the cooling for the DM particle mass $m_{\\chi } \\gtrsim 1\\, {\\rm GeV}$ , and instead of cooling, the IGM gets heated due to the DM-baryon interaction for higher DM particle mass.", "However, in our case, we need faster cooling off the IGM.", "Therefore, the friction heating always remains subdominant in our case.", "The bottom panel of Fig.", "REF shows the evolution of the residual free electron fraction, $x_e$ .", "As expected the residual free electron fraction is lower when the DM-baryon interaction comes into play.", "This is because the Hydrogen recombination rate $\\alpha _e$ is increased when the IGM temperature is lower.", "The change in $x_e$ is not significant for $m_{\\chi }=1$  GeV and $\\sigma _{45}=1$ (red curve).", "However, $x_e$ is reduced by factor of $\\sim 5$ for $m_{\\chi }=0.01\\, {\\rm GeV}$ and $\\sigma _{45}=50$ (blue curve).", "The reduced $x_e$ enhances the rate of IGM heating through the ambipolar diffusion.", "At the same time lower IGM temperature reduces the coupling co-efficient $\\gamma (T_g)$ (eq.", "REF ), which again enhances the heating rate.", "Moreover, heating due to the Compton process, which is proportional to $x_e (T_{\\gamma }-T_g)$ (second term on the rhs of eq.", "REF ), gets affected when the IGM is colder compared to the standard scenario.", "Figure: Same as Fig.", ", however both the primordial magnetic field and DM-baryon interaction are considered here." ], [ "Combined impact of primordial magnetic field and dark matter-baryon interaction", "Here we discuss results on the combined impact of the primordial magnetic field and DM-baryon interaction on the IGM temperature evolution.", "Fig.", "REF shows the evolution of the IGM temperature when both the primordial magnetic field and DM-baryon interactions are considered.", "In Table REF we have mentioned $T_{21}$ at $z=17.2$ as predicted by our models with different model parameters and shown which parameter set is allowed or not allowed by the EDGES measurements.", "We see that the differential brightness temperature $T_{21}$ at $z=17.2$ , for the parameter set $m_{\\chi }=0.001\\, {\\rm GeV}$ , $\\sigma _{45}=30$ , is within the allowed range when the primordial magnetic field with $B_0$ as high as $B_0=0.4 \\, {\\rm nG}$ is active, although $T_{21}$ is much lower when the magnetic field is kept off.", "Similarly, the parameter set $m_{\\chi }=0.1\\, {\\rm GeV}$ , $\\sigma _{45}=5$ is ruled out as it predicts much lower $T_{21}$ than what is allowed by the EDGES data.", "However, if we include the primordial magnetic field with, say, $B_0=0.1 \\, {\\rm nG}$ , the above parameter set becomes allowed.", "Contrary to this, $T_{21}$ predicted by some combinations of $m_{\\chi }$ , $\\sigma _{45}$ could be well within the allowed range when there is no primordial magnetic field, but ruled out when the magnetic field is applied.", "For example $T_{21}=-0.62 \\, {\\rm K}$ for $m_{\\chi }=1 \\, {\\rm GeV}$ , $\\sigma _{45}=1$ when $B_0=0$ , but goes to $-0.15 \\, {\\rm K}$ which is above the allowed range for $B_0=0.05 \\, {\\rm nG}$ .", "We discussed in sub-section REF that the primordial magnetic field with $B_0 \\gtrsim 0.1 \\, {\\rm nG}$ is ruled out in the standard scenario, but it can be well within the allowed range when the interaction between DM and baryon with an appropriate parameter sets comes into play.", "In general, we find that the exact upper limit on the primordial magnetic field depends on the mass of the DM particles $m_{\\chi }$ and the DM-baryonic interaction cross section $\\sigma _{45}$ .", "We see that the primordial magnetic field with $B_0 \\sim 0.4 \\, {\\rm nG}$ is allowed for an appropriate set of $m_{\\chi }$ and $\\sigma _{45}$ .", "Note that this primordial magnetic field is ruled out in the standard scenario.", "The upper panel of Fig.", "REF shows that the primordial magnetic field and DM-baryonic interaction together introduces a `plateau like feature' in the redshift evolution of the IGM temperature for a certain range of model parameters $m_{\\chi }$ , $\\sigma _{45}$ and $B_0$ .", "One such example can be seen for $m_{\\chi }=0.01\\, {\\rm GeV}$ , $\\sigma _{45}=50$ and $B_0=0.25 \\, {\\rm nG}$ where the plateau like feature is seen in redshift range $\\sim 50-150$ .", "The cooling rate due to the DM-baryonic interaction and heating rate due to the primordial magnetic field compensates each other for a certain redshift range which gives the plateau like feature.", "At lower redshifts the heating due to the primordial magnetic field, which scales as $ B^4(z)$ , becomes ineffective as the primordial magnetic field decays very fast.", "This is both due to the adiabatic expansion of universe and loss of the magnetic energy due to heating.", "We notice that this plateau like feature is not so prominent for lower primordial magnetic field.", "The `plateau like feature' is a unique signature of the DM-baryonic interaction in presence of the primordial magnetic field.", "However, it can only be probed by space based experiment as it appears at redshift range $\\sim 50-150$ .", "The middle and lower panels of Fig.", "REF show the residual electron fraction, $x_e$ and primordial magnetic field, $B(z)$ as a function of redshift respectively.", "Like in previous cases, the residual electron fraction $x_e$ is suppressed when both the DM-baryonic interactions and primordial magnetic field are active.", "The suppressed residual electron fraction enhances the heating rate occurring due to the ambipolar diffusion.", "The primordial magnetic field looses its energy (other than the adiabatic loss because of universe's expansion) due to transfer of energy to IGM heating through the ambipolar diffusion process.", "This loss starts becoming important at lower redshifts $z \\lesssim 100$ .", "As the primordial magnetic field decays very fast, the magnetic heating becomes ineffective at lower redshifts.", "The EDGES absorption spectra show that the IGM temperature is rising at redshifts $z \\lesssim 17$ .", "There are several possible mechanisms by which the IGM can be heated up such as heating due to soft X-ray, Ly-$\\alpha $ , DM decay/annihilation [27], [11], [10], [33], [8], [16].", "However, we find that the primordial magnetic field is not able to considerably heat up the IGM at the later phase of the cosmic dawn and, therefore, can not explain the heating part of the EDGES absorption profile." ], [ "Constraints on dark matter-baryon interaction in presence of the primordial magnetic field", "Fig.", "REF demonstrates the constraints on the DM-baryon interaction in presence of the primordial magnetic field.", "The top left panel presents constraints on the model parameters $m_{\\chi }$ and $\\sigma _{45}$ when there is no magnetic field i.e., $B_0=0$ .", "This is quite similar to constraints obtained by [1].", "Note that the constraints are obtained by restricting the differential brightness temperature $T_{21}$ within $-0.3 \\, {\\rm mK}$ to $-1.0 \\, {\\rm K}$ as suggested by the EDGES measurements.", "The DM particle with mass higher than a few ${\\rm GeV}$ is ruled out because the cooling due to the DM-baryonic interaction becomes inefficient and the drag heating due to the friction between the DM and baryon starts to dominate for higher DM particle mass.", "Therefore, the drag heating is found to have very negligible role in the case considered here.", "The top right, bottom left and the bottom right panels show constraints on the model parameters $m_{\\chi }$ and $\\sigma _{45}$ in presence of the primordial magnetic field with $B_0=0.05, \\, 0.1 $ and $0.2 \\, {\\rm nG}$ respectively.", "We see that the allowed range of the DM-baryon cross section $\\sigma _{45}$ gradually increases as $B_0$ is increased.", "For example, the lowest allowed $\\sigma _{45}$ moves up, from $\\sim 4 \\times 10^{-47} \\, {\\rm m^2}$ , to $\\sim 2.5 \\times 10^{-46} \\, {\\rm m^2}$ , $\\sim 1.5 \\times 10^{-45} \\, {\\rm m^2}$ and $\\sim 1.5 \\times 10^{-44} \\, {\\rm m^2}$ for $B_0=0.05, \\, 0.1 $ and $0.2 \\, {\\rm nG}$ respectively.", "On the other hand, the maximum allowed mass of the DM particle $m_{\\chi }$ gradually decreases for higher magnetic field.", "In Fig.", "REF we find that the highest allowed $m_{\\chi }$ goes down, from $\\sim 5 \\, {\\rm GeV}$ , to $ \\sim 1\\, {\\rm GeV}$ , $ \\sim 0.3 \\, {\\rm GeV}$ and $\\sim 0.1 \\, {\\rm GeV}$ for $B_0=0.05, \\, 0.1 $ and $0.2 \\, {\\rm nG}$ respectively.", "The primordial magnetic field heats up the IGM and the heating is more for higher values of $B_0$ .", "The DM-baryonic interaction needs to be more efficient to compensates for this extra heating which can be achieved either by increasing the cross section $\\sigma _{45}$ or/and lowering the mass of the Dark matter particle $m_{\\chi }$ .", "The above discussion also tells that the exact upper limit on the primordial magnetic field parameter $B_0$ depends on the mass $m_{\\chi }$ and the cross section $\\sigma _{45}$ .", "Higher primordial magnetic field is allowed if $\\sigma _{45}$ is increased and/or $m_{\\chi }$ is decreased.", "We see that the primordial magnetic field with $B_0 \\sim 0.4 \\, {\\rm nG}$ (Table REF ) is allowed for an appropriate set of $m_{\\chi }$ and $\\sigma _{45}$ .", "Note that $B_0 \\gtrsim 0.1 \\, {\\rm nG}$ is ruled out in the standard scenario.", "However, we find that the primordial magnetic field with $B_0 \\gtrsim 1 \\, {\\rm nG}$ may not be allowed as this requires very efficient cooling of the IGM which is unlikely even for very high cross section and lower DM particle mass.", "Although, we note that a recent study by [4], which has used the EDGES measurements, finds an upper limit of $\\sim 10^{-6} \\, {\\rm G}$ on the primordial magnetic field for $m_{\\chi } \\lesssim 10^{-2} \\, {\\rm GeV}$ in presence of the DM-baryonic interaction.", "We study prospects of constraining the primordial magnetic field in light of the EDGES low band 21-cm absorption spectra during the cosmic dawn.", "Our analysis is carried out on the background of `colder IGM' which is a promising avenue to explain the strong absorption signal found by the EDGES.", "We consider an interaction between baryons and cold DM particles which makes the IGM colder than in the standard scenario.", "The primordial magnetic field heats up the IGM through the ambipolar diffusion and decaying turbulence which, in turn, influences the 21-cm differential brightness temperature.", "We highlight the role of the residual electron fraction.", "We also study constraints on the DM-baryon interaction in presence of the primordial magnetic field, features in the redshift evolution of IGM temperature.", "In addition, we study redshift evolution of the primordial magnetic field during dark ages and cosmic dawn.", "In particular, we focus on the departure from the simple adiabatic scaling of the primordial magnetic field ( i.e.", "$B(z) \\propto (1+z)^2$ ) due to the transfer of magnetic energy to the IGM.", "Studying the role of the primordial magnetic field on the background of colder IGM is important for several reasons.", "First, it suppresses the abundance of the residual free electron fraction $x_e$ [7] which, in turn, enhances the rate of IGM heating through the ambipolar diffusion.", "Second, the coupling coefficient between the ionised and neutral components $\\gamma (T_g)$ decreases with the IGM temperature, which again results in the increased heating rate (see eq.", "REF ).", "Third, the heating rate due to the Compton process, which is proportional to $(T_{\\gamma } -T_g)$ and $x_e$ , too gets affected when the background IGM temperature $T_g$ is lower (eq.", "REF ).", "We find that collectively all these effects make the heating rate due the magnetic field faster in the colder background in compare to the heating rate in the standard scenario.", "Consequently, the primordial magnetic field decays, with redshift, at much faster rate compared to the simple $(1+z)^2$ scaling during the dark ages and cosmic dawn.", "The decay is particularly significant for $B_0 \\lesssim 0.5 \\, {\\rm nG}$ when the fractional change in the magnetic field due to the heating loss could be $\\sim 50 \\%$ or higher.", "This is unique in the colder IGM scenario.", "Next we find that the upper limit on the primordial magnetic field using the EDGES measurements is determined by the underlying non-standard cooling process, i.e., the DM-baryon interaction here.", "Higher primordial magnetic field may be allowed when the underlying DM-baryon interaction cross section is higher and/or the DM particle mass is lower, i.e., the exact upper limit on $B_0$ depends on the DM mass and the interaction cross section.", "For example, the primordial magnetic filed with $B_0 \\sim 0.4 \\, {\\rm nG}$ which is ruled out in the standard model [19], may be allowed if the DM-baryon interaction with $m_{\\chi }=0.01 \\, {\\rm GeV}$ and $\\sigma _{45}=100$ is included.", "However, we find that the primordial magnetic field with $B_0 \\gtrsim 1 \\, {\\rm nG}$ may not be allowed as this requires very efficient cooling of the IGM which is unlikely to occur even for very strong possible DM-baryon interaction.", "Furthermore, we observe that the primordial magnetic field and DM-baryonic interaction together introduces `a plateau like feature' in the redshift evolution of the IGM temperature for a certain range of model parameters $m_{\\chi }$ , $\\sigma _{45}$ and $B_0$ .", "The cooling rate due to the DM-baryonic interaction and heating rate due to the primordial magnetic field compensates each other for a certain redshift range which produces the plateau like feature.", "However, this kind of plateau is not prominent for lower primordial magnetic field with $B_0 \\lesssim 0.1 \\, {\\rm nG}$ .", "The EDGES absorption spectra suggest that the IGM temperature has possibly gone up from $\\sim 3 \\, {\\rm K}$ at redshift $z \\approx 16$ to $\\sim 40 \\, {\\rm K}$ at redshift $z \\approx 14.5$ .", "There are several possible candidates such soft X-ray photons from the first generation of X-ray binaries, mini-quasars, high energy photons from DM-decay/annihilations, primordial magnetic field etc.", "which could heat up the IGM during the cosmic dawn.", "However, our study shows that the heating due the primordial magnetic field becomes very weak during the above redshift range.", "Because the magnetic energy density decreases very fast prior to the cosmic dawn both due to the adiabatic expansion of universe and the loss due to IGM heating.", "Therefore, it is unlikely that the primordial magnetic field contributes to the heating of the IGM during the late phase of the cosmic dawn as indicated by the EDGES measurements.", "Finally, we see that the allowed DM-baryon cross section $\\sigma _{45}$ gradually shifts towards higher values as $B_0$ is increases.", "On the other hand, the allowed mass of the DM particle $m_{\\chi }$ gradually decreases for higher values of the primordial magnetic field.", "Because, the DM-baryon interaction needs to be more efficient to compensate for the excess heating caused due to higher magnetic field, which can be achieved either by increasing the cross section or lowering the mass of the Dark matter particle.", "There could be various other models of the DM-baryon interactions, for which the exact upper limit on the primordial magnetic field, and all other results discussed above might change to some extent.", "However, the general conclusions regarding the role of the primordial magnetic field on a colder IGM background are likely to remain valid for any mechanism providing faster cooling off the IGM." ], [ "Acknowledgements", "AB acknowledges financial support from UGC, Govt.", "of India.", "KKD and SS acknowledge financial support from BRNS through a project grant (sanction no: 57/14/10/2019-BRNS).", "KKD thanks Somnath Bharadwaj for useful discussion.", "SS thanks Presidency University for the support through FRPDF grant." ] ]

[ [ "Simultaneous feedback control of toroidal magnetic field and plasma\n current on MST using advanced programmable power supplies" ], [ "Abstract Programmable control of the inductive electric field enables advanced operations of reversed-field pinch (RFP) plasmas in the Madison Symmetric Torus (MST) device and further develops the technical basis for ohmically heated fusion RFP plasmas.", "MST's poloidal and toroidal magnetic fields ($B_\\text{p}$ and $B_\\text{t}$) can be sourced by programmable power supplies (PPSs) based on integrated-gate bipolar transistors (IGBT).", "In order to provide real-time simultaneous control of both $B_\\text{p}$ and $B_\\text{t}$ circuits, a time-independent integrated model is developed.", "The actuators considered for the control are the $B_\\text{p}$ and $B_\\text{t}$ primary currents produced by the PPSs.", "The control system goal will be tracking two particular demand quantities that can be measured at the plasma surface ($r=a$): the plasma current, $I_\\text{p} \\sim B_\\text{p}(a)$, and the RFP reversal parameter, $F\\sim B_\\text{t}(a)/\\Phi$, where $\\Phi$ is the toroidal flux in the plasma.", "The edge safety factor, $q(a)\\propto B_t(a)$, tends to track $F$ but not identically.", "To understand the responses of $I_\\text{p}$ and $F$ to the actuators and to enable systematic design of control algorithms, dedicated experiments are run in which the actuators are modulated, and a linearized dynamic data-driven model is generated using a system identification method.", "We perform a series of initial real-time experiments to test the designed feedback controllers and validate the derived model predictions.", "The feedback controllers show systematic improvements over simpler feedforward controllers." ], [ "Introduction", "The reversed field pinch (RFP) is a toroidal magnetic confinement configuration that has the potential to achieve an ohmically heated and inductively sustained steady-state fusion plasma.", "In contrast to the tokamak configuration, the RFP is magnetized primarily by plasma current.", "The magnetic equilibrium has low safety factor, $|q(r)|\\lesssim a/2R_0$ , allowing the current density and ohmic heating to be much larger than for a tokamak plasma of the same size and magnetic field strength [1].", "Furthermore, the RFP plasma exhibits magnetic relaxation that is subject to conservation of magnetic helicity [2], [3].", "This allows the possibility for using AC magnetic helicity injection, also called oscillating field current drive (OFCD), to sustain a steady-state plasma current using purely AC inductive loop voltages [4], [5].", "An ohmically heated and inductively sustained plasma could greatly simplify a toroidal magnetic fusion reactor by eliminating the need for auxiliary heating and non-inductive current drive.", "Key to achieving a steady-state, ohmically heated RFP will be advanced, programmable control of the poloidal and toroidal field magnets and their power supplies.", "Programmable power supplies are used in many fusion experiments, but the RFP has the special challenge of large power flow between the poloidal and toroidal magnetic field circuits via nonlinear relaxation processes regulated by the plasma.", "This is particularly acute with OFCD, where megawatts of reactive power oscillate between the circuits and regulated by the plasma [6].", "Precise phase control of the AC toroidal and poloidal inductive loop voltages is essential for OFCD.", "The relaxation process appearing in RFP plasmas happens through nonlinear interactions of tearing instabilities that cause magnetic turbulence, which tends to degrade energy confinement.", "This turbulence decreases with increasing plasma current and commensurate higher plasma temperature, but it is still uncertain if energy confinement scaling will be sufficient to reach ohmic ignition.", "Inductive pulsed parallel current drive (PPCD) control shows that the RFP plasma can achieve similar confinement to a tokamak of the same size and magnetic field strength when tearing instabilities are reduced [7], [8].", "The self-organized, quasi-single-helicity (QSH) regime that appears spontaneously in high current RFP plasmas also has improved confinement from reduced stochastic magnetic transport [9].", "An inductive control method called self-similar ramp-down (SSRD) has been shown theoretically to completely stabilize tearing in the RFP [10], but the required inductive programming is yet to be demonstrated in experiments.", "In SSRD, both $B_p(t)$ and $B_t(t)$ are ramped down using simultaneous programming of both circuits at a characteristic rate somewhat faster than the plasma's natural $L/R$ time.", "This inductively sustains a tearing-stable magnetic equilibrium having constant $q(r)$ without the need for dynamo relaxation and its concomitant magnetic fluctuations, whereas PPCD imparts a large change in $q(r)$ .", "Further, advanced inductive control capable of transitioning between OFCD and SSRD programming on demand and with minimal delay could yield a hybrid, nearly-steady-state scenario that combines the advantages of efficient current sustainment via OFCD and stability control via SSRD using robust inductive current drive [11].", "The development of real-time programmability is essential to achieve such advanced inductive control for an RFP plasma.", "Programmable power supplies are being developed for the MST facility, in which advanced control scenarios can be deployed and tested.", "Partial power supplies exist for low-current operation, which are used to begin the development of advanced control.", "Power supplies capable of high-current advanced inductive control experiments in MST are presently under construction.", "Immediate advantages accrue, since inductive control enables a wide range of new capabilities spanning the breadth of MST's fusion and basic plasma science missions.", "In a broader context, active control of MHD instabilities is important for magnetically confined plasmas.", "For example, in tokamak experiments, toroidal rotation is believed to have an important effect on MHD stability, where altering the plasma profile and speed can increase the stability of tearing, kink/ballooning, and resistive wall modes (RWM) [12], [13], [14], [15], [16].", "Feedback systems consisting of real-time computation, arrays of magnetic sensors (to measure the toroidal angular plasma momentum) and external actively actuated coils and beam injectors (to drive or drag the rotation) have the potential to maintain plasma stability which is an important factor in avoiding disruptions in tokamaks [17], [18].", "Physics-based feedback controllers have been successfully applied, both in the tokamak community with the real-time control schemes based on RAPTOR [19], and in the RFP community with the `clean mode control' technique for RMW stabilization in RFX-mod [20].", "In the context of the present work, two other prominent examples are classical controllers for $I_p$  [21] and $F$  [22] in RFX-mod.", "Modern control theory approaches have been taken in both RFPs and tokamaks, where advanced feedback control algorithms have been used in RWM control problems for different devices: [23], [24] for the EXTRAP T2R and RFX-mod RFPs and [25], [26], [27] for the DIII-D and NSTX tokamaks are good examples.", "This paper reports the first systematic tests of real-time control of the programmable power supplies presently available on MST.", "The initial focus is simultaneous control of the toroidal plasma current, $I_p$ , and the dimensionless reversal parameter, $F= B_T(a)(\\pi a^2/\\Phi )$ , where $B_T(a)$ is the toroidal field at the plasma surface, and $\\Phi $ is the toroidal flux within the plasma.", "The reversal parameter tends to track the edge safety factor, $q(a)\\propto B_T(a)$ , but $\\Phi $ is determined primarily by poloidal plasma current, since the applied toroidal field is small for the RFP.", "Therefore, $F$ is influenced by the nonlinear relaxation process occurring in the plasma and is not simply proportional to circuit current in a power supply.", "Controlling $F$ represents a first step toward understanding the influence of nonlinear effects in advanced control of the RFP magnetic equilibrium.", "The work begins by building linear models through a system identification method based on experimental data, and then tests these models on independent sets of experimental data.", "A systematic linear control theory procedure is then used to design a model-based controller which is eventually applied to the MST device control experiment through a fully integrated control software, here dubbed the MST Control System (MCS).", "We find that the model-based feedback controllers show some aspects of improved performance relative to simple feed-forward controllers, but the work reveals additional development is required to achieve the desired advanced controllers.", "The paper is organized as follows.", "Section  introduces the data-driven models to predict MST reversal parameter $F$ and plasma current $I_\\text{p}$ separately then simultaneously.", "Section  describes the general design of the used controllers.", "Experimental results of $F$ and $I_\\text{p}$ control are shown in Section .", "Section  concludes the paper." ], [ "The data-driven modeling", "The MST facility [28] produces RFP plasmas with current $I_p<0.6$  MA.", "Its major and minor radii are $R_0=1.5$  m and $a=0.5$  m respectively.", "For the work in this paper, low-current plasmas ($I_p\\approx 75$  kA, $T_e\\approx 60$  eV, density $n_e\\approx 1\\times 10^{19}$  m$^{-3}$ ) were studied using programmable switching power supplies [29], [30] attached to the poloidal field transformer and toroidal magnet.", "These supplies presently have limited current capability, but upgrades capable of much larger current are under construction.", "The facility also produces low-current tokamak plasmas.", "Waveforms for $F(t)$ and $I_p(t)$ in a typical MST discharge are shown in figure REF .", "The waveforms are punctuated by abrupt quasi-periodic events, called sawteeth.", "These events result from the magnetic relaxation process related to tearing instabilities that maintain the current profile near marginal stability.", "Our designed controller is not aiming to control the fast sawtooth dynamics but rather control the slower trend of $F$ and $I_\\text{p}$ .", "Therefore sawtooth dynamics will appear and affect our controlled results.", "Figure: Experimental data (Shot # 1161209010) exhibiting sawteeth effect on the dynamics of the reversal parameter FF (a) and the plasma current I p I_\\text{p} (b).", "This shot is a standard MST plasma produced using legacy non-programmable power supplies that attain an approximate “flattop” using pulse-forming-network techniques.The basic idea of system identification is characterizing a dynamical system from sampled input and output data [31].", "We assume that our MST device can be approximated by a linear time invariant (LTI) discrete-time system.", "Convergence results show that it is possible, for a particular class of models, to asymptotically obtain an accurate LTI system description as the size of the sample dataset $N$ and the model order $n$ both goes to infinity, but with $n/N$ vanishing [32].", "Three models are constructed here: A first single input single output (SISO) model of the reversal parameter $F$ , a second (SISO) model of the plasma current $I_\\text{p}$ and finally a multiple inputs multiple outputs (MIMO) model for the coupled dynamics of $F$ and $I_\\text{p}$ .", "Each model is going to have a corresponding controller and be applied in real time experimentally to the MST device.", "The schemes are depicted in Figure REF .", "The procedures are detailed in the following subsections.", "Figure: Schematic of the MIMO and the two-loop control design approaches." ], [ " The $F$ model", "The collected data come from a hundred shots (estimation data set) where the input data (actuator) is the primary current $I_{tg}$ of the $B_\\text{t}$ programmable power supply and the output data (sensor measurement) is the reversal parameter $F$ ranging from 0 to $-0.4$ .", "Figure REF represents one example of the data collected.", "we can see that up to the end of the plateau period ($t=0.04$  s), the dynamic shows a linear behavior.", "This will be the focus of our modeling.", "A system identification process was used to develop a linear state-space response model to the system.", "This model will then be used to design an optimal control law.", "The discrete linear state-space response model can be written of the form $\\begin{split}{x}_{k+1} &= A x_k + B u_k , \\\\y_k &= C x_k,\\end{split}$ where the physical actuator value $u_k=I_{tg}$ is the primary current at a certain iteration and the measurement of the system output $y= F$ is the corresponding reversal parameter.", "$A \\in \\mathbb {R}^{\\, (n_x) \\times (n_x)}$ , $B \\in \\mathbb {R}^{\\,(n_x) \\times 1}$ , and $C \\in \\mathbb {R}^{\\,1 \\times (n_x)}$ which respectively are called the dynamics, control and sensor matrices, identified through system identification.", "The subspace method [31] for state-space model identification, part of the Matlab System Identification Toolbox, was used to find the optimal system matrices for a prescribed number of states $n_x$ (model-order) that best fitted the estimation data set.", "The optimal choice of model-order was then found by identifying a set of models for a small range of $n_x$ , simulating the identified models using the inputs from the validation dataset (a different set of shots), and comparing how well each model predicted the output of the validation dataset.", "Models with too low number of states fail to capture the main dynamics of the system, while models with excessive number of states overfit the noise in the estimation data set, degrading prediction of the validation dataset.", "Figure: Comparison of output (FF) predicted by the identified model to the actual reversal parameter FF of the validation data.A comparison of the outputs of the optimal model, which was found to be of the order of three, to the validation data is shown in figure REF , showing good agreement in $F$ (root mean square error of 15%).", "The benchmarking of the model against several real data shots is a necessary first step as this model will be used for our control design testing.", "An exact plasma model is not a major concern as feedback control can be performed to tolerate errors in the model.", "The key is to ensure that the $F$ model does not deviate drastically from the actual time evolution in order to prevent control system instabilities from dominating plasma physics dynamics." ], [ " The $I_p$ model", "The dedicated data come from about 50 shots where the input data (actuator) is the primary current $I_{pg}$ of the $B_\\text{p}$ programmable power supply and the output data (sensor measurement) is the plasma current $I_\\text{p}$ .", "Figure REF represents an example of the data collected.", "We can see that during this shot period, the dynamics shows a linear behavior but with a slight decay of plasma current around $t=0.035$  s that persists until the end of the shot.", "This phenomenon is due to plasma resistance.", "Our model does not intend to capture this current decrease as it is assumed linear.", "Figure: An experimental data (Shot # 1180207047) showing the relationship between the input (I pg I_{pg}) and the output (I p I_\\text{p}) of our plasma current model.Figure: A comparisons of output (I p I_\\text{p}) predicted by the identified model to the actual plasma current of the validation dataA comparison of the outputs of the optimal model to the validation data is shown in figure REF , showing good agreement in $I_\\text{p}$ up to the time where the plasma resistance effect starts to become noticeable ($t=0.035$  s).", "The model is linear, so for a constant input ($I_{pg}$ ), it predicts a steady plasma current whereas the experiments show a little decrease towards the end due to the plasma resistance." ], [ " The coupled $F$ and {{formula:fec807c4-4f90-4adb-8e57-aa65768d6940}} model", "During the modeling, we started by a SISO model for $F$ and $I_\\text{p}$ , as the purpose was to test and control each variable independently on MST.", "Naturally we split the system into two single-input-single-output loops, and we use the models of $F$ and $I_\\text{p}$ described above and its controllers to operate it.", "Once these types of controllers work on MST, we use the fact that these two entities are dynamically coupled to build a coupled MIMO model that has the two primary currents $I_{tg}$ and $I_{pg}$ as inputs and both $F$ and $I_\\text{p}$ as outputs.", "Figure: A comparisons of outputs (FF (a) and I p I_\\text{p} (b)) predicted by the identified model to the actual reversal parameter and plasma current of the validation dataA comparison of the outputs of the optimal model to the validation data is shown in figure REF , showing good agreement in $F$ and $I_\\text{p}$ .", "We can notice that the $F$ prediction of the coupled model loses some precision (lower fitting) compared to its prediction in the individual $F$ model.", "This is due to the coupling consideration.", "It will produce though a better controller design due to the additional dynamics information it encapsulates (if we compare it to the split system).", "Once the identified models are built, we use them to design Linear Quadratic Gaussian (LQG) controllers.", "This type of controllers minimizes a cost function of the form [33], [34], [35] $J = \\int _0^T { x(t)^\\mathsf {T}\\!", "Q x(t) + u(t)^\\mathsf {T}\\!", "R u(t) + x_i(t)^\\mathsf {T}\\!", "Q_i x_i(t) \\, dt},$ where $x(t)$ is the internal state of the system at time $t$ , $u(t)$ is the control input, $x_i(t)$ is the integral of the tracking error (between the targeted and actual values).", "The controller optimizes the use of actuators according to the weights in $Q$ and $R$ , which are free design parameters, and also ensures reference tracking with the integral action tailored by choice of the free design parameters in $Q_i$ .", "A Kalman filter [33], [34], [35] is embedded in the resulting control law, which optimally estimates the unmeasured states $x(t)$ based on the measurements $y(t)$ , taking into account the process and measurement noise levels.", "An anti-windup scheme is implemented to keep the actuator requests from winding up far beyond their saturation levels by feeding back a signal proportional to an integral of the unrealized actuation.", "Figure REF represents the schematic of this controller design.", "Figure: Global schematic of the controller that combine a feedforward (F)(F), a LQR (K)(K), an observer, an integrator (K I )(K_I) and an anti-windup (AW).The same control design will be used for both One-Input One-output models that control $F$ using $I_{tg}$ and $I_\\text{p}$ using $I_{pg}$ , and the Two-input Two-output model (MIMO) that controls simultaneously $F$ and $I_\\text{p}$ using primary currents $I_{tg}$ and $I_{pg}$ .", "The only difference will be the dimension of the model inputs and outputs which change from a single variable to a vector variable; the controller dimensions will adapt accordingly.", "More details about the control theory and design can be found in [36], [37] but will be summarized succinctly in this section.", "As shown in Figure REF , the controller design has five main components:" ], [ "Feedforward design $F$", "The purpose here is to force the plasma current $I_p$ and the revesal parameter $F$ to reach a target state $x_d$ such that the sensor output $y$ matches a reference signal $y_d$ .", "In the final implementation, all one should have to prescribe is $y_d$ (e.g., the desired plasma current value $I_p$ and the desired reversal parameter $F$ ).", "The target state $x_d$ and the corresponding input $u_d$ are found by solving equations (REF ) at steady state: $ \\begin{split}0 &= A x_d + B u_d, \\\\y_d &= C x_d.\\end{split}$ We then solve for $x_d$ and $u_d$ by writing (REF ) in matrix form $\\left(\\!", "\\begin{array}{c} x_{d} \\\\ u_{d}\\end{array}\\!\\right)={ \\left(\\!", "\\begin{array}{cc} A & B \\\\ C & 0 \\end{array} \\!", "\\right)}^{-1} \\left(\\!", "\\begin{array}{c} 0 \\\\ I \\end{array} \\!\\right) y_{d} = \\left(\\!", "\\begin{array}{c} F_x \\\\ F_u \\end{array} \\!\\right) y_{d}.$ Once the desired target states $\\left( x_{d} , u_{d} \\right)$ are established, the controller is designed based on the model then tested on the MST device to determine if the controller can track and reach the desired $F$ and $I_p$ values in the vicinity of the equilibrium.", "If the model of the dynamics has no errors or uncertainties (which is never the case) and is stable, a feedforward controller is enough to reach the target.", "$ F_u$ and $ F_x $ are the feedforward gains corresponding to the input and state respectively.", "The total feedforward gain $F$ depends on the matrices $A$ , $B$ , $C$ and $K$ (explained in the following subsection)." ], [ "Linear quadratic regulator (LQR) design $K$", "The feedback control law links the input $u$ to the state $x$ by $u = u_{d} - K(x - x_{d}) = - Kx + Fy_{d},$ where $K$ is the feedback control gain to be determined from control design and $F = F_u + K F_x$ is the total feedforward gain.", "Therefore, the resulting closed-loop system of equations (REF ) can be written as $\\begin{aligned}\\dot{x} &= (A-BK) x + BF y_{d}, \\\\y &= C x.\\end{aligned}$ A standard linear control technique (linear-quadratic regulators) [33], [35] is used in order to determine the gains $K$ while minimizing a quadratic cost function." ], [ "Observer design", "The feedback law (REF ) requires the knowledge of the full state $x$ .", "However, in our actual system-identified model, we don't know the state; we don't even know what the state represents, we only measure the inputs-outputs.", "However, we may reconstruct an estimate of the state from the available sensor measurements using an observer.", "The observer will then reconstruct the state estimate $\\hat{x}$ , with dynamics given by $\\dot{\\hat{x}} = A \\hat{x} + B u + L (y - C \\hat{x})= (A- L C) \\hat{x} + B u + L y,$ where the matrices $A,B$ and $C$ are the same as those in the model (REF ), and $L$ is a matrix of gains chosen such that the state estimate converges quickly relative to the system's dynamics.", "Using our linear model, we design an optimal observer (Kalman filter)[33], [35] to find $L$ .", "The observer generates an estimate of the state from the physics model as represented by the state matrix, the inputs and outputs, and once combined to the feedback controller, it forms a linear quadratic Gaussian compensator [33], [35]." ], [ "Integrator design $K_I$", "The goal is to track both the desired plasma current and reversal parameter values (reference tracking).", "In order to do that, the steady state error between the output (measured) and the target profile has to be minimized by using an integrator and introducing a new state variable $z$ that is the integral of the error: $\\dot{z} = y_{d} - y = y_{d} - C x.$ The new feedback law can be then written as $u = u_d + K (x_d - x) + K_I \\!\\!\\int (y_d - y)$ where $K_I$ be the gain of the integrator." ], [ "Anti-windup design $AW$", "A drawback of integral control is that if the actuator values are limited to some range as in our case, then the integrator can accumulate error when the actuator is “saturated,” resulting in poor transient performance, a phenomenon known as “integrator windup.” We use a standard anti-windup scheme [35], [34] in which one feeds back the difference between the desired value of $u$ and its actual (possibly saturated) value to eliminate this effect." ], [ "Hardware and software setup", "In PPS operation on MST, a real-time Linux host for the MST Control System (MCS) provides demand waveforms clocked to a 10 kHz sampling rate via a 16-bit digital-to-analog converter (D-tAcq ACQ196).", "The analog output voltages are fed to one or both of the two programmable power supplies $B_\\text{t}$ -PPS and $B_\\text{p}$ -PPS.", "These supplies are IGBT-based switching supplies with bipolar outputs and a switching frequency of $10 \\text{kHZ}$  [29], [30].", "Each supply is powered by its own capacitor bank.", "The supplies have small-signal bandwidths of a few $\\text{kHZ}$ or less.", "Each supply uses local feedback to provide an output current proportional to the demand voltage from the controller.", "The supplies have current limiting to clamp their outputs at a safe level independent of the demand voltage.", "The $B_\\text{t}$ -PPS drives the primary of a 40:1 transformer whose secondary is the aluminum vacuum vessel, generating the toroidal field Bt.", "The $B_\\text{t}$ -PPS is capable of an output current up to +/- 25 kA at a voltage up to +/- 1800 V. The $B_\\text{p}$ -PPS drives the primary of a 20:1 transformer whose secondary is the plasma, generating the plasma current $I_p$ .", "The experiments described here use a prototype version of the BP PPS which has a limited output capability of +/- 4.8 kA at a voltage up to +/- 2700 V. Although they are referred to as programmable power supplies, the supplies themselves are not pre-programmed, but act as voltage-controlled, current-output amplifiers to produce the current demanded by their input voltages.", "The simplest control method uses the MCS to generate a preprogrammed PPS input voltage waveform which yields the desired PPS output current.", "Although this is closed-loop control with respect to the supply itself, since it is open-loop with respect to the transformer and plasma impedances, we refer to it as 'open-loop' (feedforward) control, labeled 'MST FF data' in the results presented below.", "By contrast, the more comprehensive 'closed-loop' (feedback) control is implemented ('MST CL data') to respond in real time to changes resulting from transformer and plasma impedances.", "During each clock cycle in such closed-loop experiments, MST data is input to the MCS software, processed, and the real-time demand value output to the PPS digitizers before the next clock cycle.", "For example, while in open-loop control, one only approximately controls $B_\\text{t}(a)$ , in closed-loop control, one is able to control the field reversal parameter $F = B_\\text{t} (a)/<B_\\text{t}>$ after startup.", "The output current is automatically adjusted in real time to do so.", "A demonstration of open-loop control will be superposed to the closed loop feedback control when the results are shown as a way of comparing the two controllers.", "The controller shown in Figure REF is implemented as routine running single-threaded on one dedicated core.", "Changes to the control algorithm itself appear as matrix operations whose coefficients are calculated prior to the experiment (system identification model and control design) and fixed during the experiment." ], [ " $F$ control results", "The $F$ feedback controller is implemented in the MCS and tested on MST.", "The control time occurs between $t_1 = 0.02$  s and $t_2 = 0.05$  s where we choose to track square oscillations.", "Before time $t_1$ we are in an open loop mode.", "At time $t_1$ we activate the control mode through the MCS and at time $t_2$ we release the system to open loop again.", "Figure: Comparison of outputs of FF control with and without feedback control during a tracking task.Figure REF represents the output response of MST through $F$ measurements with and without feedback control.", "As expected, the time-dependent results of the closed loop experiments successfully track the target during the control period.", "This tracking is better if compared to the open loop case (dashed line) where steady state errors appear.", "It is important to notice that despite the control effect, in both cases, sawtooth crashes are still occurring throughout the run.", "Note that for many operational purposes in the RFP, open-loop, feed-forward control is adequate, since waveforms can be optimized empirically, shot-to-shot.", "Often, however, particularly in situations where the plasma response is both important and difficult or inconvenient to predict in advance, closed-loop feedback control is needed.", "In the context of $F$ control, although it is possible to tune a pre-programmed $B_T$ waveform shot-to-shot to approximately achieve the desired waveform, as in the black signal in Figure REF , shot-to-shot changes in the time evolution of $B_T$ still affect the ratio $F$ , and better control can be achieved with feedback on the real-time signals, as discussed above.", "The superiority of closed-loop over open-loop control is expected to be especially important in the cases of OFCD [6], PPCD [38], and SSRD [10]." ], [ " $I_\\text{p}$ control results", "The $I_\\text{p}$ controller is implemented in the MCS and tested on MST.", "The control time occurs between $t_1 = 0.02$  s and $t_2 = 0.06$  s. We chose to take a longer control time frame so we can observe the plasma resistivity effects that act as a drag in the plasma current value towards the end of the shots.", "Figure REF represents the output response of MST through $I_\\text{p}$ measurements with and without control.", "The time-dependent results of the closed loop experiment successfully track the flat target during the control period.", "This tracking is better if compared to the open loop (FF) case (black line) where an important steady state error appears.", "The plasma resistance add more steady state error to both cases towards the end, but the controller does a better tracking despite the drag.", "Figure: Comparison of outputs of I p I_\\text{p} measurements with and without feedback control during a flat tracking.Figure REF is similar to Figure REF but for a different target: a square wave.", "We notice the same observations as before where there is a little overshoot at the beginning of the control period that dissipates slowly through the shot until the plasma resistance becomes important enough to start dragging the current down.", "One can argue that the controller is not aggressive enough to overcome this drag or not fast enough to get to the target.", "The tuning of the LQE control gains is indeed critical in this case but we have found that increasing the integrator or the feedback gain too much results in disruption of the plasma or introduces oscillations that we do want to avoid.", "We found some trade off values that would allow us to get to the target in less than 20 ms and minimize the steady state error.", "Figure REF is an example where we emphasized the importance of zero steady state error at the expense of the system stability.", "In this example we pushed the controller (integrator) to its limits so we can beat the plasma resistivity.", "The resulted plasma current was extremely oscillatory which cannot be considered a possible solution of the controller.", "Figure: Comparison of outputs of I p I_\\text{p} measurements with and without feedback control (a) during a square wave tracking and its corresponding input (I pg I_{pg}) (b).Figure: Resulting I p I_\\text{p} measurements with feedback control during a flat tracking" ], [ " Coupling control results", "For the two inputs two outputs control results, we studied two independently designed controllers for $F$ and $I_\\text{p}$ .", "The first case is called parallel control in the MCS where we connected the two independent controllers designed from the two independent models of $F$ and $I_\\text{p}$ in parallel without sharing any knowledge between each other.", "The second case is the coupled control case where the controller is designed directly from the coupled system of $F$ and $I_\\text{p}$ .", "Figure: Comparison of outputs of FF and I p I_\\text{p} measurements with and without parallel feedback control during a square wave tracking.Figure REF shows an example of parallel control results where we compare measurements of reversal parameter $F$ and plasma current $I_\\text{p}$ when we track a square wave using the double closed loops controller and the feedforward controller between the times $t_1 = 0.02$  s and $t_2 = 0.06$  s. We notice that despite the overshoot, we are able to successfully track the wave in both $F$ and $I_\\text{p}$ outputs.", "Designing the coupled model had a goal of improving the controller by giving it access to the coupling dynamics.", "Figure REF shows a different example of coupled control results where we compare measurements of reversal parameter $F$ and plasma current $I_\\text{p}$ when we track the same square wave using the closed loop controller and the feedforward controller between the same times $t_1 = 0.02$  s and $t_2 = 0.06$  s. We can notice a slight improvement in the $F$ control but also an overshoot elimination in the plasma current $I_\\text{p}$ control.", "This feature is important in our experiments, in that we are presently restricted by an IGBT current limit to 80 kA of maximum plasma current.", "Figure: Comparison of outputs of FF and I p I_\\text{p} measurements with and without Coupled feedback control during a square wave tracking." ], [ "Discussion", "From the experimental results shown in this paper, good performance was obtained using either the MIMO optimal controller or two SISO loop control design.", "From the perspective of an operator, this is desirable, as we showed that the two loop structure, with a small number of free parameters that can be adjusted intuitively between shots, works well in experiments.", "However, the optimal design provides a more systematic algorithm for designing a stabilizing controller.", "It is well suited for handling systems with strong cross-coupling, and can be easily extended to include additional controlled variables and actuators.", "As scenarios that exhibit stronger coupling are explored, or as additional actuators and controlled outputs are considered, the tuning of separate PID loops will become more difficult, while the MIMO control design approach will still be appropriate.", "As mentioned in Section , classical, physics-based approaches have been taken for controlling $I_p$  [21] and $F$  [22] in the RFX-mod RFP, using linear controllers with advanced feedforward or PIDs adjusted via pole placement based on transfer functions derived from simplified physical models of the plasma dynamics.", "While it is too early for precise comparisons between that work and our system-identification approach, we can nonetheless compare the merits of the approaches in a more general context.", "Physics-based models have the advantage of being immediately interpretable as the states of the system have meaningful physical representations.", "This desirable property makes the manual tuning of parameters easier and can provide insights into the science.", "On the other hand, system identification, while more opaque, has the benefit of being based on machine response rather than presumed physics models, and providing some freedom on how accurately to capture the dynamics by choosing the size of the state vector.", "And modern MIMO controllers offer systematic ways of designing robust stabilizing controllers with any number of sensors and actuators without necessitating as much manual tuning.", "A novel system has been implemented at MST to systematically control $F$ and $I_\\text{p}$ individually or simultaneously in RFP plasmas via the two primary current actuators $I_{pg}$ and $I_{tg}$ .", "Initial experiments with the closed feedback control loop show promising results and motivate future work of continued testing and design improvements.", "In addition, manipulation of $I_{pg}$ and $I_{tg}$ can be integrated into a more complex control scheme, for example by including cycle-averaged plasma current or loop voltage as controlled quantities for cases with OFCD.", "As part of this work, a flexible framework for performing feedback control design and experimentation on the MST has been developed.", "This framework will aid in the creation of advanced control algorithms by providing means for conducting system identification simulations and high-fidelity tests of proposed algorithms prior to and during experimental implementation and testing.", "In the longer term, the same framework could be extended to include additional actuators and measurements on MST, including density control through gas puffing or loop voltage control based on magnetic fluctuation amplitudes.", "These methodologies are a key element of research toward advanced inductive control of an ohmically heated RFP fusion plasma." ], [ "Acknowledgement", "The authors acknowledge helpful discussions with Dr. Brett Chapman.", "This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences program under Award Numbers DE-FC02-05ER54814 and DE-SC0018266.", "Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344." ] ]

[ [ "Cohomology and deformations of hom-dendriform algebras and coalgebras" ], [ "Abstract Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras.", "In this paper, we define a cohomology and deformation for hom-dendriform algebras.", "We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras.", "We also describe similar results for the twisted analog of dendriform coalgebras." ], [ "Introduction", "Hom-type algebras first arise in the work of Hartwig, Larsson and Silvestrov in their study of the deformations of the Witt and Virasoro algebras using $\\sigma $ -derivations [10].", "They observed that the space of $\\sigma $ -derivations satisfy a variation of the Jacobi identity, twisted by $\\sigma $ .", "An algebra satisfying such an identity is called a hom-Lie algebra.", "Other types of algebras (such as associative, Leibniz, Poisson, Hopf,$ \\ldots )$ twisted by homomorphisms have also been studied in the last few years.", "See [2], [16] and references therein for more details about such structures.", "In this paper, we deal with the twisted version of another type algebras, called dendriform algebras.", "These algebras were introduced by Loday as Koszul dual of associative dialgebras [11].", "Free dendriform algebra over a vector space has been constructed by using planar binary trees.", "Dendriform algebras also arise from Rota-Baxter operators on some associative algebra [1].", "Recently, an explicit cohomology theory for dendriform algebras has been introduced and the formal deformation theory for dendriform algebras (as well as coalgebras) has been studied in [5], [6].", "The cohomology involves certain combinatorial maps.", "See the references therein for more details about dendriform structures.", "The twisted version of dendriform structures, called hom-dendriform structures was introduced in [15].", "These algebras can be thought of as splitting of hom-associative algebras.", "They also arise from Rota-Baxter operator on hom-associative algebras.", "In [17] the authors study hom-dendriform algebras from the point of view of monoidal categories.", "Our aim in this paper is to twist the construction of [5] by a homomorphism to formulate a cohomology for hom-dendriform algebras.", "The cochain groups defining the cohomology inherits a structure of an operad with a multiplication.", "Hence by a result of Gerstenhaber and Voronov [9], the cohomology carries a Gerstenhaber structure.", "We show that there is a morphism from the cohomology of a hom-dendriform algebra to the Hochschild-type cohomology of the corresponding hom-associative algebra.", "We also study formal one-parameter deformation of a hom-dendriform algebra following the approach of Gerstenhaber [8].", "Our results about deformation are similar to the standard ones.", "The vanishing of the second cohomology implies the rigidity of the structure and the vanishing of the third cohomology ensures that a finite order deformation can be extended to a deformation of the next order.", "Finally, we consider a dual version of the above results.", "Namely, we consider hom-dendriform coalgebras, their cohomology, and deformations.", "We show that there is a morphism from the cohomology of a hom-dendriform coalgebra to the coHochschild-type cohomology of hom-associative coalgebra.", "In our knowledge, the coHochschild-type cohomology of a hom-associative coalgebra has not been mentioned before.", "Throughout the paper, we freely use the Hopf algebra and coalgebra terminology introduced in [3] and [18].", "All vector spaces, linear maps are over a field $\\mathbb {K}$ of characteristic zero, unless specified otherwise." ], [ "Hom-associative and hom-dendriform algebras", "In this section, we recall the hom-analog of associative and dendriform algebras.", "Our main references are [2], [15], [16].", "A hom-vector space is a pair $(A, \\alpha )$ consists of a vector space $A$ with a linear map $\\alpha : A \\rightarrow A$ .", "Hom-associative or hom-dendriform structures are defined on hom-vector spaces, rather than vector spaces.", "Definition 2.1 A hom-associative algebra is a hom-vector space $(A, \\alpha )$ together with a bilinear map $\\mu : A \\times A \\rightarrow A,~ (a, b) \\mapsto a \\cdot b$ , that satisfies $\\alpha (a) \\cdot (b \\cdot c) = (a \\cdot b) \\cdot \\alpha (c), ~~~ \\text{ for all } ~ a, b, c \\in A.$ A hom-associative algebra is called multiplicative if $\\alpha (a \\cdot b) = \\alpha (a) \\cdot \\alpha (b)$ .", "By a hom-associative algebra, we shall always mean a multiplicative one, unless specified otherwise.", "In [2], [16] the authors define a Hochschild-type cohomology of a hom-associative algebra and study the formal deformation theory for these type of algebras.", "Let $(A, \\alpha , \\mu )$ be a hom-associative algebra.", "For each $n \\ge 1$ , they define the group of $n$ -cochains as $C^n_{\\alpha , \\mathrm {Ass}}(A, A):= \\lbrace f : A^{\\otimes n} \\rightarrow A |~ \\alpha \\circ f (a_1, \\ldots , a_n ) = f (\\alpha (a_1), \\ldots , \\alpha (a_n)) \\rbrace $ and the differential $\\delta _{\\alpha , \\mathrm {Ass}} : C^n_{\\alpha , \\mathrm {Ass}} (A, A) \\rightarrow C^{n+1}_{\\alpha , \\mathrm {Ass}} (A, A)$ is given by $(\\delta _{\\alpha , \\mathrm {Ass}} f)(a_1, \\ldots , a_{n+1}) =~& \\alpha ^{n-1} (a_1) \\cdot f (a_2, \\ldots , a_{n+1}) \\\\~&+ \\sum _{i=1}^n (-1)^{i} f ( \\alpha (a_1), \\ldots , \\alpha (a_{i-1}), a_i \\cdot a_{i+1}, \\alpha (a_{i+2}), \\ldots , \\alpha (a_{n+1}) ) \\nonumber \\\\~&+ (-1)^{n+1} f (a_1, \\ldots , a_n) \\cdot \\alpha ^{n-1} (a_{n+1}).", "\\nonumber $ for $a_1, \\ldots , a_{n+1} \\in A$ .", "It has been shown in [4] that the above cochain groups can be endowed with a structure of an operad with a multiplication.", "Hence by a result of Gerstenhaber and Voronov [9], the cohomology inherits a Gerstenhaber structure.", "More precisely, the partial compositions of the operad are given by $(f \\bullet _i g ) (a_1, \\ldots , a_{m+n-1}) = f (\\alpha ^{n-1} a_1, \\ldots , \\alpha ^{n-1} a_{i-1}, g(a_i, \\ldots , a_{i+n-1}), \\alpha ^{n-1} a_{i+n}, \\ldots , \\alpha ^{n-1} a_{m+n-1}),$ for $f \\in C^m_{\\alpha , \\mathrm {Ass}} (A, A), ~g \\in C^n_{\\alpha , \\mathrm {Ass}} (A, A)$ and $1 \\le i \\le m$ .", "The multiplication $\\pi \\in C^2_{\\alpha , \\mathrm {Ass}} (A, A)$ is given by $\\pi (a, b) = a \\cdot b$ .", "One observes that the differential induced from the above multiplication is exactly given by (REF ).", "See [9], [4], [12] for details about operads.", "We will follow their convensions throughout.", "Definition 2.2 A hom-dendriform algebra consists of a hom-vector space $(A, \\alpha )$ together with linear maps $\\prec , \\succ : A \\otimes A \\rightarrow A$ satisfying the following identities: $(a \\prec b ) \\prec \\alpha (c) =~& \\alpha (a) \\prec ( b \\prec c + b \\succ c ), \\\\(a \\succ b ) \\prec \\alpha (c) =~& \\alpha (a) \\succ (b \\prec c),\\\\(a \\prec b + a \\succ b) \\succ \\alpha (c) =~& \\alpha (a) \\succ (b \\succ c), $ for all $a, b, c \\in A$ .", "A hom-dendriform algebra as above is denoted by the quadruple $(A, \\alpha , \\prec , \\succ )$ .", "A hom-dendriform algebra is said to be multiplicative if $\\alpha ( a \\prec b ) = \\alpha (a) \\prec \\alpha (b)$ and $\\alpha (a \\succ b ) = \\alpha (a) \\succ \\alpha (b)$ , for $a, b \\in A$ .", "When $\\alpha = \\text{id}_A$ , one obtains the definition of a dendriform algebra [11].", "In the rest of the paper, by a hom-dendriform algebra, we shall always mean a multiplicative hom-dendriform algebra unless otherwise stated.", "Example 2.3 Let $(A, \\prec , \\succ )$ be a dendriform algebra and $\\alpha : A \\rightarrow A$ be a morphism between dendriform algebras.", "Then the quadruple $(A, \\alpha , \\alpha \\circ \\prec ,~ \\alpha \\circ \\succ )$ is a hom-dendriform algebra.", "Example 2.4 (Rota-Baxter operator on hom-associative algebra) Let $(A, \\alpha , \\mu )$ be a hom-associative algebra.", "A linear map $R: A \\rightarrow A$ is said to be a Rota-Baxter operator (of weight zero) on $A$ if $\\alpha \\circ R = R \\circ \\alpha $ and the following holds $R(a) \\cdot R(b) = R(a \\cdot R(b) + R(a) \\cdot b ), ~~ \\text{ for all } a, b \\in A.$ Then it follows that the hom-vector space $(A, \\alpha )$ together with the operations $a \\prec b = a \\cdot R(b)$ and $a \\succ b = R(a) \\cdot b$ forms a hom-dendriform algebra [15].", "Remark 2.5 In [13] the authors define Rota-Baxter operator on bihom-associative algebras (associative algebras twisted by two homomorphisms) and show that such an operator induces a bihom-dendriform structure thus generalizing Example REF .", "In the next, we generalize the above example of hom-dendriform algebra in the setting of $\\mathcal {O}$ -operators on hom-associative algebras.", "Example 2.6 ($\\mathcal {O}$ -operator on hom-associative algebra) Let $A = ( A, \\mu , \\alpha )$ be a hom-associative algebra and $(M, \\beta )$ be a hom-vector space together with maps $\\mu ^l : A \\otimes M \\rightarrow M, ~ (a,m) \\rightarrow am$ and $\\mu ^r : M \\otimes A \\rightarrow M, ~ (m, a) \\rightarrow ma$ satisfying $\\beta (am) = \\alpha (a) \\beta (m)$ , $\\beta (ma ) = \\beta (m) \\alpha (a)$ and the followings $(a \\cdot b) \\beta (m) = \\alpha (a) (bm), \\quad (am) \\alpha (b) = \\alpha (a) (mb), \\quad (ma) \\alpha (b) = \\beta (m) (a \\cdot b).$ Such a tuple $(M, \\beta , \\mu ^l, \\mu ^r)$ is called a representation of $A$ .", "An $\\mathcal {O}$ -operator on $A$ with respect to the above representation is a linear map $R: M \\rightarrow A$ satisfying $\\alpha \\circ R = R \\circ \\beta $ and $R(m) \\cdot R(n) = R (m R(n) + R(m)n),$ for all $m, n \\in M$ .", "In this case, $(M, \\beta )$ together with operations $m \\prec n = m R(n)$ and $m \\succ n = R(m)n$ forms a hom-dendriform algebra.", "Like dendriform algebras are related to several other algebras, hom-dendriform algebras are related to hom analog of various algebras [15].", "Here we mention the relation with hom-associative algebras and hom-Lie algebras.", "Let $(A, \\alpha , \\prec , \\succ )$ be a hom-dendriform algebra.", "Then the new operation $a \\ast b$ defined by $a \\ast b = a \\prec b + a \\succ b$ makes $(A, \\alpha , \\ast )$ into a hom-associative algebra.", "Therefore, a hom-dendriform algebra can be thought of as an splitting of a hom-associative algebra.", "Note that the corresponding hom-Lie bracket obtained from the skew-symmetrization of the hom-associative product is given by $[a, b ] = a \\ast b - b \\ast a$ .", "On the other hand, a hom-dendriform algebra $(A, \\alpha , \\prec , \\succ )$ also induces a left hom-preLie product $ a \\diamond b = a \\succ b - b \\prec a$ , that is, the following holds $( a \\diamond b ) \\diamond \\alpha (c) - \\alpha (a) \\diamond ( b \\diamond c) = ( b \\diamond a ) \\diamond \\alpha (c) - \\alpha (b) \\diamond ( a \\diamond c).$ See [15] for details.", "The skew-symmetrization of a left hom-preLie product is a hom-Lie bracket and here it is given by $[a, b]^{\\prime } = a \\diamond b - b \\diamond a$ , for $a, b \\in A$ .", "Observe that the hom-Lie brackets on $A$ induced from the hom-associative product and hom-preLie product are same." ], [ "Cohomology", "Our aim in this section is to introduce a cohomology theory for hom-dendriform algebras as a twisted analog of the cohomology introduced in [5].", "This cohomology can be thought of as a splitting of the cohomology of hom-associative algebras.", "We will show that the cochain groups defining the cohomology of a hom-dendriform algebra carries a structure of an operad with a multiplication.", "(This construction is the twisted analog of the construction defined for dendriform algebras.)", "Hence the cohomology inherits a Gerstenhaber structure.", "In the next section, we show that this cohomology governs the formal deformation of the hom-dendriform structure.", "We recall certain combinatorial maps which are defined in [5].", "Let $C_n$ be the set of first $n$ natural numbers.", "For convenience, we denote them by $\\lbrace [1], \\ldots , [n] \\rbrace .$ For each $m, n \\ge 1$ and $1 \\le i \\le m$ , there are maps $R_0 (m; 1, \\ldots , \\underbrace{n}_{i\\text{-th}} , \\ldots , 1 ) : C_{m+n-1} \\rightarrow C_m$ and $R_i (m; 1, \\ldots , \\underbrace{n}_{i\\text{-th}}, \\ldots , 1 ) : C_{m+n-1} \\rightarrow \\mathbb {K}[C_n]$ which are given by $ R_0 (m; 1, \\ldots , 1, n, 1, \\ldots , 1) ([r]) ~=~{\\left\\lbrace \\begin{array}{ll} [r] ~~~ &\\text{ if } ~~ r \\le i-1 \\\\ [i] ~~~ &\\text{ if } i \\le r \\le i +n -1 \\\\[r -n + 1] ~~~ &\\text{ if } i +n \\le r \\le m+n -1, \\end{array}\\right.", "}$ $ R_i (m; 1, \\ldots , 1, n, 1, \\ldots , 1) ([r]) ~=~{\\left\\lbrace \\begin{array}{ll} [1] + [2] + \\cdots + [n] ~~~ &\\text{ if } ~~ r \\le i-1 \\\\ [r - (i-1)] ~~~ &\\text{ if } i \\le r \\le i +n -1 \\\\[1]+ [2] + \\cdots + [n] ~~~ &\\text{ if } i +n \\le r \\le m+n -1.", "\\end{array}\\right.", "}$ Let $(A, \\alpha , \\prec , \\succ )$ be a hom-dendriform algebra.", "We define the group of $n$ -cochains by $C^0_{\\alpha , \\mathrm {Dend}} (A, A) := 0$ and $C^n_{\\alpha , \\mathrm {Dend}} (A, A) := \\big \\lbrace f \\in \\text{Hom}_\\mathbb {K} ( \\mathbb {K}[C_n] \\otimes A^{\\otimes n}, A ) |~ \\alpha ( f ([r]; a_1, \\ldots , a_n )) = f ([r]; \\alpha (a_1), \\ldots , \\alpha (a_n) ),\\\\ \\text{ for all } [r] \\in C_n \\big \\rbrace ,$ for $n \\ge 1.$ Then the collection of spaces $\\mathcal {O}(n) = C^n_{\\alpha , \\mathrm {Dend}} (A, A), n \\ge 1$ , inherits a structure of an operad with partial compositions $\\bullet _i : \\mathcal {O}(m) \\otimes \\mathcal {O}(n) \\rightarrow \\mathcal {O}(m+n-1)$ given by $&( f \\bullet _i g) ([r]; a_1, \\ldots , a_{m+n-1})\\\\&= f \\big ( R_0 (m; 1, \\ldots , n, \\ldots , 1)[r];~ \\alpha ^{n-1} a_1, \\ldots , \\alpha ^{n-1} a_{i-1}, g (R_i (m; 1, \\ldots , n, \\ldots , 1)[r]; a_i, \\ldots , a_{i+n-1}),\\\\& \\hspace*{284.52756pt} ~~ \\alpha ^{n-1} a_{i+n}, \\ldots , \\alpha ^{n-1} a_{m+n-1} \\big ),$ for $f \\in \\mathcal {O}(m), ~ g \\in \\mathcal {O}(n),~ [r] \\in C_{m+n-1}$ and $a_1, \\ldots , a_{m+n-1} \\in A$ .", "The proof is a combination of [5] where the case $\\alpha = \\text{id}_A$ has been considered, and [4] where the linear map $\\alpha $ has been used to twist the endomorphism operad.", "Note that, in this case, there are operations $\\bullet :~& \\mathcal {O}(m ) \\otimes \\mathcal {O}(n) \\rightarrow \\mathcal {O}(m+n-1),~~~ f \\bullet g := \\sum _{i=1}^m (-1)^{(i-1)(n-1)} ~f \\bullet _i g,\\\\[~, ~]:~& \\mathcal {O}(m ) \\otimes \\mathcal {O}(n) \\rightarrow \\mathcal {O}(m+n-1), ~~~ [f, g] := f \\bullet g - (-1)^{(m-1)(n-1)} g \\bullet f.$ Moreover, one can define an element $\\pi \\in \\mathcal {O}(2) = C^2_{\\alpha , \\mathrm {Dend}} (A, A)$ by $\\pi ([1]; a, b ) = a \\prec b ~~~ \\text{ and } ~~~ \\pi ( [2]; a , b) = a \\succ b.$ It is a straightforward calculation that $(\\pi \\bullet \\pi ) ([r]; a, b, c) ={\\left\\lbrace \\begin{array}{ll} (a \\prec b) \\prec \\alpha (c) - \\alpha (a) \\prec (b \\prec c + b \\succ c) ~~~ &\\text{ if} ~~ [r] = [1] \\\\(a \\succ b) \\prec \\alpha (c) - \\alpha (a) \\succ (b \\prec c) ~~~ &\\text{ if } [r] = [2] \\\\(a \\prec b + a \\succ b ) \\prec \\alpha (c) - \\alpha (a) \\succ ( b \\succ c) ~~~ &\\text{ if } [r] = [3].", "\\end{array}\\right.", "}$ Thus, it follows from the definition of a hom-dendriform algebra that $\\pi $ defines a multiplication on the above operad $(\\lbrace \\mathcal {O}(n) \\rbrace _{n \\ge 1}, \\bullet _i ).$ The multiplication $\\pi \\in \\mathcal {O}(2)$ induces an associative product $\\mathcal {O}(m) \\otimes \\mathcal {O}(n) \\rightarrow \\mathcal {O}(m+n)$ given by $f \\cdot g = (-1)^m~ (\\pi \\bullet _2 g) \\bullet _1 f$ and a degree $+1$ differential $\\delta _\\pi : \\mathcal {O}(n) \\rightarrow \\mathcal {O}(n+1), ~f \\mapsto \\pi \\bullet f - (-1)^{n-1} f \\bullet \\pi $ .", "We also denote this differential by $\\delta _{\\alpha , \\mathrm {Dend}}$ and it is explicitly given by $&(\\delta _{\\mathrm {Dend}} f) ([r]; a_1 , \\ldots , a_{n+1}) \\\\&= \\pi \\big ( R_0 (2;1,n) [r]; ~\\alpha ^{n-1} a_1, f (R_2 (2;1,n)[r]; a_2, \\ldots , a_{n+1}) \\big ) \\\\&+ \\sum _{i=1}^n (-1)^i~ f \\big ( R_0 (n; 1, \\ldots , 2, \\ldots , 1)[r]; \\alpha (a_1), \\ldots , \\alpha (a_{i-1}), \\pi (R_i (1, \\ldots , 2, \\ldots , 1)[r]; a_i, a_{i+1}), \\alpha (a_{i+2}), \\ldots , \\alpha (a_{n+1}) \\big ) \\\\&+ (-1)^{n+1} ~\\pi \\big ( R_0 (2; n, 1) [r];~ f (R_1 (2;n,1)[r]; a_1, \\ldots , a_n), \\alpha ^{n-1}(a_{n+1}) \\big ),$ for $f \\in C^n_{\\alpha , \\mathrm {Dend}}(A, A), ~[r] \\in C_{n+1}$ and $a_1, \\ldots , a_{n+1} \\in A.$ The corresponding cohomology groups are denoted by $H^n_{\\alpha , \\mathrm {Dend}}(A, A)$ , for $n \\ge 1$ .", "When $\\alpha = \\mathrm {id}_A$ , the above cohomology coincides with the one constructed in [5].", "Hence it follows from Gerstenhaber and Voronov [9] that the cochain groups of a hom-dendriform algebra inherit a homotopy $G$ -algebra structure.", "As a consequence, the cohomology carries a Gerstenhaber structure.", "It follows from the definition that $H^1_{\\alpha , \\text{Dend}} = \\big \\lbrace f : A \\rightarrow A |~ \\alpha \\circ f = f \\circ \\alpha ~~~ \\text{ and }~&f ( a \\prec b) = a \\prec f (b) + f(a) \\prec b,\\\\~&f ( a \\succ b) = a \\succ f (b) + f(a) \\succ b \\big \\rbrace .$ Thus, $H^1_{\\alpha , \\text{Dend}} (A, A)$ is the space of all derivations for the products $\\prec , \\succ $ which commute with $\\alpha $ .", "In the next section, we interpret the second cohomology group $H^2_{\\alpha , \\text{Dend}} (A, A)$ as the equivalence classes of infinitesimal deformations of $A$ .", "Remark 3.1 One may also define cohomology of a hom-dendriform algebra with coefficients in a suitable representation.", "In such a case, the second cohomology with coefficients in a representation can also be represented by the equivalence classes of abelian extensions.", "See [5] for the case of dendriform algebras, that is, when $\\alpha = \\text{id}_A$ .", "In the next, we relate the cohomology of a hom-dendriform algebra (as defined above) with the cohomology of the corresponding hom-associative algebra.", "More precisely, we have the following whose proof is similar to [5].", "Theorem 3.2 Let $(A, \\alpha , \\prec , \\succ )$ be a hom-dendriform algebra with the corresponding hom-associative algebra $(A, \\alpha , \\star )$ .", "Then the map $S : C^n_{\\alpha , \\mathrm {Dend}} (A, A) \\rightarrow C^n_{\\alpha , \\mathrm {Ass}}(A, A), ~ f \\mapsto f_{[1]} + \\cdots + f_{[n]}, ~~~ \\text{ for } n \\ge 1$ defines a morphism between operads which preserve the respective multiplications.", "Hence the induced map $S : H^n_{\\alpha , \\mathrm {Dend}} (A, A) \\rightarrow H^n_{\\alpha , \\mathrm {Ass}} (A, A)$ on cohomology is a morphism between Gerstenhaber algebras." ], [ "Formal deformations", "Our aim in this section is to study the formal deformation theory for hom-dendriform algebras along the line of Gerstenhaber [8].", "We show that the cohomology of hom-dendriform algebra governs the corresponding deformation.", "Let $(A, \\alpha , \\prec , \\succ )$ be a hom-dendriform algebra.", "Consider the space $A[[t]]$ of formal power series in $t$ with coefficients in $A$ .", "Then $A[[t]]$ is a $\\mathbb {K}[[t]]$ -module and $A[[t]] \\cong A \\otimes _\\mathbb {K} \\mathbb {K}[[t]]$ when $A$ is finite dimensional.", "Definition 4.1 A formal deformation of a hom-dendriform algebra $(A, \\alpha , \\prec , \\succ )$ consists of formal power series $\\prec _t = \\sum _{i=0}^\\infty \\prec _i t^i$ and $\\succ _t = \\sum _{i=0}^\\infty \\succ _i t^i$ of bilinear maps on $A$ (with $\\prec _0 = \\prec $ and $\\succ _0 = \\succ )$ such that $(A[[t]], \\alpha , \\prec _t, \\succ _t)$ is a hom-dendriform algebra over $\\mathbb {K}[[t]].$ Thus, if $(\\prec _t, \\succ _t)$ is a deformation, then for each $n \\ge 0$ and $a, b, c \\in A$ , one must have $\\sum _{i+j = n} (a \\prec _i b ) \\prec _j \\alpha (c) =~& \\sum _{i+j = n} \\alpha (a) \\prec _i ( b \\prec _j c + b \\succ _j c ), \\\\\\sum _{i+j = n} (a \\succ _i b ) \\prec _j \\alpha (c) =~& \\sum _{i+j = n} \\alpha (a) \\succ _i (b \\prec _j c), \\\\\\sum _{i+j = n} (a \\prec _i b + a \\succ _i b) \\succ _j \\alpha (c) =~& \\sum _{i+j = n} \\alpha (a) \\succ _i (b \\succ _j c).", "$ The identities (REF )-() are called deformation equations.", "For each $i \\ge 0$ , we define $\\pi _i \\in \\mathcal {O}(2) = C^2_{\\alpha , \\mathrm {dend}} (A, A)$ by $\\pi _i ([1]; a, b) = a \\prec _i b ~~~~ \\text{ and } ~~~~ \\pi _i ([2]; a, b) = a \\succ _i b.$ Then the deformation equations can be simply expressed as $\\sum _{i+j = n} \\pi _i \\bullet \\pi _j = 0, ~~~ \\text{ for } n \\ge 0.$ The above identity automatically holds for $n = 0$ as $\\pi _0 = \\pi $ defines a multiplication on the operad.", "For $n = 1$ , we have $ \\pi \\bullet \\pi _1 + \\pi _1 \\bullet \\pi = 0$ which implies that $\\delta _{\\alpha , \\text{Dend}} (\\pi _1) = 0$ .", "Hence $\\pi _1 \\in C^2_{\\alpha , \\mathrm {Dend}} (A, A)$ defines a 2-cocycle in the hom-dendriform algebra cohomology of $A$ .", "It is called the infinitesimal of the deformation.", "Definition 4.2 Two deformations $(\\prec _t, \\succ _t)$ and $(\\prec _t^{\\prime }, \\succ _t^{\\prime })$ of a hom-dendriform algebra $(A, \\alpha , \\prec , \\succ )$ are said to be equivalent if there is a formal isomorphism $\\Phi _t = \\sum _{i=0}^\\infty \\Phi _i t^i : A[[t]] \\rightarrow A[[t]]$ with each $\\Phi _i \\in \\text{Hom} (A, A)$ that commute with $\\alpha $ and $\\Phi _0 = \\text{id}_A$ such that $\\Phi _t ( a \\prec _t b ) = \\Phi _t (a) \\prec _t^{\\prime } \\Phi _t (b) ~~~~ \\text{ ~~and ~~} ~~~~ \\Phi _t ( a \\succ _t b ) = \\Phi _t (a) \\succ _t^{\\prime } \\Phi _t (b).$ The above condition of equivalence can be expressed as $\\sum _{i+j = n} \\Phi _i \\bullet \\pi _j ([r]; a, b) = \\sum _{i+j+k = n} \\pi _k^{\\prime } ([r]; \\Phi _i (a), \\Phi _j (b)), ~~~ \\text{ for all } n \\ge 0.$ This always holds for $n = 0$ (as $\\prec _0 = \\prec _0^{\\prime } = \\prec $ , $\\succ _0 = \\succ _0^{\\prime } = \\succ $ and $\\Phi _0 = \\text{id}_A$ ), whereas for $n =1$ , we obtain $\\pi _1 - \\pi _1^{\\prime } = \\pi \\bullet \\Phi _1 - \\Phi _1 \\bullet \\pi = \\delta _{\\alpha , \\text{Dend}} (\\Phi _1).$ This shows that equivalent deformations have cohomologous infinitesimals, hence, they corresponds to same cohomology class in $H^2_{\\alpha , \\mathrm {Dend}}(A, A).$ To obtain a one-to-one correspondence between the second cohomology group and equivalence classes of certain type deformations, we have to use the following truncated version of formal deformations.", "Definition 4.3 An infinitesimal deformation of a hom-dendriform algebra $A = ( A, \\alpha , \\prec , \\succ )$ is a deformation of $A$ over $\\mathbb {K}[[t]]/ (t^2)$ (the local Artinian ring of dual numbers).", "In other words, an infinitesimal deformation of $A$ is given by a pair $(\\prec _t, \\succ _t)$ in which $\\prec _t = \\prec + \\prec _1 t$ and $\\succ _t = \\succ + \\succ _1 t$ such that $\\pi _1 = (\\prec _1, \\succ _1)$ defines a 2-cocycle in the cohomology of $A$ .", "More precisely, we have the following.", "Proposition 4.4 There is a one-to-one correspondence between the space of equivalence classes of infinitesimal deformations and the second cohomology group $H^2_{\\alpha , \\mathrm {Dend}} (A, A).$ It is already shown that if two infinitesimal deformations $(\\prec _t =~ \\prec + \\prec _1 t , ~ \\succ _t = ~ \\succ + \\succ _1 t)$ and $(\\prec _t^{\\prime } =~ \\prec + \\prec _1^{\\prime } t , ~ \\succ _t^{\\prime } =~ \\succ + \\succ _1^{\\prime } t)$ are equivalent, then the 2-cocycles $\\pi _1 = ( \\prec _1, \\succ _1)$ and $\\pi _1^{\\prime } = ( \\prec _1^{\\prime }, \\succ _1^{\\prime })$ are cohomologous.", "Hence, the map $\\text{infinitesimal deformations}/\\sim ~~\\longrightarrow ~ H^2_{\\alpha , \\mathrm {Dend}} ( A, A)$ defined by $[(\\prec _t, \\succ _t)] \\mapsto [\\pi _1]$ is well defined.", "This map turns out to be bijective with the inverse given as follows.", "For any 2-cocycle $\\pi _1 = (\\prec _1, \\succ _1) \\in C^2_{\\alpha , \\mathrm {Dend}} (A, A)$ , the pair $(\\prec _t =~ \\prec + \\prec _1 t , ~ \\succ _t =~ \\succ + \\succ _1 t)$ defines an infinitesimal deformation.", "If $\\pi _1^{\\prime } = (\\prec _1^{\\prime }, \\succ _1^{\\prime }) \\in C^2_{\\alpha , \\mathrm {Dend}} (A, A)$ is another 2-cocycle cohomologous to $\\pi _1$ , then we have $\\pi _1 - \\pi _1^{\\prime } = \\delta _{\\alpha , \\mathrm {Dend}} (\\Phi _1)$ , for some $\\Phi _1 \\in C^1_{\\alpha , \\mathrm {Dend}} (A, A) = \\mathrm {Hom}( \\mathbb {K}[C_1] \\otimes A , A ) \\simeq \\mathrm {Hom} (A, A)$ .", "In such a case, $\\Phi _t = \\text{id}_A + \\Phi _1 t $ defines an equivalence between infinitesimal deformations $(\\prec _t =~ \\prec + \\prec _1 t , ~ \\succ _t =~ \\succ + \\succ _1 t)$ and $(\\prec _t^{\\prime } =~ \\prec + \\prec _1^{\\prime } t , ~ \\succ _t^{\\prime } =~ \\succ + \\succ _1^{\\prime } t)$ .", "Hence the inverse map is also well defined.", "Definition 4.5 A deformation $(\\prec _t, \\succ _t)$ is said to be trivial if it is equivalent to the deformation $(\\prec _t^{\\prime } = \\prec ,~ \\succ _t^{\\prime } = \\succ ).$ Lemma 4.6 Let $(\\prec _t, \\succ _t)$ be a non-trivial deformation of a hom-dendriform algebra $A$ .", "Then it is equivalent to some deformation $(\\prec _t^{\\prime }, \\succ _t^{\\prime })$ in which $\\prec _t^{\\prime } =~ \\prec + \\sum _{i \\ge p} \\prec _i t^i$ and $\\succ _t^{\\prime } =~ \\succ + \\sum _{i \\ge p} \\succ _i t^i$ , where the first non-zero term $\\pi _p = ( \\prec _p, \\succ _p )$ is a 2-cocycle but not a coboundary.", "Let $(\\prec _t, \\succ _t)$ be a non-trivial deformation of a hom-dendriform algebra such that $\\pi _1 := (\\prec _1, \\succ _1) = 0, \\ldots , \\pi _{n-1} := (\\prec _{n-1}, \\succ _{n-1}) = 0$ and $\\pi _n := (\\prec _n, \\succ _n)$ is the first non-zero term.", "Then it has been shown that $\\pi _n = (\\prec _n, \\succ _n)$ is a 2-cocycle.", "If $\\pi _n$ is not a 2-coboundary, we are done.", "If $\\pi _n $ is a 2-coboundary, say $\\pi _n = - \\delta _{\\alpha , \\mathrm {Dend}} (\\Phi _n)$ , for some $\\Phi _n \\in C^1_{\\alpha , \\mathrm {Dend}} (A, A) = \\mathrm {Hom} (A, A)$ , then setting $\\Phi _t = \\text{id}_A + t^n \\Phi _n$ .", "We define $\\prec _t^{\\prime } = \\Phi _t^{-1} \\circ \\prec _t \\circ \\Phi _t$ and $\\succ _t^{\\prime } = \\Phi _t^{-1} \\circ \\succ _t \\circ \\Phi _t$ .", "Then $(\\prec _t^{\\prime }, \\succ _t^{\\prime })$ defines a formal deformation of the form $\\prec _t^{\\prime } =~ \\prec +~ t^{n+1} \\prec _{n+1}^{\\prime } + \\cdots \\quad \\mathrm { and } \\quad \\succ _t^{\\prime } =~ \\succ +~ t^{n+1} \\succ _{n+1}^{\\prime } + \\cdots .$ Thus, it follows that $\\pi _{n+1}^{\\prime } = (\\prec _{n+1}^{\\prime }, \\succ _{n+1}^{\\prime })$ is a 2-cocycle in $C^2_{\\alpha , \\mathrm {Dend}} (A, A)$ .", "If it is not a coboundary, we are done.", "If it is a coboundary, we can apply the same method again.", "In this way, we get a required type of equivalent deformation.", "As a consequence, we get the following.", "Theorem 4.7 If $H^2_{\\alpha , \\mathrm {Dend}} (A, A) = 0$ then every deformation of $A$ is equivalent to a trivial deformation.", "Let $(\\prec _t, \\succ _t)$ be a formal deformation of $A$ .", "If it is a trivial deformation, it is equivalent to itself.", "On the other hand, if it is non-trivial, then by Lemma REF and the fact that $H^2_{\\alpha , \\mathrm {Dend}} (A, A) = 0$ , we have $(\\prec _t, \\succ _t)$ is equivalent to $(\\prec _t^{\\prime } =~ \\prec , \\succ _t^{\\prime } =~ \\succ )$ .", "Hence the proof.", "A hom-dendriform algebra $A$ is said to be rigid if every deformation of $A$ is equivalent to a trivial deformation.", "It follows from Theorem REF that $H^2 = 0$ is a sufficient condition for the rigidity of a hom-dendriform algebra." ], [ "Extensions of finite order deformation", "A deformation $(\\prec _t, \\succ _t)$ of a hom-dendriform algebra $A$ is said to be of order $n$ if $\\prec _t$ and $\\succ _t$ are of the form $\\prec _t = \\sum _{i=0}^n \\prec _i t^i$ and $\\succ _t = \\sum _{i=0}^n \\succ _i t^i$ .", "Here we discuss the problem of extension of a deformation of order $n$ to a deformation of next order.", "Suppose there is an element $\\pi _{n+1} = (\\prec _{n+1}, \\succ _{n+1})$ such that $(\\overline{\\prec }_t = \\prec _t + \\prec _{n+1} t^{n+1} ,~ \\overline{\\succ }_t = \\succ _t + \\succ _{n+1} t^{n+1})$ is a deformation of order $n+1$ .", "Therefore, one additional deformation equation need to be satisfy $\\sum _{i+j = n+1} \\pi _i \\bullet \\pi _j = 0.$ This is equivalent to $\\delta _{\\alpha , \\text{Dend}} ( \\pi _{n+1} ) = - \\sum _{i+j = n+1, i, j \\ge 1} \\pi _i \\bullet \\pi _j.$ The right hand side of the above equation is called the obstruction to extend the deformation.", "Thus, if an extension is possible, the obstruction is always given by a coboundary.", "However, in any case, we have the following.", "Lemma 4.8 The obstruction is a 3-cocycle in the hom-dendriform algebra cohomology of $A$ , i.e $\\delta _{\\alpha , \\mathrm {Dend}} (- \\sum _{i+j = n+1, i, j \\ge 1} \\pi _i \\bullet \\pi _j) = 0$ For any $\\pi , \\pi ^{\\prime } \\in C^2_{\\alpha , \\text{Dend}} (A, A)$ , it is easy to see that $\\delta _{\\alpha , \\mathrm {Dend}} (\\pi \\bullet \\pi ^{\\prime }) = \\pi \\bullet \\delta _{\\alpha , \\mathrm {Dend}} (\\pi ^{\\prime }) - \\delta _{\\alpha , \\mathrm {Dend}} (\\pi ) \\bullet \\pi ^{\\prime } ~+~ \\pi ^{\\prime } \\cdot \\pi -~ \\pi \\cdot \\pi ^{\\prime }.$ Therefore, $\\delta _{\\alpha , \\mathrm {Dend}} \\big (- \\sum _{i+j =n+1, i, j \\ge 1} \\pi _i \\bullet \\pi _{j} \\big ) =~& - \\sum _{i+j =n+1, i, j \\ge 1} \\big ( \\pi _i \\bullet \\delta _{\\alpha , \\mathrm {Dend}} (\\pi _j) - \\delta _{\\alpha , \\mathrm {Dend}} (\\pi _i) \\bullet \\pi _j \\big ) \\\\=~& \\sum _{p+q+r = n+1, p, q, r \\ge 1} \\big ( \\pi _p \\bullet (\\pi _q \\bullet \\pi _r) - (\\pi _p \\bullet \\pi _q) \\bullet \\pi _r \\big )\\\\ =~& \\sum _{p+q+r = n+1, p, q, r \\ge 1} A_{p, q, r} \\qquad \\mathrm {(say)}.$ The product $\\circ $ is not associative, however, they satisfy the pre-Lie identities [8].", "This in particular implies that $A_{p, q, r} = 0$ whenever $q = r$ .", "Finally, if $q \\ne r$ then $A_{p, q, r} + A_{p, r, q} = 0$ by the pre-Lie identities.", "Hence we have $\\sum _{p+q+r = n+1, p, q, r \\ge 1} A_{p, q, r} = 0$ .", "Thus, we have the following.", "Theorem 4.9 If $H^3_{\\alpha , \\mathrm {Dend}} (A, A) = 0$ then every finite order deformation of $A$ extends to a deformation of next order." ], [ "Hom-dendriform coalgebras and deformations", "In this section, we consider the dual picture of the results as described in previous sections.", "Namely, we consider hom-dendriform coalgebras and study their deformations via a cohomology theory.", "Definition 5.1 A hom-dendriform coalgebra is a hom-vector space $(C, \\alpha )$ together with two linear maps $\\triangle _\\prec , \\triangle _\\succ : C \\rightarrow C \\otimes C$ satisfying the following identities $(\\triangle _\\prec \\otimes \\alpha ) \\circ \\triangle _\\prec =~& (\\alpha \\otimes (\\triangle _\\prec + \\triangle _\\succ )) \\circ \\triangle _\\prec , \\\\(\\triangle _\\succ \\otimes \\alpha ) \\circ \\triangle _\\prec =~& (\\alpha \\otimes \\triangle _\\prec ) \\circ \\triangle _\\succ , \\\\((\\triangle _\\prec + \\triangle _\\succ ) \\otimes \\alpha ) \\circ \\triangle _\\succ =~& (\\alpha \\otimes \\triangle _\\succ ) \\circ \\triangle _\\succ .", "$ Note that the above three identities are dual to the identities (REF )-().", "A hom-dendriform coalgebra as above is said to be multiplicative if $(\\alpha \\otimes \\alpha ) \\circ \\triangle _\\prec = \\triangle _\\prec \\circ \\alpha $ and $(\\alpha \\otimes \\alpha ) \\circ \\triangle _\\succ = \\triangle _\\succ \\circ \\alpha $ .", "In either case, when $\\alpha = \\text{id}_C$ , we obtain dendriform coalgebras [7], [5].", "A hom-dendriform coalgebra is a splitting of an associative coalgebra in the sense that if $(C, \\alpha , \\triangle _\\prec , \\triangle _\\succ )$ is a hom-dendriform coalgebra, then $(C, \\alpha , \\triangle _\\prec + \\triangle _\\succ )$ is a hom-associative coalgebra.", "See [14] for details about hom-associative coalgebras.", "Any hom-associative coalgebra is a hom-dendriform coalgebra with either $\\triangle _\\prec = 0$ or $\\triangle _\\succ = 0$ .", "One can construct a hom-dendriform coalgebra out of a dendriform coalgebra and a morphism of it.", "Similarly, dual to the Examples REF and REF , one can define Rota-Baxter operator and $\\mathcal {O}$ -operator on hom-associative coalgebras which induce hom-dendriform coalgebras.", "In the next, we define a cohomology for multiplicative hom-dendriform coalgebras.", "This cohomology can be thought of as a splitting of the Cartier (coHochschild-type) cohomology of hom-associative coalgebras.", "We start with the following twisted analog of the coendomorphism operad.", "Let $(C, \\alpha )$ be a hom-vector space.", "We define $C^0_{\\alpha , \\mathrm {coAss}} (C, C) = 0$ and $C^n_{\\alpha , \\mathrm {coAss}} (C, C) = \\lbrace \\sigma : C \\rightarrow C^{\\otimes n} |~ \\alpha ^{\\otimes n} \\circ \\sigma = \\sigma \\circ \\alpha \\rbrace , ~~~ \\text{ for } n \\ge 1.$ Proposition 5.2 The collection of vector spaces $\\lbrace C^n_{\\alpha , \\mathrm {coAss}} (C, C) \\rbrace _{n \\ge 1}$ forms an operad with partial compositions $\\sigma \\bullet _i \\tau = (({\\alpha ^{n-1}})^{\\otimes (i-1)} \\otimes \\tau \\otimes ({\\alpha ^{n-1}})^{\\otimes (m-i)}) \\circ \\sigma ,$ for $\\sigma \\in C^m_{\\alpha , \\mathrm {coAss}} (C, C),~\\tau \\in C^n_{\\alpha , \\mathrm {coAss}} (C, C)$ and $[r] \\in C_{[m+n-1]}$ .", "The proof of the above proposition is dual to the proof of [4].", "When $\\alpha = \\text{id}_C,$ one obtain the coendomorphism operad associated to the vector space $C$ .", "Note that a multiplication on the operad of Proposition REF is given by an element $\\triangle \\in C^2_{\\alpha , \\mathrm {coAss}} (C, C)$ such that $\\triangle \\bullet _1 \\triangle = \\triangle \\bullet _2 \\triangle $ .", "In other words, $\\triangle \\in C^2_{\\alpha , \\mathrm {coAss}} (C, C)$ satisfies $(\\triangle \\otimes \\alpha ) \\circ \\triangle = (\\alpha \\otimes \\triangle ) \\circ \\triangle .$ Therefore, $\\triangle $ defines a multiplicative hom-associative coalgebra on $(C, \\alpha ).$ The induced differential $\\delta _\\triangle : C^n_{\\alpha , \\mathrm {coAss}} (C, C) \\rightarrow C^{n+1}_{\\alpha , \\mathrm {coAss}} (C, C)$ (also denoted by $\\delta _{\\alpha , \\mathrm {coAss}}$ ) is given by $\\delta _{\\alpha , \\mathrm {coAss}} (f) = \\triangle \\bullet f - (-1)^{n+1} f \\bullet \\triangle , ~~~ \\text{ for } f \\in C^n_{\\alpha , \\mathrm {coAss}} (C, C).$ This cohomology is called the Cartier (coHochschild-type) cohomology of the multiplicative hom-associative coalgebra $(C, \\alpha , \\triangle )$ .", "Since this cohomology is induced from an operad with a multiplication, it follows from [9] that this cohomology inherits a Gerstenhaber structure.", "Next, we consider a new operad associated to a hom-vector space dual to the operad given in Section .", "Let $(C, \\alpha )$ be a hom-vector space.", "Define $C^0_{\\alpha , \\text{coDend}} (C, C) = 0$ and $C^n_{\\alpha , \\text{coDend}} (C, C) = \\lbrace \\sigma : \\mathbb {K}[C_n] \\otimes C \\rightarrow C^{\\otimes n} |~ \\alpha ^{\\otimes n} \\circ \\sigma ([r]; a) = \\sigma ([r]; \\alpha (a) ), \\forall ~[r] \\in C_n \\rbrace .$ Then we have the following.", "Theorem 5.3 The collections of vector spaces $\\lbrace C^n_{\\alpha , \\mathrm {coDend}} (C,C) \\rbrace _{n \\ge 1}$ forms an operad with partial compositions $& (\\sigma \\bullet _i \\tau ) ([r]; ~\\_~) \\\\~& = ((\\alpha ^{n-1})^{\\otimes (i-1)} \\otimes g (R_i (m; 1, \\ldots , n, \\ldots , 1)[r]; ~\\_~) \\otimes (\\alpha ^{n-1})^{\\otimes (m-i)}) \\circ f (R_0 (m; 1, \\ldots , n, \\ldots , 1)[r]; ~\\_~)$ for $\\sigma \\in C^m_{ \\alpha , \\mathrm {coDend}} (C, C),~ \\tau \\in C^n_{\\alpha , \\mathrm {coDend}} (C, C), ~[r] \\in C_{m+n-1},$ and the identity element $\\mathrm {id} \\in C^1_{\\alpha , \\mathrm {coDend}} (C,C)$ given by $\\mathrm {id} ([1]; c) = c,$ for all $c \\in C.$ Moreover, the collection of maps $\\lbrace \\Phi _n : C^n_{\\alpha , \\mathrm {coDend}} (C, C) \\rightarrow C^n_{ \\alpha , \\mathrm {coAss}} (C, C) \\rbrace _{n \\ge 1}$ given by $\\Phi _n (\\sigma ) = \\sigma ([1]; ~) + \\cdots + \\sigma ([n]; ~ ), ~~~~ \\text{ for } \\sigma \\in C^n_{\\alpha , \\mathrm {coDend}} (C, C)$ is a morphism between operads.", "The proof of the above theorem is along the same line of [5], [6].", "Hence we will not repeat it here.", "Note that a multiplication in the operad $\\lbrace C^n_{\\alpha , \\mathrm {coDend}} (C,C) , \\bullet _i \\rbrace _{n \\ge 1}$ is given by an element $\\triangle \\in C^2_{\\alpha , \\mathrm {coDend}} (C, C)$ satisfying $\\triangle \\bullet _1 \\triangle = \\triangle \\bullet _2 \\triangle $ .", "The element $\\triangle $ is equivalent to two maps $\\triangle _\\prec , \\triangle _\\succ : C \\rightarrow C \\otimes C$ given by $\\triangle _\\prec = \\triangle ([1]; ~) \\quad \\text{ and } \\quad \\triangle _\\succ = \\triangle ([2]; ~).$ The condition $\\triangle \\bullet _1 \\triangle = \\triangle \\bullet _2 \\triangle $ is equivalent to the fact that $(\\triangle _\\prec , \\triangle _\\succ )$ satisfy the identities (REF )-().", "Therefore, a multiplication is given by a multiplicative hom-dendriform structure on $(C, \\alpha )$ .", "We define a differential $\\delta _{\\alpha , \\text{coDend}} : C^n_{\\alpha , \\text{coDend}} (C,C) \\rightarrow C^{n+1}_{\\alpha , \\text{coDend}} (C,C)$ to be the one induced from the multiplication $\\triangle $ on this operad, i.e.", "$\\delta _{\\alpha , \\text{coDend}} (\\sigma ) = \\triangle \\bullet \\sigma - (-1)^{n-1} \\sigma \\bullet \\triangle $ , for $\\sigma \\in C^n_{\\alpha , \\text{coDend}}(C, C)$ .", "Explicitly, it is given by $( \\delta _{\\alpha , \\mathrm {coDend}} (\\sigma ) )& ([r] \\otimes c) \\\\=~& (\\alpha ^{n-1} \\otimes \\sigma (R_2 (2;1,n)[r];~ \\_~ )) \\circ \\triangle (R_0 (2;1, n)[r]; c) \\nonumber \\\\+~& \\sum _{i=1}^n (-1)^i ~({\\alpha }^{\\otimes (i-1)} \\otimes \\triangle _{R_i (n; 1, \\ldots , 2, \\ldots , 1)[r]} \\otimes {\\alpha }^{\\otimes (n-i)}) \\circ \\sigma (R_0 (n;1, \\ldots , 2, \\ldots , 1)[r]; c) \\nonumber \\\\+~& (-1)^{n+1}~ (\\sigma (R_1 (2;n,1)[r]; ~\\_ ~) \\otimes \\alpha ^{n-1} ) \\circ \\triangle (R_0 (2;n,1)[r]; c), \\nonumber $ for $\\sigma \\in C^n_{\\alpha , \\text{coDend}}(C, C)$ , $[r] \\in C_{n+1}$ and $c \\in C$ .", "We denote the corresponding cohomology by $H^*_{\\alpha , \\mathrm {coDend}} (C, C)$ .", "When $\\alpha = \\text{id}_C$ , this cohomology coincides with the one constructed in [6].", "Note that, since the above cohomology is induced from an operad with a multiplication, the cohomology inherits a Gerstenhaber structure." ], [ "Deformation", "Here we study deformations of multiplicative hom-dendriform coalgebras.", "This is dual to the deformation of hom-dendriform algebras.", "Therefore, we will only state the results.", "A deformation of a multiplicative hom-dendriform coalgebra $(C, \\alpha , \\triangle _\\prec , \\triangle _\\succ )$ is given by two formal sums $\\triangle _{\\prec , t} = \\sum _{i \\ge 0} t^i \\triangle _{\\prec , i} $ and $\\triangle _{\\succ , t} = \\sum _{i \\ge 0} t^i \\triangle _{\\succ , i}$ (with $\\triangle _{\\prec , 0} = \\triangle _\\prec $ and $\\triangle _{\\succ , 0} = \\triangle _\\succ $ ) such that $(C[[t]], \\alpha , \\triangle _{\\prec , t}, \\triangle _{\\succ , t})$ is a multiplicative hom-dendriform coalgebra over $\\mathbb {K}[[t]].$ Therefore, we must have the following identities hold $\\sum _{i+j= n} (\\triangle _{\\prec , i} \\otimes \\alpha ) \\circ \\triangle _{\\prec , j} =~& \\sum _{i+j= n} (\\alpha \\otimes (\\triangle _{\\prec , i} + \\triangle _{\\succ , i})) \\circ \\triangle _{\\prec , j}, \\\\\\sum _{i+j= n} (\\triangle _{\\succ , i} \\otimes \\alpha ) \\circ \\triangle _{\\prec , j} =~& \\sum _{i+j= n}(\\alpha \\otimes \\triangle _{\\prec , i} ) \\circ \\triangle _{\\succ , j}, \\\\\\sum _{i+j= n} ((\\triangle _{\\prec , i} + \\triangle _{\\succ , i} ) \\otimes \\alpha ) \\circ \\triangle _{\\succ , j} =~& (\\alpha \\otimes \\triangle _{\\succ , i} ) \\circ \\triangle _{\\succ , j} $ for each $n \\ge 0$ .", "These equations are called deformation equations for the hom-dendriform coalgebra.", "To write these equations in a compact form, we use the following notations.", "Define $\\triangle _i : \\mathbb {K}[C_2] \\otimes C \\rightarrow C \\otimes C$ by $\\triangle _i ([1]; ~) = \\triangle _{\\prec , i}$ and $\\triangle _{i} ([2]; ~) = \\triangle _{\\succ , i}$ , for $i \\ge 0$ .", "Then the equations (REF )-() can be simply reads as $\\sum _{i+j = n } \\triangle _i \\bullet _1 \\triangle _j = \\sum _{i+j = n } \\triangle _i \\bullet _2 \\triangle _j , ~~~ n \\ge 0.$ The above identity holds automatically for $n = 0$ as we know $\\triangle \\bullet _1 \\triangle = \\triangle \\bullet _2 \\triangle .$ For $n=1$ , we get $\\triangle \\bullet _1 \\triangle _1 + \\triangle _1 \\bullet _1 \\triangle = \\triangle \\bullet _2 \\triangle _1 + \\triangle _1 \\bullet _2 \\triangle $ or equivalently, $\\delta _{\\alpha , \\mathrm {coDend}} (\\triangle _1) = 0.$ Therefore, $\\triangle _1 \\in C^2_{\\alpha , \\mathrm {coDend}} (C, C)$ defines a 2-cocycle.", "It is called the infinitesimal of the deformation.", "Thus, the infinitesimal of a deformation is a 2-cocycle in the cohomology of the hom-dendriform coalgebra.", "Definition 5.4 Two deformations $(\\triangle _{\\prec , t}, \\triangle _{\\succ , t})$ and $(\\triangle ^{\\prime }_{\\prec , t}, \\triangle ^{\\prime }_{\\succ , t})$ of a hom-dendriform coalgebra $(C, \\alpha , \\triangle _\\prec , \\triangle _\\succ )$ are said to be equivalent if there exists a formal isomorphism $\\Phi _t = \\sum _{i \\ge 0} t^i \\Phi _i : C[[t]] \\rightarrow C[[t]]$ with each $\\Phi _i$ commute with $\\alpha $ and $\\Phi _0 = \\text{id}_C$ such that $\\triangle ^{\\prime }_{\\prec , t} \\circ \\Phi _t =~& (\\Phi _t \\otimes \\Phi _t) \\circ \\triangle _{\\prec , t}, \\\\\\triangle ^{\\prime }_{\\succ , t} \\circ \\Phi _t =~& (\\Phi _t \\otimes \\Phi _t) \\circ \\triangle _{\\succ , t}.$ Note that each $\\Phi _i : C \\rightarrow C$ can be thought of as an element in $C^1_{\\alpha , \\text{coDend}} (C, C)$ Then the above conditions of the equivalence can be simply expressed as $\\sum _{i+j=n} \\triangle _i^{\\prime } ([r]; ~\\_~) \\circ \\Phi _j = \\sum _{i+j +k = n} ( \\Phi _i \\otimes \\Phi _j ) \\circ \\triangle _k ([r]; ~\\_~),$ for $n \\ge 0$ and $[r] = [1], [2].$ The condition for $n = 0$ holds automatically as $\\Phi _0 = \\text{id}_C$ .", "For $n = 1$ , we have $\\triangle _1^{\\prime } ([r]; ~\\_~) ~+~ \\triangle ([r]; ~\\_~) \\circ \\Phi _1 =~ \\triangle _1 ([r]; ~\\_~) ~+ ~ (\\text{id} \\otimes \\Phi _1 ) \\circ \\triangle ([r]; ~\\_~) ~+~ (\\Phi _1 \\otimes \\text{id}) \\circ \\triangle ([r]; ~\\_~).$ This shows that the difference $\\triangle _1^{\\prime } - \\triangle _1$ is a coboundary $\\delta _{\\alpha , \\text{coDend}} { (\\Phi _1)}$ .", "Therefore, the infinitesimals corresponding to equivalent deformations are cohomologous, hence, they gives rise to same cohomology class in $H^2_{\\alpha , \\text{coDend}} (C, C).$ A deformation $(\\triangle _{\\prec , t}, \\triangle _{\\succ , t})$ of a multiplicative hom-dendriform coalgebra $(C, \\alpha , \\triangle _\\prec , \\triangle _\\succ )$ is said to be trivial if it is equivalent to the deformation $(\\triangle ^{\\prime }_{\\prec , t} = \\triangle _\\prec , \\triangle ^{\\prime }_{\\succ , t} = \\triangle _\\succ )$ .", "Theorem 5.5 Let $(C, \\alpha , \\triangle _\\prec , \\triangle _\\succ )$ be a multiplicative hom-dendriform coalgebra.", "If $H^2_{\\alpha , \\mathrm {coDend}} (C, C) = 0$ then every deformation of $C$ is equivalent to a trivial deformation.", "The proof of the above theorem is similar to Lemma REF and Theorem REF .", "Hence we omit the details.", "It follows that $H^2_{\\alpha , \\mathrm {coDend}} (C, C) = 0$ implies that the hom-dendriform coalgebra $C$ is rigid.", "Similarly, one may study an extension of a finite order deformation to the next order.", "The vanishing of the third cohomology ensures such an extension.", "Acknowledgement.", "The author would like to thank the referee for his/her comments on the earlier version of the manuscript." ] ]

[ [ "A Spectral Triple for a Solenoid Based on the Sierpinski Gasket" ], [ "Abstract The Sierpinski gasket admits a locally isometric ramified self-covering.", "A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed." ], [ "Introduction", "In this note, we introduce a semifinite spectral triple on the $C^*$ -algebra of continuous functions on the solenoid associated with a self-covering of the Sierpinski gasket.", "Such triple is finitely summable, its metric dimension coincides with the Hausdorff dimension of the gasket, and the associated non-commutative integral coincides up to a constant with a Bohr–Følner mean on the solenoid, hence reproduces the suitably normalized Hausdorff measure on periodic functions.", "The open infinite Sierpinski fractafold with a unique boundary point considered by Teplyaev [51] embeds continuously as a dense subspace of the solenoid, and the Connes distance restricted to such subspace reproduces the geodesic distance on such fractafold.", "On the one hand, this shows that our spectral triple describes aspects of both local and coarse geometry [45].", "On the other hand, this implies that the topology induced by the Connes distance, being non compact, does not coincide with the weak$^*$ -topology on the states of the solenoid algebra, as we call the $C^*$ -algebra of continuous functions on the solenoid.", "This means that the solenoid, endowed with our spectral triple, is not a quantum metric space in the sense of Rieffel [41].", "Related research concerning projective limits of (possibly quantum) spaces and the associated solenoids appeared recently in the literature.", "In the framework of noncommutative geometry, we mention: [37], where projective families of compact quantum spaces have been studied, showing their convergence to the solenoid w.r.t.", "the Gromov–Hausdorff propinquity distance; [1], where, in the same spirit as in this note, a semifinite spectral triple has been associated with the projective limit generated by endomorphisms of $C^*$ -algebras associated with commutative and noncommutative spaces; [18], where a spectral triple on the stable Ruelle algebra for Wieler solenoids has been considered and its unboundedd KK-theory has been studied, based on the Morita equivalence between the stable Ruelle algebra and a Cuntz–Pimsner algebra.", "In the same paper these techniques are used for the study of limit sets of regular self-similar groups (cf. [39]).", "When fractals are concerned, we mention the projective family of finite coverings of the octahedron gasket considered in [50], where, as in our present situation, an intermediate infinite fractafold between the tower of coverings and the projective limit is considered.", "Periodic and almost periodic functions on the infinite fractafold are considered, and a Fourier series description for the periodic functions is given, based on periodic eigenfunctions of the Laplacian (cf.", "also [46] for higher-dimensional examples).", "Let us remark that such coverings, as the ones considered in this paper, are not associated with groups of deck transformations.", "The starting point for the construction of this paper is the existence of a locally isometric ramified three-fold self-covering of the Sierpinski gasket with trivial group of deck transformations.", "Such self-covering gives rise to a projective family of coverings, whose projective limit is by definition a solenoid.", "Dually, the algebras of continuous functions on the coverings form an injective family, whose direct limit (in the category of $C^*$ -algebras) is the solenoid algebra.", "In [1] we already considered various examples of self-coverings or, dually, of endomorphisms of some $C^*$ -algebras, most of which were regular finite self-coverings.", "There we constructed a spectral triple on the solenoid algebra as a suitable limit of spectral triples on the algebras of continuous functions on the coverings.", "Given a spectral triple on the base space, attaching a spectral triple to a finite covering is not a difficult task, and in our present case consists simply in “dilating” the triple on the base gasket so that the projections are locally isometric.", "However, there is no commonly accepted procedure to define a limit of spectral triples.", "Since the method used in [1] cannot be used here (see below), we follow another route, in a sense spatializing the construction, namely showing that there exists an open fractafold which is intermediate between the projective family of coverings and the solenoid.", "More precisely, such fractafold space turns out to be an infinite covering of each of the finite coverings of the family, and embeds in a continuous way in the solenoid.", "In this way all the algebras (and their direct limit) will act on a suitable $L^2$ -space of the open fractafold, as do the Dirac operators of the associated spectral triples.", "In this way the limiting Dirac operator is well defined, but the compact resolvent property will be lost.", "Let us notice here that we are not constructing a spectral triple on the open fractafold, where a weaker compact resolvent property (cf.", "[16]) is retained, namely $f(D^2+I)^{-1/2}$ is a compact operator, where $D$ is the Dirac operator and $f$ is any function with compact support on the fractafold.", "Since we are constructing a spectral triple on the solenoid, which is a compact space, the weaker form does not help.", "In order to recover the needed compactness of the resolvent, we use a procedure first proposed by J. Roe for open manifolds with an amenable exhaustion in [42], [43], where, based on the observation that the von Neumann trace used by Atiyah [4] for his index theorem for covering manifolds can be reformulated in the case of amenable groups via the Følner condition, he considered amenable exhaustions on open manifolds and constructed a trace for finite-propagation operators acting on sections of a fiber bundle on the manifold via a renormalization procedure.", "Unfortunately such trace is not canonical, since it depends on a generalized limit procedure.", "However, in the case of infinite self-similar CW-complexes, it was observed in [13] that such trace becomes canonical when restricted to the $C^*$ -algebra of geometric operators.", "We adapt these results to our present context, namely we replace the usual trace with a renormalized trace associated with an exhaustion of the infinite fractafold.", "Such trace comes together with a noncommutative $C^*$ -algebra, the algebra of geometric operators, which is similar in spirit to the Roe $C^*$ -algebras of coarse geometry [31], [42], [43], [44], [52].", "This algebra contains the solenoid algebra, and the limiting Dirac operator is affiliated to it in a suitable sense.", "Such Dirac operator turns out to be $\\tau $ -compact w.r.t.", "the renormalized trace.", "We refer to [13], [26] for an analogous construction of the $C^*$ -algebra and of a canonical trace based on the self-similarity structure.", "As discussed above, the starting point for the construction of a spectral triple on the solenoid algebra is the association of a spectral triple to the fractal known as the Sierpinski gasket [47].", "The study of fractal spaces from a spectral, or noncommutative, point of view has now a long history, starting from the early papers of Kigami and Lapidus [33], [35], [36].", "As for the spectral triples, various constructions have been considered in the literature, mainly based on “small” triples attached to specific subsets of the fractal, following a general procedure first introduced by Connes, then considered in [27], [28], and subsequently abstracted in [10].", "More precisely, the spectral triple on the Cantor set described by Connes [16] inspired two kinds of spectral triples for various families of fractals in [27], [28].", "These triples were further analysed in [29] for the class of nested fractals.", "Such specral triples are obtained as direct sums of triples on two points (boundary points of an edge in some cases), and we call them discrete spectral triples.", "We then mention some spectral triples obtained as direct sums of spectral triples on 1-dimensional subsets, such as those considered in [3], [11], [12], [15], [34], where the 1-dimensional subsets are segments, circles or quasi-circles.", "Discrete spectral triples give a good description of metric aspects of the fractal, such as Hausdorff dimension and measure and geodesic distance, and, as shown in [24], may also reconstruct the energy functional (Dirichlet form) on the fractal, but are not suited for the study of K-theoretical properties since the pairing with K-theory is trivial.", "Conversely, spectral triples based on segments or circles describe both metric and K-theoretic properties of the fractal but can't be used for describing the Dirichlet form.", "Finally, the spectral triple based on quasi-circles considered in [15] describes metric and K-theoretic aspects together with the energy form, but requires a rather technical approach.", "In the present paper, we make use of the simple discrete spectral triple on the gasket as described in [29], thus obtaining a semifinite spectral triple on the solenoid algebra which recovers the metric dimension and the Bohr–Følner mean of the solenoid, and the geodesic distance on the infinite fractafold.", "Further analysis on the solenoid is possible, e.g., the construction of a Dirichlet form via noncommutative geometry or the study of K-theoretic properties.", "As explained above, the latter step will require a different choice of the spectral triple on the base gasket, such as the triples considered in [11], [12], [15], which admit a non-trivial pairing with the K-theory of the gasket.", "As already mentioned, our aim here is to show that the family of spectral triples on the finite coverings produces a spectral triple on the solenoidal space.", "In the examples considered in [1], the family of spectral triples had a simple tensor product structure, namely the Hilbert spaces were a tensor product of the Hilbert space ${\\mathcal {H}}$ for the base space and a finite dimensional Hilbert space, and the Dirac operators could be described as (a finite sum of) tensor product operators.", "Then the ambient $C^*$ -algebra turned out to be a product of ${\\mathcal {B}}({\\mathcal {H}})$ and a UHF algebra, allowing a GNS representation w.r.t.", "a semifinite trace.", "In the example treated here we choose a different approach since two problems forbid such simple description.", "The first is a local problem, due to the ramification points.", "This implies that the algebra of a covering is not a free module on the algebra of the base space; in particular, functions on a covering space form a proper sub-algebra of the direct sum of finitely many copies of the algebra for the base space.", "The second is a non-local problem which concerns the Hilbert spaces, which are $\\ell ^2$ -spaces on edges, and the associated operator algebras.", "Indeed, the Hilbert spaces of the coverings cannot be described as finite sums of copies of the Hilbert space on the base space due to the appearance of longer and longer edges on larger and larger coverings.", "We conclude this introduction by mentioning two further developments of the present analysis.", "First, the construction of the spectral triple on the solenoid algebra allows the possibility of lifting a spectral triple from a $C^*$ -algebra to the crossed product of the C*-algebra with a single endomorphism [2], thus generalising the results on crossed products with an automorphism group considered in [6], [30], [40].", "Second, we observe that the construction given in the present paper goes in the direction of possibly defining a $C^*$ -spectral triple, in which the semifinite von Neumann algebra is replaced by a $C^*$ -algebra with a trace to which both the Dirac operator and the “functions” on the non-commutative space are affiliated, where the compactness of the resolvent of the Dirac operator is measured by the trace on the $C^*$ -algebra, cf. also [25].", "This paper is divided in six sections.", "After this introduction, Section  contains some preliminary notions on fractals and spectral triples, Section  describes the geometry of the ramified covering and the corresponding inductive structure, together with its functional counterpart given by a family of compatible spectral triples.", "Section  concerns the self-similarity structure of the Sierpinski solenoid, whence the description of the inductive family of $C^*$ -algebras as algebras of bounded functions on the fractafold.", "The Section  describes the algebra of geometric operators and the construction of a semicontinuous semifinite trace on it.", "Finally, the semifinite spectral triple together with its main features are contained in Section ." ], [ "Preliminaries", "In this section we shall briefly recall various notions that will be used in the paper.", "Though these notions are well known among the experts, our note concerns different themes, namely spectral triples in noncommutative geometry and nested fractals (the Sierpinski gasket in particular), so that we decided to write this section with the aim of helping readers with different background to follow the various arguments, by collecting here the main notions and results that will be useful in the following." ], [ "Spectral triples", "The notion of spectral triple plays a key role in Alain Connes'noncommutative geometry [16], [23].", "Basically, it consists of a triple $({\\mathcal {L}},{\\mathcal {H}},D)$ , where ${\\mathcal {L}}$ is a *-algebra acting faithfully on the Hilbert space ${\\mathcal {H}}$ , and $D$ is an unbounded self-adjoint operator on ${\\mathcal {H}}$ satisfying the properties $(1)$ $\\big (1+D^2\\big )^{-1/2}$ is a compact operator, $(2)$ $\\pi (a)\\mathcal {D} (D) \\subset \\mathcal {D} (D)$ , and $[D,\\pi (a)]$ is bounded for all $a\\in {\\mathcal {L}}$ .", "We shall also say that $({\\mathcal {L}},{\\mathcal {H}},D)$ is a spectral triple on the $C^*$ -algebra ${\\mathcal {A}}$ generated by ${\\mathcal {L}}$ .", "Such triple is meant as a generalization of a compact smooth manifold, the algebra ${\\mathcal {L}}$ replacing the algebra of smooth functions, the Hilbert space describing a vector bundle (a spin bundle indeed) on which the algebra of functions acts, and the operator $D$ generalizing the notion of Dirac operator.", "Further structure may be added to the properties above, allowing deeper analysis of the geometric features of the noncommutative manifold, but these are not needed in this paper.", "Property $(2)$ above allows the definition of a (possibily infinite) distance (Connes distance) on the state space of the $C^*$ -algebra ${\\mathcal {A}}$ generated by ${\\mathcal {L}}$ , defined as $d(\\varphi ,\\psi )=\\sup \\lbrace |\\varphi (a)-\\psi (a)|\\colon \\Vert [D,a]\\Vert \\le 1,\\, a\\in {\\mathcal {L}}\\rbrace .$ When the Connes distance induces the weak$^*$ -topology on the state space, the seminorm $\\Vert [D,a]\\Vert $ on ${\\mathcal {A}}$ is called a Lip-norm (cf.", "[41]) and the algebra ${\\mathcal {A}}$ endowed with the Connes distance is a quantum metric space.", "A spectral triple is called finitely summable if $\\big (1+D^2\\big )^{-s}$ has finite trace for some $s>0$ , in this case the abscissa of convergence $d$ of the function $\\operatorname{tr}\\big (1+D^2\\big )^{-s}$ is called the metric dimension of the triple.", "Then the logarithmic singular trace introduced by Dixmier [19] may be used to define a noncommmutative integral on ${\\mathcal {A}}$ .", "Let us denote by $\\lbrace \\mu _n(T)\\rbrace $ the sequence (with multiplicity) of singular values of the compact operator $T$ , arranged in decreasing order.", "Then, on the positive compact operators for which the sequence $\\sum _{k=1}^n\\mu _n(T)$ , is at most logarithimically divergent, we may consider the positive functional $\\operatorname{tr}_\\omega (T)=\\operatorname{Lim}_\\omega \\frac{\\sum _{k=1}^n\\mu _n(T)}{\\log n},$ where $\\operatorname{Lim}_\\omega $ is a suitable generalized limit.", "Such functional extends to a positive trace on ${\\mathcal {B}}({\\mathcal {H}})$ which vanishes on trace class operators, and is called Dixmier (logarithmic) trace.", "If $\\big (1+D^2\\big )^{-d}$ is in the domain of the Dixmier trace, one defines the following noncommutative integral: $\\oint a=\\operatorname{tr}_\\omega \\big (a\\big (I+D^2\\big )^{-d/2}\\big ),\\qquad a\\in {\\mathcal {A}}.$ When the function $\\big (1+D^2\\big )^{-s}$ has a finite residue for $s=d$ , such residue turns out to coincide, up to a constant, with the Dixmier trace, which therefore does not depend on the generalized limit procedure (cf.", "[16], and [9]): $d\\cdot \\operatorname{tr}_\\omega \\big (a\\big (I+D^2\\big )^{-d/2}\\big )={\\operatorname{Res}}_{s=d} \\operatorname{tr}(a|D|^{-s}).$ We note in passing that spectral triples may also describe non-compact smooth manifolds, with the algebra ${\\mathcal {L}}$ describing smooth functions with compact support and property (1) replaced by $a\\big (1+D^2\\big )^{-1/2}$ is a compact operator for any $a\\in {\\mathcal {L}}$ ." ], [ "Semifinite spectral triples", "The notion of spectral triple has been generalized to the semifinite case, by replacing the ambient algebra ${\\mathcal {B}}({\\mathcal {H}})$ with a semifinite von Neumann algebra ${\\mathcal {M}}$ endowed with a normal semifinite faithful trace $\\tau $ .", "We recall that an operator $T$ affiliated with $({\\mathcal {M}},\\tau )$ is called $\\tau $ -compact if its generalized s-number function $\\mu _t(T)$ is infinitesimal or, equivalently, if $\\tau (e_{(t,\\infty )}(T))<\\infty $ , for any $t>0$ (cf.", "[21], [22]).", "Definition 2.1 ([8]) An odd semifinite spectral triple $({\\mathcal {L}},{\\mathcal {M}},D)$ on a unital C$^*$ -algebra ${\\mathcal {A}}$ is given by a unital, norm-dense, $^*$ -subalgebra ${\\mathcal {L}}\\subset {\\mathcal {A}}$ , a semifinite von Neumann algebra $({\\mathcal {M}},\\tau )$ , acting on a (separable) Hilbert space ${\\mathcal {H}}$ , a faithful representation $\\pi \\colon {\\mathcal {A}}\\rightarrow {\\mathcal {B}}({\\mathcal {H}})$ such that $\\pi ({\\mathcal {A}})\\subset {\\mathcal {M}}$ , and an unbounded self-adjoint operator $D \\widehat{\\in } {\\mathcal {M}}$ such that $(1)$ $\\big (1+D^2\\big )^{-1/2}$ is a $\\tau $ -compact operator, $(2)$ $\\pi (a)\\mathcal {D} (D) \\subset \\mathcal {D} (D)$ , and $[D,\\pi (a)] \\in {\\mathcal {M}}$ , for all $a\\in {\\mathcal {L}}$ .", "As in the type $I$ case, such triple is called finitely summable if $\\big (1+D^2\\big )^{-s}$ has finite trace for some $s>0$ , and $d$ denotes the abscissa of convergence of the function $\\tau \\big (1+D^2\\big )^{-s}$ , and is called the metric dimension of the triple.", "The logarithmic Dixmier trace associated with the normal trace $\\tau $ may be defined in this case too, (cf.", "[9], [24]) and, when the function $\\big (1+D^2\\big )^{-s}$ has a finite residue for $s=d$ , the equality $d\\cdot \\operatorname{tr}_\\omega \\big (a|D|^{-d}\\big )={\\operatorname{Res}}_{s=d} \\operatorname{tr}\\big (a|D|^{-s}\\big )$ still holds [9]." ], [ "Self-similar fractals", "Let $\\Omega := \\lbrace w_i \\colon i=1,\\ldots ,k \\rbrace $ be a family of contracting similarities of ${\\mathbb {R}}^{N}$ , with scaling parameters $\\lbrace \\lambda _i\\rbrace $ .", "The unique non-empty compact subset $K$ of ${\\mathbb {R}}^{N}$ such that $K = \\bigcup _{i=1}^{k} w_i(K)$ is called the self-similar fractal defined by $\\lbrace w_i \\rbrace _{i=1,\\ldots ,k}$ .", "For any $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ , let $p_i\\in {\\mathbb {R}}^N$ be the unique fixed-point of $w_i$ , and say that $p_i$ is an essential fixed-point of $\\Omega $ if there are $i^{\\prime },j,j^{\\prime }\\in \\lbrace 1,\\ldots ,k\\rbrace $ such that $i^{\\prime }\\ne i$ , and $w_j(p_i)=w_{j^{\\prime }}(p_{i^{\\prime }})$ .", "Denote by $V_0(K)$ the set of essential fixed-points of $\\Omega $ , and let $E_0(K):=\\lbrace (p,q)\\colon p,q\\in V_0,\\ p\\ne q\\rbrace $ .", "Observe that $(V_0,E_0)$ is a directed finite graph whose edges are in $1:1$ correspondence with ordered pairs of distinct vertices.", "Definition 2.2 We call an element of the family $\\lbrace w_{i_1}\\cdots w_{i_k}(K)\\colon k\\ge 0\\rbrace $ a cell, and call its diameter the size of the cell.", "We call an element of the family $E(K)=\\lbrace w_{i_1}\\cdots w_{i_k}(e)\\colon k\\ge 0$ , $e\\in E_{0}(K)\\rbrace $ an $($ oriented$)$ edge of $K$ .", "We denote by $e^-$ resp.", "$e^+$ the source, resp.", "the target of the oriented edge $e$ .", "As an example, the Sierpinski gasket is the self-similar fractal determined by 3 similarities with scaling parameter 1/2 centered in the vertices of an equilateral triangle (see Fig.", "REF ).", "Figure: The first four steps of the construction of the gasket.Under suitable conditions, the Hausdorff dimension $d_H$ of a self-similar fractal coincides with its scaling dimension, namely with the only positive number $d$ such that $\\sum _{i=1}^k\\lambda _i^d=1$ , therefore when all scaling parameters coincide with $\\lambda $ we have $d_H=\\frac{\\log k}{\\log (1/\\lambda )}$ .", "In particular, the Hausdorff dimension of the Sierpinski gasket is $\\frac{\\log 3}{\\log 2}$ .", "We note in passing that one of the most important aspects of the Sierpinski gasket and of more general classes of fractals is the existence of a self-similar diffusion, associated with a Dirichlet form, see, e.g., [32].", "Even though Dirichlet forms on fractals can be recovered in the noncommutative geometry framework [29], and in particular by means of the spectral triples which we use in this paper, we do not analyse this aspect in the present note.", "In [28] discrete spectral triples have been introduced on some classes of fractals, generalizing an example of Connes in [16].", "Such triples have been further studied in [29] for nested fractals.", "On a self-similar fractal $K$ , the triple $ ({\\mathcal {L}},{\\mathcal {H}},D)$ on the $C^*$ -algebra ${\\mathcal {A}}={\\mathcal {C}}(K)$ is defined as follows: Definition 2.3        $(a)$ ${\\mathcal {H}}=\\ell ^2(E(K))$ , $(b)$ ${\\mathcal {A}}$ acts on the Hilbert space as $\\rho (f)e=f(e^+)e$ , $f\\in {\\mathcal {A}}_n$ , $e\\in E_n$ , $(c)$ $F$ is the orientation-reversing map on edges, $(d)$ $D$ maps an edge $e\\in E(K)$ to $\\mathrm {length}(e)^{-1}Fe$ , $(e)$ ${\\mathcal {L}}$ is given by the elements $ f\\in {\\mathcal {A}}$ such that $\\Vert [D,\\rho (f)]\\Vert <\\infty $ .", "It turns out that ${\\mathcal {L}}$ coincides with the algebra of Lipschitz functions on $K$ , hence is dense in ${\\mathcal {A}}$ , and the seminorm $L(f):=\\Vert [D,f]\\Vert $ is a Lip-norm.", "By Theorem 3.3 in [29], see also Remark 2.11 in [28], the triple $({\\mathcal {L}},{\\mathcal {H}},D)$ is a finitely summable spectral triple on ${\\mathcal {A}}$ , its metric dimension coincides with the Hausdorff dimension, and the noncommutative integral recovers the Hausdorff measure up to a constant: $\\oint f=\\operatorname{tr}_\\omega \\big (f|D|^{-d}\\big ) = \\frac{1}{ \\log k} \\sum _{e\\in E_0(K)} \\ell (e)^d \\int _K f\\, {\\rm d}H_d, \\qquad f\\in C(K),$ where $H_d$ denotes the normalized Hausdorff measure on the fractal $K$ .", "Moreover, in some cases, and in particular for the Sierpinski gasket, the Connes distance induced by the Lip-norm $L(f):=\\Vert [D,f]\\Vert $ coincides with the geodesic distance on the points of the gasket $K$ , see [29]." ], [ "Covering fractafolds and solenoids", "Generally speaking, a solenoid is the inverse limit of a projective family of coverings of a given space [38].", "Dually, the solenoid algebra is the direct limit of the family of algebras of continuous functions on the spaces of the projective family.", "In this sense the notion of solenoid makes sense for injective families of $C^*$ -algebras, cf., e.g., [1] for sequences generated by a single endomorphism and [37] for sequences of compact quantum spaces.", "Other examples of the treatment of solenoids in the recent literature have been mentioned in the introduction.", "The notion of fractafold as a connected Hausdorff topological space such that every point has a neighborhood homeomorphic to a neighborhood in a given fractal has been introduced in [49], even though examples of such notion were already considered before, e.g., in [5], [48], [51].", "In some cases projective families of covering fractafold spaces related to the Sierpinski gasket have been considered.", "Since the gasket does not admit a simply connected covering, one may consider coverings where more and more cycles are unfolded, in particular consider the regular infinite abelian covering $S_n$ where all the cycles of size at least $2^{-n}$ are unfolded.", "Each of those is a closed fractafold (with boundary) and they form a projective family.", "The associated solenoid $S_\\infty $ , i.e., the projective limit, which turns out to be an abelian counterpart of the Uniform Universal Cover introduced by Berestovskii and Plaut [7], has been considered in [14], where it is shown that any locally exact 1-form on the gasket possesses a potential on $S_\\infty $ .", "Another projective family of covering fractafolds has been considered in [50], each element of the family being a compact finite covering of the octahedral fractafold modeled on the gasket.", "Any element of the family is covered by the infinite Sierpinski gasket with a unique boundary point, which we call $K_\\infty $ here (see Fig.", "REF ), considered in [51].", "The solenoid associated with the projective family is also mentioned explicitly in [50], together with the dense embedding of $K_\\infty $ in it, and also a Bohr–Følner mean on the solenoid is considered (p. 1199).", "Figure: The gasket and its infinite blowup.In the present paper a self-covering of the gasket gives rise to a projective family of finite ramified coverings, the fractafold $K_\\infty $ projects onto each element of the family and embeds densely in the solenoid, and we recover the Bohr–Følner mean on the solenoid via a noncommutative integral." ], [ "A ramified covering of the Sierpinski gasket", "Let us choose an equilateral triangle of side 1 in the Euclidean plane with vertices $v_0$ , $v_1$ , $v_2$ (numbered in a counterclockwise order) and consider the associated Sierpinski gasket as in the previous section, namely the set $K$ such that $K=\\bigcup _{j=0,1,2}w_j(K),$ where $w_j$ is the dilation around $v_j$ with contraction parameter $1/2$ .", "Clearly, for the cell $C=w_{i_1}\\cdots w_{i_k}(K)$ , $\\mathrm {size}(C)=2^{-k}$ and, if $e_0\\in E_{0}(K)$ and $e=w_{i_1}\\cdots w_{i_k}(e_0)$ , $\\mathrm {length}(e)=2^{-k}$ .", "5In the following we shall set $K_0:=K$ , $E_0=E_0(K)$ , $K_n=w_0^{-n}K_0$ .", "Let us now consider the middle point $x_{i,i+1}$ of the segment $\\big (w_0^{-1}v_i,w_0^{-1}v_{i+1}\\big )$ , $i=0,1,2$ , the map $R_{i+1,i}\\colon w_0^{-1}w_{i}K\\rightarrow w_0^{-1}w_{i+1}K$ consisting of the rotation of $\\frac{4}{3}\\pi $ around the point $x_{i,i+1}$ , $i=0,1,2$ , and observe that $R_{i,i+2}\\circ R_{i+2,i+1}\\circ R_{i+1,i} = {\\rm id}_{w_0^{-1}w_i K},\\qquad i=0,1,2.$ Setting $R_{i,i+1} = R_{i+1,i}^{-1}$ , the previous identities may also be written as $R_{i+2,i+1}\\circ R_{i+1,i} = R_{i+2,i},\\qquad i=0,1,2.$ We then construct the map $p\\colon K_1\\rightarrow K$ given by $p(x)={\\left\\lbrace \\begin{array}{ll}x,&x\\in K,\\\\R_{0,1}(x),&x\\in w_0^{-1}w_1 K,\\\\R_{0,2}(x),&x\\in w_0^{-1}w_2 K,\\end{array}\\right.", "}$ and observe that this map, which appears to be doubly defined in the points $x_{i,i+1}$ , $i=0,1,2$ , is indeed well defined (see Fig.", "REF ).", "Figure: The covering map p:K 1 →Kp\\colon K_1\\rightarrow K.The following result is easily verified.", "Proposition 3.1 The map $p$ is a well defined continuous map which is a ramified covering, with ramification points given by $\\lbrace x_{i,i+1},i=0,1,2\\rbrace $ .", "Moreover, the covering map is isometric on suitable neighbourhoods of the non-ramification points.", "Since $K_1$ and $K$ are homeomorphic, this map may be seen as a self-covering of the gasket.", "The map $p$ gives rise to an embedding $\\alpha _{1,0}\\colon {\\mathcal {C}}(K)\\rightarrow {\\mathcal {C}}(K_1)$ , hence, following [17], to an inductive family of C$^*$ -algebras ${\\mathcal {A}}_n={\\mathcal {C}}(K_n)$ , whose inductive limit ${\\mathcal {A}}_\\infty $ consists of continuous function on the solenoidal space based on the gasket.", "As in Definition REF , we consider the triple $ ({\\mathcal {L}}_n,{\\mathcal {H}}_n,D_n)$ on the $C^*$ -algebra ${\\mathcal {A}}_n$ , $n\\ge 0$ , where ${\\mathcal {H}}_n=\\ell ^2(E_n)$ , $E_n=\\lbrace w_0^{-n}e,\\, e\\in E_0\\rbrace $ (the set of oriented edges in $K_n$ ).", "Let us also note that, since the covering projections are locally isometric and any Lip-norm $L_m(f)=\\Vert [D_m,f]\\Vert $ associated with the triple $({\\mathcal {A}}_m,{\\mathcal {H}}_m,D_m)$ produces the geodesic distance on $K_m$ , we get $L_{m+q}(\\alpha _{m+q,m}(f))=L_m(f)$ , namely we obtain a seminorm on the algebraic inductive limit of the ${\\mathcal {A}}_n$ 's." ], [ "A groupoid of local isometries on the infinite Sierpinski fractafold", "A groupoid of local isometries on the infinite Sierpinski fractafold Let us consider the infinite fractafold $K_\\infty =\\cup _{n\\ge 0}K_n$  [51] endowed with the Hausdorff measure $\\operatorname{\\mu _d}$ of dimension $d=\\frac{\\log 3}{\\log 2}$ normalized to be 1 on $K=K_0$ , with the exhaustion $\\lbrace K_n\\rbrace _{n\\ge 0}$ , and with the family of local isometries $R=\\big \\lbrace R^n_{i+1,i},R^n_{i,i+1}\\colon i=0,1,2, n\\ge 0\\big \\rbrace $ , where $R^n_{i,j} = w_0^{-n}R_{i,j}w_0^{n}\\colon C^n_j \\rightarrow C^n_i$ , and $C^n_i := w_0^{-n-1}w_iK$ , $n\\ge 0$ , $i,j\\in \\lbrace 0,1,2\\rbrace $ .", "We also denote by $s(\\gamma )$ and $r(\\gamma )$ the domain and range of the local isometry $\\gamma $ .", "Such local isometries act on points and on oriented edges of $K_\\infty $ .", "We say that the product of the two local isometries $\\gamma _1$ , $\\gamma _2\\in R$ is defined if $\\gamma _2^{-1}(s(\\gamma _1))\\cap $ $s(\\gamma _2) \\ne \\varnothing $ .", "In this case we consider the product $\\gamma _1\\cdot \\gamma _2\\colon \\ \\gamma _2^{-1}(s(\\gamma _1))\\cap s(\\gamma _2)\\rightarrow r\\big (\\gamma _1|_{s(\\gamma _1))\\cap r(\\gamma _2)}\\big ).$ We then consider the family ${\\mathcal {G}}$ consisting of all (the well-defined) finite products of isometries in $R$ .", "Clearly, any $\\gamma $ in ${\\mathcal {G}}$ is a local isometry, and its domain and range are cells of the same size.", "We set ${\\mathcal {G}}_n=\\lbrace g\\in {\\mathcal {G}}\\colon s(\\gamma )\\,\\&\\,r(\\gamma )$ are cells of size $2^n\\rbrace $ , $n\\ge 0$ .", "Proposition 4.1 For any $n\\ge 0$ , $C_1$ , $C_2$ cells of size $2^n$ , $\\exists !\\, \\gamma \\in {\\mathcal {G}}_n$ such that $s(\\gamma )=C_1$ , $r(\\gamma )=C_2$ .", "In particular, if $C$ has size $2^n$ , the identity map of $C$ belongs to ${\\mathcal {G}}_n$ , $n\\ge 0$ .", "It is enough to show that for any cell $C$ of size $2^n$ there exists a unique $\\gamma \\in {\\mathcal {G}}_n$ such that $\\gamma \\colon C\\rightarrow K_n$ .", "For any cell $C$ , let $m=\\mathrm {level}(C)$ be the minimum number such that $C\\subset K_m$ .", "We prove the existence: if $C$ has size $2^n$ and $\\mathrm {level}(C)=m>n$ , then $C\\subset C^{m-1}_i$ , for some $i=1,2$ , hence $R^{m-1}_{0,i}(C) \\subset K_{m-1}$ .", "Iterating, the result follows.", "The second statement follows directly by equation (REF ).", "As for the uniqueness, $\\forall \\, n\\ge 0$ , we call $R^n_{i,0}$ ascending, $i=1,2$ , $R^n_{0,i}$ descending, $i=1,2$ , $R^n_{i,j}$  constant-level, $i,j\\in \\lbrace 1,2\\rbrace $ .", "Indeed, if $C \\subset s(R^n_{i,0})$ , then $\\mathrm {level}(C)\\le n$ and $\\mathrm {level}(R^n_{i,0}(C)) = n+1$ ; if $C \\subset s(R^n_{0,i})$ , then $\\mathrm {level}(C)=n+1$ , $\\mathrm {level}(R^n_{0,i}(C))\\le n$ and $\\mathrm {level}(R^n_{j,i}(C))= n+1$ , $i,j\\in \\lbrace 1,2\\rbrace $ , $n\\ge 0$ .", "The following facts hold: The product $R^n_{l,k} \\cdot R^m_{j,i}$ of two constant-level elements $R^n_{l,k}$ , $R^m_{j,i}$ is defined iff $n=m$ and $k=j$ , therefore any product of constant-level elements in $R$ is either the identity map on the domain or coincides with a single constant-level element.", "Any product of constant level elements in $R$ followed by a descending element coincides with a single descending element: indeed, if the product of constant level elements is the identity, the statement is trivially true; if it coincides with a single element, say $R^n_{i,j}$ with $i,j\\in \\lbrace 1,2\\rbrace $ , then, by compatibility, the descending element should be $R^n_{0,i}$ so that the product is $R^n_{0,i},$ by equation (REF ).", "Given a cell $C$ with $\\mathrm {size}(C)=2^n$ and $\\mathrm {level}(C)>n$ , the exists a unique descending element $\\gamma \\in R$ such that $C \\subset s(\\gamma )$ : indeed, if $m=\\mathrm {level}(C)$ , then $C\\subset C^{m-1}_i$ , for some $i\\in \\lbrace 1,2\\rbrace $ .", "The only descending element is then $\\gamma =R^{m-1}_{0,i}$ .", "Any product of an ascending element followed by a descending one is the identity on the domain: indeed if the ascending element is $R^n_{i,0}$ , then, by compatibility, the descending element should be $R^n_{0,i}$ .", "Now let $\\mathrm {size}(C)=2^n$ , $\\gamma \\in {\\mathcal {G}}_n$ such that $\\gamma \\colon C\\rightarrow K_n$ , $\\gamma =\\gamma _p\\cdot \\gamma _{p-1}\\cdots \\gamma _2\\cdot \\gamma _1$ , where $\\gamma _j\\in R$ , $1\\le j\\le p$ .", "Since $\\mathrm {level}(C) \\ge \\mathrm {level}(K_n) = n$ , for any possible ascending element $\\gamma _i$ there should be a $j>i$ such that $\\gamma _j$ is descending.", "If $i+q$ is the minimum among such $j$ 's, all terms $\\gamma _j$ , $i<j<i+q$ , are constant-level, hence the product $\\gamma _{i+q}\\cdot \\gamma _{i+q-1}\\cdots \\gamma _{i}={\\rm id}_{s(\\gamma _i)}$ .", "Then, we note that $\\gamma _p$ can only be descending.", "As a consequence, $\\gamma $ can be reduced to a product of descending elements, and, by the uniqueness of the descending element acting on a given cell, we get the result.", "Let us observe that each ${\\mathcal {G}}_n$ , and so also ${\\mathcal {G}}$ , is a groupoid under the usual composition rule, namely two local isometries are composable if the domain of the first coincides with the range of the latter.", "We now consider the action on points of the local isometries in ${\\mathcal {G}}$ .", "Proposition 4.2 Let us define $\\widetilde{{\\mathcal {A}}}_n$ as the algebra $\\widetilde{{\\mathcal {A}}}_n=\\lbrace f\\in {\\mathcal {C}}_b(K_\\infty )\\colon f( \\gamma (x))= f(x),\\, x\\in s(\\gamma ),\\, \\gamma \\in {\\mathcal {G}}_n\\rbrace .$ Then, for any $n\\ge 0$ , the following diagram commutes, $\\begin{matrix}\\widetilde{{\\mathcal {A}}}_n & \\subset & \\widetilde{{\\mathcal {A}}}_{n+1} \\\\\\Big \\downarrow \\iota _n & & \\Big \\downarrow \\iota _{n+1} \\\\{\\mathcal {A}}_n & \\mathop {\\longrightarrow }\\limits ^{\\alpha _{n+1,n}} & {\\mathcal {A}}_{n+1},\\end{matrix}$ where $\\iota _n\\colon f\\in \\widetilde{{\\mathcal {A}}}_n \\rightarrow f|_{K_n} \\in {\\mathcal {A}}_n$ are isomorphisms.", "Hence the inductive limit ${\\mathcal {A}}_\\infty $ is isomorphic to a $C^*$ -subalgebra of ${\\mathcal {C}}_b(K_\\infty )$ .", "The request in the definition of $\\widetilde{{\\mathcal {A}}}_n$ means that the value of $f$ in any point of $K_\\infty $ is determined by the value on $K_n$ , while such request gives no restrictions on the values of $f$ on $K_n$ .", "The other assertions easily follow.", "As shown above, we may identify the algebra ${\\mathcal {A}}_n$ , $0\\le n\\le \\infty $ , with its isomorphic copy $\\widetilde{{\\mathcal {A}}}_n$ in ${\\mathcal {C}}_b(K_\\infty )$ , so that the embeddings $\\alpha _{k,j}$ become inclusions.", "Moreover, we may consider the operator $\\widetilde{D}_n$ on $\\ell ^2(E_\\infty )$ , with $E_\\infty =\\cup _{n\\ge 0}E_n$ , given by $\\widetilde{D}_n e=\\mathrm {length}(e)^{-1}Fe$ , if $\\mathrm {length}(e)\\le 2^n$ , and $\\widetilde{D}_n e=0$ , if $\\mathrm {length}(e) >2^n$ , where $F$ is defined as in Definition REF $(c)$ .", "Then the spectral triples $({\\mathcal {A}}_n,{\\mathcal {H}}_n,D_n)$ are isomorphic to the spectral triples $(\\widetilde{{\\mathcal {A}}}_n,{\\mathcal {H}}_n,\\widetilde{D}_n)$ , where ${\\mathcal {C}}_b(K_\\infty )$ acts on the space $\\ell ^2(E_\\infty )$ through the representation $\\rho $ given by $\\rho (f)e=f(e^+)e$ .", "Remark 4.3 Because of the isomorphism above, from now on we shall remove the tildes and denote by ${\\mathcal {A}}_n$ the subalgebras of ${\\mathcal {C}}_b(K_\\infty )$ and by $D_n$ the operators acting on $\\ell ^2(E_\\infty )$ ." ], [ "The ${C^*}$ -algebra of geometric operators and a tracial weight on it", "The C*-algebra of geometric operators and a tracial weight on it We now come to the action of local isometries on edges.", "We shall use the following notation, where in the table below to any subset of edges listed on the left we indicate on the right the projection on the closed subspace spanned by the same subset: Table: Edges and projections.Let us note that any local isometry $\\gamma \\in {\\mathcal {G}}$ , $\\gamma \\colon s(\\gamma ) \\rightarrow r(\\gamma )$ , gives rise to a partial isometry $V_\\gamma $ defined as $V_\\gamma e={\\left\\lbrace \\begin{array}{ll}\\gamma (e), & e\\subset s(\\gamma ),\\\\0, & \\mathrm {elsewhere}.\\end{array}\\right.", "}$ In particular, if $C$ is a cell, and $\\gamma ={\\rm id}_C$ , $V_\\gamma =P_C$ .", "We then consider the subalgebras ${\\mathcal {B}}_n$ of $B(\\ell ^2(E_\\infty ))$ , ${\\mathcal {B}}_n=\\lbrace V_\\gamma \\colon \\gamma \\in {\\mathcal {G}}_m,\\,m\\ge n\\rbrace ^{\\prime },\\qquad {\\mathcal {B}}_{\\text{fin}}= \\bigcup _n{\\mathcal {B}}_n,\\qquad {\\mathcal {B}}_\\infty =\\overline{{\\mathcal {B}}_{\\text{fin}}},$ and note that the elements of ${\\mathcal {B}}_n$ commute with the projections $P_C$ , for all cells $C$ s.t.", "$\\mathrm {size}(C)\\ge 2^n$.", "By definition, the sequence ${\\mathcal {B}}_n$ is increasing, therefore, since the ${\\mathcal {B}}_n$ 's are von Neumann algebras, ${\\mathcal {B}}_\\infty $ is a $C^*$ -algebra.", "Let us observe that, $\\forall \\, n\\ge 0$ , $\\rho ({\\mathcal {A}}_n)\\subset {\\mathcal {B}}_n$ .", "Definition 5.1 The elements of the $C^*$ -algebra ${\\mathcal {B}}_\\infty $ are called geometric operators.", "Now consider the hereditary positive cone $\\mathcal {I}_0^+ = \\big \\lbrace T\\in {\\mathcal {B}}_{\\text{fin}}^+\\colon \\exists \\, c_T\\in {\\mathbb {R}}\\mathrm {\\ such\\ that\\ }\\operatorname{tr}(P_{m}T) \\le c_T \\operatorname{\\mu _d}(K_m), \\, \\forall \\, m \\ge 0 \\big \\rbrace .$ Lemma 5.2 For any $T\\in \\mathcal {I}_0^+$ , the sequence $\\frac{\\operatorname{tr}(P_{m} T) }{\\operatorname{\\mu _d}(K_m)}$ is eventually increasing, hence convergent.", "In particular $\\operatorname{tr}(P_{p}^{p} T)=0\\qquad \\forall \\, p>m\\Rightarrow \\tau _0(T)=\\frac{\\operatorname{tr}(P_{m} T) }{\\operatorname{\\mu _d}(K_{m})}.$ Let $T\\in {\\mathcal {B}}_n^+$ .", "Then we have, for $m\\ge n$ , $\\operatorname{tr}(P_{{m+1}} T) = \\!\\!\\sum _{e\\subset K_{m+1}} (e,Te) = \\!\\!\\sum _{i=0,1,2} \\sum _{e\\in C^m_i }(e,Te) + \\!\\sum _{e\\in E_{m+1}^{m+1}}\\!", "(e,Te)= 3\\operatorname{tr}(P_{{m}} T) + \\operatorname{tr}(P_{m+1}^{m+1} T),$ hence $\\frac{\\operatorname{tr}(P_{m+1} T) }{\\operatorname{\\mu _d}(K_{m+1})}=\\frac{\\operatorname{tr}(P_{m} T) }{\\operatorname{\\mu _d}(K_m)}+\\frac{\\operatorname{tr}(P_{m+1}^{m+1} T)}{\\operatorname{\\mu _d}(K_{m+1})},$ from which the thesis follows.", "We then define the weight $\\tau _0$ on ${\\mathcal {B}}_\\infty ^+$ as follows: $\\tau _0(T) ={\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\lim _{m\\rightarrow \\infty }\\frac{\\operatorname{tr}(P_{m} T)}{\\operatorname{\\mu _d}(K_m)}},& T\\in {\\mathcal {I}_0^+},\\\\0,& \\mathrm {elsewhere}.\\end{array}\\right.", "}$ The next step is to regularize the weight $\\tau _0$ in order to obtain a semicontinuous semifinite tracial weight $\\tau $ on ${\\mathcal {B}}_\\infty $ .", "Lemma 5.3 For any $T\\in \\mathcal {I}_0^+$ , $A\\in {\\mathcal {B}}_{\\text{fin}}$ , it holds $ATA^* \\in \\mathcal {I}_0^+$ , and $\\tau _0(ATA^*) \\le \\Vert A\\Vert ^2 \\tau _0(T)$ .", "Let $A\\in {\\mathcal {B}}_n$ .", "Then, for any $m>n$ , we have $\\operatorname{tr}(P_mATA^*) = \\operatorname{tr}(A^*AP_mT) \\le \\Vert A^*A \\Vert \\operatorname{tr}(P_mT) \\le \\Vert A\\Vert ^2 c_T \\operatorname{\\mu _d}(K_m),$ and the thesis follows.", "Proposition 5.4 For all $p\\in \\mathbb {N}$ , recall that $P^{-p,\\infty }$ is the orthogonal projection onto the closed vector space generated by $\\big \\lbrace e\\in \\ell ^2(E_\\infty )\\colon \\mathrm {length}(e) \\ge 2^{-p} \\big \\rbrace $ , and let $\\varphi _p(T) := \\tau _0(P^{-p,\\infty }TP^{-p,\\infty })$ , $\\forall \\, T\\in {\\mathcal {B}}_\\infty ^+$ .", "Then $P^{-p,\\infty }\\in {\\mathcal {B}}_0$ , $\\varphi _p$ is a positive linear functional, and $\\varphi _p(T) \\le \\varphi _{p+1}(T) \\le \\tau _0(T)$ , $\\forall \\, T\\in {\\mathcal {B}}_\\infty ^+$ .", "We first observe that $\\operatorname{tr}(P^j_n)=\\#\\big \\lbrace e\\in K_n\\colon \\mathrm {length}(e)=2^j\\big \\rbrace =6 \\cdot 3^{n-j},\\qquad j\\le n.$ Then it is easy to verify that $P^{-p,\\infty }\\in {\\mathcal {B}}_0$ .", "Since $\\varphi _p(I)= \\tau _0(P^{-p,\\infty })= \\lim _{n\\rightarrow \\infty } \\frac{\\operatorname{tr}P^{-p,n}_n}{\\mu _d(K_n)}= \\lim _{n\\rightarrow \\infty } 3^{-n} \\sum _{j=-p}^n \\operatorname{tr}(P^j_n)=\\sum _{j=-p}^\\infty 6\\cdot 3^{-j}= 3^{p+2},$ $\\varphi _p$ extends by linearity to a positive functional on ${\\mathcal {B}}_\\infty $ .", "Moreover, by Lemma REF , $\\varphi _p(T) \\le \\tau _0(T)$ , $\\forall \\, T\\in {\\mathcal {B}}_\\infty ^+$ .", "Finally, since $P^{-p,\\infty }P_n=P_nP^{-p,\\infty }=P^{-\\infty ,n}_n$ , $\\forall \\, n\\in \\mathbb {N}$ , we get, for all $T\\in {\\mathcal {B}}_\\infty ^+$ , p+1(T) - p(T) = 0(P-(p+1),TP-(+1)p,) - 0(P-p,TP-p,) = n tr((P-(p+1),n)n -P-p,nn)T)d(Kn) =n tr(P-(p+1))n T)d(Kn) 0.", "* Proposition 5.5 Let $\\tau (T) := \\lim \\limits _{p\\rightarrow \\infty } \\varphi _p(T)$ , $\\forall \\, T\\in {\\mathcal {B}}_\\infty ^+$ .", "Then $(i)$ $\\tau $ is a lower semicontinuous weight on ${\\mathcal {B}}_\\infty $ , $(ii)$ $\\tau (T) = \\tau _0(T)$ , $\\forall \\, T\\in \\mathcal {I}_0^+$ .", "$(i)$ Let $T\\in {\\mathcal {B}}_\\infty ^+$ .", "Since $\\lbrace \\varphi _p(T) \\rbrace _{p\\in \\mathbb {N}}$ is an increasing sequence, there exists $\\lim \\limits _{p\\rightarrow \\infty } \\varphi _p(T) = \\sup \\limits _{p\\in \\mathbb {N}} \\varphi _p(T)$ .", "Then $\\tau $ is a weight on ${\\mathcal {B}}_\\infty ^+$ .", "Since $\\varphi _p$ is continuous, $\\tau $ is lower semicontinuous.", "$(ii)$ Let us prove that, $\\forall \\, T\\in {\\mathcal {B}}_n^+$ , $ \\frac{\\operatorname{tr}(P^j_mT)}{\\mu _d(K_m)} = \\frac{\\operatorname{tr}(P^j_nT)}{\\mu _d(K_n)}, \\qquad j\\le n \\le m.$ Indeed, tr(Pjm+1T) = e Km+1 length(e)=2j ( e , Te ) = i=02 e Cmi length(e)=2j ( e , Te ) = i=02 e Km length(e)=2j ( VRmie , TVRmie ) = i=02 e Km length(e)=2j ( e , Te ) = 3 tr(PjmT), from which (REF ) follows.", "Let us now prove that $ \\tau (T) = \\sup _{p\\in \\mathbb {N}} \\varphi _p(T) = \\tau _0(T), \\qquad T\\in \\mathcal {I}_0^+ .$ Let $T\\in {\\mathcal {B}}_n^+ \\cap \\mathcal {I}_0^+$ , and $\\varepsilon >0$ .", "From the definition of $\\tau _0(T)$ , there exists $r\\in \\mathbb {N}$ , $r>n$ , such that $\\frac{\\operatorname{tr}(P_rT)}{\\mu _d(K_r)} > \\tau _0(T)-\\varepsilon $ .", "Since $\\frac{\\operatorname{tr}(P_rT)}{\\mu _d(K_r)} = \\sum _{j=-\\infty }^r \\frac{\\operatorname{tr}(P^j_rT)}{\\mu _d(K_r)}$ , there exists $p\\in \\mathbb {N}$ such that $\\sum _{j=-p}^r \\frac{\\operatorname{tr}(P^j_rT)}{\\mu _d(K_r)} > \\frac{\\operatorname{tr}(P_rT)}{\\mu _d(K_r)} -\\varepsilon > \\tau _0(T)-2\\varepsilon $ .", "Then, for any $s\\in \\mathbb {N}$ , $s>r$ , we have tr(PsP-p,TP-p,Ps)d(Ks) = j=-ps tr(PjsT)d(Ks) = j=-pr tr(PjsT)d(Ks) + j=r+1s tr(PjsT)d(Ks) (REF )= j=-pr tr(PjrT)d(Kr) + j=r+1s tr(PjsT)d(Ks) > 0(T)-2, and, passing to the limit for $s\\rightarrow \\infty $ , we get $\\varphi _p(T) = \\tau _0(P^{-p,\\infty }TP^{-p,\\infty }) = \\lim _{s\\rightarrow \\infty } \\frac{\\operatorname{tr}(P_sP^{-p,\\infty }TP^{-p,\\infty }P_s)}{\\mu _d(K_s)} \\ge \\tau _0(T)-\\varepsilon ,$ and equation (REF ) follows.", "We want to prove that $\\tau $ is a tracial weight.", "Definition 5.6 An operator $U\\in B\\big (\\ell ^2(E_\\infty )\\big )$ is called $\\delta $ -unitary, $\\delta >0$ , if $\\Vert U^*U-1\\Vert <\\delta $ , and $\\Vert UU^*-1\\Vert <\\delta $ .", "Let us denote with ${\\mathcal {U}}_\\delta $ the set of $\\delta $ -unitaries in ${\\mathcal {B}}_{\\text{fin}}$ and observe that, if $\\delta <1$ , ${\\mathcal {U}}_\\delta $ consists of invertible operators, and $U\\in {\\mathcal {U}}_\\delta $ implies $U^{-1}\\in {\\mathcal {U}}_{\\delta /(1-\\delta )}$ .", "Proposition 5.7 The weight $\\tau _0$ is $\\varepsilon $ -invariant for $\\delta $ -unitaries in ${\\mathcal {B}}_{\\text{fin}}$ , namely, for any $\\varepsilon \\in (0,1)$ , there is $\\delta >0$ s.t., for any $U\\in {\\mathcal {U}}_\\delta $ , and $T\\in {\\mathcal {B}}^+_\\infty $ , $(1-\\varepsilon )\\tau _0(T) \\le \\tau _0(UTU^*) \\le (1+\\varepsilon )\\tau _0(T) .$ We first observe that, if $\\delta \\in (0,1)$ and $U\\in {\\mathcal {U}}_\\delta $ , $T\\in \\mathcal {I}_0^+\\Leftrightarrow UTU^*\\in \\mathcal {I}_0^+$ .", "Indeed, choose $n$ such that $U,T\\in {\\mathcal {B}}_n$ .", "Then $\\operatorname{tr}(P_{n} UTU^*) = \\operatorname{tr}(U^* UP_{n}TP_{n}) \\le \\Vert U^*U\\Vert \\operatorname{tr}(P_{n}T) \\le (1+\\delta ) c_T\\operatorname{\\mu _d}(K_n)$ , $\\forall \\, n\\in \\mathbb {N}$ , so that $UTU^* \\in \\mathcal {I}_0^+$ .", "Moreover, $\\tau _0(UTU^*) = \\lim _{n\\rightarrow \\infty } \\frac{\\operatorname{tr}(P_nUTU^*)}{\\operatorname{\\mu _d}(K_n)} \\le \\Vert U^*U \\Vert \\lim _{n\\rightarrow \\infty } \\frac{\\operatorname{tr}(P_nT)}{\\operatorname{\\mu _d}(K_n)} = \\Vert U^*U \\Vert \\tau _0(T) < (1+\\delta ) \\tau _0(T).$ Conversely, $UTU^*\\in \\mathcal {I}_0^+$ , and $U^{-1}\\in {\\mathcal {U}}_{\\delta /(1-d)} \\Rightarrow T \\in \\mathcal {I}_0^+$ .", "Moreover, $\\tau _0(T) \\le \\big \\Vert \\big (U^{-1}\\big )^*U^{-1} \\big \\Vert \\tau _0(UTU^*) < \\frac{1}{1-\\delta } \\tau _0(UTU^*).$ The result follows by the choice $\\delta =\\varepsilon $ .", "Theorem 5.8 The lower semicontinuous weight $\\tau $ in Proposition $\\ref {def.tau}$ is a trace on ${\\mathcal {B}}_\\infty $ , that is, setting ${\\mathcal {J}}^+:= \\lbrace A\\in {\\mathcal {B}}_\\infty ^+\\colon \\tau (A)<\\infty \\rbrace $ , and extending $\\tau $ to the vector space ${\\mathcal {J}}$ generated by ${\\mathcal {J}}^+$ , we get $(i)$ ${\\mathcal {J}}$ is an ideal in ${\\mathcal {B}}_\\infty $ , $(ii)$ $\\tau (AB)=\\tau (BA)$ , for any $A\\in {\\mathcal {J}}$ , $B\\in {\\mathcal {B}}_\\infty $ .", "$(i)$ Let us prove that ${\\mathcal {J}}^+$ is a unitarily-invariant face in ${\\mathcal {B}}_\\infty ^+$ , and suffices it to prove that $A\\in {\\mathcal {J}}^+$ implies that $UAU^*\\in {\\mathcal {J}}^+$ , for any $U\\in {\\mathcal {U}}({\\mathcal {B}}_\\infty )$ , the set of unitaries in ${\\mathcal {B}}_\\infty $ .", "To reach a contradiction, assume that there exists $U\\in {\\mathcal {U}}({\\mathcal {B}}_\\infty )$ such that $\\tau (UAU^*)=\\infty $ .", "Then there is $p\\in \\mathbb {N}$ such that $\\varphi _p(UAU^*) > 2\\tau (A)+2$ .", "Let $\\delta <3$ be such that $V\\in {\\mathcal {U}}_\\delta $ implies $\\tau (VAV^*)\\le 2\\tau (A)$ , and let $U_0\\in {\\mathcal {B}}_{\\text{fin}}$ be such that $\\Vert U-U_0\\Vert < \\min \\big \\lbrace \\frac{\\delta }{3}, \\frac{1}{3\\Vert A\\Vert \\Vert \\varphi _p\\Vert } \\big \\rbrace $ .", "The inequalities $\\Vert U_0U_0^*-1\\Vert = \\Vert U^*U_0U_0^*-U^*\\Vert \\le \\Vert U^*U_0-1\\Vert \\Vert U_0^*\\Vert +\\Vert U_0^*-U^*\\Vert < \\delta $ and $\\Vert U_0^*U_0-1\\Vert <\\delta $ , prove that $U_0\\in {\\mathcal {U}}_\\delta $ .", "Since $ |\\varphi _p(U_0AU_0^*)-\\varphi _p(UAU^*)|\\le 3\\Vert \\varphi _p \\Vert \\Vert A\\Vert \\Vert U-U_0\\Vert <1,$ we get $2\\tau (A)\\ge \\tau (U_0AU_0^*) \\ge \\varphi _p(U_0AU_0^*) \\ge \\varphi _p(UAU^*) - 1 \\ge 2\\tau (A)+1$ which is absurd.", "$(ii)$ We only need to prove that $\\tau $ is unitarily-invariant.", "Let $A\\in {\\mathcal {J}}^+$ , $U\\in {\\mathcal {U}}({\\mathcal {B}}_\\infty )$ .", "For any $\\varepsilon >0$ , there is $p\\in \\mathbb {N}$ such that $\\varphi _p(UAU^*)>\\tau (UAU^*)-\\varepsilon $ , since, by $(1)$ , $\\tau (UAU^*)$ is finite.", "Then, arguing as in the proof of $(1)$ , we can find $U_0\\in {\\mathcal {B}}_{\\text{fin}}$ , so close to $U$ that $|\\varphi _p(U_0AU_0^*)-\\varphi _p(UAU^*)|<\\varepsilon , \\\\(1-\\varepsilon )\\tau (A)\\le \\tau (U_0AU_0^*) \\le (1+\\varepsilon )\\tau (A).$ Then (A) 11+ (U0AU0*) 11+ p(U0AU0*) 11+ (p(UAU*) -) 11+ ((UAU*) -2).", "By the arbitrariness of $\\varepsilon >0$ , we get $\\tau (A)\\ge \\tau (UAU^*)$ .", "Exchanging $A$ with $UAU^*$ , we get the thesis.", "Proposition 5.9 The lower semicontinuous tracial weight $\\tau $ defined in Proposition $\\ref {def.tau}$ is semifinite and faithful.", "Let us recall that, for any $p\\in \\mathbb {N}$ , $P^{-p,\\infty } \\in \\mathcal {I}_0^+$ by Proposition REF .", "From Proposition REF follows that $\\tau (P^{-p,\\infty })=\\tau _0(P^{-p,\\infty })<\\infty $ , hence $P^{-p,\\infty }\\in {\\mathcal {J}}^+$ .", "Then, for any $T\\in {\\mathcal {B}}_\\infty ^+$ , $S_p := T^{1/2}P^{-p,\\infty }T^{1/2} \\in {\\mathcal {J}}^+$ , and $0 \\le S_p \\le T$ .", "Moreover, (Sp) = (T1/2P-p,T1/2) = (P-p,TP-p,) = qN 0(QqP-p,TP-p,Qq) = 0(P-p,TP-p,) = p(T), so that $\\sup \\limits _{p\\in \\mathbb {N}} \\tau (S_p) = \\tau (T)$ , and $\\tau $ is semifinite.", "Finally, if $T\\in {\\mathcal {B}}_\\infty ^+$ is such that $\\tau (T)=0$ , then $\\sup \\limits _{p\\in \\mathbb {N}} \\varphi _p(T)=0$ .", "Since $\\lbrace \\varphi _p(T) \\rbrace _{p\\in \\mathbb {N}}$ is an increasing sequence, $\\varphi _p(T)=0$ , $\\forall \\, p\\in \\mathbb {N}$ .", "Then, for a fixed $p\\in \\mathbb {N}$ , we get $0 = \\tau _0(P^{-p,\\infty }TP^{-p,\\infty }) = \\lim \\limits _{n\\rightarrow \\infty } \\frac{ \\operatorname{tr}(P_nP^{-p,\\infty }TP^{-p,\\infty }P_n) }{\\mu _d(K_n)}$ .", "Since the sequence $\\big \\lbrace \\frac{ \\operatorname{tr}(P_nP^{-p,\\infty }TP^{-p,\\infty }P_n) }{\\mu _d(K_n)} \\big \\rbrace _{n\\in \\mathbb {N}}$ is definitely increasing, we get $\\operatorname{tr}(P_nP^{-p,\\infty }TP^{-p,\\infty }P_n) = 0$ definitely, that is $TP^{-p,\\infty }P_n = 0$ definitely, so that $TP^{-p,\\infty }=0$ .", "By the arbitrariness of $p\\in \\mathbb {N}$ , we get $T=0$." ], [ "A semifinite spectral triple on the inductive limit ${{\\mathcal {A}}_\\infty }$", "A semifinite spectral triple on the inductive limit A infty Since the covering we are studying is ramified, the family $\\lbrace {\\mathcal {A}}_n,{\\mathcal {H}}_n,D_n\\rbrace $ does not have a simple tensor product structure, contrary to what happened in [1].", "We therefore use a different approach to construct a semifinite spectral triple on ${\\mathcal {A}}_\\infty $ : our construction is indeed based on the pair $({\\mathcal {B}}_\\infty ,\\tau )$ of the $C^*$ -algebra of geometric operators and the semicontinuous semifinite weight on it.", "The Dirac operator will be defined below (Definition REF ) through its phase and the functional calculi of its modulus with continuous functions vanishing at $\\infty $ .", "More precisely we shall use the following Definition 6.1 Let $(\\mathfrak {C},\\tau )$ be a $C^*$ -algebra with unit endowed with a semicontinuous semifinite faithful trace.", "A selfadjoint operator $T$ affiliated to $(\\mathfrak {C},\\tau )$ is defined as a pair given by a closed subset $\\sigma (T)$ in ${\\mathbb {R}}$ and a $*$ homomorphism $\\phi \\colon {\\mathcal {C}}_0(\\sigma (T))\\rightarrow \\mathfrak {C}$ , $f(T)\\mathop {=}\\limits ^{\\mathrm {def}}\\phi (f)$ , provided that the support of such homomorphism is the identity in the GNS representation $\\pi _\\tau $ induced by the trace $\\tau $ .", "The previous definition was inspired by that in [20] appendix A, and should not be confused with that of Woronowicz for $C^*$ -algebras without identity.", "Remark 6.2 The $*$ -homomorphism $\\phi _\\tau =\\pi _\\tau \\circ \\phi $ extends to bounded Borel functions on ${\\mathbb {R}}$ and $e_{(-\\infty ,t]}\\mathop {=}\\limits ^{\\mathrm {def}}\\phi _\\tau (\\chi _{(-\\infty ,t]})$ tends strongly to the identity when $t\\rightarrow \\infty $ , hence it is a spectral family.", "We shall denote by $\\pi _\\tau (T)$ the selfadjoint operator affiliated to $\\pi _\\tau (\\mathfrak {C})^{\\prime \\prime }$ given by $\\pi _\\tau (T)\\mathop {=}\\limits ^{\\mathrm {def}}\\int _{\\mathbb {R}}t\\, {\\rm d} e_{(-\\infty ,t]}.$ Proposition 6.3 Let $T$ be a selfadjoint operator affiliated to $(\\mathfrak {C},\\tau )$ as above.", "$(a)$ Assume that for any $n\\in \\mathbb {N}$ , there is $\\varphi _n\\in {\\mathcal {C}}({\\mathbb {R}})\\colon 0\\le \\varphi _n\\le 1,\\varphi _n=1$ for $|t|\\le a_n$ , $\\varphi _n(t)=0$ for $|t|\\ge b_n$ with $0<a_n<b_n$ and $\\lbrace a_n\\rbrace $ , $\\lbrace b_n\\rbrace $ increasing to $\\infty $ .", "Then, for any $A\\in \\mathfrak {C}$ , if $\\sup \\limits _n\\Vert [T\\cdot \\varphi _n(T),A]\\Vert =C<\\infty $ then $[\\pi _t(T),\\pi _\\tau (A)]$ is bounded and $\\Vert [\\pi _t(T),\\pi _\\tau (A)]\\Vert =C$ .", "$(b)$ If $\\tau (f(T))<\\infty $ for any positive function $f$ with compact support on the spectrum of $T$ then $\\pi _\\tau (T)$ has $\\tau $ -compact resolvent.", "$(a)$ Let ${\\mathcal {D}}$ be the domain of $\\pi _\\tau (T)$ , ${\\mathcal {D}}_0$ the space of vectors in ${\\mathcal {D}}$ with bounded support w.r.t.", "to $\\pi _\\tau (T)$ , and consider the sesquilinear form $F(y,x)=(\\pi _\\tau (T)y,\\pi _\\tau (A)x)-(y,\\pi _\\tau (A)\\pi _\\tau (T)x)$ defined on ${\\mathcal {D}}$ .", "By hypothesis, for any $x,y\\in {\\mathcal {D}}_0$ there exists $n$ such that $\\pi _\\tau (\\varphi _n(T))x=x$ and $\\pi _\\tau ((\\varphi _n(T))y=y$ , hence $F(y,x)=(y,\\pi _\\tau ([T\\cdot \\varphi _n(T),A])x)\\le C\\Vert x\\Vert \\ \\Vert y\\Vert $ .", "By the density of ${\\mathcal {D}}_0$ in ${\\mathcal {D}}$ w.r.t.", "the graph norm of $\\pi _\\tau (T)$ , the same bound holds on ${\\mathcal {D}}$ .", "Then for $y,x\\in {\\mathcal {D}}$ , $|(\\pi _\\tau (T)y,\\pi _\\tau (A)x)|\\le |(y,\\pi _\\tau (A)\\pi _\\tau (T)x)|+|F(y,x)|\\le (\\Vert \\pi _\\tau (A)\\pi _\\tau (T)x\\Vert +C\\Vert x\\Vert )\\Vert y\\Vert $ which implies $\\pi _\\tau (A)x$ belongs to the domain of $\\pi _\\tau (T)^*=\\pi _\\tau (T)$ .", "Therefore $\\pi _\\tau (T)\\pi _\\tau (A)-\\pi _\\tau (A)\\pi _\\tau (T)$ is defined on ${\\mathcal {D}}$ and its norm is bounded by $C$ .", "Since $C$ is the optimal bound for the sesquilinear form $F$ it is indeed the norm of the commutator.", "$(b)$ Let $\\lambda $ be in the resolvent of $|T|$ .", "We then note that for any $f$ positive and zero on a neighbourhood of the origin there is a $g$ positive and with compact support such that $f\\big ((|T|-\\lambda I)^{-1}\\big )=g(|T|)$ .", "Therefore $\\tau \\big (f\\big ((|T|-\\lambda I)^{-1}\\big )\\big )<\\infty $ , hence $\\tau \\big (e_{(t,+\\infty )}\\big (\\pi _\\tau \\big ((|T|-\\lambda I)^{-1}\\big )\\big )\\big )<\\infty $ for any $t>0$ , i.e., $\\pi _\\tau \\big ((|T|-\\lambda I)^{-1}\\big )$ is $\\tau $ -compact (cf.", "Section REF ).", "Definition 6.4 We consider the Dirac operator $D=F|D|$ on $\\ell ^2(E_\\infty )$ , where $F$ is the orientation reversing operator on edges and $|D|=\\sum _{n\\in {\\mathbb {Z}}}2^{-n}P^n,\\qquad \\sigma (|D|)=\\lbrace 2^{-n},\\ n\\in {\\mathbb {Z}}\\rbrace \\cup \\lbrace 0\\rbrace .$ Proposition 6.5 The following hold: $(a)$ The elements $D$ and $|D|$ are affiliated to $({\\mathcal {B}}_\\infty ,\\tau )$ .", "$(b)$ The following formulas hold: $\\tau (P^n)=6\\cdot 3^{-n}$ , $\\tau (P^{-p,\\infty })=3^{p+2}$ , as a consequence the operator $D$ has $\\tau $ -compact resolvents $(c)$ The trace $\\tau (I+ D^{2})^{-s/2}<\\infty $ if and only if $s>d=\\frac{\\log 3}{\\log 2}$ and $\\operatorname{Res}_{s=d}\\tau \\big (I+ D^{2}\\big )^{-s/2}=\\frac{6}{\\log 2}.$ ($a$ ) We first observe that the $*$ -homomorphisms for $D$ and $|D|$ have the same support projection, then note that since $F$ and $P_n$ belong to ${\\mathcal {B}}_0$ (which is a von Neumann algebra) for any $n\\in \\mathbb {N}$ , then $f(D)$ and $f(|D|)$ belong to ${\\mathcal {B}}_0$ for any $f\\in {\\mathcal {C}}_0({\\mathbb {R}})$ ; therefore it is enough to show that the support of $f\\mapsto f( |D|)$ is the identity in the representation $\\pi _\\tau $ .", "In order to prove this, it is enough to show that $\\pi _\\tau (e_{|D|} [0,2^p])$ tends to the identity strongly when $p\\rightarrow \\infty $ , that is to say that $\\pi _\\tau (e_{|D|}(2^p,\\infty ))$ tends to 0 strongly when $p\\rightarrow \\infty $ .", "We consider then the projection $P^{-\\infty ,0}$ which projects on the space generated by the edges with $\\mathrm {length}(e)\\le 1$ .", "Clearly, such projection belongs to ${\\mathcal {B}}_0$ , we now show that it is indeed central there.", "In fact, if $c$ is a cell with $\\mathrm {size}(c)=1$ , $P_c$ commutes with ${\\mathcal {B}}_0$ .", "Since $P^{-\\infty ,0}=\\sum _{\\mathrm {size}(c)=1}P_c$ , then $P^{-\\infty ,0}$ commutes with ${\\mathcal {B}}_0$ .", "On the one hand, the von Neumann algebra $P^{-\\infty ,0}{\\mathcal {B}}_0$ is isomorphic to ${\\mathcal {B}}\\big (\\ell ^2(K)\\big )$ and the restriction of $\\tau $ to $P^{-\\infty ,0}{\\mathcal {B}}_0$ coincides with the usual trace on ${\\mathcal {B}}\\big (\\ell ^2(K)\\big )$ , therefore the representation $\\pi _\\tau $ is normal when restricted to $P^{-\\infty ,0}{\\mathcal {B}}_0$ .", "On the other hand, since $e_{|D|}(2^p,\\infty )=P^{-\\infty ,-p-1}$ is, for $-p\\le 1$ , a sub-projection of $P^{-\\infty ,0}$ , and $P^{-\\infty ,-p-1}$ tends to 0 strongly in the given representation, the same holds of the representation $\\pi _\\tau $ .", "($b$ ) We prove the first equation.", "Indeed $\\tau (P^n)=\\lim _m \\frac{\\operatorname{tr}P_{m}^n}{\\operatorname{\\mu _d}(K_m)}=\\operatorname{tr}P_{0}^n+\\lim _m\\sum _{j=1}^m\\frac{\\operatorname{tr}P_j^j P^n}{\\operatorname{\\mu _d}(K_j)}.$ The first summand is non-zero iff $n\\le 0$ , while the second vanishes exactly for such $n$ .", "Since $\\lim _m\\sum _{j=1}^m\\frac{\\operatorname{tr}P_j^j P^n}{\\operatorname{\\mu _d}(K_j)}=\\frac{\\operatorname{tr}P_n^n }{\\operatorname{\\mu _d}(K_n)},$ the result in (REF ) shows that in both cases we obtain $6\\cdot 3^{-n}$ .", "We already proved in (REF ) that $\\tau _0(P^{-p,\\infty })=3^{p+2}$ .", "Since $P^{-p,\\infty }\\in {\\mathcal {B}}_0$ , the same holds for $\\tau $ by Proposition REF $(ii)$ .", "Then the thesis follows by condition $(b)$ in Proposition REF .", "($c$ ) We have $\\tau \\big (I+ D^{2}\\big )^{-s/2}=\\tau \\big (P^{-\\infty ,0}\\big (I+ D^{2}\\big )^{-s/2}\\big )+\\tau \\big (P^{1,+\\infty }\\big (I+ D^{2}\\big )^{-s/2}\\big )$ .", "A straightforward computation and (REF ) give $\\tau \\big (P^{-\\infty ,0}\\big (I+ D^{2}\\big )^{-s/2}\\big )=\\operatorname{tr}\\big (P_0\\big (I+ D^{2}\\big )^{-s/2}\\big )=6\\sum _{n\\ge 0}\\big (1+2^{2n}\\big )^{-s/2}3^{n},$ which converges iff $s>d$ .", "As for the second summand, we have (P1,+(I+ D2)-s/2) =0(P1,+(I+ D2)-s/2) =mtr(P1,mm(I+ D2)-s/2)d(Km) =mj=1m 3-mtr(P1,mm(I+ D2)-s/2) =6j=13-j(1+2-2j)-s/2, which converges for any $s$ hence does not contribute to the residue.", "Finally Ress=d(I+ D2)-s/2 =sd+(s-d)(I+ D2)-s/2 =sd+(s-32)6n0(1+2-2n)-s/2 en(3-s2) =62sd+s2-31-e-(s2-3) =62.", "* Proposition 6.6 For any $f\\in {\\mathcal {A}}_n$ $\\sup \\limits _{t>0}\\big \\Vert \\big [ e_{[-t,t]}(D)\\, D,\\rho (f)\\big ]\\big \\Vert =\\Vert [ D_n,\\rho (f|_{K_n})]\\Vert $ .", "We observe that $| D|$ is a multiplication operator on $\\ell ^2(E_\\infty )$ , therefore it commutes with $\\rho (f)$ .", "Hence, $\\big \\Vert \\big [ D\\,e_{[-2^p,2^p]}(D),\\rho (f)\\big ]\\big \\Vert =\\big \\Vert |D|\\,e_{[0,2^p]}(|D|)\\, (\\rho (f)-F\\rho (f)F)\\big \\Vert \\!=\\!\\!\\sup _{\\mathrm {length}(e)\\ge 2^{-p}}\\!\\!\\!\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}.$ As a consequence, $\\sup _{p\\in {\\mathbb {Z}}}\\big \\Vert \\big [ D\\,e_{[-2^p,2^p]}(D),\\rho (f)\\big ]\\big \\Vert =\\sup _{e\\in E_\\infty }\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}.$ Recall now that any edge $e$ of length $2^{n+1}$ is the union of two adjacent edges $e_1$ and $e_2$ of length $2^n$ such that $e_1^+=e_2^-$ , therefore $\\frac{|f(e^+)-f(e^-)|}{2^{n+1}}\\le \\frac{1}{2}\\bigg (\\frac{|f(e_1^+)-f(e_1^-)|}{2^{n}}+\\frac{|f(e_2^+)-f(e_2^-)|}{2^{n}}\\bigg )\\le \\sup _{\\mathrm {length}(e)=2^n}\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}.$ Iterating, we get $\\sup _{e\\in E_\\infty }\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}=\\sup _{\\mathrm {length}(e)\\le 2^{n}}\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}.$ Since $f\\in {\\mathcal {A}}_n$ , $\\sup _{\\mathrm {length}(e)\\le 2^{n}}\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}=\\sup _{e\\in K_n}\\frac{|f(e^+)-f(e^-)|}{\\mathrm {length}(e)}=\\Vert [ D_n,\\rho (f|_{K_n})]\\Vert .\\qquad \\mathrm {}$ In the following Theorem we identify ${\\mathcal {B}}_\\infty $ with $\\pi _\\tau ({\\mathcal {B}}_\\infty )$ , the trace $\\tau $ on $\\pi _\\tau ({\\mathcal {B}}_\\infty )$ with its extension to $\\pi _\\tau ({\\mathcal {B}}_\\infty )^{\\prime \\prime }$ , and $D_n$ and $D$ as unbounded operators affiliated with $({\\mathcal {B}}_\\infty ,\\tau )$ with $\\pi _\\tau (D_n)$ and $\\pi _\\tau (D)$ as unbounded operators affiliated with $(\\pi _\\tau ({\\mathcal {B}}_\\infty )^{\\prime \\prime },\\tau )$ .", "Theorem 6.7 The triple $({\\mathcal {L}},\\pi _\\tau (B_\\infty )^{\\prime \\prime },D)$ on the unital C$^*$ -algebra ${\\mathcal {A}}_\\infty $ is an odd semifinite spectral triple, where ${\\mathcal {L}}=\\cup _n\\lbrace f\\in {\\mathcal {A}}_n, f\\text{\\,Lipschitz}\\rbrace $ .", "The spectral triple has metric dimension $d=\\frac{\\log 3}{\\log 2}$ , the functional $\\oint f=\\tau _\\omega \\big (\\rho (f)\\big (I+D^2\\big )^{-d/2}\\big ),$ is a finite trace on ${\\mathcal {A}}_\\infty $ where $\\tau _\\omega $ is the logarithmic Dixmier trace associated with $\\tau $ , and $\\oint f=\\frac{6}{\\log 3}\\frac{\\int _{K_n} f\\,{\\rm d}\\operatorname{\\mu _d}}{\\operatorname{\\mu _d}(K_n)},\\qquad f\\in {\\mathcal {A}}_n,$ where $\\operatorname{\\mu _d}$ is the Hausdorff measure of dimension $d$ normalized as above.", "As a consequence, $\\oint f$ is a Bohr–Følner mean on the solenoid: $\\oint f=\\frac{6}{\\log 3}\\,\\lim _{n\\in \\mathbb {N}}\\frac{\\int _{K_n} f\\,{\\rm d}\\operatorname{\\mu _d}}{\\operatorname{\\mu _d}(K_n)},\\qquad f\\in {\\mathcal {A}}_\\infty .$ The Connes distance $d(\\varphi ,\\psi )=\\sup \\lbrace |\\varphi (f)-\\psi (f)|\\colon f\\in {\\mathcal {L}},\\, \\Vert [ D,\\rho (f) ] \\Vert = 1 \\rbrace ,\\qquad \\varphi ,\\psi \\in {\\mathcal {S}}({\\mathcal {A}}_\\infty )$ between states on ${\\mathcal {A}}_\\infty $ verifies $d(\\delta _x,\\delta _y)=d_{\\rm geo}(x,y),\\qquad x,y\\in K_\\infty ,$ where $d_{\\rm geo}$ is the geodesic distance on $K_\\infty $ .", "The properties of a semifinite spectral triple follow by the properties proved above, in particular property $(1)$ of Definition REF follows by Propositions REF $(a)$ and REF , while property $(2)$ follows by Proposition REF $(b)$ .", "The functional in equality (REF ) is a finite trace by Proposition REF $(c)$ .", "Equations (REF ) and (REF ) only remain to be proved.", "We observe that $(I+D^2)^{-d/2}-|D_n|^{-d}$ have finite trace.", "Indeed $\\big ((I+D^2)^{-d/2}-|D_n|^{-d}\\big )e={\\left\\lbrace \\begin{array}{ll}\\big (1+4^{-k}\\big )^{-d/2}e, & \\mathrm {length}(e)=2^k,\\quad k>n,\\\\\\big (\\big (1+4^{-k}\\big )^{-d/2}-2^{dk}\\big )e, & \\mathrm {length}(e)=2^k,\\quad k>n k\\le n,\\end{array}\\right.", "}$ hence, makig use of a formula in Theorem REF $(b)$ , we get |((I+D2)-d/2-|Dn|-d)| k>n(1+4-k)-d/2(Pk)+kn|(1+4-k)-d/2-3k|(Pk) 6(1+4-(n+1))-d/2k>n3-k +6kn|(1+4k)-d/2-1| and both series are convergent.", "Since the Dixmier trace vanishes on trace class operators, this implies that $\\tau _\\omega (\\rho (f)\\big (I+D^2\\big )^{-d/2})=\\tau _\\omega \\big (\\rho (f)|D_n|^{-d}\\big )=\\frac{1}{d} \\operatorname{Res}_{s=d}\\tau \\big (\\rho (f)|D_n|^{-s}\\big ),$ therefore, if $f\\in {\\mathcal {A}}_n$ , $\\oint f=\\frac{1}{d} \\operatorname{Res}_{s=d}\\tau \\big (\\rho (f)|D_n|^{-s}\\big )=\\frac{1}{d} \\operatorname{Res}_{s=d}\\frac{\\operatorname{tr}(\\rho (f_{K_n})|D_n|^{-s})}{\\operatorname{\\mu _d}(K_n)}=\\frac{\\operatorname{tr}_\\omega \\big (\\rho (f)|D_n|^{-d}\\big )}{\\operatorname{\\mu _d}(K_n)}.$ Now, by formula (REF ) applied to $K_n$ , $\\operatorname{tr}_\\omega (\\rho (f)|D_n|^{-d}) = \\frac{6\\cdot \\ell (e)^d}{ \\log 3} \\int _{K_n} f\\, {\\rm d}H_d$ , where $H_d$ is the Hausdorff measure normalized on $K_n$ , hence $H_d=(\\mu _d(K_d))^{-1}\\mu _d=3^{-n}\\mu _d$ , and $e\\in E_0(K_n)$ , hence $\\ell (e)^d=3^n$ .", "Therefore $\\operatorname{tr}_\\omega \\big (\\rho (f)|D_n|^{-d}\\big ) = \\frac{6}{\\log 3} \\int _{K_n} f\\, {\\rm d}\\mu _d$ and formula (REF ) follows.", "As for equation (REF ), given $x,y\\in K_\\infty $ let $n$ such that $x,y\\in K_n$ , $m\\ge n$ .", "Then, combining Propositions REF $(a)$ and REF , we have, for $f\\in {\\mathcal {A}}_m$ , $\\Vert [ D, \\rho (f)]\\Vert =\\Vert [ D_m,\\rho (f|_{K_m})]\\Vert ,$ and, by Theorem 5.2 and Corollary 5.14 in [29], $\\sup \\lbrace |f(x)-f(y)|\\colon f\\in {\\mathcal {A}}_m,\\Vert [D_m,\\rho (f|_{K_n})]\\Vert =1\\rbrace =d_{\\rm geo}(x,y),\\qquad m\\ge n.$ Therefore d(x,y) ={|f(x)-f(y)|fL,  [ D,(f)]=1} =m {|f(x)-f(y)|fAm,  [ D,(f)]=1} =m {|f(x)-f(y)|fAm,  [Dm,(f|Kn)]=1}=dgeo(x,y).", "* Remark 6.8 The last statement in Theorem REF shows that the triple $({\\mathcal {L}},{\\mathcal {M}},D_\\infty )$ recovers two incompatible aspects of the space ${\\mathcal {A}}_\\infty $ : on the one hand the compact space given by the spectrum of the unital algebra ${\\mathcal {A}}_\\infty $ , with the corresponding finite integral, and on the other hand the open fractafold $K_\\infty $ with its geodesic distance.", "In particular, the functional on ${\\mathcal {L}}$ given by $L(f)=\\Vert [D,\\rho (f)]\\Vert $ is not a Lip-norm in the sense of Rieffel [41] because it does not give rise to the weak$^*$ topology on ${\\mathcal {S}}({\\mathcal {A}}_\\infty )$ .", "In fact, such seminorm produces a distance which is unbounded on points, therefore the induced topology cannot be compact." ], [ "Acknowledgements", "We thank the referees of this paper for many interesting observations and suggestions.", "V.A.", "is supported by the Swiss National Science Foundation.", "D.G.", "and T.I.", "are supported in part by GNAMPA-INdAM and the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, and acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.", "[1]Referencesref" ] ]

[ [ "Super-Penrose process for extremal rotating neutral white holes" ], [ "Abstract We consider collision of two particles 1 and 2 near the horizon of the extremal rotating axially symmetric neutral generic black hole producing particles 3 and 4.", "We discuss the scenario in which both particles 3 and 4 fall into a black hole and move in a white hole region.", "If particle 1 is fine-tuned, the energy $E_{c.m.", "}$ in the centre of mass grows unbounded (the Ba\\~{n}ados-Silk-West effect).", "Then, particle 3 \\ can, in principle, reach a flat infinity in another universe.", "If not only $E_{c.m.", "}$ but also the corresponding Killing energy $E$ is unbounded, this gives a so-called super-Penrose process (SPP).", "We show that the SPP\\ is indeed possible.", "Thus white holes turn out to be potential sources of high energy fluxes that transfers from one universe to another.", "This generalizes recent observaitons made by Patil and Harada for the Kerr metric.", "We analyze two different regimes of the process on different scales." ], [ "Introduction", "During last decade a lot of work has been made for investigation of properties of high energy processes near black holes.", "This was stimulated by the paper [1], where it was found that collision of two particles near rapidly rotating black holes can lead to unbounded energies $E_{c.m.", "}$ in the centre of mass frame.", "This was called the Bañados-Silk-West (BSW) effect, after the authors' names.", "After its publication, it turned out, that there are also earlier works [2], [3] in which near-horizon particle collisions in the Kerr metric were investigated.", "Meanwhile, a typical process considered there, includes head-on collision between two arbitrary particles, where particle 1 arrives from infinity while particle 2 comes from the horizon (see eq.", "2.57 of [3]).", "But as far as particle 2 is concerned, this is nothing else than a typical behavior of a particle near a white hole.", "If such a region is allowed in the complete space-time, the effect of unbounded $E_{c.m.", "}$ for head-on collisions exists even in the Schwarzschild metric [4], [5].", "Thus white holes can be an alternative to black ones as a source of high energy collisions.", "More important question is whether it is possible to gain not only unbounded $E_{c.m.", "}$ but also unbounded conserved Killing energies $E$ since it is the latter quantity which can be measured in the Earth laboratory, at least in principle.", "The collisions in the Schwarzschild background are useless for this purpose since energy cannot be extracted at all.", "For such an extraction, the existence of negative energies and ergoregion are required that makes it possible the Penrose process [6] or its collisional analogue [7].", "If the energy gain is unbounded, this is called the super-Penrose process (SPP).", "For black holes, the energy gain is finite, so the SPP is impossible for them (see [8] and references therein).", "In this context, there is a scenario with participation of white holes, different from those in [2] - [5].", "Now, both particles collide near the black hole horizon in \"our\" part of Universe but afterwards the products of reaction leave it.", "Passing though the horizon, they appear inside a white whole region and, eventually, transfer energy to another universe.", "Or, vice verse, collision in another universe can give rise to high energy in our one.", "If such a process is possible, this would give astrophysical realization of high energy transfer with white holes as a source that was suggested earlier [9], [10].", "The concrete process of this kind in the Kerr background was considered recently in [11].", "The authors showed that the conserved energy of produced particles can be as large as one like.", "In other words, the SPP is possible.", "Our aim is to extend consideration to generic rotating axially symmetric stationary white holes.", "In doing so, we exploit the approach that, in our view, is simpler and was already used for examination of the energy extraction from generic black holes of the aforementioned type including the Kerr metric [12], [13].", "In particular, we do not use transformation between the three frames (center of mass, locally non-rotating and stationary ones) and work in the original frame.", "Although there are reasons to believe that white holes are unstable [14] (see also Sec.", "15 of [15]), motivation for consideration of such objects stems from different roots.", "(i) High energy process, if they are confirmed, can themselves contribute to the instability of white holes, so they are important for elucidation of the fate of such objects.", "(ii) The complete theory of the BSW effect should take into consideration all possible configurations and scenarios, at least for better understanding the phenomenon.", "We use the system of units in which the fundamental constants $G=c=1$ ." ], [ "Basic equations", "Let us consider the metric $ds^{2}=-N^{2}dt^{2}+\\frac{dr^{2}}{A}+g_{\\phi }(d\\phi -\\omega dt)^{2}+g_{\\theta }d\\theta ^{2}\\text{,} $ where $g_{\\phi }\\equiv g_{\\phi \\phi }$ , $g_{\\theta }\\equiv g_{\\phi \\phi }$ , all coefficients do not depend on $t$ and $\\phi $ .", "Before turning to the analysis of the collisions between two particles in the background of a white hole, we describe main features of motion inherent to an individual particle.", "To simplify formulas, we assume that in the equatorial plane $A=N^{2}$ .", "Otherwise, we can always achieve this equality by redefining the radial coordinate according to $r\\rightarrow \\tilde{r}$ , where $\\frac{dr}{\\sqrt{A}}=\\frac{d\\tilde{r}}{N}\\text{,}$ so $\\tilde{r}=\\int ^{r}\\frac{dr^{\\prime }N}{\\sqrt{A^{\\prime }}}\\text{.", "}$ In general, for an arbitrary $\\theta $ , this transformation does not work since $d\\tilde{r}$ would not be a total differential, if $A$ and $N$ depend on $\\theta $ .", "However, for our purposes (for motion within the equatorial plane, so $\\theta =\\frac{\\pi }{2}$ is fixed), this is a quite legitimate operation.", "It is valid for any metric of the type (REF ) including the Kerr one.", "For a given energy $E$ , angular momentum $L$ and mass $m$ the equation of motion in the equatorial plane read $m\\dot{t}=\\frac{X}{N^{2}}\\text{,}$ $X=E-\\omega L\\text{,} $ $m\\dot{\\phi }=\\frac{L}{g_{\\phi }}$ $m\\dot{r}=\\sigma P\\text{, }\\sigma =\\pm 1\\text{,}$ $P=\\sqrt{X^{2}-\\tilde{m}^{2}N^{2}}\\text{,} $ $\\tilde{m}^{2}=m^{2}+\\frac{L^{2}}{g_{\\phi }}\\text{,}$ dot denotes differentiation with respect to the proper time $\\tau $ .", "TThe forward-in-time condition requires $X\\ge 0\\text{.}", "$ In what follows, we will use the standard classification of particles.", "If $X_{H}=0$ , a particle is called critical.", "If $X_{H}=O(1)$ , it is called usual.", "If $X_{H}=O(N_{c}),$ it is called near-critical.", "Here, subscripts \"H\" and \"c\" refer to the quantities calculated on the horizon and the point of collision, respectively.", "For the near-critical particle, we use presentation $L=\\frac{E}{\\omega _{H}}(1+\\delta ) $ exploited in [13].", "Here, $\\delta =C_{1}N_{c} $ is a small quantity for collisions near the horizon, $C_{1}=O(1)$ is a constant.", "Near the horizon, we assume the Taylor expansion that for the extremal case reads [16] $\\omega =\\omega _{H}-B_{1}N+O(N^{2})\\text{.", "}$ Then, we have the following approximate expressions there.", "The critical particle: $X=\\frac{b}{h}EN+O(N^{2})\\text{,} $ $P=N\\sqrt{E^{2}\\left( \\frac{b^{2}-1}{h^{2}}\\right) -\\frac{1}{h^{2}}-m^{2}}.$ A usual particle: $X=X_{H}+B_{1}LN+O(N^{2})\\text{, }$ $X_{H}=E-\\omega _{H}L\\text{,}$ $P=X+O(N^{2}).", "$ The near-critical particle: $X=E(\\frac{b}{h}-C_{1})N $ $P=N\\sqrt{E^{2}[(\\frac{b}{h}-C_{1})^{2}-\\frac{1}{h^{2}}]-m^{2}}+O(N^{2})\\text{.}", "$ We introduced notations $b=B_{1}\\sqrt{g_{H}}$ , $h=\\omega _{H}\\sqrt{g_{H}}$ .", "Here, for shortness, we also use notation $g\\equiv g_{\\phi }$ , where $g_{\\phi }$ is defined in (REF ).", "To give an example, we list the concrete expressions for these relevant characteristic for the physically relevant case of the extremal Kerr-Newman metric: $g_{H}=\\frac{(M^{2}+a^{2})^{2}}{M^{2}}\\text{,}$ $B_{1}=\\frac{2a}{M^{2}+a^{2}}\\text{,}$ $\\omega _{H}=\\frac{a}{M^{2}+a^{2}}\\text{,}$ $b=\\frac{2a}{M}\\text{, }h=\\frac{a}{M}\\text{.}", "$ Here, $a$ is the standard parameter of the Kerr-Newman metric that characterizes its angular momentum, $M$ being the mass.", "It is interesting that the electric charge $Q$ drops out from the explicit expressions for $b$ and $h$ .", "If $Q=0$ , we our extremal metric transforms to the Kerr one with $M=a$ and the values (REF ) return to $b=2$ and $h=1$ in accordance with eq.", "(46) of [13].", "Meanwhile, below we operate with relevant quantities in a general form, without specifying the metric.", "This is justified by the fact that, as we will see, in the cases under discussion there is no concrete upper bound on the maximum possible energy, this result being model-independent.", "It is worth noting that the coordinates in (REF ) generalize the Boyer-Lindquiste ones for the Kerr metric.", "It is known that such coordinates do not cover the whole space-time.", "Therefore, one is led to introduce an infinite set of different coordinate patches, the space-time structure includes an infinite set of black hole and white hole regions.", "For the, say, extremal Kerr-Newman metric for the equatorial plane the Carter-Penrose diagrame is similar to that for the Reissner-Nordström metric and is represented on Fig.", "1 (see, e.g.", "detailed description of these metrics in [26], especially Ch.", "11).", "We schematically showed a trajectory of a test particle Fig.", "1.", "Figure: The Carter-Penrose diagrame of the extremal Kerr-Newman space-time in the equatorial plane.The concrete descreiption of motion inside the horizon was done in [11] with the help of ligh-like coordinates for the Kerr metric.", "Although such details are of interest on their own right, for our goal (to elucidate the absence or prsence of the upper bound on $E$ ) they are not necessary, so we use more simple and straightforward approach.", "It is based on already elaborated scheme exploited for the analysis of collisions in the black hole background [13], [25]." ], [ "Scenario of collision", "In this section, we give brief set-up for the description of the process under consideration.", "We assume that particles 1 and 2 collide producing particles 3 and 4.", "We want to elucidate, whether the resulting energy in the center f mass frame $E_{c.m.", "}$ can grow unbounded, if we take into account the processes in the white hole region.", "Before elucidating this issue, we (i) describe basics of analysis of collision, (ii) possible scenarios of collision that can be realized near the horizon Afterwards, we discuss, which types of scenario correspond to the black hole region and which ones are relevant for the white hole one.", "The general analysis of collisions relies on the restrictions that come from the conservation laws.", "We assume that such laws are valid in the point of collision.", "This includes the conservation laws for the energy and angular momentum: $E_{0}\\equiv E_{1}+E_{2}=E_{3}+E_{4}\\text{,} $ $L_{0}=L_{1}+L_{2}=L_{3}+L_{4}.", "$ It follows from (REF ), (REF ) that $X_{0}\\equiv X_{1}+X_{2}=X_{3}+X_{4}.", "$ There is also the conservation law for the radial momentum: $\\sigma _{1}P_{1}+\\sigma _{2}P_{2}=\\sigma _{3}P_{3}+\\sigma _{4}P_{4}\\text{.", "}$ We assume that particles 1 and 2 fall from infinity, so $\\sigma _{1}=\\sigma _{2}=-1$ .", "We are interested in high energy processes in which $E_{c.m.", "}$ is unbounded since this is the necessary condition for $E$ to be unbounded as well [17], [18].", "To this end, we choose particle 1 to be the critical, particle 2 being usual since this gives rise to the unbounded $E_{c.m.", "}$ [1], [19].", "Then, one of particles (say, 3) is near-critical and the other one (4) is usual [12], [13].", "All possible scenarios can be described by two parameters - the sign of $C_{1}$ in (REF ) and the value of $\\sigma _{3}$ immediately after collision (OUT for $\\sigma _{3}=+1$ and IN for $\\sigma _{3}=-1$ ).", "As a result, we have 4 scenarios OUT$+$ , OUT$-$ , IN$+$ , IN$-$ .", "The first three were already analyzed in [13].", "In scenarios OUT$+$ and OUT$-$ particle 3 after collision escapes immediately to infinity.", "In scenario IN$+$ particle 3 continues to move inward after collision, bounces back from the potential barrier and also returns to infinity.", "It turned out [12], [13] that the later scenario is especially effective for the energy extraction.", "As far as IN$-$ is concerned, both particles do not encounter a potential barrier and, therefore, fall into a black hole.", "For this reason, scenario IN$-$ was rejected in [13] since no energy returns to infinity.", "However, now it is just this scenario which we focus on.", "It corresponds to high energy propagation in the white hole region (see below).", "Thus we have $\\sigma _{3}=\\sigma _{4}=-1$ .", "We must analyze the process under discussion for $N_{c}\\rightarrow 0$ on the basic of the conservation law (REF ).", "In doing so, we follow the lines of Ref.", "[13] applying the corresponding approach to the case that was not considered there." ], [ "Lower bounds on energy", "If we collect the terms of the zeroth and first order in $N_{c}$ and take into account the approximate expressions (REF ) - (REF ), we obtain $F=-\\sqrt{E_{3}^{2}[(\\frac{b}{h}-C_{1})^{2}-\\frac{1}{h^{2}}]-m_{3}^{2}},$ where $F\\equiv A+E_{3}(C_{1}-\\frac{b}{h}), $ $A=\\frac{E_{1}b-\\sqrt{E_{1}^{2}(b^{2}-1)-m_{1}^{2}h^{2}}}{h}, $ $C_{1}=\\frac{b}{h}-\\frac{A^{2}+m_{3}^{2}+\\frac{E_{3}^{2}}{h^{2}}}{2E_{3}A},$ $F=\\frac{A^{2}-m_{3}^{2}-\\frac{E_{3}^{2}}{h^{2}}}{2A}.$ We are interested in scenario IN$-$ .", "Then, $C_{1}<0$ gives us $E_{3}^{2}-2E_{3}hA_{1}b+h^{2}(A^{2}+m_{3}^{2})>0, $ that can be rewritten as $\\left( E_{3}-\\lambda _{+}\\right) (E_{3}-\\lambda _{+})>0,$ $\\lambda _{\\pm }=h[A_{1}b\\pm \\sqrt{A^{2}(b^{2}-1)-m_{3}^{2}}]\\text{.", "}$ The condition $F<0$ gives us $E_{3}^{2}>h^{2}(A^{2}-m_{3}^{2})\\equiv \\lambda _{0}^{2}\\text{.}", "$ If $\\lambda _{\\pm }$ are real, both bounds $E_{3}>\\lambda _{+}$ and $E_{3}>\\lambda _{0}$ are quite compatible with each other.", "If $\\lambda _{\\pm } $ are complex, (REF ) and (REF ) are mutually consistent as well.", "Thus there is no upper bound on $E_{3}$ and the SPP is possible.", "As far as particle 4 is concerned, it has $E_{4}<0$ .", "To obey the forward-in-time condition (REF ), it must have $L_{4}=-\\left|L_{4}\\right|<0$ .", "Then, $X_{4}=\\left|L_{4}\\right|\\omega -\\left|E_{4}\\right|$ .", "Assuming that there is a flat infinity, where $\\omega \\rightarrow 0$ , we see that particle 4 either falls into singularity or oscillates between turning points $r_{1}$ and $r_{2}$ .", "In doing so, it can intersect the horizons, thus appearing in new \"universes\" due to a potentially rich space-time structure inside similarly to what takes place for the Kerr metric [20].", "However, under a rather weak and reasonable restrictions on the properties of the metric, it cannot have more than 1 turning point in the outer region, so the situation when $r_{+}<r_{1}\\le r\\le r_{2}$ is impossible.", "This was shown for the Kerr metric in [21] and generalized in [22].", "For more information about trajectories of particles 3 and 4, the metric should be specified.", "The above treatment changes only slightly if we consider the Schnittman process [23] when the critical particle 1 does not come from infinity but moves from the horizon.", "Then, instead of (REF ), we should take $A=\\frac{E_{1}b+\\sqrt{E_{1}^{2}(b^{2}-1)-m_{1}^{2}h^{2}}}{h}$ ." ], [ "Superenergetic particles", "In the above treatment, we tacitly assumed that all energies and angular momenta are finite and do not grow unbounded when $N_{c}\\rightarrow 0$ .", "The only place where $N_{c}$ appear in the relation between them are equalities (REF ), (REF ), where it gives only small corrections.", "The above approximate expressions for particle characteristics (REF ) - (REF ) take into account this circumstances.", "In particular, for a usual particle, $X=O(1)$ , the second term in the radical in (REF ) has the order $N_{c}^{2}$ .", "For a near-critical one, both terms in $P$ have the order $O(N_{c})$ .", "Meanwhile, it turns out that there exists self-consistent scenario, in which $L_{3}=\\frac{l_{3}}{\\sqrt{N_{c}}}\\text{,} $ where $l_{3}$ is some coefficient not containing $N_{c}$ .", "For small $N_{c}$ , $L_{4}=L_{0}-L_{3}\\approx -\\frac{l_{3}}{\\sqrt{N_{c}}}$ .", "It was found in [24], where it was pointed out that it corresponds to falling both particles in a black hole, so it was put aside since we were interested in particles returning to infinity.", "But now, it is this case that came into play.", "Therefore, we take advantage of formulas already derived in [24] but exploit them in a new context - see eqs.", "(REF ), (REF ) below.", "If (REF ) is satisfied, the previous consideration fails and the conservation law (REF ) is to be analyzed anew.", "Now, for particles 3 and 4 the second term in (REF ) gives a small correction (whereas for finite $L_{3}$ both terms for particle 3 would have the same order), so $P_{3,4}\\approx \\sqrt{X_{3,4}^{2}-N_{c}\\frac{l_{3,4}^{2}}{\\left( g_{\\phi }\\right) _{H}}}\\approx X_{3,4}-\\frac{N_{c}}{2X_{3,4}}\\frac{l_{3}^{2}}{\\left(g_{\\phi }\\right) _{H}}\\text{.", "}$ Then, taking into account (REF ) - (REF ) for particles 1 and 2, (REF ) - (REF ) and discarding the terms $O(N_{c}^{2}$ ) and higher, one can show after algebraic manipulations that the following equation holds: $\\frac{l_{3}^{2}}{2\\left( g_{\\phi }\\right) _{H}}(\\frac{1}{X_{3}}+\\frac{1}{X_{4}})=A\\text{.", "}$ Using again (REF ), one can obtain $\\left( X_{3,4}\\right) _{c}\\approx \\frac{\\left( X_{0}\\right) _{c}}{2}(1\\mp \\sqrt{1-b})\\text{,} $ where $b\\equiv \\frac{2l_{3}^{2}}{\\left( g_{\\phi }\\right) _{H}X_{0}A}\\text{.", "}$ It is implied that $b<1$ .", "It follows from definition (REF ) that now $E_{3}=\\left( X_{3}\\right) _{H}+\\omega _{H}L_{3}\\approx \\left( X_{3}\\right)_{c}+\\omega _{H}\\frac{l_{3}}{\\sqrt{N_{c}}}\\text{.", "}$ Thus we have two usual particles which come down into a black hole.", "This is contrasted with the standard case when particle 3 is near-critical and returns to infinity.", "We see that there are two energy scales for the SPP.", "On the first scale, $E_{3}$ can be as large as we like but with reservation that $E_{3}\\ll \\frac{const}{\\sqrt{N_{c}}}$ .", "On the second scale, $E_{3}\\sim \\frac{1}{\\sqrt{N_{c}}}.$ In doing so, $E_{4}=E_{0}-E_{3}$ is negative having the same order $N_{c}^{-1/2}$ .", "Discussion about the properties of the trajectories of such particles from Section applies now as well." ], [ "Conclusions", "Thus we showed that particle collision on our side of Universe (near the black hole horizon) can lead to high energy fluxes on the other side.", "If, vice verse, collision occurs in \"another world\", we can detect its consequences in our one.", "We did not resort to the transformation between the original stationary frame and the center of mass one.", "We would like to stress that high energy behavior is found for the energies $E_{3}$ that can be in principle detected in a laboratory.", "These results qualitatively agree with claims made in [11] for the Kerr metric.", "It is instructive to compare the situation for static charged and neutral rotating black/white holes collecting the results of the present and previous works [25], [27].", "Table: NO_CAPTIONTable 1.", "Conditions of the existence of the super-Penrose process.", "We see that for the Reissner-Nordström metric, if we compare collisions near black and white holes, the situation is partially complementary to each other.", "For finite $L_{i}$ (for example, with all $L_{i}=0$ ), there is the SPP in the black hole case.", "However, it fails to exist near white holes.", "As shown in [27], only if particles with angular momenta $L_{3,4}=O(N_{c}^{-1/2})$ come into play, we obtain the SPP.", "Meanwhile, for the rotating case, the SPP does not exist for black holes at all.", "Instead, the white hole scenario opens new possibilities for the SPP, in which $E$ and $L$ of particles at infinity are unbounded.", "This happens on two scales: on the first one $E$ and $L$ do not contain the parameter $N_{c}^{-1}$ , on the second scale they have the order $O(N_{c}^{-1/2})$ similarly to the static charged case.", "The concrete properties of collisions are described by somewhat different formulas and the type of energetic particles are different: in the first case particle 3 is near-critical, in the second one it is usual.", "However, in both cases the conserved energy $E_{3}$ is unbounded.", "The phenomenon under discussion is two-faced.", "On one hand, it shows that high energy processes are indeed possible due to white holes and poses anew the question about their potential role in nature.", "From the other hand, it poses also a question about backreaction of such collisions on the metric itself, including the fate of white holes.", "This work was supported by the Russian Government Program of Competitive Growth of Kazan Federal University." ] ]

[ [ "Notes on ridge functions and neural networks" ], [ "Abstract These notes are about ridge functions.", "Recent years have witnessed a flurry of interest in these functions.", "Ridge functions appear in various fields and under various guises.", "They appear in fields as diverse as partial differential equations (where they are called plane waves), computerized tomography and statistics.", "These functions are also the underpinnings of many central models in neural networks.", "We are interested in ridge functions from the point of view of approximation theory.", "The basic goal in approximation theory is to approximate complicated objects by simpler objects.", "Among many classes of multivariate functions, linear combinations of ridge functions are a class of simpler functions.", "These notes study some problems of approximation of multivariate functions by linear combinations of ridge functions.", "We present here various properties of these functions.", "The questions we ask are as follows.", "When can a multivariate function be expressed as a linear combination of ridge functions from a certain class?", "When do such linear combinations represent each multivariate function?", "If a precise representation is not possible, can one approximate arbitrarily well?", "If well approximation fails, how can one compute/estimate the error of approximation, know that a best approximation exists?", "How can one characterize and construct best approximations?", "If a smooth function is a sum of arbitrarily behaved ridge functions, can it be expressed as a sum of smooth ridge functions?", "We also study properties of generalized ridge functions, which are very much related to linear superpositions and Kolmogorov's famous superposition theorem.", "These notes end with a few applications of ridge functions to the problem of approximation by single and two hidden layer neural networks with a restricted set of weights.", "We hope that these notes will be useful and interesting to both researchers and students." ], [ " To the Memory of My Parents These notes are about ridge functions.", "Recent years have witnessed a flurry of interest in these functions.", "Ridge functions appear in various fields and under various guises.", "They appear in fields as diverse as partial differential equations (where they are called plane waves), computerized tomography and statistics.", "These functions are also the underpinnings of many central models in neural networks.", "We are interested in ridge functions from the point of view of approximation theory.", "The basic goal in approximation theory is to approximate complicated objects by simpler objects.", "Among many classes of multivariate functions, linear combinations of ridge functions are a class of simpler functions.", "These notes study some problems of approximation of multivariate functions by linear combinations of ridge functions.", "We present here various properties of these functions.", "The questions we ask are as follows.", "When can a multivariate function be expressed as a linear combination of ridge functions from a certain class?", "When do such linear combinations represent each multivariate function?", "If a precise representation is not possible, can one approximate arbitrarily well?", "If well approximation fails, how can one compute/estimate the error of approximation, know that a best approximation exists?", "How can one characterize and construct best approximations?", "If a smooth function is a sum of arbitrarily behaved ridge functions, is it true that it can be expressed as a sum of smooth ridge functions?", "We also study properties of generalized ridge functions, which are very much related to linear superpositions and Kolmogorov's famous superposition theorem.", "These notes end with a few applications of ridge functions to the problem of approximation by single and two hidden layer neural networks with a restricted set of weights.", "We hope that these notes will be useful and interesting to both researchers and graduate students.", "tocchapterIntroduction Recent years have seen a growing interest in the study of special multivariate functions called ridge functions.", "A ridge function, in its simplest format, is a multivariate function of the form $g\\left( \\mathbf {a}\\cdot \\mathbf {x}\\right) $ , where $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ , $\\mathbf {a}=\\left( a_{1},...,a_{d}\\right) $ is a fixed vector (direction) in $\\mathbb {R}^{d}\\backslash \\left\\lbrace \\mathbf {0}\\right\\rbrace $ , $\\mathbf {x}=\\left(x_{1},...,x_{d}\\right) $ is the variable and $\\mathbf {a}\\cdot \\mathbf {x}$ is the standard inner product.", "In other words, a ridge function is a multivariate function constant on the parallel hyperplanes $\\mathbf {a}\\cdot \\mathbf {x}=c$ , $c\\in \\mathbb {R}$ .", "These functions arise naturally in various fields.", "They arise in computerized tomography (see, e.g., [85], [86], [87], [113], [121], [128]), statistics (see, e.g., [22], [23], [38], [43], [55]) and neural networks (see, e.g., [32], [77], [79], [109], [117], [135], [140]).", "These functions are also used in modern approximation theory as an effective and convenient tool for approximating complicated multivariate functions (see, e.g., [49], [71], [75], [103], [118], [131], [134], [155]).", "It should be remarked that long before the appearance of the name “ridge\", these functions were used in PDE theory under the name of plane waves.", "For example, see the book by F. John [82].", "In general, sums of ridge functions with fixed directions occur in the study of hyperbolic constant coefficient partial differential equations.", "As an example, assume that $(\\alpha _{i},\\beta _{i}),~i=1,...,r,$ are pairwise linearly independent vectors in $\\mathbb {R}^{2}$ .", "Then the general solution to the homogeneous partial differential equation $\\prod \\limits _{i=1}^{r}\\left( \\alpha {_{i}{\\frac{\\partial }{\\partial {x}}}+\\beta _{i}{\\frac{\\partial }{\\partial {y}}}}\\right) {u}\\left( {x,y}\\right) =0$ are all functions of the form $u(x,y)=\\sum \\limits _{i=1}^{r}g_{i}\\left( \\beta {_{i}x-\\alpha _{i}y}\\right)$ for arbitrary continuous univariate functions $g_{i}$ , $i=1,...,r$ .", "Here the derivatives are understood in the sense of distributions.", "The term “ridge function\" was coined by Logan and Shepp in their seminal paper [113] devoted to the basic mathematical problem of computerized tomography.", "This problem consists of reconstructing a given multivariate function from values of its integrals along certain straight lines in the plane.", "The integrals along parallel lines can be considered as a ridge function.", "Thus, the problem is to reconstruct $f$ from some set of ridge functions generated by the function $f$ itself.", "In practice, one can consider only a finite number of directions along which the above integrals are taken.", "Obviously, reconstruction from such data needs some additional conditions to be unique, since there are many functions $g$ having the same integrals.", "For uniqueness, Logan and Shepp [113] used the criterion of minimizing the $L_{2}$ norm of $g$ .", "That is, they found a function $g(x,y)$ with the minimum $L_{2}$ norm among all functions, which has the same integrals as $f$ .", "More precisely, let $D$ be the unit disk in the plane and an unknown function $f(x,y)$ be square integrable and supported on $D.$ We are given projections $P_{f}(t,\\theta )$ (integrals of $f$ along the lines $x\\cos \\theta +y\\sin \\theta =t$ ) and looking for a function $g=g(x,y)$ of minimum $L_{2}$ norm, which has the same projections as $f:$ $P_{g}(t,\\theta _{j})=P_{f}(t,\\theta _{j}),$ $j=0,1,...,n-1$ , where the angles $\\theta _{j}$ generate equally spaced directions, i.e.", "$\\theta _{j}=\\frac{j\\pi }{n},$ $j=0,1,...,n-1.$ The authors of [113] showed that this problem of tomography is equivalent to the problem of $L_{2}$ -approximation of the function $f$ by sums of ridge functions with the equally spaced directions $(\\cos \\theta _{j},\\sin \\theta _{j})$ , $j=0,1,...,n-1.$ They gave a closed-form expression for the unique function $g(x,y)$ and showed that the unique polynomial $P(x,y)$ of degree $n-1$ which best approximates $f$ in $L_{2}(D)$ is determined from the above $n$ projections of $f$ and can be represented as a sum of $n$ ridge functions.", "Kazantsev [85] solved the above problem of tomography without requiring that the considered directions are equally spaced.", "Marr [121] considered the problem of finding a polynomial of degree $n-2$ , whose projections along lines joining each pair of $n$ equally spaced points on the circumference of $D$ best matches the given projections of $f$ in the sense of minimizing the sum of squares of the differences.", "Thus we see that the problems of tomography give rise to an independent study of approximation theoretic properties of the following set of linear combinations of ridge functions: $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right):g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},i=1,...,r\\right\\rbrace ,$ where directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ are fixed and belong to the $d$ -dimensional Euclidean space.", "Note that the set $\\mathcal {R}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ is a linear space.", "Ridge function approximation also appears in statistics in Projection Pursuit.", "This term was introduced by Friedman and Tukey [42] to name a technique for the explanatory analysis of large and multivariate data sets.", "This technique seeks out “interesting\" linear projections of the multivariate data onto a line or a plane.", "Projection Pursuit algorithms approximate a multivariate function $f$ by sums of ridge functions with variable directions, that is, by functions from the set $\\mathcal {R}_{r}=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) :\\mathbf {a}^{i}\\in \\mathbb {R}^{d}\\setminus \\lbrace \\mathbf {0}\\rbrace ,\\ g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},i=1,...,r\\right\\rbrace .$ Here $r$ is the only fixed parameter, directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ and functions $g_{1},...,g_{r}$ are free to choose.", "The first method of such approximation was developed by Friedman and Stuetzle [43].", "Their approximation process called Projection Pursuit Regression (PPR) operates in a stepwise and greedy fashion.", "The process does not find a best approximation from $\\mathcal {R}_{r}$ , it algorithmically constructs functions $g_{r}\\in \\mathcal {R}_{r},$ such that $\\left\\Vert g_{r}-f\\right\\Vert _{L_{2}}\\rightarrow 0,$ as $r\\rightarrow \\infty $ .", "At stage $m$ , PPR looks for a univariate function $g_{m}$ and direction $\\mathbf {a}^{m}$ such that the ridge function $g_{m}\\left( \\mathbf {a}^{m}\\cdot \\mathbf {x}\\right) $ best approximates the residual $f(x)-\\sum \\limits _{j=1}^{m-1}g_{j}\\left( \\mathbf {a}^{j}\\cdot \\mathbf {x}\\right) $ .", "Projection pursuit regression has been proposed as an approach to bypass the curse of dimensionality and now is applied to prediction in applied sciences.", "In [22], [23], Candes developed a new approach based not on stepwise construction of approximation but on a new transform called the ridgelet transform.", "The ridgelet transform represents general functions as integrals of ridgelets – specifically chosen ridge functions.", "The significance of approximation by ridge functions is well understood from its role in the theory of neural networks.", "Ridge functions appear in the definitions of many central neural network models.", "It is a broad knowledge that neural networks are being successfully applied across an extraordinary range of problem domains, in fields as diverse as finance, medicine, engineering, geology and physics.", "Generally speaking, neural networks are being introduced anywhere that there are problems of prediction, classification or control.", "Thus not surprisingly, there is a great interest to this powerful and very popular area of research (see, e.g., [135] and a great deal of references therein).", "An artificial neural network is a way to perform computations using networks of interconnected computational units vaguely analogous to neurons simulating how our brain solves them.", "An artificial neuron, which forms the basis for designing neural networks, is a device with $d$ real inputs and an output.", "This output is generally a ridge function of the given inputs.", "In mathematical terms, a neuron may be described as $y=\\sigma (\\mathbf {w\\cdot x}-\\theta ),$ where $\\mathbf {x=(x}_{1},...,x_{d})\\in \\mathbb {R}^{d}$ are the input signals, $w=(w_{1},...,w_{d})\\in \\mathbb {R}^{d}$ are the synaptic weights, $\\theta \\in \\mathbb {R}$ is the bias, $\\sigma $ is the activation function and $y$ is the output signal of the neuron.", "In a layered neural network the neurons are organized in the form of layers.", "We have at least two layers: an input and an output layer.", "The layers between the input and the output layers (if any) are called hidden layers, whose computation nodes are correspondingly called hidden neurons or hidden units.", "The output signals of the first layer are used as inputs to the second layer, the output signals of the second layer are used as inputs to the third layer, and so on for the rest of the network.", "Neural networks with this kind of architecture is called a Multilayer Feedforward Perceptron (MLP).", "This is the most popular model among other neural network models.", "In this model, a neural network with a single hidden layer and one output represents a function of the form $\\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i}).$ Here the weights $\\mathbf {w}^{i}$ are vectors in $\\mathbb {R}^{d}$ , the thresholds $\\theta _{i}$ and the coefficients $c_{i}$ are real numbers and the activation function $\\sigma $ is a univariate function.", "We fix only $\\sigma $ and $r$ .", "Note that the functions $\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i})$ are ridge functions.", "Thus it is not surprising that some approximation theoretic problems related to neural networks have strong association with the corresponding problems of approximation by ridge functions.", "It is clear that in the special case, linear combinations of ridge functions turn into sums of univariate functions.", "This is also the simplest case.", "The simplicity of the approximation apparatus itself guarantees its utility in applications where multivariate functions are constant obstacles.", "In mathematics, this type of approximation has arisen, for example, in connection with the classical functional equations [17], the numerical solution of certain PDE boundary value problems [14], dimension theory [149], [148], etc.", "In computer science, it arises in connection with the efficient storage of data in computer databases (see, e.g., [158]).", "There is an interesting interconnection between the theory of approximation by univariate functions and problems of equilibrium construction in economics (see [154]).", "Linear combinations of ridge functions with fixed directions allow a natural generalization to functions of the form $g(\\alpha _{1}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}(x_{d}))$ , where $\\alpha _{i}(x_{i})$ , $i=\\overline{1,d},$ are real univariate functions.", "Such a generalization has a strong association with linear superpositions.", "A linear superposition is a function expressed as the sum $\\sum \\limits _{i=1}^{r}g_{i}(h_{i}(x)), \\; x \\in X,$ where $X$ is any set (in particular, a subset of $\\mathbb {R}^{d}$ ), $h_{i}:X\\rightarrow {{\\mathbb {R}}},~i=1,...,r,$ are arbitrarily fixed functions, and $g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r.$ Note that here we deal with more complicated composition than the composition of a univariate function with the inner product.", "A starting point in the study of linear superpositions was the well known superposition theorem of Kolmogorov [97] (see also the paper on Kolmogorov's works by Tikhomirov [157]).", "This theorem states that for the unit cube $\\mathbb {I}^{d},~\\mathbb {I}=[0,1],~d\\ge 2,$ there exist $2d+1$ functions $\\lbrace s_{q}\\rbrace _{q=1}^{2d+1}\\subset C(\\mathbb {I}^{d})$ of the form $s_{q}(x_{1},...,x_{d})=\\sum _{p=1}^{d}\\varphi _{pq}(x_{p}),~\\varphi _{pq}\\in C(\\mathbb {I}),~p=1,...,d,~q=1,...,2d+1$ such that each function $f\\in C(\\mathbb {I}^{d})$ admits the representation $f(x)=\\sum _{q=1}^{2d+1}g_{q}(s_{q}(x)),~x=(x_{1},...,x_{d})\\in \\mathbb {I}^{d},~g_{q}\\in C({{\\mathbb {R)}}}.$ Thus, any continuous function on the unit cube can be represented as a linear superposition with the fixed inner functions $s_{1},...,s_{2d+1}$ .", "In literature, these functions are called universal functions or the Kolmogorov functions.", "Note that all the functions $g_{q}(s_{q}(x))$ in the Kolmogorov superposition formula are generalized ridge functions, since each $s_{q}$ is a sum of univariate functions.", "In these notes, we consider some problems of approximation and/or representation of multivariate functions by linear combinations of ridge functions, generalized ridge functions and feedforward neural networks.", "The notes consist of five chapters.", "Chapter 1 is devoted to the approximation from some sets of ridge functions with arbitrarily fixed directions in $C$ and $L_{2}$ metrics.", "First, we study problems of representation of multivariate functions by linear combinations of ridge functions.", "Then, in case of two fixed directions and under suitable conditions, we give complete solutions to three basic problems of uniform approximation, namely, problems on existence, characterization, and construction of a best approximation.", "We also study problems of well approximation (approximation with arbitrary accuracy) and representation of continuous multivariate functions by sums of two continuous ridge functions.", "The reader will see the main difficulties and remained open problems in the uniform approximation by sums of more than two ridge functions.", "For $L_{2}$ approximation, a number of summands does not play such an essential role as it plays in the uniform approximation.", "In this case, it is known that a best approximation always exists and unique.", "For some special domains in $\\mathbb {R}^{d}$ , we characterize and then construct the best approximation.", "We also give an explicit formula for the approximation error.", "Chapter 2 explores the following open problem raised in Buhmann and Pinkus [18], and Pinkus [137].", "Assume we are given a function $f(\\mathbf {x})=f(x_{1},...,x_{n})$ of the form $f(\\mathbf {x})=\\sum _{i=1}^{k}f_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),$ where the $\\mathbf {a}^{i},$ $i=1,...,k,$ are pairwise linearly independent vectors (directions) in $\\mathbb {R}^{d}$ , $f_{i}$ are arbitrarily behaved univariate functions and $\\mathbf {a}^{i}\\cdot \\mathbf {x}$ are standard inner products.", "Assume, in addition, that $f$ is of a certain smoothness class, that is, $f\\in C^{s}(\\mathbb {R}^{d})$ , where $s\\ge 0$ (with the convention that $C^{0}(\\mathbb {R}^{d})=C(\\mathbb {R}^{d})$ ).", "Is it true that there will always exist $g_{i}\\in C^{s}(\\mathbb {R})$ such that $f(\\mathbf {x})=\\sum _{i=1}^{k}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})\\text{ ?", "}$ In this chapter, we solve this problem up to some multivariate polynomial.", "We find various conditions on the directions $\\mathbf {a}^{i}$ allowing to express this polynomial as a sum of smooth ridge functions with these directions.", "We also consider the question of constructing $g_{i}$ using the information about the known functions $f_{i}$ .", "Chapter 3 is devoted to the simplest type of ridge functions – univariate functions.", "Note that a ridge function depends only on one variable if its direction coincides with the coordinate direction.", "Thus, in case of coincidence of all given directions with the coordinate directions, the problem of ridge function approximation turns into the problem of approximation of multivariate functions by sums of univariate functions.", "In this chapter, we first consider the approximation of a bivariate function $f(x,y)$ by sums $\\varphi (x)+\\psi (y)$ on a rectangular domain $R$ .", "We construct special classes of continuous functions depending on a numerical parameter and characterize each class in terms of the approximation error calculation formulas.", "This parameter will show which points of $R$ the calculation formula involves.", "We will also construct a best approximating sum $\\varphi _{0}(x)+\\psi _{0}(y)$ to a function from constructed classes.", "Then we develop a method for obtaining explicit formulas for the error of approximation of bivariate functions, defined on a union of rectangles, by sums of univariate functions.", "It should be remarked that formulas of such type were known only for functions defined on a rectangle with sides parallel to the coordinate axes.", "Our method, based on a maximization process over certain objects, called “closed bolts\", allows the consideration of functions defined on hexagons, octagons and stairlike polygons with sides parallel to the coordinate axes.", "At the end of this chapter we discuss one important result from Golomb's paper [48].", "This paper, published in 1959, made a start of a systematic study of approximation of multivariate functions by various compositions, including sums of univariate functions.", "In [48], along with many other results, Golomb obtained a duality formula for the error of approximation to a multivariate function from the set of sums of univariate functions.", "Unfortunately, his proof had a gap, which was 24 years later pointed out by Marshall and O'Farrell [123].", "But the question if Golomb's formula was correct, remained unsolved.", "In Chapter 3, we show that Golomb's formula is correct, and moreover it holds in a stronger form.", "Chapter 4 tells us about some problems concerning generalized ridge functions $g(\\alpha _{1}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}(x_{d}))$ and linear superpositions.", "We consider the problem of representation of general functions by linear superpositions.", "We show that if some representation by linear superpositions, in particular by linear combinations of generalized ridge functions, holds for continuous functions, then it holds for all functions.", "This leads us to extensions of many superpositions theorems (such as the well-known Kolmogorov superposition theorem, Ostrand's superposition theorem, etc.)", "from continuous to arbitrarily behaved multivariate functions.", "Concerning generalized ridge functions, we see that every multivariate function can be written as a generalized ridge function or as a sum of finitely many such functions.", "We also study the uniqueness of representation of functions by linear superpositions.", "Chapter 5 is about neural network approximation.", "The analysis in this chapter is based on properties of ordinary and generalized ridge functions.", "We consider a single and two hidden layer feedforward neural network models with a restricted set of weights.", "Such network models are important from the point of view of practical applications.", "We study approximation properties of single hidden layer neural networks with weights varying on a finite set of directions and straight lines.", "We give several necessary and sufficient conditions for well approximation by such networks.", "For a set of weights consisting of two directions (and two straight lines), we show that there is a geometrically explicit solution to the problem.", "Regarding two hidden layer feedforward neural networks, we prove that two hidden layer neural networks with $d$ inputs, $d$ neurons in the first hidden layer, $2d+2$ neurons in the second hidden layer and with a specifically constructed sigmoidal, infinitely differentiable and almost monotone activation function can approximate any continuous multivariate function with arbitrary precision.", "We show that for this approximation only a finite number of fixed weights (precisely, $d$ fixed weights) suffice.", "There are topics related to ridge functions that are not presented here.", "The glaring omission is that of interpolation at points and on straight lines by ridge functions.", "We also do not address, for example, questions of linear independence and spanning by linear combinations of ridge monomials in the spaces of homogeneous and algebraic polynomials of a fixed degree, integral representations of functions where the kernel is a ridge function, approximation algorithms for finding best approximations from spaces of linear combinations of ridge functions.", "These and similar topics may be found in the monograph by Pinkus [137].", "The reader may also consult the survey articles [75], [102], [134].", "In this chapter, we consider approximation-theoretic problems arising in ridge function approximation.", "First we briefly review some results on approximation by sums of ridge functions with both fixed and variable directions.", "Then we analyze the problem of representability of an arbitrary multivariate function by linear combinations of ridge functions with fixed directions.", "In the special case of two fixed directions, we characterize a best uniform approximation from the set of sums of ridge functions with these directions.", "For a class of bivariate functions we use this result to construct explicitly a best approximation.", "Questions on existence of a best approximation are also studied.", "We also study problems of well approximation (approximation with arbitrary accuracy) and representation of continuous multivariate functions by sums of two continuous ridge functions.", "The reader will see the main difficulties and remained open problems in the uniform approximation by sums of more than two ridge functions.", "For $L_{2}$ approximation, a number of summands does not play such an essential role as it plays in the uniform approximation.", "In this case, it is known that a best approximation always exists and unique.", "For some special domains in $\\mathbb {R}^{d}$ , we characterize and then construct the best approximation.", "We also give an explicit formula for the approximation error.", "In this section we briefly review some results on approximation properties of the sets $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ and $\\mathcal {R}_{r}$ .", "These results are presented without proofs but with discussions and complete references.", "We hope this section will whet the reader's appetite for the rest of these notes, where a more comprehensive study of concrete mathematical problems is provided.", "It is clear that well approximation of a multivariate function $f:X\\rightarrow \\mathbb {R}$ from some normed space by using elements of the set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ is not always possible.", "The value of the approximation error depends not only on the approximated function $f$ but also on geometrical structure of the given set $X$ .", "This poses challenging research problems on computing the error of approximation and constructing best approximations from $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ .", "Serious difficulties arise when one attempts to solve these problems in continuous function spaces endowed with the uniform norm.", "For example, let us consider the algorithm for finding best approximations, called the Diliberto-Straus algorithm (see [111]).", "The essence of this algorithm is as follows.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ and $A_{i}$ be a best approximation operator from the space of continuous functions $C(X)$ to the subspace of ridge functions $G_{i}=\\lbrace g_{i}\\left(\\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) :~g_{i}\\in C(\\mathbb {R)},~\\mathbf {x}\\in X\\rbrace $ , $i=1,...,r.$ That is, for each function $f$ $\\in C(X)$ , the function $A_{i}f$ is a best approximation to $f$ from $G_{i}.$ Set $Tf=(I-A_{r})(I-A_{r-1})\\cdot \\cdot \\cdot (I-A_{1})f,$ where $I$ is the identity operator.", "It is clear that $Tf=f-g_{1}-g_{2}-\\cdot \\cdot \\cdot -g_{r},$ where $g_{k}$ is a best approximation from $G_{k}$ to the function $f-g_{1}-g_{2}-\\cdot \\cdot \\cdot -g_{k-1}$ , $k=1,...,r.$ Consider powers of the operator $T$ : $T^{2},T^{3}$ and so on.", "Is the sequence $\\lbrace T^{n}f\\rbrace _{n=1}^{\\infty }$ convergent?", "In case of an affirmative answer, which function is the limit of $T^{n}f,$ as $n\\rightarrow \\infty $ ?", "One may expect that the sequence $\\lbrace T^{n}f\\rbrace _{n=1}^{\\infty }$ converges to $f-g^{\\ast },$ where $g^{\\ast }$ is a best approximation from $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ to $f$ .", "This conjecture was first stated by Diliberto and Straus [36] in 1951 for the uniform approximation of a multivariate function, defined on the unit cube, by sums of univariate functions (that is, sums of ridge functions with the coordinate directions).", "But later it was shown by Aumann [9] that the sequence generated by this algorithm may not converge if $r>2$ .", "For $r=2$ and certain convex compact sets $X$ , the sequence $\\lbrace \\Vert T^{n}f\\Vert \\rbrace _{n=1}^{\\infty }$ converges to the approximation error $\\Vert f-g_{0}\\Vert $ , where $g_{0}$ is a best approximation from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right)$ (see [73], [137]).", "However, it is not yet clear whether $\\Vert T^{n}f-(f-g_{0})\\Vert $ converges to zero as $n\\rightarrow \\infty $ .", "In the case $r>2$ no efficient algorithm is known for finding a best uniform approximation from $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ .", "Note that in the $L_{2}$ metric, the Diliberto-Straus algorithm converges as desired for an arbitrary number of distinct directions.", "This also holds in the $L_{p}$ space setting, provided that $p>1$ and $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ is closed (see [134]).", "But in the $L_{1}$ space setting, the alternating algorithm does not work even in the case of two directions (see [137]).", "One of the basic problems concerning the approximation by sums of ridge functions with fixed directions is the problem of verifying if a given function $f$ belongs to the space $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ .", "This problem has a simple solution if the space dimension $d=2$ and a given function $f(x,y)$ has partial derivatives up to $r$ -th order.", "For the representation of $f(x,y)$ in the form $f(x,y)=\\sum _{i=1}^{r}g_{i}(a_{i}x+b_{i}y),$ it is necessary and sufficient that $\\prod \\limits _{i=1}^{r}\\left( b_{i}\\frac{\\partial }{\\partial x}-a_{i}\\frac{\\partial }{\\partial y}\\right) f=0.", "(1.1)$ This recipe is also valid for continuous bivariate functions provided that the derivatives are understood in the sense of distributions.", "Unfortunately such a simple characterization does not carry over to the case of more than two variables.", "Below we provide two results concerning the general case of arbitrarily many variables.", "Proposition 1.1 (Diaconis, Shahshahani [35]).", "Let $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ be pairwise linearly independent vectors in $\\mathbb {R}^{d}.", "$ Let for $i=1,2,...,r$ , $H^{i}$ denote the hyperplane $\\lbrace \\mathbf {c}\\in \\mathbb {R}^{d}$ : $\\mathbf {c\\cdot a}^{i}=0\\rbrace .$ Then a function $f\\in C^{r}(\\mathbb {R}^{d})$ can be represented in the form $f(\\mathbf {x})=\\sum \\limits _{i=1}^{r}g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) +P(\\mathbf {x}),$ where $P(\\mathbf {x})$ is a polynomial of degree not more than $r$ , if and only if $\\prod \\limits _{i=1}^{r}\\sum _{s=1}^{d}c_{s}^{i}\\frac{\\partial f}{\\partial x_{s}}=0,$ for all vectors $\\mathbf {c}^{i}=(c_{1}^{i},c_{2}^{i},...,c_{d}^{i})\\in H^{i}, $ $i=1,2,...,r.$ There are examples showing that one cannot simply dispense with the polynomial $P(\\mathbf {x})$ in the above proposition (see [35]).", "In fact, a polynomial term appears in the sufficiency part of the proof of this proposition.", "Lin and Pinkus [112] obtained more general result on the representation by sums of ridge functions with fixed directions.", "We need some notation to present their result.", "Each polynomial $p(x_{1},...,x_{d})$ generates the differential operator $p(\\frac{\\partial }{\\partial x_{1}},...,\\frac{\\partial }{\\partial x_{d}}).$ Let $P(\\mathbf {a}^{1},...,\\mathbf {a}^{r})$ denote the set of polynomials which vanish on all the lines $\\lbrace \\lambda \\mathbf {a}^{i},\\lambda \\in \\mathbb {R}\\rbrace ,$ $i=1,...,r.$ Obviously, this is an ideal in the ring of all polynomials.", "Let $Q$ be the set of polynomials $q=q(x_{1},...,x_{d})$ such that $p(\\frac{\\partial }{\\partial x_{1}},...,\\frac{\\partial }{\\partial x_{d}})q=0$ , for all $p(x_{1},...,x_{d})\\in P(\\mathbf {a}^{1},...,\\mathbf {a}^{r}).$ Proposition 1.2 (Lin, Pinkus [112]).", "Let $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ be pairwise linearly independent vectors in $\\mathbb {R}^{d}.$ A function $f\\in C(\\mathbb {R}^{d})$ can be expressed in the form $f(\\mathbf {x})=\\sum \\limits _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x)},$ if and only if $f$ belongs to the closure of the linear span of $Q.$ In [136], A.Pinkus considered the problems of smoothness and uniqueness in ridge function representation.", "For a given function $f$ $\\in $ $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ , he posed and answered the following questions.", "If $f$ belongs to some smoothness class, what can we say about the smoothness of the functions $g_{i}$ ?", "How many different ways can we write $f$ as a linear combination of ridge functions?", "These and similar problems will be extensively discussed in Chapter 2.", "The above problem of representation of fixed functions by sums of ridge functions gives rise to the problem of representation of some classes of functions by such sums.", "For example, one may consider the following problem.", "Let $X$ be a subset of the $d$ -dimensional Euclidean space.", "Let $C(X),$ $B(X),$ $T(X)$ denote the set of continuous, bounded and all real functions defined on $X$ , respectively.", "In the first case, we additionally suppose that $X$ is a compact set.", "Let $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ and $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ denote the subspaces of $\\mathcal {R}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ comprising only sums with continuous and bounded terms $g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ , $i=1,...,r$ , respectively.", "The following questions naturally arise: For which sets $X$ , $(1)$ $C(X)=\\mathcal {R}_{c}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ ?", "$(2)$ $B(X)=\\mathcal {R}_{b}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ ?", "$(3)$ $T(X)=\\mathcal {R}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ ?", "The first two questions in a more general setting were answered in Sternfeld [149], [152].", "The third question will be answered in the next section.", "Let us briefly discuss some results of Sternfeld concerning ridge function representation.", "These results have been mostly overlooked in the corresponding ridge function literature, as they have to do with more general superpositions of functions and do not directly mention ridge functions.", "Assume we are given directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\in \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ and a set $X\\subseteq \\mathbb {R}^{d}.$ Following Sternfeld, we say that a family $F=\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ uniformly separates points of $X$ if there exists a number $0<\\lambda \\le 1$ such that for each pair $\\lbrace \\mathbf {x}_{j}\\rbrace _{j=1}^{m}$ , $\\lbrace \\mathbf {z}_{j}\\rbrace _{j=1}^{m}$ of disjoint finite sequences in $X$ , there exists some direction $\\mathbf {a}^{k}\\in F$ so that if from the two sequences $\\lbrace \\mathbf {a}^{k}\\cdot \\mathbf {x}_{j}\\rbrace _{j=1}^{m}$ and $\\lbrace \\mathbf {a}^{k}\\cdot \\mathbf {z}_{j}\\rbrace _{j=1}^{m}$ we remove a maximal number of pairs of points $\\mathbf {a}^{k}\\cdot \\mathbf {x}_{j_{1}}$ and $\\mathbf {a}^{k}\\cdot \\mathbf {z}_{j_{2}}$ with $\\mathbf {a}^{k}\\cdot \\mathbf {x}_{j_{1}}=\\mathbf {a}^{k}\\cdot \\mathbf {z}_{j_{2}},$ then there remains at least $\\lambda m$ points in each sequence (or, equivalently, at most $(1-\\lambda )m$ pairs can be removed).", "Sternfeld [149], in particular, proved that a family of directions $F=\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ uniformly separates points of $X$ if and only if $\\mathcal {R}_{b}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =B(X)$ .", "In [149], he also obtained a practically convenient sufficient condition for the equality $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =B(X).$ To describe his condition, define the set functions $\\tau _{i}(Z)=\\lbrace \\mathbf {x}\\in Z:~|p_{i}^{-1}(p_{i}(\\mathbf {x}))\\bigcap Z|\\ge 2\\rbrace ,$ where $Z\\subset X,~p_{i}(\\mathbf {x})=\\mathbf {a}^{i}\\cdot \\mathbf {x}$ , $i=1,\\ldots ,r,$ and $|Y|$ denotes the cardinality of a set $Y$ .", "Define $\\tau (Z)$ to be $\\bigcap _{i=1}^{k}\\tau _{i}(Z)$ and define $\\tau ^{2}(Z)=\\tau (\\tau (Z))$ , $\\tau ^{3}(Z)=\\tau (\\tau ^{2}(Z))$ and so on inductively.", "Proposition 1.3 (Sternfeld [149]).", "If $\\tau ^{n}(X)=\\emptyset $ for some $n$ , then $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =B(X)$ .", "If $X$ is a compact subset of $\\mathbb {R}^{d}$ , and $\\tau ^{n}(X)=\\emptyset $ for some $n$ , then $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =C(X)$ .", "If $r=2$ , the sufficient condition “$\\tau ^{n}(X)=\\emptyset $ for some $n$ \" turns out to be also necessary.", "In this case, the equality $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right)=B(X)$ is equivalent to the equality $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) =C(X)$ .", "In another work [152], Sternfeld obtained a measure-theoretic necessary and sufficient condition for the equality $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)=C(X)$ .", "Let $p_{i}(\\mathbf {x})=\\mathbf {a}^{i}\\cdot \\mathbf {x}$ , $i=1,\\ldots ,r$ , $X$ be a compact set in $\\mathbb {R}^{d}$ and $M(X)$ be a class of measures defined on some field of subsets of $X$ .", "Following Sternfeld, we say that a family $F=\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ uniformly separates measures of the class $M(X)$ if there exists a number $0<\\lambda \\le 1$ such that for each measure $\\mu $ in $M(X)$ the equality $\\left\\Vert \\mu \\circ p_{k}^{-1}\\right\\Vert \\ge \\lambda \\left\\Vert \\mu \\right\\Vert $ holds for some direction $\\mathbf {a}^{k}\\in F$ .", "Sternfeld [150], [152], in particular, proved that the equality $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =C(X)$ holds if and only if the family of directions $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ uniformly separates measures of the class $C(X)^{\\ast }$ (that is, the class of regular Borel measures).", "In addition, he proved that $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =B(X)$ if and only if the family of directions $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ uniformly separates measures of the class $l_{1}(X)$ (that is, the class of finite measures defined on countable subsets of $X$ ).", "Since $l_{1}(X)\\subset C(X)^{\\ast },$ the first equality $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =C(X)$ implies the second equality $\\mathcal {R}_{b}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) =B(X).$ The inverse is not true (see [152]).", "We emphasize again that the above results of Sternfeld were obtained for more general functions, than linear combinations of ridge functions, namely for functions of the form $\\sum _{i=1}^{r}g_{i}(h_{i}(x))$ , where $h_{i}$ arbitrarily fixed functions (bounded or continuous) defined on $X.$ Such functions will be discussed in Chapter 4.", "Obviously, the set $\\mathcal {R}_{c}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ is not dense in $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compact subsets of $\\mathbb {R}^{d}.$ Density here does not hold because the number of considered directions is finite.", "If consider all the possible directions, then the set $\\mathcal {R}=span\\lbrace g(\\mathbf {a}\\cdot \\mathbf {x)}:~g\\in C(\\mathbb {R)},~\\mathbf {a}\\in \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace \\rbrace $ will certainly be dense in the space $C(\\mathbb {R}^{d})$ in the above mentioned topology.", "In order to be sure, it is enough to consider only the functions $e^{\\mathbf {a}\\cdot \\mathbf {x}}\\in \\mathcal {R}$ , the linear span of which is dense in $C(\\mathbb {R}^{d})$ by the Stone-Weierstrass theorem.", "In fact, for density it is not necessary to comprise all directions.", "The following theorem shows how many directions in totality satisfy the density requirements.", "Proposition 1.4 (Vostrecov and Kreines [160], Lin and Pinkus [112]).", "For density of the set $\\mathcal {R(A)}=span\\lbrace g(\\mathbf {a}\\cdot \\mathbf {x)}:~g\\in C(\\mathbb {R)},~\\mathbf {a}\\in \\mathcal {A}\\subset \\mathbb {R}^{d}\\rbrace $ in $C(\\mathbb {R}^{d})$ (in the topology of uniform convergence on compact sets) it is necessary and sufficient that the only homogeneous polynomial which vanishes identically on $\\mathcal {A}$ is the zero polynomial.", "Since in the definition of $\\mathcal {R(A)}$ we vary over all univariate functions$~g,$ allowing one direction $\\mathbf {a}$ is equivalent to allowing all directions $k\\mathbf {a}$ for every real $k$ .", "Thus it is sufficient to consider only the set $\\mathcal {A}$ of directions normalized to the unit sphere $S^{n-1}.$ For example, if $\\mathcal {A}$ is a subset of the sphere $S^{n-1},$ which contains an interior point (interior point with respect to the induced topology on $S^{n-1}$ ), then $\\mathcal {R(A)}$ is dense in the space $C(\\mathbb {R}^{d}).$ The proof of Proposition 1.4 highlights an important fact that the set $\\mathcal {R(A)}$ is dense in $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compact subsets if and only if $\\mathcal {R(A)}$ contains all the polynomials (see [112]).", "Representability of polynomials by sums of ridge functions is a building block for many results.", "In many works (see, e.g., [135]), the following fact is fundamental: Every multivariate polynomial $h(\\mathbf {x})=h(x_{1},...,x_{d})$ of degree $k $ can be represented in the form $h(\\mathbf {x})=\\sum \\limits _{i=1}^{l}p_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x),}$ where $p_{i}$ is a univariate polynomial, $\\mathbf {a}^{i}\\in \\mathbb {R}^{d}$ , and $l=$ $\\binom{d-1+k}{k}$ .", "For example, for the representation of a bivariate polynomial of degree $k$ , it is needed $k+1$ univariate polynomials and $k+1$ directions (see [113]).", "The proof of this fact is organized so that the directions $\\mathbf {a}^{i}$ , $i=1,...,k+1$ , are chosen once for all multivariate polynomials of $k$ -th degree.", "At one of the seminars in the Technion – Israel Institute of Technology in 2007, A. Pinkus posed two problems: 1) Can every multivariate polynomial of degree $k$ be represented by less than $l$ ridge functions?", "2) How large is the set of polynomials represented by $l-1,$ $l-2,...$ ridge functions?", "Note that for bivariate polynomials the 1-st problem is solved positively, that is, the number $l=k+1$ can be reduced.", "Indeed, for a bivariate polynomial $P(x,y)$ of $k$ -th degree, there exist many combinations of real numbers $c_{0},...,c_{k}$ such that $\\sum _{i=0}^{k}c_{i}\\frac{\\partial ^{k}}{\\partial x^{i}\\partial y^{k-i}}P(x,y)=0.$ Further the numbers $c_{i}$ , $i=0,...,k$ , can be selected to enjoy the property that the polynomial $\\sum _{i=0}^{k}c_{i}t^{i}$ has distinct real zeros.", "Then it is not difficult to verify that the differential operator $\\sum _{i=0}^{k}c_{i}\\frac{\\partial ^{k}}{\\partial x^{i}\\partial y^{k-i}}$ can be written in the form $\\prod \\limits _{i=1}^{k}\\left( b_{i}\\frac{\\partial }{\\partial x}-a_{i}\\frac{\\partial }{\\partial y}\\right) ,$ for some pairwise linearly independent vectors $(a_{i},b_{i})$ , $i=1,...,k$ .", "Now from the above criterion (1.1) we obtain that the polynomial $P(x,y)$ can be represented as a sum of $k$ ridge functions.", "Note that the problem of representation of a multivariate algebraic polynomial $P(\\mathbf {x})$ in the form $\\sum _{i=0}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})$ with minimal $r$ was extensively studied in the monograph by Pinkus [137].", "In connection with the 2-nd problem of Pinkus, V. Maiorov [116] studied certain geometrical properties of the manifold $\\mathcal {R}_{r}$ .", "Namely, he estimated the $\\varepsilon $ -entropy numbers in terms of smaller $\\varepsilon $ -covering numbers of the compact class formed by the intersection of the class $\\mathcal {R}_{r}$ with the unit ball in the space of polynomials of degree at most $s$ on $\\mathbb {R}^{d}$ .", "Let $E$ be a Banach space and let for $x\\in E$ and $\\delta >0,$ $S(x,\\delta )$ denote the ball of radius $\\delta $ centered at the point $x$ .", "For any positive number $\\varepsilon $ , the $\\varepsilon $ -covering number of a set $F$ in the space $E$ represents the quantity $L_{\\varepsilon }(F,E)=\\min \\left\\lbrace N:~\\exists x_{1},...,x_{N}\\in F\\text{such that }F\\subset \\bigcup _{i=1}^{N}S(x_{i},\\varepsilon )\\text{ }\\right\\rbrace .$ The $\\varepsilon $ -entropy of $F$ is defined as the number $H_{\\varepsilon }(F,E)\\overset{def}{=}$ $\\log _{2}L_{\\varepsilon }(F,E)$ .", "The notion of $\\varepsilon $ -entropy has been devised by A.N.Kolmogorov (see [96], [98]) to classify compact metric sets according to their massivity.", "In order to formulate Maiorov's result, let $\\mathcal {P}_{s}^{d}$ be the space of all polynomials of degree at most $s$ on $\\mathbb {R}^{d}$ , $L_{q}=L_{q}(I)$ , $1\\le q\\le \\infty $ , be the space of $q$ -integrable functions on the unit cube $I=[0,1]^{d}$ with the norm $\\left\\Vert f\\right\\Vert _{q}=\\left( \\int _{I}\\left|f(x)\\right|^{q}dx\\right)^{1/q}$ , $BL_{q}$ be the unit ball in the space $L_{q},$ and $B_{q}\\mathcal {P}_{s}^{d}=$ $BL_{q}\\cap $ $\\mathcal {P}_{s}^{d}$ be the unit ball in the space $\\mathcal {P}_{s}^{d}$ equipped with the $L_{q}$ metric.", "Proposition 1.5 (Maiorov [116]).", "Let $r,s\\in \\mathbb {N}$ , $1\\le q\\le \\infty $ , $0 < \\varepsilon < 1$ .", "The $\\varepsilon $ -entropy of the class $B_{q}\\mathcal {P}_{s}^{d}\\cap \\mathcal {R}_{r}$ in the space $L_{q}$ satisfies the inequalities 1) $c_{1}rs\\le \\frac{H_{\\varepsilon }(B_{q}\\mathcal {P}_{s}^{d}\\cap \\mathcal {R}_{r},L_{q})}{\\log _{2}\\frac{1}{\\varepsilon }}\\le c_{2}rs\\log _{2}\\frac{2es^{d-1}}{r},$ for $r\\le s^{d-1}.$ 2) $c_{1}^{^{\\prime }}s^{d}\\le \\frac{H_{\\varepsilon }(B_{q}\\mathcal {P}_{s}^{d}\\cap \\mathcal {R}_{r},L_{q})}{\\log _{2}\\frac{1}{\\varepsilon }}\\le c_{2}^{^{\\prime }}s^{d},$ for $r>s^{d-1}.$ In these inequalities $c_{1},c_{2},c_{1}^{^{\\prime }},c_{2}^{^{\\prime }}$ are constants depending only on $d$ .", "Let us consider $\\mathcal {R}_{r}$ as a subspace of some normed linear space $X$ endowed with the norm $\\left\\Vert \\cdot \\right\\Vert _{X}.$ The error of approximation of a given function $f\\in X$ by functions $g\\in \\mathcal {R}_{r} $ is defined as follows $E(f,\\mathcal {R}_{r},X)\\overset{def}{=}\\underset{g\\in \\mathcal {R}_{r}}{\\inf }\\left\\Vert f-g\\right\\Vert _{X}.$ Let $B^{d}$ denote the unit ball in the space $\\mathbb {R}^{d}.$ Besides, let $\\mathbb {Z}_{+}^{d}$ denote the lattice of nonnegative multi-integers in $\\mathbb {R}^{d}.$ For $k=(k_{1},...,k_{d})\\in \\mathbb {Z}_{+}^{d},$ set $\\left|k\\right|=k_{1}+\\cdot \\cdot \\cdot +k_{d}$ , $\\mathbf {x}^{\\mathbf {k}}=x_{1}^{k_{1}}\\cdot \\cdot \\cdot x_{d}^{k_{d}}$ and $D^{\\mathbf {k}}=\\frac{\\partial ^{\\left|k\\right|}}{\\partial ^{k_{1}}x_{1}\\cdot \\cdot \\cdot \\partial ^{k_{d}}x_{d}}$ The Sobolev space $W_{p}^{m}(B^{d})$ is the space of functions defined on $B^{d}$ with the norm $\\left\\Vert f\\right\\Vert _{m,p}=\\left\\lbrace \\begin{array}{c}\\left( \\sum _{0\\le \\left|\\mathbf {k}\\right|\\le m}\\left\\Vert D^{\\mathbf {k}}f\\right\\Vert _{p}^{p}\\right) ^{1/p},\\text{ if }1\\le p<\\infty \\\\\\max _{0\\le \\left|\\mathbf {k}\\right|\\le m}\\left\\Vert D^{\\mathbf {k}}f\\right\\Vert _{\\infty },\\text{ if }p=\\infty .\\end{array}\\right.$ Here $\\left\\Vert h(\\mathbf {x})\\right\\Vert _{p}=\\left\\lbrace \\begin{array}{c}\\left( \\int _{B^{n}}\\left|h(\\mathbf {x})\\right|^{p}d\\mathbf {x}\\right) ^{1/p},\\text{ if }1\\le p<\\infty \\\\ess\\sup _{\\mathbf {x}\\in B^{d}}\\left|h(\\mathbf {x})\\right|,\\text{ if }p=\\infty .\\end{array}\\right.$ Let $S_{p}^{m}(B^{d})$ be the unit ball in $W_{p}^{m}(B^{d})$ : $S_{p}^{m}(B^{d})=\\lbrace f\\in W_{p}^{m}(B^{d}):\\left\\Vert f\\right\\Vert _{m,p}\\le 1~\\rbrace .$ In 1999, Maiorov [115] proved the following result Proposition 1.6 (Maiorov [115]).", "Assume $m\\ge 1$ and $d\\ge 2$ .", "Then for each $r\\in \\mathbb {N}$ there exists a function $f\\in S_{2}^{m}(B^{d})$ such that $E(f,\\mathcal {R}_{r},L_{2})\\ge Cr^{-m/(d-1)},(1.2)$ where $C$ is a constant independent of $f$ and $r.$ For $d=2,$ this inequality was proved by Oskolkov [131].", "In [115], Maiorov also proved that for each function $f\\in S_{2}^{m}(B^{d})$ $E(f,\\mathcal {R}_{r},L_{2})\\le Cr^{-m/(d-1)}.", "(1.3)$ Thus he established the following order for the error of approximation to functions in $S_{2}^{m}(B^{d})$ from the class $\\mathcal {R}_{r}$ : $E(S_{2}^{m}(B^{d}),\\mathcal {R}_{r},L_{2})\\overset{def}{=}\\sup _{f\\in S_{2}^{m}(B^{d})}E(f,\\mathcal {R}_{r},L_{2})\\asymp r^{-m/(d-1)}.$ Pinkus [135] revealed that the upper bound (1.3) is also valid in the $L_{p}$ metric ($1\\le p\\le \\infty $ ).", "In other words, for every function $f\\in S_{p}^{m}(B^{d})$ $E(f,\\mathcal {R}_{r},L_{p})\\le Cr^{-m/(d-1)}.$ These inequalities were successfully applied to some problems of approximation of multivariate functions by neural networks with a single hidden layer.", "Recall that such networks are given by the formula $\\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i}).$ By $\\mathcal {M}_{r}(\\sigma )$ let us denote the set of all single hidden layer networks with the activation function $\\sigma $ .", "That is, $\\mathcal {M}_{r}(\\sigma )=\\left\\lbrace \\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i}):~c_{i},\\theta _{i}\\in \\mathbb {R},~\\mathbf {w}^{i}\\in \\mathbb {R}^{d}\\right\\rbrace .$ The above results on ridge approximation from $\\mathcal {R}_{r}$ enable us to estimate the rate with which the approximation error $E(f,\\mathcal {M}_{r}(\\sigma ),L_{2})$ tends to zero.", "First note that $\\mathcal {M}_{r}(\\sigma )\\subset \\mathcal {R}_{r},$ since each function of the form $\\sigma (\\mathbf {w\\cdot x}-\\theta )$ is a ridge function with the direction $\\mathbf {w}$ .", "Thus the lower bound (1.2) holds also for the set $\\mathcal {M}_{r}(\\sigma )$ : there exists a function $f\\in S_{2}^{m}(B^{d})$ for which $E(f,\\mathcal {M}_{r}(\\sigma ),L_{2})\\ge Cr^{-m/(d-1)}.$ It remains to see whether the upper bound (1.3) is valid for $\\mathcal {M}_{r}(\\sigma )$ .", "Clearly, it cannot be valid if $\\sigma $ is an arbitrary continuous function.", "Here we are dealing with the question if there exists a function $\\sigma ^{\\ast }\\in C(\\mathbb {R})$ , for which $E(f,\\mathcal {M}_{r}(\\sigma ^{\\ast }),L_{2})\\le Cr^{-m/(d-1)}.$ This question is answered affirmatively by the following result.", "Proposition 1.7 (Maiorov, Pinkus [119]).", "There exists a function $\\sigma ^{\\ast }\\in C(\\mathbb {R})$ with the following properties 1) $\\sigma ^{\\ast }$ is infinitely differentiable and strictly increasing; 2) $\\lim _{t\\rightarrow \\infty }\\sigma ^{\\ast } (t)=1$ and $\\lim _{t\\rightarrow -\\infty }\\sigma ^{\\ast } (t)=0;$ 3) for every $g\\in \\mathcal {R}_{r}$ and $\\varepsilon >0$ there exist $c _{i},\\theta _{i}\\in \\mathbb {R}$ and $\\mathbf {w}^{i}\\in \\mathbb {R}^{d}$ satisfying $\\sup _{\\mathbf {x}\\in B^{d}}\\left|g(\\mathbf {x})-\\sum _{i=1}^{r+d+1}c_{i}\\sigma ^{\\ast } (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i})\\right|<\\varepsilon .$ Temlyakov [156] considered the approximation from some certain subclass of $\\mathcal {R}_{r}$ in $L_{2}$ metric.", "More precisely, he considered the approximation of a function $f\\in L_{2}(D),$ where $D$ is the unit disk in $\\mathbb {R}^{2}$ , by functions $\\sum \\limits _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x)}\\in \\mathcal {R}_{r}\\cap L_{2}(D)$ , which satisfy the additional condition $\\left\\Vert g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x)}\\right\\Vert _{2}\\le B\\left\\Vert f\\right\\Vert _{2},$ $i=1,...,r$ ($B$ is a given positive number).", "Let $\\sigma _{r}^{B}(f)$ be the error of this approximation.", "For this approximation error, the author of [156] obtained upper and lower bounds.", "Let, for $\\alpha >0,$ $H^{\\alpha }(D)$ denote the set of all functions $f\\in L_{2}(D)$ , which can be represented in the form $f=\\sum _{n=1}^{\\infty }P_{n},$ where $P_{n}$ are bivariate algebraic polynomials of total degree $2^{n}-1$ satisfying the inequalities $\\left\\Vert P_{n}\\right\\Vert _{2}\\le 2^{-\\alpha n},\\text{ }n=1,2,...$ Proposition 1.8 (Temlyakov [156]).", "1) For every $f\\in H^{\\alpha }(D) $ , we have $\\sigma _{r}^{1}(f)\\le C(\\alpha )r^{-\\alpha }.$ 2) For any given $\\alpha >0$ , $B>0$ , $r>1$ , there exists a function $f\\in H^{\\alpha }(D)$ such that $\\sigma _{r}^{B}(f)\\ge C(\\alpha ,B)(r\\ln r)^{-\\alpha }.$ Petrushev [133] proved the following interesting result: Let $X_{k}$ be the $k$ dimensional linear space of univariate functions in $L_{2}[-1,1],$ $k=1,2,...$ .", "Besides, let $B^{d}$ and $S^{d-1}$ denote correspondingly the unit ball and unit sphere in the space $\\mathbb {R}^{d}$ .", "If $X_{k}$ provides order of approximation $O(k^{-m})$ for univariate functions with $m$ derivatives in$\\ L_{2}[-1,1]$ and $\\Omega _{k}$ are appropriately chosen finite sets of directions distributed on $S^{d-1}$ , then the space $Y_{k}=span\\lbrace p_{k}(\\mathbf {a}\\cdot \\mathbf {x}):~p_{k}\\in X_{k},~\\mathbf {a}\\in \\Omega _{k}\\rbrace $ will provide approximation of order $O(k^{-m-d/2+1/2})$ for every function $f\\in L_{2}(B^{d})$ with smoothness of order $m+d/2-1/2$ .", "Thus, Petrushev showed that the above form of ridge approximation has the same efficiency of approximation as the traditional multivariate polynomial approximation.", "Many other results concerning the approximation of multivariate functions by functions from the set $\\mathcal {R}_{r}$ and their applications in neural network theory may be found in [69], [109], [119], [135], [140].", "In this section we develop a technique for verifying if a multivariate function can be expressed as a sum of ridge functions with given directions.", "We also obtain a necessary and sufficient condition for the representation of all multivariate functions on a subset $X$ of $\\mathbb {R}^{d}$ by sums of ridge functions with fixed directions.", "Let $X$ be a subset of ${{\\mathbb {R}}}^{d}$ and $\\lbrace \\mathbf {a}^{i}\\rbrace _{i=1}^{r} $ be arbitrarily fixed nonzero directions (vectors) in ${{\\mathbb {R}}}^{d}$ .", "Consider the following set of linear combinations of ridge functions.", "$\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),~\\mathbf {x}\\in X,~g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r\\right\\rbrace $ In this section, we are going to deal with the following two problems: Problem 1.", "What conditions imposed on $f:X\\rightarrow \\mathbb {R}$ are necessary and sufficient for the inclusion $f\\in \\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)$ ?", "Problem 2.", "What conditions imposed on $X$ are necessary and sufficient that every function defined on $X$ belongs to the space $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)$ ?", "As noticed in Section 1.1, Problem 1 was considered for continuous functions in [112] and a theoretical result was obtained.", "It was also noticed there that the similar problem of representation of $f$ in the form $\\sum _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})+P(\\mathbf {x})$ with polynomial $P(\\mathbf {x})$ was solved for continuously differentiable functions in [35].", "Problem 2 was solved in [16] for finite subsets $X$ of ${{\\mathbb {R}}}^{d}$ and in [94] for the case when $r=d$ and $\\mathbf {a}^{i}$ are the coordinate directions.", "Here we consider both Problem 1 and Problem 2 without imposing on $X$ , $f$ and $r$ any conditions.", "In fact, we solve these problems for more general, than $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)$ , set of functions.", "Namely, we solve them for the set $\\mathcal {B}(X)=\\mathcal {B}(h_{1},...,h_{r};X)=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}(h_{i}(x)),~x\\in X,~g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r\\right\\rbrace ,$ where $h_{i}:X\\rightarrow {{\\mathbb {R}}},~i=1,...,r,$ are arbitrarily fixed functions.", "In particular, the functions $h_{i},~i=1,...,r$ , may be equal to scalar products of the variable $\\mathbf {x}$ with some vectors $\\mathbf {a}^{i}$ , $i=1,...,r$ .", "Only in this special case, we have $\\mathcal {B}(h_{1},...,h_{r};X)=\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X).$ The main idea leading to solutions of the above problems is in using new objects called cycles with respect to $r$ functions $h_{i}:X\\rightarrow \\mathbb {R},~i=1,...,r$ (and in particular, with respect to $r$ directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ ).", "In the sequel, by $\\delta _{A}$ we will denote the characteristic function of a set $\\ A\\subset \\mathbb {R}.$ That is, $\\delta _{A}(y)=\\left\\lbrace \\begin{array}{c}1,~if~y\\in A \\\\0,~if~y\\notin A.\\end{array}\\right.$ Definition 1.1.", "Given a subset $X\\subset \\mathbb {R}^{d}$ and functions $h_{i}:X\\rightarrow \\mathbb {R},~i=1,...,r$ .", "A set of points $\\lbrace x_{1},...,x_{n}\\rbrace \\subset X$ is called a cycle with respect to the functions $h_{1},...,h_{r}$ (or, concisely, a cycle if there is no confusion), if there exists a vector $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ with the nonzero real coordinates $\\lambda _{i},~i=1,...,n,$ such that $\\sum _{j=1}^{n}\\lambda _{j}\\delta _{h_{i}(x_{j})}=0,~i=1,...,r.(1.4)$ If $h_{i}=\\mathbf {a}^{i}\\cdot \\mathbf {x}$ , $i=1,...,r$ , where $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ are some directions in $\\mathbb {R}^{d}$ , a cycle, with respect to the functions $h_{1},...,h_{r}$ , is called a cycle with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}.$ Let for $i=1,...,r,$ the set $\\lbrace h_{i}(x_{j}),~j=1,...,n\\rbrace $ have $k_{i}$ different values.", "Then it is not difficult to see that Eq.", "(1.4) stands for a system of $\\sum _{i=1}^{r}k_{i}$ homogeneous linear equations in unknowns $\\lambda _{1},...,\\lambda _{n}.$ If this system has any solution with the nonzero components, then the given set $\\lbrace x_{1},...,x_{n}\\rbrace $ is a cycle.", "In the last case, the system has also a solution $m=(m_{1},...,m_{n})$ with the nonzero integer components $m_{i},~i=1,...,n.$ Thus, in Definition 1.1, the vector $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ can be replaced with a vector $m=(m_{1},...,m_{n})$ with $m_{i}\\in \\mathbb {Z}\\backslash \\lbrace 0\\rbrace .$ For example, the set $l=\\lbrace (0,0,0),~(0,0,1),~(0,1,0),~(1,0,0),~(1,1,1)\\rbrace $ is a cycle in $\\mathbb {R}^{3}$ with respect to the functions $h_{i}(z_{1},z_{2},z_{3})=z_{i},~i=1,2,3.$ The vector $\\lambda $ in Definition 1.1 can be taken as $(2,1,1,1,-1).$ In case $r=2,$ the picture of cycles becomes more clear.", "Let, for example, $h_{1}$ and $h_{2}$ be the coordinate functions on $\\mathbb {R}^{2}.$ In this case, a cycle is the union of some sets $A_{k}$ with the property: each $A_{k}$ consists of vertices of a closed broken line with the sides parallel to the coordinate axis.", "These objects (sets $A_{k}$ ) have been exploited in practically all works devoted to the approximation of bivariate functions by univariate functions, although under various different names (see “bolt of lightning\" in Section 1.3).", "If the functions $h_{1}$ and $h_{2}$ are arbitrary, the sets $A_{k}$ can be described as a trace of some point traveling alternatively in the level sets of $h_{1}$ and $h_{2},$ and then returning to its primary position.", "It should be remarked that in the case $r>2,$ cycles do not admit such a simple geometric description.", "We refer the reader to Braess and Pinkus [16] for the description of cycles when $r=3$ and $h_{i}(\\mathbf {x})=\\mathbf {a}^{i}\\cdot \\mathbf {x},$ $\\mathbf {x}\\in \\mathbb {R}^{2},~\\mathbf {a}^{i}\\in \\mathbb {R}^{2}\\backslash \\lbrace \\mathbf {0}\\rbrace ,~i=1,2,3.$ Let $T(X)$ denote the set of all functions on $X.$ With each pair $\\left\\langle p,\\lambda \\right\\rangle ,$ where $p=\\lbrace x_{1},...,x_{n}\\rbrace $ is a cycle in $X$ and $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ is a vector known from Definition 1.1, we associate the functional $G_{p,\\lambda }:T(X)\\rightarrow \\mathbb {R},~~G_{p,\\lambda }(f)=\\sum _{j=1}^{n}\\lambda _{j}f(x_{j}).$ In the following, such pairs $\\left\\langle p,\\lambda \\right\\rangle $ will be called cycle-vector pairs of $X.$ It is clear that the functional $G_{p,\\lambda }$ is linear and $G_{p,\\lambda }(g)=0$ for all functions $g\\in \\mathcal {B}(h_{1},...,h_{r};X).$ Lemma 1.1.", "Let $X$ have cycles and $h_{i}(X)\\cap h_{j}(X)=\\varnothing ,$ for all $i,j\\in \\lbrace 1,...,r\\rbrace ,~i\\ne j.$ Then a function $f:X\\rightarrow \\mathbb {R}$ belongs to the set $\\mathcal {B}(h_{1},...,h_{r};X)$ if and only if $G_{p,\\lambda }(f)=0$ for any cycle-vector pair $\\left\\langle p,\\lambda \\right\\rangle $ of $X.$ The necessity is obvious, since the functional $G_{p,\\lambda }$ annihilates all members of $\\mathcal {B}(h_{1},...,h_{r};X)$ .", "Let us prove the sufficiency.", "Introduce the notation $Y_{i} &=&h_{i}(X),~i=1,...,r; \\\\\\Omega &=&Y_{1}\\cup ...\\cup Y_{r}.$ Consider the following set.", "$\\mathcal {L}=\\lbrace Y=\\lbrace y_{1},...,y_{r}\\rbrace :\\text{if there exists }x\\in X\\text{ suchthat }h_{i}(x)=y_{i},~i=1,...,r\\rbrace (1.5)$ Note that $\\mathcal {L}$ is not a subset of $\\Omega $ .", "It is a set of some certain subsets of $\\Omega .$ Each element of $\\mathcal {L}$ is a set $Y=\\lbrace y_{1},...,y_{r}\\rbrace \\subset \\Omega $ with the property that there exists $x\\in X$ such that $h_{i}(x)=y_{i},~i=1,...,r.$ In what follows, all the points $x$ associated with $Y$ by (1.5) will be called $(\\ast )$ -points of $Y.$ It is clear that the number of such points depends on $Y$ as well as on the functions $h_{1},...,h_{r}$ , and may be greater than 1.", "But note that if any two points $x_{1}$ and $x_{2}$ are $(\\ast )$ -points of $Y$ , then the set $\\lbrace x_{1}$ , $x_{2}\\rbrace $ necessarily forms a cycle with the associated vector $\\lambda _{0}=(1;-1).$ Indeed, if $x_{1}$ and $x_{2}$ are $(\\ast )$ -points of $Y$ , then $h_{i}(x_{1})=h_{i}(x_{2})$ , $i=1,...,r,$ whence $1\\cdot \\delta _{h_{i}(x_{1})}+(-1)\\cdot \\delta _{h_{i}(x_{2})}\\equiv 0,~i=1,...,r.$ The last identity means that the set $p_{0}=\\lbrace x_{1},$ $x_{2}\\rbrace $ forms a cycle and $\\lambda _{0}=(1;-1)$ is an associated vector.", "Then by the the sufficiency condition, $G_{p_{0},\\lambda _{0}}(f)=0$ , whcih yields that $f(x_{1})=f(x_{2})$ .", "Let now $Y^{\\ast }$ be the set of all $(\\ast )$ -points of $Y.$ Since we have already known that $f(Y^{\\ast })$ is a single number, we can define the function $t:\\mathcal {L}\\rightarrow \\mathbb {R},~t(Y)=f(Y^{\\ast }).$ Or, equivalently, $t(Y)=f(x),$ where $x$ is an arbitrary $(\\ast )$ -point of $Y$ .", "Consider now a class $\\mathcal {S}$ of functions of the form $\\sum _{j=1}^{k}r_{j}\\delta _{D_{j}},$ where $k$ is a positive integer, $r_{j}$ are real numbers and $D_{j}$ are elements of $\\mathcal {L},~j=1,...,k.$ We fix neither the numbers $\\ k,~r_{j},$ nor the sets $D_{j}.$ Clearly, $\\mathcal {S\\ }$ is a linear space.", "Over $\\mathcal {S}$ , we define the functional $F:\\mathcal {S}\\rightarrow \\mathbb {R},~F\\left( \\sum _{j=1}^{k}r_{j}\\delta _{D_{j}}\\right) =\\sum _{j=1}^{k}r_{j}t(D_{j}).$ First of all, we must show that this functional is well defined.", "That is, the equality $\\sum _{j=1}^{k_{1}}r_{j}^{\\prime }\\delta _{D_{j}^{\\prime }}=\\sum _{j=1}^{k_{2}}r_{j}^{\\prime \\prime }\\delta _{D_{j}^{\\prime \\prime }}$ always implies the equality $\\sum _{j=1}^{k_{1}}r_{j}^{\\prime }t(D_{j}^{\\prime })=\\sum _{j=1}^{k_{2}}r_{j}^{\\prime \\prime }t(D_{j}^{\\prime \\prime }).$ In fact, this is equivalent to the implication $\\sum _{j=1}^{k}r_{j}\\delta _{D_{j}}=0\\Longrightarrow \\sum _{j=1}^{k}r_{j}t(D_{j})=0,~\\text{for all }k\\in \\mathbb {N}\\text{, }r_{j}\\in \\mathbb {R}\\text{, }D_{j}\\subset \\mathcal {L}\\text{.", "}(1.6)$ Suppose that the left-hand side of the implication (1.6) be satisfied.", "Each set $D_{j}$ consists of $r$ real numbers $y_{1}^{j},...,y_{r}^{j}$ , $j=1,...,k.$ By the hypothesis of the lemma, all these numbers are different.", "Therefore, $\\delta _{D_{j}}=\\sum _{i=1}^{r}\\delta _{y_{i}^{j}},~j=1,...,k.(1.7)$ Eq.", "(1.7) together with the left-hand side of (1.6) gives $\\sum _{i=1}^{r}\\sum _{j=1}^{k}r_{j}\\delta _{y_{i}^{j}}=0.", "(1.8)$ Since the sets $\\lbrace y_{i}^{1},y_{i}^{2},...,y_{i}^{k}\\rbrace $ , $i=1,...,r,$ are pairwise disjoint, we obtain from (1.8) that $\\sum _{j=1}^{k}r_{j}\\delta _{y_{i}^{j}}=0,\\text{ }i=1,...,r.(1.9)$ Let now $x_{1},...,x_{k}$ be some $(\\ast )$ -points of the sets $D_{1},...,D_{k}$ respectively.", "Since by (1.5), $y_{i}^{j}=h_{i}(x_{j})$ , for $i=1,...,r$ and $j=1,...,k,$ it follows from (1.9) that the set $\\lbrace x_{1},...,x_{k}\\rbrace $ is a cycle.", "Then by the condition of the sufficiency, $\\sum _{j=1}^{k}r_{j}f(x_{j})=0.$ Hence $\\sum _{j=1}^{k}r_{j}t(D_{j})=0.$ We have proved the implication (1.6) and hence the functional $F$ is well defined.", "Note that the functional $F$ is linear (this can be easily seen from its definition).", "Consider now the following space: $\\mathcal {S}^{\\prime }=\\left\\lbrace \\sum _{j=1}^{k}r_{j}\\delta _{\\omega _{j}}\\right\\rbrace ,$ where $k\\in \\mathbb {N}$ , $r_{j}\\in \\mathbb {R}$ , $\\omega _{j}\\subset \\Omega .$ As above, we do not fix the parameters $k$ , $r_{j}$ and $\\omega _{j}.$ Clearly, the space $\\mathcal {S}^{\\prime }$ is larger than $\\mathcal {S}$ .", "Let us prove that the functional $F$ can be linearly extended to the space $\\mathcal {S}^{\\prime }$ .", "So, we must prove that there exists a linear functional $F^{\\prime }:\\mathcal {S}^{\\prime }\\rightarrow \\mathbb {R}$ such that $F^{\\prime }(x)=F(x)$ , for all $x\\in \\mathcal {S}$ .", "Let $H$ denote the set of all linear extensions of $F$ to subspaces of $\\mathcal {S}^{\\prime }$ containing $\\mathcal {S}$ .", "The set $H$ is not empty, since it contains a functional $F.$ For each functional $v\\in H$ , let $dom(v)$ denote the domain of $v$ .", "Consider the following partial order in $H$ : $v_{1}\\le v_{2}$ , if $v_{2}$ is a linear extension of $v_{1}$ from the space $dom(v_{1})$ to the space $dom(v_{2}).$ Let now $P$ be any chain (linearly ordered subset) in $H$ .", "Consider the following functional $u$ defined on the union of domains of all functionals $p\\in P$ : $u:\\bigcup \\limits _{p\\in P}dom(p)\\rightarrow \\mathbb {R},~u(x)=p(x),\\text{ if }x\\in dom(p)$ Obviously, this functional is well defined and linear.", "Besides, the functional $u$ provides an upper bound for $P.$ We see that the arbitrarily chosen chain $P$ has an upper bound.", "Then by Zorn's lemma, there is a maximal element $F^{\\prime }\\in H$ .", "We claim that the functional $F^{\\prime } $ must be defined on the whole space $\\mathcal {S}^{\\prime }$ .", "Indeed, if $F^{\\prime }$ is defined on a proper subspace $\\mathcal {D\\subset }$ $\\mathcal {S}^{\\prime }$ , then it can be linearly extended to a space larger than $\\mathcal {D}$ by the following way: take any point $x\\in \\mathcal {S}^{\\prime }\\backslash \\mathcal {D}$ and consider the linear space $\\mathcal {D}^{\\prime }=\\lbrace \\mathcal {D}+\\alpha x\\rbrace $ , where $\\alpha $ runs through all real numbers.", "For an arbitrary point $y+\\alpha x\\in \\mathcal {D}^{\\prime }$ , set $F^{^{\\prime \\prime }}(y+\\alpha x)=F^{\\prime }(y)+\\alpha b$ , where $b$ is any real number considered as the value of $F^{^{\\prime \\prime }}$ at $x$ .", "Thus, we constructed a linear functional $F^{^{\\prime \\prime }}\\in H$ satisfying $F^{\\prime }\\le F^{^{\\prime \\prime }}.$ The last contradicts the maximality of $F^{\\prime }.$ This means that the functional $F^{\\prime }$ is defined on the whole $\\mathcal {S}^{\\prime }$ and $F\\le F^{\\prime }$ ($F^{\\prime }$ is a linear extension of $F$ ).", "Define the following functions by means of the functional $F^{\\prime }$ : $g_{i}:Y_{i}\\rightarrow \\mathbb {R},\\text{ }g_{i}(y_{i})\\overset{def}{=}F^{\\prime }(\\delta _{y_{i}}),\\text{ }i=1,...,r.$ Let $x$ be an arbitrary point in $X.$ Obviously, $x$ is a $(\\ast )$ -point of some set $Y=\\lbrace y_{1},...,y_{r}\\rbrace \\subset \\mathcal {L}.$ Thus, $f(x) &=&t(Y)=F(\\delta _{Y})=F\\left( \\sum _{i=1}^{r}\\delta _{y_{i}}\\right)=F^{\\prime }\\left( \\sum _{i=1}^{r}\\delta _{y_{i}}\\right) = \\\\\\sum _{i=1}^{r}F^{\\prime }(\\delta _{y_{i}})&=&\\sum _{i=1}^{r}g_{i}(y_{i})=\\sum _{i=1}^{r}g_{i}(h_{i}(x)).$ Definition 1.2.", "A cycle $p=\\lbrace x_{1},...,x_{n}\\rbrace $ is said to be minimal if $p$ does not contain any cycle as its proper subset.", "For example, the set $l=\\lbrace (0,0,0),~(0,0,1),~(0,1,0),~(1,0,0),~(1,1,1)\\rbrace $ considered above is a minimal cycle with respect to the functions $h_{i}(z_{1},z_{2},z_{3})=z_{i},~i=1,2,3.$ Adding the point $(0,1,1)$ to $l$ , we will have a cycle, but not minimal.", "The vector $\\lambda $ associated with $l\\cup \\lbrace (0,1,1)\\rbrace $ can be taken as $(3,-1,-1,-2,2,-1).$ A minimal cycle $p=\\lbrace x_{1},...,x_{n}\\rbrace $ has the following obvious properties: (a) The vector $\\lambda $ associated with $p$ through Eq.", "(1.4) is unique up to multiplication by a constant; (b) If in (1.4), $\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1,$ then all the numbers $\\lambda _{j},~j=1,...,n,$ are rational.", "The vector $\\lambda $ associated with $p$ through Eq.", "(1.4) is unique up to multiplication by a constant; If in (1.4), $\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1,$ then all the numbers $\\lambda _{j},~j=1,...,n,$ are rational.", "Thus, a minimal cycle $p$ uniquely (up to a sign) defines the functional $~G_{p}(f)=\\sum _{j=1}^{n}\\lambda _{j}f(x_{j}),\\text{ \\ }\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1.$ Lemma 1.2.", "The functional $G_{p,\\lambda }$ is a linear combination of functionals $G_{p_{1}},...,G_{p_{k}},$ where $p_{1},...,p_{k}$ are minimal cycles in $p.$ Let $\\left\\langle p,\\lambda \\right\\rangle $ be a cycle-vector pair of $X$ , where $p=\\lbrace x_{1},...,x_{n}\\rbrace $ and $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ .", "Let $p_{1}=$ $\\lbrace y_{1}^{1},...,y_{s_{1}}^{1}\\rbrace ,$ $s_{1}<n$ , be a minimal cycle in $p$ and $G_{p_{1}}(f)=\\sum _{j=1}^{s_{1}}\\nu _{j}^{1}f(y_{j}^{1}),\\text{ }\\sum _{j=1}^{s_{1}}\\left|\\nu _{j}^{1}\\right|=1.$ Without loss of generality, we may assume that $y_{1}^{1}=x_{1}.$ Put $t_{1}=\\frac{\\lambda _{1}}{\\nu _{1}^{1}}.$ Then the functional $G_{p,\\lambda }-t_{1}G_{p_{1}}$ has the form $G_{p,\\lambda }-t_{1}G_{p_{1}}=\\sum _{j=1}^{n_{1}}\\lambda _{j}^{1}f(x_{j}^{1}),$ where $x_{j}^{1}\\in p$ , $\\lambda _{j}^{1}\\ne 0$ , $j=1,...,n_{1}$ .", "Clearly, the set $l_{1}=\\lbrace x_{1}^{1},...,x_{n_{1}}^{1}\\rbrace $ is a cycle in $p$ with the associated vector $\\lambda ^{1}=(\\lambda _{1}^{1},...,\\lambda _{n_{1}}^{1})$ .", "Besides, $x_{1}\\notin l_{1}$ .", "Thus, $n_{1}<n$ and $G_{l_{1},\\lambda ^{1}}=$ $G_{p,\\lambda }-t_{1}G_{p_{1}}$ .", "If $l_{1}$ is minimal, then the proof is completed.", "Assume $l_{1}$ is not minimal.", "Let $p_{1}=$ $\\lbrace y_{1}^{2},...,y_{s_{2}}^{2}\\rbrace ,$ $s_{2}<n_{1},$ be a minimal cycle in $l_{1} $ and $G_{p_{2}}(f)=\\sum _{j=1}^{s_{2}}\\nu _{j}^{2}f(y_{j}^{2}),\\text{ }\\sum _{j=1}^{s_{2}}\\left|\\nu _{j}^{2}\\right|=1.$ Without loss of generality, we may assume that $y_{1}^{2}=x_{1}^{1}.$ Put $t_{2}=\\frac{\\lambda _{1}^{1}}{\\nu _{1}^{2}}.$ Then the functional $G_{l_{1},\\lambda ^{1}}-t_{2}G_{p_{2}}$ has the form $G_{l_{1},\\lambda ^{1}}-t_{2}G_{p_{2}}=\\sum _{j=1}^{n_{2}}\\lambda _{j}^{2}f(x_{j}^{2}),$ where $x_{j}^{2}\\in l_{1}$ , $\\lambda _{j}^{2}\\ne 0$ , $j=1,...,n_{2}$ .", "Clearly, the set $l_{2}=\\lbrace x_{1}^{2},...,x_{n_{2}}^{2}\\rbrace $ is a cycle in $l_{1} $ with the associated vector $\\lambda ^{2}=(\\lambda _{1}^{2},...,\\lambda _{n_{2}}^{2})$ .", "Besides, $x_{1}^{1}\\notin l_{2}$ .", "Thus, $n_{2}<n_{1}$ and $G_{l_{2},\\lambda ^{2}}=$ $G_{l_{1},\\lambda ^{1}}-t_{2}G_{p_{2}}.$ If $l_{2}$ is minimal, then the proof is completed.", "Let $l_{2}$ be not minimal.", "Repeating the above process for $l_{2}$ , then for $l_{3}$ , etc., after some $k-1$ steps we will come to a minimal cycle $l_{k-1}$ and the functional $G_{l_{k-1},\\lambda ^{k-1}}=G_{l_{k-2},\\lambda ^{k-2}}-t_{k-1}G_{p_{k-1}}=\\sum _{j=1}^{n_{k-1}}\\lambda _{j}^{k-1}f(x_{j}^{k-1}).$ Since the cycle $l_{k-1}$ is minimal, $G_{l_{k-1},\\lambda ^{k-1}}=t_{k}G_{l_{k-1}},\\text{ \\ where }t_{k}=\\sum _{j=1}^{n_{k-1}}\\left|\\lambda _{j}^{k-1}\\right|.$ Now putting $p_{k}=l_{k-1}$ and considering the above chain relations between the functionals $G_{l_{i},\\lambda ^{i}}$ , $i=1,...,k-1,$ we obtain that $G_{p,\\lambda }=\\sum _{i=1}^{k}t_{i}G_{p_{i}}.$ Theorem 1.1.", "Assume $X\\subset \\mathbb {R}^{d}$ and $h_{1},...,h_{r}$ are arbitrarily fixed real functions on $X.$ The following assertions are valid.", "1) Let $X$ have cycles with respect to the functions $h_{1},...,h_{r} $ .", "A function $f:X\\rightarrow \\mathbb {R}$ belongs to the space $\\mathcal {B}(h_{1},...,h_{r};X)$ if and only if $G_{p}(f)=0$ for any minimal cycle $p\\subset X$ .", "2) Let $X$ have no cycles.", "Then $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).", "$ 1) The necessity is clear.", "Let us prove the sufficiency.", "On the strength of Lemma 1.2, it is enough to prove that if $G_{p,\\lambda }(f)=0$ for any cycle-vector pair $\\left\\langle p,\\lambda \\right\\rangle $ of $X$ , then $f\\in \\mathcal {B}(X).$ Consider a system of intervals $\\lbrace (a_{i},b_{i})\\subset \\mathbb {R}\\rbrace _{i=1}^{r} $ such that $(a_{i},b_{i})\\cap (a_{j},b_{j})=\\varnothing $ for all the indices $i,j\\in \\lbrace 1,...,r\\rbrace $ , $~i\\ne j.$ For $i=1,...,r$ , let $\\tau _{i}$ be one-to-one mappings of $\\mathbb {R}$ onto $(a_{i},b_{i}).$ Introduce the following functions on $X$ : $h_{i}^{^{\\prime }}(x)=\\tau _{i}(h_{i}(x)),\\text{ }i=1,...,r.$ It is clear that any cycle with respect to the functions $h_{1},...,h_{r}$ is also a cycle with respect to the functions $h_{1}^{^{\\prime }},...,h_{r}^{^{\\prime }}$ , and vice versa.", "Besides, $h_{i}^{\\prime }(X)\\cap h_{j}^{\\prime }(X)=\\varnothing ,$ for all $i,j\\in \\lbrace 1,...,r\\rbrace ,~i\\ne j.$ Then by Lemma 1.1, $f(x)=g_{1}^{\\prime }(h_{1}^{\\prime }(x))+\\cdots +g_{r}^{\\prime }(h_{r}^{\\prime }(x)),$ where $g_{1}^{\\prime },...,g_{r}^{\\prime }$ are univariate functions depending on $f$ .", "From the last equality we obtain that $f(x)=g_{1}^{\\prime }(\\tau _{1}(h_{1}(x)))+\\cdots +g_{r}^{\\prime }(\\tau _{r}(h_{r}(x)))=g_{1}(h_{1}(x))+\\cdots +g_{r}(h_{r}(x)).$ That is, $f\\in \\mathcal {B}(X)$ .", "2) Let $f:X\\rightarrow \\mathbb {R}$ be an arbitrary function.", "First suppose that $h_{i}(X)\\cap h_{j}(X)=\\varnothing ,$ for all $i,j\\in \\lbrace 1,...,r\\rbrace $ ,$~i\\ne j.$ In this case, the proof is similar to and even simpler than that of Lemma 1.1.", "Indeed, the set of all $(\\ast )$ -points of $Y$ consists of a single point, since otherwise we would have a cycle with two points, which contradicts the hypothesis of the 2-nd part of the theorem.", "Further, well definition of the functional $F$ becomes obvious, since the left-hand side of (1.6) also contradicts the nonexistence of cycles.", "Thus, as in the proof of Lemma 1.1, we can extend $F$ to the space $\\mathcal {S}^{\\prime }$ and then obtain the desired representation for the function $f$ .", "Since $f$ is arbitrary, $T(X)=\\mathcal {B}(X).$ Using the techniques from the proof of the 1-st part of the theorem, one can easily generalize the above argument to the case when the functions $h_{1},...,h_{r}$ have arbitrary ranges.", "Theorem 1.2.", "$\\mathcal {B}(h_{1},...,h_{r};X)=T(X)$ if and only if $X$ has no cycles with respect to the functions $h_{1},...,h_{r}$.", "The sufficiency immediately follows from Theorem 1.1.", "To prove the necessity, assume that $X$ has a cycle $p=\\lbrace x_{1},...,x_{n}\\rbrace $ .", "Let $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ be a vector associated with $p$ by Eq.", "(1.4).", "Consider a function $f_{0}$ on $X$ with the property: $f_{0}(x_{i})=1,$ for indices $i$ such that $\\lambda _{i}\\,>0$ and $f_{0}(x_{i})=-1,$ for indices $i$ such that $\\lambda _{i}\\,<0$ .", "For this function, $G_{p,\\lambda }(f_{0})\\ne 0$ .", "Then by Theorem 1.1, $f_{0}\\notin \\mathcal {B}(X)$ .", "Hence $\\mathcal {B}(X)\\ne T(X)$ .", "The contradiction shows that $X$ does not admit cycles.", "From Theorems 1.1 and 1.2 we obtain the following corollaries for the ridge function representation.", "Corollary 1.1.", "Assume $X\\subset \\mathbb {R}^{d}$ and $\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\in \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ .", "The following assertions are valid.", "1) Let $X$ have cycles with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ .", "A function $f:X\\rightarrow \\mathbb {R}$ belongs to the space $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)$ if and only if $G_{p}(f)=0$ for any minimal cycle $p\\subset X$ .", "2) Let $X$ have no cycles.", "Then every function $f:X\\rightarrow \\mathbb {R}$ belongs to the space $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)$ .", "Corollary 1.2.", "$\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=T(X)$ if and only if $X$ has no cycles with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ .", "Note that solutions to Problems 1 and 2 are given by Corollaries 1.1 and 1.2, correspondingly.", "Although it is not always easy to find all cycles of a given set $X$ and even to know if $X$ possesses a single cycle, Corollaries 1.1 and 1.2 are of more practical than theoretical character.", "Particular cases of Problems 1 and 2 evidence in favor of our opinion.", "For example, for the problem of representation by sums of two ridge functions, the picture of cycles is completely describable (see the beginning of this section).", "The interpretation of cycles with respect to three directions in the plane can be found in Braess and Pinkus [16].", "A geometric description of cycles with respect to 4 and more directions is quite complicated and requires deep techniques from geometry and graph theory.", "This is not within the aim of our study.", "From the last corollary, it follows that if representation by sums of ridge functions with fixed directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ is valid in the class of continuous functions (or in the class of bounded functions), then such representation is valid in the class of all functions.", "For a rigid mathematical formulation of this result, let us introduce the notation: $\\mathcal {R}_{c}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),~\\mathbf {x}\\in X,~g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})\\in C(X\\mathbb {)},~i=1,...,r\\right\\rbrace $ and $\\mathcal {R}_{b}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),~\\mathbf {x}\\in X,~g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})\\in B(X\\mathbb {)},~i=1,...,r\\right\\rbrace $ Here $C(X)$ and $B(X)$ denote the spaces of continuous and bounded functions defined on $X\\subset \\mathbb {R}^{d}$ correspondingly (for the first space, the set $X$ is supposed to be compact).", "As we know (see Section 1.1) from the results of Sternfeld it follows that the equality $\\mathcal {R}_{c}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=C(X)$ implies the equality $\\mathcal {R}_{b}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=B(X).$ In other words, if every continuous function is represented by sums of ridge functions (with fixed directions!", "), then every bounded function also obeys such representation (with bounded summands).", "Corollaries 1.1 and 1.2 allow us to obtain the following result.", "Corollary 1.3.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ and $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ be given directions in $\\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ .", "If $\\mathcal {R}_{c}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=C(X),$ then $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=T(X).$ If every continuous function defined on $X\\subset \\mathbb {R}^{d}$ is represented by sums of ridge functions with the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ , then it can be shown by applying the same idea (as in the proof of Theorem 1.2) that the set $X$ has no cycles with respect to the given directions.", "Only, because of continuity, Urysohn's great lemma should be taken into account.", "That is, it should be taken into account that, by assuming the existence of a cycle $p_{0}=\\lbrace x_{1},...,x_{n}\\rbrace $ with an associated vector $\\lambda _{0}=(\\lambda _{1},...,\\lambda _{n})$ , we can deduce from Urysohn's great lemma the existence of a continuous function $u:X\\rightarrow \\mathbb {R}$ satisfying 1) $u(x_{i})=1,$ for indices $i$ such that $\\lambda _{i}\\,>0$ 2) $u(x_{j})=-1,$ for indices $j$ such that $\\lambda _{j}\\,<0$ , 3) $-1<u(x)<1,$ for all $x\\in X\\backslash p_{0}.$ These properties mean that $G_{p_{0},\\lambda _{0}}(u)\\ne 0\\Longrightarrow u\\notin \\mathcal {R}_{c}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)\\Longrightarrow \\mathcal {R}_{c}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)\\ne C(X).$ But if $X$ has no cycles with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ , then by Corollary 1.2, $\\mathcal {R}(\\mathbf {a}^{1},...,\\mathbf {a}^{r};X)=T(X).$ Let us now give some examples of sets over which the representation by linear combinations of ridge functions is possible.", "(1) Let $r=2$ and $X$ be the union of two parallel lines not perpendicular to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Then $X$ has no cycles with respect to $\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace $ .", "Therefore, by Corollary 1.2, $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2};X\\right) =T(X).$ (2) Let $r=2,$ $\\mathbf {a}^{1}=(1,1)$ , $\\mathbf {a}^{2}=(1,-1)$ and $X$ be the graph of the function $y=\\arcsin (\\sin x)$ .", "Then $X$ has no cycles and hence $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2};X\\right) =T(X).$ (3) Assume now we are given $r$ directions $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ and $r+1$ points $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{r+1}\\subset \\mathbb {R}^{d}$ such that $\\mathbf {a}^{1}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{1}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{1}\\cdot \\mathbf {x}^{2}\\text{, \\ for }1\\le i,j\\le r+1\\text{, }i,j\\ne 2 \\\\\\mathbf {a}^{2}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{2}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{2}\\cdot \\mathbf {x}^{3}\\text{, \\ for }1\\le i,j\\le r+1\\text{, }i,j\\ne 3 \\\\&&\\mathbf {......................................} \\\\\\mathbf {a}^{r}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{r}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{r}\\cdot \\mathbf {x}^{r+1}\\text{, \\ for }1\\le i,j\\le r.$ The simplest data realizing these equations are the basis directions in $\\mathbb {R}^{d}$ and the points $(0,0,...,0)$ , $(1,0,...,0)$ , $(0,1,...,0)$ ,..., $(0,0,...,1)$ .", "From the first equation we obtain that $\\mathbf {x}^{2}$ cannot be a point of any cycle in $X=\\lbrace \\mathbf {x}^{1},...,\\mathbf {x}^{r+1}\\rbrace $ .", "Sequentially, from the second, third, ..., $r$ -th equations it follows that the points $\\mathbf {x}^{3},\\mathbf {x}^{4},...,\\mathbf {x}^{r+1}$ also cannot be points of cycles in $X$ , respectively.", "Thus the set $X$ does not contain cycles at all.", "By Corollary 1.2, $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};X\\right) =T(X).$ (4) Assume we are given directions $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ and a curve $\\gamma $ in $\\mathbb {R}^{d}$ such that for any $c\\in \\mathbb {R}$ , $\\gamma $ has at most one common point with at least one of the hyperplanes $\\mathbf {a}^{j}\\cdot \\mathbf {x}=c$ , $j=1,...,r.$ Clearly, the curve $\\gamma $ has no cycles and hence $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};\\gamma \\right) =T(\\gamma ).$ Let $r=2$ and $X$ be the union of two parallel lines not perpendicular to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Then $X$ has no cycles with respect to $\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace $ .", "Therefore, by Corollary 1.2, $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2};X\\right) =T(X).$ Let $r=2,$ $\\mathbf {a}^{1}=(1,1)$ , $\\mathbf {a}^{2}=(1,-1)$ and $X$ be the graph of the function $y=\\arcsin (\\sin x)$ .", "Then $X$ has no cycles and hence $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2};X\\right) =T(X).$ Assume now we are given $r$ directions $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ and $r+1$ points $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{r+1}\\subset \\mathbb {R}^{d}$ such that $\\mathbf {a}^{1}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{1}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{1}\\cdot \\mathbf {x}^{2}\\text{, \\ for }1\\le i,j\\le r+1\\text{, }i,j\\ne 2 \\\\\\mathbf {a}^{2}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{2}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{2}\\cdot \\mathbf {x}^{3}\\text{, \\ for }1\\le i,j\\le r+1\\text{, }i,j\\ne 3 \\\\&&\\mathbf {......................................} \\\\\\mathbf {a}^{r}\\cdot \\mathbf {x}^{i} &=&\\mathbf {a}^{r}\\cdot \\mathbf {x}^{j}\\ne \\mathbf {a}^{r}\\cdot \\mathbf {x}^{r+1}\\text{, \\ for }1\\le i,j\\le r.$ The simplest data realizing these equations are the basis directions in $\\mathbb {R}^{d}$ and the points $(0,0,...,0)$ , $(1,0,...,0)$ , $(0,1,...,0)$ ,..., $(0,0,...,1)$ .", "From the first equation we obtain that $\\mathbf {x}^{2}$ cannot be a point of any cycle in $X=\\lbrace \\mathbf {x}^{1},...,\\mathbf {x}^{r+1}\\rbrace $ .", "Sequentially, from the second, third, ..., $r$ -th equations it follows that the points $\\mathbf {x}^{3},\\mathbf {x}^{4},...,\\mathbf {x}^{r+1}$ also cannot be points of cycles in $X$ , respectively.", "Thus the set $X$ does not contain cycles at all.", "By Corollary 1.2, $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};X\\right) =T(X).$ Assume we are given directions $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ and a curve $\\gamma $ in $\\mathbb {R}^{d}$ such that for any $c\\in \\mathbb {R}$ , $\\gamma $ has at most one common point with at least one of the hyperplanes $\\mathbf {a}^{j}\\cdot \\mathbf {x}=c$ , $j=1,...,r.$ Clearly, the curve $\\gamma $ has no cycles and hence $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};\\gamma \\right) =T(\\gamma ).$ Braess and Pinkus [16] considered the partial case of Problem 2: characterize a set of points $\\left( \\mathbf {x}^{1},...,\\mathbf {x}^{k}\\right) \\subset \\mathbb {R}^{d}$ such that for any data $\\lbrace \\alpha _{1},...,\\alpha _{k}\\rbrace \\subset \\mathbb {R}$ there exists a function $g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};\\mathbb {R}^{d}\\right) $ satisfying $g(\\mathbf {x}^{i})=\\alpha _{i},$ $i=1,...,k$ .", "In connection with this problem, they introduced the notion of the NI-property (non interpolation property) and MNI-property (minimal non interpolation property) of a finite set of points as follows: Given directions $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}\\subset \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ , we say that a set of points $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}\\subset \\mathbb {R}^{d}$ has the NI-property with respect to $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ , if there exists $\\lbrace \\alpha _{i}\\rbrace _{i=1}^{k}\\subset \\mathbb {R}$ such that we cannot find a function $g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};\\mathbb {R}^{d}\\right) $ satisfying $g(\\mathbf {x}^{i})=\\alpha _{i},$ $i=1,...,k$ .", "We say that the set $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}\\subset \\mathbb {R}^{d}$ has the MNI-property with respect to $\\lbrace \\mathbf {a}^{j}\\rbrace _{j=1}^{r}$ , if $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}$ but no proper subset thereof has the NI-property.", "It follows from Corollary 1.2 that a set $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}$ has the NI-property if and only if $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}$ contains a cycle with respect to the functions $h_{i}=\\mathbf {a}^{i}\\cdot \\mathbf {x},$ $i=1,...,r$ (or, simply, to the directions $\\mathbf {a}^{i},$ $i=1,...,r$ ) and the MNI-property if and only if the set $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}$ itself is a minimal cycle with respect to the given directions.", "Taking into account this argument and Definitions 1.1 and 1.2, we obtain that the set $\\lbrace \\mathbf {x}^{i}\\rbrace _{i=1}^{k}$ has the NI-property if and only if there is a vector $\\mathbf {m}=(m_{1},...,m_{k})\\in \\mathbb {Z}^{k}\\backslash \\lbrace \\mathbf {0}\\rbrace $ such that $\\sum _{j=1}^{k}m_{j}g(\\mathbf {a}^{i}\\cdot \\mathbf {x}^{j})=0,$ for $i=1,...,r$ and all functions $g:\\mathbb {R\\rightarrow R}$ .", "This set has the MNI-property if and only if the vector $\\mathbf {m}$ has the additional properties: it is unique up to multiplication by a constant and all its components are different from zero.", "This special consequence of Corollary 1.2 was proved in [16].", "The approximation problem considered in this section is to approximate a continuous multivariate function $f\\left( \\mathbf {x}\\right) =f\\left( {x_{1},...,x_{d}}\\right) $ by sums of two ridge functions in the uniform norm.", "We give a necessary and sufficient condition for a sum of two ridge functions to be a best approximation to $f\\left( \\mathbf {x}\\right) .$ This main result is next used in a special case to obtain an explicit formula for the approximation error and to construct a best approximation.", "The problem of well approximation by such sums is also considered.", "Consider the following set of sums of ridge functions $\\mathcal {R}=\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) ={\\left\\lbrace {g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) +g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) :g}_{i}{\\in C\\left( {\\mathbb {R}}\\right) ,i=1,2}\\right\\rbrace }.$ That is, we fix directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ and consider linear combinations of ridge functions with these directions.", "Assume $f\\left( \\mathbf {x}\\right) $ is a continuous function on a compact subset $Q$ of $\\mathbb {R}^{d}$ .", "We want to find conditions that are necessary and sufficient for a function $g_{_{0}}\\in \\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ to be an extremal element (or a best approximation) to $f$ .", "In other words, we want to characterize such sums $g_{0}\\left( \\mathbf {x}\\right) =g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) +g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) $ of ridge functions that ${\\left\\Vert {f-g_{0}}\\right\\Vert }={\\max \\limits _{{\\mathbf {x}\\in Q}}}{\\left|{f\\left( \\mathbf {x}\\right) -g}_{{0}}{\\left( \\mathbf {x}\\right) }\\right|}=E\\left( {f}\\right) ,$ where $E\\left( {f}\\right) =E(f,\\mathcal {R})\\overset{def}{=}{\\inf _{g \\in \\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right)}}{\\left\\Vert {f-g}\\right\\Vert }$ is the error in approximating from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) .$ The other related problem is how to construct these sums of ridge functions.", "We also want to know if we can approximate well, i.e.", "for which compact sets $Q,$ $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is dense in $C\\left( {Q}\\right) $ in the topology of uniform convergence.", "It should be remarked that solutions to these problems may be useful in connection with the study of partial differential equations.", "For example, assume that $\\left( {a_{1},b_{1}}\\right) $ and $\\left( {a_{2},b_{2}}\\right) $ are linearly independent vectors in $\\mathbb {R}^{2}.$ Then the general solution to the homogeneous partial differential equation $\\left( {a_{1}{\\frac{\\partial }{\\partial {x}}}+b_{1}{\\frac{\\partial }{\\partial {y}}}}\\right) \\left( {a_{2}{\\frac{\\partial }{\\partial {x}}}+b_{2}{\\frac{\\partial }{\\partial {y}}}}\\right) {u}\\left( {x,y}\\right) =0(1.10)$ are all functions of the form $u\\left( {x,y}\\right) =g_{1}\\left( {b_{1}x-a_{1}y}\\right) +g_{_{2}}\\left( {b_{2}x-a_{2}y}\\right) (1.11)$ for arbitrary $g_{1}$ and $g_{2}.$ In [47], Golitschek and Light described an algorithm that computes the error of approximation of a continuous function $f\\left( {x,y}\\right) $ by solutions of equation (1.10), provided that $a_{1}=b_{2}=1$ , $a_{2}=b_{1}=0.$ Using our result (see Theorem 1.3), one can characterize those solutions (1.11) that are extremal to a given function $f(x,y)$ .", "For a certain class of functions $f(x,y)$ , one can also easily calculate the approximation error and construct an extremal solution (see Theorems 1.5 and 1.6 below).", "The problem of approximating by functions from the set $\\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ arises in other contexts too.", "Buck [17] studied the classical functional equation: given $\\beta (t)\\in C[0,1]$ , $0\\le \\beta (t)\\le 1$ , for which $u\\in C[0,1]$ does there exist $\\varphi \\in C[0,1]$ such that $\\varphi (t)=\\varphi \\left( \\beta (t)\\right) +u(t)?$ He proved that the set of all $u$ satisfying this condition is dense in the set $\\lbrace v\\in C[0,1]:\\ v(t)=0\\ \\mbox{whenever}\\ \\beta (t)=t\\rbrace $ if and only if $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ with the unit directions $\\mathbf {a}^{1}=(1;0)$ and $\\mathbf {a}^{2}=(0,1)$ is dense in $C(K)$ , where $K=\\lbrace (x,y):y=x\\ \\mbox{or}\\ y=\\beta (x),\\ 0\\le x\\le 1\\rbrace $ .", "Although there are enough reasons to consider approximation problems associated with the set $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ in an independent way, one may ask why sums of only two ridge functions are considered instead of sums with an arbitrary number of terms.", "We will try to answer this fair question in Section 1.3.4.", "Let $Q$ be a compact subset of $\\mathbb {R}^{d}$ and $\\mathbf {a}^{1},\\mathbf {a}^{2}\\in \\mathbb {R}^{d}\\backslash {\\left\\lbrace \\mathbf {0}\\right\\rbrace }.$ Definition 1.3.", "A finite or infinite ordered set $p=\\left(\\mathbf {p}{_{1},\\mathbf {p}_{2},...}\\right) \\subset Q$ with $\\mathbf {p}_{i}\\ne \\mathbf {p}_{i+1},$ and either $\\mathbf {a}^{1}\\cdot \\mathbf {p}_{1}=\\mathbf {a}^{1}\\cdot \\mathbf {p}_{2},\\mathbf {a}^{2}\\cdot \\mathbf {p}_{2}=\\mathbf {a}^{2}\\cdot \\mathbf {p}_{3},\\mathbf {a}^{1}\\cdot \\mathbf {p}_{3}=\\mathbf {a}^{1}\\cdot \\mathbf {p}_{4},...$ or $\\mathbf {a}^{2}\\cdot \\mathbf {p}_{1}=\\mathbf {a}^{2}\\cdot \\mathbf {p}_{2},~\\mathbf {a}^{1}\\cdot \\mathbf {p}_{2}=\\mathbf {a}^{1}\\cdot \\mathbf {p}_{3},\\mathbf {a}^{2}\\cdot \\mathbf {p}_{3}=\\mathbf {a}^{2}\\cdot \\mathbf {p}_{4},...$ is called a path with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "This notion (in the two-dimensional case) was introduced by Braess and Pinkus [16].", "They showed that paths give geometric means of deciding if a set of points ${\\left\\lbrace {\\mathbf {x}}^{i}\\right\\rbrace }_{i=1}^{m}\\subset \\mathbb {R}^{2}$ has the NI property (see Section 1.2.4).", "Ismailov and Pinkus [78] used these objects to study the problem of interpolation on straight lines by linear combinations of a finite number of ridge functions with fixed directions.", "In [60], [62], [70] paths were generalized to those with respect to two functions.", "The last objects turned out to be useful in problems of approximation and representation by sums of compositions of fixed multivariate functions with univariate functions.", "If $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ are the coordinate vectors in $\\mathbb {R}^{2}$ , then Definition 1.3 defines a bolt of lightning.", "The idea of bolts was first introduced in Diliberto and Straus [36], where these objects are called permissible lines.", "They appeared further in a number of papers, although under several different names (see, e.g., [39], [45], [47], [58], [59], [89], [90], [92], [93], [110], [122], [123], [130]).", "Note that the term “bolt of lightning\" is due to Arnold [8].", "For the sake of brevity, we use the term “path\" instead of the long expression “path with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ \".", "The length of a path is the number of its points.", "A single point is a path of the unit length.", "A finite path $\\left( \\mathbf {p}_{1} ,\\mathbf {p}_{2},...,\\mathbf {p}_{2n} \\right)$ is said to be closed if $\\left(\\mathbf {p}_{1} ,\\mathbf {p}_{2} ,...,\\mathbf {p}_{2n}, \\mathbf {p}_{1}\\right)$ is a path.", "We associate each closed path $p=\\left(\\mathbf {p}_{1}, \\mathbf {p}_{2} ,...,\\mathbf {p}_{2n} \\right) $ with the functional $G_{p} (f)=\\frac{1}{2n} \\sum \\limits _{k=1}^{2n}(-1)^{k+1} f(\\mathbf {p}_{k}).$ This functional has the following obvious properties: (a) If $g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right)$ , then $G_{p}(g)=0$ .", "(b) $\\left\\Vert G_{p} \\right\\Vert \\le 1$ and if $\\mathbf {p}_{i} \\ne \\mathbf {p}_{j}$ for all $i\\ne j,$ $1\\le i,j\\le 2n$ , then $\\left\\Vert G_{p} \\right\\Vert =1$ .", "Lemma 1.3.", "Let a compact set $Q$ have closed paths.", "Then $\\sup \\limits _{p\\subset Q}\\left|G_{p}(f)\\right|\\le E\\left( f\\right),(1.12)$ where the sup is taken over all closed paths.", "Moreover, inequality (1.12) is sharp, i.e.", "there exist functions for which (1.12) turns into equality.", "Let $p$ be a closed path in $Q$ and $g$ be any function from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ .", "By the linearity of $G_{p}$ and properties (a) and (b), $\\left|G_{p}(f)\\right|=\\left|G_{p}(f-g)\\right|\\le \\left\\Vert f-g\\right\\Vert .", "(1.13)$ Since the left-hand and the right-hand sides of (1.13) do not depend on $g$ and $p$ respectively, it follows from (1.13) that $\\sup _{p\\subset Q}\\left|G_{p}(f)\\right|\\le \\inf _{g \\in \\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) }\\left\\Vert f-g\\right\\Vert .", "(1.14)$ Now we prove the sharpness of (1.12).", "By assumption $Q$ has closed paths.", "Then $Q$ has a closed path $p^{\\prime }=\\left( \\mathbf {p}_{1}^{\\prime },...,\\mathbf {p}_{2m}^{\\prime }\\right)$ with distinct points $\\mathbf {p}_{1}^{\\prime },...,\\mathbf {p}_{2m}^{\\prime }$ .", "In fact, such a special path can be obtained from any closed path $p=\\left( \\mathbf {p}_{1},...,\\mathbf {p}_{2n}\\right) $ by the following simple algorithm: if the points of the path $p$ are not all distinct, let $i$ and $k>0$ be the minimal indices such that $\\mathbf {p}_{i}=\\mathbf {p}_{i+2k}$ ; delete from $p$ the subsequence $\\mathbf {p}_{i+1},...,\\mathbf {p}_{i+2k}$ and call $p$ the obtained path; repeat the above step until all points of $p$ are all distinct; set $p^{\\prime }:=p$ .", "On the other hand there exist continuous functions $h=h(\\mathbf {x})$ on $Q$ such that $h(\\mathbf {p}_{i}^{\\prime })=1$ , $i=1,3,...,2m-1$ , $h(\\mathbf {p}_{i}^{\\prime })=-1$ , $i=2,4,...,2m$ and $-1<h(\\mathbf {x})<1$ elsewhere.", "For such functions we have $G_{p^{\\prime }}(h)=\\Vert h\\Vert =1(1.15)$ and $E(h)\\le \\Vert h\\Vert ,(1.16)$ where the last inequality follows from the fact that $0\\in \\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) .$ From (1.14)-(1.16) it follows that $\\sup _{p\\subset Q}\\left|G_{p}(h)\\right|=E\\left( h\\right) .$ Lemma 1.4.", "Let $Q$ be a convex compact subset of $\\mathbb {R}^{d}$ and $f \\in C(Q)$ .", "For a vector $\\mathbf {e}\\in \\mathbb {R}^{d}\\backslash \\mathbf {\\lbrace 0\\rbrace }$ and a real number $t$ set $Q_{t}=\\left\\lbrace {\\mathbf {x}}\\in Q:\\mathbf {e}\\cdot \\mathbf {x}=t\\right\\rbrace ,\\ \\ \\ T_{h}=\\left\\lbrace t\\in \\mathbb {R}:Q_{t}\\ne \\emptyset \\right\\rbrace .$ The functions $g_{1} (t)=\\max _{\\mathbf {x} \\in Q_t} f(\\mathbf {x} ),\\ \\ t\\in T_{h}\\ \\ \\mbox{and}\\ \\ g_{2} (t)=\\min \\limits _{\\mathbf {x} \\in Q_t} f(\\mathbf {x} ),\\ \\ \\ t\\in T_{h}$ are defined and continuous on $T_{h} $ .", "The proof of this lemma is not difficult and can be obtained by the well-known elementary methods of mathematical analysis.", "Definition 1.4.", "A finite or infinite path $(\\mathbf {p}_{1},\\mathbf {p}_{2},...)$ is said to be extremal for a function $u \\in C(Q)$ if $u(\\mathbf {p}_{i})=(-1)^{i}\\left\\Vert u\\right\\Vert ,i=1,2,...$ or $u(\\mathbf {p}_{i})=(-1)^{i+1}\\left\\Vert u\\right\\Vert ,$ $i=1,2,...$ Theorem 1.3.", "Let $Q\\subset \\mathbb {R}^{d}$ be a convex compact set satisfying the following condition Condition (A): For any path $q=(\\mathbf {q}_{1},\\mathbf {q}_{2},...,\\mathbf {q}_{n})\\subset Q$ there exist points $\\mathbf {q}_{n+1},\\mathbf {q}_{n+2},...,\\mathbf {q}_{n+s}\\in Q$ such that $(\\mathbf {q}_{1},\\mathbf {q}_{2},...,\\mathbf {q}_{n+s})$ is a closed path and $s$ is not more than some positive integer $N_{0}$ independent of $q$ .", "Then a necessary and sufficient condition for a function $g_{0}\\in \\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ to be an extremal element to the given function $f \\in C(Q)$ is the existence of a closed or infinite path $l=(\\mathbf {p}_{1},\\mathbf {p}_{2},...)$ extremal for the function $f_{1}=f-g_{0}$ .", "It should be remarked that the above condition (A) strongly depends on the fixed directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "For example, in the familiar case of a square $S\\subset \\mathbb {R}^2$ there are many directions which are not allowed.", "If it is possible to reach a corner of $S$ with not more than one of the two directions orthogonal to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , respectively (we don't differentiate between directions $\\mathbf {c}$ and $-\\mathbf {c}$ ), the triple $(S,\\mathbf {a}^{1}, \\mathbf {a}^{2})$ does not satisfy condition (A) of the theorem.", "Here are simple examples: Let $S=[0;1]^2$ , $\\mathbf {a}^{1}=(1;0)$ , $\\mathbf {a}^{2}=(1;1)$ .", "Then the ordered set $\\lbrace (0;1), (1;0), (1;1)\\rbrace $ is a path in $S$ which can not be made closed.", "In this case, $(1;1)$ is not reached with the direction orthogonal to $\\mathbf {b}$ .", "Let now $\\mathbf {a}^{1}=\\left(1;\\frac{1}{2}\\right)$ , $\\mathbf {a}^{2}=(1;1)$ .", "Then the corner $(1;1)$ is reached with none of the directions orthogonal to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ respectively.", "In this case, for any positive integer $N_0$ and any point $\\mathbf {q}_0$ in $S$ one can chose a point $\\mathbf {q}_1\\in S$ from a sufficiently small neighborhood of the corner $(1;1)$ so that any path containing $\\mathbf {q}_0$ and $\\mathbf {q}_1$ has the length more than $N_0$ .", "These examples and a little geometry show that if a convex compact set $Q\\subset \\mathbb {R}^2$ satisfies condition (A) of Theorem 1.3, then any point in the boundary of $Q$ must be reached with each of the two directions orthogonal to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ respectively.", "If $Q\\subset \\mathbb {R}^d, \\mathbf {a}^{1}, \\mathbf {a}^{2}\\in \\mathbb {R}^d\\backslash \\lbrace \\mathbf {0}\\rbrace $ , $d>2$ , there are many directions orthogonal to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2} $ .", "In this case, condition (A) requires that any point in the boundary of $Q$ should be reached with at least two directions orthogonal to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , respectively.", "Necessity.", "Let $g_{0}(\\mathbf {x}) =g_{1,0} \\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) +g_{2,0} \\left(\\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right)$ be an extremal element from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2} \\right)$ to $f$ .", "We must show that if there is not a closed path extremal for $f_{1} $ , then there exists a path extremal for $f_{1} $ with the infinite length (number of points).", "Suppose the contrary.", "Suppose that there exists a positive integer $N$ such that the length of each path extremal for $f_{1} $ is not more than $N$ .", "Set the following functions: $f_{n} =f_{n-1} -g_{1,n-1} -g_{2,n-1} ,\\ \\ n=2, 3, ...,$ where $g_{1,n-1} =g_{1,n-1} \\left(\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right)=\\frac{1}{2}\\left(\\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}} f_{n-1}(\\mathbf {y}) +\\min \\limits _{\\begin{array}{c}\\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}} f_{n-1}(\\mathbf {y})\\right)$ $g_{2,n-1} =g_{2,n-1} (\\mathbf {a}^{2}{\\cdot }\\mathbf {x})=\\frac{1}{2} \\left( \\max _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}{\\cdot }\\mathbf {y}=\\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\end{array}} \\left( f_{n-1} (\\mathbf {y})-g_{1,n-1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {y})\\right)\\right.$ $\\left.+\\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}{\\cdot }\\mathbf {y}=\\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\end{array}} \\left( f_{n-1}(\\mathbf {y}) -g_{1,n-1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {y}) \\right) \\right).$ By Lemma 1.4, all the functions $f_{n}(\\mathbf {x}),$ $n=2,3,...,$ are continuous on $Q$ .", "By assumption $g_{0}$ is a best approximation to $f$ .", "Hence $\\left\\Vert f_{1}\\right\\Vert =E\\left( f\\right) $ .", "Now let us show that $\\left\\Vert f_{2}\\right\\Vert =E\\left( f\\right) $ .", "Indeed, for any $\\mathbf {x}\\in Q$ $f_{1}(\\mathbf {x})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})\\le \\frac{1}{2}\\left( \\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}}f_{1}(\\mathbf {y})-\\min \\limits _{\\begin{array}{c}\\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}}f_{1}(\\mathbf {y})\\right) \\le E(f)(1.17)$ and $f_{1}(\\mathbf {x})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})\\ge \\frac{1}{2}\\left( \\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}}f_{1}(\\mathbf {y})-\\max \\limits _{\\begin{array}{c}\\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\end{array}}f_{1}(\\mathbf {y})\\right) \\ge -E(f).", "(1.18)$ Using the definition of $g_{2,1}(\\mathbf {a}^{2}\\cdot \\mathbf {x})$ , for any $\\mathbf {x}\\in Q$ we have $f_{1}(\\mathbf {x})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {x})-g_{2,1}(\\mathbf {a}^{2}\\cdot \\mathbf {x})$ $\\le \\frac{1}{2}\\left( \\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}\\cdot \\mathbf {y}=\\mathbf {a}^{2}\\cdot \\mathbf {x}\\end{array}}\\left( f_{1}(\\mathbf {y})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {y})\\right) -\\min \\limits _{ _{\\begin{array}{c}\\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}\\cdot \\mathbf {y}=\\mathbf {a}^{2}\\cdot \\mathbf {x}\\end{array}}}\\left( f_{1}(\\mathbf {y})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {y})\\right) \\right)$ and $f_{1}(\\mathbf {x})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {x})-g_{2,1}(\\mathbf {a}^{2}\\cdot \\mathbf {x})$ $\\le \\frac{1}{2}\\left( \\min \\limits _{_{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}\\cdot \\mathbf {y}=\\mathbf {a}^{2}\\cdot \\mathbf {x}\\end{array}}}\\left( f_{1}(\\mathbf {y})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {y})\\right) -\\max \\limits _{\\begin{array}{c}\\mathbf {y}\\in Q \\\\ \\mathbf {a}^{2}\\cdot \\mathbf {y}=\\mathbf {a}^{2}\\cdot \\mathbf {x}\\end{array}}\\left( f_{1}(\\mathbf {y})-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {y})\\right) \\right).$ Using (1.17) and (1.18) in the last two inequalities, we obtain that for any $\\mathbf {x}\\in Q$ $-E(f)\\le f_{2}(\\mathbf {x})=f_{1}(\\mathbf {x})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})-g_{2,1}(\\mathbf {a}^{2}{\\cdot }\\mathbf {x})\\le E(f).$ Therefore, $\\left\\Vert f_{2}\\right\\Vert \\le E(f).", "(1.19)$ Since $f_{2}(\\mathbf {x})-f(\\mathbf {x})$ belongs to $\\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ , we deduce from (1.19) that $\\left\\Vert f_{2}\\right\\Vert =E(f).$ By the same way, one can show that $\\Vert f_3\\Vert =E(f)$ , $\\Vert f_4\\Vert =E(f)$ , and so on.", "Thus we can write $\\left\\Vert f_{n} \\right\\Vert =E(f), \\ \\mbox{for any}\\ n .$ Let us now prove the implications $f_{1}(\\mathbf {p}_{0})<E(f)\\Rightarrow f_{2}(\\mathbf {p}_{0})<E(f)(1.20)$ and $f_{1}(\\mathbf {p}_{0})>-E(f)\\Rightarrow f_{2}(\\mathbf {p}_{0})>-E(f),(1.21)$ where $\\mathbf {p}_{0}\\in Q$ .", "First, we are going to prove the implication $f_{1}(\\mathbf {p}_{0})<E(f)\\Rightarrow f_{1}(\\mathbf {p}_{0})-g_{1,1}(\\mathbf {a}\\cdot \\mathbf {p}_{0})<E(f).", "(1.22)$ There are two possible cases.", "1) $\\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_0\\end{array}} f_{1} (\\mathbf {y} )=E(f)$ and $\\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_0\\end{array}} f_{1} (\\mathbf {y}) = -E(f).", "$ In this case, $g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {p}_0)=0$ .", "Hence $f_1(\\mathbf {p}_0)-g_{1,1}(\\mathbf {a}^{1}\\cdot \\mathbf {p}_0)< E(f).$ 2) $\\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}\\end{array}}f_{1}(\\mathbf {y})=E(f)-\\varepsilon _{1}$ and $\\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}\\end{array}}f_{1}(\\mathbf {y})=-E(f)+\\varepsilon _{2}$ , where $\\varepsilon _{1}$ , $\\varepsilon _{2}$ are nonnegative real numbers with the sum $\\varepsilon _{1}+\\varepsilon _{2}\\ne 0$ .", "In this case, $f_{1}(\\mathbf {p}_{0})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}) &\\le &\\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}\\end{array}}f_{1}(\\mathbf {y})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0})= \\\\&=&\\frac{1}{2}\\left( \\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}\\end{array}}f_{1}(\\mathbf {y})-\\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0}\\end{array}}f_{1}(\\mathbf {y})\\right) =$ $=E(f)-\\frac{\\varepsilon _{1}+\\varepsilon _{2}}{2}<E(f).$ Thus we have proved (1.22).", "Using this method, we can also prove that $f_{1}(\\mathbf {p}_{0})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0})<E(f)\\Rightarrow f_{1}(\\mathbf {p}_{0})-g_{1,1}(\\mathbf {a}^{1}{\\cdot }\\mathbf {p}_{0})-g_{2,1}(\\mathbf {a}^{2}{\\cdot }\\mathbf {p}_{0})<E(f).", "(1.23)$ Now (1.20) follows from (1.22) and (1.23).", "By the same way we can prove (1.21).", "It follows from implications (1.20) and (1.21) that if $f_{2}(\\mathbf {p}_{0})=E(f)$ , then $f_{1}(\\mathbf {p}_{0})=E(f)$ and if $f_{2}(\\mathbf {p}_{0})=-E(f)$ , then $f_{1}(\\mathbf {p}_{0})=-E(f)$ .", "This simply means that each path extremal for $f_{2}$ will be extremal for $f_{1}$ .", "Now we show that if any path extremal for $f_{1} $ has the length not more than $N$ , then any path extremal for $f_{2} $ has the length not more than $N-1$ .", "Suppose the contrary.", "Suppose that there is a path extremal for $f_{2}$ with the length equal to $N$ .", "Denote it by $q=(\\mathbf {q}_{1} ,\\mathbf {q}_{2} ,...,\\mathbf {q}_{N} )$ .", "Without loss of generality we may assume that $\\mathbf {a}^{2}\\cdot \\mathbf {q}_{N-1} =\\mathbf {a}^{2}\\cdot \\mathbf {q}_{N}$ .", "As it has been shown above, the path $q$ is also extremal for $f_{1}$ .", "Assume that $f_{1} (\\mathbf {q}_{N} )=E(f)$ .", "Then there is not any $\\mathbf {q}_{0} \\in Q$ such that $\\mathbf {q}_{0} \\ne \\mathbf {q}_{N} $ , $\\mathbf {a}^{1}\\cdot \\mathbf {q}_{0} =\\mathbf {a}^{1}\\cdot \\mathbf {q}_{N}$ and $f_{1} (\\mathbf {q}_{0} )=-E(f)$ .", "Indeed, if there was such $\\mathbf {q}_{0}$ and $\\mathbf {q}_{0} \\notin q$ , the path $(\\mathbf {q}_{1} ,\\mathbf {q}_{2} ,...,\\mathbf {q}_{N} ,\\mathbf {q}_{0} )$ would be extremal for $f_{1} $ .", "But this would contradict our assumption that any path extremal for $f_{1} $ has the length not more than $N$ .", "Besides, if there was such $\\mathbf {q}_{0} $ and $\\mathbf {q}_{0} \\in q$ , we could form some closed path extremal for $f_{1} $ .", "This also would contradict our assumption that there does not exist a closed path extremal for $f_{1} $ .", "Hence $\\max \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {q}_N\\end{array}} f_{1} (\\mathbf {y} )=E(f),\\ \\ \\min \\limits _{\\begin{array}{c} \\mathbf {y}\\in Q \\\\ \\mathbf {a}^{1}{\\cdot }\\mathbf {y}=\\mathbf {a}^{1}{\\cdot }\\mathbf {q}_N\\end{array}} f_{1} (\\mathbf {y})>-E(f).$ Therefore, $\\left| f_{1} (\\mathbf {q}_{N} )-g_{1,1} (\\mathbf {a}^{1}{\\cdot }\\mathbf {q}_N)\\right| <E(f).$ From the last inequality it is easy to obtain that (see the proof of implications (1.20) and (1.21)) $\\left|f_{2}(\\mathbf {q}_{N})\\right|<E(f).$ This means, on the contrary to our assumption, that the path $(\\mathbf {q}_{1},\\mathbf {q}_{2},...,\\mathbf {q}_{N})$ can not be extremal for $f_{2}$ .", "Hence any path extremal for $f_{2}$ has the length not more than $N-1$ .", "By the same way, it can be shown that any path extremal for $f_{3}$ has the length not more than $N-2$ , any path extremal for $f_{4}$ has the length not more than $N-3$ and so on.", "Finally, we will obtain that there is not a path extremal for $f_{N+1}$ .", "Hence there is not a point $\\mathbf {p}_{0}\\in Q$ such that $|f_{N+1}(\\mathbf {p}_{0})|=\\Vert f_{N+1}\\Vert $ .", "But by Lemma 1.4, all the functions $f_{2}$ , $f_{3},...,f_{N+1}$ are continuous on the compact set $Q$ ; hence the norm $\\Vert f_{N+1}\\Vert $ must be attained.", "This contradiction means that there exists a path extremal for $f_{1}$ with the infinite length.", "Sufficiency.", "Let a path $p=(\\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{2n})$ be closed and extremal for $f_{1}$ .", "Then $\\left|G_{p}(f)\\right|=\\left\\Vert f-g_{0}\\right\\Vert .", "(1.24)$ By Lemma 1.3, $\\left|G_{p}(f)\\right|\\le E(f).", "(1.25)$ It follows from (1.24) and (1.25) that $g_{0}$ is a best approximation.", "Let now a path $p=(\\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{n},...)$ be infinite and extremal for $f_{1}$ .", "Consider the sequence $p_{n}=(\\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{n})$ , $n=1,2,...,$ of finite paths.", "By condition (A) of the theorem, for each $p_{n}$ there exists a closed path $p_{n}^{m_{n}}=(\\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{n},\\mathbf {q}_{n+1},...,\\mathbf {q}_{n+m_{n}})$ , where $m_{n}\\le N_{0}$ .", "Then for any positive integer $n$ , $\\left|G_{p_{n}^{m_{n}}}(f)\\right|=\\left|G_{p_{n}^{m_{n}}}(f-g_{0})\\right|\\le \\frac{n\\left\\Vert f-g_{0}\\right\\Vert +m_{n}\\left\\Vert f-g_{0}\\right\\Vert }{n+m_{n}}=\\left\\Vert f-g_{0}\\right\\Vert $ and $\\left|G_{p_{n}^{m_{n}}}(f)\\right|\\ge \\frac{n\\left\\Vert f-g_{0}\\right\\Vert -m_{n}\\left\\Vert f-g_{0}\\right\\Vert }{n+m_{n}}=\\frac{n-m_{n}}{n+m_{n}}\\left\\Vert f-g_{0}\\right\\Vert .$ It follows from the above two inequalities for $\\left|G_{p_{n}^{m_{n}}}(f)\\right|$ that $\\sup _{p_{n}^{m_{n}}}\\left|G_{p_{n}^{m_{n}}}(f)\\right|=\\left\\Vert f-g_{0}\\right\\Vert .$ This together with Lemma 1.3 give that $\\Vert f-g_{0}\\Vert \\le E(f).$ Hence $g_{0}$ is a best approximation.", "Theorem 1.3 has been proved by using only methods of classical analysis.", "By implementing more deep techniques from functional analysis we will see below that condition (A) and the convexity assumption on a compact set $Q$ can be dropped.", "Theorem 1.4.", "Assume $Q$ is a compact subset of $\\mathbb {R}^{d}$ .", "A function $g_{0}\\in \\mathcal {R}$ is a best approximation to a function $f\\in C(Q)$ if and only if there exists a closed or infinite path $p=(\\mathbf {p}_{1},\\mathbf {p}_{2},...)$ extremal for the function $f-g_{0}$ .", "Sufficiency.", "There are two possible cases.", "The first case happens when there exists a closed path $(\\mathbf {p}_{1},...,\\mathbf {p}_{2n}) $ extremal for the function $f-g_{0}.$ Let us check that in this case, $f-g_{0}$ is a best approximation.", "Indeed, on the one hand, the following equalities are valid $\\left|\\sum _{i=1}^{2n}(-1)^{i}f(\\mathbf {p}_{i})\\right|=\\left|\\sum _{i=1}^{2n}(-1)^{i}\\left[ f-g_{0}\\right] (\\mathbf {p}_{i})\\right|=2n\\left\\Vert f-g_{0}\\right\\Vert .$ On the other hand, for any function $g\\in \\mathcal {R}$ , we have $\\left|\\sum _{i=1}^{2n}(-1)^{i}f(\\mathbf {p}_{i})\\right|=\\left|\\sum _{i=1}^{2n}(-1)^{i}\\left[ f-g\\right] (\\mathbf {p}_{i})\\right|\\le 2n\\left\\Vert f-g\\right\\Vert .$ Therefore, $\\left\\Vert f-g_{0}\\right\\Vert \\le \\left\\Vert f-g\\right\\Vert $ for any $g\\in \\mathcal {R}$ .", "That is, $g_{0}$ is a best approximation.", "The second case happens when we do not have closed paths extremal for $f-g_{0}$ , but there exists an infinite path $(\\mathbf {p}_{1},\\mathbf {p}_{2},...)$ extremal for $f-g_{0}$ .", "To analyze this case, consider the following linear functional $L_{q}:C(Q)\\rightarrow \\mathbb {R}\\text{, \\ }L_{q}(F)=\\frac{1}{n}\\sum _{i=1}^{n}(-1)^{i}F(\\mathbf {q}_{i}),$ where $q=\\lbrace \\mathbf {q}_{1},...,\\mathbf {q}_{n}\\rbrace $ is a finite path in $Q$ .", "It is easy to see that the norm $\\left\\Vert L_{q}\\right\\Vert \\le 1$ and $\\left\\Vert L_{q}\\right\\Vert =1$ if and only if the set of points of $q$ with odd indices $O=\\lbrace \\mathbf {q}_{i}\\in q:$ $i$ is an odd number$\\rbrace $ do not intersect with the set of points of $q$ with even indices $E=\\lbrace \\mathbf {q}_{i}\\in q:$ $i$ is an even number$\\rbrace $ .", "Indeed, from the definition of $L_{q}$ it follows that $\\left|L_{q}(F)\\right|\\le \\left\\Vert F\\right\\Vert $ for all functions $F\\in C(Q)$ , whence $\\left\\Vert L_{q}\\right\\Vert \\le 1.$ If $O\\cap E=\\varnothing $ , then for a function $F_{0}$ with the property $F_{0}(\\mathbf {q}_{i})=-1$ if $i$ is odd, $F_{0}(\\mathbf {q}_{i})=1$ if $i$ is even and $-1<F_{0}(x)<1$ elsewhere on $Q,$ we have $\\left|L_{q}(F_{0})\\right|=\\left\\Vert F_{0}\\right\\Vert .$ Hence, $\\left\\Vert L_{q}\\right\\Vert =1$ .", "Recall that such a function $F_{0}$ exists on the basis of Urysohn's great lemma.", "Note that if $q$ is a closed path, then $L_{q}$ annihilates all members of the class $\\mathcal {R}$ .", "But in general, when $q$ is not closed, we do not have the equality $L_{q}(g)=0,$ for all members $g\\in \\mathcal {R}$ .", "Nonetheless, this functional has the important property that $\\left|L_{q}(g_{1}+g_{2})\\right|\\le \\frac{2}{n}(\\left\\Vert g_{1}\\right\\Vert +\\left\\Vert g_{2}\\right\\Vert ),(1.26)$ where $g_{1}$ and $g_{2}$ are ridge functions with the directions $\\mathbf {a}_{1}$ and $\\mathbf {a}_{2}$ , respectively, that is, $g_{1}=g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {x})$ and $g_{2}=g_{2}(\\mathbf {a}_{2}\\cdot \\mathbf {x}).$ This property is important in the sense that if $n$ is sufficiently large, then the functional $L_{q}$ is close to an annihilating functional.", "To prove (1.26), note that $\\left|L_{q}(g_{1})\\right|\\le \\frac{2}{n}\\left\\Vert g_{1}\\right\\Vert $ and $\\left|L_{q}(g_{2})\\right|\\le \\frac{2}{n}\\left\\Vert g_{2}\\right\\Vert $ .", "These estimates become obvious if consider the chain of equalities $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{1})=g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{2}),$ $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{3})=g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{4}),...$ (or $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{2})=g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{3}),$ $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{4})=g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {q}_{5}),...$ ) for $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {x})$ and the corresponding chain of equalities for $g_{2}(\\mathbf {a}_{2}\\cdot \\mathbf {x})$ .", "Now consider the infinite path $p=(\\mathbf {p}_{1},\\mathbf {p}_{2},...)$ and form the finite paths $p_{k}=(\\mathbf {p}_{1},...,\\mathbf {p}_{k}),$ $k=1,2,... $ .", "For ease of notation, let us set $L_{k}=L_{p_{k}}.$ The sequence $\\lbrace L_{_{k}}\\rbrace _{k=1}^{\\infty }$ is a subset of the unit ball of the conjugate space $C^{\\ast }(Q).$ By the Banach-Alaoglu theorem, the unit ball is weak$^{\\text{*}}$ compact in the weak$^{\\text{*}}$ topology of $C^{\\ast }(Q)$ (see [139]).", "It follows from this theorem that the sequence $\\lbrace L_{_{k}}\\rbrace _{k=1}^{\\infty }$ must have weak$^{\\text{*}}$ cluster points.", "Suppose $L^{\\ast }$ denotes one of them.", "Without loss of generality we may assume that $L_{k}\\overset{weak^{\\ast }}{\\longrightarrow }L^{\\ast },$ as $k\\rightarrow \\infty .$ From (1.26) it follows that $L^{\\ast }(g_{1}+g_{2})=0.$ That is, $L^{\\ast }\\in \\mathcal {R}^{\\bot },$ where the symbol $\\mathcal {R}^{\\bot }$ stands for the annihilator of $\\mathcal {R}$ .", "Since in addition $\\left\\Vert L^{\\ast }\\right\\Vert \\le 1,$ we can write that $\\left|L^{\\ast }(f)\\right|=\\left|L^{\\ast }(f-g)\\right|\\le \\left\\Vert f-g\\right\\Vert ,(1.27)$ for all functions $g\\in \\mathcal {R}.$ On the other hand, since the infinite bolt $p$ is extremal for $f-g_{0}$ $\\left|L_{k}(f-g_{0})\\right|=\\left\\Vert f-g_{0}\\right\\Vert ,\\text{ }k=1,2,...$ Therefore, $\\left|L^{\\ast }(f)\\right|=\\left|L^{\\ast }(f-g_{0})\\right|=\\left\\Vert f-g_{0}\\right\\Vert .", "(1.28)$ From (1.27) and (1.28) we conclude that $\\left\\Vert f-g_{0}\\right\\Vert \\le \\left\\Vert f-g\\right\\Vert ,$ for all $g\\in \\mathcal {R}.$ In other words, $g_{0}$ is a best approximation to $f$ .", "We proved the sufficiency of the theorem.", "Necessity.", "The proof of this part is mainly based on the following result of Singer [142]: Let $X$ be a compact space, $U$ be a linear subspace of $C(X)$ , $f\\in C(X)\\backslash U$ and $u_{0}\\in U.$ Then $u_{0}$ is a best approximation to $f$ if and only if there exists a regular Borel measure $\\mu $ on $X$ such that (1) The total variation $\\left\\Vert \\mu \\right\\Vert =1$ ; (2) $\\mu $ is orthogonal to the subspace $U$ , that is, $\\int _{X}ud\\mu =0$ for all $u\\in U$ ; (3) For the Jordan decomposition $\\mu =\\mu ^{+}-\\mu ^{-}$ , $f(x)-u_{0}(x)=\\left\\lbrace \\begin{array}{c}\\left\\Vert f-u_{0}\\right\\Vert \\text{ for }x\\in S^{+}, \\\\-\\left\\Vert f-u_{0}\\right\\Vert \\text{ for }x\\in S^{-},\\end{array}\\right.$ where $S^{+}$ and $S^{-}$ are closed supports of the positive measures $\\mu ^{+}$ and $\\mu ^{-}$ , respectively.", "Let us show how we use this theorem in the proof of necessity part of our theorem.", "Assume $g_{0}\\in \\mathcal {R}$ is a best approximation.", "For the subspace $\\mathcal {R},$ the existence of a measure $\\mu $ satisfying the conditions (1)-(3) is a direct consequence of Singer's result.", "Let $\\mathbf {x}_{0}$ be any point in $S^{+}.$ Consider the point $y_{0}=\\mathbf {a}_{1}\\cdot \\mathbf {x}_{0}$ and a $\\delta $ -neighborhood of $y_{0}$ .", "That is, choose an arbitrary $\\delta >0$ and consider the set $I_{\\delta }=(y_{0}-\\delta ,y_{0}+\\delta )\\cap \\mathbf {a}_{1}\\cdot Q.$ Here, $\\mathbf {a}_{1}\\cdot Q=\\lbrace \\mathbf {a}_{1}\\cdot \\mathbf {x}:$ $\\mathbf {x}\\in Q\\rbrace .$ For any subset $E\\subset \\mathbb {R}$ , put $E^{i}=\\lbrace \\mathbf {x}\\in Q:\\mathbf {a}_{i}\\cdot \\mathbf {x}\\in E\\rbrace ,\\text{ }i=1,2.\\text{ }$ Clearly, for some sets $E,$ one or both the sets $E^{i}$ may be empty.", "Since $I_{\\delta }^{1}\\cap S^{+}$ is not empty (note that $\\mathbf {x}_{0}\\in I_{\\delta }^{1}$ ), it follows that $\\mu ^{+}(I_{\\delta }^{1})>0.$ At the same time $\\mu (I_{\\delta }^{1})=0,$ since $\\mu $ is orthogonal to all functions $g_{1}(\\mathbf {a}_{1}\\cdot \\mathbf {x}).$ Therefore, $\\mu ^{-}(I_{\\delta }^{1})>0.$ We conclude that $I_{\\delta }^{1}\\cap S^{-}$ is not empty.", "Denote this intersection by $A_{\\delta }.$ Tending $\\delta $ to $0,$ we obtain a set $A$ which is a subset of $S^{-}$ and has the property that for each $\\mathbf {x}\\in A,$ we have $\\mathbf {a}_{1}\\cdot \\mathbf {x}=\\mathbf {a}_{1}\\cdot \\mathbf {x}_{0}.$ Fix any point $\\mathbf {x}_{1}\\in A$ .", "Changing $\\mathbf {a}_{1}$ , $\\mu ^{+}$ , $S^{+}$ to $\\mathbf {a}_{2}$ , $\\mu ^{-} $ and $S^{-}$ correspondingly, repeat the above process with the point $y_{1}=\\mathbf {a}_{2}\\cdot \\mathbf {x}_{1}$ and a $\\delta $ -neighborhood of $y_{1}$ .", "Then we obtain a point $\\mathbf {x}_{2}\\in S^{+}$ such that $\\mathbf {a}_{2}\\cdot \\mathbf {x}_{2}=\\mathbf {a}_{2}\\cdot \\mathbf {x}_{1}.$ Continuing this process, one can construct points $\\mathbf {x}_{3}$ , $\\mathbf {x}_{4}$ , and so on.", "Note that the set of all constructed points $\\mathbf {x}_{i}$ , $i=0,1,...,$ forms a path.", "By Singer's above result, this path is extremal for the function $f-g_{0}$ .", "We have proved the necessity and hence Theorem 1.4.", "Theorem 1.4, in a more general setting, was proven in Pinkus [137] under additional assumption that $Q$ is convex.", "Convexity assumption was made to guarantee continuity of the following functions $g_{1,i}(t)=\\max _{\\begin{array}{c} \\mathbf {x}\\in Q \\\\ \\mathbf {a}_{i}\\cdot \\mathbf {x}=t\\end{array}}F(\\mathbf {x})\\ \\ \\text{and }\\ g_{2,i}(t)=\\min \\limits _{\\begin{array}{c}\\mathbf {x}\\in Q \\\\ \\mathbf {a}_{i}\\cdot \\mathbf {x}=t\\end{array}}F(\\mathbf {x}),\\text{ }i=1,2,$ where $F$ is an arbitrary continuous function on $Q$ .", "Note that in the proof of Theorem 1.4 we did not need continuity of these functions.", "It is well known that characterization theorems of this type are very essential in approximation theory.", "Chebyshev was the first to prove a similar result for polynomial approximation.", "Khavinson [89] characterized extremal elements in the special case of the problem considered here.", "His case allows the approximation of a continuous bivariate function $f\\left( {x,y}\\right) $ by functions of the form $\\varphi \\left( {x}\\right) +\\psi \\left( {y}\\right)$ .", "In 1951, Diliberto and Straus [36] established a formula for the error of approximation of a bivariate function by sums of univariate functions.", "Their formula contains the supremum over all closed bolts (see Section 3.3.1).", "Although the mentioned formula is valid for all continuous functions, it is not easily calculable.", "Therefore, it cannot give a desired effect if one is interested in the precise value of the approximation error.", "After this general result some authors started to seek easily calculable formulas for the approximation error by considering not the whole space of continuous functions, but some subsets thereof (see, for example, [9], [12], [58], [59], [89], [138]).", "These subsets were chosen so that they could provide precise and easy computation of the approximation error.", "Since the set of ridge functions contains univariate functions, one may ask for explicit formulas for the error of approximation of a multivariate function by sums of ridge functions.", "In this section, we see how with the use of Theorem 1.3 (or 1.4) it is possible to find the approximation error and construct an extremal element in the problem of approximation by sums of ridge functions.", "We restrict ourselves to $\\mathbb {R}^{2}.$ To make the problem more precise, let $\\Omega $ be a compact set in $\\mathbb {R}^{2},$ $f \\in C\\left( {\\Omega }\\right)$ , $\\mathbf {a}=\\left( {a_{1},a_{2}}\\right)$ and $\\mathbf {b}=\\left( {b_{1},b_{2}}\\right)$ be linearly independent vectors.", "Consider the approximation of $f$ by functions from $\\mathcal {R}=\\mathcal {R}\\left(\\mathbf {a},\\mathbf {b}\\right)$ .", "We want, under some suitable conditions on $f\\;$ and $\\Omega $ , to establish a formula for an easy and direct computation of the approximation error $E\\left(f,\\mathcal {R}\\right)$ .", "Theorem 1.5.", "Let $\\Omega =\\left\\lbrace \\mathbf {x}\\in \\mathbb {R}^{2}:c_{1}\\le \\mathbf {a}\\cdot \\mathbf {x}\\le d_{1},\\ \\ c_{2}\\le \\mathbf {b}\\cdot \\mathbf {x}\\le d_{2}\\right\\rbrace ,$ where $c_{1}<d_{1}$ and $c_{2}<d_{2}$ .", "Let a function $f(\\mathbf {x})\\in C(\\Omega )$ have the continuous partial derivatives $\\frac{\\partial ^{2}f}{\\partial x_{1}^{2}},\\frac{\\partial ^{2}f}{\\partial x_{1}\\partial x_{2}},\\frac{\\partial ^{2}f}{\\partial x_{2}^{2}}$ and for any $\\mathbf {x}\\in \\Omega $ $\\frac{\\partial ^{2}f}{\\partial x_{1}\\partial x_{2}}\\left(a_{1}b_{2}+a_{2}b_{1}\\right) -\\frac{\\partial ^{2}f}{\\partial x_{1}^{2}}a_{2}b_{2}-\\frac{\\partial ^{2}f}{\\partial x_{2}^{2}}a_{1}b_{1}\\ge 0.$ Then $E\\left(f,\\mathcal {R}\\right)=\\frac{1}{4}\\left(f_{1}(c_{1},c_{2})+f_{1}(d_{1},d_{2})-f_{1}(c_{1},d_{2})-f_{1}(d_{1},c_{2})\\right) ,$ where $f_{1}(y_{1},y_{2})=f\\left( \\frac{y_{1}b_{2}-y_{2}a_{2}}{a_{1}b_{2}-a_{2}b_{1}},\\frac{y_{2}a_{1}-y_{1}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}\\right) .", "(1.29)$ Introduce the new variables $y_{1}=a_{1}x_{1}+a_{2}x_{2},\\ \\ y_{2}=b_{1}x_{1}+b_{2}x_{2}.", "(1.30)$ Since the vectors $(a_{1},a_{2})$ and $(b_{1},b_{2})$ are linearly independent, for any $(y_{1},y_{2})\\in Y$ , where $Y=[c_{1},d_{1}]\\times [c_{2},d_{2}]$ , there exists only one solution $(x_{1},x_{2})\\in \\Omega $ of the system (1.30).", "The coordinates of this solution are $x_{1}=\\frac{y_{1}b_{2}-y_{2}a_{2}}{a_{1}b_{2}-a_{2}b_{1}},\\qquad \\ x_{2}=\\frac{y_{2}a_{1}-y_{1}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}.", "(1.31)$ The linear transformation (1.31) transforms the function $f(x_{1},x_{2})$ to the function $f_{1}(y_{1},y_{2})$ .", "Consider the approximation of $f_{1}(y_{1},y_{2})$ from the set $\\mathcal {Z}=\\left\\lbrace z_{1}(y_{1})+z_{2}(y_{2}):z_{i}\\in C(\\mathbb {R}),\\ i=1,2\\right\\rbrace .$ It is easy to see that $E\\left( f,\\mathcal {R}\\right) =E\\left( f_{1},\\mathcal {Z}\\right) .", "(1.32)$ With each rectangle $S=[u_{1} ,v_{1} ]\\times [u_{2} ,v_{2}]\\subset Y$ we associate the functional $L\\left(h, S\\right) =\\frac{1}{4} \\left(h(u_{1} ,u_{2} )+h(v_{1} ,v_{2})-h(u_{1} ,v_{2} )-h (v_{1} ,u_{2})\\right),\\ \\ h\\in C(Y).$ This functional has the following obvious properties: (i) $L(z,S)=0$ for any $z\\in \\mathcal {Z}$ and $S\\subset Y$ .", "(ii) For any point $(y_{1} ,y_{2} )\\in Y$ , $L(f_1,Y)=\\sum \\limits _{i=1}^{4}L(f_1, S_{i} ) $ , where $S_{1} =[c_{1} ,y_{1}]\\times [c_{2} ,y_{2}],$ $S_{2} =[y_{1} ,d_{1}]\\times [y_{2} ,d_{2}],$ $S_{3}=[c_{1} ,y_{1}]\\times [y_{2} ,d_{2}],$ $S_{4} =[y_{1} ,d_{1}]\\times [c_{2},y_{2}]$ .", "By the conditions of the theorem, it is not difficult to verify that $\\frac{\\partial ^{2} f_1}{\\partial y_{1} \\partial y_{2} } \\ge 0\\ \\ \\mbox{forany}\\ \\ (y_{1} ,y_{2} )\\in Y.$ Integrating both sides of the last inequality over arbitrary rectangle $S=[u_{1},v_{1}]\\times [u_{2},v_{2}]\\subset Y$ , we obtain that $L\\left( f_{1},S\\right) \\ge 0.", "(1.33)$ Set the function $f_{2}(y_{1},y_{2})=L\\left( f_{1},S_{1}\\right) +L\\left( f_{1},S_{2}\\right)-L\\left( f_{1},S_{3}\\right) -L\\left( f_{1},S_{4}\\right) .", "(1.34)$ It is not difficult to verify that the function $f_{1}-f_{2}$ belongs to $\\mathcal {Z}$ .", "Hence $E\\left( f_{1},\\mathcal {Z}\\right) =E\\left( f_{2},\\mathcal {Z}\\right) .", "(1.35)$ Calculate the norm $\\left\\Vert f_{2}\\right\\Vert $ .", "From the property (ii), it follows that $f_{2}(y_{1},y_{2})=L(f_{1},Y)-2(L(f_{1},S_{3})+L(f_{1},S_{4}))$ and $f_{2}(y_{1},y_{2})=2\\left( L\\left( f_{1},S_{1}\\right) +L\\left(f_{1},S_{2}\\right) \\right) -L\\left( f_{1},Y\\right) .$ From the last equalities and (1.33), we obtain that $\\left|f_{2}(y_{1},y_{2})\\right|\\le L\\left( f_{1},Y\\right) ,\\ \\mbox{for any}\\ (y_{1},y_{2})\\in Y.$ On the other hand, one can check that $f_{2}(c_{1},c_{2})=f_{2}(d_{1},d_{2})=L\\left( f_{1},Y\\right) (1.36)$ and $f_{2}(c_{1},d_{2})=f_{2}(d_{1},c_{2})=-L\\left( f_{1},Y\\right) .", "(1.37)$ Therefore, $\\left\\Vert f_{2}\\right\\Vert =L\\left( f_{1},Y\\right) .", "(1.38)$ Note that the points $(c_{1},c_{2}),(c_{1},d_{2}),(d_{1},d_{2}),(d_{1},c_{2}) $ in the given order form a closed path with respect to the directions $(0;1) $ and $(1;0)$ .", "We conclude from (1.36)-(1.38) that this path is extremal for $f_{2}$ .", "By Theorem 1.3, $z_{0}=0$ is a best approximation to $f_{2}$ .", "Hence $E\\left( f_{2},\\mathcal {Z}\\right) =L\\left( f_{1},Y\\right) .", "(1.39)$ Now from (1.32),(1.35) and (1.39) we finally conclude that $E\\left( f,\\mathcal {R}\\right) =L\\left( f_{1},Y\\right) =\\frac{1}{4}\\left(f_{1}(c_{1},c_{2})+f_{1}(d_{1},d_{2})-f_{1}(c_{1},d_{2})-f_{1}(d_{1},c_{2})\\right) ,$ which is the desired result.", "Corollary 1.4.", "Let all the conditions of Theorem 1.5 hold and $f_{1}(y_{1},y_{2})$ is the function defined in (1.29).", "Then the function $g_{0}(y_{1},y_{2})=g_{1,0}(y_{1})+g_{2,0}(y_{2})$ , where $g_{1,0}(y_{1})=\\frac{1}{2}f_{1}(y_{1},c_{2})+\\frac{1}{2}f_{1}(y_{1},d_{2})-\\frac{1}{4}f_{1}(c_{1},c_{2})-\\frac{1}{4}f_{1}(d_{1},d_{2}),$ $g_{2,0}(y_{2})=\\frac{1}{2}f_{1}(c_{1},y_{2})+\\frac{1}{2}f_{1}(d_{1},y_{2})-\\frac{1}{4}f_{1}(c_{1},d_{2})-\\frac{1}{4}f_{1}(d_{1},c_{2})$ and $y_{1}=a_{1}x_{1}+a_{2}x_{2}$ , $y_{2}=b_{1}x_{1}+b_{2}x_{2}$ , is a best approximation from the set $\\mathcal {R}(a,b)$ to the function $f$ .", "It is not difficult to verify that the function $f_{2}(y_{1},y_{2})$ defined in (1.34) has the form $f_{2}(y_{1},y_{2})=f_{1}(y_{1},y_{2})-g_{1,0}(y_{1})-g_{2,0}(y_{2}).$ On the other hand, we know from the proof of Theorem 1.5 that $E(f_{1},\\mathcal {Z})=\\left\\Vert f_{2}\\right\\Vert .$ Therefore, the function $g_{1,0}(y_{1})+g_{2,0}(y_{2})$ is a best approximation to $f_{1}$ .", "Then the function $g_{1,0}(\\mathbf {a}\\cdot \\mathbf {x})+g_{2,0}(\\mathbf {b}\\cdot \\mathbf {x})$ is an extremal element from $\\mathcal {R}(\\mathbf {a},\\mathbf {b})$ to $f(\\mathbf {x})$ .", "Remark 1.1.", "Rivlin and Sibner [138], and Babaev [12] proved Theorem 1.5 for the case in which $\\mathbf {a}$ and $\\mathbf {b}$ are the coordinate vectors.", "Our proof of Theorem 1.5 is different, short and elementary.", "Moreover, it has turned out to be useful in constructing an extremal element (see the proof of Corollary 1.4).", "Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be nonzero directions in $\\mathbb {R}^{d}$ .", "One may ask the following question: are there cases in which the set $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is dense in the space of all continuous functions?", "Undoubtedly, a positive answer depends on the geometrical structure of compact sets over which all the considered functions are defined.", "This problem may be interesting in the theory of partial differential equations.", "Take, for example, equation (1.10).", "A positive answer to the problem means that for any continuous function $f$ there exist solutions of the given equation uniformly converging to $f$ .", "It should be remarked that our problem is a special case of the problem considered by Marshall and O'Farrell.", "In [123], they obtained a necessary and sufficient condition for a sum $A_{1}+A_{2}$ of two subalgebras to be dense in $C(U)$ , where $C(U)$ denotes the space of real-valued continuous functions on a compact Hausdorff space $U$ .", "Below we describe Marshall and O' Farrell's result for sums of ridge functions.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}.$ The relation on $X$ , defined by setting $\\mathbf {x}\\approx \\mathbf {y}$ if $\\mathbf {x}$ and $\\mathbf {y}$ belong to some path in $X$ , is an equivalence relation.", "The equivalence classes we call orbits.", "Theorem 1.6.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ with all its orbits closed.", "The set $\\mathcal {\\ R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is dense in $C(X)$ if and only if $X$ contains no closed path with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "The proof immediately follows from proposition 2 in [122] established for the sum of two algebras.", "Since that proposition was given without proof, for completeness of the exposition we give the proof of Theorem 1.6.", "Necessity.", "If $X$ has closed paths, then $X$ has a closed path $p^{\\prime }=\\left( \\mathbf {p}_{1}^{\\prime },...,\\mathbf {p}_{2m}^{\\prime }\\right) $ such that all points $\\mathbf {p}_{1}^{\\prime },...,\\mathbf {p}_{2m}^{\\prime }$ are distinct.", "In fact, such a special path can be obtained from any closed path $p=\\left( \\mathbf {p}_{1},...,\\mathbf {p}_{2n}\\right) $ by the following simple algorithm: if the points of the path $p$ are not all distinct, let $i$ and $k>0$ be the minimal indices such that $\\mathbf {p}_{i}=\\mathbf {p}_{i+2k}$ ; delete from $p$ the subsequence $\\mathbf {p}_{i+1},...,\\mathbf {p}_{i+2k}$ and call $p$ the obtained path; repeat the above step until all points of $p$ are all distinct; set $p^{\\prime }:=p$ .", "By Urysohn's great lemma, there exist continuous functions $h=h(\\mathbf {x})$ on $X$ such that $h(\\mathbf {p}_{i}^{\\prime })=1$ , $i=1,3,...,2m-1$ , $h(\\mathbf {p}_{i}^{\\prime })=-1$ , $i=2,4,...,2m$ and $-1<h(\\mathbf {x})<1$ elsewhere.", "Consider the measure $\\mu _{p^{\\prime }}=\\frac{1}{2m}\\sum _{i=1}^{2m}(-1)^{i-1}\\delta _{\\mathbf {p}_{i}^{\\prime }}\\text{ ,}$ where $\\delta _{\\mathbf {p}_{i}^{\\prime }}$ is a point mass at $\\mathbf {p}_{i}^{\\prime }$ .", "For this measure, $\\int \\limits _{X}hd\\mu _{p^{\\prime }}=1$ and $\\int \\limits _{X}gd\\mu _{p^{\\prime }}=0$ for all functions $g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ .", "Thus the set $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ cannot be dense in $C(X)$ .", "Sufficiency.", "We are going to prove that the only annihilating regular Borel measure for $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is the zero measure.", "Suppose, contrary to this assumption, there exists a nonzero annihilating measure on $X$ for $\\mathcal {R}\\left(\\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ .", "The class of such measures with total variation not more than 1 we denote by $S.$ Clearly, $S$ is weak-* compact and convex.", "By the Krein-Milman theorem, there exists an extreme measure $\\mu $ in $S.$ Since the orbits are closed, $\\mu $ must be supported on a single orbit.", "Denote this orbit by $T.$ For $i=1,2,$ let $X_{i}$ be the quotient space of $X$ obtained by identifying the points $\\mathbf {y}$ and $\\mathbf {z}$ whenever $\\mathbf {a}^{i}{\\cdot }\\mathbf {y}=\\mathbf {a}^{i}{\\cdot }\\mathbf {z}$ .", "Let $\\pi _{i}$ be the natural projection of $X$ onto $X_{i}$ .", "For a fixed point $t\\in X$ set $T_{1}=\\lbrace t\\rbrace $ , $T_{2}=\\pi _{1}^{-1}\\left( \\pi _{1}T_{1}\\right) $ , $T_{3}=\\pi _{2}^{-1}\\left( \\pi _{2}T_{2}\\right) $ , $T_{4}=\\pi _{1}^{-1}\\left( \\pi _{1}T_{3}\\right) $ , $...$ Obviously, $T_{1}\\subset T_{2}\\subset T_{3}\\subset \\cdot \\cdot \\cdot $ .", "Therefore, for some $k\\in \\mathbb {N}$ , $\\left|\\mu \\right|(T_{2k})>0$ , where $\\left|\\mu \\right|$ is a total variation measure of $\\mu $ .", "Since $\\mu $ is orthogonal to every continuous function of the form ${g\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) }$ , $\\mu (T_{2k})=0$ .", "From the Haar decomposition $\\mu (T_{2k})=\\mu ^{+}(T_{2k})-\\mu ^{-}(T_{2k})$ it follows that $\\mu ^{+}(T_{2k})=\\mu ^{-}(T_{2k})>0$ .", "Fix a Borel subset $S_{0}\\subset T_{2k}$ such that $\\mu ^{+}(S_{0})>0$ and $\\mu ^{-}(S_{0})=0$ .", "Since $\\mu $ is orthogonal to every continuous function of the form ${g\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) }$ , $\\mu (\\pi _{2}^{-1}\\left( \\pi _{2}S_{0}\\right) )=0.$ Therefore, one can chose a Borel set $S_{1}$ such that $S_{1}\\subset \\pi _{2}^{-1}\\left( \\pi _{2}S_{0}\\right) \\subset T_{2k+1}$ , $S_{1}\\cap S_{0}=\\varnothing $ , $\\mu ^{+}(S_{1})=0$ , $\\mu ^{-}(S_{1})\\geqslant \\mu ^{+}(S_{0})$ .", "By the same way one can chose a Borel set $S_{2}$ such that $S_{2}\\subset \\pi _{1}^{-1}\\left( \\pi _{1}S_{1}\\right) \\subset T_{2k+2}$ , $S_{2}\\cap S_{1}=\\varnothing $ , $\\mu ^{-}(S_{2})=0$ , $\\mu ^{+}(S_{2})\\geqslant \\mu ^{-}(S_{1})$ , and so on.", "The sets $S_{0},S_{1},S_{2},...$ are pairwise disjoint.", "For otherwise, there would exist positive integers $n$ and $m,$ with $n<m$ and a path $(y_{n},y_{n+1},...,y_{m})$ such that $y_{i}\\in S_{i}$ for $i=n,...,m$ and $y_{m}\\in S_{m}\\cap S_{n}$ .", "But then there would exist paths $(z_{1},z_{2},...,z_{n-1},y_{n})$ and $(z_{1},z_{2}^{^{\\prime }},...,z_{n-1}^{^{\\prime }},y_{m})$ with $z_{i}$ and $z_{i}^{^{\\prime }}$ in $T_{i}$ for $i=2,...,n-1.$ Hence, the set $\\lbrace z_{1},z_{2},...,z_{n-1},y_{n},y_{n+1},...,y_{m},z_{n-1}^{^{\\prime }},...,z_{2}^{^{\\prime }},z_{1}\\rbrace $ would contain a closed path.", "This would contradict our assumption on $X.$ Now, since the sets $S_{0},S_{1},S_{2},...,$ are pairwise disjoint and $\\left|\\mu \\right|(S_{i})\\geqslant \\mu ^{+}(S_{0})>0$ for each $i=1,2,...,$ it follows that the total variation of $\\mu $ is infinite.", "This contradiction completes the proof.", "The following corollary concerns the problem considered by Colitschek and Light [47].", "Corollary 1.5.", "Let $D$ be a compact subset of $\\mathbb {R}^{2}$ with all its orbits closed.", "Let $W$ denote the set of all solutions of the wave equation $\\frac{\\partial ^{2}w}{\\partial s\\partial t}(s,t)=0,\\;\\ \\ \\ \\ (s,t)\\in D.$ Then $\\inf \\limits _{w\\in W}\\left\\Vert f-w\\right\\Vert =0$ for any continuous function $f(s,t)$ on $D$ if and only if $D$ contains no closed bolt of lightning.", "Let $\\pi _{1}$ and $\\pi _{2}$ denote the usual coordinate projections, viz: $\\pi _{1}(s,t)=s$ and $\\pi _{2}(s,t)=t$ , $(s,t)\\in \\mathbb {R}^{2}$ .", "Set $S=\\pi _{1}(D)$ and $T=\\pi _{2}(D)$ .", "It is easy to see that $W=\\left\\lbrace w\\in C(D):w(s,t)=x(s)+y(t),\\;\\ \\ x\\in C^{2}(S),\\;\\ y\\in C^{2}(T)\\right\\rbrace .$ Set $\\widetilde{W}=\\left\\lbrace w\\in C(D):w(s,t)=x(s)+y(t),\\;\\ \\ x\\in C(S),\\;\\ y\\in C(T)\\right\\rbrace .$ Since the set $W$ is dense in $\\widetilde{W},$ $\\inf \\limits _{w\\in W}\\left\\Vert f-w\\right\\Vert =\\inf \\limits _{w\\in \\widetilde{W}}\\left\\Vert f-w\\right\\Vert .$ But by Theorem 1.6, the equality $\\inf \\limits _{w\\in \\widetilde{W}}\\left\\Vert f-w\\right\\Vert =0$ holds for any $f\\in C(D)$ if and only if $D$ contains no closed bolt of lightning.", "Let us discuss some difficulties that arise when studying sums of more than two ridge functions.", "Consider the set $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) ={\\left\\lbrace \\sum \\limits _{i=1}^{r}{g}_{i}{{{\\left( \\mathbf {a}{^{i}\\cdot }\\mathbf {x}\\right) ,g}}}_{{i}}{{{\\ \\in C\\left( \\mathbb {R}\\right) ,i=1,...,r}}}\\right\\rbrace },$ where $\\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}$ are pairwise linearly independent vectors in $\\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ .", "Let $r\\ge 3$ .", "How can one define a path in this general case?", "Recall that in the case when $r=2$ , a path is an ordered set of points $\\left( \\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{n}\\right) $ in $\\mathbb {R}^{d}$ with edges $\\mathbf {p}_{i}\\mathbf {p}_{i+1}$ in alternating hyperplanes.", "The first, the third, the fifth,... hyperplanes (also the second, the fourth, the sixth,... hyperplanes) are parallel.", "If not differentiate between parallel hyperplanes, the path $\\left( \\mathbf {p}_{1},\\mathbf {p}_{2},...,\\mathbf {p}_{n}\\right) $ can be considered as a trace of some point traveling in two alternating hyperplanes.", "In this case, if the point starts and stops at the same location (i.e., if $\\mathbf {p}_{n}=\\mathbf {p}_{1})$ and $n$ is an odd number, then the path functional $G(f)=\\frac{1}{n-1}\\sum \\limits _{i=1}^{n-1}(-1)^{i+1}f(\\mathbf {p}_{i}),$ annihilates sums of ridge functions with the corresponding two fixed directions.", "The picture becomes quite different and more complicated when the number of directions more than two.", "The simple generalization of the above-mentioned arguments demands a point traveling in three or more alternating hyperplanes.", "But in this case the appropriate generalization of the functional $G$ does not annihilate functions from $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ .", "There were several attempts to fill this gap in the special case when $r=d$ and $\\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}$ are the coordinate vectors.", "Unfortunately, all these attempts failed (see, for example, the attempts in [36], [48] and the refutations in [9], [126]).", "At the end of this subsection we want to draw the readers attention to the following problems.", "All these problems are open and cannot be solved by the methods presented here.", "Let $Q$ be a compact subset of $\\mathbb {R}^{d}$ .", "Consider the approximation of a continuous function defined on $Q$ by functions from $\\mathcal {R}\\left(\\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right)$ .", "Let $r\\ge 3$ .", "Problem 3.", "Characterize those functions from $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ that are extremal to a given continuous function.", "Problem 4.", "Establish explicit formulas for the error in approximating from $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ and construct a best approximation.", "Problem 5.", "Find necessary and sufficient geometrical conditions for the set $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ to be dense in $C(Q)$ .", "It should be remarked that in [122], Problem 5 was set up for the sum of $r $ subalgebras of $C(Q)$ .", "Lin and Pinkus [112] proved that the set $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ ($r$ may be very large) is not dense in $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compact subsets of $\\mathbb {R}^{d}$ .", "That is, there are compact sets $Q\\subset \\mathbb {R}^{d}$ such that $\\mathcal {R}\\left( \\mathbf {a}{^{1},...,}\\mathbf {a}{^{r}}\\right) $ is not dense in $C(Q)$ .", "In the case $r=2$ , Theorem 1.6 complements this result, by describing compact sets $Q\\subset \\mathbb {R}^{2}$ , for which $\\mathcal {R}\\left( \\mathbf {a}{^{1},}\\mathbf {a}{^{2}}\\right) $ is dense in $C(Q)$ .", "In this section, we find geometric means of deciding if any continuous multivariate function can be represented by a sum of two continuous ridge functions.", "In this section, we will consider the following representation problem associated with the set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) .$ Problem 6.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}.$ Give geometrical conditions that are necessary and sufficient for $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)=C\\left( X\\right),$ where $C\\left( X\\right) $ is the space of continuous functions on $X$ furnished with the uniform norm.", "We solve this problem for $r=2$ .", "Problem 6, like Problems 3–5 from the previous section, is open in the case $r\\ge 3$ .", "Geometrical characterization of compact sets $X \\subset \\mathbb {R}^{d}$ with the property $\\mathcal {R}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)=C\\left(X\\right)$ , $r\\ge 3$ , seems to be beyond the scope of the methods discussed herein.", "Nevertheless, recall that this problem in a quite abstract form, which involves regular Borel measures on $X$ , was solved by Sternfeld (see Section 1.1.1).", "In the sequel, we will use the notation $H_{1}=H_{1}\\left( X\\right) =\\left\\lbrace g_{1}\\left( \\mathbf {a}^{1}\\cdot \\mathbf {x}\\right) :g_{1}\\in C\\left( \\mathbb {R}\\right) \\right\\rbrace ,$ $H_{2}=H_{2}\\left( X\\right) =\\left\\lbrace g_{2}\\left( \\mathbf {a}^{2}\\cdot \\mathbf {x}\\right) :g_{2}\\in C\\left( \\mathbb {R}\\right) \\right\\rbrace .$ Note that by this notation, $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) =H_{1}+H_{2}.$ At the end of this section, we generalize the obtained result from $H_{1}+H_{2}$ to the set of sums $g_{1}\\left( h_{1}\\left( \\mathbf {x}\\right)\\right) +g_{2}\\left( h_{2}\\left( \\mathbf {x}\\right) \\right)$ , where $h_{1},h_{2}$ are fixed continuous functions on $X$ .", "Theorem 1.7.", "Let $X$ be a compact subset of $\\mathbb {R}^{d} $ .", "The equality $H_{1}\\left( X\\right) +H_{2}\\left( X\\right) =C\\left( X\\right)$ holds if and only if $X$ contains no closed path and there exists a positive integer $n_{0}$ such that the lengths of paths in $X$ are bounded by $n_{0}$ .", "Necessity.", "Let $H_{1}+H_{2}=C\\left( X\\right) $ .", "Consider the linear operator $A:H_{1}\\times H_{2}\\rightarrow C\\left( X\\right), ~~~ A\\left[ \\left(g_{1},g_{2}\\right) \\right] =g_{1}+g_{2},$ where $g_{1}\\in H_{1},g_{2}\\in H_{2}.$ The norm on $H_{1}\\times H_{2}$ we define as $\\left\\Vert \\left( g_{1},g_{2}\\right) \\right\\Vert =\\left\\Vert g_{1}\\right\\Vert +\\left\\Vert g_{2}\\right\\Vert .$ It is obvious that the operator $A$ is continuous with respect to this norm.", "Besides, since $C\\left( X\\right) =H_{1}+H_{2},$ $A$ is a surjection.", "Consider the conjugate operator $A^*:C\\left( X\\right) ^{\\ast }\\rightarrow \\left[ H_{1}\\times H_{2}\\right]^{\\ast }, ~~~ A^{\\ast }\\left[ G\\right] =\\left( G_{1},G_{2}\\right) ,$ where the functionals $G_{1}$ and $G_{2}$ are defined as follows $G_{1}\\left( g_{1}\\right) =G\\left( g_{1}\\right) ,g_{1}\\in H_{1}; ~~~G_{2}\\left( g_{2}\\right) =G\\left( g_{2}\\right) ,g_{2}\\in H_{2}.$ An element $\\left( G_{1},G_{2}\\right) $ from $\\left[ H_{1}\\times H_{2}\\right]^{\\ast }$ has the norm $\\left\\Vert \\left( G_{1},G_{2}\\right) \\right\\Vert =\\max \\left\\lbrace \\left\\Vert G_{1}\\right\\Vert ,\\left\\Vert G_{2}\\right\\Vert \\right\\rbrace .", "(1.40)$ Let now $p=\\left( p_{1},...,p_{m}\\right) $ be any path with different points: $p_{i}\\ne p_{j}$ for any $i\\ne j$ , $1\\le i,~j\\le m$ .", "We associate with $p$ the following functional over $C\\left( X\\right) $ $L\\left[ f\\right] =\\frac{1}{m}\\sum \\limits _{i=1}^{m}\\left( -1\\right)^{i-1}f\\left( p_{i}\\right) .$ Since $\\left|L(f)\\right|\\le \\left\\Vert f\\right\\Vert $ and $\\left|L(g)\\right|=\\left\\Vert g\\right\\Vert $ for a continuous function $g(\\mathbf {x})$ such that $g(p_{i})=1,\\ $ for odd indices $i,\\ g(p_{j})=-1,$ for even indices$\\ j\\ $ and $-1<g(\\mathbf {x})<1$ elsewhere, we obtain that $\\left\\Vert L\\right\\Vert =1$ .", "Let $A^{\\ast }\\left[ L\\right]=\\left( L_{1},L_{2}\\right) $ .", "One can easily verify that $\\left\\Vert L_{i}\\right\\Vert \\le \\frac{2}{m},i=1,2.$ Therefore, from (1.40) we obtain that $\\left\\Vert A^{\\ast }\\left[ L\\right] \\right\\Vert \\le \\frac{2}{m}.", "(1.41)$ Since $A$ is a surjection, there exists $\\delta >0$ such that $\\left\\Vert A^{\\ast }\\left[ G\\right] \\right\\Vert \\ge \\delta \\left\\Vert G\\right\\Vert ~~~~\\;\\mbox{for any functional}\\;\\ G\\in C\\left( X\\right) ^{\\ast }$ Hence $\\left\\Vert A^{\\ast }\\left[ L\\right] \\right\\Vert \\ge \\delta .", "(1.42)$ Now from (1.41) and (1.42) we conclude that $m\\le \\frac{2}{\\delta }.$ This means that for a path with different points, $n_{0}$ can be chosen as $\\left[ \\frac{2}{\\delta }\\right] +1$ .", "Let now $p=\\left( p_{1},...,p_{m}\\right) $ be a path with at least two coinciding points.", "Then we can form a closed path with different points.", "This may be done by the following way: let $i\\ $ and $j\\ $ be indices such that $p_{i}=\\ p_{j}\\ $ and $j-i\\ $ takes its minimal value.", "Note that in this case all the points $p_{i},p_{i+1},...,p_{j-1}\\ $ are distinct.", "Now if $j-i\\ $ is an even number, then the path $(p_{i},p_{i+1},...,p_{j-1})\\ $ , and if $\\ j-i\\ $ is an odd number, then the path $(p_{i+1},...,p_{j-1})$ is a closed path with different points.", "It remains to show that $X$ can not possess closed paths with different points.", "Indeed, if $q=\\left(q_{1},...,q_{2k}\\right) $ is a path of this type, then the functional $L,$ associated with $q,$ annihilates all functions from $H_{1}+H_{2}$ .", "On the other hand, $L\\left[ f\\right] =1$ for a continuous function $f$ on $X$ satisfying the conditions $f\\left( t\\right) =1$ if $t\\in \\left\\lbrace q_{1},q_{3},...,q_{2k-1}\\right\\rbrace ;$ $f\\left( t\\right) =-1$ if $t\\in \\left\\lbrace q_{2},q_{4},...,q_{2k}\\right\\rbrace ;$ $f\\left( t\\right) \\in \\left( -1;1\\right) $ if $t\\in X\\backslash q$ .", "This implies on the contrary to our assumption that $H_{1}+H_{2}\\ne C\\left( X\\right) $ .", "The necessity has been proved.", "Sufficiency.", "Let $X$ contains no closed path and the lengths of all paths are bounded by some positive integer $n_{0}$ .", "We may suppose that any path has different points.", "Indeed, in other case we can form a closed path, which contradicts our assumption.", "For $i=1,2,$ let $X_{i}$ be the quotient space of $X$ obtained by identifying the points $a$ and $b$ whenever $g\\left( a\\right) =g\\left(b\\right) $ for each $g$ in $H_{i}$ .", "Let $\\pi _{i}$ be the natural projection of $X$ onto $X_{i}$ .", "For a point $t\\in X$ set $T_{1}=\\pi _{1}^{-1}\\left( \\pi _{1}t\\right) ,T_{2}=\\pi _{2}^{-1}\\left( \\pi _{2}T_{1}\\right) ,\\ldots .$ By $O\\left( t\\right) $ denote the orbit of $X$ containing $t.$ Since the length of any path in $X$ is not more than $n_{0}$ , we conclude that $O\\left(t\\right) =T_{n_{0}}$ .", "Since $X\\ $ is compact, the sets $T_{1},T_{2},...,T_{n_{0}},\\ $ hence $O(t),$ are compact.", "By Theorem 1.6, $\\overline{H_{1}+H_{2}}=C\\left( X\\right)$ .", "Now let us show that $H_{1}+H_{2}$ is closed in $C\\left( X\\right) $ .", "Set $H_{3}=H_{1}\\cap H_{2}.$ Let $X_{3}$ and $\\pi _{3}$ be the associated quotient space and projection.", "Fix some $a\\in X_{3}$ .", "Show, within conditions of our theorem, that if $t\\in \\pi _{3}^{-1}\\left( a\\right) ,$ then $O\\left( t\\right) =\\pi _{3}^{-1}\\left(a\\right) $ .", "The inclusion $O\\left( t\\right) \\subset \\pi _{3}^{-1}\\left(a\\right) $ is obvious.", "Suppose that there exists a point $t_{1}\\in \\pi _{3}^{-1}\\left( a\\right) $ such that $t_{1}\\notin O\\left( t\\right)$ .", "Then $O\\left( t\\right) \\cap O\\left( t_{1}\\right) =\\emptyset $ .", "By $X|O$ denote the factor space generated by orbits of $X$ .", "$X|O$ is a normal topological space with its natural factor topology.", "Hence we can construct a continuous function $u\\in C\\left( X|O\\right) $ such that $u\\left( O\\left(t\\right) \\right) =0,$ $u\\left( O\\left( t_{1}\\right) \\right) =1$ .", "The function $\\upsilon \\left( x\\right) =u\\left( O\\left( x\\right) \\right) ,\\;\\ x\\in X,$ is continuous on $X$ and belongs to $H_{3}$ as a function being constant on each orbit.", "But, since $O\\left( t\\right) \\subset \\pi _{3}^{-1}\\left( a\\right) $ and $O\\left( t_{1}\\right) \\subset \\pi _{3}^{-1}\\left( a\\right) $ , the function $\\upsilon \\left( x\\right) $ can not take different values on $O\\left( t\\right) $ and $O\\left( t_{1}\\right) $ .", "This contradiction means that there is not a point $t_{1}\\in \\pi _{3}^{-1}\\left( a\\right) $ such that $t_{1}\\notin O\\left( t\\right) $ .", "Thus, $O\\left( t\\right) =\\pi _{3}^{-1}\\left( a\\right) (1.43)$ for any $a\\in X_{3}$ and $t\\in \\pi _{3}^{-1}\\left( a\\right) $ .", "Now prove that there exists a positive real number $c$ such that $\\sup \\limits _{z\\in X_{3}}\\underset{\\pi _{3}^{-1}\\left( z\\right) }{var}f\\le c\\sup \\limits _{y\\in X_{2}}\\underset{\\pi _{2}^{-1}\\left( y\\right) }{var}f(1.44)$ for all $f$ in $H_{1}$ .", "Note that for $Y\\subset X,\\ \\;\\underset{Y}{var}f$ is the variation of $f$ on the set $Y.$ That is, $\\;$ $\\underset{Y}{var}f=\\sup \\limits _{x,y\\in Y}\\left|f\\left( x\\right)-f\\left( y\\right) \\right|.$ Due to (1.43), inequality (1.44) can be written in the following form $\\sup _{t\\in X}\\underset{O\\left( t\\right) }{var}f\\le c\\sup _{t\\in X}\\underset{\\pi _{2}^{-1}\\left( \\pi _{2}\\left( t\\right) \\right) }{var}f(1.45)$ for all $f\\in H_{1}$ .", "Let $t\\in X$ and $t_{1},t_{2}$ be arbitrary points of $O\\left( t\\right) $ .", "Then there is a path $\\left( b_{1},b_{2},...,b_{m}\\right) $ with $b_{1}=t_{1} $ and $b_{m}=t_{2}$ .", "Besides, by the condition, $m\\le n_{0}$ .", "Let first $\\mathbf {a}^{2}\\cdot b_{1}=\\mathbf {a}^{2}\\cdot b_{2},$ $\\mathbf {a}^{1}\\cdot b_{2}=\\mathbf {a}^{1}\\cdot b_{3},...,\\mathbf {a}^{2}\\cdot b_{m-1}=\\mathbf {a}^{2}\\cdot b_{m}$ .", "Then for any function $f\\in H_{1}$ $\\left|f\\left( t_{1}\\right) -f\\left( t_{2}\\right) \\right|=\\left|f\\left( b_{1}\\right) -f\\left( b_{2}\\right) +...-f\\left(b_{m}\\right) \\right|\\le $ $\\le \\left|f\\left( b_{1}\\right) -f\\left( b_{2}\\right) \\right|+...+\\left|f\\left( b_{m-1}\\right) -f\\left( b_{m}\\right) \\right|\\le \\frac{n_{o}}{2}\\sup _{t\\in X}\\underset{\\pi _{2}^{-1}\\left( \\pi _{2}\\left( t\\right) \\right) }{var}f.(1.46)$ It is not difficult to verify that inequality (1.46) holds in all other possible cases of the path $\\left( b_{1},...,b_{m}\\right) $ .", "Now from (1.46) we obtain (1.45), hence (1.44), where $c=\\frac{n_{0}}{2}$ .", "In [122], Marshall and O'Farrell proved the following result (see [122]): Let $A_{1}\\ $ and $A_{2}\\ $ be closed subalgebras of $C(X)\\ $ that contain the constants.", "Let $(X_{1},\\pi _{1}),\\ (X_{2},\\pi _{2})\\ $ and $(X_{3},\\pi _{3})\\ $ be the quotient spaces and projections associated with the algebras $A_{1},$ $A_{2}\\ $ and $A_{3}=A_{1}\\cap A_{2}\\ $ respectively.", "Then $A_{1}+A_{2}\\ $ is closed in $C(X)\\ $ if and only if there exists a positive real number $c$ such that $\\sup \\limits _{z\\in X_{3}}\\underset{\\pi _{3}^{-1}\\left( z\\right) }{var}f\\le c\\sup \\limits _{y\\in X_{2}}\\underset{\\pi _{2}^{-1}\\left( y\\right) }{var}f$ for all $f\\ $ in $A_{1}.$ By this proposition, (1.44) implies that $H_{1}+H_{2}$ is closed in $C\\left(X\\right) $ .", "Thus we finally obtain that $H_{1}+H_{2}=C\\left( X\\right) $ .", "Paths with respect to two directions are explicit objects and give geometric means of deciding if $H_{1}+H_{2}=C\\left( X\\right) $ .", "Let us show this in the example of the bivariate ridge functions $g_{1}=x_{1}+x_{2}\\ $ and $g_{2}=x_{1}-x_{2}.$ If $X$ is the union of two parallel line segments in $\\mathbb {R}^{2},$ not parallel to any of the lines $x_{1}+x_{2}=0$ and $x_{1}-x_{2}=0,\\ $ then Theorem 1.7 holds.", "If $X$ is any bounded part of the graph of the function $x_{2}=\\arcsin (\\sin x_{1}),$ then Theorem 1.7 also holds.", "Let now $X\\ $ be the set $\\begin{array}{c}\\lbrace (0,0),(1,-1),(0,-2),(-1\\frac{1}{2},-\\frac{1}{2}),(0,1),(\\frac{3}{4},\\frac{1}{4}),(0,-\\frac{1}{2}), \\\\(-\\frac{3}{8},-\\frac{1}{8}),(0,\\frac{1}{4}),(\\frac{3}{16},\\frac{1}{16}),...\\rbrace .\\end{array}$ In this case, there is no positive integer bounding lengths of all paths.", "Thus Theorem 1.7 fails.", "Note that since orbits of all paths are closed, Theorem 1.6 from the previous section shows $H_{1}+H_{2}$ is dense in $C\\left( X\\right) .$ If $X$ is any set with interior points, then both Theorem 1.6 and Theorem 1.7 fail, since any such set contains the vertices of some parallelogram with sides parallel to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , that is a closed path.", "Theorem 1.7 admits a direct generalization to the representation by sums $g_{1}\\left( h_{1}\\left( \\mathbf {x}\\right) \\right) +g_{2}\\left( h_{2}\\left(\\mathbf {x}\\right) \\right) $ , where $h_{1}\\left( \\mathbf {x}\\right) $ and $h_{2}\\left( \\mathbf {x}\\right) $ are fixed continuous functions on $X$ .", "This generalization needs consideration of new objects – paths with respect to two continuous functions.", "Definition 1.5.", "Let $X$ be a compact set in $\\mathbb {R}^{d}$ and $h_{1},h_{2}\\in C\\left( X\\right)$ .", "A finite ordered subset $\\left( p_{1},p_{2},...,p_{m}\\right)$ of $X$ with $p_{i}\\ne p_{i+1}\\left(i=1,...,m-1\\right)$ , and either $h_{1}\\left( p_{1}\\right) =h_{1}\\left(p_{2}\\right)$ , $h_{2}\\left( p_{2}\\right) =h_{2}\\left( p_{3}\\right)$ , $h_{1}\\left( p_{3}\\right) =h_{1}\\left( p_{4}\\right),...,$ or $h_{2}\\left(p_{1}\\right) =h_{2}\\left( p_{2}\\right)$ , $h_{1}\\left( p_{2}\\right)=h_{1}\\left( p_{3}\\right)$ , $h_{2}\\left( p_{3}\\right) =h_{2}\\left(p_{4}\\right),...,$ is called a path with respect to the functions $h_{1}$ and $h_{2}$ or, shortly, an $h_{1}$ -$h_{2}$ path.", "Theorem 1.8.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ .", "All functions $f \\in C(X)$ admit a representation $f(\\mathbf {x})=g_{1}\\left( h_{1}\\left( \\mathbf {x}\\right) \\right) +g_{2}\\left(h_{2}\\left( \\mathbf {x}\\right) \\right) ,~g_{1},g_{2}\\in C(\\mathit {\\mathbb {R}})$ if and only if the set $X$ contains no closed $h_{1}$ -$h_{2}$ path and there exists a positive integer $n_{0}$ such that the lengths of $h_{1}$ -$h_{2}$ paths in $X$ are bounded by $n_{0}$ .", "The proof can be carried out by the same arguments as above.", "It should be noted that Theorem 1.8 was first proved by Khavinson in his monograph [92].", "Khavinson's proof (see [92]) used theorems of Sternfeld [149] and Medvedev [92], whereas our proof, which generalizes the ideas of Khavinson, was based on the above proposition of Marshall and O'Farrell.", "In this section, using two results of Garkavi, Medvedev and Khavinson [46], we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions, respectively.", "In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for certain subsets of $\\mathbb {R}^{n}$ .", "In the second case, we obtain also a necessary condition for proximinality.", "All the results are furnished with plenty of examples.", "The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions.", "Let $E$ be a normed linear space and $F$ be its subspace.", "We say that $F$ is proximinal in $E$ if for any element $e\\in E$ there exists at least one element $f_{0}\\in F$ such that $\\left\\Vert e-f_{0}\\right\\Vert =\\inf _{f\\in F}\\left\\Vert e-f\\right\\Vert .$ In this case, the element $f_{0}$ is said to be extremal to $e$ .", "We are interested in the problem of proximinality of the set of linear combinations of ridge functions in the spaces of bounded and continuous functions respectively.", "This problem will be considered in the simplest case when the class of approximating functions is the set $\\mathcal {R}=\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) ={\\ \\left\\lbrace {g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) +g_{2}\\left(\\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) :g}_{i}:{{\\mathbb {R\\rightarrow R}},i=1,2}\\right\\rbrace }.$ Here $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ are fixed directions and we vary over ${g}_{i}$ .", "It is clear that this is a linear space.", "Consider the following three subspaces of $\\mathcal {R}$ .", "The first is obtained by taking only bounded sums ${g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right)+g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) }$ over some set $X$ in $\\mathbb {R}^{n}.$ We denote this subspace by $\\mathcal {R}_{a}(X)$ .", "The second and the third are subspaces of $\\mathcal {R}$ with bounded and continuous summands $g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right),~i=1,2,$ on $X$ respectively.", "These subspaces will be denoted by $\\mathcal {R }_{b}(X)$ and $\\mathcal {R}_{c}(X).$ In the case of $\\mathcal {R}_{c}(X),$ the set $X$ is considered to be compact.", "Let $B(X)$ and $C(X)$ be the spaces of bounded and continuous multivariate functions over $X$ respectively.", "What conditions must one impose on $X$ in order that the sets $\\mathcal {R}_{a}(X)$ and $\\mathcal {R}_{b}(X)$ be proximinal in $B(X)$ and the set $\\mathcal {R}_{c}(X)$ be proximinal in $C(X)$ ?", "We are also interested in necessary conditions for proximinality.", "It follows from one result of Garkavi, Medvedev and Khavinson (see [46]) that $\\mathcal {R}_{a}(X)$ is proximinal in $B(X)$ for all subsets $X$ of $\\mathbb {R}^{n}$ .", "There is also an answer (see [46]) for proximinality of $\\mathcal {R}_{b}(X)$ in $B(X)$ .", "This will be discussed in Section 1.5.2.", "Is the set $\\mathcal {R}_{b}(X)$ always proximinal in $B(X)$ ?", "There is an an example of a set $X\\subset \\mathbb {R}^{n}$ and a bounded function $f$ on $X$ for which there does not exist an extremal element in $\\mathcal {R}_{b}(X)$ .", "In Section 1.5.3, we will obtain sufficient conditions for the existence of extremal elements from $\\mathcal {R}_{c}(X)$ to an arbitrary function $f$ $\\in $ $C(X)$ .", "Based on one result of Marshall and O'Farrell [122], we will also give a necessary condition for proximinality of $\\mathcal {R}_{c}(X)$ in $C(X)$ .", "All the theorems, following discussions and examples of the paper will lead us naturally to a conjecture on the proximinality of the subspaces $\\mathcal {R}_{b}(X)$ and $\\mathcal {R}_{c}(X)$ in the spaces $B(X)$ and $C(X)$ respectively.", "The reader may also be interested in the more general case with the set $\\mathcal {R}=\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ .", "In this case, the corresponding sets $\\mathcal {R}_{a}(X)$ , $\\mathcal {R}_{b}(X)$ and $\\mathcal {R}_{c}(X)$ are defined similarly.", "Using the results of [46], one can obtain sufficient (but not necessary) conditions for proximinality of these sets.", "This needs, besides paths, the consideration of some additional and more complicated relations between points of $X$ .", "Here we will not consider the case $r\\ge 3$ , since our main purpose is to draw the reader's attention to the arisen problems of proximinality in the simplest case of approximation.", "For the existing open problems connected with the set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ , where $r\\ge 3$ , see [62] and [134].", "Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two different directions in $\\mathbb {R}^{n}$ .", "In the sequel, we will use paths with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Recall that a length of a path is the number of its points and can be equal to $\\infty $ if the path is infinite.", "A singleton is a path of the unit length.", "We say that a path $\\left( \\mathbf {x}^{1},...,\\mathbf {x}^{m}\\right) $ belonging to some subset $X$ of $\\mathbb {R}^{n}$ is irreducible if there is not another path $\\left(\\mathbf {y}^{1},...,\\mathbf {y}^{l}\\right) \\subset X$ with $\\mathbf {y}^{1}=\\mathbf {x}^{1},~\\mathbf {y}^{l}=\\mathbf {x}^{m}$ and $l<m$ .", "The following theorem follows from [46].", "Theorem 1.9.", "Let $X\\subset $ $\\mathbb {R}^{n}$ and the lengths of all irreducible paths in $X$ be uniformly bounded by some positive integer.", "Then each function in $B(X)$ has an extremal element in $\\mathcal {R}_{b}(X)$ .", "There are a large number of sets in $\\mathbb {R}^{n}$ satisfying the hypothesis of this theorem.", "For example, if a set $X$ has a cross section according to one of the directions $\\mathbf {a}^{1}$ or $\\mathbf {a}^{2}$ , then the set $X$ satisfies the hypothesis of Theorem 1.9.", "By a cross section according to the direction $\\mathbf {a}^{1}$ we mean any set $X_{\\mathbf {a}^{1}}=\\lbrace x\\in X:\\ \\mathbf {a}^{1}\\cdot \\mathbf {x}=c\\rbrace $ , $c\\in \\mathbb {R}$ , with the property: for any $\\mathbf {y}\\in X$ there exists a point $\\mathbf {y}^{1}\\in X_{\\mathbf {a}^{1}}$ such that $\\mathbf {a}^{2}\\cdot \\mathbf {y}=\\mathbf {a}^{2}\\cdot \\mathbf {y}^{1}$ .", "By the similar way, one can define a cross section according to the direction $\\mathbf {a}^{2}$ .", "For more on cross sections in problems of proximinality of sums of univariate functions see [45], [91].", "Regarding Theorem 1.9 one may ask if the condition of the theorem is necessary for proximinality of $\\mathcal {R}_{b}(X)$ in $B(X)$ .", "While we do not know a complete answer to this question, we are going to give an example of a set $X $ for which Theorem 1.9 fails.", "Let $\\mathbf {a}^{1}=(1;-1),\\ \\mathbf {a}^{2}=(1;1).$ Consider the set $X &=&\\lbrace (2;\\frac{2}{3}),(\\frac{2}{3};-\\frac{2}{3}),(0;0),(1;1),(1+\\frac{1}{2};1-\\frac{1}{2}),(1+\\frac{1}{2}+\\frac{1}{4};1-\\frac{1}{2}+\\frac{1}{4}), \\\\&&(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8};1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{8}),...\\rbrace .$ In what follows, the elements of $X$ in the given order will be denoted by $\\mathbf {x}^{0},\\mathbf {x}^{1},\\mathbf {x}^{2},...$ .", "It is clear that $X$ is a path of the infinite length and $\\mathbf {x}^{n}\\rightarrow \\mathbf {x}^{0}$ as $n\\rightarrow \\infty $ .", "Let $\\sum _{n=1}^{\\infty }c_{n}$ be any divergent series with the terms $c_{n}>0$ and $c_{n}\\rightarrow 0$ as $n\\rightarrow \\infty $ .", "Besides let $f_{0}$ be a function vanishing at the points $\\mathbf {x}^{0},\\mathbf {x}^{2},\\mathbf {x}^{4},...,$ and taking values $c_{1},c_{2},c_{3},...$ at the points $\\mathbf {x}^{1},\\mathbf {x}^{3},\\mathbf {\\ x}^{5},...$ , respectively.", "It is obvious that $f_{0}$ is continuous on $X$ .", "The set $X$ is compact and satisfies all the conditions of Theorem 1.6.", "By that theorem, $\\overline{\\mathcal {R}_{c}(X)}=C(X).$ Therefore, for any continuous function on $X$ , thus for $f_{0}$ , $\\inf _{g\\in \\mathcal {R}_{c}(X)}\\left\\Vert f_{0}-g\\right\\Vert _{C(X)}=0.", "(1.47)$ Since $\\mathcal {R}_{c}(X)\\subset \\mathcal {R}_{b}(X),$ we obtain from (1.47) that $\\inf _{g\\in \\mathcal {R}_{b}(X)}\\left\\Vert f_{0}-g\\right\\Vert _{B(X)}=0.", "(1.48)$ Suppose that $f_{0}$ has an extremal element ${g_{1}^{0}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) +g_{2}^{0}\\left( \\mathbf {a}^{2}{\\ \\cdot }\\mathbf {x}\\right) }$ in $\\mathcal {R}_{b}(X).$ By the definition of $\\mathcal {R}_{b}(X)$ , the ridge functions ${g_{i}^{0},i=1,2}$ , are bounded on $X.$ From (1.48) it follows that $f_{0}={g_{1}^{0}\\left( \\mathbf {a}^{1}{\\ \\cdot }\\mathbf {x}\\right) +g_{2}^{0}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) .", "}$ Since $\\mathbf {a}^{1}\\cdot \\mathbf {x}^{2n}=\\mathbf {a}^{1}\\cdot \\mathbf {x}^{2n+1}$ and $\\mathbf {a}^{2}\\cdot \\mathbf {x}^{2n+1}=\\mathbf {a}^{2}\\cdot \\mathbf {x}^{2n+2},$ for $n=0,1,...,$ we can write that $\\sum _{n=0}^{k}c_{n+1}=\\sum _{n=0}^{k}\\left[ f(\\mathbf {x}^{2n+1})-f(\\mathbf {x}^{2n})\\right]$ $=\\sum _{n=0}^{k}\\left[ {g_{2}^{0}}(\\mathbf {x}^{2n+1})-{g_{2}^{0}}(\\mathbf {x}^{2n})\\right] ={g_{2}^{0}(}\\mathbf {a}^{2}\\cdot \\mathbf {x}^{2k+1})-{g_{2}^{0}(}\\mathbf {a}^{2}\\cdot \\mathbf {x}^{0}).", "(1.49)$ Since $\\sum _{n=1}^{\\infty }c_{n}=\\infty ,$ we deduce from (1.49) that the function ${g_{2}^{0}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) }$ is not bounded on $X.$ This contradiction means that the function $f_{0}$ does not have an extremal element in $\\mathcal {R}_{b}(X).$ Therefore, the space $\\mathcal {R}_{b}(X)$ is not proximinal in $B(X).$ In this section, we give a sufficient condition and also a necessary condition for proximinality of $\\mathcal {R}_{c}(X)$ in $C(X)$ .", "Theorem 1.10.", "Let the system of linearly independent vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ have a complement to a basis $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{n}\\rbrace $ in $\\mathbb {R}^{n}$ with the property: for any point $\\mathbf {x}^{0}\\in X$ and any positive real number $\\delta $ there exist a number $\\delta _{0}\\in (0,\\delta ]$ and a point $\\mathbf {x}^{\\sigma } $ in the set $\\sigma =\\lbrace \\mathbf {x}\\in X:\\mathbf {a}^{2}\\cdot \\mathbf {x}^{0}-\\delta _{0}\\le \\mathbf {a}^{2}\\cdot \\mathbf {x}\\le \\mathbf {a}^{2}\\cdot \\mathbf {x}^{0}+\\delta _{0}\\rbrace ,$ such that the system $\\left\\lbrace \\begin{array}{c}\\mathbf {a}^{2}\\cdot \\mathbf {x}^{\\prime }=\\mathbf {a}^{2}\\cdot \\mathbf {x}^{\\sigma } \\\\\\mathbf {a}^{1}\\cdot \\mathbf {x}^{\\prime }=\\mathbf {a}^{1}\\cdot \\mathbf {x} \\\\\\sum _{i=3}^{n}\\left|\\mathbf {a}^{i}\\cdot \\mathbf {x}^{\\prime }-\\mathbf {a}^{i}\\cdot \\mathbf {x}\\right|<\\delta \\end{array}\\right.", "(1.50)$ has a solution $\\mathbf {x}^{\\prime }\\in \\sigma $ for all points $\\mathbf {x}\\in \\sigma .$ Then the space $\\mathcal {R}_{c}(X)$ is proximinal in $C(X).$ Introduce the following mappings and sets: $\\pi _{i}:X\\rightarrow \\mathbb {R}\\text{, }\\pi _{i}(\\mathbf {x)=a}^{i}\\cdot \\mathbf {x}\\text{, }Y_{i}=\\pi _{i}(X\\mathbf {)}\\text{, }i=1,...,n.$ Since the system of vectors $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{n}\\rbrace $ is linearly independent, the mapping $\\pi =(\\pi _{1},...\\pi _{n})$ is an injection from $X$ into the Cartesian product $Y_{1} \\times ...\\times Y_{n} $ .", "Besides, $\\pi $ is linear and continuous.", "By the open mapping theorem, the inverse mapping $\\pi ^{-1}$ is continuous from $Y=\\pi (X)$ onto $X.$ Let $f$ be a continuous function on $X$ .", "Then the composition $f\\circ \\pi ^{-1}(y_{1},...y_{n})$ will be continuous on $Y,$ where $y_{i}=\\pi _{i}(\\mathbf {x),}\\ i=1,...,n,$ are the coordinate functions.", "Consider the approximation of the function $f\\circ \\pi ^{-1}$ by elements from $G_{0}=\\lbrace g_{1}(y_{1})+g_{2}(y_{2}):\\ g_{i}\\in C(Y_{i}),\\ i=1,2\\rbrace $ over the compact set $Y$ .", "Then one may observe that the function $f$ has an extremal element in $\\mathcal {R}_{c}(X)$ if and only if the function $f\\circ \\pi ^{-1}$ has an extremal element in $G_{0}$ .", "Thus the problem of proximinality of $\\mathcal {R}_{c}(X)$ in $C(X)$ is reduced to the problem of proximinality of $G_{0}$ in $C(Y).$ Let $T,T_{1},...,T_{m+1}$ be metric compact spaces and $T\\subset $ $T_{1}\\times ...\\times T_{m+1}.$ For $i=1,...,m,$ let $\\varphi _{i}$ be the continuous mappings from $T$ onto $T_{i}.$ In [46], the authors obtained sufficient conditions for proximinality of the set $C_{0}=\\lbrace \\sum _{i=1}^{n}g_{i}\\circ \\varphi _{i}:\\ g_{i}\\in C(T_{i}),\\ i=1,...m\\rbrace $ in the space $C(T)$ of continuous functions on $T.$ Since $Y\\subset $ $Y_{1}\\times Y_{2}\\times Z_{3},$ where $Z_{3}=Y_{3}\\times ...\\times Y_{n},$ we can use this result in our case, for the approximation of the function $f\\circ \\pi ^{-1}$ by elements from $G_{0}$ .", "By this theorem, the set $G_{0}$ is proximinal in $C(Y)$ if for any $y_{2}^{0}\\in Y_{2}$ and $\\delta >0$ there exists a number $\\delta _{0}\\in (0,$ $\\delta )$ such that the set $\\sigma (y_{2}^{0},\\delta _{0})=[y_{2}^{0}-\\delta _{0},y_{2}^{0}+\\delta _{0}]\\cap Y_{2}$ has $(2,\\delta )$ maximal cross section.", "The last means that there exists a point $y_{2}^{\\sigma }\\in \\sigma (y_{2}^{0},\\delta _{0})$ with the property: for any point $(y_{1},y_{2},z_{3})\\in Y,$ with the second coordinate $y_{2}$ from the set $\\sigma (y_{2}^{0},\\delta _{0}),$ there exists a point $(y_{1}^{\\prime },y_{2}^{\\sigma },z_{3}^{\\prime })\\in Y$ such that $y_{1}=y_{1}^{\\prime }$ and $\\rho (z_{3},z_{3}^{\\prime })<\\delta ,$ where $\\rho $ is a metrics in $Z_{3}.$ Since these conditions are equivalent to the conditions of Theorem 1.10, the space $G_{0}$ is proximinal in the space $C(Y).$ Then by the above conclusion, the space $\\mathcal {R}_{c}(X)$ is proximinal in $C(X).$ Let us give some simple examples of compact sets satisfying the hypothesis of Theorem 1.10.", "For the sake of brevity, we restrict ourselves to the case $n=3.$ Assume $X$ is a closed ball in $\\mathbb {R}^{3}$ and $\\mathbf {a}^{1}$ , $\\mathbf {a}^{2}$ are orthogonal directions.", "Then Theorem 1.10 holds.", "Note that in this case, we can take $\\delta _{0}=\\delta $ and $\\mathbf {a}^{3}$ as an orthogonal vector to both the vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}.$ Let $X$ be the unite cube, $\\mathbf {a}^{1}=(1;1;0),\\ a^{2}=(1;-1;0).$ Then Theorem 1.10 also holds.", "In this case, we can take $\\delta _{0}=\\delta $ and $\\mathbf {a}^{3}=(0;0;1).$ Note that the unit cube does not satisfy the hypothesis of the theorem for many directions (take, for example, $\\mathbf {a}^{1}=(1;2;0)$ and $\\mathbf {a}^{2}=(2;-1;0)$ ).", "Assume $X$ is a closed ball in $\\mathbb {R}^{3}$ and $\\mathbf {a}^{1}$ , $\\mathbf {a}^{2}$ are orthogonal directions.", "Then Theorem 1.10 holds.", "Note that in this case, we can take $\\delta _{0}=\\delta $ and $\\mathbf {a}^{3}$ as an orthogonal vector to both the vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}.$ Let $X$ be the unite cube, $\\mathbf {a}^{1}=(1;1;0),\\ a^{2}=(1;-1;0).$ Then Theorem 1.10 also holds.", "In this case, we can take $\\delta _{0}=\\delta $ and $\\mathbf {a}^{3}=(0;0;1).$ Note that the unit cube does not satisfy the hypothesis of the theorem for many directions (take, for example, $\\mathbf {a}^{1}=(1;2;0)$ and $\\mathbf {a}^{2}=(2;-1;0)$ ).", "In the following example, one can not always chose $\\delta _{0}$ as equal to $\\delta $ .", "Let $X=\\lbrace (x_{1},x_{2},x_{3}):\\ (x_{1},x_{2})\\in Q,\\ 0\\le x_{3}\\le 1\\rbrace ,$ where $Q$ is the union of two triangles $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$ with the vertices $A_{1}=(0;0),\\ B_{1}=(1;2),\\ C_{1}=(2;0),\\ A_{2}=(1\\frac{1}{2};1),\\ B_{2}=(2\\frac{1}{2};-1),\\ C_{2}=(3\\frac{1}{2};1).$ Let $\\mathbf {a}^{1}=(0;1;0)$ and $\\mathbf {a}^{2}=(1;0;0).$ Then it is easy to see that Theorem 1.10 holds (the vector $\\mathbf {a}^{3}$ can be chosen as $(0;0;1)$ ).", "In this case, $\\delta _{0}$ can not be always chosen as equal to $\\delta $ .", "Take, for example, $\\mathbf {x}^{0}=(1\\frac{3}{4};0;0)$ and $\\delta =1\\frac{3}{4}.$ If $\\delta _{0}=\\delta ,$ then the second equation of the system (1.50) has not a solution for a point $(1;2;0)$ or a point $(2\\frac{1}{2};-1;0).$ But if we take $\\delta _{0}$ not more than $\\frac{1}{4}$ , then for $\\mathbf {x}^{\\sigma }=\\mathbf {x}^{0}$ the system has a solution.", "Note that the last inequality $\\left|\\mathbf {a}^{3}\\cdot \\mathbf {x}^{\\prime }-\\mathbf {a}^{3}\\cdot \\mathbf {x}\\right|<\\delta $ of the system can be satisfied with the equality $\\mathbf {a}^{3}\\cdot \\mathbf {x}^{\\prime }=\\mathbf {a}^{3}\\cdot \\mathbf {x}$ if $\\mathbf {a}^{3}=(0;0;1).$ Let $X=\\lbrace (x_{1},x_{2},x_{3}):\\ (x_{1},x_{2})\\in Q,\\ 0\\le x_{3}\\le 1\\rbrace ,$ where $Q$ is the union of two triangles $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$ with the vertices $A_{1}=(0;0),\\ B_{1}=(1;2),\\ C_{1}=(2;0),\\ A_{2}=(1\\frac{1}{2};1),\\ B_{2}=(2\\frac{1}{2};-1),\\ C_{2}=(3\\frac{1}{2};1).$ Let $\\mathbf {a}^{1}=(0;1;0)$ and $\\mathbf {a}^{2}=(1;0;0).$ Then it is easy to see that Theorem 1.10 holds (the vector $\\mathbf {a}^{3}$ can be chosen as $(0;0;1)$ ).", "In this case, $\\delta _{0}$ can not be always chosen as equal to $\\delta $ .", "Take, for example, $\\mathbf {x}^{0}=(1\\frac{3}{4};0;0)$ and $\\delta =1\\frac{3}{4}.$ If $\\delta _{0}=\\delta ,$ then the second equation of the system (1.50) has not a solution for a point $(1;2;0)$ or a point $(2\\frac{1}{2};-1;0).$ But if we take $\\delta _{0}$ not more than $\\frac{1}{4}$ , then for $\\mathbf {x}^{\\sigma }=\\mathbf {x}^{0}$ the system has a solution.", "Note that the last inequality $\\left|\\mathbf {a}^{3}\\cdot \\mathbf {x}^{\\prime }-\\mathbf {a}^{3}\\cdot \\mathbf {x}\\right|<\\delta $ of the system can be satisfied with the equality $\\mathbf {a}^{3}\\cdot \\mathbf {x}^{\\prime }=\\mathbf {a}^{3}\\cdot \\mathbf {x}$ if $\\mathbf {a}^{3}=(0;0;1).$ It should be remarked that the results of [46] tell nothing about necessary conditions for proximinality of the spaces considered there.", "To fill this gap in our case, we want to give a necessary condition for proximinality of $\\mathcal {R}_{c}(X)$ in $C(X)$ .", "First, let us introduce some notation.", "By $\\mathcal {R}_{c}^{i},\\ i=1,2,$ we will denote the set of continuous ridge functions $g\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ on the given compact set $X\\subset \\mathbb {R}^{n}.$ Note that $\\mathcal {R}_{c}=\\mathcal {R}_{c}^{1}+\\mathcal {R}_{c}^{2}.$ Besides, let $\\mathcal {R}_{c}^{3}=\\mathcal {R}_{c}^{1}\\cap \\mathcal {R}_{c}^{2}.$ For $i=1,2,3,$ let $X_{i}$ be the quotient space obtained by identifying points $y_{1}$ and $y_{2}$ in $X$ whenever $f(y_{1})=f(y_{2})$ for each $f$ in $\\mathcal {R}_{c}^{i}.$ By $\\pi _{i}$ denote the natural projection of $X$ onto $X_{i},$ $i=1,2,3.$ Note that we have already dealt with the quotient spaces $X_{1}$ , $X_{2}$ and the projections $\\pi _{1},\\pi _{2}$ in the previous section.", "Recall that the relation on $X$ , defined by setting $\\ y_{1}\\approx y_{2}$ if $y_{1}$ and $y_{2}$ belong to some path, is an equivalence relation and the equivalence classes are called orbits.", "By $O(t)$ denote the orbit of $X$ containing $t.$ For $Y\\subset X,$ let $var_{Y}\\ f$ be the variation of a function $f$ on the set $Y.$ That is, $\\underset{Y}{var}f=\\sup \\limits _{x,y\\in Y}\\left|f\\left( x\\right)-f\\left( y\\right) \\right|.$ The following theorem is valid.", "Theorem 1.11.", "Suppose that the space $\\mathcal {R}_{c}(X)$ is proximinal in $C(X).$ Then there exists a positive real number c such that $\\sup _{t\\in X}\\underset{O\\left( t\\right) }{var}\\mathit {f\\le c}\\sup _{t\\in X}\\underset{\\pi _{2}^{-1}\\left( \\pi _{2}\\left( t\\right) \\right) }{var}\\mathit {f}(1.51)$ for all $f$ in $\\mathcal {R}_{c}^{1}.$ The proof is based on the following result of Marshall and O'Farrell (see [122]): Let $A_{1}\\ $ and $A_{2}\\ $ be closed subalgebras of $C(X)\\ $ that contain the constants.", "Let $(X_{1},\\pi _{1}),\\ (X_{2},\\pi _{2})\\ $ and $(X_{3},\\pi _{3})\\ $ be the quotient spaces and projections associated with the algebras $A_{1},$ $A_{2}\\ $ and $A_{3}=A_{1}\\cap A_{2}\\ $ respectively.", "Then $A_{1}+A_{2}\\ $ is closed in $C(X)\\ $ if and only if there exists a positive real number $c$ such that $\\sup \\limits _{z\\in X_{3}}\\underset{\\pi _{3}^{-1}\\left( z\\right) }{var}f\\le c\\sup \\limits _{y\\in X_{2}}\\underset{\\pi _{2}^{-1}\\left( y\\right) }{var}f(1.52)$ for all $f\\ $ in $A_{1}.$ If $\\mathcal {R}_{c}(X)$ is proximinal in $C(X),$ then it is necessarily closed and therefore, by the above proposition, (1.52) holds for the algebras $A_{1}^{i}=\\mathcal {R}_{c}^{i},\\ i=1,2,3.$ The right-hand side of (1.52) is equal to the right-hand side of (1.51).", "Let $t$ be some point in $X $ and $z=\\pi _{3}(t).$ Since each function $\\ f\\in \\mathcal {R}_{c}^{3}$ is constant on the orbit of $t$ (note that $f$ is both of the form ${\\ g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) }$ and of the form ${\\ g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) }$ ), $O(t)\\subset \\pi _{3}^{-1}(z).$ Hence, $\\sup _{t\\in X}\\underset{O\\left( t\\right) }{var}f\\le c\\sup \\limits _{z\\in X_{3}}\\underset{\\pi _{3}^{-1}\\left( z\\right) }{var}f(1.53)$ From (1.52) and (1.53) we obtain (1.51).", "Note that the inequality (1.52) provides not worse but less practicable necessary condition for proximinality than the inequality (1.51) does.", "On the other hand, there are many cases in which both the inequalities are equivalent.", "For example, assume the lengths of irreducible paths of $X$ are bounded by some positive integer $n_{0}$ .", "In this case, it can be shown that the inequality (1.52), hence (1.51), holds with the constant $c=\\frac{n_{0}}{2}$ and moreover $O(t)=\\pi _{3}^{-1}(z)$ for all $t\\in X$ , where $z=\\pi _{3}(t)$ (see the proof of [62]).", "Therefore, the inequalities (1.51) and (1.52) are equivalent for the considered class of sets $X.$ The last argument shows that all the compact sets $X\\subset $ $\\mathbb {R}^{n}$ over which $\\mathcal {R}_{c}(X)$ is not proximinal in $C(X)$ should be sought in the class of sets having irreducible paths consisting of sufficiently many points.", "For example, let $I=[0;1]^{2}$ be the unit square, $\\mathbf {a}^{1}=(1;1)$ , $\\mathbf {a}^{2}=(1;\\frac{1}{2}).$ Consider the path $l_{k}=\\lbrace (1;0),(0;1),(\\frac{1}{2};0),(0;\\frac{1}{2}),(\\frac{1}{4};0),...,(0;\\frac{1}{2^{k}})\\rbrace .$ It is clear that $l_{k}$ is an irreducible path with the length $2k+2 $ , where $k$ may be very large.", "Let $g_{k}$ be a continuous univariate function on $\\mathbb {R}$ satisfying the conditions: $g_{k}(\\frac{1}{2^{k-i}} )=i,\\ i=0,...,k,$ $g_{k}(t)=0$ if $t<\\frac{1}{2^{k}},\\ i-1\\le g_{k}(t)\\le i $ if $t\\in (\\frac{1}{2^{k-i+1}},\\frac{1}{2^{k-i}}),\\ i=1,...,k,$ and $g_{k}(t)=k$ if $t>1.$ Then it can be easily verified that $\\sup _{t\\in X}\\underset{\\pi _{2}^{-1}\\left( \\pi _{2}\\left( t\\right) \\right) }{var}g_{k}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})\\le 1.", "(1.54)$ Since $\\max _{\\mathbf {x}\\in I}g_{k}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x} )=k,$ $\\min _{\\mathbf {x}\\in I}g_{k}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})=0$ and $var_{\\mathbf {x}\\in O\\left( t_{1}\\right) }g_{k}(\\mathbf {a}^{1}{\\cdot }\\mathbf {\\ x})=k $ for $t_{1}=(1;0),$ we obtain that $\\sup _{t\\in X}\\underset{O\\left( t\\right) }{var}g_{k}(\\mathbf {a}^{1}{\\cdot }\\mathbf {x})=k.", "(1.55)$ Since $k$ may be very large, from (1.54) and (1.55) it follows that the inequality (1.51) cannot hold for the function $g_{k}(\\mathbf {a}^{1}{\\ \\cdot }\\mathbf {x})\\in \\mathcal {R}_{c}^{1}.$ Thus the space $\\mathcal {R}_{c}(I)$ with the directions $\\mathbf {a}^{1}=(1;1)$ and $\\mathbf {a}^{2}=(1;\\frac{1}{2})$ is not proximinal in $C(I)$ .", "It should be remarked that if a compact set $X\\subset $ $\\mathbb {R}^{n}$ satisfies the hypothesis of Theorem 1.10, then the length of all irreducible paths are uniformly bounded (see the proof of Theorem 1.10 and lemma in [46]).", "We have already seen that if the last condition does not hold, then the proximinality of both $\\mathcal {R}_{c}(X)$ in $C(X)$ and $\\mathcal {R}_{b}(X)$ in $B(X)$ fail for some sets $X$ .", "In addition to the examples given above and in Section 1.5.2, one can easily construct many other examples of such sets.", "All these examples, Theorems 1.9–1.11 and the subsequent remarks justify the statement of the following conjecture: Conjecture.", "Let $X$ be some subset of $\\mathbb {R} ^{n}.$ The space $\\mathcal {R}_{b}(X)$ is proximinal in $B(X)$ and the space $\\mathcal {R}_{c}(X)$ is proximinal in $C(X)$ (in this case, $X$ is considered to be compact) if and only if the lengths of all irreducible paths of $X$ are uniformly bounded.", "Remark 1.2.", "Medvedev's result (see [92]), which later came to our attention, in particular, says that the set $R_{c}(X)$ is closed in $C(X)$ if and only if the lengths of all irreducible paths of $X$ are uniformly bounded.", "Thus, in the case of $C(X)$ , the necessity of the above conjecture was proved by Medvedev.", "Remark 1.3.", "Note that there are situations in which a continuous function (a specific function on a specially constructed set) has an extremal element in $\\mathcal {R}_{b}(X)$ , but not in $\\mathcal {R}_{c}(X)$ (see [92]).", "One subsection of [92] (see p.68 there) was devoted to the proximinality of sums of two univariate functions with continuous and bounded summands in the spaces of continuous and bounded bivariate functions, respectively.", "If $X\\subset \\mathbb {R}^{2}$ and $\\mathbf {a}^{1},\\mathbf {a}^{2} $ be linearly independent directions in $\\mathbb {R}^{2}$ , then the linear transformation $y_{1}=$ $\\mathbf {a}^{1}\\cdot \\mathbf {x\\,}$ , $y_{2}=$ $\\mathbf {a}^{2}\\cdot \\mathbf {x}$ reduces the problems of proximinality of $\\mathcal {R}_{b}(X)$ in $B(X)$ and $\\mathcal {R}_{c}(X)$ in $C(X)$ to the problems considered in that subsection.", "But in general, when $X\\subset \\mathbb {R}^{n}$ and $n>2$ , they cannot be reduced to those in [92].", "In this section, we characterize the best $L_{2}$ -approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions.", "In the special case, when the weight functions are constants, we obtain explicit formulas for both the best approximation and approximation error.", "Ridge approximation in $L_{2}$ started to be actively studied in the late 90's by K.I.", "Oskolkov [131], V.E.", "Maiorov [115], A. Pinkus [134], V.N.", "Temlyakov [156], P. Petrushev [133] and other researchers.", "Let $D$ be the unit disk in $\\mathbb {R}^{2}$ .", "In [113], Logan and Shepp along with other results gave a closed-form expression for the best $L_{2}$ -approximation to a function $f \\in L_{2}\\left(D\\right)$ from the set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right)$ .", "Their solution requires that the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ be equally-spaced and involves finite sums of convolutions with explicit kernels.", "In the $n$ -dimensional case, we obtained an expression of simpler form for the best $L_{2}$ -approximation to square-integrable multivariate functions over a certain domain, provided that $r=n$ and the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ are linearly independent (see [61]).", "In this section, we consider the approximation by functions from the following more general set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right)=\\left\\lbrace \\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) :g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r\\right\\rbrace ,$ where $w_{1},...,w_{r}$ are fixed multivariate functions.", "We characterize the best $L_{2}$ -approximation from this set in the case $r\\le n.$ Then, in the special case when the weight functions $w_{1},...,w_{r}$ are constants, we will prove two theorems on explicit formulas for the best approximation and the approximation error, respectively.", "At present, we do not yet know how to approach these problems in other possible cases of $r.$ Let $X$ be a subset of $\\mathbb {R}^{n}$ with a finite Lebesgue measure.", "Consider the approximation of a function $f\\left( \\mathbf {x}\\right) =f\\left(x_{1},...,x_{n}\\right) $ in $L_{2}\\left( X\\right) $ by functions from the manifold $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right) $ , where $r\\le n.$ We suppose that the functions $w_{i}(\\mathbf {x})$ and the products $w_{i}(\\mathbf {x})\\cdot g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) ,~i=1,...,r$ , belong to the space $L_{2}\\left( X\\right) .$ Besides, we assume that the vectors $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ are linearly independent.", "We say that a function $g_{w}^{0}=\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ in $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right) $ is the best approximation (or extremal) to $f$ if $\\left\\Vert f-g_{w}^{0}\\right\\Vert _{L_{2}\\left( X\\right) }=\\inf \\limits _{g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right)}\\left\\Vert f-g\\right\\Vert _{L_{2}\\left( X\\right) }.$ Let the system of vectors $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r},\\mathbf {a}^{r+1},...,\\mathbf {a}^{n}\\rbrace $ be a completion of the system $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\rbrace $ to a basis in $\\mathbb {R}^{n}.$ Let $J:X\\rightarrow \\mathbb {R}^{n}$ be the linear transformation given by the formulas $y_{i}=\\mathbf {a}^{i}\\cdot \\mathbf {x,}\\quad \\,i=1,...,n.(1.56)$ Since the vectors $\\mathbf {a}^{i},$ $i=1,...,n$ , are linearly independent, it is an injection.", "The Jacobian $\\det J$ of this transformation is a constant different from zero.", "Let the formulas $x_{i}=\\mathbf {b}^{i}\\cdot \\mathbf {y},\\;\\;i=1,...,n,$ stand for the solution of linear equations (1.56) with respect to $x_{i},\\;i=1,...,n.$ Introduce the notation $Y=J\\left( X\\right)$ and $Y_{i}=\\left\\lbrace y_{i}\\in \\mathbb {R}:\\;\\;y_{i}=\\mathbf {a}^{i}\\cdot \\mathbf {x},\\;\\;\\mathbf {x}\\in X\\right\\rbrace ,\\,i=1,...,n.$ For any function $u\\in L_{2}\\left( X\\right) ,$ put $u^{\\ast }=u^{\\ast }\\left( \\mathbf {y}\\right) \\overset{def}{=}u\\left( \\mathbf {b}^{1}\\cdot \\mathbf {y},...,\\mathbf {b}^{n}\\cdot \\mathbf {y}\\right) .$ It is obvious that $u^{\\ast }\\in L_{2}\\left( Y\\right) .$ Besides, $\\int \\limits _{Y}u^{\\ast }\\left( \\mathbf {y}\\right) d\\mathbf {y}=\\left|\\det J\\right|\\cdot \\int \\limits _{X}u\\left( \\mathbf {x}\\right) d\\mathbf {x}(1.57)$ and $\\left\\Vert u^{\\ast }\\right\\Vert _{L_{2}\\left( Y\\right) }=\\left|\\det J\\right|^{1/2}\\cdot \\left\\Vert u\\right\\Vert _{L_{2}\\left( X\\right) }.", "(1.58)$ Set $L_{2}^{i}=\\lbrace w_{i}^{\\ast }(\\mathbf {y})g\\left( y_{i}\\right) \\in L_{2}(Y)\\rbrace ,~i=1,...,r.$ We need the following auxiliary lemmas.", "Lemma 1.5.", "Let $f\\left( \\mathbf {x}\\right) \\in L_{2}\\left( X\\right) $ .", "A function $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is extremal to the function $f\\left( \\mathbf {x}\\right) $ if and only if $\\sum \\limits _{i=1}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) $ is extremal from the space $L_{2}^{1}\\mathit {\\oplus }...\\oplus L_{2}^{r}$ to the function $f^{\\ast }\\left( \\mathbf {y}\\right) $ .", "Due to (1.58) the proof of this lemma is obvious.", "Lemma 1.6.", "Let $f\\left( \\mathbf {x}\\right) \\in L_{2}\\left( X\\right) $ .", "A function $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is extremal to the function $f\\left( \\mathbf {x}\\right) $ if and only if $\\int \\limits _{X}\\left( f\\left( \\mathbf {x}\\right) -\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) \\right)w_{j}(\\mathbf {x})h\\left( \\mathbf {a}^{j}\\cdot \\mathbf {x}\\right) d\\mathbf {x}=0\\ $ for any ridge function $h\\left( \\mathbf {a}^{j}\\cdot \\mathbf {x}\\right) $ such that $\\mathit {w}_{j}\\mathit {(x)h}\\left( \\mathbf {a}^{j}\\cdot \\mathbf {x}\\right) $$\\in L_{2}\\left( X\\right)$ , $j=1,...,r$ .", "Lemma 1.7.", "The following formula is valid for the error of approximation to a function $f\\left( \\mathbf {x}\\right) $ in $L_{2}\\left(X\\right) $ from $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right) $ : $E\\left( f\\right) =\\left( \\left\\Vert f\\left( \\mathbf {x}\\right) \\right\\Vert _{L_{2}\\left( X\\right) }^{2}-\\left\\Vert \\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) \\right\\Vert _{L_{2}\\left( X\\right) }^{2}\\right) ^{\\frac{1}{2}},$ where $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is the best approximation to $f\\left(\\mathbf {x}\\right) $ .", "Lemmas 1.6 and 1.7 follow from the well-known facts of functional analysis that the best approximation of an element $x$ in a Hilbert space $H$ from a linear subspace $Z$ of $H$ must be the image of $x$ via the orthogonal projection onto $Z$ and the sum of squares of norms of orthogonal vectors is equal to the square of the norm of their sum.", "We say that $Y$ is an $r$ -set if it can be represented as $Y_{1}\\times ...\\times Y_{r}\\times Y_{0},$ where $Y_{0}$ is some set from the space $\\mathbb {R}^{n-r}.$ In a special case, $Y_{0}$ may be equal to $Y_{r+1}\\times ...\\times Y_{n},$ but it is not necessary.", "By $Y^{\\left( i\\right) },$ we denote the Cartesian product of the sets $Y_{1},...,Y_{r},Y_{0}$ except for $Y_{i},\\;i=1,...,r$ .", "That is, $Y^{\\left( i\\right) }=Y_{1}\\times ...\\times Y_{i-1}\\times Y_{i+1}\\times ...\\times Y_{r}\\times Y_{0},\\,\\ i=1,...,r$ .", "Theorem 1.12.", "Let $Y$ be an $r$ -set.", "A function $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is the best approximation to $f(\\mathbf {x)}$ if and only if $g_{j}^{0}\\left( y_{j}\\right) =\\frac{1}{\\int \\limits _{Y^{\\left( j\\right)}}w_{j}^{\\ast 2}(\\mathbf {y})d\\mathbf {y}^{\\left( j\\right) }}\\int \\limits _{Y^{\\left( j\\right) }}\\left( f^{\\ast }\\left( \\mathbf {y}\\right)-\\sum \\limits _{\\begin{array}{c} i=1 \\\\ i\\ne j\\end{array}}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) \\right) w_{j}^{\\ast }(\\mathbf {y})d\\mathbf {y}^{\\left( j\\right) },(1.59)$ for $j=1,...,r$ .", "Necessity.", "Let a function $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ be extremal to $f$ .", "Then by Lemma 1.5, the function $\\sum \\limits _{i=1}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) $ in $L_{2}^{1}\\oplus ...\\oplus L_{2}^{r}$ is extremal to $f^{\\ast }$ .", "By Lemma 1.6 and equality (1.57), $\\int \\limits _{Y}f^{\\ast }\\left( \\mathbf {y}\\right) w_{j}^{\\ast }(\\mathbf {y})h\\left( y_{j}\\right) d\\mathbf {y}=\\int \\limits _{Y}w_{j}^{\\ast }(\\mathbf {y})h\\left( y_{j}\\right) \\sum \\limits _{i=1}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) d\\mathbf {y}(1.60)$ for any product $w_{j}^{\\ast }(\\mathbf {y})h\\left( y_{j}\\right) $ in $L_{2}^{j},\\;\\;j=1,...,r$ .", "Applying Fubini's theorem to the integrals in (1.60), we obtain that $&&\\int \\limits _{Y_{j}}h\\left( y_{j}\\right) \\left[ \\int \\limits _{Y^{\\left(j\\right) }}f^{\\ast }\\left( \\mathbf {y}\\right) w_{j}^{\\ast }(\\mathbf {y})d\\mathbf {y}^{\\left( j\\right) }\\right] dy_{j} \\\\&=&\\int \\limits _{Y_{j}}h\\left( y_{j}\\right) \\left[ \\int \\limits _{Y^{\\left(j\\right) }}w_{j}^{\\ast }(\\mathbf {y})\\sum \\limits _{i=1}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) d\\mathbf {y}^{\\left( j\\right) }\\right]dy_{j}.$ Since $h\\left( y_{j}\\right) $ is an arbitrary function such that $w_{j}^{\\ast }(\\mathbf {y})h\\left( y_{j}\\right) \\in L_{2}^{j}$ , $\\int \\limits _{Y^{\\left( j\\right) }}f^{\\ast }\\left( \\mathbf {y}\\right)w_{j}^{\\ast }(\\mathbf {y})d\\mathbf {y}^{(j)}=\\int \\limits _{Y^{\\left( j\\right)}}w_{j}^{\\ast }(\\mathbf {y})\\sum \\limits _{i=1}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) d\\mathbf {y}^{\\left( j\\right) },\\;\\;j=1,...,r.$ Therefore, $\\int \\limits _{Y^{\\left( j\\right) }}w_{j}^{\\ast 2}(\\mathbf {y})g_{j}^{0}\\left( {y_{j}}\\right) d\\mathbf {y}^{\\left( j\\right) }=\\int \\limits _{Y^{\\left( j\\right)}}\\left( f^{\\ast }\\left( \\mathbf {y}\\right) -\\sum \\limits _{\\begin{array}{c} i=1 \\\\ i\\ne j\\end{array}}^{r}w_{i}^{\\ast }(\\mathbf {y})g_{i}^{0}\\left( y_{i}\\right) \\right)w_{j}^{\\ast }(\\mathbf {y})d\\mathbf {y}^{\\left( j\\right) },$ for $j=1,...,r.$ Now, since $y_{j}\\notin Y^{\\left( j\\right) }$ , we obtain (1.59).", "Sufficiency.", "Note that all the equalities in the proof of the necessity can be obtained in the reverse order.", "Thus, (1.60) can be obtained from (1.59).", "Then by (1.57) and Lemma 1.6, we finally conclude that the function $\\sum \\limits _{i=1}^{r}w_{i}(\\mathbf {x})g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is extremal to $f\\left( \\mathbf {x}\\right) $ .", "In the following, $\\left|Q\\right|$ will denote the Lebesgue measure of a measurable set $Q.$ The following corollary is obvious.", "Corollary 1.6.", "Let $Y$ be an $r$-set.", "A function $\\sum \\limits _{i=1}^{r}g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ in $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ is the best approximation to $f(\\mathbf {x)}$ if and only if $g_{j}^{0}\\left( y_{j}\\right) =\\frac{1}{\\left|Y^{\\left( j\\right)}\\right|}\\int \\limits _{Y^{\\left( j\\right) }}\\left( f^{\\ast }\\left(\\mathbf {y}\\right) -\\sum \\limits _{\\begin{array}{c} i=1 \\\\ i\\ne j\\end{array}}^{r}g_{i}^{0}\\left( y_{i}\\right) \\right) d\\mathbf {y}^{\\left( j\\right)},\\;\\;j=1,...,r.$ In [61], this corollary was proven for the case $r=n.$ In this section, we establish explicit formulas for both the best approximation and approximation error, provided that the weight functions are constants.", "In this case, since we vary over $g_{i},$ the set $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};~w_{1},...,w_{r}\\right) $ coincides with $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) .$ Thus, without loss of generality, we may assume that $w_{i}(\\mathbf {x})=1$ for $i=1,...,r.$ For brevity of the further exposition, introduce the notation $A=\\int \\limits _{Y}f^{\\ast }\\left( \\mathbf {y}\\right) d\\mathbf {y}\\text{ and \\ }f_{i}^{\\ast }=f_{i}^{\\ast }(y_{i})=\\int \\limits _{Y^{\\left( i\\right) }}f^{\\ast }\\left( \\mathbf {y}\\right) d\\mathbf {y}^{\\left( i\\right) },~i=1,...,r.$ The following theorem is a generalization of the main result of [61] from the case $r=n$ to the cases $r<n.$ Theorem 1.13.", "Let $Y$ be an $r$-set.", "Set the functions $g_{1}^{0}\\left( y_{1}\\right) =\\frac{1}{\\left|Y^{\\left( 1\\right)}\\right|}f_{1}^{\\ast }-\\left( r-1\\right) \\frac{A}{\\left|Y\\right|}$ and $g_{j}^{0}\\left( y_{j}\\right) =\\frac{1}{\\left|Y^{\\left( j\\right)}\\right|}f_{j}^{\\ast },\\;j=2,...,r.$ Then the function $\\sum \\limits _{i=1}^{r}g_{i}^{0}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) $ is the best approximation from $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ to $f\\left( \\mathbf {x}\\right) $ .", "The proof is simple.", "It is sufficient to verify that the functions $g_{j}^{0}\\left( y_{j}\\right) ,\\;j=1,...,r$ , satisfy the conditions of Corollary 1.6.", "This becomes obvious if note that $\\sum \\limits _{\\underset{i\\ne j}{i=1}}^{r}\\frac{1}{\\left|Y^{\\left(j\\right) }\\right|}\\frac{1}{\\left|Y^{\\left( i\\right) }\\right|}\\int \\limits _{Y^{\\left( j\\right) }}\\left[ \\int \\limits _{Y^{\\left( i\\right)}}f^{\\ast }\\left( \\mathbf {y}\\right) d\\mathbf {y}^{\\left( i\\right) }\\right] d\\mathbf {y}^{\\left( j\\right) }=\\left( r-1\\right) \\frac{1}{\\left|Y\\right|}\\int \\limits _{Y}f^{\\ast }\\left( \\mathbf {y}\\right) d\\mathbf {y}$ for $j=1,...,r$ .", "Theorem 1.14.", "Let $Y$ be an $r$ -set.", "Then the error of approximation to a function $f(x)$ from the set $\\mathcal {R}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{r}\\right) $ can be calculated by the formula $E(f)=\\left|\\det J\\right|^{-1/2}\\left( \\left\\Vert f^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}-\\sum _{i=1}^{r}\\frac{1}{\\left|Y^{\\left(i\\right) }\\right|^{2}}\\left\\Vert f_{i}^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}+(r-1)\\frac{A^{2}}{\\left|Y\\right|}\\right) ^{1/2}.$ From Eq.", "(1.58), Lemma 1.7 and Theorem 1.13, it follows that $E(f)=\\left|\\det J\\right|^{-1/2}\\left( \\left\\Vert f^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}-I\\right) ^{1/2},(1.61)$ where $I=\\left\\Vert \\sum _{i=1}^{r}\\frac{1}{\\left|Y^{\\left( i\\right)}\\right|}f_{i}^{\\ast }-(r-1)\\frac{A}{\\left|Y\\right|}\\right\\Vert _{L_{2}(Y)}^{2}.$ The integral $I$ can be written as a sum of the following four integrals: $I_{1} &=&\\sum _{i=1}^{r}\\frac{1}{\\left|Y^{\\left( i\\right) }\\right|^{2}}\\left\\Vert f_{i}^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2},~I_{2}=\\sum _{i=1}^{r}\\sum \\limits _{\\begin{array}{c} j=1 \\\\ j\\ne i\\end{array}}^{r}\\frac{1}{\\left|Y^{\\left( i\\right) }\\right|}\\frac{1}{\\left|Y^{\\left( j\\right) }\\right|}\\int \\limits _{Y}f_{i}^{\\ast }f_{j}^{\\ast }d\\mathbf {y,} \\\\I_{3} &=&-2(r-1)\\frac{1}{\\left|Y\\right|}A\\sum _{i=1}^{r}\\frac{1}{\\left|Y^{\\left( i\\right) }\\right|}\\int \\limits _{Y}f_{i}^{\\ast }d\\mathbf {y,}~I_{4}=(r-1)^{2}\\frac{A^{2}}{\\left|Y\\right|}.$              It is not difficult to verify that $\\int \\limits _{Y}f_{i}^{\\ast }f_{j}^{\\ast }d\\mathbf {y=}\\left|Y_{0}\\times \\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne i,j\\end{array}}^{r}Y_{k}\\right|A^{2},\\text{for }i,j=1,...,r,~i\\ne j,(1.62)$ and $\\int \\limits _{Y}f_{i}^{\\ast }d\\mathbf {y}=\\left|Y_{0}\\times \\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne i\\end{array}}^{r}Y_{k}\\right|A,\\text{ for }i=1,...,r.(1.63)$ Considering (1.62) and (1.63) in the expressions of $I_{2}$ and $I_{3}$ respectively, we obtain that $I_{2}=r(r-1)\\frac{A^{2}}{\\left|Y\\right|}\\text{ and }I_{3}=-2r(r-1)\\frac{A^{2}}{\\left|Y\\right|}.$ Therefore, $I=I_{1}+I_{2}+I_{3}+I_{4}=\\sum _{i=1}^{r}\\frac{1}{\\left|Y^{\\left(i\\right) }\\right|^{2}}\\left\\Vert f_{i}^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}-(r-1)\\frac{A^{2}}{\\left|Y\\right|}.$ Now the last equality together with (1.61) complete the proof.", "Example.", "Consider the following set $X=\\lbrace \\mathbf {x}\\in \\mathbb {R}^{4}:y_{i}=y_{i}(\\mathbf {x})\\in [0;1],~i=1,...,4\\rbrace ,$ where $\\left\\lbrace \\begin{array}{c}y_{1}=x_{1}+x_{2}+x_{3}-x_{4} \\\\y_{2}=x_{1}+x_{2}-x_{3}+x_{4} \\\\y_{3}=x_{1}-x_{2}+x_{3}+x_{4} \\\\y_{4}=-x_{1}+x_{2}+x_{3}+x_{4}\\end{array}\\right.", "(1.64)$ Let the function $f=8x_{1}x_{2}x_{3}x_{4}-\\sum _{i=1}^{4}x_{i}^{4}+2\\sum _{i=1}^{3}\\sum _{j=i+1}^{4}x_{i}^{2}x_{j}^{2}$ be given on $X.$ Consider the approximation of this function by functions from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2},\\mathbf {a}^{3}\\right) ,\\mathcal {\\ }$ where $\\mathbf {a}^{1}=(1;1;1;-1),~\\mathbf {a}^{2}=(1;1;-1;1),~\\mathbf {a}^{3}=(1;-1;1;1).$ Putting $\\mathbf {a}^{4}=(-1;1;1;1),$ we complete the system of vectors $\\mathbf {a}^{1},\\mathbf {a}^{2},\\mathbf {a}^{3}$ to the basis $\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2},\\mathbf {a}^{3},\\mathbf {a}^{4}\\rbrace $ in $\\mathbb {R}^{4}.$ The linear transformation $J$ defined by (1.64) maps the set $X$ onto the set $Y=[0;1]^{4}.$ The inverse transformation is given by the formulas $\\left\\lbrace \\begin{array}{c}x_{1}=\\frac{1}{4}y_{1}+\\frac{1}{4}y_{2}+\\frac{1}{4}y_{3}-\\frac{1}{4}y_{4} \\\\x_{2}=\\frac{1}{4}y_{1}+\\frac{1}{4}y_{2}-\\frac{1}{4}y_{3}+\\frac{1}{4}y_{4} \\\\x_{3}=\\frac{1}{4}y_{1}-\\frac{1}{4}y_{2}+\\frac{1}{4}y_{3}+\\frac{1}{4}y_{4} \\\\x_{4}=-\\frac{1}{4}y_{1}+\\frac{1}{4}y_{2}+\\frac{1}{4}y_{3}+\\frac{1}{4}y_{4}\\end{array}\\right.$ It can be easily verified that $f^{\\ast }=y_{1}y_{2}y_{3}y_{4}$ and $Y$ is a 3-set with $Y_{i}=[0;1],$ $i=1,2,3.$ Besides, $Y_{0}=[0;1].$ After easy calculations we obtain that $A=\\frac{1}{16};~$ $f_{i}^{\\ast }=\\frac{1}{8}y_{i}$ for $i=1,2,3;$ $\\det J=-16;$ $\\left\\Vert f^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}=\\frac{1}{81};$ $\\left\\Vert f_{i}^{\\ast }\\right\\Vert _{L_{2}(Y)}^{2}=\\frac{1}{192},$ $i=1,2,3.$ Now from Theorems 1.13 and 1.14 it follows that the function $\\frac{1}{8}\\sum _{i=1}^{3}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right) -\\frac{1}{8}$ is the best approximation from $\\mathcal {R}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2},\\mathbf {a}^{3}\\right) $ to $f$ and $E(f)=\\frac{1}{576}\\sqrt{2}\\sqrt{47}.$ Remark 1.4.", "Most of the material in this chapter is to be found in [61], [62], [63], [64], [65], [66], [71], [76].", "This chapter discusses the following open problem raised in Buhmann and Pinkus [18], and Pinkus [137].", "Assume we are given a function $f(\\mathbf {x})=f(x_{1},...,x_{n})$ of the form $f(\\mathbf {x})=\\sum _{i=1}^{k}f_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),(2.1)$ where the $\\mathbf {a}^{i},$ $i=1,...,k,$ are pairwise linearly independent vectors (directions) in $\\mathbb {R}^{n}$ , $f_{i}$ are arbitrarily behaved univariate functions and $\\mathbf {a}^{i}\\cdot \\mathbf {x}$ are standard inner products.", "Assume, in addition, that $f$ is of a certain smoothness class, that is, $f\\in C^{s}(\\mathbb {R}^{n})$ , where $s\\ge 0$ (with the convention that $C^{0}(\\mathbb {R}^{n})=C(\\mathbb {R}^{n})$ ).", "Is it true that there will always exist $g_{i}\\in C^{s}(\\mathbb {R})$ such that $f(\\mathbf {x})=\\sum _{i=1}^{k}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})\\text{ ?", "}(2.2)$ In this chapter, we solve this problem up to some multivariate polynomial.", "In the special case $n=2$ , we see that this multivariate polynomial can be written as a sum of polynomial ridge functions with the given directions $\\mathbf {a}^{i}$ .", "In addition, we find various conditions on the directions $\\mathbf {a}^{i}$ guaranteeing a positive solution to the problem.", "We also consider the question on constructing $g_{i}$ using the information about the known functions $f_{i}$ .", "Most of the material of this chapter may be found in [2], [3], [4], [5], [136].", "In this section, we solve the above problem up to a multivariate polynomial.", "That is, we show that if (2.1) holds for $f\\in C^{s}(\\mathbb {R}^{n})$ and arbitrarily behaved $f_{i}$ , then there exist $g_{i}\\in C^{s}(\\mathbb {R})$ such that $f(\\mathbf {x})=\\sum _{i=1}^{k}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})+P(\\mathbf {x}),$ where $P(\\mathbf {x})$ is a polynomial of degree at most $k-1$ .", "In the special case $n=2$ , we see that this multivariate polynomial can be written as a sum of polynomial ridge functions with the given directions $\\mathbf {a}^{i}$ and thus (2.2) holds with $g_{i}\\in C^{s}(\\mathbb {R})$ .", "We start this subsection with the simple observation that for $k=1$ and $k=2$ the smoothness problem is easily solved.", "Indeed for $k=1$ by choosing $\\mathbf {c}\\in \\mathbb {R}^{n}$ satisfying $\\mathbf {a}^{1}\\cdot \\mathbf {c}=1$ , we have that $f_{1}(t)=f(t\\mathbf {c)}$ is in $C^{s}(\\mathbb {R})$ .", "The same argument can be carried out for the case $k=2.$ In this case, since the vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ are linearly independent, there exists a vector $\\mathbf {c}\\in \\mathbb {R}^{n}$ satisfying $\\mathbf {a}^{1}\\cdot \\mathbf {c}=1$ and $\\mathbf {a}^{2}\\cdot \\mathbf {c}=0.$ Therefore, we obtain that the function $f_{1}(t)=f(t\\mathbf {c)}-f_{2}(0)$ is in the class $C^{s}(\\mathbb {R})$ .", "Similarly, one can verify that $f_{2}\\in C^{s}(\\mathbb {R})$ .", "The above cases with one and two ridge functions in (2.1) show that the functions $f_{i}$ inherit smoothness properties of the given $f$ .", "The picture is absolutely different if the number of directions $k\\ge 3$ .", "For $k=3$ , there are ultimately smooth functions which decompose into sums of very badly behaved ridge functions.", "This phenomena comes from the classical Cauchy Functional Equation (CFE).", "This equation, $h(x+y)=h(x)+h(y),\\text{ }h:\\mathbb {R\\rightarrow R},(2.3)$ looks very simple and has a class of simple solutions $h(x)=cx,$ $c\\in \\mathbb {R}$ .", "However, it easily follows from Hamel basis theory that CFE also has a large class of wild solutions.", "These solutions are called “wild\" because they are extremely pathological.", "They are, for example, not continuous at a point, not monotone on an interval, not bounded on any set of positive measure (see, e.g., [1]).", "Let $h_{1}$ be any wild solution of the equation (2.3).", "Then the zero function can be represented as $0=h_{1}(x)+h_{1}(y)-h_{1}(x+y).", "(2.4)$ Note that the functions involved in (2.4) are bivariate ridge functions with the directions $(1,0)$ , $(0,1)$ and $(1,1)$ , respectively.", "This example shows that for $k\\ge 3$ the functions $f_{i}$ in (2.1) may not inherit smoothness properties of the function $f$ , which in the case of (2.4) is the identically zero function.", "Thus the above problem arises naturally.", "However, it was shown by some authors that, additional conditions on $f_{i}$ or the directions $\\mathbf {a}^{i}$ guarantee smoothness of the representation (2.1).", "It was first proved by Buhmann and Pinkus [18] that if in (2.1) $f\\in C^{s}(\\mathbb {R}^{n})$ , $s\\ge k-1$ and $f_{i}\\in L_{loc}^{1}(\\mathbb {R)}$ for each $i$ , then $f_{i}\\in C^{s}(\\mathbb {R)}$ for $i=1,...,k.$ Later Pinkus [136] found a strong relationship between CFE and the problem of smoothness in ridge function representation.", "He generalized extensively the previous result of Buhmann and Pinkus [18].", "He showed that the solution is quite simple and natural if the functions $f_{i}$ are taken from a certain class $\\mathcal {B}$ of real-valued functions defined on $\\mathbb {R}$ .", "$\\mathcal {B}$ includes, for example, the set of continuous functions, the set of bounded functions, the set of Lebesgue measurable functions (for the precise definition of $\\mathcal {B}$ see the next subsection).", "The result of Pinkus [136] states that if in (1.1) $f\\in C^{s}(\\mathbb {R}^{n})$ and each $f_{i}\\in \\mathcal {B}$ , then necessarily $f_{i}\\in C^{s}(\\mathbb {R)}$ for $i=1,...,k$ .", "Note that severe restrictions on the directions $\\mathbf {a}^{i}$ also guarantee smoothness of the representation (2.1).", "For example, in (2.1) the inclusions $f_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k,$ are automatically valid if the directions $\\mathbf {a}^{i}$ are linearly independent and if these directions are not linearly independent, then there exists $f\\in C^{s}(\\mathbb {R}^{n})$ of the form (2.1) such that the $f_{i}\\notin C^{s}(\\mathbb {R}),$ $i=1,...,k$ (see [100]).", "Indeed, if the directions $\\mathbf {a}^{i}$ are linearly independent, then for each $i=1,...,k,$ we can choose a vector $\\mathbf {b}^{i}$ such that $\\mathbf {b}^{i}\\cdot \\mathbf {a}^{i}=1,$ but at the same time $\\mathbf {b}^{i}\\cdot \\mathbf {a}^{j}=0,$ for all $j=1,...,k,$ $j\\ne i$ .", "Putting $\\mathbf {x}=\\mathbf {b}^{i}t$ in (2.1) yields that $f(\\mathbf {b}^{i}t)=f_{i}(t)+\\sum _{j=1,j\\ne i}^{k}f_{j}(0),\\text{ }i=1,...,k.$ This shows that all the functions $f_{i}$ and $f$ belong to the same smoothness class.", "If the directions $\\mathbf {a}^{i}$ are not linearly independent, then there exist numbers $\\lambda _{1},...,\\lambda _{k}$ such that $\\sum _{i=1}^{k}\\left|\\lambda _{i}\\right|>0$ and $\\sum _{i=1}^{k}\\lambda _{i}\\mathbf {a}^{i}=\\mathbf {0}$ .", "Let $h$ be any wild solution of CFE.", "Then it is not difficult to verify that $0=\\sum _{i=1}^{k}h_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),$ where $h_{i}(t)=h(\\lambda _{i}t),$ $i=1,...,k.$ Note that in the last representation, the zero function is an ultimately smooth function, while all the functions $h_{i}$ are highly nonsmooth.", "The above result of Pinkus was a starting point for further research on continuous and smooth sums of ridge functions.", "Much work in this direction was done by Konyagin and Kuleshov [100], [101], and Kuleshov [106].", "They mainly analyze the continuity of $f_{i}$ , that is, the question of if and when continuity of $f$ guarantees the continuity of $f_{i}$ .", "There are also other results concerning different properties, rather than continuity, of $f_{i}$ .", "Most results in [100], [101], [106] involve certain subsets (convex open sets, convex bodies, etc.)", "of $\\mathbb {R}^{n}$ instead of only $\\mathbb {R}^{n}$ itself.", "In [3], Aliev and Ismailov gave a partial solution to the smoothness problem.", "Their solution comprises the cases in which $s\\ge 2$ and $k-1$ directions of the given $k$ directions are linearly independent.", "Kuleshov [105] generalized Aliev and Ismailov's result [3] to all possible cases of $s$ .", "That is, he proved that if a function $f\\in C^{s}(\\mathbb {R}^{n})$ , where $s\\ge 0$ , is of the form (2.1) and $(k-1) $ -tuple of the given set of $k$ directions $\\mathbf {a}^{i}$ forms a linearly independent system, then there exist $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k $ , such that (2.2) holds (see [105]).", "In Section 2.2 we give a new constructive proof of Kuleshov's result.", "In [136], A. Pinkus considered the smoothness problem in ridge function representation.", "For a given function $f$ $:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ , he posed and partially answered the following question.", "If $f$ belongs to some smoothness class and (2.1) holds, what can we say about the smoothness of the functions $f_{i}$ ?", "He proved that for a large class of representing functions $f_{i}$ , these $f_{i}$ are smooth.", "That is, if apriori we assume that in the representation (2.1) the functions $f_{i}$ is of a certain class of “reasonably well behaved functions\", then they have the same degree of smoothness as the function $f.$ As the mentioned class of “reasonably well behaved functions\" one may take, e.g., the set of functions that are continuous at a point, the set of Lebesgue measurable functions, etc.", "All these classes arise from the class $\\mathcal {B}$ considered by Pinkus [136] and the classical theory of CFE.", "In [136], $\\mathcal {B}$ denotes any linear space of real-valued functions $u$ defined on $\\mathbb {R}$ , closed under translation, such that if there is a function $v\\in C(\\mathbb {R)}$ for which $u-v$ satisfies CFE, then $u-v$ is necessarily linear, i.e.", "$u(x)-v(x)=cx,$ for some constant $c\\in \\mathbb {R}$ .", "Such a definition of $\\mathcal {B}$ is required in the proof of the following theorem.", "Theorem 2.1 (Pinkus [136]).", "Assume $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.1).", "Assume, in addition, that each $f_{i}\\in \\mathcal {B}$ .", "Then necessarily $f_{i}\\in C^{s}(\\mathbb {R)}$ for $i=1,...,k.$ We prove this theorem by induction on $k.$ The result is valid when $k=1$ .", "Indeed, taking any direction $\\mathbf {c}$ such that $\\mathbf {a}^{1}\\cdot \\mathbf {c}=1$ and putting $x=\\mathbf {c}t$ in (2.1), we obtain that $f_{1}(t)=f(\\mathbf {c}t)\\in C^{s}(\\mathbb {R})$ .", "Assume that the result is valid for $k-1.$ Let us show that it is valid for $k$ .", "Chose any vector $\\mathbf {e}\\in \\mathbb {R}^{n}$ satisfying $\\mathbf {e\\cdot a}^{k}=0$ and $\\mathbf {e\\cdot a}^{i}=b_{i}\\ne 0$ , for $i=1,...,k-1.$ Clearly, there exists a vector with this property.", "The property of $\\mathbf {e}$ enables us to write that $f(\\mathbf {x+e}t)-f(\\mathbf {x)=}\\sum _{i=1}^{k-1}f_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}+b_{i}t)-f_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}).$ Thus $F(\\mathbf {x}):=f(\\mathbf {x+e}t)-f(\\mathbf {x})=\\sum _{i=1}^{k-1}h_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}),$ where $h_{i}(y)=f_{i}(y+b_{i}t)-f_{i}(y)\\text{, }i=1,...,k-1.$ Since $f_{i}\\in \\mathcal {B}$ and $\\mathcal {B}$ is translation invariant, $h_{i}\\in \\mathcal {B}$ .", "In addition, since $F\\in C^{s}(\\mathbb {R}^{n})$ , it follows by our induction assumption that $h_{i}\\in C^{s}(\\mathbb {R})$ .", "Note that this inclusion is valid for all $t\\in \\mathbb {R}$ .", "In [19], de Bruijn proved that if for any $c\\in \\mathbb {R}$ the difference $u(y+c)-u(y)$ ($u$ is any real function on $\\mathbb {R}$ ) belongs to the class $C^{s}(\\mathbb {R})$ , then $u$ is necessarily of the form $u=v+r$ , where $v\\in $ $C^{s}(\\mathbb {R})$ and $r$ satisfies CFE.", "Thus each function $f_{i}$ is of the form $f_{i}=v_{i}+r_{i}$ , where $v_{i}\\in $ $C^{s}(\\mathbb {R})$ and $r_{i}$ satisfies CFE.", "By our assumption, each $f_{i}$ is in $\\mathcal {B}$ , and from the definition of $\\mathcal {B}$ it follows that $r_{i}=f_{i}-v_{i}$ is a linear function.", "Thus $f_{i}=v_{i}+r_{i}$ , where both $v_{i},r_{i}\\in $ $C^{s}(\\mathbb {R})$ , implying that $f_{i}\\in C^{s}(\\mathbb {R})$ .", "This is valid for $i=1,...,k-1$ , and hence also for $i=k$ .", "Remark 2.1.", "In de Bruijn [19], [20], there are delineated various classes of real-valued functions $\\mathcal {D}$ with the property that if $\\bigtriangleup _{t}f=f(\\cdot +t)-f(\\cdot )\\in \\mathcal {D}$ for all $t\\in \\mathbb {R}$ , then $f-s\\in \\mathcal {D}$ , for some $s$ satisfying CFE (for such classes see the next subsection).", "Some translation invariant classes among them are $C^{\\infty }(\\mathbb {R})$ functions; analytic functions; algebraic polynomials; trigonometric polynomials.", "Theorem 2.1 can be suitably restated for any of these classes.", "Given $h_{1},...,h_{k}\\in \\mathbb {R}$ , we define inductively the difference operator $\\Delta _{h_{1}...h_{k}}$ as follows $\\Delta _{h_{1}}f(x) &:&=f(x+h_{1})-f(x), \\\\\\Delta _{h_{1}...h_{k}}f &:&=\\Delta _{h_{k}}(\\Delta _{h_{1}...h_{k-1}}f),\\text{ }f:\\mathbb {R\\rightarrow R}.$ If $h_{1}=\\cdots =h_{k}=h,$ then we write briefly $\\Delta _{h}^{k}f$ instead of $\\Delta _{\\underset{n\\text{ times}}{\\underbrace{h...h}}}f$ .", "For various properties of difference operators see [104].", "Definition 2.1 (see [104]).", "A function $f:\\mathbb {R\\rightarrow R}$ is called a polynomial function of order $k$ ($k\\in \\mathbb {N}$ ) if for every $x\\in \\mathbb {R}$ and $h\\in \\mathbb {R}$ we have $\\Delta _{h}^{k+1}f(x)=0.$ It can be shown that if $\\Delta _{h}^{k+1}f=0$ for any $h\\in \\mathbb {R}$ , then $\\Delta _{h_{1}...h_{k+1}}f=0$ for any $h_{1},...,h_{k+1}\\in \\mathbb {R}$ (see [104]).", "A polynomial of degree at most $k$ is a polynomial function of order $k$ (see [104]).", "The polynomial functions generalize ordinary polynomials, and reduce to the latter under mild regularity assumptions.", "For example, if a polynomial function is continuous at one point, or bounded on a set of positive measure, then it continuous at all points (see [28], [107]), and therefore is a polynomial of degree $k$ (see [104]).", "Basic results concerning polynomial functions are due to S. Mazur-W. Orlicz [124], McKiernan [125], Djoković [37].", "The following theorem, which we will use in the sequel, yield implicitly the general construction of polynomial functions.", "Theorem 2.2 (see [104]).", "A function $f:\\mathbb {R\\rightarrow R}$ is a polynomial function of order $k$ if and only if it admits a representation $f=f_{0}+f_{1}+...+f_{k},$ where $f_{0}$ is a constant and $f_{j}:\\mathbb {R\\rightarrow R}$ , $j=1,...,k$ , are diagonalizations of $j$ -additive symmetric functions $F_{j}:\\mathbb {R}^{j}\\mathbb {\\rightarrow R}$ , i.e., $f_{j}(x)=F_{j}(x,...,x).$ Note that a function $F_{p}:\\mathbb {R}^{p}\\mathbb {\\rightarrow R}$ is called $p$ -additive if for every $j,$ $1\\le j\\le p,$ and for every $x_{1},...,x_{p},y_{j}\\in \\mathbb {R}$ $F(x_{1},...,x_{j}+y_{j},...,x_{p})=F(x_{1},...,x_{p})+F(x_{1},...,x_{j-1},y_{j},x_{j+1},...,x_{p}),$ i.e., $F$ is additive in each of its variables $x_{j}$ (see [104]).", "A simple example of a $p$ -additive function is given by the product $f_{1}(x_{1})\\times \\cdots \\times f_{p}(x_{p}),$ where the univariate functions $f_{j},$ $j=1,...,p$ , are additive.", "Following de Bruijn, we say that a class $\\mathcal {D}$ of real functions has the difference property if any function $f:\\mathbb {R\\rightarrow R}$ such that $\\bigtriangleup _{h}f\\in \\mathcal {D}$ for all $h\\in \\mathbb {R}$ , admits a decomposition $f=g+S$ , where $g\\in \\mathcal {D}$ and $S$ satisfies the Cauchy Functional Equation (2.3).", "Several classes with the difference property are investigated in de Bruijn [19], [20].", "Some of these classes are: 1) $C(\\mathbb {R)}$ , continuous functions; 2) $C^{s}(\\mathbb {R)}$ , functions with continuous derivatives up to order $s$ ; 3) $C^{\\infty }(\\mathbb {R)}$ , infinitely differentiable functions; 4) analytic functions; 5) functions which are absolutely continuous on any finite interval; 6) functions having bounded variation over any finite interval; 7) algebraic polynomials; 8) trigonometric polynomials; 9) Riemann integrable functions.", "A natural generalization of classes with the difference property are classes of functions with the difference property of $k$ -th order.", "Definition 2.2 (see [44]).", "A class $\\mathcal {F}$ is said to have the difference property of $k$ -th order if any function $f:\\mathbb {R\\rightarrow R}$ such that $\\bigtriangleup _{h}^{k}f\\in \\mathcal {F}$ for all $h\\in \\mathbb {R}$ , admits a decomposition $f=g+H$ , where $g\\in \\mathcal {F}$ and $H$ is a polynomial function of $k$ -th order.", "It is not difficult to see that the class $\\mathcal {F}$ has the difference property of first order if and only if it has the difference property in de Bruijn's sense.", "There arises a natural question: which of the above classes have difference properties of higher orders?", "Gajda [44] considered this question in its general form, for functions defined on a locally compact Abelian group and showed that for any $k\\in \\mathbb {N}$ , continuous functions have the difference property of $k$ -th order (see [44]).", "The proof of this result is based on several lemmas, in particular, on the following lemma, which we will also use in the sequel.", "Lemma 2.1.", "(see [44]).", "For each $k\\in \\mathbb {N}$ the class of all continuous functions defined on $\\mathbb {R}$ has the difference property of $k$ -th order.", "In fact, Gajda [44] proved this lemma for Banach space valued functions, but the simplest case with the space $\\mathbb {R}$ has all the difficulties.", "Unfortunately, the proof of the lemma has an essential gap.", "The author of [44] tried to reduce the proof to $\\mod {1}$ periodic functions, but made a mistake in proving the continuity of the difference $\\Delta _{h_{1}...h_{k-1}}(f-f^{\\ast })$ .", "Here $f^{\\ast }:\\mathbb {R\\rightarrow R}$ is a $\\mod {1}$ periodic function defined on the interval $[0,1)$ as $f^{\\ast }(x)=f(x)$ and extended to the whole $\\mathbb {R}$ with the period 1.", "That is, $f^{\\ast }(x)=f(x)$ for $x\\in [0,1)$ and $f^{\\ast }(x+1)=f^{\\ast }(x)$ for $x\\in \\mathbb {R}$ .", "In the proof, the author of [44] takes a point $x\\in [m,m+1)$ and writes that $\\Delta _{h_{1}...h_{k-1}}(f-f^{\\ast })(x)=\\Delta _{h_{1}...h_{k-1}}(f(x)-f(x-m)),$ which is not valid.", "Even though $f^{\\ast }(x)=f(x-m)$ for any $x\\in [m,m+1)$ , the differences $\\Delta _{h_{1}...h_{k-1}}f^{\\ast }(x)$ and $\\Delta _{h_{1}...h_{k-1}}f(x-m)$ are completely different, since the latter may involve values of $f$ at points outside $[0,1)$ , which have no relationship with the definition of $f^{\\ast }$ .", "In the next section, we give a new proof for Lemma 2.1 (see Theorem 2.3 below).", "We hope that our proof is free from mathematical errors and thus the above lemma itself is valid.", "In this section, we do further research on polynomial functions and prove some auxiliary results.", "Lemma 2.2.", "If $f:\\mathbb {R\\rightarrow R}$ is a polynomial function of order $k$ , then for any $p\\in $ $\\mathbb {N}$ and any fixed $\\xi _{1},...,\\xi _{p}\\in \\mathbb {R}$ , the function $g(x_{1},...,x_{p})=f(\\xi _{1}x_{1}+\\cdots +\\xi _{p}x_{p}),$ considered on the $p$ dimensional space $\\mathbb {Q}^{p}$ of rational vectors, is an ordinary polynomial of degree at most $k$ .", "By Theorem 2.2, $f=\\sum _{m=0}^{k}f_{m},(2.5)$ where $f_{0}$ is a constant and $f_{m}:\\mathbb {R\\rightarrow R}$ , $1,...,m$ , are diagonalizations of $m$ -additive symmetric functions $F_{m}:\\mathbb {R}^{m}\\mathbb {\\rightarrow R}$ , i.e., $f_{m}(x)=F_{m}(x,...,x).$ For a $m$ -additive function $F_{m}$ the equality $F_{m}(\\xi _{1},...,\\xi _{i-1},r\\xi _{i},\\xi _{i+1},...,\\xi _{m})=rF_{m}(\\xi _{1},...,\\xi _{m})$ holds for all $i=1,...,m$ and any $r\\in \\mathbb {Q}$ , $\\xi _{i}\\in $ $\\mathbb {R}$ , $i=1,...,m$ (see [104]).", "Using this, it is not difficult to verify that for any $(x_{1},...,x_{p})\\in \\mathbb {Q}^{p}$ , $f_{m}(\\xi _{1}x_{1}+\\cdots +\\xi _{p}x_{p}) &=&F_{m}(\\xi _{1}x_{1}+\\cdots +\\xi _{p}x_{p},...,\\xi _{1}x_{1}+\\cdots +\\xi _{p}x_{p}) \\\\&=&\\sum _{\\begin{array}{c} 0\\le s_{i}\\le m,~\\overline{i=1,p} \\\\ s_{1}+\\cdots +s_{p}=m\\end{array}}A_{s_{1}...s_{p}}F_{m}(\\underset{s_{1}}{\\underbrace{\\xi _{1},...,\\xi _{1}}},...,\\underset{s_{p}}{\\underbrace{\\xi _{p},...,\\xi _{p}}})x_{1}^{s_{1}}...x_{p}^{s_{p}}.$ Here $A_{s_{1}...s_{p}}$ are some coefficients, namely $A_{s_{1}...s_{p}}=m!/(s_{1}!...s_{p}!", ").$ Considering the last formula in (2.5), we conclude that the function $g(x_{1},...,x_{p})$ , restricted to $\\mathbb {Q}^{p}$ , is a polynomial of degree at most $k$ .", "Lemma 2.3.", "Assume $f$ is a polynomial function of order $k$ .", "Then there exists a polynomial function $H$ of order $k+1$ such that $H(0)=0$ and $f(x)=H(x+1)-H(x).", "(2.6)$ Consider the function $H(x):=xf(x)+\\sum _{i=1}^{k}(-1)^{i}\\frac{x(x+1)...(x+i)}{(i+1)!", "}\\Delta _{1}^{i}f(x).", "(2.7)$ Clearly, $H(0)=0.$ We are going to prove that $H$ is a polynomial function of order $k+1$ and satisfies (2.6).", "Let us first show that for any polynomial function $g$ of order $m$ the function $G_{1}(x)=xg(x)$ is a polynomial function of order $m+1.$ Indeed, for any $h_{1},...,h_{m+2}\\in \\mathbb {R}$ we can write that $\\Delta _{h_{1}...h_{m+2}}G_{1}(x)=(x+h_{1}+\\cdots +h_{m+2})\\Delta _{h_{1}...h_{m+2}}g(x)$ $+\\sum _{i=1}^{m+2}h_{i}\\Delta _{h_{1}...h_{i-1}h_{i+1...}h_{m+2}}g(x).", "(2.8)$ The last formula is verified directly by using the known product property of differences, that is, the equality $\\Delta _{h}(g_{1}g_{2})=g_{1}\\Delta _{h}g_{2}+g_{2}\\Delta _{h}g_{1}+\\Delta _{h}g_{1}\\Delta _{h}g_{2}.", "(2.9)$ Now since $g$ is a polynomial function of order $m$ , all summands in (2.8) is equal to zero; hence we obtain that $G_{1}(x)$ is a polynomial function of order $m+1$ .", "By induction, we can prove that the function $G_{p}(x)=x^{p}g(x)$ is a polynomial function of order $m+p.$ Since $\\Delta _{1}^{i}f(x)$ in (2.7) is a polynomial function of order $k-i$ , it follows that all summands in (2.7) are polynomial functions of order $k+1$ .", "Therefore, $H(x)$ is a polynomial function of order $k+1$ .", "Now let us prove (2.6).", "Considering the property (2.9) in (2.7) we can write that $\\Delta _{1}H(x)=\\left[ f(x)+(x+1)\\Delta _{1}f(x)\\right]$ $+\\sum _{i=1}^{k}(-1)^{i}\\left[ \\frac{(x+1)...(x+i+1)}{(i+1)!", "}\\Delta _{1}^{i+1}f(x)+\\Delta _{1}\\left( \\frac{x(x+1)...(x+i)}{(i+1)!", "}\\right) \\Delta _{1}^{i}f(x)\\right] .", "(2.10)$ Note that in (2.10) $\\Delta _{1}\\left( \\frac{x(x+1)...(x+i)}{(i+1)!", "}\\right) =\\frac{(x+1)...(x+i)}{i!", "}.$ Considering this and the assumption $\\Delta _{1}^{k+1}f(x)=0$ , it follows from (2.10) that $\\Delta _{1}H(x)=f(x),$ that is, (2.6) holds.", "The next lemma is due to Gajda [44].", "Lemma 2.4 (see [44]).", "Let $f:$ $\\mathbb {R\\rightarrow R}$ be a $\\mod {1}$ periodic function such that, for any $h_{1},...,h_{k}\\in \\mathbb {R}$ , $\\Delta _{h_{1}...h_{k}}f$ is continuous.", "Then there exist a continuous function $g:$ $\\mathbb {R\\rightarrow R}$ and a polynomial function $H$ of $k$ -th order such that $f=g+H$ .", "The following theorem generalizes de Bruijn's theorem (see [19]) on the difference property of continuous functions and shows that Gajda's above lemma (see Lemma 2.1) is valid.", "Note that the main result of [44] also uses this theorem.", "Theorem 2.3.", "Assume for any $h_{1},...,h_{k}\\in \\mathbb {R}$ , the difference $\\Delta _{h_{1}...h_{k}}f(x)$ is a continuous function of the variable $x$ .", "Then there exist a function $g\\in C(\\mathbb {R})$ and a polynomial function $H$ of $k$ -th order with the property $H(0)=0$ such that $f=g+H.$ We prove this theorem by induction.", "For $k=1$ , the theorem is the result of de Bruijn: if $f$ is such that, for each $h$ , $\\Delta _{h}f(x)$ is a continuous function of $x$ , then it can be written in the form $g+H$ , where $g$ is continuous and $H$ is additive (that is, satisfies the Cauchy Functional Equation).", "Assume that the theorem is valid for $k-1.$ Let us prove it for $k$ .", "Without loss of generality we may assume that $f(0)=f(1)$ .", "Otherwise, we can prove the theorem for $f_{0}(x)=f(x)-\\left[f(1)-f(0)\\right] x$ and then automatically obtain its validity for $f$ .", "Consider the function $F_{1}(x)=f(x+1)-f(x)\\text{, }x\\in \\mathbb {R}.", "(2.11)$ Since for any $h_{1},...,h_{k}\\in \\mathbb {R}$ , $\\Delta _{h_{1}...h_{k}}f(x)$ is a continuous function of $x$ and $\\Delta _{h_{1}...h_{k-1}}F_{1}=\\Delta _{h_{1}...h_{k-1}1}f$ , the difference $\\Delta _{h_{1}...h_{k-1}}F_{1}(x)$ will be a continuous function of $x$ , as well.", "By assumption, there exist a function $g_{1}\\in C(\\mathbb {R})$ and a polynomial function $H_{1}$ of $(k-1) $ -th order with the property $H_{1}(0)=0$ such that $F_{1}=g_{1}+H_{1}.", "(2.12)$ It follows from Lemma 2.3 that there exists a polynomial function $H_{2}$ of order $k$ such that $H_{2}(0)=0$ and $H_{1}(x)=H_{2}(x+1)-H_{2}(x).", "(2.13)$ Substituting (2.13) in (2.12) we obtain that $F_{1}(x)=g_{1}(x)+H_{2}(x+1)-H_{2}(x).", "(2.14)$ It follows from (2.11) and (2.14) that $g_{1}(x)=\\left[ f(x+1)-H_{2}(x+1)\\right] -\\left[ f(x)-H_{2}(x)\\right] .", "(2.15)$ Consider the function $F_{2}=f-H_{2}.", "(2.16)$ Since $H_{2}$ is a polynomial function of order $k$ and for any $h_{1},...,h_{k}\\in \\mathbb {R}$ the difference $\\Delta _{h_{1}...h_{k}}f(x)$ is a continuous function of $x$ , we obtain that $\\Delta _{h_{1}...h_{k}}F_{2}(x)$ is also a continuous function of $x$ .", "In addition, since $f(0)=f(1)$ and $H_{2}(0)=H_{2}(1)=0$ , it follows from (2.16) that $F_{2}(0)=F_{2}(1)$ .", "We will use these properties of $F_{2}$ below.", "Let us write (2.15) in the form $g_{1}(x)=F_{2}(x+1)-F_{2}(x),(2.17)$ and define the following $\\mod {1}$ periodic function $F^{\\ast }(x) &=&F_{2}(x)\\text{ for }x\\in [0,1), \\\\F^{\\ast }(x+1) &=&F^{\\ast }(x)\\text{ for }x\\in \\mathbb {R}.$ Consider the function $F=F_{2}-F^{\\ast }.", "(2.18)$ Let us show that $F\\in C(\\mathbb {R)}$ .", "Indeed since $F(x)=0$ for $x\\in [0,1)$ , $F$ is continuous on $(0,1)$ .", "Consider now the interval $[1,2) $ .", "For any $x\\in [1,2)$ by the definition of $F^{\\ast }$ and (2.17) we can write that $F(x)=F_{2}(x)-F_{2}(x-1)=g_{1}(x-1).", "(2.19)$ Since $g_{1}\\in C(\\mathbb {R)}$ , it follows from (2.19) that $F$ is continuous on $(1,2)$ .", "Note that by (2.17) $g_{1}(0)=0$ ; hence $F(1)=g_{1}(0)=0$ .", "Since $F\\equiv 0$ on $[0,1)$ , $F(1)=0$ and $F\\in C(1,2),$ we obtain that $F$ is continuous on $(0,2)$ .", "Consider the interval $[2,3)$ .", "For any $x\\in [2,3)$ we can write that $F(x)=F_{2}(x)-F_{2}(x-2)=g_{1}(x-1)+g_{1}(x-2).", "(2.20)$ Since $g_{1}\\in C(\\mathbb {R)}$ , $F$ is continuous on $(2,3)$ .", "Note that by (2.19) $\\lim _{x\\rightarrow 2-}F(x)=g_{1}(1)$ and by (2.20) $F(2)=g_{1}(1).$ We obtain from these arguments that $F$ is continuous on $(0,3)$ .", "In the same way, we can prove that $F$ is continuous on $(0,m)$ for any $m\\in \\mathbb {N}$ .", "Similar arguments can be used to prove the continuity of $F$ on $(-m,0)$ for any $m\\in \\mathbb {N}$ .", "We show it for the first interval $[-1,0)$ .", "For any $x\\in [-1,0)$ by the definition of $F^{\\ast }$ and (2.17) we can write that $F(x)=F_{2}(x)-F_{2}(x+1)=-g_{1}(x).$ Since $g_{1}\\in C(\\mathbb {R)}$ , it follows that $F$ is continuous on $(-1,0)$ .", "Besides, $\\lim _{x\\rightarrow 0-}F(x)=-g_{1}(0)=0.$ This shows that $F$ is continuous on $(-1,1)$ , since $F\\equiv 0$ on $[0,1).$ Combining all the above arguments we conclude that $F\\in C(\\mathbb {R)}$ .", "Since $F\\in C(\\mathbb {R)}$ and $\\Delta _{h_{1}...h_{k}}F_{2}(x)$ is a continuous function of $x$ , we obtain from (2.18) that $\\Delta _{h_{1}...h_{k}}F^{\\ast }(x)$ is also a continuous function of $x.$ By Lemma 2.4, there exist a function $g_{2}\\in C(\\mathbb {R)}$ and a polynomial function $H_{3}$ of order $k$ such that $F^{\\ast }=g_{2}+H_{3}.", "(2.21)$ It follows from (2.16), (2.18) and (2.21) that $f=F+g_{2}+H_{2}+H_{3}.", "(2.22)$ Introduce the notation $H(x) &=&H_{2}(x)+H_{3}(x)-H_{3}(0), \\\\g(x) &=&F(x)+g_{2}(x)+H_{3}(0).$ Obviously, $g\\in C(\\mathbb {R)}$ and $H(0)=0$ .", "It follows from (2.22) and the above notation that $f=g+H.$ This completes the proof of the theorem.", "We start this subsection with the following lemma.", "Lemma 2.5.", "Assume we are given pairwise linearly independent vectors $\\mathbf {a}^{i},$ $i=1,...,k,$ and a function $f\\in C(\\mathbb {R}^{n})$ of the form (2.1) with arbitrarily behaved univariate functions $f_{i}$ .", "Then for any $h_{1},...,h_{k-1}\\in \\mathbb {R}$ , and all indices $i=1,...,k$ , $\\Delta _{h_{1}...h_{k-1}}f_{i}\\in C(\\mathbb {R})$ .", "We prove this lemma for the function $f_{k}.$ It can be proven for the other functions $f_{i}$ in the same way.", "Let $h_{1},...,h_{k-1}\\in \\mathbb {R}$ be given.", "Since the vectors $\\mathbf {a}^{i}$ are pairwise linearly independent, for each $j=1,...,k-1,$ there is a vector $\\mathbf {b}^{j}$ such that $\\mathbf {b}^{j}\\cdot \\mathbf {a}^{j}=0$ and $\\mathbf {b}^{j}\\cdot \\mathbf {a}^{k}\\ne 0$ .", "It is not difficult to see that for any $\\lambda \\in \\mathbb {R}$ , $\\Delta _{\\lambda \\mathbf {b}^{j}}f_{j}(\\mathbf {a}^{j}\\cdot \\mathbf {x})=0.$ Therefore, for any $\\lambda _{1},...,\\lambda _{k-1}\\in \\mathbb {R}$ , we obtain from (2.1) that $\\Delta _{\\lambda _{1}\\mathbf {b}^{1}...\\lambda _{k-1}\\mathbf {b}^{k-1}}f(\\mathbf {x})=\\Delta _{\\lambda _{1}\\mathbf {b}^{1}...\\lambda _{k-1}\\mathbf {b}^{k-1}}f_{k}(\\mathbf {a}^{k}\\cdot \\mathbf {x}).", "(2.23)$ Note that in multivariate setting the difference operator $\\Delta _{\\mathbf {h}^{1}...\\mathbf {h}^{k}}f(\\mathbf {x})$ is defined similarly as in the previous section.", "If in (2.23) we take $\\mathbf {x} &\\mathbf {=}&\\frac{\\mathbf {a}^{k}}{\\left\\Vert \\mathbf {a}^{k}\\right\\Vert ^{2}}t\\text{, }t\\in \\mathbb {R}, \\\\\\lambda _{j} &=&\\frac{h_{j}}{\\mathbf {a}^{k}\\cdot \\mathbf {b}^{j}}\\text{, }j=1,...,k-1,$ we will obtain that $\\Delta _{h_{1}...h_{k-1}}f_{k}\\in C(\\mathbb {R})$ .", "The following theorem is valid.", "Theorem 2.4.", "Assume a function $f\\in C(\\mathbb {R}^{n})$ is of the form (2.1).", "Then there exist continuous functions $g_{i}:\\mathbb {R\\rightarrow R}$ , $i=1,...,k$ , and a polynomial $P(\\mathbf {x})$ of degree at most $k-1$ such that $f(\\mathbf {x})=\\sum _{i=1}^{k}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})+P(\\mathbf {x}).", "(2.24)$ By Lemma 2.5 and Theorem 2.3, for each $i=1,...,k$ , there exists a function $g_{i}\\in C(\\mathbb {R})$ and a polynomial function $H_{i}$ of $(k-1)$ -th order with the property $H_{i}(0)=0$ such that $f_{i}=g_{i}+H_{i}.", "(2.25)$ Consider the function $F(\\mathbf {x})=f(\\mathbf {x})-\\sum _{i=1}^{k}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}).", "(2.26)$ It follows from (2.1), (2.25) and (2.26) that $F(\\mathbf {x})=\\sum _{i=1}^{k}H_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}).", "(2.27)$ Denote the restrictions of the multivariate functions $H_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})$ to the space $\\mathbb {Q}^{n}$ by $P_{i}(\\mathbf {x})$ , respectively.", "By Lemma 2.2, the functions $P_{i}(\\mathbf {x})$ are ordinary polynomials of degree at most $k-1$ .", "Since the space $\\mathbb {Q}^{n}$ is dense in $\\mathbb {R}^{n}$ , and the functions $F(\\mathbf {x})$ , $P_{i}(\\mathbf {x})$ , $i=1,...,k$ , are continuous on $\\mathbb {R}^{n}$ , and the equality $F(\\mathbf {x})=\\sum _{i=1}^{k}P_{i}(\\mathbf {x}),(2.28)$ holds for all $\\mathbf {x}\\in \\mathbb {Q}^{n}$ , we obtain that (2.28) holds also for all $\\mathbf {x}\\in \\mathbb {R}^{n}$ .", "Now (2.24) follows from (2.26) and (2.28) by putting $P=\\sum _{i=1}^{k}P_{i}$ .", "Now we generalize Theorem 2.4 from $C(\\mathbb {R}^{n})$ to any space $C^{s}(\\mathbb {R}^{n})$ of $s$ -th order continuously differentiable functions.", "Theorem 2.5.", "Assume $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.1).", "Then there exist functions $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k$ , and a polynomial $P(\\mathbf {x})$ of degree at most $k-1$ such that (2.24) holds.", "The proof is based on Theorems 2.1 and 2.4.", "On the one hand, it follows from Theorem 2.4 that the $s$ -th order continuously differentiable function $f-P$ can be expressed as $\\sum _{i=1}^{k}g_{i}$ with continuous $g_{i}$ .", "On the other hand, since the class $\\mathcal {B}$ in Theorem 2.1, in particular, can be taken as $C(\\mathbb {R}),$ it follows that $g_{i}\\in C^{s}(\\mathbb {R})$ .", "Note that Theorem 2.5 solves the problem posed in Buhmann and Pinkus [18] and Pinkus [137] up to a polynomial.", "The following theorem shows that in the two dimensional setting $n=2$ it solves the problem completely.", "Theorem 2.6.", "Assume a function $f\\in C^{s}(\\mathbb {R}^{2})$ is of the form $f(x,y)=\\sum _{i=1}^{k}f_{i}(a_{i}x+b_{i}y),$ where $(a_{i},b_{i})$ are pairwise linearly independent vectors in $\\mathbb {R}^{2}$ and $f_{i}$ are arbitrary univariate functions.", "Then there exist functions $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k$ , such that $f(x,y)=\\sum _{i=1}^{k}g_{i}(a_{i}x+b_{i}y).", "(2.29)$ The proof of this theorem is not difficult.", "First we apply Theorem 2.5 and obtain that $f(x,y)=\\sum _{i=1}^{k}\\overline{g}_{i}(a_{i}x+b_{i}y)+P(x,y),(2.30)$ where $\\overline{g}_{i}\\in C^{s}(\\mathbb {R})$ and $P(x,y)$ is a bivariate polynomial of degree at most $k-1$ .", "Then we use the known fact that a bivariate polynomial $P(x,y)$ of degree $k-1$ is decomposed into a sum of ridge polynomials with any given $k$ pairwise linearly independent directions $(a_{i},b_{i}),$ $i=1,...,k$ (see e.g.", "[113]).", "That is, $P(x,y)=\\sum _{i=1}^{k}p_{i}(a_{i}x+b_{i}y),$ where $p_{i}$ are univariate polynomials of degree at most $k-1$ .", "Considering this in (2.30) gives the desired representation (2.29).", "Remark 2.2.", "Theorem 2.5 can be restated also for the classes $C^{\\infty }(\\mathbb {R})$ of infinitely differentiable functions and $D(\\mathbb {R})$ of analytic functions.", "That is, if under the conditions of Theorem 2.5, we have $f\\in C^{\\infty }(\\mathbb {R}^{n})$ (or $f\\in D(\\mathbb {R}^{n})$ ), then this function can be represented also in the form (2.24) with $g_{i}\\in C^{\\infty }(\\mathbb {R})$ (or $g_{i}\\in D(\\mathbb {R})$ ).", "This follows, similarly to the case $C^{s}(\\mathbb {R})$ above, from Theorem 2.4 and Remark 2.1.", "These arguments are also valid for Theorem 2.6.", "Assume we are given a function $f\\in C^{s}(\\mathbb {R}^{n})$ of the form (2.1).", "In this section, we discuss various conditions on the directions $\\mathbf {a}^{i}$ guaranteeing the validity of (2.2) with $g_{i}\\in C^{s}(\\mathbb {R})$ .", "The following theorems, in particular, show that if directions of ridge functions have only rational coordinates then no polynomial term appears in Theorems 2.4 and 2.5.", "Theorem 2.7.", "Assume a function $f\\in C(\\mathbb {R}^{n})$ is of the form (2.1) and there is a nonsingular linear transformation $T:$ $\\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$ such that $T\\mathbf {a}^{i}\\in \\mathbb {Q}\\mathit {^{n}},$ $i=1,...,k$ .", "Then there exist continuous functions $g_{i}:\\mathbb {R\\rightarrow R}$ , $i=1,...,k$ , such that (2.2) holds.", "Applying the coordinate change $\\mathbf {x\\rightarrow y}$ , given by the formula $\\mathbf {x}=T\\mathbf {y}$ , to both sides of (2.1) we obtain that $\\tilde{f}(\\mathbf {y})=\\sum _{i=1}^{k}f_{i}(\\mathbf {b}^{i}\\cdot \\mathbf {y}),$ where $\\tilde{f}(\\mathbf {y})=f(T\\mathbf {y})$ and $\\mathbf {b}^{i}=T\\mathbf {a}^{i},$ $i=1,...,k.$ Let us repeat the proof of Theorem 2.4 for the function $\\tilde{f}$ .", "Since the vectors $\\mathbf {b}^{i}$ , $i=1,...,k,$ have rational coordinates, it is not difficult to see that the restrictions of the functions $H_{i}$ to $\\mathbb {Q}$ are univariate polynomials.", "Indeed, for each $\\mathbf {b}^{i}$ we can choose a vector $\\mathbf {c}^{i}$ with rational coordinates such that $\\mathbf {b}^{i}\\cdot \\mathbf {c}^{i}=1$ .", "If in the equality $H_{i}(\\mathbf {b}^{i}\\cdot \\mathbf {x})=P_{i}(\\mathbf {x}),$ $\\mathbf {x}\\in \\mathbb {Q}^{n}$ , we take $\\mathbf {x=c}^{i}t$ with $t\\in \\mathbb {Q}$ , we obtain that $H_{i}(t)=P_{i}(\\mathbf {c}^{i}t)$ for all $t\\in \\mathbb {Q}$ .", "Now since $P_{i}$ is a multivariate polynomial on $\\mathbb {Q}^{n}$ , $H_{i}$ is a univariate polynomial on $\\mathbb {Q}$ .", "Denote this univariate polynomial by $L_{i}$ .", "Thus the formula $P_{i}(\\mathbf {x})=L_{i}(\\mathbf {b}^{i}\\cdot \\mathbf {x})(2.31)$ holds for each $i=1,...,k$ , and all $\\mathbf {x}\\in \\mathbb {Q}^{n}$ .", "Since $\\mathbb {Q}^{n}$ is dense in $\\mathbb {R}^{n}$ , we see that (2.31) holds, in fact, for all $\\mathbf {x}\\in \\mathbb {R}^{n}$ .", "Thus the polynomial $P(\\mathbf {x})$ in (2.24) can be expressed as $\\sum _{i=1}^{k}L_{i}(\\mathbf {b}^{i}\\cdot \\mathbf {x})$ .", "Considering this in Theorem 2.4, we obtain that $\\tilde{f}(\\mathbf {y})=\\sum _{i=1}^{k}g_{i}(\\mathbf {b}^{i}\\cdot \\mathbf {y}),(2.32)$ where $g_{i}$ are continuous functions.", "Using the inverse transformation $\\mathbf {y}=T^{-1}\\mathbf {x}$ in (2.32) we arrive at (2.2).", "Theorem 2.8.", "Assume a function $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.1) and there is a nonsingular linear transformation $T:$ $\\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$ such that $T\\mathbf {a}^{i}\\in \\mathbb {Q}\\mathit {^{n}},$ $i=1,...,k$ .", "Then there exist functions $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k$ , such that (2.2) holds.", "The proof of this theorem easily follows from Theorem 2.7 and Theorem 2.5.", "We already know that if the given directions $\\mathbf {a}^{i}$ form a linearly independent set, then the smoothness problem has a positive solution (see Section 2.1.1).", "What can we say if all $\\mathbf {a}^{i}$ are not linearly independent?", "In the sequel, we show that if $k-1$ of the directions $\\mathbf {a}^{i}$ , $i=1,...,k,$ are linearly independent, then in (2.1) $f_{i}$ can be replaced with $g_{i}\\in C^{s}(\\mathbb {R})$ .", "We will also estimate the modulus of continuity of $g_{i}$ in terms of the modulus of continuity of a function generated from $f$ under a linear transformation.", "Let $F:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ , $n\\ge 1,$ be any function and $\\Omega \\subset \\mathbb {R}^{n}$ .", "The function $\\omega (F;\\delta ;\\Omega )=\\sup \\left\\lbrace \\left|F(\\mathbf {x})-F(\\mathbf {y})\\right|:\\mathbf {x},\\mathbf {y}\\in \\Omega ,\\text{ }\\left|\\mathbf {x}-\\mathbf {y}\\right|\\le \\delta \\right\\rbrace ,\\text{ }0\\le \\delta \\le diam\\Omega ,$ is called the modulus of continuity of the function $F(\\mathbf {x})=F(x_{1},...,x_{n})$ on the set $\\Omega .$ We will also use the notation $\\omega _{\\mathbb {Q}}(F;\\delta ;\\Omega )$ , which stands for the function $\\omega (F;\\delta ;\\Omega \\cap \\mathbb {Q}^{n})$ .", "Here $\\mathbb {Q}$ denotes the set of rational numbers.", "Note that $\\omega _{\\mathbb {Q}}(F;\\delta ;\\Omega )$ makes sense if the set $\\Omega \\cap \\mathbb {Q}^{n}$ is not empty.", "Clearly, $\\omega _{\\mathbb {Q}}(F;\\delta ;\\Omega )\\le \\omega (F;\\delta ;\\Omega )$ .", "The equality $\\omega _{\\mathbb {Q}}(F;\\delta ;\\Omega )=\\omega (F;\\delta ;\\Omega )$ holds for continuous $F$ and certain sets $\\Omega $ .", "For example, it holds if for any $\\mathbf {x},\\mathbf {y}\\in \\Omega $ with $\\left|\\mathbf {x}-\\mathbf {y}\\right|\\le \\delta $ there exist sequences $\\left\\lbrace \\mathbf {x}_{m}\\right\\rbrace ,\\left\\lbrace \\mathbf {y}_{m}\\right\\rbrace \\subset \\Omega \\cap \\mathbb {Q}^{n}$ such that $\\mathbf {x}_{m}\\rightarrow \\mathbf {x}$ , $\\mathbf {y}_{m}\\rightarrow \\mathbf {y}$ and $\\left|\\mathbf {x}_{m}-\\mathbf {y}_{m}\\right|\\le \\delta ,$ for all $m$ .", "There are many sets $\\Omega $ , which satisfy this property.", "The following lemma is valid.", "Lemma 2.6.", "Assume a function $G\\in C(\\mathbb {R}^{n})$ has the form $G(x_{1},...,x_{n})=\\sum _{i=1}^{n}g(x_{i})-g(x_{1}+\\cdot \\cdot \\cdot +x_{n}),(2.33)$ where $g$ is an arbitrarily behaved function.", "Then the following inequality holds $\\omega _{\\mathbb {Q}}(g;\\delta ;[-M,M])\\le 2\\delta \\left|g(1)-g(0)\\right|+3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) ,(2.34)$ where $\\delta \\in \\left( 0,\\frac{1}{2}\\right) \\cap \\mathbb {Q}$ and $M\\ge 1$ .", "Consider the function $f(t)=g(t)-g(0)$ and write (2.33) in the form $F(x_{1},...,x_{n})=\\sum _{i=1}^{n}f(x_{i})-f(x_{1}+\\cdot \\cdot \\cdot +x_{n}),(2.35)$ where $F(x_{1},...,x_{n})=G(x_{1},...,x_{n})-(n-1)g(0).$ Note that the functions $f$ and $g$ , as well as the functions $F$ and $G,$ have a common modulus of continuity.", "Thus we prove the lemma if we prove it for the pair $\\left\\langle F,f\\right\\rangle .$ Since $f(0)=0,$ it follows from (2.35) that $F(x_{1},0,...,0)=F(0,x_{2},0,...,0)=\\cdot \\cdot \\cdot =F(0,0,...,x_{n})=0.", "(2.36)$ For the sake of brevity, introduce the notation $\\mathcal {F}(x_{1},x_{2})=$ $F(x_{1},x_{2},0,...,0)$ .", "Obviously, for any real number $x,$ $\\mathcal {F}(x,x) &=&2f(x)-f(2x); \\\\\\mathcal {F}(x,2x) &=&f(x)+f(2x)-f(3x); \\\\&&\\cdot \\cdot \\cdot \\\\\\mathcal {F}(x,(k-1)x) &=&f(x)+f((k-1)x)-f(kx).$ We obtain from the above equalities that $f(2x) &=&2f(x)-\\mathcal {F}(x,x), \\\\f(3x) &=&3f(x)-\\mathcal {F}(x,x)-\\mathcal {F}(x,2x), \\\\&&\\cdot \\cdot \\cdot \\\\f(kx) &=&kf(x)-\\mathcal {F}(x,x)-\\mathcal {F}(x,2x)-\\cdot \\cdot \\cdot -\\mathcal {F}(x,(k-1)x).$ Thus for any nonnegative integer $k$ , $f(x)=\\frac{1}{k}f(kx)+\\frac{1}{k}\\left[ \\mathcal {F}(x,x)+\\mathcal {F}(x,2x)+\\cdot \\cdot \\cdot +\\mathcal {F}(x,(k-1)x)\\right] .", "(2.37)$ Consider now the simple fraction $\\frac{p}{m}\\in (0,\\frac{1}{2})$ and set $m_{0}=\\left[ \\frac{m}{p}\\right] .$ Here $[r]$ denotes the whole number part of $r$ .", "Clearly, $m_{0}\\ge 2$ and the remainder $p_{1}=m-m_{0}p<p.$ Taking $x=\\frac{p}{m}$ and $k=m_{0}$ in (2.37) gives us the following equality $f\\left( \\frac{p}{m}\\right) =\\frac{1}{m_{0}}f\\left( 1-\\frac{p_{1}}{m}\\right)$ $+\\frac{1}{m_{0}}\\left[ \\mathcal {F}\\left( \\frac{p}{m},\\frac{p}{m}\\right) +\\mathcal {F}\\left( \\frac{p}{m},\\frac{2p}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p}{m},(m_{0}-1)\\frac{p}{m}\\right) \\right] .", "(2.38)$ On the other hand, since $\\mathcal {F}\\left( \\frac{p_{1}}{m},1-\\frac{p_{1}}{m}\\right) =f\\left( \\frac{p_{1}}{m}\\right) +f\\left( 1-\\frac{p_{1}}{m}\\right) -f(1),$ it follows from (2.38) that $f\\left( \\frac{p}{m}\\right) =\\frac{f(1)}{m_{0}}$ $+\\frac{1}{m_{0}}\\left[\\mathcal {F}\\left( \\frac{p}{m},\\frac{p}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p}{m},(m_{0}-1)\\frac{p}{m}\\right) +\\mathcal {F}\\left(\\frac{p_{1}}{m},1-\\frac{p_{1}}{m}\\right) \\right]$ $-\\frac{1}{m_{0}}f\\left( \\frac{p_{1}}{m}\\right) .", "(2.39)$ Put $m_{1}=\\left[ \\frac{m}{p_{1}}\\right] $ , $p_{2}=m-m_{1}p_{1}.$ Clearly, $0\\le p_{2}<p_{1}$ .", "Similar to (2.39), we can write that $f\\left( \\frac{p_{1}}{m}\\right) =\\frac{f(1)}{m_{1}}$ $+\\frac{1}{m_{1}}\\left[\\mathcal {F}\\left( \\frac{p_{1}}{m},\\frac{p_{1}}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p_{1}}{m},(m_{1}-1)\\frac{p_{1}}{m}\\right) +\\mathcal {F}\\left( \\frac{p_{2}}{m},1-\\frac{p_{2}}{m}\\right) \\right]$ $-\\frac{1}{m_{1}}f\\left( \\frac{p_{2}}{m}\\right) .", "(2.40)$ Let us make a convention that (2.39) is the 1-st and (2.40) is the 2-nd formula.", "One can continue this process by defining the chain of pairs $(m_{2},p_{3}),$ $(m_{3},p_{4})$ until the pair $(m_{k-1},p_{k})$ with $p_{k}=0$ and writing out the corresponding formulas for each pair.", "For example, the last $k$ -th formula will be of the form $f\\left( \\frac{p_{k-1}}{m}\\right) =\\frac{f(1)}{m_{k-1}}$ $+\\frac{1}{m_{k-1}}\\left[ \\mathcal {F}\\left( \\frac{p_{k-1}}{m},\\frac{p_{k-1}}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p_{k-1}}{m},(m_{k-1}-1)\\frac{p_{k-1}}{m}\\right) +\\mathcal {F}\\left( \\frac{p_{k}}{m},1-\\frac{p_{k}}{m}\\right) \\right]$ $-\\frac{1}{m_{k-1}}f\\left( \\frac{p_{k}}{m}\\right) .", "(2.41)$ Note that in (2.41), $f\\left( \\frac{p_{k}}{m}\\right) =0$ and $\\mathcal {F}\\left( \\frac{p_{k}}{m},1-\\frac{p_{k}}{m}\\right) =0$ .", "Considering now the $k$ -th formula in the $(k-1)$ -th formula, then the obtained formula in the $(k-2)$ -th formula, and so forth, we will finally arrive at the equality $f\\left( \\frac{p}{m}\\right) =f(1)\\left[ \\frac{1}{m_{0}}-\\frac{1}{m_{0}m_{1}}+\\cdot \\cdot \\cdot +\\frac{(-1)^{k-1}}{m_{0}m_{1}\\cdot \\cdot \\cdot m_{k-1}}\\right]$ $+\\frac{1}{m_{0}}\\left[ \\mathcal {F}\\left( \\frac{p}{m},\\frac{p}{m}\\right)+\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p}{m},(m_{0}-1)\\frac{p}{m}\\right) +\\mathcal {F}\\left( \\frac{p_{1}}{m},1-\\frac{p_{1}}{m}\\right) \\right]$ $-\\frac{1}{m_{0}m_{1}}\\left[ \\mathcal {F}\\left( \\frac{p_{1}}{m},\\frac{p_{1}}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p_{1}}{m},(m_{1}-1)\\frac{p_{1}}{m}\\right) +\\mathcal {F}\\left( \\frac{p_{2}}{m},1-\\frac{p_{2}}{m}\\right) \\right]$ $+\\cdot \\cdot \\cdot +$ $\\frac{(-1)^{k-1}}{m_{0}m_{1}\\cdot \\cdot \\cdot m_{k-1}}\\left[ \\mathcal {F}\\left( \\frac{p_{k-1}}{m},\\frac{p_{k-1}}{m}\\right) +\\cdot \\cdot \\cdot +\\mathcal {F}\\left( \\frac{p_{k-1}}{m},(m_{k-1}-1)\\frac{p_{k-1}}{m}\\right) \\right] .", "(2.42)$ Taking into account (2.36) and the definition of $\\mathcal {F}$ , for any point of the form $\\left( \\frac{p_{i}}{m},c\\right) $ , $i=0,1,...,k-1,$ $p_{0}=p$ , $c\\in [0,1]$ , we can write that $\\left|\\mathcal {F}\\left( \\frac{p_{i}}{m},c\\right) \\right|=\\left|\\mathcal {F}\\left( \\frac{p_{i}}{m},c\\right) -\\mathcal {F}\\left(0,c\\right) \\right|\\le \\omega \\left( F;\\frac{p_{i}}{m};[0,1]^{n}\\right)\\le \\omega \\left( F;\\frac{p}{m};[0,1]^{n}\\right) .$ Applying this inequality to each term $\\mathcal {F}\\left( \\frac{p_{i}}{m},\\cdot \\right) $ in (2.42), we obtain that $\\left|f\\left( \\frac{p}{m}\\right) \\right|\\le \\left[ \\frac{1}{m_{0}}-\\frac{1}{m_{0}m_{1}}+\\cdot \\cdot \\cdot +\\frac{(-1)^{k-1}}{m_{0}m_{1}\\cdot \\cdot \\cdot m_{k-1}}\\right] \\left|f(1)\\right|$ $+\\left[ 1+\\frac{1}{m_{0}}+\\cdot \\cdot \\cdot +\\frac{1}{m_{0}\\cdot \\cdot \\cdot m_{k-2}}\\right] \\omega \\left( F;\\frac{p}{m};[0,1]^{n}\\right) .", "(2.43)$ Since $m_{0}\\le m_{1}\\le \\cdot \\cdot \\cdot \\le m_{k-1},$ it is not difficult to see that in (2.43) $\\frac{1}{m_{0}}-\\frac{1}{m_{0}m_{1}}+\\cdot \\cdot \\cdot +\\frac{(-1)^{k-1}}{m_{0}m_{1}\\cdot \\cdot \\cdot m_{k-1}}\\le \\frac{1}{m_{0}}$ and $1+\\frac{1}{m_{0}}+\\cdot \\cdot \\cdot +\\frac{1}{m_{0}\\cdot \\cdot \\cdot m_{k-2}}\\le \\frac{m_{0}}{m_{0}-1}.$ Considering the above two inequalities in (2.43) we obtain that $\\left|f\\left( \\frac{p}{m}\\right) \\right|\\le \\frac{\\left|f(1)\\right|}{m_{0}}+\\frac{m_{0}}{m_{0}-1}\\omega \\left( F;\\frac{p}{m};[0,1]^{n}\\right) .", "(2.44)$ Since $m_{0}=\\left[ \\frac{m}{p}\\right] \\ge 2,$ it follows from (2.44) that $\\left|f\\left( \\frac{p}{m}\\right) \\right|\\le \\frac{2p\\left|f(1)\\right|}{m}+2\\omega \\left( F;\\frac{p}{m};[0,1]^{n}\\right) .", "(2.45)$ Let now $\\delta \\in \\left( 0,\\frac{1}{2}\\right) \\cap \\mathbb {Q}$ be a rational increment, $M\\ge 1$ and $x,x+\\delta $ be two points in $\\left[ -M,M\\right] \\cap \\mathbb {Q}.$ By (2.45) we can write that $\\left|f(x+\\delta )-f(x)\\right|\\le \\left|f(\\delta )\\right|+\\left|F(x,\\delta ,0,...,0)\\right|\\le 2\\delta \\left|f(1)\\right|+3\\omega \\left( F;\\delta ;[-M,M]^{n}\\right) .", "(2.46)$ Now (2.34) follows from (2.46) and the definitions of $f$ and $F$ .", "Remark 2.3.", "The above lemma shows that the restriction of $g$ to the set of rational numbers $\\mathbb {Q}$ is uniformly continuous on any interval $[-M,M]\\cap \\mathbb {Q}$ .", "To prove the main result of this section we need the following lemma.", "Lemma 2.7.", "Assume a function $G\\in C(\\mathbb {R}^{n})$ has the form $G(x_{1},...,x_{n})=\\sum _{i=1}^{n}g(x_{i})-g(x_{1}+\\cdot \\cdot \\cdot +x_{n}),$ where $g$ is an arbitrary function.", "Then there exists a function $F\\in C(\\mathbb {R})$ such that $G(x_{1},...,x_{n})=\\sum _{i=1}^{n}F(x_{i})-F(x_{1}+\\cdot \\cdot \\cdot +x_{n})(2.47)$ and the following inequality holds $\\omega (F;\\delta ;[-M,M])\\le 3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) ,(2.48)$ where $0\\le \\delta \\le \\frac{1}{2}$ and $M\\ge 1$ .", "Consider the function $u(t)=g(t)-\\left[ g(1)-g(0)\\right] t.$ Obviously, $u(1)=u(0)$ and $G(x_{1},...,x_{n})=\\sum _{i=1}^{n}u(x_{i})-u(x_{1}+\\cdot \\cdot \\cdot +x_{n}).", "(2.49)$ By Lemma 2.6, the restriction of $u$ to $\\mathbb {Q}$ is continuous and uniformly continuous on every interval $[-M,M]\\cap \\mathbb {Q}$ .", "Denote this restriction by $v$ .", "Let $y$ be any real number and $\\lbrace y_{k}\\rbrace _{k=1}^{\\infty }$ be any sequence of rational numbers converging to $y$ .", "We can choose $M>0$ so that $y_{k}\\in [-M,M]$ for any $k\\in \\mathbb {N}$ .", "It follows from the uniform continuity of $v$ on $[-M,M]\\cap \\mathbb {Q}$ that the sequence $\\lbrace v(y_{k})\\rbrace _{k=1}^{\\infty }$ is Cauchy.", "Thus there exits a finite limit $\\lim _{k\\rightarrow \\infty }v(y_{k})$ .", "It is not difficult to see that this limit does not depend on the choice of $\\lbrace y_{k}\\rbrace _{k=1}^{\\infty }$ .", "Let $F$ denote the following extension of $v$ to the set of real numbers.", "$F(y)=\\left\\lbrace \\begin{array}{c}v(y),\\text{ if }y\\in \\mathbb {Q}\\text{;} \\\\\\lim _{k\\rightarrow \\infty }v(y_{k}),\\text{ if }y\\in \\mathbb {R}\\backslash \\mathbb {Q}\\text{ and }\\lbrace y_{k}\\rbrace \\text{ is a sequence in }\\mathbb {Q}\\text{tending to }y.\\end{array}\\right.$ In view of the above arguments, $F$ is well defined on the whole real line.", "Let us prove that for this function (2.47) is valid.", "Consider an arbitrary point $(x_{1},...,x_{n})\\in \\mathbb {R}^{n}$ and sequences of rationale numbers $\\lbrace y_{k}^{i}\\rbrace _{k=1}^{\\infty },i=1,...,n,$ tending to $x_{1},...,x_{n},$ respectively.", "Taking into account (2.49), we can write that $G(y_{k}^{1},...,y_{k}^{n})=\\sum _{i=1}^{n}v(y_{k}^{i})-v(y_{k}^{1}+\\cdot \\cdot \\cdot +y_{k}^{n}),\\text{ for all }k=1,2,...,(2.50)$ since $v$ is the restriction of $u$ to $\\mathbb {Q}$ .", "Tending $k\\rightarrow \\infty $ in both sides of (2.50) we obtain (2.47).", "Let us now prove that $F\\in C(\\mathbb {R})$ and (2.48) holds.", "Since $v(1)=v(0) $ we obtain from (2.49) and (2.34) that for $\\delta \\in \\left( 0,\\frac{1}{2}\\right) \\cap \\mathbb {Q}$ , $M\\ge 1$ and any numbers $a,b\\in [-M,M]\\cap \\mathbb {Q}$ , $\\left|a-b\\right|\\le \\delta ,$ the following inequality holds $\\left|v(a)-v(b)\\right|\\le 3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) .", "(2.51)$ Consider any real numbers $r_{1}$ and $r_{2}$ satisfying $r_{1},r_{2}\\in [-M,M]$ , $\\left|r_{1}-r_{2}\\right|\\le \\delta $ and take sequences $\\lbrace a_{k}\\rbrace _{k=1}^{\\infty }\\subset [-M,M]\\cap \\mathbb {Q}$ , $\\lbrace b_{k}\\rbrace _{k=1}^{\\infty }\\subset [-M,M]\\cap \\mathbb {Q}$ with the property $\\left|a_{k}-b_{k}\\right|\\le \\delta ,$ $k=1,2,...,$ and tending to $r_{1}$ and $r_{2}$ , respectively.", "By (2.51), $\\left|v(a_{k})-v(b_{k})\\right|\\le 3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) .$ If we take limits on both sides of the above inequality, we obtain that $\\left|F(r_{1})-F(r_{2})\\right|\\le 3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) ,$ which means that $F$ is uniformly continuous on $[-M,M]$ and $\\omega \\left( F;\\delta ;[-M,M]\\right) \\le 3\\omega \\left( G;\\delta ;[-M,M]^{n}\\right) .$ Note that in the last inequality $\\delta $ is a rational number from the interval $\\left( 0,\\frac{1}{2}\\right) .$ It is well known that the modulus of continuity $\\omega (f;\\delta ;\\Omega )$ of a continuous function $f$ is continuous from the right for any compact set $\\Omega \\subset \\mathbb {R}^{n}$ and it is continuous from the left for certain compact sets $\\Omega $ , in particular for rectangular sets (see [99]).", "It follows immediately that (2.48) is valid for all $\\delta \\in [0,\\frac{1}{2}].$ The following theorem was first obtained by Kuleshov [105].", "Below, we prove this using completely different ideas.", "Our proof, which is taken from [2], contains a theoretical method for constructing the functions $g_{i}\\in C^{s}(\\mathbb {R})$ in (2.2).", "Using this method, we will also estimate the modulus of continuity of $g_{i}$ in terms of the modulus of continuity of $f$ (see Remark 2.4 below).", "Theorem 2.9.", "Assume we are given $k$ directions $\\mathbf {a}^{i}$ , $i=1,...,k$ , in $\\mathbb {R}^{n}\\backslash \\lbrace \\mathbf {0}\\rbrace $ and $k-1$ of them are linearly independent.", "Assume that a function $f\\in C(\\mathbb {R}^{n})$ is of the form (2.1).", "Then $f$ can be represented also in the form (2.2) with $g_{i}\\in C(\\mathbb {R})$ , $i=1,...,k$ .", "Without loss of generality, we may assume that the first $k-1 $ vectors $\\mathbf {a}^{1},\\mathbf {...},\\mathbf {a}^{k-1}$ are linearly independent.", "Thus there exist numbers $\\lambda _{1},...,\\lambda _{k-1}\\in \\mathbb {R}$ such that $\\mathbf {a}^{k}=\\lambda _{1}\\mathbf {a}^{1}+\\cdot \\cdot \\cdot +\\lambda _{k-1}\\mathbf {a}^{k-1}$ .", "We may also assume that the first $p$ numbers $\\lambda _{1},...,\\lambda _{p}$ , $1\\le p\\le k-1$ , are nonzero and the remaining $\\lambda _{j}$ s are zero.", "Indeed, if necessary, we can rearrange the vectors $\\mathbf {a}^{1},\\mathbf {...},\\mathbf {a}^{k-1}$ so that this assumption holds.", "Complete the system $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{k-1}\\rbrace $ to a basis $\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{k-1},\\mathbf {b}^{k},...,\\mathbf {b}^{n}\\rbrace $ and consider the linear transformation $\\mathbf {y}=A\\mathbf {x,}$ where $\\mathbf {x}=(x_{1},...,x_{n})^{T},$ $\\mathbf {y}=(y_{1},...,y_{n})^{T}$ and $A$ is the matrix, rows of which are formed by the coordinates of the vectors $\\mathbf {a}^{1},...,\\mathbf {a}^{k-1},\\mathbf {b}^{k},...,\\mathbf {b}^{n}.$ Using this transformation, we can write (2.1) in the form $f(A^{-1}\\mathbf {y})=f_{1}(y_{1})+\\cdot \\cdot \\cdot +f_{k-1}(y_{k-1})+f_{k}(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p}).", "(2.52)$ For the brevity of exposition in the sequel, we put $l=k-1$ and use the notation $w=f_{l+1},\\text{ }\\Phi (y_{1},...,y_{l})=f(A^{-1}\\mathbf {y})\\text{ and }Y_{j}=(y_{1},...,y_{j-1},y_{j+1},...,y_{l}), j=1,...,l.$ Using this notation, we can write (2.52) in the form $\\Phi (y_{1},...,y_{l})=f_{1}(y_{1})+\\cdot \\cdot \\cdot +f_{l}(y_{l})+w(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p}).", "(2.53)$ In (2.53), taking sequentially $Y_{1}=0,$ $Y_{2}=0,$ ..., $Y_{l}=0$ we obtain that $f_{j}(y_{j})=\\Phi (y_{1},...,y_{l})|_{Y_{j}=\\mathbf {0}}-w(\\lambda _{j}y_{j})-\\sum _{\\begin{array}{c} i=1 \\\\ i\\ne j\\end{array}}^{l}f_{i}(0),\\text{ }j=1,...,l.(2.54)$ Substituting (2.54) in (2.53), we obtain the equality $\\left.\\begin{array}{c}w(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p})-\\sum _{j=1}^{p}w(\\lambda _{j}y_{j})-(l-p)w(0)= \\\\=\\Phi (y_{1},...,y_{l})-\\sum _{j=1}^{l}\\Phi (y_{1},...,y_{l})|_{Y_{j}=\\mathbf {0}}+(l-1)\\sum _{j=1}^{l}f_{j}(0).\\end{array}\\right.", "(2.55)$ We see that the right hand side of (2.55) depends only on the variables $y_{1},y_{2},...,y_{p}.$ Denote the right hand side of (2.55) by $H(y_{1},...,y_{p}).$ That is, set $H(y_{1},...,y_{p})\\overset{def}{=}\\Phi (y_{1},...,y_{l})-\\sum _{j=1}^{l}\\Phi (y_{1},...,y_{l})|_{Y_{j}=\\mathbf {0}}+(l-1)\\sum _{j=1}^{l}f_{j}(0).", "(2.56)$ We will use the following identity, which follows from (2.55) and (2.56) $H(y_{1},...,y_{p})=w(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p})-\\sum _{j=1}^{p}w(\\lambda _{j}y_{j})-(l-p)w(0).", "(2.57)$ It follows from (2.56) and the continuity of $f$ that the function $H$ is continuous on $\\mathbb {R}^{p}$ .", "Then, defining the function $G(y_{1},...,y_{p})=H(\\frac{y_{1}}{\\lambda _{1}},...,\\frac{y_{p}}{\\lambda _{p}})+(l-p)w(0)(2.58)$ and applying Lemma 2.7, we obtain that there exists a function $F\\in C(\\mathbb {R})$ such that $G(y_{1},...,y_{p})=F(y_{1}+\\cdot \\cdot \\cdot +y_{p})-\\sum _{j=1}^{p}F(y_{p}).", "(2.59)$ It follows from the formulas (2.57)-(2.59) that $w(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p})-\\sum _{j=1}^{p}w(\\lambda _{j}y_{j})=F(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p})-\\sum _{j=1}^{p}F(\\lambda _{j}y_{j}).", "(2.60)$ Let us introduce the following functions $\\left.\\begin{array}{c}g_{j}(y_{j})=\\Phi (y_{1},...,y_{l})|_{Y_{j}=\\mathbf {0}}-F(\\lambda _{j}y_{j})-\\sum _{\\begin{array}{c} i=1 \\\\ i\\ne j\\end{array}}^{l}f_{i}(0),\\text{ }j=1,...,p,\\\\g_{j}(y_{j})=\\Phi (y_{1},...,y_{l})|_{Y_{j}=\\mathbf {0}}-\\sum _{\\begin{array}{c} i=1\\\\ i\\ne j\\end{array}}^{l}f_{i}(0)-w(0),\\text{ }j=p+1,...,l.\\end{array}\\right.", "(2.61)$ Note that $g_{j}\\in C(\\mathbb {R}),$ $j=1,...,l.$ Considering (2.55), (2.60) and (2.61) it is not difficult to verify that $\\Phi (y_{1},...,y_{l})=g_{1}(y_{1})+\\cdot \\cdot \\cdot +g_{l}(y_{l})+F(\\lambda _{1}y_{1}+\\cdot \\cdot \\cdot +\\lambda _{p}y_{p}).", "(2.62)$ In (2.62), denoting $F=g_{k}$ , recalling the definition of $\\Phi $ and going back to the variable $\\mathbf {x}=(x_{1},...,x_{n})$ by using again the linear transformation $\\mathbf {y}=A\\mathbf {x}$ , we finally obtain (2.2).", "Remark 2.4.", "Using Theorem 2.9 and Lemma 2.7, one can estimate the modulus of continuity of the functions $g_{i}$ in representation (2.2) in terms of the modulus of continuity of $\\Phi $ .", "To show how one can do this, assume $C>0,$ $M\\ge 1,$ $0\\le \\delta \\le \\frac{1}{2\\max \\lbrace \\left|\\lambda _{j}\\right|\\rbrace }$ and introduce the following sets $\\mathcal {M}_{j}=\\left\\lbrace (x_{1},...,x_{l})\\in \\mathbb {R}^{l}:x_{j}\\in [-M/\\left|\\lambda _{j}\\right|,M/\\left|\\lambda _{j}\\right|]\\text{, }x_{i}=0\\text{ for }i\\ne j\\right\\rbrace ,$ $j=1,...,p;$ $\\mathcal {C}_{j}=\\left\\lbrace (x_{1},...,x_{l})\\in \\mathbb {R}^{l}:x_{j}\\in [-C,C]\\text{, }x_{i}=0\\text{ for }i\\ne j\\right\\rbrace ,\\text{ }j=p+1,...,l.$ It can be easily obtained from (2.61) that $\\omega (g_{j};\\delta ;[-M/\\left|\\lambda _{j}\\right|,M/\\left|\\lambda _{j}\\right|])\\le \\omega \\left( \\Phi ;\\delta ;\\mathcal {M}_{j}\\right) +\\omega (F;\\delta _{1};[-M,M]),\\text{ }j=1,...,p,(2.63)$ $\\omega (g_{j};\\delta ;[-C,C])\\le \\omega \\left( \\Phi ;\\delta ;\\mathcal {C}_{j}\\right) ,\\text{ }j=p+1,...,l,(2.64)$ where $\\delta _{1}=\\delta \\cdot \\max \\lbrace \\left|\\lambda _{j}\\right|\\rbrace .", "$ To estimate $\\omega (F;\\delta _{1};[-M,M])$ in (2.63), we refer to Lemma 2.7.", "Applying Lemma 2.7 to the function $G$ in (2.58) we obtain that in addition to (2.59) the following inequality holds.", "$\\omega (F;\\delta _{1};[-M,M])\\le 3\\omega \\left( G;\\delta _{1};[-M,M]^{p}\\right) .", "(2.65)$ Note that here $0\\le \\delta _{1}\\le \\frac{1}{2}$ as in Lemma 2.7.", "It follows from (2.58) and (2.65) that $\\omega (F;\\delta _{1};[-M,M])\\le 3\\omega \\left( H;\\delta _{2};\\mathcal {K}\\right) ,(2.66)$ where $\\mathcal {K}=[-M/\\left|\\lambda _{1}\\right|,M/\\left|\\lambda _{1}\\right|]\\times \\cdot \\cdot \\cdot \\times [-M/\\left|\\lambda _{p}\\right|,M/\\left|\\lambda _{p}\\right|] $ and $\\delta _{2}=\\delta _{1}/\\min \\lbrace \\left|\\lambda _{j}\\right|\\rbrace $ .", "Further, (2.66) and (2.56) together yield that $\\omega (F;\\delta _{1};[-M,M])\\le (3l+3)\\omega \\left( \\Phi ;\\delta _{2};\\mathcal {S}\\right) ,(2.67)$ where $\\mathcal {S}=\\left\\lbrace (x_{1},...,x_{l})\\in \\mathbb {R}^{l}:(x_{1},...,x_{p})\\in \\mathcal {K}\\text{, }x_{i}=0\\text{ for }i>p\\right\\rbrace $ .", "Now it follows from (2.63) and (2.67) that $\\omega (g_{j};\\delta ;[-M/\\left|\\lambda _{j}\\right|,M/\\left|\\lambda _{j}\\right|])\\le \\omega \\left( \\Phi ;\\delta ;\\mathcal {M}_{j}\\right) +(3l+3)\\omega \\left( \\Phi ;\\delta _{2};\\mathcal {S}\\right) ,\\text{}j=1,...,p.(2.68)$ Formulas (2.64), (2.67) and (2.68) provide us with upper estimates for the modulus of continuity of the functions $g_{j},$ $j=1,...,k,$ in terms of the modulus of continuity of $\\Phi $ .", "Recall that in these estimates $l=k-1$ , $F=g_{k}$ and $\\lambda _{j}$ are coefficients in the expression $\\mathbf {a}^{k}=\\lambda _{1}\\mathbf {a}^{1}+\\cdot \\cdot \\cdot +\\lambda _{p}\\mathbf {a}^{p}$ .", "Theorems 2.1 and 2.9 together give the following result.", "Theorem 2.10.", "Assume we are given $k$ directions $\\mathbf {a}^{i}$ , $i=1,...,k$ , in $\\mathbb {R}^{n}\\backslash \\lbrace \\mathbf {0}\\rbrace $ and $k-1$ of them are linearly independent.", "Assume that a function $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.1).", "Then $f$ can be represented also in the form (2.2), where the functions $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,k$ .", "Indeed, on the one hand, it follows from Theorem 2.9 that $f$ can be expressed as (2.2) with continuous $g_{i}$ .", "On the other hand, since the class $\\mathcal {B}$ in Theorem 2.1, in particular, can be taken as $C(\\mathbb {R}),$ it follows that $g_{i}\\in C^{s}(\\mathbb {R})$ .", "Remark 2.5.", "In addition to the above $C^{s}(\\mathbb {R})$ , Theorems 2.9 can be restated also for some other subclasses of the space of continuous functions.", "These are $C^{\\infty }(\\mathbb {R})$ functions; analytic functions; algebraic polynomials; trigonometric polynomials.", "More precisely, assume $\\mathcal {H}(\\mathbb {R})$ is any of these subclasses and $\\mathcal {H}(\\mathbb {R}^{n})$ is the $n$ -variable analog of the $\\mathcal {H}(\\mathbb {R})$ .", "If under the conditions of Theorem 2.9, we have $f\\in \\mathcal {H}(\\mathbb {R}^{n})$ , then this function can be represented in the form (2.2) with $g_{i}\\in \\mathcal {H}(\\mathbb {R}).$ This follows, similarly to the case $C^{s}(\\mathbb {R})$ above, from Theorem 2.9 and Remark 2.1.", "Note that Theorems 2.4-2.10 are generally existence results.", "They tell about existence of smooth ridge functions $g_{i}$ in the corresponding representation formula (2.2) or (2.24).", "They are uninformative if we want to construct explicitly these functions.", "In this section, we give two theorems which do not only address the smoothness problem, but also are useful in constructing the mentioned $g_{i}$ .", "We start with the constructive analysis of the smoothness problem for bivariate functions.", "We show that if a bivariate function of a certain smoothness class is represented by a sum of finitely many, arbitrarily behaved ridge functions, then, under suitable conditions, it also can be represented by a sum of ridge functions of the same smoothness class and these ridge functions can be constructed explicitely.", "Theorem 2.11.", "Assume $(a_{i},b_{i})$ , $i=1,...,n$ are pairwise linearly independent vectors in $\\mathbb {R}^{2}$ .", "Assume that a function $f\\in C^{s}(\\mathbb {R}^{2})$ has the form $f(x,y)=\\sum _{i=1}^{n}f_{i}(a_{i}x+b_{i}y),$ where $f_{i}$ are arbitrary univariate functions and $s\\ge n-2.$ Then $f$ can be represented also in the form $f(x,y)=\\sum _{i=1}^{n}g_{i}(a_{i}x+b_{i}y),(2.69)$ where the functions $g_{i}\\in C^{s}(\\mathbb {R})$ , $i=1,...,n$ .", "In (2.69), the functions $g_{i}$ , $i=1,...,n,$ can be constructed by the formulas $g_{p} &=&\\varphi _{p,n-p-1},\\text{ }p=1,...,n-2; \\\\g_{n-1} &=&h_{1,n-1};\\text{ }g_{n}=h_{2,n-1}.$ Here all the involved functions $\\varphi _{p,n-p-1}$ , $h_{1,n-1}$ and $h_{2,n-1}$ can be found inductively as follows $h_{1,1}(t) &=&\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f^{\\ast }(t,0),~ \\\\h_{2,1}(t) &=&\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f^{\\ast }(0,t)-\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f^{\\ast }(0,0); \\\\h_{1,k+1}(t) &=&\\frac{1}{e_{1}\\cdot l_{k}}\\int _{0}^{t}h_{1,k}(z)dz,\\text{ }k=1,...,n-2; \\\\h_{2,k+1}(t) &=&\\frac{1}{e_{2}\\cdot l_{k}}\\int _{0}^{t}h_{2,k}(z)dz,\\text{ }k=1,...,n-2;$ and $\\varphi _{p,1}(t)=\\frac{\\partial ^{n-p-2}f^{\\ast }}{\\partial l_{p+1}\\cdot \\cdot \\cdot \\partial l_{n-2}}\\left( \\frac{\\widetilde{a}_{p}t}{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}},\\frac{\\widetilde{b}_{p}t}{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}\\right) -h_{1,p+1}\\left( \\frac{\\widetilde{a}_{p}t}{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}\\right)$ $-h_{2,p+1}\\left( \\frac{\\widetilde{b}_{p}t}{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}\\right)-\\sum _{j=1}^{p-1}\\varphi _{j,p-j+1}\\left( \\frac{\\widetilde{a}_{j}\\widetilde{a}_{p}+\\widetilde{b}_{j}\\widetilde{b}_{p}}{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}t\\right),$ $p=1,...,n-2\\left( \\text{for }p=n-2\\text{, }\\frac{\\partial ^{n-p-2}f^{\\ast }}{\\partial l_{p+1}\\cdot \\cdot \\cdot \\partial l_{n-2}}:=f^{\\ast }\\right) ;$ $\\varphi _{p,k+1}(t)=\\frac{1}{(\\widetilde{a}_{p},\\widetilde{b}_{p})\\cdot l_{k+p}}\\int _{0}^{t}\\varphi _{p,k}(z)dz,\\text{ }p=1,...,n-3,\\text{ }k=1,...,n-p-2.$ In the above formulas $\\widetilde{a}_{p}=\\frac{a_{p}b_{n}-a_{n}b_{p}}{a_{n-1}b_{n}-a_{n}b_{n-1}};~\\widetilde{b}_{p}=\\frac{a_{n-1}b_{p}-a_{p}b_{n-1}}{a_{n-1}b_{n}-a_{n}b_{n-1}},~p=1,...,n-2,$ $l_{p}=\\left( \\frac{\\widetilde{b}_{p}}{\\sqrt{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}},\\frac{-\\widetilde{a}_{p}}{\\sqrt{\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2}}}\\right) ,\\text{ }p=1,...,n-2.$ $f^{\\ast }(x,y)=f\\left( \\frac{b_{n}x-b_{n-1}y}{a_{n-1}b_{n}-a_{n}b_{n-1}},\\frac{a_{n}x-a_{n-1}y}{a_{n}b_{n-1}-a_{n-1}b_{n}}\\right) .$ Since the vectors $(a_{n-1},b_{n-1})$ and $(a_{n},b_{n})$ are linearly independent, there is a nonsingular linear transformation $S:(x,y)\\rightarrow (x^{^{\\prime }},y^{^{\\prime }})$ such that $S:(a_{n-1},b_{n-1})\\rightarrow (1,0)$ and $S:(a_{n},b_{n})\\rightarrow (0,1).$ Thus, without loss of generality we may assume that the vectors $(a_{n-1},b_{n-1})$ and $(a_{n},b_{n})$ coincide with the coordinate vectors $e_{1}=(1,0)$ and $e_{2}=(0,1)$ respectively.", "Therefore, to prove the first part of the theorem it is enough to show that if a function $f\\in C^{s}(\\mathbb {R}^{2})$ is expressed in the form $f(x,y)=\\sum _{i=1}^{n-2}f_{i}(a_{i}x+b_{i}y)+f_{n-1}(x)+f_{n}(y),$ with arbitrary $f_{i}$ , then there exist functions $g_{i}$ $\\in C^{s}(\\mathbb {R})$ , $i=1,...,n$ , such that $f$ is also expressed in the form $f(x,y)=\\sum _{i=1}^{n-2}g_{i}(a_{i}x+b_{i}y)+g_{n-1}(x)+g_{n}(y).", "(2.70)$ By $\\Delta _{l}^{(\\delta )}F$ we denote the increment of a function $F$ in a direction $l=(l^{\\prime },l^{\\prime \\prime }).$ That is, $\\Delta _{l}^{(\\delta )}F(x,y)=F(x+l^{\\prime }\\delta ,y+l^{\\prime \\prime }\\delta )-F(x,y).$ We also use the notation $\\frac{\\partial F}{\\partial l}$ which denotes the derivative of $F$ in the direction $l$ .", "It is easy to check that the increment of a ridge function $g(ax+by)$ in a direction perpendicular to $(a,b)$ is zero.", "Let $l_{1},...,l_{n-2}$ be unit vectors perpendicular to the vectors $(a_{1},b_{1}),...,(a_{n-2},b_{n-2})$ correspondingly.", "Then for any set of numbers $\\delta _{1},...,\\delta _{n-2}\\in \\mathbb {R}$ we have $\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(x,y)=\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}\\left[ f_{n-1}(x)+f_{n}(y)\\right] .", "(2.71)$ Denote the left hand side of (2.71) by $S(x,y).$ That is, set $S(x,y)\\overset{def}{=}\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(x,y).$ Then from (2.71) it follows that for any real numbers $\\delta _{n-1}$ and $\\delta _{n}$ , $\\Delta _{e_{1}}^{(\\delta _{n-1})}\\Delta _{e_{2}}^{(\\delta _{n})}S(x,y)=0,$ or in expanded form, $S(x+\\delta _{n-1},y+\\delta _{n})-S(x,y+\\delta _{n})-S(x+\\delta _{n-1},y)+S(x,y)=0.$ Putting in the last equality $\\delta _{n-1}=-x,$ $\\delta _{n}=-y$ , we obtain that $S(x,y)=S(x,0)+S(0,y)-S(0,0).$ This means that $\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(x,y)$ $=\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(x,0)+\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(0,y)-\\Delta _{l_{1}}^{(\\delta _{1})}\\cdot \\cdot \\cdot \\Delta _{l_{n-2}}^{(\\delta _{n-2})}f(0,0).$ By the hypothesis of the theorem, the derivative $\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f(x,y)$ exists at any point $(x,y)\\in $ $\\mathbb {R}^{2}$ .", "Thus, it follows from the above formula that $\\frac{\\partial ^{n-2}f}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}(x,y)=h_{1,1}(x)+h_{2,1}(y),(2.72)$ where $h_{1,1}(x)=\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f(x,0)$ and $h_{2,1}(y)=\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f(0,y)-\\frac{\\partial ^{n-2}}{\\partial l_{1}\\cdot \\cdot \\cdot \\partial l_{n-2}}f(0,0)$ .", "Note that $h_{1,1}$ and $h_{2,1}$ belong to the class $C^{s-n+2}(\\mathbb {R}).$ By $h_{1,2}$ and $h_{2,2}$ denote the antiderivatives of $h_{1,1}$ and $h_{2,1}$ satisfying the condition $h_{1,2}(0)=h_{2,2}(0)=0$ and multiplied by the numbers $1/(e_{1}\\cdot l_{1})$ and $1/(e_{2}\\cdot l_{1})$ correspondingly.", "That is, $h_{1,2}(x) &=&\\frac{1}{e_{1}\\cdot l_{1}}\\int _{0}^{x}h_{1,1}(z)dz; \\\\h_{2,2}(y) &=&\\frac{1}{e_{2}\\cdot l_{1}}\\int _{0}^{y}h_{2,1}(z)dz.$ Here $e\\cdot l$ denotes the scalar product between vectors $e$ and $l$ .", "Obviously, the function $F_{1}(x,y)=h_{1,2}(x)+h_{2,2}(y)$ obeys the equality $\\frac{\\partial F_{1}}{\\partial l_{1}}(x,y)=h_{1,1}(x)+h_{2,1}(y).", "(2.73)$ From (2.72) and (2.73) we obtain that $\\frac{\\partial }{\\partial l_{1}}\\left[ \\frac{\\partial ^{n-3}f}{\\partial l_{2}\\cdot \\cdot \\cdot \\partial l_{n-2}}-F_{1}\\right] =0.$ Hence, for some ridge function $\\varphi _{1,1}(a_{1}x+b_{1}y),$ $\\frac{\\partial ^{n-3}f}{\\partial l_{2}\\cdot \\cdot \\cdot \\partial l_{n-2}}(x,y)=h_{1,2}(x)+h_{2,2}(y)+\\varphi _{1,1}(a_{1}x+b_{1}y).", "(2.74)$ Here all the functions $h_{2,1},h_{2,2}(y),\\varphi _{1,1}\\in C^{s-n+3}(\\mathbb {R}).$ Set the following functions $h_{1,3}(x) &=&\\frac{1}{e_{1}\\cdot l_{2}}\\int _{0}^{x}h_{1,2}(z)dz; \\\\h_{2,3}(y) &=&\\frac{1}{e_{2}\\cdot l_{2}}\\int _{0}^{y}h_{2,2}(z)dz; \\\\\\varphi _{1,2}(t) &=&\\frac{1}{(a_{1},b_{1})\\cdot l_{2}}\\int _{0}^{t}\\varphi _{1,1}(z)dz.$ Note that the function $F_{2}(x,y)=h_{1,3}(x)+h_{2,3}(y)+\\varphi _{1,2}(a_{1}x+b_{1}y)$ obeys the equality $\\frac{\\partial F_{2}}{\\partial l_{2}}(x,y)=h_{1,2}(x)+h_{2,2}(y)+\\varphi _{1,1}(a_{1}x+b_{1}y).", "(2.75)$ From (2.74) and (2.75) it follows that $\\frac{\\partial }{\\partial l_{2}}\\left[ \\frac{\\partial ^{n-4}f}{\\partial l_{3}\\cdot \\cdot \\cdot \\partial l_{n-2}}-F_{2}\\right] =0.$ The last equality means that for some ridge function $\\varphi _{2,1}(a_{2}x+b_{2}y),$ $\\frac{\\partial ^{n-4}f}{\\partial l_{3}\\cdot \\cdot \\cdot \\partial l_{n-2}}(x,y)=h_{1,3}(x)+h_{2,3}(y)+\\varphi _{1,2}(a_{1}x+b_{1}y)+\\varphi _{2,1}(a_{2}x+b_{2}y).", "(2.76)$ Here all the functions $h_{1,3},$ $h_{2,3},$ $\\varphi _{1,2},$ $\\varphi _{2,1}\\in C^{s-n+4}(\\mathbb {R}).$ Note that in the left hand sides of (2.72), (2.74) and (2.76) we have the mixed directional derivatives of $f$ and the order of these derivatives is decreased by one in each consecutive step.", "Continuing the above process, until it reaches the function $f$ , we obtain the desired representation (2.70).", "The formulas for $g_{i}$ are obtained in the process of the above proof.", "These formulas involve certain functions which can be found inductively as described in the proof.", "The validity of the formulas for the functions $h_{1,k}$ and $h_{2,k}$ , $k=1,...,n-1,$ is obvious.", "The formulas for $\\varphi _{p,1}$ and $\\varphi _{p,k+1}$ can be obtained from (2.74), (2.76) and the subsequent (assumed but not written) equations if we put $x=\\widetilde{a}_{p}t/(\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2})$ and $y=\\widetilde{b}_{p}t/(\\widetilde{a}_{p}^{2}+\\widetilde{b}_{p}^{2})$ .", "Note that $(\\widetilde{a}_{p},\\widetilde{b}_{p}),$ $p=1,...,n-2,$ are the images of vectors $(a_{p},b_{p})$ under the linear transformation $S$ which takes the vectors $(a_{n-1},b_{n-1})$ and $(a_{n},b_{n})$ to the coordinate vectors $e_{1}=(1,0)$ and $e_{2}=(0,1),$ respectively.", "Besides, note that for $p=1,...,n-2,$ the vectors $l_{p}$ are perpendicular to the vectors $(\\widetilde{a}_{p},\\widetilde{b}_{p})$ , respectively and $f^{\\ast }$ is the function generated from $f$ by the above liner transformation.", "Theorem 2.11 can be applied to some higher order partial differential equations in two variables, e.g., to the following homogeneous equation $\\prod \\limits _{i=1}^{r}\\left( \\alpha _{i}\\frac{\\partial }{\\partial x}+\\beta _{i}\\frac{\\partial }{\\partial y}\\right) u(x,y)=0,(2.77)$ where $(\\alpha _{i},\\beta _{i}),~i=1,...,r,$ are pairwise linearly independent vectors in $\\mathbb {R}^{2}$ .", "Clearly, the general solution to this equation are all functions of the form $u(x,y)=\\sum \\limits _{i=1}^{r}v_{i}(\\beta _{i}x-\\alpha _{i}y),(2.78)$ where $v_{i}\\in C^{r}(\\mathbb {R})$ , $i=1,...,r$ .", "Based on Theorem 2.11, for the general solution, one can demand only smoothness of the sum $u$ and dispense with smoothness of the summands $v_{i}$ .", "More precisely, the following corollary is valid.", "Corollary 2.1.", "Assume a function $u\\in C^{r}(\\mathbb {R}^{2}) $ is of the form (2.78) with arbitrarily behaved $v_{i}$ .", "Then $u$ is a solution to Equation (2.77).", "Remark 2.6.", "If in Theorem 2.11 $s\\ge n-1,$ then the functions $g_{i}$ , $i=1,...,n,$ can be constructed (up to polynomials) by the method discussed in Buhmann and Pinkus [18].", "This method is based on the fact that for a direction $\\mathbf {c}=(c_{1},...,c_{m})$ orthogonal to a given direction $\\mathbf {a}\\in \\mathbb {R}^{m}\\backslash \\lbrace \\mathbf {0}\\rbrace ,$ the operator $D_{\\mathbf {c}}=\\sum _{k=1}^{m}c_{k}\\frac{\\partial }{\\partial x_{k}}$ acts on $m$ -variable ridge functions $g(\\mathbf {a}\\cdot \\mathbf {x})$ as follows $D_{\\mathbf {c}}g(\\mathbf {a}\\cdot \\mathbf {x})=\\left( \\mathbf {c}\\cdot \\mathbf {a}\\right) g^{\\prime }(\\mathbf {a}\\cdot \\mathbf {x}).$ Thus, if in our case for fixed $r\\in \\lbrace 1,...,n\\rbrace ,$ vectors $l_{k},$ $k\\in \\lbrace 1,...,n\\rbrace $ , $k\\ne r$ , are perpendicular to the vectors $(a_{k},b_{k})$ , then $\\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne r\\end{array}}^{n}D_{l_{k}}f(x,y)=\\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne r\\end{array}}^{n}D_{l_{k}}\\sum _{i=1}^{n}g_{i}(a_{i}x+b_{i}y)$ $=\\sum _{i=1}^{n}\\left( \\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne r\\end{array}}^{n}\\left((a_{i},b_{i})\\cdot l_{k}\\right) \\right)g_{i}^{(n-1)}(a_{i}x+b_{i}y)=\\prod \\limits _{\\begin{array}{c} k=1 \\\\ k\\ne r\\end{array}}^{n}\\left( (a_{r},b_{r})\\cdot l_{k}\\right) g_{r}^{(n-1)}(a_{r}x+b_{r}y).$ Now $g_{r}$ can be easily constructed from the above formula (up to a polynomial of degree at most $n-2$ ).", "Note that this method is not feasible if in Theorem 2.11 the function $f$ is of the class $C^{n-2}(\\mathbb {R}^{2})$ .", "In this subsection, we generalize ideas from the previous subsection to prove constructively that if a multivariate function of a certain smoothness class is represented by a sum of $k$ arbitrarily behaved ridge functions, then, under suitable conditions, it can be represented by a sum of ridge functions of the same smoothness class and some polynomial of a certain degree.", "The appearance of a polynomial term is mainly related to the fact that in $\\mathbb {R}^{n}$ ($n\\ge 3)$ there are many directions orthogonal to a given direction.", "Such a result was proved nonconstructively in Section 2.1.5 (see Theorem 2.5), but here under a mild hypothesis on the degree of smoothness, we give a new proof for this theorem, which will provide us with a recipe for constructing the functions $g_{i}$ in (2.24).", "The following theorem is valid.", "Theorem 2.12.", "Assume $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.1).", "Let $s\\ge k-p+1,$ where $p$ is the number of vectors $\\mathbf {a}^{i}$ forming a maximal linearly independent system.", "Then there exist functions $g_{i}\\in C^{s}(\\mathbb {R})$ and a polynomial $P(\\mathbf {x})$ of total degree at most $k-p+1$ such that (2.24) holds and $g_{i}$ can be constructed algorithmically.", "We start the proof by choosing a maximal linearly independent system in $\\lbrace \\mathbf {a}^{1},....,\\mathbf {a}^{k}\\rbrace $ .", "The case when the system $\\lbrace \\mathbf {a}^{1},....,\\mathbf {a}^{k}\\rbrace $ itself is linearly independent is obvious (see Section 2.1.1).", "Thus we omit this special case here.", "Without loss of generality we may assume that the first $p$ vectors $\\mathbf {a}^{1},....,\\mathbf {a}^{p}$ , $p<k$ , are linearly independent.", "Thus, the vectors $\\mathbf {a}^{j},$ $j=p+1,...,k,$ can be expressed as linear combinations $\\lambda _{1}^{j}\\mathbf {a}^{1}+\\cdot \\cdot \\cdot +\\lambda _{p}^{j}\\mathbf {a}^{p}$ , where $\\lambda _{1}^{j},...,\\lambda _{p}^{j}$ are real numbers.", "In addition, we can always apply a nonsingular linear transformation $S$ of the coordinates such that $S:\\mathbf {a}^{i}\\rightarrow \\mathbf {e}_{i},$ $i=1,...,p,$ where $\\mathbf {e}_{i}$ denotes the $i$ -th unit vector.", "This reduces the initial representation (2.1) to the following simpler form $f(\\mathbf {x})=f_{1}(x_{1})+\\cdot \\cdot \\cdot +f_{p}(x_{p})+\\sum _{i=1}^{m}f_{p+i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}).", "(2.79)$ Note that we keep the notation of (2.1), but here $\\mathbf {x}=(x_{1},...,x_{p}),$ $\\mathbf {a}^{i}=(\\lambda _{1}^{i}\\mathbf {,...,}\\lambda _{p}^{i})\\in \\mathbb {R}^{p}$ and $m=k-p.$ Obviously, we prove Theorem 2.12 if we prove it for the representation (2.79).", "Thus, in the sequel, we prove that if $f\\in C^{s}(\\mathbb {R}^{n})$ is of the form (2.79) and $s\\ge m+1,$ then there exist functions $g_{i}\\in C^{s}(\\mathbb {R})$ and a polynomial $P(\\mathbf {x})$ of total degree at most $m+1$ such that $f(\\mathbf {x})=g_{1}(x_{1})+\\cdot \\cdot \\cdot +g_{p}(x_{p})+\\sum _{i=1}^{m}g_{p+i}(\\mathbf {a}^{i}\\cdot \\mathbf {x})+P(\\mathbf {x}).$ In the process of the proof, we also see how these $g_{i}$ are constructed.", "For each $i=1,...,m,$ let $\\lbrace \\mathbf {e}_{1}^{(i)},...,\\mathbf {e}_{p-1}^{(i)}\\rbrace $ denote an orthonormal basis in the hyperplane perpendicular to $\\mathbf {a}^{i}.$ By $\\Delta _{\\mathbf {e}}^{(\\delta )}F$ we denote the increment of a function $F$ in a direction $\\mathbf {e}$ of length $\\delta .$ That is, $\\Delta _{\\mathbf {e}}^{(\\delta )}F(\\mathbf {x})=F(\\mathbf {x}+\\delta \\mathbf {e})-F(\\mathbf {x}).$ We also use the notation $\\frac{\\partial F}{\\partial \\mathbf {e}}$ to denote the derivative of $F$ in a direction $\\mathbf {e}$ .", "It is easy to check that the increment of a ridge function $g(\\mathbf {a\\cdot x})$ in any direction perpendicular to $\\mathbf {a}$ is zero.", "For example, $\\Delta _{\\mathbf {e}_{j}^{(i)}}^{(\\delta )}g(\\mathbf {a}^{i}\\mathbf {\\cdot x)}=0,$ for all $i=1,...,m,$ $j=1,...,p-1.$ Therefore, for any indices $i_{1},...,i_{m}\\in \\lbrace 1,...,p-1\\rbrace $ , $q\\in \\lbrace 1,...,p\\rbrace $ and numbers $\\delta _{1},...,\\delta _{m},\\delta \\in \\mathbb {R}$ we have the formula $\\Delta _{\\mathbf {e}_{i_{1}}^{(1)}}^{(\\delta _{1})}\\Delta _{\\mathbf {e}_{i_{2}}^{(2)}}^{(\\delta _{2})}\\cdot \\cdot \\cdot \\Delta _{\\mathbf {e}_{i_{m}}^{(m)}}^{(\\delta _{m})}\\Delta _{\\mathbf {e}_{q}}^{(\\delta )}f(\\mathbf {x})=\\Delta _{\\mathbf {e}_{i_{1}}^{(1)}}^{(\\delta _{1})}\\Delta _{\\mathbf {e}_{i_{2}}^{(2)}}^{(\\delta _{2})}\\cdot \\cdot \\cdot \\Delta _{\\mathbf {e}_{i_{m}}^{(m)}}^{(\\delta _{m})}\\Delta _{\\mathbf {e}_{q}}^{(\\delta )}f_{q}(x_{q}),$ where $\\mathbf {e}_{q}$ denotes the $q$ -th unit vector.", "This means that for each $q=1,...,p,$ the mixed directional derivative $\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m}}^{(m)}\\partial x_{q}}(\\mathbf {x})$ depends only on the variable $x_{q}.$ Denote this derivative by $h_{i_{1},...,i_{m}}^{0,q}(x_{q})$ : $h_{i_{1},...,i_{m}}^{0,q}(x_{q})=\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m}}^{(m)}\\partial x_{q}}(\\mathbf {x}),\\text{ }q=1,...,p.(2.80)$ Since $f\\in C^{s}(\\mathbb {R}^{p}),$ we obtain that $h_{i_{1},...,i_{m}}^{0,q}\\in C^{s-m-1}(\\mathbb {R}).$ It follows from (2.80) that $d\\left( \\frac{\\partial ^{m}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m}}^{(m)}}\\right)=h_{i_{1},...,i_{m}}^{0,1}(x_{1})dx_{1}+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m}}^{0,p}(x_{p})dx_{p}.", "(2.81)$ We conclude from (2.81) that $\\frac{\\partial ^{m}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m}}^{(m)}}(\\mathbf {x})=h_{i_{1},...,i_{m}}^{1,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m}}^{1,p}(x_{p})+c_{i_{1},...,i_{m}},(2.82)$ where the functions $h_{i_{1},...,i_{m}}^{1,q}(x_{q})$ , $q=1,...,p,$ are antiderivatives of $h_{i_{1},...,i_{m}}^{0,q}(x_{q})$ satisfying the condition $h_{i_{1},...,i_{m}}^{1,q}(0)=0$ and $c_{i_{1},...,i_{m}}$ is a constant.", "Note that $h_{i_{1},...,i_{m}}^{1,q}\\in C^{s-m}(\\mathbb {R}),$ $q=1,...,p.$ Obviously, for any pair $k,t\\in \\lbrace 1,...,p-1\\rbrace $ , $\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}\\partial \\mathbf {e}_{k}^{(m)}\\partial \\mathbf {e}_{t}^{(m)}}=\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}\\partial \\mathbf {e}_{t}^{(m)}\\partial \\mathbf {e}_{k}^{(m)}}(2.83)$ It follows from (2.82) and (2.83) that $(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m)})\\left(h_{i_{1},...,i_{m-1},t}^{1,q}\\right) ^{^{\\prime }}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m)})\\left( h_{i_{1},...,i_{m-1},k}^{1,q}\\right)^{^{\\prime }}(x_{q})+c,$ where $c$ is a constant depending on the parameters $i_{1},...,i_{m-1},k,t$ and $q.$ Recall that by construction, $h_{i_{1},...,i_{m}}^{1,q}(0)=0.$ Hence $(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m)})h_{i_{1},...,i_{m-1},t}^{1,q}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m)})h_{i_{1},...,i_{m-1},k}^{1,q}(x_{q})+cx_{q}.", "(2.84)$ Since for each $q=1,...,p,$ the vectors $\\mathbf {e}_{q}$ and $\\mathbf {a}^{m}$ are linearly independent, there exists an index $i_{m}(q)\\in \\lbrace 1,...,p-1\\rbrace $ such that the vector $\\mathbf {e}_{i_{m}(q)}^{(m)}$ is not orthogonal to $\\mathbf {e}_{q}.$ That is, $\\mathbf {e}_{q}\\cdot \\mathbf {e}_{i_{m}(q)}^{(m)}$ $\\ne 0.$ For each $q=1,...,p,$ fix the index $i_{m}(q)$ and define the following functions $h_{i_{1},...,i_{m-1}}^{2,q}(x_{q})=\\frac{1}{\\mathbf {e}_{q}\\cdot \\mathbf {e}_{i_{m}(q)}^{(m)}}\\int _{0}^{x_{q}}h_{i_{1},...,i_{m-1},i_{m}(q)}^{1,q}(z)dz.", "(2.85)$ and $F_{i_{1},...,i_{m-1}}(\\mathbf {x})=h_{i_{1},...,i_{m-1}}^{2,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m-1}}^{2,p}(x_{p}).$ It is easy to obtain from (2.84) and (2.85) that for any $i_{m}\\in \\lbrace 1,...,p-1\\rbrace ,$ $\\frac{\\partial F_{i_{1},...,i_{m-1}}}{\\partial \\mathbf {e}_{i_{m}}^{(m)}}(\\mathbf {x})=h_{i_{1},...,i_{m}}^{1,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m}}^{1,p}(x_{p})+P_{i_{1},...,i_{m}}^{(1)},(2.86)$ where $P_{i_{1},...,i_{m}}^{(1)}$ is a polynomial of total degree not greater than 1.", "It follows from (2.82) and (2.86) that $\\frac{\\partial }{\\partial \\mathbf {e}_{i_{m}}^{(m)}}\\left[ \\frac{\\partial ^{m-1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}}-F_{i_{1},...,i_{m-1}}\\right] (\\mathbf {x})=c_{i_{1},...,i_{m}}-P_{i_{1},...,i_{m}}^{(1)}(\\mathbf {x}).", "(2.87)$ Note that the last equality is valid for all vectors $\\mathbf {e}_{i_{m}}^{(m)},$ which form a basis in the hyperplane orthogonal to $\\mathbf {a}^{m}$ .", "Thus from (2.87) we conclude that the following expansion is valid $\\left.\\begin{array}{c}\\frac{\\partial ^{m-1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}}(\\mathbf {x})=h_{i_{1},...,i_{m-1}}^{2,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m-1}}^{2,p}(x_{p}) \\\\+\\varphi _{i_{1},...,i_{m-1}}^{2,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x})+P_{i_{1},...,i_{m-1}}^{(2)}(\\mathbf {x}).\\end{array}\\right.", "(2.88)$ Here all the functions $h_{i_{1},...,i_{m-1}}^{2,1},...,h_{i_{1},...,i_{m-1}}^{2,p}(x_{p}),\\varphi _{i_{1},...,i_{m-1}}^{2,1}\\in C^{s-m+1}(\\mathbb {R})$ and $P_{i_{1},...,i_{m-1}}^{(2)}$ is a polynomial of total degree not greater than $2.$ Since for each $q=1,...,p,$ the vector $\\mathbf {e}_{q}$ is not collinear to $\\mathbf {a}^{m-1},$ there is an index $i_{m-1}(q)\\in \\lbrace 1,...,p-1\\rbrace $ such that $\\mathbf {e}_{i_{m-1}(q)}^{(m-1)}$ is not orthogonal to $\\mathbf {e}_{q}$ .", "Similarly, since $\\mathbf {a}^{m-1}$ is not collinear to $\\mathbf {a}^{m},$ there is an index $i_{m-1}(m)\\in \\lbrace 1,...,p-1\\rbrace $ such that $\\mathbf {e}_{i_{m-1}(m)}^{(m-1)}$ is not orthogonal to $\\mathbf {a}^{m}$ .", "Fix the indices $i_{m-1}(q)$ , $i_{m-1}(m)$ and consider the following functions $h_{i_{1},...,i_{m-2}}^{3,q}(x_{q})=\\frac{1}{\\mathbf {e}_{q}\\cdot \\mathbf {e}_{i_{m-1}(q)}^{(m-1)}}\\int _{0}^{x_{q}}h_{i_{1},...,i_{m-2},i_{m-1}(q)}^{2,q}(z)dz,\\text{ }q=1,...,p,(2.89)$ $\\varphi _{i_{1},...,i_{m-2}}^{3,1}(t)=\\frac{1}{\\mathbf {a}^{m}\\cdot \\mathbf {e}_{i_{m-1}(m)}^{(m-1)}}\\int _{0}^{t}\\varphi _{i_{1},...,i_{m-2},i_{m-1}(m)}^{2,1}(z)dz,(2.90)$ and $F_{i_{1},...,i_{m-2}}(\\mathbf {x})=h_{i_{1},...,i_{m-2}}^{3,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m-2}}^{3,p}(x_{p})+\\varphi _{i_{1},...,i_{m-2}}^{3,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x}).", "(2.91)$ Similar to (2.84), one can easily verify that for any pair $k,t\\in \\lbrace 1,...,p-1\\rbrace $ and for all $q=1,...,p,$ the following equalities are valid.", "$\\left.\\begin{array}{c}(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m-1)})h_{i_{1},...,i_{m-2},t}^{2,q}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m-1)})h_{i_{1},...,i_{m-2},k}^{2,q}(x_{q})+H_{q}(x_{q}), \\\\(\\mathbf {a}^{m}\\cdot \\mathbf {e}_{k}^{(m-1)})\\varphi _{i_{1},...,i_{m-2},t}^{2,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x})=(\\mathbf {a}^{m}\\cdot \\mathbf {e}_{t}^{(m-1)})\\varphi _{i_{1},...,i_{m-2},k}^{2,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x})+\\Phi (\\mathbf {x}),\\end{array}\\right.", "(2.92)$ where $H_{q}$ and $\\Phi $ are univariate and $n$ -variable polynomials of degree not greater than $2.$ Indeed, applying the Schwarz formula $\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-2}}^{(m-2)}\\partial \\mathbf {e}_{k}^{(m-1)}\\partial \\mathbf {e}_{t}^{(m-1)}\\partial \\mathbf {e}_{i_{m}(q)}^{(m)}}=\\frac{\\partial ^{m+1}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-2}}^{(m-2)}\\partial \\mathbf {e}_{t}^{(m-1)}\\partial \\mathbf {e}_{k}^{(m-1)}\\partial \\mathbf {e}_{i_{m}(q)}^{(m)}}$ on the symmetry of derivatives, it follows from (2.82) that for any $k,t\\in \\lbrace 1,...,p-1\\rbrace $ $(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m-1)})\\left(h_{i_{1},...,i_{m-2},t,i_{m}(q)}^{1,q}\\right) ^{^{\\prime }}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m-1)})\\left(h_{i_{1},...,i_{m-2},k,i_{m}(q)}^{1,q}\\right) ^{^{\\prime }}(x_{q})+d,$ where $d$ is a constant depending on the parameters $i_{1},...,i_{m-2},k,t$ and $i_{m}(q).$ Since, by construction, $h_{i_{1},...,i_{m}}^{1,q}(0)=0,$ we obtain that $(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m-1)})h_{i_{1},...,i_{m-2},t,i_{m}(q)}^{1,q}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m-1)})h_{i_{1},...,i_{m-2},k,i_{m}(q)}^{1,q}(x_{q})+dx_{q}.$ The last equality together with (2.85) yield that $(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{k}^{(m-1)})\\left(h_{i_{1},...,i_{m-2},t}^{2,q}\\right) ^{^{\\prime }}(x_{q})=(\\mathbf {e}_{q}\\cdot \\mathbf {e}_{t}^{(m-1)})\\left( h_{i_{1},...,i_{m-2},k}^{2,q}\\right)^{^{\\prime }}(x_{q})+dx_{q}.$ Therefore, the first equality in (2.92) holds.", "Considering this and applying the corresponding Schwarz formula to (2.88) we obtain the second equality in (2.92).", "Taking into account the definitions (2.89), (2.90) and the relations (2.92), we obtain from (2.91) that for any $i_{m-1}\\in \\lbrace 1,...,p-1\\rbrace ,$ $\\frac{\\partial F_{i_{1},...,i_{m-2}}}{\\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}}(\\mathbf {x})$ $=h_{i_{1},...,i_{m-1}}^{2,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m-1}}^{2,p}(x_{p})+\\varphi _{i_{1},...,i_{m-1}}^{2,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x})+\\widetilde{P}_{i_{1},...,i_{m-1}}^{(2)}(\\mathbf {x}),(2.93)$ where $\\widetilde{P}_{i_{1},...,i_{m-1}}^{(2)}$ is a polynomial of degree not greater than $2.$ It follows from (2.88) and (2.93) that $\\frac{\\partial }{\\partial \\mathbf {e}_{i_{m-1}}^{(m-1)}}\\left[ \\frac{\\partial ^{m-2}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-2}}^{(m-2)}}-F_{i_{1},...,i_{m-2}}\\right] (\\mathbf {x})=P_{i_{1},...,i_{m-1}}^{(2)}(\\mathbf {x})-\\widetilde{P}_{i_{1},...,i_{m-1}}^{(2)}(\\mathbf {x}).", "(2.94)$ Note that the last equality is valid for all vectors $\\mathbf {e}_{i_{m-1}}^{(m-1)},$ which form a basis in the hyperplane orthogonal to $\\mathbf {a}^{m-1}$ .", "Considering this, from (2.94) we derive the following representation $\\left.\\begin{array}{c}\\frac{\\partial ^{m-2}f}{\\partial \\mathbf {e}_{i_{1}}^{(1)}\\cdot \\cdot \\cdot \\partial \\mathbf {e}_{i_{m-2}}^{(m-2)}}(\\mathbf {x})=h_{i_{1},...,i_{m-2}}^{3,1}(x_{1})+\\cdot \\cdot \\cdot +h_{i_{1},...,i_{m-2}}^{3,p}(x_{p}) \\\\+\\varphi _{i_{1},...,i_{m-2}}^{3,1}(\\mathbf {a}^{m}\\cdot \\mathbf {x})+\\varphi _{i_{1},...,i_{m-2}}^{3,2}(\\mathbf {a}^{m-1}\\cdot \\mathbf {x})+P_{i_{1},...,i_{m-2}}^{(3)}(\\mathbf {x}).\\end{array}\\right.", "(2.95)$ Here all the functions $h_{i_{1},...,i_{m-2}}^{3,1},...,h_{i_{1},...,i_{m-2}}^{3,p}(x_{p}),$ $\\varphi _{i_{1},...,i_{m-2}}^{3,1},$ $\\varphi _{i_{1},...,i_{m-2}}^{3,2}\\in C^{s-m+2}(\\mathbb {R})$ and $P_{i_{1},...,i_{m-2}}^{(3)}$ is a polynomial of total degree not greater than $3.$ Note that in the left hand sides of (2.82), (2.88) and (2.95) we have the mixed directional derivatives of $f$ and the order of these derivatives is decreased by one at each consecutive step.", "Continuing the above process, until it reaches the function $f$ , we obtain the desired representation.", "Note that the above proof gives a recipe for constructing the smooth ridge functions $g_{i}$ .", "Writing out explicit recurrent formulas for $g_{i}$ , as in Theorem 2.11, is technically cumbersome here and hence is avoided.", "Remark 2.7.", "Note that using Theorem 2.12, the degree of polynomial $P(\\mathbf {x})$ in Theorem 2.5 can be reduced.", "Indeed, it follows from (2.27) and (2.28) that the the above polynomial $P(\\mathbf {x})$ is of the form (2.1).", "On the other hand, by Theorem 2.12 there exist functions $g_{i}^{\\ast }\\in C^{s}(\\mathbb {R})$ , $i=1,...,k$ , and a polynomial $G(\\mathbf {x})$ of degree at most $k-p+1$ such that $P(\\mathbf {x})=\\sum _{i=1}^{k}g_{i}^{\\ast }(\\mathbf {a}^{i}\\cdot \\mathbf {x})+G(\\mathbf {x}).$ Now considering this in (2.24) we see that our assertion is true.", "At the end of this chapter, we want to draw the reader's attention to the following uniqueness question.", "Assume we are given pairwise linearly independent vectors $\\mathbf {a}^{i},$ $i=1,...,k,$ in $\\mathbb {R}^{n}$ and a function $f:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ of the form (2.1).", "How many different ways can $f$ be written as a sum of ridge functions with the directions $\\mathbf {a}^{i}$ ?", "Clearly, representation (2.1) is not unique, since we can always add some constants $c_{i}$ to $f_{i}$ without changing the resulting sum in (2.1) provided that $\\sum _{i=1}^{k}c_{i}=0$ .", "It turns out that under minimal requirements representation (2.1) is unique up to polynomials of degree at most $k-2$ .", "More precisely, if, in addition to (2.1), $f$ also has the form (2.2) and $f_{i},g_{i}\\in \\mathcal {B}$ , $i=1,...,k$ , then the functions $f_{i}-g_{i}$ are univariate polynomials of degree at most $k-2$ .", "This result is due to Pinkus [137].", "It follows immediately from this result that in Theorems 2.6–2.10 the functions $g_{i}$ is unique up to a univariate polynomial.", "This is also valid for $g_{i}$ in Theorems 2.4 and 2.5, but in this case for the proof we must apply a slightly different result of Pinkus [137]: Assume a multivariate polynomial $f$ of degree $m$ is of the form (2.1) and $f_{i}\\in \\mathcal {B}$ for $i=1,...,k$ .", "Then $f_{i}$ are univariate polynomials of degree at most $l=\\max \\left\\lbrace m,k-2\\right\\rbrace $ .", "A different uniqueness problem, in a more general setting, will be analyzed in Chapter 4.", "In that problem we will look for sets $Q\\subset \\mathbb {R}^{n}$ for which representation (2.1), considered on $Q$ , is unique.", "It is clear that in the special case, when directions of ridge functions coincide with the coordinate directions, the problem of approximation by linear combinations of these functions turn into the problem of approximation by sums of univariate functions.", "This is also the simplest case in ridge function approximation.", "The simplicity of the approximation guarantees its practicability in application areas, where complicated multivariate functions are main obstacles.", "In mathematics, this type of approximation has arisen, for example, in connection with the classical functional equations [17], the numerical solution of certain PDE boundary value problems [14], dimension theory [148], [149], etc.", "In this chapter, we obtain some results concerning the problem of best approximation by sums of univariate functions.", "Most of the material of this chapter is taken from [57], [58], [59], [67].", "This section is devoted to calculation formulas for the error of approximation of bivariate functions by sums of univariate functions.", "Certain classes of bivariate functions depending on some numerical parameter are constructed and characterized in terms of the approximation error calculation formulas.", "The approximation problem considered here is to approximate a continuous and real-valued function of two variables by sums of two continuous functions of one variable.", "To make the problem precise, let $Q$ be a compact set in the $xOy$ plane.", "Consider the approximation of a continuous function $f \\in C(Q)$ by functions from the manifold $D=\\left\\lbrace \\varphi (x)+\\psi (y)\\right\\rbrace ,$ where $\\varphi (x),\\psi (y)$ are defined and continuous on the projections of $Q$ into the coordinate axes $x$ and $y$ , respectively.", "The approximation error is defined as the distance from $f$ to $D:$ $E(f)=dist(f,D)=\\inf \\limits _{D}\\left\\Vert f-\\varphi -\\psi \\right\\Vert _{C(Q)}=$ $=\\inf \\limits _{D}\\max \\limits _{(x,y)\\in Q}\\left|f(x,y)-\\varphi (x)-\\psi (y)\\right|.$ A function $\\varphi _{0}(x)+\\psi _{0}(y)$ from $D$ , if it exists, is called an extremal element or a best approximating sum if $E(f)=\\left\\Vert f-\\varphi _{0}-\\psi _{0}\\right\\Vert _{C(Q)}.$ To show that $E(f)$ depends also on $Q$ , in some cases to avoid confusion, we will write $E(f,Q)$ instead of $E(f)$ .", "In this section we deal with calculation formulas for $E(f)$ .", "In 1951 Diliberto and Straus published a paper [36], in which along with other results they established a formula for $E(f,R)$ , where $R$ here and throughout this section is a rectangle with sides parallel to the coordinate axes, containing supremum over all closed lightning bolts.", "Later the same formula was established by other authors differently, in cases of both rectangle (see [130]) and more general sets (see [89], [123]).", "Although the formula was valid for all continuous functions, it was not easily calculable.", "Some authors started to seek easily calculable formulas for the approximation error for some subsets of continuous functions.", "Rivlin and Sibner [138] proved a result, which allow one to find the exact value of $E(f,R)$ for a function $f(x,y)$ having the continuous and nonnegative derivative $\\frac{\\partial ^{2}f}{\\partial x\\partial y}$ .", "This result in a more general case (for functions of $n$ variables) was proved by Flatto [40].", "Babaev [10] generalized Rivlin and Sibner's result (as well as Flatto's result, see [12]).", "More precisely, he considered the class $M(R)$ of continuous functions $f(x,y) $ with the property $\\Delta _{h_{1},h_{1}}f=f(x,y)+f(x+h_{1},y+h_{2})-f(x,y+h_{2})-f(x+h_{1},y)\\ge 0$ for each rectangle $\\left[ x,x+h_{1}\\right] \\times \\left[ y,y+h_{2}\\right]\\subset R$ , and proved that if $f(x,y)$ belongs to $M(R)$ , where $R=\\left[a_{1},b_{1}\\right] \\times \\left[ a_{2},b_{2}\\right] $ , then $E(f,R)=\\frac{1}{4}\\left[f(a_{1},a_{2})+f(b_{1},b_{2})-f(a_{1},b_{2})-f(b_{1},a_{2})\\right] .$ As seen from this formula, to calculate $E(f)$ it is sufficient to find only values of $f(x,y)$ at the vertices of $R$ .", "One can see that the formula also gives a sufficient condition for membership in the class $M(R)$ , i.e.", "if $E(f,S)=\\frac{1}{4}\\left[f(x_{1},y_{1})+f(x_{2},y_{2})-f(x_{1},y_{2})-f(x_{2},y_{1})\\right] ,$ for a given $f$ and for each $S=\\left[ x_{1},x_{2}\\right] \\times \\left[y_{1},y_{2}\\right] \\subset R$ , then the function $f(x,y)$ is from $M(R)$ .", "Our purpose is to construct new classes of continuous functions, which will depend on a numerical parameter, and characterize each class in terms of the approximation error calculation formulas.", "The mentioned parameter will show which points of $R$ the calculation formula involves.", "We will also construct a best approximating sum $\\varphi _{0}+\\psi _{0}$ to a function from constructed classes.", "Let throughout this section $R=\\left[ a_{1},b_{1}\\right] \\times \\left[a_{2},b_{2}\\right] $ be a rectangle and $c\\in (a_{1},b_{1}]$ .", "Denote $R_{1}=\\left[ a_{1},c\\right] \\times \\left[ a_{2},b_{2}\\right] $ and $R_{2}=\\left[c,b_{1}\\right] \\times \\left[ a_{2},b_{2}\\right] $ .", "It is clear that $R=R_{1}\\cup R_{2}$ and if $c=b_{1}$ , then $R=R_{1}$ .", "We associate each rectangle $S=\\left[ x_{1},x_{2}\\right] \\times \\left[y_{1},y_{2}\\right] $ lying in $R$ with the following functional: $L(f,S)=\\frac{1}{4}\\left[f(x_{1},y_{1})+f(x_{2},y_{2})-f(x_{1},y_{2})-f(x_{2},y_{1})\\right] .$ Definition 3.1.", "We say that a continuous function $f(x,y)$ belongs to the class $V_{c}(R)$ if 1) $L(f,S)\\ge 0$ , for each $S\\subset R_{1}$ ; 2) $L(f,S)\\le 0$ , for each $S\\subset R_{2}$ ; 3) $L(f,S)\\ge 0$ , for each $S=\\left[ a_{1},b_{1}\\right] \\times \\left[ y_{1},y_{2}\\right] ,~\\ S\\subset R$ .", "It can be shown that for any $c\\in (a_{1},b_{1}]$ the class $V_{c}(R)$ is not empty.", "Indeed, one can easily verify that the function $v _{c}(x,y)=\\left\\lbrace \\begin{array}{c}w(x,y)-w(c,y),\\;\\ (x,y)\\in R_{1} \\\\w(c,y)-w(x,y),\\;\\ (x,y)\\in R_{2}\\end{array}\\right.$ where $w(x,y)=\\left( \\frac{x-a_{1}}{b_{1}-a_{1}}\\right) ^{\\frac{1}{n}}\\cdot y $ and $n\\ge \\log _{2}\\frac{b_{1}-a_{1}}{c-a_{1}}$ , satisfies conditions 1)-3) and therefore belongs to $V_{c}(R)$ .", "The class $V_{c}(R)$ has the following obvious properties: a) For given functions $f_{1},f_{2}\\in V_{c}(R)$ and numbers $\\alpha _{1},\\alpha _{2}\\ge 0$ , $\\alpha _{1}f_{1}+\\alpha _{2}f_{2}\\in V_{c}(R)$ .", "$V_{c}(R)$ is a closed subset of the space of continuous functions.", "b) $V_{b_{1}}(R)=M(R)$ .", "c) If $f$ is a common element of $V_{c_{1}}(R)$ and $V_{c_{2}}(R)$ , $a_{1}<c_{1}<c_{2}\\le b_{1}$ then $f(x,y)=\\varphi (x)+\\psi (y)$ on the rectangle $\\left[ c_{1},c_{2}\\right] \\times \\left[ a_{2},b_{2}\\right] $ .", "The properties a) and b) are clear.", "The property c) also becomes clear if note that according to the definition of the classes $V_{c_{1}}(R)$ and $V_{c_{2}}(R)$ , for each rectangle $S\\subset \\left[ c_{1},c_{2}\\right] \\times \\left[ a_{2},b_{2}\\right]$ we have $L(f,S)\\le 0\\;\\; \\mbox{and}\\;\\; L(f,S)\\ge 0,$ respectively.", "Hence $L(f,S)=0\\;\\; \\mbox{for each}\\;\\; S\\subset \\left[ c_{1},c_{2}\\right] \\times \\left[ a_{2},b_{2}\\right].$ Thus it is not difficult to understand that $f$ is of the form $\\varphi (x)+\\psi (y)$ on the rectangle $\\left[ c_{1},c_{2}\\right] \\times \\left[a_{2},b_{2}\\right] $ .", "Lemma 3.1.", "Assume a function $f(x,y)$ has the continuous derivative $\\frac{\\partial ^{2}f}{\\partial x\\partial y}$ on the rectangle $R$ and satisfies the following conditions 1) $\\frac{\\partial ^{2}f}{\\partial x\\partial y}\\ge 0$ , for all $(x,y)\\in R_{1}$ ; 2) $\\frac{\\partial ^{2}f}{\\partial x\\partial y}\\le 0$ , for all $(x,y)\\in R_{2}$ ; 3) $\\frac{df(a_{1},y)}{dy}\\le \\frac{df(b_{1},y)}{dy}$ , for all $y\\in \\left[ a_{2},b_{2}\\right]$ .", "Then $f(x,y)$ belongs to $V_{c}(R)$ .", "The proof of this lemma is very simple and can be obtained by integrating both sides of inequalities in conditions 1)-3) through sets $\\left[x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] \\subset R_{1}$ , $\\left[x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] \\subset R_{2}$ and $\\left[ y_{1},y_{2}\\right] \\subset \\left[ a_{2},b_{2}\\right]$ , respectively.", "Example 3.1.", "Consider the function $f(x,y)=y\\sin \\pi x$ on the unit square $K=\\left[ 0,1\\right] \\times \\left[ 0,1\\right] $ and rectangles $K_{1}=\\left[ 0,\\frac{1}{2}\\right] \\times \\left[ 0,1\\right] ,K_{2}=\\left[ \\frac{1}{2},1\\right] \\times \\left[ 0,1\\right] $ .", "It is not difficult to verify that this function satisfies all conditions of the lemma and therefore belongs to $V_{\\frac{1}{2}}(K)$ .", "The following theorem is valid.", "Theorem 3.1.", "The approximation error of a function $f(x,y)$ from the class $V_{c}(R)$ can be calculated by the formula $E(f,R)=L(f,R_{1})=\\frac{1}{4}\\left[f(a_{1},a_{2})+f(c,b_{2})-f(a_{1},b_{2})-f(c,a_{2})\\right] .$ Let $y_{0}$ be any solution from $\\left[ a_{2},b_{2}\\right] $ of the equation $L(f,Y)=\\frac{1}{2}L(f,R_{1}),\\;\\;Y=\\left[ a_{1},c\\right] \\times \\left[a_{2},y\\right] .$ Then the function $\\varphi _{0}(x)+\\psi _{0}(y)$ , where $\\varphi _{0}(x)=f(x,y_{0}),$ $\\psi _{0}(y)=\\frac{1}{2}\\left[ f(a_{1},y)+f(c,y)-f(a_{1},y_{0})-f(c,y_{0})\\right]$ is a best approximating sum from the manifold $D$ to $f$ .", "To prove this theorem we need the following lemma.", "Lemma 3.2.", "Let $f(x,y)$ be a function from $V_{c}(R)$ and $X=\\left[ a_{1},x\\right] \\times \\left[ y_{1},y_{2}\\right] $ be a rectangle with fixed $y_{1},y_{2}\\in \\left[ a_{2},b_{2}\\right] $ .", "Then the function $h(x)=L(f,X)$ has the properties: 1) $h(x)\\ge 0$ , for any $x\\in \\left[ a_{1},b_{1}\\right] $ ; 2) $\\max \\limits _{\\left[ a_{1},b_{1}\\right] }h(x)=h(c)$ and $\\min \\limits _{\\left[ a_{1},b_{1}\\right] }h(x)=h(a_{1})=0$ .", "Proof.", "If $X\\subset R_{1}$ , then the validity of $h(x)\\ge 0$ follows from the definition of $V_{c}(R)$ .", "If $X$ is from $R$ but not lying in $R_{1}$ , then by denoting $X^{\\prime }=\\left[ x,b_{1}\\right] \\times \\left[y_{1},y_{2}\\right] ,S=X\\cup X^{\\prime }$ and using the obvious equality $L(f,S)=L(f,X)+L(f,X^{\\prime })$ we deduce from the definition of $V_{c}(R)$ that $h(x)\\ge 0$ .", "To prove the second part of the lemma, it is enough to show that $h(x)$ increases on the interval $\\left[ a_{1},c\\right] $ and decreases on the interval $\\left[ c,b_{1}\\right] $ .", "Indeed, if $a_{1}\\le x_{1}\\le x_{2}\\le c$ , then $h(x_{2})=L(f,X_{2})=L(f,X_{1})+L(f,X_{12}),(3.1)$ where $X_{1}=\\left[ a_{1},x_{1}\\right] \\times \\left[ y_{1},y_{2}\\right] ,$ $X_{2}=\\left[ a_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] ,$ $X_{12}=\\left[ x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] $ .", "Taking into consideration that $L(f,X_{1})=h(x_{1})$ and $X_{12}$ lies in $R_{1}$ we obtain from (3.1) that $h(x_{2})\\ge h(x_{1})$ .", "If $c\\le x_{1}\\le x_{2}\\le b_{1}$ , then $X_{12}$ lies in $R_{2}$ and we obtain from (3.1) that $h(x_{2})\\le h(x_{1})$ .", "Proof of Theorem 3.1.", "It is obvious that $L(f,R_{1})=L(f-\\varphi -\\psi ,R_{1})$ for each sum $\\varphi (x)+\\psi (y)$ .", "Hence $L(f,R_{1})\\le \\left\\Vert f-\\varphi -\\psi \\right\\Vert _{C(R_{1})}\\le \\left\\Vert f-\\varphi -\\psi \\right\\Vert _{C(R)}.$ Since a sum $\\varphi (x)+\\psi (y)$ is arbitrary, $L(f,R_{1})\\le E(f,R)$ .", "To complete the proof it is sufficient to construct a sum $\\varphi _{0}(x)+\\psi _{0}(y)$ for which the equality $\\left\\Vert f-\\varphi _{0}-\\psi _{0}\\right\\Vert _{C(R)}=L(f,R_{1})(3.2)$ holds.", "Consider the function $g(x,y)=f(x,y)-f(x,a_{2})-f(a_{1},y)+f(a_{1},a_{2}).$ This function has the following obvious properties 1) $g(x,a_{2})=g(a_{1},y)=0$ ; 2) $L(f,R_{1})=L(g,R_{1})=\\frac{1}{4}g(c,b_{2})$ ; 3) $E(f,R)=E(g,R)$ ; 4) The function of one variable $g(c,y)$ increases on the interval $\\left[a_{2},b_{2}\\right] $ .", "The last property of $g$ allows us to write that $0=g(c,a_{2})\\le \\frac{1}{2}g(c,b_{2})\\le g(c,b_{2}).$ Since $g(x,y)$ is continuous, there exists at least one solution $y=y_{0}$ of the equation $g(c,y)=\\frac{1}{2}g(c,b_{2})$ or, in other notation, of the equation $L(f,Y)=\\frac{1}{2}L(f,R_{1}),\\;\\;\\text{where}\\;\\;Y=\\left[ a_{1},c\\right]\\times \\left[ a_{2},y\\right] ,$ Introduce the functions $\\varphi _{1}(x)=g(x,y_{0}),$ $\\psi _{1}(y)=\\frac{1}{2}\\left( g(c,y)-g(c,y_{0})\\right) ,$ $G(x,y)=g(x,y)-\\varphi _{1}(x)-\\psi _{1}(y).$ Calculate the norm of $G(x,y)$ on $R$ .", "Consider the rectangles $R^{\\prime }=\\left[ a_{1},b_{1}\\right] \\times \\left[ y_{0},b_{2}\\right] $ and $R^{\\prime \\prime }=\\left[ a_{1},b_{1}\\right] \\times \\left[ a_{2},y_{0}\\right] $ .", "It is clear that $\\left\\Vert G\\right\\Vert _{C(R)}=\\max \\left\\lbrace \\left\\Vert G\\right\\Vert _{C(R^{\\prime })},\\left\\Vert G\\right\\Vert _{C(R^{\\prime \\prime })}\\right\\rbrace .$ First calculate the norm $\\left\\Vert G\\right\\Vert _{C(R^{\\prime })}$ : $\\left\\Vert G\\right\\Vert _{C(R^{\\prime })}=\\max \\limits _{(x,y)\\in R^{\\prime }}\\left|G(x,y)\\right|=\\max \\limits _{y\\in \\left[ y_{0},b_{2}\\right]}\\max \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }\\left|G(x,y)\\right|.", "(3.3)$ For a fixed point $y$ (we keep it fixed until (3.6)) from the interval $\\left[ y_{0},b_{2}\\right] $ we can write that $\\max \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }G(x,y)=\\max \\limits _{x\\in \\left[a_{1},b_{1}\\right] }\\left( g(x,y)-g(x,y_{0})\\right) -\\psi _{1}(y)(3.4)$ and $\\min \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }G(x,y)=\\min \\limits _{x\\in \\left[a_{1},b_{1}\\right] }\\left( g(x,y)-g(x,y_{0})\\right) -\\psi _{1}(y).", "(3.5)$ By Lemma 3.2, the function $h_{1}(x)=4L(f,X)=g(x,y)-g(x,y_{0}),\\;\\;\\mbox{where}\\;\\;X=\\left[ a_{1},x\\right] \\times \\left[ y_{0},y\\right] ,$ reaches its maximum on $x=c$ and minimum on $x=a_{1}$ : $\\max \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }h_{1}(x)=g(c,y)-g(c,y_{0})$ $\\min \\limits _{x\\in \\left[ a_{1},b_{1}\\right]}h_{1}(x)=g(a_{1},y)-g(a_{1},y_{0})=0.$ Considering these facts in (3.4) and (3.5) we obtain that $\\max \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }G(x,y)=g(c,y)-g(c,y_{0})-\\psi _{1}(y)=\\frac{1}{2}\\left( g(c,y)-g(c,y_{0})\\right) ,$ $\\min \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }G(x,y)=-\\psi _{1}(y)=-\\frac{1}{2}\\left( g(c,y)-g(c,y_{0})\\right) .$ Consequently, $\\max \\limits _{x\\in \\left[ a_{1},b_{1}\\right] }\\left|G(x,y)\\right|=\\frac{1}{2}\\left( g(c,y)-g(c,y_{0})\\right) .", "(3.6)$ Taking (3.6) and the 4-th property of $g$ into account in (3.3) yields $\\left\\Vert G\\right\\Vert _{C(R^{\\prime })}=\\frac{1}{2}\\left(g(c,b_{2})-g(c,y_{0})\\right) =\\frac{1}{4}g(c,b_{2}).$ Similarly it can be shown that $\\left\\Vert G\\right\\Vert _{C(R^{\\prime \\prime })}=\\frac{1}{4}g(c,b_{2}).$ Hence $\\left\\Vert G\\right\\Vert _{C(R)}=\\frac{1}{4}g(c,b_{2})=L(f,R_{1}).$ But by the definition of $G$ , $G(x,y)=g(x,y)-\\varphi _{1}(x)-\\psi _{1}(y)=f(x,y)-\\varphi _{0}(x)-\\psi _{0}(y),$ where $\\varphi _{0}(x)=\\varphi _{1}(x)+f(x,a_{2})-f(a_{1},a_{2})+f(a_{1},y_{0})=f(x,y_{0}),$ $\\psi _{0}(y)=\\psi _{1}(y)+f(a_{1},y)-f(a_{1},y_{0})=$ $=\\frac{1}{2}\\left( f(a_{1},y)+f(c,y)-f(a_{1},y_{0})-f(c,y_{0})\\right) .$ Therefore, $\\left\\Vert f-\\varphi _{0}-\\psi _{0}\\right\\Vert _{C(R)}=L(f,R_{1}).$ We proved (3.2) and hence Theorem 3.1.", "Note that the function $\\varphi _{0}(x)+\\psi _{0}(y)$ is a best approximating sum from the manifold ${D}$ to $f$ .", "Remark 3.1.", "In the special case $c=b_{1}$ , Theorem 3.1 turns into Babaev's result from [10].", "Corollary 3.1.", "Let a function $f(x,y)$ have the continuous derivative $\\frac{\\partial ^{2}f}{\\partial x\\partial y}$ on the rectangle $R$ and satisfy the following conditions 1) $\\frac{\\partial ^{2}f}{\\partial x\\partial y}\\ge 0$ , for all $(x,y)\\in R_{1}$ ; 2) $\\frac{\\partial ^{2}f}{\\partial x\\partial y}\\le 0$ , for all $(x,y)\\in R_{2}$ ; 3) $\\frac{df(a_{1},y)}{dy}\\le \\frac{df(b_{1},y)}{dy}$ , for all $y\\in \\left[ a_{2},b_{2}\\right] $ .", "Then $E(f,R)=L(f,R_{1})=\\frac{1}{4}\\left[f(a_{1},a_{2})+f(c,b_{2})-f(a_{1},b_{2})-f(c,a_{2})\\right] .$ The proof of this corollary can be obtained directly from Lemma 3.1 and Theorem 3.1.", "Remark 3.2.", "Rivlin and Sibner [138] proved Corollary 3.1 in the special case $c=b_{1}$ .", "Example 3.2.", "As we know (see Example 3.1) the function $f=y\\sin \\pi x$ belongs to $V_{\\frac{1}{2}}(K)$ , where $K=\\left[ 0,1\\right] \\times \\left[0,1\\right] $ .", "By Theorem 3.1, $E(f,K)=\\frac{1}{4}$ and the function $\\frac{1}{2}\\sin \\pi x+\\frac{1}{2}y-\\frac{1}{4}$ is a best approximating sum.", "The following theorem shows that in some cases the approximation error formula in Theorem 3.1 is valid for more general sets than rectangles with sides parallel to the coordinate axes.", "Theorem 3.2.", "Let $f(x,y)$ be a function from $V_{c}(R)$ and $Q\\subset R$ be a compact set which contains all vertices of $R_{1}$ (points $(a_{1},a_{2}),(a_{1},b_{2}),(c,a_{2}),(c,b_{2})$ ).", "Then $E(f,Q)=L(f,R_{1})=\\frac{1}{4}\\left[f(a_{1},a_{2})+f(c,b_{2})-f(a_{1},b_{2})-f(c,a_{2})\\right] .$ Proof.", "Since $Q\\subset R,$ $E(f,Q)\\le E(f,R)$ .", "On the other hand by Theorem 3.1, $E(f,R)=L(f,R_{1})$ .", "Hence $E(f,Q)\\le L(f,R_{1})$ .", "It can be shown, as it has been shown in the proof of Theorem 3.1, that $L(f,R_{1})\\le E(f,Q)$ .", "But then automatically $E(f,Q)=L(f,R_{1})$ .", "Example 3.3.", "Calculate the approximation error of the function $f(x,y)=-(x-2)^{2n}y^{m}$ ($n$ and $m$ are positive integers) on the domain $Q=\\left\\lbrace (x,y):0\\le x\\le 2,0\\le y\\le (x-1)^{2}+1\\right\\rbrace .$ It can be easily verified that $f \\in V_{2}(R)$ , where $R=\\left[ 0,4\\right]\\times \\left[ 0,2\\right] $ .", "Besides, $Q$ contains all vertices of $R_{1}=\\left[ 0,2\\right] \\times \\left[ 0,2\\right] $ .", "Consequently, by Theorem 3.2, $E(f,Q)=L(f,R_{1})=2^{2(n-1)+m}$ .", "The following theorem characterizes the class $V_{c}(R)$ in terms of the approximation error calculation formulas.", "Theorem 3.3.", "The following conditions are necessary and sufficient for a continuous function $f(x,y)$ belong to $V_{c}(R):$ 1) $E(f,S)=L(f,S)$ , for each rectangle $S=\\left[ x_{1},x_{2}\\right]\\times \\left[ y_{1},y_{2}\\right] ,S\\subset R_{1}$ ; 2) $E(f,S)=-L(f,S)$ , for each rectangle $S=\\left[ x_{1},x_{2}\\right]\\times \\left[ y_{1},y_{2}\\right] ,S\\subset R_{2}$ ; 3) $E(f,S)=L(f,S_{1})$ , for each rectangle $S=\\left[ a_{1},b_{1}\\right] \\times \\left[ y_{1},y_{2}\\right] ,S\\subset R$ and $S_{1}=\\left[a_{1},c\\right] \\times \\left[ y_{1},y_{2}\\right] $ .", "Proof.", "The necessity easily follows from the definition of $V_{c}(R)$ , Babaev's above-mentioned result (see Section 3.1.1) and Theorem 3.1.", "The sufficiency is clear if pay attention to the fact that $E(f,S)\\ge 0 $ .", "By $V_{c}^{-}(R)$ we denote the class of functions $f(x,y)$ such that $-f\\in V_{c}(R)$ .", "It is clear that $E(f,R)=-L(f,R_{1})$ for each $f\\in V_{c}^{-}(R)$ .", "We define $U_{c}(R),a_{1}\\le c< b_{1}$ , as a class of continuous functions $f(x,y)$ with the properties 1) $L(f,S)\\le 0$ , for each rectangle $S=\\left[ x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] ,\\;S\\subset R_{1};$ 2) $L(f,S)\\ge 0$ , for each rectangle $S=\\left[ x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] ,\\;S\\subset R_{2};$ 3) $L(f,S)\\ge 0$ , for each rectangle $S=\\left[ a_{1},b_{1}\\right] \\times \\left[ y_{1},y_{2}\\right] ,\\;S\\subset R.$ Using the same techniques in the proof of Theorem 3.1 it can be shown that the following theorem is valid: Theorem 3.4.", "The approximation error of a function $f(x,y)$ from the class $U_{c}(R)$ can be calculated by the formula $E(f,R)=L(f,R_{2})=\\frac{1}{4}\\left[f(c,a_{2})+f(b_{1},b_{2})-f(c,b_{2})-f(b_{1},a_{2})\\right] .$ Let $y_{0}$ be any solution from $\\left[ a_{2},b_{2}\\right] $ of the equation $L(f,Y)=\\frac{1}{2}L(f,R_{2}),\\qquad Y=\\left[ c,b_{1}\\right] \\times \\left[a_{2},y\\right] .$ Then the function $\\varphi _{0}(x)+\\psi _{0}(y)$ , where $\\varphi _{0}(x)=f(x,y_{0}),\\quad \\psi _{0}(y)=\\frac{1}{2}\\left[f(c,y)+f(b_{1},y)-f(c,y_{0})-f(b_{1},y_{0})\\right] ,$ is a best approximating sum from the manifold $D$ to $f$ .", "By $U_{c}^{-}(R)$ denote the class of functions $f(x,y)$ such that $-f \\in U_{c}(R)$ .", "It is clear that $E(f,R)=-L(f,R_{2})$ for each $f\\in U_{c}^{-}(R)$ .", "Remark 3.3.", "The correspondingly modified versions of Theorems 2.2, 2.3 and Corollary 3.1 are valid for the classes $V_{c}^{-}(R),U_{c}(R)$ and $U_{c}^{-}(R)$ .", "Example 3.4.", "Consider the function $f(x,y)=\\left( x-\\frac{1}{2}\\right) ^{2}y$ on the unit square $K=\\left[ 0,1\\right] \\times \\left[ 0,1\\right] $ .", "It can be easily verified that $f\\in U_{\\frac{1}{2}}(K)$ .", "Hence, by Theorem 3.4, $E(f,K)=\\frac{1}{16}$ and the function $\\frac{1}{2}\\left( x-\\frac{1}{2}\\right) ^{2}+\\frac{1}{8}y-\\frac{1}{16}$ is a best approximating function.", "The purpose of this section is to develop a method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions.", "It should be remarked that formulas of this type were known only for functions defined on a rectangle with sides parallel to the coordinate axes.", "Our method, based on a maximization process over closed bolts, allows the consideration of functions defined on hexagons, octagons and stairlike polygons with sides parallel to the coordinate axes.", "Let $Q$ be a compact set in $\\mathbb {R}^2$ .", "Consider the approximation of a continuous function $f \\in C(Q)$ by functions from the set $D=\\left\\lbrace \\varphi (x)+\\psi (y)\\right\\rbrace ,$ where $\\varphi (x),\\psi (y)$ are defined and continuous on the projections of $Q$ into the coordinate axes $x$ and $y$ , respectively.", "The approximation error is defined as follows $E(f,Q)=\\inf \\limits _{\\varphi +\\psi \\in D}\\left\\Vert f-\\varphi -\\psi \\right\\Vert _{C(Q)}.$ Our purpose is to develop a method for obtaining explicit formulas providing precise and easy computation of $E(f,Q)$ for polygons $Q$ with sides parallel to the coordinate axes.", "This method will be based on the herein developed closed bolts maximization process and can be used in alternative proofs of the known results from [10], [57] and [138].", "First, we show efficiency of the method in the example of a hexagon with sides parallel to the coordinate axes.", "Then we formulate an analogous theorem for staircase polygons and two theorems for octagons, which can be proved in a similar way, and touch some aspects of the question about the case of an arbitrary polygon with sides parallel to the coordinate axes.", "The condition posed on sides of polygons (being parallel to the coordinate axes) is essential for our method.", "This has several reasons, which get clear through the proof of Theorem 3.5.", "Here we are able to explain one of these reasons: by [45], a continuous function $f(x,y)$ defined on a polygon with sides parallel to the coordinate axes has an extremal element, the existence of which is required in our method.", "Now let $K$ be a rectangle (not speaking about polygons) with sides not parallel to the coordinate axes.", "Does any function $f\\in C(K)$ have an extremal element?", "No one knows (see [45]).", "In the sequel, all the considered polygons are supposed to have sides parallel to the coordinate axes.", "Let $H$ be a closed hexagon.", "It is clear that $H$ can be uniquely represented in the form $H=R_{1}\\cup R_{2},(3.7)$ where $R_{1},R_{2}$ are rectangles and there does not exist any rectangle $R$ such that $R_{1}\\subset R\\subset H$ or $R_{2}\\subset R\\subset H$ .", "We associate each closed bolt $p=\\left\\lbrace p_{1},p_{2},\\cdots p_{2n}\\right\\rbrace $ with the following functional $l(f,p)=\\frac{1}{2n}\\sum \\limits _{k=1}^{2n}(-1)^{k-1}f(p_{k}).$ Denote by $M(H)$ the class of bivariate continuous functions $f$ on $H$ satisfying the condition $f(x_{1},y_{1})+f(x_{2},y_{2})-f(x_{1},y_{2})-f(x_{2},y_{1})\\ge 0$ for any rectangle $\\left[ x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right]\\subset H.$ Theorem 3.5.", "Let $H$ be a hexagon and (3.7) be its representation.", "Let $f \\in M(H)$ .", "Then $E(f,H)=\\max \\left\\lbrace \\left|l(f,h)\\right|,\\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|\\right\\rbrace ,(3.8)$ where $h,r_{1},r_{2}$ are closed bolts formed by vertices of the polygons $H,R_{1},R_{2}$ respectively.", "Without loss of generality, we may assume that the rectangles $R_{1}$ and $R_{2}$ are of the following form $R_{1}=\\left[ a_{1},a_{2}\\right] \\times \\left[ b_{1},b_{3}\\right] ,~\\ R_{2}=\\left[ a_{1},a_{3}\\right] \\times \\left[ b_{1},b_{2}\\right] ,~\\ a_{1}<a_{2}<a_{3},\\;b_{1}<b_{2}<b_{3}.$ Introduce the notation $\\begin{array}{c}f_{11}=f\\left( a_{1},b_{1}\\right) ,~\\ f_{12}=-f\\left( a_{1},b_{2}\\right),\\;f_{13}=-f\\left( a_{1},b_{3}\\right) ; \\\\f_{21}=-f\\left( a_{2},b_{1}\\right) ,\\;f_{22}=-f\\left( a_{2},b_{2}\\right) ,~\\ f_{23}=f\\left( a_{2},b_{3}\\right) ; \\\\f_{31}=-f\\left( a_{3},b_{1}\\right) ,\\;f_{32}=f\\left( a_{3},b_{2}\\right) .\\end{array}(3.9)$ It is clear that $\\begin{array}{c}\\left|l(f,r_{1})\\right|=\\dfrac{1}{4}\\left(f_{11}+f_{13}+f_{23}+f_{21}\\right) , \\\\\\left|l(f,r_{2})\\right|=\\dfrac{1}{4}\\left(f_{11}+f_{12}+f_{32}+f_{31}\\right) , \\\\\\left|l(f,h)\\right|=\\dfrac{1}{6}\\left(f_{11}+f_{13}+f_{23}+f_{22}+f_{32}+f_{31}\\right) .\\end{array}(3.10)$ Let $p=\\left\\lbrace p_{1},p_{2},\\cdots p_{2n}\\right\\rbrace $ be any closed bolt.", "We group the points $p_{1},p_{2},\\cdots p_{2n}$ by putting $p_{+}=\\left\\lbrace p_{1},p_{3},\\cdots p_{2n-1}\\right\\rbrace ,\\;p_{-}=\\left\\lbrace p_{2},p_{4},\\cdots p_{2n}\\right\\rbrace .$ First, assume that $l(f,p)\\ge 0$ .", "We apply the following algorithm, which we call the maximization process over closed bolts, to $p$ .", "Step 1.", "Consider sequentially the units $p_ip_{i+1}$ $\\left(i=\\overline{1,2n}, p_{2n+1}=p_1\\right)$ with the vertices $p_{i}\\left(x_{i},y_{i}\\right) ,~\\ p_{i+1}\\left( x_{i+1},y_{i+1}\\right) $ having equal abscissae: $x_{i}=x_{i+1}$ .", "Four cases are possible.", "1) $p_{i}\\in p_{+}$ and $y_{i+1}>y_{i}$ .", "In this case, replace the unit $p_{i}p_{i+1}$ by a new unit $q_{i}q_{i+1}$ with the vertices $q_{i}=(a_{1},y_{i}),\\;\\ q_{i+1}=(a_{1},y_{i+1})$ .", "2) $p_{i}\\in p_{+}$ and $y_{i+1}<y_{i}$ .", "In this case, replace the unit $p_{i}p_{i+1}$ by a new unit $q_{i}q_{i+1}$ with the vertices $q_{i}=\\left(a_{2},y_{i}\\right) ,\\;q_{i+1}=(a_{2},y_{i+1})$ if $b_{2}<y_{i}\\le b_{3}$ or with the vertices $q_{i}=\\left( a_{3},y_{i}\\right),q_{i+1}=(a_{3},y_{i+1})$ if $b_{1}\\le y_{i}\\le b_{2}$ .", "3) $p_{i}\\in p_{-}$ and $y_{i+1}<y_{i}$ .", "In this case, replace $p_{i}p_{i+1}$ by a new unit $q_{i}q_{i+1}$ with the vertices $q_{i}=(a_{1},y_{i}),~q_{i+1}=(a_{1},y_{i+1})$ .", "4) $p_{i}\\in p_{-}$ and $y_{i+1}>y_{i}$ .", "In this case, replace $p_{i}p_{i+1}$ by a new unit $q_{i}q_{i+1}$ with the vertices $q_{i}=(a_{2},y_{i}),~q_{i+1}=(a_{2},y_{i+1})$ if $b_{2}<y_{i+1}\\le b_{3}$ or with the vertices $q_{i}=(a_{3},y_{i}),~q_{i+1}=(a_{3},y_{i+1})$ if $b_{1}\\le y_{i+1}\\le b_{2}$ .", "Since $f\\in M(H)$ , it is not difficult to verify that $\\begin{array}{c}f(p_{i})-f(p_{i+1})\\le f(q_{i})-f(q_{i+1})\\ \\ \\mbox{for cases 1)and 2)}, \\\\-f(p_{i})+f(p_{i+1})\\le -f(q_{i})+f(q_{i+1})\\ \\ \\mbox{for cases 3) and 4)}\\end{array}(3.11)$ It is clear that after Step 1 the bolt $p$ will be replaced by the ordered set $q=\\left\\lbrace q_{1},q_{2},\\cdots ,q_{2n}\\right\\rbrace $ .", "We do not say a bolt but an ordered set because of a possibility of coincidence of some successive points $q_{i}, q_{i+1}$ (this, for example, may happen if the 1-st case takes place for the units $p_{i-1}p_{i}$ and $p_{i+1}p_{i+2}$ ).", "Let us exclude simultaneously successive and coincident points from $q$ .", "Then we obtain some closed bolt, which we denote by $q^{\\prime }=\\left\\lbrace q_{1}^{\\prime },q_{2}^{\\prime },\\cdots ,q_{2m}^{\\prime }\\right\\rbrace $ .", "It is not difficult to understand that all points of the bolt $q^{\\prime }$ are located on straight lines $x=a_{1},~x=a_{2},~x=a_{3}$ .", "From inequalities (3.11) and the fact that $2m\\le 2n,$ we deduce that $l(f,p)\\le l(f,q^{\\prime }).", "(3.12)$ Step 2.", "Consider sequentially units $q_{i}^{\\prime }q_{i+1}^{\\prime }\\;\\left( i=\\overline{1,2m},q_{2m+1}^{\\prime }=q_{1}^{\\prime }\\right) $ with the vertices $q_{i}^{\\prime }=\\left( x_{i}^{\\prime },y_{i}^{\\prime }\\right),~\\ q_{i+1}^{\\prime }\\left( x_{i+1}^{\\prime },y_{i+1}^{\\prime }\\right) $ having equal ordinates: $y_{i}^{\\prime }=y_{i+1}^{\\prime }$ .", "The following four cases are possible.", "1) $q_{i}^{\\prime }\\in q_{+}^{\\prime }$ and $x_{i+1}^{\\prime }>x_{i}^{\\prime }$ .", "In this case, replace the unit $q_{i}^{\\prime }q_{i+1}^{\\prime }$ by a new unit $p_{i}^{\\prime }p_{i+1}^{\\prime }$ with the vertices $p_{i}^{\\prime }=\\left( x_{i}^{\\prime },b_{1}\\right) ,~\\ p_{i+1}^{\\prime }=\\left(x_{i+1}^{\\prime },b_{1}\\right) $ .", "2) $q_{i}^{\\prime }\\in q_{+}^{\\prime }$ and $x_{i+1}^{\\prime }<x_{i}^{\\prime }$ .", "In this case, replace the unit $q_{i}^{\\prime }q_{i+1}^{\\prime }$ by a new unit $p_{i}^{\\prime }p_{i+1}^{\\prime }$ with the vertices $p_{i}^{\\prime }=\\left( x_{i}^{\\prime },b_{2}\\right) ,$ $\\ p_{i+1}^{\\prime }=\\left(x_{i+1}^{\\prime },b_{2}\\right) $ if $x_{i}^{\\prime }=a_{3}$ and with the vertices $p_{i}^{\\prime }=\\left( x_{i}^{\\prime },b_{3}\\right) ,$ $p_{i+1}^{\\prime }=\\left( x_{i+1}^{\\prime },b_{3}\\right) $ if $x_{i}^{\\prime }=a_{2}$ .", "3) $q_{i}^{\\prime }\\in q_{-}^{\\prime }$ and $x_{i+1}^{\\prime }<x_{i}^{\\prime }$ .", "In this case, replace $q_{i}^{\\prime }q_{i+1}^{\\prime }$ by a new unit $p_{i}^{\\prime }p_{i+1}^{\\prime }$ with the vertices $p_{i}^{\\prime }=\\left(x_{i}^{\\prime },b_{1}\\right) ,~\\ p_{i+1}^{\\prime }=\\left( x_{i+1}^{\\prime },b_{1}\\right)$ .", "4) $q_{i}^{\\prime }\\in q_{-}^{\\prime }$ and $x_{i+1}^{\\prime }>x_{i}^{\\prime }$ .", "In this case, replace $q_{i}^{\\prime }q_{i+1}^{\\prime }$ by a new unit $p_{i}^{\\prime }p_{i+1}^{\\prime }$ with the vertices $p_{i}^{\\prime }=\\left(x_{i}^{\\prime },b_{2}\\right) ,~\\ p_{i+1}^{\\prime }=\\left( x_{i+1}^{\\prime },b_{2}\\right)$ if $x_{i+1}^{\\prime }=a_{3}$ and with the vertices $p_{i}^{\\prime }=\\left( x_{i}^{\\prime },b_{3}\\right) ,~\\ p_{i+1}^{\\prime }=\\left( x_{i+1}^{\\prime },b_{3}\\right) $ if $x_{i+1}^{\\prime }=a_{2}$ .", "It is easy to see that after Step 2 the bolt $q^{\\prime }$ will be replaced by the bolt $p^{\\prime }=\\left\\lbrace p_{1}^{\\prime },p_{2}^{\\prime },\\cdots p_{2m}^{\\prime }\\right\\rbrace $ and $l(f,q^{\\prime })\\le l(f,p^{\\prime }).", "(3.13)$ From (3.12) and (3.13) we obtain that $l(f,p)\\le l(f,p^{\\prime }).", "(3.14)$ It is clear that each point of the set $p_{+}^{\\prime }$ coincides with one of the points $\\left( a_{1},b_{1}\\right) ,~\\left( a_{2},b_{3}\\right) ,$ $\\left( a_{3},b_{2}\\right) $ and each point of the set $p_{-}^{\\prime }$ coincides with one of the points $\\left( a_{1},b_{2}\\right) ,~\\left(a_{1},b_{3}\\right) ,$ $~\\left( a_{2},b_{1}\\right) ,~\\left(a_{2},b_{2}\\right) ,~\\left( a_{3},b_{1}\\right) .$ Denote by $m_{ij}$ the number of points of the bolt $p^{\\prime }$ coinciding with the point $\\left(a_{i},b_{j}\\right) ,~i,j=\\overline{1,3},~i+j\\ne 6$ .", "By (3.9), we can write that $l(f,p^{\\prime })=\\frac{1}{2m}\\sum \\limits _{\\begin{array}{c} i,j=\\overline{1,3} \\\\ i+j\\le 5\\end{array}}m_{ij}f_{ij}.", "(3.15)$ On the straight line $x=a_{i}~\\ $ or $\\ y=b_{i},~i=\\overline{1,3}$ , the number of points of the set $p_{+}^{\\prime }$ is equal to the number of points of the set $p_{-}^{\\prime }$ .", "Hence $m_{11}=m_{12}+m_{13}=m_{21}+m_{31};\\ m_{23}=m_{22}+m_{21}=m_{13};\\ m_{32}=m_{31}=m_{12}+m_{22}.$ From these equalities we deduce that $m_{11}=m_{12}+m_{21}+m_{22};\\ m_{13}=m_{21}+m_{22};\\ m_{23}=m_{21}+m_{22};\\ m_{31}=m_{12}+m_{22}.", "(3.16)$ Consequently, $2m=\\sum \\limits _{\\begin{array}{c} i,j=\\overline{1,3} \\\\ i+j\\le 5\\end{array}}m_{ij}=4m_{12}+4m_{21}+6m_{22}.", "(3.17)$ Considering (3.16) and (3.17) in (3.15) and taking (3.10) into account, we obtain that $l(f,p^{\\prime })=\\dfrac{4m_{12}\\left|l(f,r_{2})\\right|+4m_{21}\\left|l(f,r_{1})\\right|+6m_{22}\\left|l(f,h)\\right|}{4m_{12}+4m_{21}+6m_{22}}$ $\\le \\max \\left\\lbrace \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|,\\left|l(f,h)\\right|\\right\\rbrace .$ Therefore, due to (3.14), $l(f,p)\\le \\max \\left\\lbrace \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|,\\left|l(f,h)\\right|\\right\\rbrace .", "(3.18)$ Note that in the beginning of the proof the bolt $p$ has been chosen so that $l(f,p)\\ge 0$ .", "Let now $p=\\left\\lbrace p_{1},p_{2},\\cdots p_{2n}\\right\\rbrace $ be any closed bolt such that $l(f,p)\\le 0$ .", "Since $l(f,p^{\\prime \\prime })=$ $-l(f,p)\\ge 0$ for the bolt $p^{\\prime \\prime }=\\left\\lbrace p_{2},p_{3},\\cdots ,p_{2n},p_{1}\\right\\rbrace $ ,we obtain from (3.18) that $-l(f,p)\\le \\max \\left\\lbrace \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|,\\left|l(f,h)\\right|\\right\\rbrace .", "(3.19)$ From (3.18) and (3.19) we deduce on the strength of arbitrariness of $p$ that $\\sup \\limits _{p\\subset H}\\left\\lbrace \\left|l(f,p)\\right|\\right\\rbrace =\\max \\left\\lbrace \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|,\\left|l(f,h)\\right|\\right\\rbrace ,(3.20)$ where the $sup$ is taken over all closed bolts of the hexagon $H$ .", "The hexagon $H$ satisfies the conditions of Theorem 1.10 on the existence of a best approximation.", "By [89] (see Section 3.3), we obtain that $E(f,H)=\\sup \\limits _{p\\subset H}\\left\\lbrace \\left|l(f,p)\\right|\\right\\rbrace .", "(3.21)$ From (3.20) and (3.21) we finally conclude that $E(f,H)=\\max \\left\\lbrace \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|,\\left|l(f,h)\\right|\\right\\rbrace .$ Corollary 3.2.", "Let a function $f(x,y)$ have the continuous nonnegative derivative $\\dfrac{\\partial ^{2}f}{\\partial x\\partial y}$ on $H$ .", "Then the formula (3.8) is valid.", "The proof is very simple and can be obtained by integrating the inequality $\\dfrac{\\partial ^{2}f}{\\partial x\\partial y}\\!\\ge \\!0$ over an arbitrary rectangle $\\left[ x_{1},x_{2}\\right] \\times \\left[ y_{1},y_{2}\\right] \\subset H$ and applying Theorem 3.5.", "The method used in the proof of Theorem 3.5 can be generalized to obtain similar results for stairlike polygons.", "For example, let $S$ be a closed polygon of the following form $S=\\bigcup \\limits _{i=1}^{N-1}P_{i},$ where $N\\ge 2,$ $P_{i}=\\left[ a_{i},a_{i+1}\\right] \\times \\left[b_{1},b_{N+1-i}\\right] ,$ $i=\\overline{1,N-1},$ $a_{1}<a_{2}<\\dots <a_{N},$ $b_{1}<b_{2}<\\dots <b_{N}$ .", "Such polygons will be called stairlike polygons (see [59]).", "A closed $2m$ -gon $F$ with sides parallel to the coordinate axes is called a maximal $2m$ -gon of the polygon $S$ if $F\\subset S$ and there is no another $2m$ -gon $F^{\\prime }$ such that $F\\subset F^{\\prime }\\subset S$ .", "Clearly, if $F$ is a maximal $2m$ -gon of the polygon $S$ , then $m\\le N.$ A closed bolt formed by the vertices of a maximal polygon $F$ is called a maximal bolt of $S$ .", "By $S^{B}$ denote the set of all maximal bolts of the stairlike polygon $S.$ Theorem 3.6.", "Let $S$ be a stairlike polygon.", "The approximation error of a function $f \\in M(S)$ can be computed by the formula $E\\left( f,S\\right) =\\max \\left\\lbrace \\left|r(f,h)\\right|,\\;h\\in S^{B}\\right\\rbrace .$ For the proof of this theorem see [59].", "The main idea in the proof of Theorem 3.5 can be successfully used in obtaining formulas of type (3.8) for functions $f(x,y)$ defined on another simple polygons.", "The following two theorems include cases of some octagons and can be proved in a similar way.", "Theorem 3.7.", "Let $a_{1}<a_{2}<a_{3}<a_{4},$ $b_{1}<b_{2}<b_{3}$ and $Q$ be an octagon of the following form $Q=\\bigcup \\limits _{i=1}^{4}R_{i},\\ \\ \\ where$ $R_{1}=\\left[ a_{1},a_{2}\\right] \\times \\left[ b_{1},b_{2}\\right] ,R_{2}=\\left[ a_{2},a_{3}\\right] \\times \\left[ b_{1},b_{2}\\right] ,R_{3}=\\left[a_{3},a_{4}\\right] \\times \\left[ b_{1},b_{2}\\right] ,R_{4}=\\left[ a_{2},a_{3}\\right] \\times \\left[ b_{2},b_{3}\\right] $ .", "Let $f\\in M(Q)$ .", "Then the following formula holds $E(f,Q)=\\max \\left\\lbrace \\left| l(f,q)\\right| ,\\left| l(f,r_{123} )\\right|,\\left| l(f,r_{124} )\\right| ,\\left| l(f,r_{234} )\\right| ,\\left| l(f,r_{24})\\right| \\right\\rbrace ,$ where $q,$ $r_{123},$ $r_{124},$ $r_{234},$ $r_{24} $ are closed bolts formed by the vertices of the polygons $Q,$ $R_{1} \\cup R_{2}\\cup R_{3},R_{1}\\cup R_{2}\\cup R_{4} ,R_{2}\\cup R_{3}\\cup R_{4} $ and $R_{2}\\cup R_{4}$ , respectively.", "Theorem 3.8.", "Let $a_{1}<a_{2}<a_{3}<a_{4},\\ b_{1}<b_{2}<b_{3}$ and $Q$ be an octagon of the following form $Q=\\bigcup _{i=1}^{3}R_{i},$ where $R_{1}=\\left[ a_{1},a_{4}\\right] \\times \\left[ b_{1},b_{2}\\right],R_{2}=\\left[ a_{1},a_{2}\\right] \\times \\left[ b_{2},b_{3}\\right] ,R_{3}=\\left[ a_{3},a_{4}\\right] \\times \\left[ b_{2},b_{3}\\right] $ .", "Let $f\\in M(Q)$ .", "Then $E(f,Q)=\\max \\left\\lbrace \\left| l(f,r)\\right| ,\\left| l(f,r_{12} )\\right| ,\\left|l(f,r_{13} )\\right| \\right\\rbrace ,$ where $r,r_{12} ,r_{13} $ are closed bolts formed by the vertices of the polygons $R=\\left[ a_{1} ,a_{4} \\right] \\times \\left[ b_{1} ,b_{3} \\right],$ $R_{1}\\cup R_{2} ,R_{1}\\cup R_{3}$ , respectively.", "Although the closed bolts maximization process can be applied to bolts of an arbitrary polygon, some combinatorial difficulties arise when grouping values at points of maximized bolts (bolts obtained after the maximization process, see (3.15)-(3.18)).", "While we do not know a complete answer to this problem, we can describe points of a polygon $F$ with which points of maximized bolts coincide and state a conjecture concerning the approximation error.", "Let $F=A_{1}A_{2}...A_{2n}$ be any polygon with sides parallel to the coordinate axes.", "The vertices $A_{1},$ $A_{2},$ $...,$ $A_{2n}$ in the given order form a closed bolt, which we denote by $r_{F}$ .", "By $\\left[ r_{F}\\right]$ denote the length of $r_{F}$ .", "In our case, $\\left[ r_{F}\\right] =2n$ .", "Definition 3.2.", "Let $F$ and $S$ be polygons with sides parallel to the coordinate axes.", "We say that the closed bolt $r_{F}$ is an $e $ -bolt (extended bolt) of $S$ if $r_{F}\\subset S$ and there does not exist any polygon $F^{^{\\prime }}$ such that $F\\subset F^{^{\\prime }},\\ \\ r_{F^{^{\\prime }}}\\subset S,\\ \\ \\left[ r_{F^{^{\\prime }}}\\right] \\le \\left[r_{F}\\right] .$ For example, in Theorem 3.8 the octagon $Q$ has 3 $e$ -bolts.", "They are $r,r_{12}$ and $r_{13}$ .", "In Theorem 3.7, the octagon $Q$ has 5 $e$ -bolts, which are $q,r_{123},r_{124},r_{234}$ and $r_{24}$ .", "The polygon $S_{2n}=\\bigcup \\limits _{i=1}^{n-1}R_{i}$ , where $R_{i}=\\left[ a_{i},a_{i+1}\\right] \\times \\left[ b_{1},b_{n+1-i}\\right] ,i=\\overline{1,n-1},a_{1}<a_{2}<...<a_{n},b_{1}<b_{2}<...<b_{n}$ has exactly $2^{n-1}-1$ $e$ -bolts.", "It is not difficult to observe that the set of points of a closed bolt obtained after the maximization process is a subset of the set of points of all $e$ -bolts.", "This condition and Theorems 2.5-2.8 justify the statement of the following conjecture: Let $S$ be any polygon with sides parallel to the coordinate axes and $f \\in M(S)$ .", "Then $E(f,S)=\\max _{h\\in S^{E}}\\left\\lbrace \\left|l(f,h)\\right|\\right\\rbrace ,$ where $S^{E}$ is a set of all $e$ -bolts of the polygon $S$ .", "Theorem 3.5 allows us to consider classes wider than $M(H)$ and establish sharp estimates for the approximation error.", "Theorem 3.9.", "Let $H$ be a hexagon and (3.7) be its representation.", "The following sharp estimates are valid for a function $f(x,y)$ having the continuous derivative $\\dfrac{\\partial ^{2}f}{\\partial x\\partial y}$ on $H$ : $A\\le E(f,H)\\le BC+\\frac{3}{2}\\left( B\\left|l(g,h)\\right|-\\left|l(f,h)\\right|\\right) ,(3.22)$ where $B=\\max _{(x,y)\\in H}\\left|\\frac{\\partial ^{2}f(x,y)}{\\partial x\\partial y}\\right|,\\ \\ \\ g=g(x,y)=x\\cdot y,$ $A=\\max \\left\\lbrace \\left|l(f,h)\\right|,\\ \\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|\\right\\rbrace ,\\ C=\\max \\left\\lbrace \\left|l(g,h)\\right|,\\left|l(g,r_{1})\\right|,\\ \\left|l(g,r_{2})\\right|\\right\\rbrace ,$ where $h,r_{1},r_{2}$ are closed bolts formed by vertices of the polygons $H,R_{1}$ and $R_{2}$ , respectively.", "Remark 3.4.", "Inequalities similar to (3.22) were established in Babaev [11] for the approximation of a function $f(x)=f(x_{1},...,x_{n})$ , defined on a parallelepiped with sides parallel to the coordinate axes, by sums $\\sum \\limits _{i=1}^{n}\\varphi _{i}(x\\backslash x_{i})$ .", "For the approximation of bivariate functions, Babaev's result contains only rectangular case.", "Remark 3.5.", "Estimates (3.22) are easily calculable in contrast to those established in [13] for continuous functions defined on certain domains, which are different from polygons.", "To prove Theorem 3.9 we need the following lemmas.", "Lemma 3.3.", "Let $X$ be a normed space, $F$ be a subspace of $X$ .", "The following inequality is valid for an element $x=x_{1}+x_{2}$ from $X$ : $\\left|E(x_{1})-E(x_{2})\\right|\\le E(x)\\le E(x_{1})+E(x_{2}),$ where $E(x)=E(x,F)=\\inf _{y\\in F}\\left\\Vert x-y\\right\\Vert .$ Lemma 3.4.", "If $f\\in M(H)$ , then $\\left|l(f,r_{i})\\right|\\le \\frac{3}{2}\\left|l(f,h)\\right|,i=1,2.$ Lemma 3.3 is obvious.", "To prove Lemma 3.4, note that for any $f\\in M(H)$ $6\\left|l(f,h)\\right|=4\\left|l(f,r_{i})\\right|+4\\left|l(f,r_{3})\\right|,\\ \\ i=1,2,$ where $r_{3}$ is a closed bolt formed by the vertices of the rectangle $R_{3}=H\\backslash R_{i}.$ Now let us prove Theorem 3.9.", "It is not difficult to verify that if $\\frac{\\partial ^{2}u}{\\partial x\\partial y}\\ge 0$ on $H$ for some $u(x,y),$ $\\frac{\\partial ^{2}u(x,y)}{\\partial x\\partial y}\\in C(H)$ , then $u\\in M(H)$ (see the proof of Corollary 3.2).", "Set $f_{1}=f+Bg$ .", "Since $\\frac{\\partial ^{2}f_{1}}{\\partial x\\partial y}\\ge 0$ on $H$ , $f_{1}\\in M(H)$ .", "By Lemma 3.4, $\\left|l(f_{1},r_{i})\\right|\\le \\frac{3}{2}\\left|l(f_{1},h)\\right|,i=1,2.", "(3.23)$ Theorem 3.5 implies that $E(f_{1},H)=\\max \\left\\lbrace \\left|l(f_{1},h)\\right|,\\left|l(f_{1},r_{1})\\right|,\\left|l\\left( f_{1},r_{2}\\right) \\right|\\right\\rbrace .", "(3.24)$ We deduce from (3.23) and (3.24) that $E(f_{1},H)\\le \\frac{3}{2}\\left|l(f_{1},h)\\right|.$ First, let the closed bolt $h$ start at the point $(a_{1},b_{1})$ .", "Then it is clear that $E(f_{1},H)\\le \\frac{3}{2}l(f_{1},h).", "(3.25)$ By Lemma 3.3, $E(f,H)-E(Bg,H)\\le E(f_{1},H).", "(3.26)$ Inequalities (3.25) and (3.26) yield $E(f,H)\\le BE(g,H)+\\frac{3}{2}l(f_{1},h).", "(3.27)$ Since the functional $l(f,h)$ is linear, $l(f_{1},h)=l(f,h)+Bl(g,h).$ Considering this expression of $l(f_{1},h)$ in (3.27), we obtain that $E(f,H)\\le BE(g,H)+\\frac{3}{2}Bl(g,h)+\\frac{3}{2}l(f,h).", "(3.28)$ Now consider the function $f_{2}=Bg-f$ .", "Obviously, $\\frac{\\partial ^{2}f_{2}}{\\partial x\\partial y}\\ge 0$ on $H$ .", "It can be shown, in the same way as (3.28) has been obtained, that $E(f,H)\\le BE(g,H)+\\frac{3}{2}Bl(g,h)-\\frac{3}{2}l(f,h).", "(3.29)$ From (3.28) and (3.29) it follows that $E(f,H)\\le BE(g,H)+\\frac{3}{2}Bl(g,h)-\\frac{3}{2}\\left|l(f,h)\\right|.", "(3.30)$ Since $g\\in M(H)$ and $h$ starts at the point $(a_{1},b_{1}),$ we have $l(g,h)\\ge 0$ .", "Let now $h$ start at a point such that $l(u,h)\\le 0$ for any $u\\in M(H)$ .", "Then in a similar way as above we can prove that $E(f,H)\\le BE(g,H)-\\frac{3}{2}Bl(g,h)-\\frac{3}{2}\\left|l(f,h)\\right|,(3.31)$ where $l(g,h)\\le 0$ .", "From (3.30), (3.31) and the fact that $E(g,H)=C$ (in view of Theorem 3.5), it follows that $E(f,H)\\le BC+\\frac{3}{2}\\left( B\\left|l(g,h)\\right|-\\left|l(f,h)\\right|\\right) .$ The upper bound in (3.22) has been established.", "Note that it is attained by $f=g=xy$ .", "The proof of the lower bound in (3.22) is simple.", "One of the obvious properties of the functional $l(f,p)$ is that $\\left|l(f,p)\\right|\\le E(f,H)$ for any continuous function $f$ on $H$ and a closed bolt $p$ .", "Hence, $A=\\max \\left\\lbrace \\left|l(f,h)\\right|,\\left|l(f,r_{1})\\right|,\\left|l(f,r_{2})\\right|\\right\\rbrace \\le E(f,H).$ Note that by Theorem 3.5 the lower bound in (3.22) is attained by an arbitrary function from $M(H)$ .", "Remark 3.6.", "Using Theorems 2.7 and 2.8 one can obtain sharp estimates of type (3.22) for bivariate functions defined on the corresponding simple polygons with sides parallel to the coordinate axes.", "Let $X_{1},...,X_{n}$ be compact spaces and $X=X_{1}\\times \\cdots \\times X_{n}.$ Consider the approximation of a function $f\\in C(X)$ by sums $g_{1}(x_{1})+\\cdots +g_{n}(x_{n}),$ where $g_{i}\\in C(X_{i}),$ $i=1,...,n.$ In [48], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$ , called “projection cycles\".", "However, his proof had a gap, which was pointed out later by Marshall and O'Farrell [123].", "But the question if the formula was correct, remained open.", "The purpose of this section is to prove that Golomb's formula is valid, and moreover it holds in a stronger form.", "Let $X_{i},i=1,...,n,$ be compact Hausdorff spaces.", "Consider the approximation to a continuous function $f$ , defined on $X=X_{1}\\times \\cdots \\times X_{n}$ , from the manifold $M=\\left\\lbrace \\sum _{i=1}^{n}g_{i}(x_{i}):g_{i}\\in C(X_{i}),~~i=1,...,n\\right\\rbrace .$ The approximation error is defined as the distance from $f$ to $M$ : $E(f)\\overset{def}{=}dist(f,M)=\\underset{g\\in M}{\\inf }\\left\\Vert f-g\\right\\Vert _{C(X)}.$ The well-known duality relation says that $E(f)=\\underset{\\left\\Vert \\mu \\right\\Vert \\le 1}{\\underset{\\mu \\in M^{\\bot }}{\\sup }}\\left|\\int \\limits _{X}fd\\mu \\right|,(3.32)$ where $M^{\\bot }$ is the space of regular Borel measures annihilating all functions in $M$ and $\\left\\Vert \\mu \\right\\Vert $ stands for the total variation of a measure $\\mu $ .", "It should be noted that the $\\sup $ in (3.32) is attained by some measure $\\mu ^{\\ast }$ with total variation $\\left\\Vert \\mu ^{\\ast }\\right\\Vert =1.$ We are interested in the problem: is it possible to replace in (3.32) the class $M^{\\bot }$ by some subclass of it consisting of measures of simple structure?", "For the case $n=2,$ this problem was first considered by Diliberto and Straus [36].", "They showed that the measures generated by closed bolts are sufficient for the equality (3.32).", "In case of general topological spaces, a lightning bolt is defined similarly to the case $\\mathbb {R}^2$ .", "Let $X=X_{1}\\times X_{2}$ and $\\pi _{i}$ be the projections of $X$ onto $X_{i},$ $i=1,2.$ A lightning bolt (or, simply, a bolt) is a finite ordered set $\\lbrace a_{1},...,a_{k}\\rbrace $ contained in $X$ , such that $a_{i}\\ne a_{i+1}$ , for $i=1,2,...,k-1$ , and either $\\pi _{1}(a_{1})=\\pi _{1}(a_{2}),$ $\\pi _{2}(a_{2})=\\pi _{2}(a_{3})$ , $\\pi _{1}(a_{3})=\\pi _{1}(a_{4}),...,$ or $\\pi _{2}(a_{1})=\\pi _{2}(a_{2}),$ $\\pi _{1}(a_{2})=\\pi _{1}(a_{3})$ , $\\pi _{2}(a_{3})=\\pi _{2}(a_{4}),...$ A bolt $\\lbrace a_{1},...,a_{k}\\rbrace $ is said to be closed if $k$ is an even number and the set $\\lbrace a_{2},...,a_{k},a_{1}\\rbrace $ is also a bolt.", "Let $l=\\lbrace a_{1},...,a_{2k}\\rbrace $ be a closed bolt.", "Consider a measure $\\mu _{l}$ having atoms $\\pm \\frac{1}{2k}$ with alternating signs at the vertices of $l$ .", "That is, $\\mu _{l}=\\frac{1}{2k}\\sum _{i=1}^{2k}(-1)^{i-1}\\delta _{a_{i}}\\text{ \\ or \\ }\\mu _{l}=\\frac{1}{2k}\\sum _{i=1}^{2k}(-1)^{i}\\delta _{a_{i}},$ where $\\delta _{a_{i}}$ is a point mass at $a_{i}.$ It is clear that $\\mu _{l}\\in M^{\\bot }$ and $\\left\\Vert \\mu _{l}\\right\\Vert \\le 1$ .", "$\\left\\Vert \\mu _{l}\\right\\Vert =1$ if and only if the set of vertices of the bolt $l$ having even indices does not intersect with that having odd indices.", "The following duality relation was first established by Diliberto and Straus [36] $E(f)=\\underset{l\\subset X}{\\sup }\\left|\\int \\limits _{X}fd\\mu _{l}\\right|,(3.33)$ where $X=X_{1}\\times X_{2}$ and the $\\sup $ is taken over all closed bolts of $X$ .", "In fact, Diliberto and Straus obtained the formula (3.33) for the case when $X$ is a rectangle in $\\mathbb {R}^{2}$ with sides parallel to the coordinate axis.", "The same result was independently proved by Smolyak (see [130]).", "Yet another proof of (3.33), in the case when $X$ is a Cartesian product of two compact Hausdorff spaces, was given by Light and Cheney [110].", "For $X$ 's other than a rectangle in $\\mathbb {R}^{2}$ , the theorem under some additional assumptions appeared in the works [56], [89], [123].", "But we shall not discuss these works here.", "Golomb's paper [48] made a start to a systematic study of approximation of multivariate functions by various compositions, including sums of univariate functions.", "Golomb generalized the notion of a closed bolt to the $n$ -dimensional case and obtained the analogue of formula (3.33) for the error of approximation from the manifold $M$ .", "The objects introduced in [48] were called projection cycles and they are defined as sets of the form $p=\\lbrace b_{1},...,b_{k};~c_{1},...,c_{k}\\rbrace \\subset X,(3.34)$ with the property that $b_{i}\\ne c_{j}$ , $i,j=1,...,k$ and for all $\\nu =1,...,n,$ the group of the $\\nu $ -th coordinates of $c_{1},...,c_{k}$ is a permutation of that of the $\\nu $ -th coordinates of $b_{1},...,b_{k}.$ Some points in the $b$ -part $\\left( b_{1},...,b_{k}\\right) $ or $c$ -part $\\left(c_{1},...,c_{k}\\right) $ of $p$ may coincide.", "The measure associated with $p$ is $\\mu _{p}=\\frac{1}{2k}\\left( \\sum _{i=1}^{k}\\delta _{b_{i}}-\\sum _{i=1}^{k}\\delta _{c_{i}}\\right).$ It is clear that $\\mu _{p}\\in M^{\\bot }$ and $\\left\\Vert \\mu _{p}\\right\\Vert =1.$ Besides, if $n=2,$ then a projection cycle is the union of closed bolts after some suitable permutation of its points.", "Golomb's result states that $E(f)=\\underset{p\\subset X}{\\sup }\\left|\\int \\limits _{X}fd\\mu _{p}\\right|,(3.35)$ where $X=X_{1}\\times \\cdots \\times X_{n}$ and the $\\sup $ is taken over all projection cycles of $X$ .", "It can be proved that in the case $n=2,$ the formulas (3.33) and (3.35) are equivalent.", "Unfortunately, the proof of (3.35) had a gap, which was pointed out many years later by Marshall and O'Farrell [123].", "But the question if the formula (3.35) was correct, remained unsolved (see also the monograph by Khavinson [92]).", "Note that Golomb's result was used and cited in the literature, for example, in works [88], [144].", "In the following subsection, we will construct families of normalized measures (that is, measures with the total variation equal to 1) on projection cycles.", "Each measure $\\mu _{p}$ defined above will be a member of some family.", "We will also consider minimal projection cycles and measures constructed on them.", "By properties of these measures, we show that Golomb's formula (3.35) is valid in a stronger form.", "Let us give an equivalent definition of a projection cycle.", "This will be useful in constructing of certain measures having simple structure and capability of approximating arbitrary measures in $M^{\\bot }$ .", "In the sequel, $\\chi _{a}$ will denote the characteristic function of a single point set $\\lbrace a\\rbrace \\subset \\mathbb {R}$ .", "Definition 3.3.", "Let $X=X_{1}\\times \\cdots \\times X_{n}$ and $\\pi _{i}$ be the projections of $X$ onto the sets $X_{i},$ $i=1,...,n.$ We say that a set $p=\\lbrace x_{1},...,x_{m}\\rbrace \\subset X$ is a projection cycle if there exists a vector $\\lambda =(\\lambda _{1},...,\\lambda _{m})$ with nonzero real coordinates such that $\\sum _{j=1}^{m}\\lambda _{j}\\chi _{\\pi _{i}(x_{j})}=0,\\text{ \\ }i=1,...,n.(3.36)$ Let us give some explanatory remarks concerning Definition 3.3.", "Fix the subscript $i.$ Let the set $\\lbrace \\pi _{i}(x_{j})$ , $j=1,...,m\\rbrace $ have $s_{i}$ different values, which we denote by $\\gamma _{1}^{i},\\gamma _{2}^{i},...,\\gamma _{s_{i}}^{i}.$ Then (3.36) implies that $\\sum _{j}\\lambda _{j}=0,$ where the sum is taken over all $j$ such that $\\pi _{i}(x_{j})=\\gamma _{k}^{i},$ $k=1,...,s_{i}.$ Thus for fixed $i$ , we have $s_{i}$ homogeneous linear equations in $\\lambda _{1},...,\\lambda _{m}.$ The coefficients of these equations are the integers 0 and $1.$ By varying $i$ , we obtain $s=\\sum _{i=1}^{n}s_{i}$ such equations.", "Hence (3.36), in its expanded form, stands for the system of these equations.", "One can observe that if this system has a solution $(\\lambda _{1},...,\\lambda _{m})$ with nonzero real components $\\lambda _{i},$ then it also has a solution $(n_{1},...,n_{m})$ with nonzero integer components $n_{i},$ $i=1,...,m.$ This means that in Definition 3.3, we can replace the vector $\\lambda $ by the vector $n=(n_{1},...,n_{m})\\,$ , where $n_{i}\\in \\mathbb {Z}\\backslash \\lbrace 0\\rbrace ,$ $i=1,...,m.$ Thus, Definition 3.3 is equivalent to the following definition.", "Definition 3.4.", "A set $p=\\lbrace x_{1},...,x_{m}\\rbrace \\subset X$ is called a projection cycle if there exist nonzero integers $n_{1},...,n_{m}$ such that $\\sum _{j=1}^{m}n_{j}\\chi _{\\pi _{i}(x_{j})}=0,\\text{ \\ }i=1,...,n.(3.37)$ Lemma 3.5.", "Definition 3.4 is equivalent to Golomb's definition of a projection cycle.", "Let $p=\\lbrace x_{1},...,x_{m}\\rbrace $ be a projection cycle with respect to Definition 3.4.", "By $b$ and $c$ denote the set of all points $x_{i}$ such that the integers $n_{i}$ associated with them in (3.37) are positive and negative correspondingly.", "Write out each point $x_{i}$ $n_{i}$ times if $n_{i}>0$ and $-n_{i}$ times if $n_{i}<0.$ Then the set $\\lbrace b;c\\rbrace $ is a projection cycle with respect to Golomb's definition.", "The inverse is also true.", "Let a set $p_{1}=\\lbrace b_{1},...,b_{k};~c_{1},...,c_{k}\\rbrace $ be a projection cycle with respect to Golomb's definition.", "Here, some points $b_{i}$ or $c_{i}$ may be repeated.", "Let $p=\\lbrace x_{1},...,x_{m}\\rbrace $ stand for the set $p_{1}$ , but with no repetition of its points.", "Let $n_{i}$ show how many times $x_{i}$ appear in $p_{1}.$ We take $n_{i}$ positive if $x_{i}$ appears in the $b$ -part of $p_{1}$ and negative if it appears in the $c$ -part of $p_{1}.$ Clearly, the set $\\lbrace x_{1},...,x_{m}\\rbrace $ is a projection cycle with respect to Definition 3.4, since the integers $n_{i},$ $i=1,...,m, $ satisfy (3.37).", "In the sequel, we will use Definition 3.3.", "A pair $\\left\\langle p,\\lambda \\right\\rangle ,$ where $p$ is a projection cycle in $X$ and $\\lambda $ is a vector associated with $p$ by (3.36), will be called a “projection cycle-vector pair\" of $X.$ To each such pair $\\left\\langle p,\\lambda \\right\\rangle $ with $p=\\lbrace x_{1},...,x_{m}\\rbrace $ and $\\lambda =(\\lambda _{1},...,\\lambda _{m})$ , we correspond the measure $\\mu _{p,\\lambda }=\\frac{1}{\\sum _{j=1}^{m}\\left|\\lambda _{j}\\right|}\\sum _{j=1}^{m}\\lambda _{j}\\delta _{x_{j}}.", "(3.38)$ Clearly, $\\mu _{p,\\lambda }\\in M^{\\bot }$ and $\\left\\Vert \\mu _{p,\\lambda }\\right\\Vert =1$ .", "We will also deal with measures supported on some certain subsets of projection cycles called minimal projection cycles.", "A projection cycle is said to be minimal if it does not contain any projection cycle as its proper subset.", "For example, the set $p=\\lbrace (0,0,0),~(0,0,1),~(0,1,0),~(1,0,0),~(1,1,1)\\rbrace $ is a minimal projection cycle in $\\mathbb {R}^{3},$ since the vector $\\lambda =(2,-1,-1,-1,1)$ satisfies Eq.", "(3.36) and there is no such vector for any other subset of $p$ .", "Adding one point $(0,1,1)$ from the right to $p$ , we will also have a projection cycle, but not minimal.", "Note that in this case, $\\lambda $ can be taken as $(3,-1,-1,-2,2,-1).$ Remark 3.7.", "A minimal projection cycle under the name of a loop was introduced and used in the works of Klopotowski, Nadkarni, Rao [94], [95].", "To prove our main result we need some auxiliary facts.", "Lemma 3.6.", "(1) The vector $\\lambda =(\\lambda _{1},...,\\lambda _{m})$ associated with a minimal projection cycle $p=(x_{1},...,x_{m})$ is unique up to multiplication by a constant.", "(2) If in (1), $\\sum _{j=1}^{m}\\left|\\lambda _{j}\\right|=1, $ then all the numbers $\\lambda _{j}$ , $j=1,...,m,$ are rational.", "Let $\\lambda ^{1}=(\\lambda _{1}^{1},...,\\lambda _{m}^{1})$ and $\\lambda ^{2}=(\\lambda _{1}^{2},...,\\lambda _{m}^{2})$ be any two vectors associated with $p.$ That is, $\\sum _{j=1}^{m}\\lambda _{j}^{1}\\chi _{\\pi _{i}(x_{j})}=0\\text{ and }\\sum _{j=1}^{m}\\lambda _{j}^{2}\\chi _{\\pi _{i}(x_{j})}=0,\\text{ \\ }i=1,...,n.$ After multiplying the second equality by $c=\\frac{\\lambda _{1}^{1}}{\\lambda _{1}^{2}}$ and subtracting from the first, we obtain that $\\sum _{j=2}^{m}(\\lambda _{j}^{1}-c\\lambda _{j}^{2})\\chi _{\\pi _{i}(x_{j})}=0\\text{, \\ }i=1,...,n.$ Now since the cycle $p$ is minimal, $\\lambda _{j}^{1}=c\\lambda _{j}^{2},$ for all $j=1,...,m.$ The second part of the lemma is a consequence of the first part.", "Indeed, let $n=(n_{1},...,n_{m})$ be a vector with the nonzero integer coordinates associated with $p.$ Then the vector $\\lambda ^{^{\\prime }}=(\\lambda _{1}^{^{\\prime }},...,\\lambda _{m}^{^{\\prime }}),$ where $\\lambda _{j}^{^{\\prime }}=\\frac{n_{j}}{\\sum _{j=1}^{m}\\left|n_{j}\\right|},$ $j=1,...,m,$ is also associated with $p.$ All coordinates of $\\lambda ^{^{\\prime }}$ are rational and therefore by the first part of the lemma, it is the unique vector satisfying $\\sum _{j=1}^{m}\\left|\\lambda _{j}^{^{\\prime }}\\right|=1.$ By this lemma, a minimal projection cycle $p$ uniquely (up to a sign) defines the measure $~\\mu _{p}=\\sum _{j=1}^{m}\\lambda _{j}\\delta _{x_{j}},\\text{ \\ }\\sum _{j=1}^{m}\\left|\\lambda _{j}\\right|=1.$ Lemma 3.7.", "Let $\\mu $ be a normalized orthogonal measure on a projection cycle $l\\subset X$ .", "Then it is a convex combination of normalized orthogonal measures on minimal projection cycles of $l$ .", "That is, $\\mu =\\sum _{i=1}^{s}t_{i}\\mu _{l_{i}},\\text{ }\\sum _{i=1}^{s}t_{i}=1,~t_{i}>0,$ where $l_{i},$ $i=1,...,s,$ are minimal projection cycles in $l.$ This lemma follows from the result of Navada (see [129]): Let $S\\subset X_{1}\\times \\cdots \\times X_{n}$ be a finite set.", "Then any extreme point of the convex set of measures $\\mu $ on $S$ , $\\mu \\in M^{\\bot }$ , $\\left\\Vert \\mu \\right\\Vert \\le 1$ , has its support on a minimal projection cycle contained in $S$ .", "Remark 3.8.", "In the case $n=2$ , Lemma 3.7 was proved by Medvedev (see [92]).", "Lemma 3.8 (see [92]).", "Let $X=X_{1}\\times \\cdots \\times X_{n}$ and $\\pi _{i}$ be the projections of $X$ onto the sets $X_{i},$ $i=1,...,n.$ In order that a measure $\\mu \\in C(X)^{\\ast }$ be orthogonal to the subspace $M$ , it is necessary and sufficient that $\\mu \\circ \\pi _{i}^{-1}=0,\\text{ }i=1,...,n.$ Lemma 3.9 (see [92]).", "Let $\\mu \\in M^{\\bot }$ and $\\left\\Vert \\mu \\right\\Vert =1.$ Then there exist a net of measures $\\lbrace \\mu _{\\alpha }\\rbrace \\subset M^{\\bot }$ weak$^{\\text{*}}$ converging in $C(X)^{\\ast }$ to $\\mu $ and satisfying the following properties: 1) $\\left\\Vert \\mu _{\\alpha }\\right\\Vert =1;$ 2) The closed support of each $\\mu _{\\alpha }$ is a finite set.", "Our main result is the following theorem.", "Theorem 3.10.", "The error of approximation from the manifold $M$ obeys the equality $E(f)=\\underset{l\\subset X}{\\sup }\\left|\\int \\limits _{X}fd\\mu _{l}\\right|,$ where the $\\sup $ is taken over all minimal projection cycles of $X.$ Let $\\overset{\\sim }{\\mu }$ be a measure with finite support $\\lbrace x_{1},...,x_{m}\\rbrace $ and orthogonal to the space $M.$ Put $\\lambda _{j}=\\overset{\\sim }{\\mu }(x_{j}),$ $j=1,...m.$ By Lemma 3.8, $\\overset{\\sim }{\\mu }(\\pi _{i}^{-1}(\\pi _{i}(x_{j})))=0,$ for all $i=1,...,n,$ $j=1,...,m.$ Fix the indices $i$ and $j.$ Then we have the equation $\\sum _{k}\\lambda _{k}=0,$ where the sum is taken over all indices $k$ such that $\\pi _{i}(x_{k})=\\pi _{i}(x_{j}).$ Varying $i$ and $j,$ we obtain a system of such equations, which concisely can be written as $\\sum _{k=1}^{m}\\lambda _{k}\\chi _{\\pi _{i}(x_{k})}=0,\\text{ \\ }i=1,...,n.$ This means that the finite support of $\\overset{\\sim }{\\mu }$ forms a projection cycle.", "Therefore, a net of measures approximating the given measure $\\mu $ in Lemma 3.9 are all of the form (3.38).", "Let now $\\mu _{p,\\lambda }$ be any measure of the form (3.38).", "Since $\\mu _{p,\\lambda }\\in M^{\\bot }$ and $\\left\\Vert \\mu _{p,\\lambda }\\right\\Vert =1,$ we can write $\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|=\\left|\\int \\limits _{X}(f-g)d\\mu _{p,\\lambda }\\right|\\le \\left\\Vert f-g\\right\\Vert ,(3.39)$ where $g$ is an arbitrary function in $M$ .", "It follows from (3.39) that $\\underset{\\left\\langle p,\\lambda \\right\\rangle }{\\sup }\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|\\le E(f),(3.40)$ where the $\\sup $ is taken over all projection cycle-vector pairs of $X.$ Consider the general duality relation (3.32).", "Let $\\mu _{0}$ be a measure attaining the supremum in (3.32) and $\\left\\lbrace \\mu _{p,\\lambda }\\right\\rbrace $ be a net of measures of the form (3.38) approximating $\\mu _{0}$ in the weak$^{\\text{*}}$ topology of $C(X)^{\\ast }.$ We already know that this is possible.", "For any $\\varepsilon >0,$ there exists a measure $\\mu _{p_{0},\\lambda _{0}}$ in $\\left\\lbrace \\mu _{p,\\lambda }\\right\\rbrace $ such that $\\left|\\int \\limits _{X}fd\\mu _{0}-\\int \\limits _{X}fd\\mu _{p_{0},\\lambda _{0}}\\right|<\\varepsilon .$ From the last inequality we obtain that $\\left|\\int \\limits _{X}fd\\mu _{p_{0},\\lambda _{0}}\\right|>\\left|\\int \\limits _{X}fd\\mu _{0}\\right|-\\varepsilon =E(f)-\\varepsilon .$ Hence, $\\underset{\\left\\langle p,\\lambda \\right\\rangle }{\\sup }\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|\\ge E(f).", "(3.41)$ From (3.40) and (3.41) it follows that $\\underset{\\left\\langle p,\\lambda \\right\\rangle }{\\sup }\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|=E(f).", "(3.42)$ By Lemma 3.7, $\\mu _{p,\\lambda }=\\sum _{i=1}^{s}t_{i}\\mu _{l_{i}},$ where $l_{i}$ , $i=1,...,s,$ are minimal projection cycles in $p$ and $\\sum _{i=1}^{s}t_{i}=1,~t_{i}>0.$ Let $k$ be an index in the set $\\lbrace 1,...,s\\rbrace $ such that $\\left|\\int \\limits _{X}fd\\mu _{l_{k}}\\right|=\\max \\left\\lbrace \\left|\\int \\limits _{X}fd\\mu _{l_{i}}\\right|,\\text{ }i=1,...,s\\right\\rbrace .$ Then $\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|\\le \\left|\\int \\limits _{X}fd\\mu _{l_{k}}\\right|.", "(3.43)$ Now since $\\left|\\int \\limits _{X}fd\\mu _{l}\\right|\\le E(f),$ for any minimal cycle $l,$ from (3.42) and (3.43) we obtain the assertion of the theorem.", "Remark 3.9.", "Theorem 3.10 not only proves Golomb's formula, but also improves it.", "Indeed, based on Lemma 3.5, one can easily observe that the formula (3.35) is equivalent to the formula $E(f)=\\underset{\\left\\langle p,\\lambda \\right\\rangle }{\\sup }\\left|\\int \\limits _{X}fd\\mu _{p,\\lambda }\\right|,$ where the $\\sup $ is taken over all projection cycle-vector pairs $\\left\\langle p,\\lambda \\right\\rangle $ of $X$ provided that all the numbers $\\lambda _{i}\\diagup \\sum _{j=1}^{m}\\left|\\lambda _{j}\\right|$ , $i=1,...,m,$ are rational.", "But by Lemma 3.6, minimal projection cycles enjoy this property.", "A ridge function $g(\\mathbf {a}\\cdot \\mathbf {x})$ with a direction $\\mathbf {a}\\in \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ admits a natural generalization to a multivariate function of the form $g(\\alpha _{1}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}(x_{d}))$ , where $\\alpha _{i}(x_{i})$ , $i=\\overline{1,d},$ are real, presumably well behaved, fixed univariate functions.", "We know from Chapter 1 that finitely many directions $\\mathbf {a}^{j}$ are not enough for sums $\\sum g_{j}\\left( \\mathbf {a}^{j}\\cdot \\mathbf {x}\\right) $ to approximate multivariate functions.", "However, we will see in this chapter that sums of the form $\\sum g_{j}(\\alpha _{1}^{j}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}^{j}(x_{d}))$ with finitely many $\\alpha _{i}^{j}(x_{i})$ is capable not only approximating multivariate functions but also precisely representing them.", "First we study the problem of representation of a function $f:X\\rightarrow \\mathbb {R}$ , where $X$ is any set, as a linear superposition $\\sum _{j}g_{j}(h_{j}(x))$ with arbitrary but fixed functions $h_{j}:X\\rightarrow {{\\mathbb {R}}}$ .", "Then we apply the obtained result and the famous Kolmogorov superposition theorem to prove representability of an arbitrarily behaved multivariate function in the form of a generalized ridge function $\\sum g_{j}(\\alpha _{1}^{j}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}^{j}(x_{d}))$ .", "We also study the uniqueness of representation of functions by linear superpositions.", "The material of this chapter is taken from [64], [74].", "In this section, we study some problems of representation of real functions by linear superpositions and linear combinations of generalized ridge functions.", "Let $X$ be any set and $h_{i}:X\\rightarrow {{\\mathbb {R}}},~i=1,...,r,$ be arbitrarily fixed functions.", "Consider the set $\\mathcal {B}(X)=\\mathcal {B}(h_{1},...,h_{r};X)=\\left\\lbrace \\sum \\limits _{i=1}^{r}g_{i}(h_{i}(x)),~x\\in X,~g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r\\right\\rbrace (4.1)$ Members of this set will be called linear superpositions with respect to the functions $h_{1},...,h_{r}$ (see [159]).", "For a detailed study of linear superpositions and their approximation-theoretic properties we refer the reader to the monograph by Khavinson [92].", "Note that sums of generalized ridge functions $\\sum g_{j}(\\alpha _{1}^{j}(x_{1})+\\cdot \\cdot \\cdot +\\alpha _{d}^{j}(x_{d}))$ with fixed $\\alpha _{i}^{j}(x_{i})$ are a special case of linear superpositions.", "In Section 1.2, we considered linear superpositions defined on a subset of the $d$ -dimensional Euclidean space, while here $X$ is a set of arbitrary nature.", "As in Section 1.2, we are interested in the question: what conditions on $X$ guarantee that each function on $X$ will be in the set $\\mathcal {B}(X)$ ?", "The simplest case $X\\subset \\mathbb {R}^{d},~r=d$ and $h_{i}$ are the coordinate functions was solved in [94].", "See also [92] for the case $r=2.$ By $\\mathcal {B}_{c}(X)$ and $\\mathcal {B}_{b}(X)$ denote the right hand side of (4.1) with continuous and bounded $g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,r,$ respectively.", "Our starting point is the well-known superposition theorem of Kolmogorov [97].", "It states that for the unit cube $\\mathbb {I}^{d},~\\mathbb {I}=[0,1],~d\\ge 2,$ there exists $2d+1$ functions $\\lbrace s_{q}\\rbrace _{q=1}^{2d+1}\\subset C(\\mathbb {I}^{d})$ of the form $s_{q}(x_{1},...,x_{d})=\\sum _{p=1}^{d}\\varphi _{pq}(x_{p}),~\\varphi _{pq}\\in C(\\mathbb {I}),~p=1,...,d,~q=1,...,2d+1(4.2)$ such that each function $f\\in C(\\mathbb {I}^{d})$ admits the representation $f(x)=\\sum _{q=1}^{2d+1}g_{q}(s_{q}(x)),~x=(x_{1},...,x_{d})\\in \\mathbb {I}^{d},~g_{q}\\in C({{\\mathbb {R)}}}.", "(4.3)$ Note that the functions $g_{q}(s_{q}(x))$ , involved in the right hand side of (4.3), are generalized ridge functions.", "In our notation, (4.3) means that $\\mathcal {B}_{c}(s_{1},...,s_{2d+1};\\mathbb {I}^{d})=C(\\mathbb {I}^{d}).$ This surprising and deep result, which solved (negatively) Hilbert's 13-th problem, was improved and generalized in several directions.", "It was first observed by Lorentz [114] that the functions $g_{q}$ can be replaced by a single continuous function $g.$ Sprecher [145] showed that the theorem can be proven with constant multiples of a single function $\\varphi $ and translations.", "Specifically, $\\varphi _{pq}$ in (4.2) can be chosen as $\\lambda ^{p}\\varphi (x_{p}+\\varepsilon q),$ where $\\varepsilon $ and $\\lambda $ are some positive constants.", "Fridman [41] succeeded in showing that the functions $\\varphi _{pq}$ can be constructed to belong to the class $Lip(1).$ Vitushkin and Henkin [159] showed that $\\varphi _{pq}$ cannot be taken to be continuously differentiable.", "Ostrand [132] extended the Kolmogorov theorem to general compact metric spaces.", "In particular, he proved that for each compact $d$ -dimensional metric space $X$ there exist continuous real functions $\\lbrace \\alpha _{i}\\rbrace _{i=1}^{2d+1}\\subset C(X)$ such that $\\mathcal {B}_{c}(\\alpha _{1},...,\\alpha _{2d+1};X)=C(X).$ Sternfeld [151] showed that the number $2d+1$ cannot be reduced for any $d$ -dimensional space $X.$ Thus the number of terms in the Kolmogorov superposition theorem is the best possible.", "Some papers of Sternfeld were devoted to the representation of continuous and bounded functions by linear superpositions.", "Let $C(X)$ and $B(X)$ denote the space of continuous and bounded functions on some set $X$ respectively (in the first case, $X$ is supposed to be a compact metric space).", "Let $F=\\lbrace h\\rbrace $ be a family of functions on $X.$ $F$ is called a uniformly separating family (u.s.f.)", "if there exists a number $0<\\lambda \\le 1$ such that for each pair $\\lbrace x_{j}\\rbrace _{j=1}^{m}$ , $\\lbrace z_{j}\\rbrace _{j=1}^{m}$ of disjoint finite sequences in $X$ , there exists some $h\\in F$ so that if from the two sequences $\\lbrace h(x_{j})\\rbrace _{j=1}^{m}$ and $\\lbrace h(z_{j})\\rbrace _{j=1}^{m}$ in $h(X)$ we remove a maximal number of pairs of points $h(x_{j_{1}})$ and $h(z_{j_{2}})$ with $h(x_{j_{1}})=h(z_{j_{2}}),$ there remains at least $\\lambda m$ points in each sequence (or , equivalently, at most $(1-\\lambda )m$ pairs can be removed).", "Sternfeld [149] proved that for a finite family $F=\\lbrace h_{1},...,h_{r}\\rbrace $ of functions on $X$ , being a u.s.f.", "is equivalent to the equality $\\mathcal {B}_{b}(h_{1},...,h_{r};X)=B(X),$ and that in the case where $X$ is a compact metric space and the elements of $F$ are continuous functions on $X$ , the equality $\\mathcal {B}_{c}(h_{1},...,h_{r};X)=C(X)$ implies that $F$ is a u.s.f.", "Thus, in particular, Sternfeld obtained that the formula (4.3) is valid for all bounded functions, where $g_{q}$ are bounded functions depending on $f$ (see also [92]).", "Let $X$ be a compact metric space.", "The family $F=\\lbrace h\\rbrace \\subset C(X)$ is said to be a measure separating family (m.s.f.)", "if there exists a number $0<\\lambda \\le 1$ such that for any measure $\\mu $ in $\\ C(X)^{\\ast },$ the inequality $\\left\\Vert \\mu \\circ h^{-1}\\right\\Vert \\ge \\lambda \\left\\Vert \\mu \\right\\Vert $ holds for some $h\\in F.$ Sternfeld [152] proved that $\\mathcal {B}_{c}(h_{1},...,h_{r};X)=C(X)$ if and only if the family $\\lbrace h_{1},...,h_{r}\\rbrace $ is a m.s.f.", "In [149], it was also shown that if $r=2,$ then the properties u.s.f.", "and m.s.f.", "are equivalent.", "Therefore, the equality $\\mathcal {B}_{b}(h_{1},h_{2};X)=B(X)$ is equivalent to $\\mathcal {B}_{c}(h_{1},h_{2};X)=C(X).$ But for $r\\,>2$ , these two properties are no longer equivalent.", "That is, $\\mathcal {B}_{b}(h_{1},...,h_{r};X)=B(X)$ does not always imply $\\mathcal {B}_{c}(h_{1},...,h_{r};X)=C(X)$ (see [152]).", "Our purpose is to consider the above mentioned problem of representation by linear superpositions without involving any topology (that of continuity or boundedness).", "We start with characterization of those sets $X$ for which $\\mathcal {B}(h_{1},...,h_{r};X)=T(X),$ where $T(X)$ is the space of all functions on $X.$ As in Section 1.2, this will be done in terms of cycles.", "We claim that nonexistence of cycles in $X$ is equivalent to the equality $\\mathcal {B}(X)=T(X)$ for an arbitrary set $X$ .", "In particular, we show that $\\mathcal {B}_{c}(X)=C(X)$ always implies $\\mathcal {B}(X)=T(X).$ This implication will enable us to obtain some new results, namely extensions of the previously known theorems from continuous to discontinuous multivariate functions.", "For example, we will prove that the formula (4.3) is valid for all discontinuous multivariate functions $f$ defined on the unite cube $\\mathbb {I}^{d},$ where $g_{q}$ are univariate functions depending on $f.$ In this subsection, we show that if some representation by linear superpositions holds for continuous functions, then it holds for all functions.", "This will lead us to natural extensions of some known superposition theorems (such as Kolmogorov's superposition theorem, Ostrand's superposition theorem, etc) from continuous to discontinuous functions.", "In the sequel, by $\\chi _{A}$ we will denote the characteristic function of a set $\\ A\\subset \\mathbb {R}.$ That is, $\\chi _{A}(y)=\\left\\lbrace \\begin{array}{c}1,~if~y\\in A \\\\0,~if~y\\notin A.\\end{array}\\right.$ The following definition is a generalized version of Definition 1.1 from Section 1.2, where in connection with ridge functions only subsets of $\\mathbb {R}^{d}$ were considered.", "Definition 4.1.", "Given an arbitrary set $X$ and functions $h_{i}:X\\rightarrow \\mathbb {R},~i=1,...,r$ .", "A set of points $\\lbrace x_{1},...,x_{n}\\rbrace \\subset X$ is called to be a cycle with respect to the functions $h_{1},...,h_{r}$ (or, concisely, a cycle if there is no confusion), if there exists a vector $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ with the nonzero real coordinates $\\lambda _{i},~i=1,...,n,$ such that $\\sum _{j=1}^{n}\\lambda _{j}\\chi _{h_{i}(x_{j})}=0,~i=1,...,r.(4.4)$ A cycle $p=\\lbrace x_{1},...,x_{n}\\rbrace $ is said to be minimal if $p$ does not contain any cycle as its proper subset.", "Note that in this definition the vector $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ can be chosen so that it has only integer components.", "Indeed, let for $i=1,...,r,$ the set $\\lbrace h_{i}(x_{j}),~j=1,...,n\\rbrace $ have $k_{i}$ different values.", "Then it is not difficult to see that Eq.", "(4.4) stands for a system of $\\sum _{i=1}^{r}k_{i}$ homogeneous linear equations in unknowns $\\lambda _{1},...,\\lambda _{n}.$ This system can be written in the matrix form $(\\lambda _{1},\\ldots ,\\lambda _{n})\\times C=0,$ where $C$ is an $n$ by $\\sum _{i=1}^{r}k_{i}$ matrix.", "The basic property of this matrix is that all of its entries are 0's and 1's and no row or column of $C$ is identically zero.", "Since Eq.", "(4.4) has a nontrivial solution $(\\lambda _{1}^{^{\\prime }},\\ldots ,\\lambda _{n}^{^{\\prime }})\\in \\mathbf {R}^{n}$ and all entries of $C$ are integers, by applying the Gauss elimination method we can see that there always exists a nontrivial solution $(\\lambda _{1},\\ldots ,\\lambda _{n})$ with the integer components $\\lambda _{i}$ , $i=1,...,n$ .", "For a number of simple examples, see Section 1.2.", "Let $T(X)$ denote the set of all functions on $X.$ With each pair $\\left\\langle p,\\lambda \\right\\rangle ,$ where $p=\\lbrace x_{1},...,x_{n}\\rbrace $ is a cycle in $X$ and $\\lambda =(\\lambda _{1},...,\\lambda _{n})$ is a vector known from Definition 4.1, we associate the functional $G_{p,\\lambda }:T(X)\\rightarrow \\mathbb {R},~~G_{p,\\lambda }(f)=\\sum _{j=1}^{n}\\lambda _{j}f(x_{j}).$ In the following, such pairs $\\left\\langle p,\\lambda \\right\\rangle $ will be called cycle-vector pairs of $X.$ It is clear that the functional $G_{p,\\lambda }$ is linear.", "Besides, $G_{p,\\lambda }(g)=0$ for all functions $g\\in \\mathcal {B}(h_{1},...,h_{r};X).$ Indeed, assume that (4.4) holds.", "Given $i\\le r$ , let $z=h_{i}(x_{j})$ for some $j$ .", "Hence, $\\sum _{j~(h_{i}(x_{j})=z)}\\lambda _{j}=0$ and $\\sum _{j~(h_{i}(x_{j})=z)}\\lambda _{j}g_{i}(h_{i}(x_{j}))=0$ .", "A summation yields $G_{p,\\lambda }(g_{i}\\circ h_{i})=0$ .", "Since $G_{p,\\lambda }$ is linear, we obtain that $G_{p,\\lambda }(\\sum _{i=1}^{r}g_{i}\\circ h_{i})=0$ .", "A minimal cycle $p=\\lbrace x_{1},...,x_{n}\\rbrace $ has the following obvious properties: (a) The vector $\\lambda $ associated with $p$ by Eq.", "(4.4) is unique up to multiplication by a constant; (b) If in (4.4), $\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1,$ then all the numbers $\\lambda _{j},~j=1,...,n,$ are rational.", "The vector $\\lambda $ associated with $p$ by Eq.", "(4.4) is unique up to multiplication by a constant; If in (4.4), $\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1,$ then all the numbers $\\lambda _{j},~j=1,...,n,$ are rational.", "Thus, a minimal cycle $p$ uniquely (up to a sign) defines the functional $~G_{p}(f)=\\sum _{j=1}^{n}\\lambda _{j}f(x_{j}),\\text{ \\ }\\sum _{j=1}^{n}\\left|\\lambda _{j}\\right|=1.$ Proposition 4.1.", "1) Let $X$ have cycles.", "A function $f:X\\rightarrow \\mathbb {R}$ belongs to the space $\\mathcal {B}(h_{1},...,h_{r};X)$ if and only if $G_{p}(f)=0$ for any minimal cycle $p\\subset X$ with respect to the functions $h_{1},...,h_{r}$ .", "2) Let $X$ has no cycles.", "Then $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ Proposition 4.2.", "$\\mathcal {B}(h_{1},...,h_{r};X)=T(X)$ if and only if $X$ has no cycles.", "These propositions are proved by the same way as Theorems 1.1 and 1.2.", "We use these propositions to obtain our main result (see Theorem 4.1 below).", "The condition whether $X$ have cycles or not, depends both on $X$ and the functions $h_{1},...,h_{r}$ .", "In the following, we see that if $h_{1},...,h_{r}$ are “nice\" functions (smooth functions with the simple structure.", "For example, ridge functions) and $X\\subset \\mathbb {R}^{d}$ is a “rich\" set (for example, the set with interior points), then $X$ has always cycles.", "Thus the representability by linear combinations of univariate functions with the fixed “nice\" multivariate functions requires at least that $X$ should not possess interior points.", "The picture is quite different when the functions $h_{1},...,h_{r}$ are not “nice\".", "Even in the case when they are continuous, we will see that many sets of $\\mathbb {R}^{d}$ (the unite cube, any compact subset of that, or even the whole space $\\mathbb {R}^{d}$ itself) may have no cycles.", "If disregard the continuity, there exists even one function $h$ such that every multivariate function is representable as $g\\circ h$ over any subset of $\\mathbb {R}^{d}$ .", "First, let us introduce the following definition.", "Definition 4.2.", "Let $X$ be a set and $h_{i}:X\\rightarrow \\mathbb {R}, $ $i=1,...,r,$ be arbitrarily fixed functions.", "A class $A(X)$ of functions on $X$ will be called a “permissible function class\" if for any minimal cycle $p\\subset X$ with respect to the functions $h_{1},...,h_{r}$ (if it exists), there is a function $f_{0}$ in $A(X)$ such that $G_{p}(f_{0})\\ne 0.", "$ Clearly, $C(X)$ and $B(X)$ are both permissible function classes (in case of $C(X),$ $X$ is considered to be a normal topological space).", "Theorem 4.1.", "Let $A(X)$ be a permissible function class.", "If $A(X) \\subset \\mathcal {B}(h_{1},...,h_{r};X)$ , then $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ The proof is simple and based on Propositions 4.1 and 4.2.", "Assume for a moment that $X$ admits a cycle $p$ .", "By Proposition 4.1, the functional $G_{p}$ annihilates all members of the set $B(h_{1},...,h_{r};X).$ By Definition 4.2 of permissible function classes, $A(X)\\ $ contains a function $f_{0}$ such that $G_{p}(f_{0})\\ne 0.$ Therefore, $f_{0}\\notin B(h_{1},...,h_{r};X)$ .", "We see that the embedding $A(X) \\subset B(h_{1},...,h_{r};X)$ is impossible if $X$ has a cycle.", "Thus $X$ has no cycles.", "Then by Proposition 4.2, $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ In the “if part\" of Theorem 4.1, instead of $\\mathcal {B}(h_{1},...,h_{r};X)$ and $A(X)$ one can take $\\mathcal {B}_{c}(h_{1},...,h_{r};X)$ and $C(X)$ (or $\\mathcal {B}_{b}(h_{1},...,h_{r};X)$ and $B(X)$ ) respectively.", "That is, the following corollaries are valid.", "Corollary 4.1.", "Let $X$ be a set and $h_{i}:X\\rightarrow \\mathbb {R},$ $i=1,...,r,$ be arbitrarily fixed bounded functions.", "If $\\mathcal {B}_{b}(h_{1},...,h_{r};X)=B(X)$ , then $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ Corollary 4.2.", "Let $X$ be a normal topological space and $h_{i}:X\\rightarrow \\mathbb {R},$ $i=1,...,r,$ be arbitrarily fixed continuous functions.", "If $\\mathcal {B}_{c}(h_{1},...,h_{r};X)=C(X)$ , then $\\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ The main advantage of Theorem 4.1 is that we need not check directly if the set $X$ has no cycles, which in many cases may turn out to be very tedious task.", "Using this theorem, we can extend free-of-charge the existing superposition theorems from the classes $B(X)$ or $C(X)$ (or some other permissible function classes) to all functions defined on $X.$ For example, this theorem allows us to extend the Kolmogorov superposition theorem from continuous to all multivariate functions.", "Theorem 4.2.", "Let $d\\ge 2$ , $\\mathbb {I}=[-1;1]$ , and $~\\varphi _{pq}, ~p=1,...,d, ~q=1,...,2d+1$ , be the universal continuous functions in (4.2).", "Then each multivariate function $f:\\mathbb {I}^{d}\\rightarrow \\mathbb {R}$ can be represented in the form $f(x)=\\sum _{q=1}^{2d+1}g_{q}(\\sum _{p=1}^{d}\\varphi _{pq}(x_{p})),~x=(x_{1},...,x_{d})\\in \\mathbb {I}^{d}.$ where $g_{q}$ are univariate functions depending on $f.$ It should be remarked that Sternfeld [149], in particular, obtained that the formula (4.3) is valid for functions $f\\in B(\\mathbb {I}^{d})$ provided that $g_{q}$ are bounded functions depending on $f$ (see [92] for more detailed information and interesting discussions).", "Let $X$ be a compact metric space and $h_{i}\\in C(X)$ , $i=1,...,r.$ The result of Sternfeld (see Section 4.1) and Corollary 4.1 give us the implications $\\mathcal {B}_{c}(h_{1},...,h_{r};X)=C(X)\\Rightarrow \\mathcal {B}_{b}(h_{1},...,h_{r};X)=B(X)$ $\\Rightarrow \\mathcal {B}(h_{1},...,h_{r};X)=T(X).$ The first implication is invertible when $r=2$ (see [149]).", "We want to show that the second is not invertible even in the case $r=2.$ The following interesting example is due to Khavinson [92].", "Let $X\\subset \\mathbb {R}^{2}$ consist of a broken line whose sides are parallel to the coordinate axis and whose vertices are $(0;0),(1;0),(1;1),(1+\\frac{1}{2^{2}};1),(1+\\frac{1}{2^{2}};1+\\frac{1}{2^{2}}),(1+\\frac{1}{2^{2}}+\\frac{1}{3^{2}};1+\\frac{1}{2^{2}}),...$ We add to this line the limit point of the vertices $(\\frac{\\pi ^{2}}{6},\\frac{\\pi ^{2}}{6})$ .", "Let $r=2$ and $h_{1},h_{2}$ be the coordinate functions.", "Then the set $X$ has no cycles with respect to $h_{1}$ and $h_{2}.", "$ By Proposition 4.1, every function $f$ on $X$ is of the form $g_{1}(x_{1})+g_{2}(x_{2})$ , $(x_{1},x_{2})\\in X$ .", "Now construct a function $f_{0}$ on $X$ as follows.", "On the link joining $(0;0)$ to $(1;0)$ $f_{0}(x_{1},x_{2})$ continuously increases from 0 to 1; on the link from $(1;0)$ to $(1;1)$ it continuously decreases from 1 to 0; on the link from $(1;1)$ to $(1+\\frac{1}{2^{2}};1)$ it increases from 0 to $\\frac{1}{2}$ ; on the link from $(1+\\frac{1}{2^{2}};1)$ to $(1+\\frac{1}{2^{2}};1+\\frac{1}{2^{2}})$ it decreases from $\\frac{1}{2}$ to 0; on the next link it increases from 0 to $\\frac{1}{3}$ , etc.", "At the point $(\\frac{\\pi ^{2}}{6},\\frac{\\pi ^{2}}{6})$ set the value of $f_{0}$ equal to $0.$ Obviously, $f_{0} $ is a continuous functions and by the above argument, $f_{0}(x_{1},x_{2})=g_{1}(x_{1})+g_{2}(x_{2}).$ But $g_{1}$ and $g_{2}$ cannot be chosen as continuous functions, since they get unbounded as $x_{1}$ and $x_{2}$ tends to $\\frac{\\pi ^{2}}{6}$ .", "Thus, $\\mathcal {B}(h_{1},h_{2};X)=T(X)$ , but at the same time $\\mathcal {B}_{c}(h_{1},h_{2};X)\\ne C(X)$ (or, equivalently, $\\mathcal {B}_{b}(h_{1},h_{2};X)\\ne B(X)$ ).", "We have seen in the previous subsection that the unit cube in $\\mathbb {R}^{d} $ has no cycles with respect to some $2d+1$ continuous functions (namely, the Kolmogorov functions $s_{q}$ (4.2)).", "From the result of Ostrand [132] (see Section 4.1) and Corollary 4.2 it follows that compact sets $X$ of finite dimension also lack cycles with respect to a certain family of finitely many continuous functions on $X$ .", "Namely, the following generalization of Ostrand's theorem is valid.", "Theorem 4.3.", "For $p=1,2,...,m$ let $X_{p}$ be a compact metric space of finite dimension $d_{p}$ and let $n=\\sum _{p=1}^{n}d_{p}.$ There exist continuous functions $\\alpha _{pq}:X_{p}\\rightarrow [0,1], $ $p=1,...,m,$ $q=1,...,2n+1,$ such that every real function $f$ defined on $\\Pi _{p=1}^{m}X_{p}$ is representable in the form $f(x_{1},...,x_{m})=\\sum _{q=1}^{2n+1}g_{q}(\\sum _{p=1}^{m}\\alpha _{pq}(x_{p})).", "(4.5)$ where $g_{q}$ are real functions depending on $f$ .", "If $f$ is continuous, then the functions $g_{q}$ can be chosen continuous.", "Note that Ostrand proved “if $f$ is continuous...\" part of Theorem 4.3, while we prove the validity of (4.5) for discontinuous $f$ .", "One may ask if there exists a finite family of functions $\\lbrace h_{i}:\\mathbb {R}^{d}\\rightarrow \\mathbb {R}\\rbrace _{i=1}^{n}$ such that any subset of $\\mathbb {R}^{d}$ does not admit cycles with respect to this family?", "The answer is positive.", "This follows from the result of Demko [33]: there exist $2d+1$ continuous functions $\\varphi _{1},...,\\varphi _{2d+1}$ defined on $\\mathbb {R}^{d}$ such that every bounded continuous function on $\\mathbb {R}^{d}$ is expressible in the form $\\sum _{i=1}^{2d+1}g\\circ \\varphi _{i}$ for some $g\\in C(\\mathbb {R})$ .", "This theorem together with Corollary 4.1 yield that every function on $\\mathbb {R}^{d}$ is expressible in the form $\\sum _{i=1}^{2d+1}g_{i}\\circ \\varphi _{i}$ for some $g_{i}:\\mathbb {R}\\rightarrow \\mathbb {R},~i=1,...,2d+1$ .", "We do not yet know if $g_{i}$ here can be replaced by a single univariate function.", "We also don't know if the number $2d+1$ can be reduced so that the whole space of $\\mathbb {R}^{d}$ (or any $d$ -dimensional compact subset of that, or at least the unit cube $\\mathbb {I}^{d}$ ) has no cycles with respect to some continuous functions $\\varphi _{1},...,\\varphi _{k}:\\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ , where $k<2d+1$ .", "One of the basic results of Sternfeld [151] says that the dimension of a compact metric space $X$ equals $d$ if and only if there exist functions $\\varphi _{1},...,\\varphi _{2d+1}\\in C(X)$ such that $\\mathcal {B}_{c}(\\varphi _{1},...,\\varphi _{2d+1};X)=C(X)$ and for any fmily $\\lbrace \\psi _{i}\\rbrace _{i=1}^{k}\\subset C(X),$ $k<2d+1$ , we have $\\mathcal {B}_{c}(\\psi _{1},...,\\psi _{k};X)\\ne C(X).$ In particular, from this result it follows that the number of terms in the Kolmogorov superposition theorem cannot be reduced.", "But since the equalities $\\mathcal {B}_{c}(X)=C(X)$ and $\\mathcal {B}(X)=T(X)$ are not equivalent, the above question on the nonexistence of cycles in $\\mathbb {R}^{d}$ with respect to less than $2d+1$ continuous functions is far from trivial.", "If disregard the continuity, one can construct even one function $\\varphi :\\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ such that the whole space $\\mathbb {R}^{d}$ will not possess cycles with respect to $\\varphi $ and therefore, every function $f:\\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ will admit the representation $f=g\\circ \\varphi $ with some univariate $g$ depending on $f$ .", "Our argument easily follows from Corollary 4.2 and the result of Sprecher [143]: for any natural number $d$ , $d\\ge 2$ , there exist functions $h_{p}:\\mathbb {I}\\rightarrow \\mathbb {R}$ , $p=1,...,d,$ such that every function $f\\in C(\\mathbb {I}^{d})$ can be represented in the form $f(x_{1},...,x_{d})=g\\left( \\sum _{p=1}^{d}h_{p}(x_{p})\\right) ,(4.6)$ where $g$ is a univariate (generally discontinuous) function depending on $f$ .", "Note that the function involved in the right hand side of (4.6) is a generalized ridge function.", "Thus, the result of Sprecher together with our result means that every multivariate function $f$ is representable as a generalized ridge function $g\\left( \\cdot \\right) $ and if $f$ is continuous, then $g$ can be chosen continuous as well.", "Remark 4.1.", "Concerning ordinary ridge functions $g(\\mathbf {a}\\cdot \\mathbf {x})$ , representation of every multivariate function by linear combinations of such functions may not be possible over many sets in $\\mathbb {R}^{d}$ .", "For example, this is not possible for sets having interior points.", "More precisely, assume we are given finitely many nonzero directions $\\mathbf {a}^{1},...,\\mathbf {a}^{r}$ in $\\mathbb {R}^{d}$ .", "Then $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};X\\right) \\ne T(X)$ for any set $X\\subset \\mathbb {R}^{d}$ with a nonempty interior.", "Indeed, let $\\mathbf {y}$ be a point in the interior of $X$ .", "Consider vectors $\\mathbf {b}^{i}$ , $i=1,...,r,$ with sufficiently small coordinates such that $\\mathbf {a}^{i}\\cdot \\mathbf {b}^{i}=0$ , $i=1,...,r$ .", "Note that the vectors $\\mathbf {b}^{i}$ , $i=1,...,r,$ can be chosen pairwise linearly independent.", "With each vector $\\mathbf {\\varepsilon }=(\\varepsilon _{1},...,\\varepsilon _{r})$ , $\\varepsilon _{i}\\in \\lbrace 0,1\\rbrace $ , $i=1,...,r,$ we associate the point $\\mathbf {x}_{\\mathbf {\\varepsilon }}=\\mathbf {y+}\\sum _{i=1}^{r}\\varepsilon _{i}\\mathbf {b}^{i}.$ Since the coordinates of $\\mathbf {b}^{i}$ are sufficiently small, we may assume that all the points $\\mathbf {x}_{\\mathbf {\\varepsilon }}$ are in the interior of $X$ .", "We correspond each point $\\mathbf {x}_{\\mathbf {\\varepsilon }} $ to the number $(-1)^{\\left|\\mathbf {\\varepsilon }\\right|}$ , where $\\left|\\mathbf {\\varepsilon }\\right|=\\varepsilon _{1}+\\cdots +\\varepsilon _{r}.$ One may easily verify that the pair $\\left\\langle \\lbrace \\mathbf {x}_{\\mathbf {\\varepsilon }}\\rbrace ,\\lbrace (-1)^{\\left|\\mathbf {\\varepsilon }\\right|}\\rbrace \\right\\rangle $ is a cycle-vector pair of $X$ .", "Therefore, by Proposition 4.2, $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{r};X\\right) \\ne T(X).$ Note that the above method of construction of the set $\\lbrace \\mathbf {x}_{\\mathbf {\\varepsilon }}\\rbrace $ is due to Lin and Pinkus [112].", "Remark 4.2.", "A different generalization of ridge functions was considered in Lin and Pinkus [112].", "This generalization involves multivariate functions of the form $g(A\\mathbf {x})$ , where $\\mathbf {x}\\in \\mathbb {R}^{d}$ is the variable, $A$ is a fixed $d\\times n$ matrix, $1\\le n<d$ , and $g$ is a real-valued function defined on $\\mathbb {R}^{n}$ .", "For $n=1,$ this reduces to a ridge function.", "Let $Q$ be a set such that every function on $Q$ can be represented by linear superpositions.", "This representation is generally not unique.", "But for some sets it may be unique provided that initial values of the representing functions are prescribed at some point of $Q$ .", "In this section, we are going to study properties of such sets.", "All the obtained results are valid, in particular, for linear combinations of generalized ridge functions.", "Assume $X$ is an arbitrary set, $h_{i}:X\\rightarrow \\mathbb {R}$ , $i=1,\\ldots ,r$ , are fixed functions and $\\mathcal {B}(X)$ is the set defined in (4.1).", "Let $T(X)$ denote the set of all real functions on $X$ .", "Obviously, $\\mathcal {B}(X)$ is a linear subspace of $T(X)$ .", "For a set $Q\\subset X$ , let $T(Q)$ and $\\mathcal {B}(Q)$ denote the restrictions of $T(X)$ and $\\mathcal {B}(X)$ to $Q$ , respectively.", "Sets $Q$ with the property $\\mathcal {B}(Q)=T(Q)$ will be called representation sets.", "Recall that Proposition 4.2 gives a complete characterization of such sets.", "For a representation set $Q$ , we will also use the notation $Q\\in RS.$ Here, $RS$ stands for the set of all representation sets in $X$ .", "Let $Q\\in RS.$ Clearly for a function $f$ defined on $Q$ the representation $f(x)=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),~x\\in Q(4.7)$ is not unique.", "We are interested in the uniqueness of such representation under some reasonable restrictions on the functions $g_{i}\\circ h_{i}$ .", "These restrictions may be various, but in this section, we require that the values of $g_{i}\\circ h_{i}$ are prescribed at some point $x_{0}\\in Q$ .", "That is, we require that $g_{i}(h_{i}(x_{0}))=a_{i},~i=1,...,r-1,(4.8)$ where $a_{i}$ are arbitrarily fixed real numbers.", "Is representation (4.7) subject to initial conditions (4.8) always unique?", "Obviously, not.", "We are going to identify those representation sets $Q$ for which representation (4.7) subject to conditions (4.8) is unique for all functions $f:Q\\rightarrow \\mathbb {R}$ .", "In the sequel, such sets $Q$ will be called unicity sets.", "From Proposition 4.2 it is easy to obtain the following set-theoretic properties of representation sets: (1) $Q\\in RS$ $\\Longleftrightarrow $ $A\\in RS$ for every finite set $A\\subset Q$ ; (2) The union of any linearly ordered (under inclusion) system of representation sets is also a representation set (3) For any representation set $Q$ there is a maximal representation set, that is, a set $M\\in RS$ such that $Q\\subset M$ and for any $P\\supset M$ , $P\\in RS$ we have $P=M$ .", "(4) If $M\\subset X$ is a maximal representation set, then $h_{i}(M)=h_{i}(X)$ , $i=1,...,r$ .", "Properties (1) and (2) are obvious, since any cycle is a finite set.", "The (3)-rd property follows from (2) and Zorn's lemma.", "To prove property (4) note that if $x_{0}\\in X$ and $h_{i}(x_{0})\\notin h_{i}(M)$ for some $i$ , one can construct the representation set $M\\cup \\lbrace x_{0}\\rbrace $ , which is bigger than $M$ .", "But this is impossible, since $M$ is maximal.", "Definition 4.3.", "A set $Q\\subset X$ is called a complete representation set if $Q$ itself is a representation set and there is no other representation set $P$ such that $Q\\subset P$ and $h_{i}(P)=h_{i}(Q)$ , $i=1,...,r$ .", "The set of all complete representation sets of $X$ will be denoted by $CRS$ .", "Obviously, every representation set is contained in a complete representation set.", "That is, if $A\\in RS$ , then there exists $B\\in CRS$ such that $h_{i}(B)=h_{i}(A),$ $i=1,...,r.$ It turns out that for the functions $h_{1},...,h_{r}$ , complete representation sets entirely characterize unicity sets.", "To prove this fact we need some auxiliary lemmas.", "Lemma 4.1.", "Let $Q\\subset X$ be a representation set and for some point $x_{0}\\in Q$ the zero function representation $0=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ \\ }x\\in Q,$ is unique, provided that $g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1$ .", "That is, all the functions $g_{i}\\equiv 0$ on the sets $h_{i}(Q)$ , $i=1,...,r.$ Then $Q\\in CRS.$ Assume that $Q\\notin CRS$ .", "Then there exists a point $p\\in X$ such that $p\\notin Q$ , $h_{i}(p)\\in h_{i}(Q)$ , for all $i=1,...,r,$ and $Q^{^{\\prime }}=Q\\cup \\lbrace p\\rbrace $ is also a representation set.", "Consider a function $f_{0}:Q^{^{\\prime }}\\rightarrow \\mathbb {R}$ such that $f_{0}(q)=0$ , for any $q\\in Q$ and $f_{0}(p)=1.$ Since $Q^{^{\\prime }}\\in RS$ , $f_{0}(x)=\\sum _{i=1}^{r}s_{i}(h_{i}(x)),\\text{ \\ }x\\in Q^{^{\\prime }}.$ Then $f_{0}(x)=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ \\ }x\\in Q^{^{\\prime }},(4.9)$ where $g_{i}(h_{i}(x))=s_{i}(h_{i}(x))-s_{i}(h_{i}(x_{0})),\\text{ }i=1,...,r-1$ and $g_{r}(h_{r}(x))=s_{r}(h_{r}(x))+\\sum _{i=1}^{r-1}s_{i}(h_{i}(x_{0})).$        A restriction of representation (4.9) to the set $Q$ gives the equality $\\sum _{i=1}^{r}g_{i}(h_{i}(x))=0,\\text{ for all }x\\in Q.", "(4.10)$ Note that $g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1.$ It follows from the hypothesis of the lemma that representation (4.10) is unique.", "Hence, $g_{i}(h_{i}(x))=0,$ for all $x\\in Q$ and $i=1,...,r.$ But from (4.9) it follows that $\\sum _{i=1}^{r}g_{i}(h_{i}(p))=f_{0}(p)=1.$ Since $h_{i}(p)\\in h_{i}(Q)$ for all $i=1,...,r,$ the above relation contradicts that the functions $g_{i}$ are identically zero on the sets $h_{i}(Q)$ , $i=1,...,r.$ This means that our assumption is not true and $Q\\in CRS.$ The following lemma is a strengthened version of Lemma 4.1.", "Lemma 4.2.", "Let $Q\\in RS$ and for some point $x_{0}\\in Q$ , numbers $c_{1},c_{2},...,c_{r-1}\\in \\mathbb {R}$ and a function $v\\in T(Q)$ the representation $v(x)=\\sum _{i=1}^{r}v_{i}(h_{i}(x))$ is unique under the initial conditions $v_{i}(h_{i}(x_{0}))=c_{i},$ $i=1,...,r-1$ .", "Then for any numbers $b_{1},b_{2}...,b_{r-1}\\in \\mathbb {R}$ and an arbitrary function $f\\in T(Q)$ the representation $f(x)=\\sum _{i=1}^{r}f_{i}(h_{i}(x))$ is also unique, provided that $f_{i}(h_{i}(x_{0}))=b_{i},$ $i=1,...,r-1$ .", "Besides, $Q\\in CRS.$ Assume the contrary.", "Assume that there is a function $f\\in T(Q)$ having two different representations subject to the same initial conditions.", "That is, $f(x)=\\sum _{i=1}^{r}f_{i}(h_{i}(x))=\\sum _{i=1}^{r}f_{i}^{^{\\prime }}(h_{i}(x))$ with $f_{i}(h_{i}(x_{0}))=f_{i}^{^{\\prime }}(h_{i}(x_{0}))=b_{i},$ $i=1,...,r-1$ and $f_{i}\\ne f_{i}^{^{\\prime }}$ for some indice $i\\in \\lbrace 1,...,r\\rbrace .$ In this case, the function $v(x)$ will possess the following two different representations $v(x)=\\sum _{i=1}^{r}v_{i}(h_{i}(x))=\\sum _{i=1}^{r}\\left[v_{i}(h_{i}(x))+f_{i}(h_{i}(x))-f_{i}^{^{\\prime }}(h_{i}(x))\\right] .$ both satisfying the initial conditions.", "The obtained contradiction and above Lemma 4.1 complete the proof.", "In the sequel, we will assume that for any points $t_{i}\\in h_{i}(X),$ $i=1,...,r,$ the system of equations $h_{i}(x)=t_{i}$ , $i=1,...,r,$ has at least one solution.", "Lemma 4.3.", "Let $Q\\in CRS.$ Then for any point $x_{0}\\in Q$ the representation $0=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ }x\\in Q,(4.11)$ subject to the conditions $g_{i}(h_{i}(x_{0}))=0,\\text{ }i=1,...,r-1,(4.12)$ is unique.", "That is, $g_{i}\\equiv 0$ on the sets $h_{i}(Q)$ , $i=1,...,r.$ Assume the contrary.", "Assume that representation (4.11) subject to (4.12) is not unique, or in other words, not all of $g_{i}$ are identically zero.", "Without loss of generality, we may suppose that $g_{r}(h_{r}(y))\\ne 0,$ for some $y\\in Q.$ Let $\\xi \\in X$ be a solution of the system of equations $h_{i}(x)=h_{i}(x_{0}),$ $i=1,...,r-1,$ and $h_{r}(x)=h_{r}(y)$ .", "Therefore, $g_{i}(h_{i}(\\xi ))=0,$ $i=1,...,r-1,$ and $g_{r}(h_{r}(\\xi ))\\ne 0.$ Obviously, $\\xi \\notin Q.$ Otherwise, we may have $g_{r}(h_{r}(\\xi ))=0.$ We are going to prove that $Q^{\\prime }=Q\\cup \\lbrace \\xi \\rbrace $ is a representation set.", "For this purpose, consider an arbitrary function $f:Q^{\\prime }\\rightarrow \\mathbb {R}$ .", "The restriction of $f$ to the set $Q$ admits a decomposition $f(x)=\\sum _{i=1}^{r}t_{i}(h_{i}(x)),\\text{ }x\\in Q.$ One is allowed to fix the values $t_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1.$ Note that then $t_{i}(h_{i}(\\xi ))=0,$ $i=1,...,r-1.$ Consider now the functions $v_{i}(h_{i}(x))=t_{i}(h_{i}(x))+\\frac{f(\\xi )-t_{r}(h_{r}(\\xi ))}{g_{r}(h_{r}(\\xi ))}g_{i}(h_{i}(x)),\\text{ }x\\in Q^{\\prime },\\text{ }i=1,...,r.$ It can be easily verified that $f(x)=\\sum _{i=1}^{r}v_{i}(h_{i}(x)),\\text{ }x\\in Q^{\\prime }.$ Since $f$ is arbitrary, we obtain that $Q^{\\prime }\\in RS,$ where $Q^{\\prime }\\supset Q$ and $h_{i}(Q^{\\prime })=h_{i}(Q),$ $i=1,...,r.$ But this contradicts the hypothesis of the lemma that $Q\\in CRS$ .", "The following theorem is valid.", "Theorem 4.4.", "$Q\\in CRS$ if and only if for any $x_{0}\\in Q,$ any $f\\in T(Q)$ and any $a_{1},...,a_{r-1}\\in \\mathbb {R}$ the representation $f(x)=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ }x\\in Q,$ subject to the conditions $g_{i}(h_{i}(x_{0}))=a_{i},$ $i=1,...,r-1,$ is unique.", "Equivalently, a set $Q\\in CRS$ if and only if it is a unicity set.", "Theorem 4.4 is an obvious consequence of Lemmas 4.2 and 4.3.", "Remark 4.3.", "In Theorem 4.4, all the words \"any\" can be replaced with the word \"some\".", "Remark 4.4.", "For the case $X=X_{1}\\times \\cdot \\cdot \\cdot \\times X_{n}$ , the possibility and uniqueness of the representation by sums $\\sum _{i=1}^{n}u_{i}(x_{i})$ ,  $u_{i}:X_{i}\\rightarrow \\mathbb {R}$ , $i=1,...,n$ , were investigated in [94], [95].", "Examples.", "Let $r=2,$ $X=\\mathbb {R}^{2},$ $h_{1}(x_{1},x_{2})=x_{1}+x_{2},$ $h_{2}(x_{1},x_{2})=x_{1}-x_{2},$ $Q$ be the graph of the function $x_{2}=\\arcsin (\\sin x_{1}).$ The set $Q$ has no cycles with respect to the functions $h_{1}$ and $h_{2}.$ Therefore, by Proposition 4.2, $Q\\in RS.$ By adding a point $p\\notin Q$ , we obtain the set $Q\\cup \\lbrace p\\rbrace ,$ which contains a cycle and hence is not a representation set.", "Thus, $Q\\in CRS$ and hence $Q$ is a unicity set.", "Let now $r=2,$ $X=\\mathbb {R}^{2},$ $h_{1}(x_{1},x_{2})=x_{1},$ $h_{2}(x_{1},x_{2})=x_{2},$ and $Q$ be the graph of the function $x_{2}=x_{1}.$ Clearly, $Q\\in RS$ and $Q\\notin CRS.$ By the definition of complete representation sets, there is a set $P\\supset Q$ such that $P\\in RS$ and for any $T\\supset P$ , $T$ is not a representation set.", "There are many sets $P$ with this property.", "One of them can be obtained by adding to $Q$ any straight line $l$ parallel to one of the coordinate axes.", "Indeed, if $y\\notin Q\\cup l,$ then the set $Q_{1}=Q\\cup l\\cup \\lbrace y\\rbrace $ contains a four-point cycle (with one vertex as $y$ , two vertices lying on $l$ and one vertex lying on $Q$ ).", "This means that $Q_{1}\\notin RS$ and hence $Q\\cup l\\in CRS.$ The following corollary can be easily obtained from Theorem 4.4 and Lemma 4.2.", "Corollary 4.3.", "$Q\\in CRS$ if and only if $Q\\in RS$ and in the representation $0=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ }x\\in Q,$ all the functions $g_{i},$ $i=1,...,r,$ are constants.", "We have seen that complete representation sets enjoy the unicity property.", "Let us study some other properties of these sets.", "The following properties are valid.", "(a) If $Q_{1},Q_{2}\\in CRS,$ $Q_{1}\\cap Q_{2}\\ne \\emptyset $ and $Q_{1}\\cup Q_{2}\\in RS$ , then $Q_{1}\\cup Q_{2}\\in CRS.$ (b) Let $\\lbrace Q_{\\alpha }\\rbrace ,$ $\\alpha \\in \\Phi ,$ be a family of complete representation sets such that $\\cap _{\\alpha \\in \\Phi }Q_{\\alpha }\\ne \\emptyset $ and $\\cup _{\\alpha \\in \\Phi }Q_{\\alpha }\\in RS.$ Then $\\cup _{\\alpha \\in \\Phi }Q_{\\alpha }\\in CRS.$ The above two properties follow from Corollary 4.3.", "Note that (b) is a generalization of (a).", "The following property is a consequence of (b) and property (2) of representation sets.", "(c) Let $\\lbrace Q_{\\alpha }\\rbrace ,$ $\\alpha \\in \\Phi ,$ be a totally ordered (under inclusion) family of complete representation sets.", "Then $\\cup _{\\alpha \\in \\Phi }Q_{\\alpha }\\in CRS.$ We know that every representation set $A$ is contained in a complete representation set $Q$ such that $h_{i}(A)=h_{i}(Q),$ $i=1,...,r.$ What can we say about the set $Q\\backslash A$ ?", "Clearly, $Q\\backslash A\\in RS.$ But can we chose $Q$ so that $Q\\backslash A\\in CRS$ ?", "The following theorem answers this question.", "Theorem 4.5.", "Let $A\\in RS$ and $A\\notin CRS.$ Then there exists a set $B\\in CRS$ such that $A\\subset B,$ $h_{i}(A)=h_{i}(B),$ $i=1,...,r,$ and $B\\backslash A\\in CRS.$ Since the representation set $A$ is not complete, there exists a point $p\\notin A$ such that $h_{i}(p)\\in h_{i}(A),$ $i=1,...,r,$ and $A^{\\prime }=A\\cup \\lbrace p\\rbrace \\in RS$ .", "By $\\mathcal {M}$ denote the collection of sets $M$ such that 1) $A\\subset M$ and $M\\in RS$ ; 2) $h_{i}(M)=h_{i}(A)$ for all $i=1,...,r$ ; 3) $M\\backslash A\\in CRS.$ Obviously, $\\mathcal {M}$ is not empty.", "It contains the above set $A^{\\prime } $ .", "Consider the partial order on $\\mathcal {M}$ defined by inclusion.", "Let $\\lbrace M_{\\beta }\\rbrace ,$ $\\beta \\in \\Gamma $ , be any chain in $\\mathcal {M}$ .", "The set $\\cup _{\\beta \\in \\Gamma }M_{\\beta }$ is an upper bound for this chain.", "To see this, let us check that $\\cup _{\\beta \\in \\Gamma }M_{\\beta }$ belongs to $\\mathcal {M}$ .", "That is, all the above conditions 1)-3) are satisfied.", "Indeed, 1) $A\\subset \\cup _{\\beta \\in \\Gamma }M_{\\beta }$ and $\\cup _{\\beta \\in \\Gamma }M_{\\beta }\\in RS.$ This follows from property (2) of representation sets; 2) $h_{i}(\\cup _{\\beta \\in \\Gamma }M_{\\beta })=\\cup _{\\beta \\in \\Gamma }h_{i}(M_{\\beta })=\\cup _{\\beta \\in \\Gamma }h_{i}(A)=h_{i}(A),$ $i=1,...,r$ ; 3) $\\cup _{\\beta \\in \\Gamma }M_{\\beta }\\backslash A\\in CRS$ .", "This follows from property (c) of complete representation sets and the facts that $M_{\\beta }\\backslash A\\in CRS$ for any $\\beta \\in \\Gamma $ and the system $\\lbrace M_{\\beta }\\backslash A\\rbrace $ , $\\beta \\in \\Gamma $ , is totally ordered under inclusion.", "Thus we see that any chain in $\\mathcal {M}$ has an upper bound.", "By Zorn's lemma, there are maximal sets in $\\mathcal {M}$ .", "Assume $B$ is one of such sets.", "Let us now prove that $B\\in CRS$ .", "Assume on the contrary that $B\\notin CRS$ .", "Then by Lemma 4.2, for any point $x_{0}\\in B$ the representation $0=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ }x\\in B,(4.13)$ subject to the conditions $g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1,$ is not unique.", "That is, there is a point $y\\in B$ such that for some index $i,$ $g_{i}(h_{i}(y))\\ne 0.$ Without loss of generality we may assume that $g_{r}(h_{r}(y))\\ne 0$ .", "Clearly, $y$ cannot belong to $B\\backslash A$ , since $B\\backslash A\\in CRS$ and over complete representation sets, the zero function has a trivial representation provided that conditions (4.12) hold.", "Thus, $y\\in A$ .", "Let $\\xi \\in X$ be a point such that $h_{i}(\\xi )=h_{i}(x_{0}),$ $i=1,...,r-1$ , and $h_{r}(\\xi )=h_{r}(y).$ The point $\\xi \\notin B,$ otherwise from (4.13) we would obtain that $g_{r}(h_{r}(y))=g_{r}(h_{r}(\\xi ))=0$ .", "Following the techniques in the proof of Lemma 4.3, it can be shown that $B_{1}=B\\cup \\lbrace \\xi \\rbrace \\in RS$ .", "Now we prove that $B_{1}\\backslash A\\in CRS$ .", "Consider the representation $0=\\sum _{i=1}^{r}g_{i}^{\\prime }(h_{i}(x)),\\text{ }x\\in B_{1}\\backslash A,(4.14)$ subject to the conditions $g_{i}^{\\prime }(h_{i}(x_{0}))=0,$ $i=1,...,r-1,$ where $x_{0}$ is some point in $B\\backslash A.$ Such representation holds uniquely on $B\\backslash A,$ since $B\\backslash A\\in CRS$ .", "That is, all the functions $g_{i}^{\\prime }$ are identically zero on $h_{i}(B\\backslash A),$ $i=1,...,r$ .", "On the other hand, since $g_{i}^{\\prime }(h_{i}(\\xi ))=g_{i}^{\\prime }(h_{i}(x_{0}))=0$ , for all $i=1,...,r-1$ , we obtain that $g_{r}^{\\prime }(h_{r}(\\xi ))=0.$ This means that representation (4.14) subject to the conditions $g_{i}^{\\prime }(h_{i}(x_{0}))=0,$ $i=1,...,r-1,$ is unique on $B_{1}\\backslash A.$ That is, all the functions $g_{i}^{\\prime }$ in (4.14) are zero functions on $h_{i}(B_{1}\\backslash A),$ $i=1,...,r.$ Hence by Lemma 4.1, $B_{1}\\backslash A\\in CRS$ .", "Thus, $B_{1}\\in \\mathcal {M}$ .", "But the set $B$ was chosen as a maximal set in $\\mathcal {M}$ .", "We see that our assumption $B\\notin CRS$ leads to the contradiction that there is a set $B_{1}\\in \\mathcal {M}$ bigger than the maximal set $B$ .", "Thus, in fact, $B\\in CRS$ .", "Let $A$ be a representation set.", "The relation on $A$ defined by setting $x\\sim y$ if there is a finite complete representation subset of $A$ containing both $x$ and $y$ , is an equivalence relation.", "Indeed, it is reflexive and symmetric.", "It is transitive by property (a) of complete representation sets.", "The equivalence classes we call $C$-orbits.", "In the case $r=2$ , $C$ -orbits turn into classical orbits considered by Marshall and O'Farrell [122], [123], which have a very nice geometric interpretation in terms of paths (see Section 1.3).", "A classical orbit consists of all possible traces of an arbitrary point in it traveling alternatively in the level sets of $h_{1}$ and $h_{2}.$ In the general setting, one partial case of $C$ -orbits were introduced by Klopotowski, Nadkarni, Rao [95] under the name of related components.", "The case considered in [95] requires that $A\\subset X=X_{1}\\times \\cdot \\cdot \\cdot \\times X_{n}$ and $h_{i}$ be the canonical projections of $X$ onto $X_{i},$ $i=1,...,r,$ respectively.", "Finite complete representation sets containing $x$ and $y$ will be called $C$-trips connecting $x$ and $y$ .", "A $C$ -trip of the smallest cardinality connecting $x$ and $y$ will be called a minimal $C$-trip.", "Theorem 4.6.", "Let $A$ be a representation set and $x$ and $y$ be any two points of some $C$ -orbit in $A$ .", "Then there is only one minimal $C$ -trip connecting them.", "Assume that $L_{1}$ and $L_{2}$ are two minimal $C$ -trips connecting $x$ and $y.$ By the definition, $L_{1}$ and $L_{2}$ are complete representation sets.", "Note that $L_{1}\\cup L_{2}$ is also complete.", "Let us prove that the set $L_{1}\\cap L_{2}$ is complete.", "Clearly, $L_{1}\\cap L_{2}\\in RS.$ Let $x_{0}\\in L_{1}\\cap L_{2}$ .", "In particular, $x_{0}$ can be one of the points $x$ and $y$ .", "Consider the representation $0=\\sum _{i=1}^{r}g_{i}(h_{i}(x)),\\text{ }x\\in L_{1}\\cap L_{2},(4.15)$ subject to $g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1$ .", "On the strength of Lemma 4.1, it is enough to prove that this representation is unique.", "For $i=1,...,r$ , let $g_{i}^{\\prime }$ be any extension of $g_{i}$ from the set $h_{i}(L_{1}\\cap L_{2})$ to the set $h_{i}(L_{1})$ .", "Construct the function $f^{\\prime }(x)=\\sum _{i=1}^{r}g_{i}^{\\prime }(h_{i}(x)),\\text{ }x\\in L_{1}.", "(4.16)$ Since $f^{\\prime }(x)=0$ on $L_{1}\\cap L_{2}$ , the following function is well defined $f(x)=\\left\\lbrace \\begin{array}{c}f^{\\prime }(x),\\text{ }x\\in L_{1}, \\\\0,\\text{ }x\\in L_{2}.\\end{array}\\right.$ Since $L_{1}\\cup L_{2}\\in CRS$ , the representation $f(x)=\\sum _{i=1}^{r}w_{i}(h_{i}(x)),\\text{ }x\\in L_{1}\\cup L_{2}.", "(4.17)$ subject to $w_{i}(h_{i}(x_{0}))=0,\\text{ }i=1,...,r-1.", "(4.18)$ is unique.", "Besides, since $L_{1}\\in CRS$ and $g_{i}^{\\prime }(h_{i}(x_{0}))=g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1$ , representation (4.16) is unique.", "This means that for each function $g_{i}$ , there is only one extension $g^{\\prime }$ .", "Note that $f(x)=f^{\\prime }(x)=\\sum _{i=1}^{r}w_{i}(h_{i}(x)),\\text{ }x\\in L_{1}.$ Now from the uniqueness of representation (4.16) we obtain that $w_{i}(h_{i}(x))=g_{i}^{\\prime }(h_{i}(x)),\\text{ }i=1,...,r,\\text{ }x\\in L_{1}.", "(4.19)$ A restriction of formula (4.17) to the set $L_{2}$ gives $0=\\sum _{i=1}^{r}w_{i}(h_{i}(x)),\\text{ }x\\in L_{2}.", "(4.20)$ Since $L_{2}\\in CRS$ , representation (4.20) subject to conditions (4.18) is unique, whence $w_{i}(h_{i}(x))=0,\\text{ }i=1,...,r\\text{, \\ }x\\in L_{2}.", "(4.21)$ From (4.19) and (4.21) it follows that $g_{i}(h_{i}(x))=g_{i}^{\\prime }(h_{i}(x))=0,\\text{ }i=1,...,r,\\text{ }x\\in L_{1}\\cap L_{2}.$ Thus, we see that representation (4.15) subject to the conditions $g_{i}(h_{i}(x_{0}))=0,$ $i=1,...,r-1$ is unique on the intersection $L_{1}\\cap L_{2}.$ Therefore by Lemma 4.1, $L_{1}\\cap L_{2}\\in CRS.$ Let the cardinalities of $L_{1}$ and $L_{2}$ be equal to $n.$ Since $x,y\\in L_{1}\\cap L_{2}$ and $L_{1}\\cap L_{2}\\in CRS$ , we obtain from the definition of minimal $C$ -trips that the cardinality of $L_{1}\\cap L_{2}$ is also $n.$ Hence, $L_{1}\\cap L_{2}=L_{1}=L_{2}.$ Let $Q$ be a representation set.", "That is, each function $f:Q\\rightarrow \\mathbb {R}$ enjoys representation (4.7).", "Can we construct the functions $g_{i},$ $i=1,...,r,$ for a given $f$ ?", "There is a procedure for constructing one certain collection of $g_{i}$ , provided that $Q$ consists of a single $C$ -orbit, that is, any two points of $Q$ can be connected by a $C$ -trip.", "To describe this procedure, take a point $x_{0}\\in Q$ and fix it.", "We are going to find $g_{i}$ from (4.7) and conditions (4.8).", "Let $y$ be any point in $Q$ .", "To find the values of $g_{i}$ at the points $h_{i}(y),$ $i=1,...,r,$ connect $x_{0}$ and $y$ by a minimal $C$ -trip $S=\\lbrace x_{1},...,x_{n}\\rbrace ,$ where $x_{1}=x_{0}$ and $x_{n}=y.$ Since $S$ is a complete representation set, equation (4.7) subject to (4.8) has a unique solution on $S$ .", "That is, we can find $g_{i}(h_{i}(y)),$ $i=1,...,r,$ by solving the system of linear equations $\\sum _{i=1}^{r}g_{i}(h_{i}(x_{j}))=f(x_{j}),\\text{ }j=1,...,n\\text{.", "}$ We see that each minimal $C$ -trip containing $x_{0}$ generates a system of linear equations, which is uniquely solvable.", "Since any point in $Q$ can be connected with $x_{0}$ by such a trip, we can find $g_{i}(t)$ at each point $t\\in h_{i}(Q),$ $i=1,...,r.$ The above procedure can still be effective for some particular representation sets $Q$ consisting of many $C$ -orbits.", "Let $\\lbrace C_{\\alpha }\\rbrace ,$ $\\alpha \\in \\Lambda ,$ denote the set of all $C$ -orbits of $Q$ .", "Fix some points $x_{\\alpha }\\in C_{\\alpha },$ $\\alpha \\in \\Lambda $ , one in each orbit.", "Let $y_{\\alpha }$ be any points of $C_{\\alpha },$ $\\alpha \\in \\Lambda ,$ respectively.", "We can apply the above procedure of finding the values of $g_{i}$ at each $y_{\\alpha }$ if $h_{i}(y_{\\alpha })\\ne h_{i}(y_{\\beta })$ for all $i$ and $\\alpha \\ne \\beta $ .", "For $h_{i}(y_{\\alpha })=h_{i}(y_{\\beta }),$ one cannot guarantee that after solving the corresponding systems of linear equations (associated with $y_{\\alpha }$ and $y_{\\beta }$ ), the solutions $g_{i}(h_{i}(y_{\\alpha })$ and $g_{i}(h_{i}(y_{\\beta }))$ will be equal.", "That is, for the case $h_{i}(y_{\\alpha })=h_{i}(y_{\\beta })$ , the constructed functions $g_{i}$ may not be well defined.", "Remark 4.5.", "All the results in this section are valid, in particular, for linear combinations of generalized ridge functions.", "Neural networks have increasingly been used in many areas of applied sciences.", "Most of the applications employ neural networks to approximate complex nonlinear functional dependencies on a high dimensional data set.", "The theoretical justification for such applications is that any continuous function can be approximated within an arbitrary precision by carefully selecting parameters in the network.", "The most commonly used model of neural networks is the multilayer feedforward perceptron (MLP) model.", "This model consists of a finite number of successive layers.", "The first and the last layers are called the input and the output layers, respectively.", "The intermediate layers are called hidden layers.", "MLP models are usually classified not by their number of layers, but by their number of hidden layers.", "In this chapter, we study approximation properties of the single and two hidden layer feedforward perceptron models.", "Our analysis is based on ridge functions and the Kolmogorov superposition theorem.", "The material of this chapter may be found in [51], [68], [69], [72].", "In this section, we consider single hidden layer neural networks with a set of weights consisting of a finite number of directions or straight lines.", "For certain activation functions, we characterize compact sets $X$ in the $d$ -dimensional space such that the corresponding neural network can approximate any continuous function on $X$ .", "Approximation capabilities of neural networks have been investigated in a great deal of works over the last 30 years (see, e.g., [6], [7], [15], [24], [26], [27], [29], [30], [31], [34], [50], [51], [52], [54], [80], [81], [83], [84], [108], [120], [127], [135], [140], [153]).", "In this section, we are interested in questions of density of a single hidden layer perceptron model.", "A typical density result shows that this model can approximate an arbitrary function in a given class with any degree of accuracy.", "A single hidden layer perceptron model with $r$ units in the hidden layer and input $\\mathbf {x}=(x_{1},...,x_{d})$ evaluates a function of the form $\\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i}),(5.1)$ where the weights $\\mathbf {w}^{i}$ are vectors in $\\mathbb {R}^{d}$ , the thresholds $\\theta _{i}$ and the coefficients $c_{i}$ are real numbers and the activation function $\\sigma $ is a univariate function, which is considered to be continuous here.", "Note that in Eq (5.1) each function $\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i})$ is a ridge function with the direction $\\mathbf {w}^{i}$ .", "For various activation functions $\\sigma $ , it has been proved in a number of papers that one can approximate arbitrarily well a given continuous function by functions of the form (5.1) ($r$ is not fixed!)", "over any compact subset of $\\mathbb {R}^{d}$ .", "In other words, the set $\\mathcal {M}(\\sigma )=span\\text{\\ }\\lbrace \\sigma (\\mathbf {w\\cdot x}-\\theta ):\\ \\theta \\in \\mathbb {R}\\text{, }\\mathbf {w\\in }\\mathbb {R}^{d}\\rbrace $ is dense in the space $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compact sets (see, e.g., [26], [31], [54], [80], [81]).", "The most general result of this type belongs to Leshno, Lin, Pinkus and Schocken [108].", "They proved that a necessary and sufficient condition for a continuous activation function to have the density property is that it not be a polynomial.", "This result shows the efficacy of the single hidden layer perceptron model within all possible choices of the activation function $\\sigma $ , provided that $\\sigma $ is continuous.", "In fact, density of the set $\\mathcal {M}(\\sigma )$ also holds for some reasonable sets of weights and thresholds.", "(see[135]).", "Some authors showed that a single hidden layer perceptron with a suitably restricted set of weights can also have the density property (or, in neural network terminology, the universal approximation property).", "For example, White and Stinchcombe [153] proved that a single layer network with a polygonal, polynomial spline or analytic activation function and a bounded set of weights has the density property.", "Ito [81] investigated this property of networks using monotone sigmoidal functions (tending to 0 at minus infinity and 1 at infinity), with only weights located on the unit sphere.", "We see that weights required for the density property are not necessary to be of an arbitrarily large magnitude.", "But what if they are too restricted.", "How can one learn approximation properties of networks with an arbitrarily restricted set of weights?", "This problem is too difficult to be solved completely in this general formulation.", "But there are some cases that deserve a special attention.", "The most interesting case is, of course, neural networks with weights varying on a finite set of directions or lines.", "To the best of our knowledge, approximation capabilities of such networks have not been studied yet.", "More precisely, let $W$ be a set of weights consisting of a finite number of vectors (or straight lines) in $\\mathbb {R}^{d}$ .", "It is clear that if $w$ varies only in $W$ , the set $\\mathcal {M}(\\sigma )$ can not be dense in $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compacta (compact sets).", "In this case, one may want to determine boundaries of efficacy of the model.", "Over which compact sets $X\\subset \\mathbb {R}^{d}$ does the model preserve its general propensity to approximate arbitrarily well every continuous multivariate function?", "In Section 5.1.2, we will consider this problem and give both sufficient and necessary conditions for well approximation (approximation with arbitrary precision) by networks with weights from a finite set of directions or lines.", "For a set $W$ of weights consisting of two vectors, we show that there is a geometrically explicit solution to the problem.", "In Section 5.1.3, we discuss some aspects of the exact representation by neural networks with weights varying on finitely many straight lines.", "In this subsection we give a sufficient and also a necessary conditions for approximation by neural networks with finitely many weights and with weights varying on a finite set of straight lines (through the origin).", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ .", "Consider the following set functions $\\tau _{i}(Z)=\\lbrace \\mathbf {x}\\in Z:~|p_{i}^{-1}(p_{i}(\\mathbf {x}))\\bigcap Z|\\ge 2\\rbrace ,\\quad Z\\subset X,~i=1,\\ldots ,k,$ where $p_{i}(\\mathbf {x})=\\mathbf {a}^{i}\\cdot \\mathbf {x}$ , $|Y|$ denotes the cardinality of a considered set $Y$ .", "Define $\\tau (Z)$ to be $\\bigcap _{i=1}^{k}\\tau _{i}(Z)$ and define $\\tau ^{2}(Z)=\\tau (\\tau (Z))$ , $\\tau ^{3}(Z)=\\tau (\\tau ^{2}(Z))$ and so on inductively.", "These functions first appeared in the work [149] by Sternfeld, where he investigated problems of representation by linear superpositions.", "Clearly, $\\tau (Z)\\supseteq \\tau ^{2}(Z)\\supseteq \\tau ^{3}(Z)\\supseteq ...$ It is possible that for some $n$ , $\\tau ^{n}(Z)=\\emptyset .$ In this case, one can see that $Z$ does not contain a cycle.", "In general, if some set $Z\\subset X$ forms a cycle, then $\\tau ^{n}(Z)=Z.$ But the reverse is not true.", "Indeed, let $Z=X=\\lbrace (0,0,\\frac{1}{2}),(0,0,1),(0,1,0),(1,0,1),(1,1,0),(\\frac{1}{2},\\frac{1}{2},0),(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})\\rbrace $ , $\\mathbf {a}^{i},i=1,2,3,$ are the coordinate directions in $\\mathbb {R}^{3}$ .", "It is not difficult to verify that $X$ does not possess cycles with respect to these directions and at the same time $\\tau (X)=X$ (and so $\\tau ^{n}(X)=X$ for every $n)$ .", "Consider the linear combinations of ridge functions with fixed directions $\\mathbf {a}^{1},...,\\mathbf {a}^{k}$ $\\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) =\\left\\lbrace \\sum \\limits _{i=1}^{k}g_{i}\\left( \\mathbf {a}^{i}\\cdot \\mathbf {x}\\right):g_{i}\\in C(\\mathbb {R)},~i=1,...,k\\right\\rbrace .", "(5.2)$ Let $K$ be a family of functions defined on $\\mathbb {R}^{d}$ and $X$ be a subset of $\\mathbb {R}^{d}.$ By $K_{X}$ we will denote the restriction of this family to $X.$ Thus $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ stands for the set of sums of ridge functions in (5.2) defined on $X$ .", "The following theorem is a particular case of the known general result of Sproston and Strauss [146] established for the sum of subalgebras of $C(X)$ .", "Theorem 5.1.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ .", "If $\\cap _{n=1,2,...}\\tau ^{n}(X)=\\emptyset $ , then the set $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ is dense in $C(X)$ .", "In our analysis, we need the following lemma.", "Lemma 5.1.", "If $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ is dense in $C(X),$ then the set $X$ does not contain a cycle with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{k}$ .", "Suppose the contrary.", "Suppose that the set $X$ contains cycles.", "Each cycle $l=(x_{1},\\ldots ,x_{n})$ and the associated vector $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ generate the functional $G_{l,\\lambda }(f)=\\sum _{j=1}^{n}\\lambda _{j}f(x_{j}),\\quad f\\in C(X).$ Clearly, $G_{l,\\lambda }$ is linear and continuous with the norm $\\sum _{j=1}^{n}|\\lambda _{j}|.$ It is not difficult to verify that $G_{l,\\lambda }(g)=0$ for all functions $g\\in \\mathcal {R}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) .$ Let $f_{0}$ be a continuous function such that $f_{0}(x_{j})=1$ if $\\lambda _{j}>0$ and $f_{0}(x_{j})=-1$ if $\\lambda _{j}<0$ , $j=1,\\ldots ,n$ .", "For this function, $G_{l,\\lambda }(f_{0})\\ne 0$ .", "Thus, we have constructed a nonzero linear functional which belongs to the annihilator of the manifold $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ .", "This means that $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ is not dense in $C(X)$ .", "The obtained contradiction proves the lemma.", "Now we are ready to step forward from ridge function approximation to neural networks.", "Let $\\sigma \\in C(\\mathbb {R)}$ be a continuous activation function.", "For a subset $W\\subset \\mathbb {R}^{d},$ let $\\mathcal {M}(\\sigma ;W,\\mathbb {R})$ stand for the set of neural networks with weights from $W.$ That is, $\\mathcal {M}(\\sigma ;W,\\mathbb {R})=span\\lbrace \\sigma (\\mathbf {w}\\cdot \\mathbf {x}-\\theta ):~\\mathbf {w}\\in W,~\\theta \\in \\mathbb {R}\\rbrace .$ Theorem 5.2.", "Let $\\sigma \\in C(\\mathbb {R})\\cap L_{p}(\\mathbb {R)}$ , where $1\\le p<\\infty $ , or $\\sigma $ be a continuous, bounded, nonconstant function, which has a limit at infinity (or minus infinity).", "Let $W=\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\rbrace \\subset \\mathbb {R}^{d}$ be the given set of weights and $X$ be a compact subset of $\\mathbb {R}^{d}$ .", "The following assertions are valid: (1) if $\\cap _{n=1,2,...}\\tau ^{n}(X)=\\emptyset $ , then the set $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ is dense in the space of all continuous functions on $X$ .", "(2) if $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ is dense in $C(X)$ , then the set $X$ does not contain cycles.", "Part (1).", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ for which $\\cap _{n=1,2,...}\\tau ^{n}(X)=\\emptyset $ .", "By Theorem 5.1, the set $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ is dense in $C(X)$ .", "This means that for any positive real number $\\varepsilon $ there exist continuous univariate functions $g_{i},$ $i=1,...,k$ such that $\\left|f(\\mathbf {x})-\\sum _{i=1}^{k}{g_{i}\\left( \\mathbf {a}^{i}{\\cdot }\\mathbf {x}\\right) }\\right|<\\frac{\\varepsilon }{k+1}(5.3)$ for all $\\mathbf {x}\\in X$ .", "Since $X$ is compact, the sets $Y_{i}=\\lbrace \\mathbf {a}^{i}{\\cdot }\\mathbf {x:\\ x}\\in X\\rbrace ,\\ i=1,2,...,k$ are also compacts.", "In 1947, Schwartz [141] proved that continuous and $p$ -th degree Lebesgue integrable univariate functions or continuous, bounded, nonconstant functions having a limit at infinity (or minus infinity) are not mean-periodic.", "Note that a function $f\\in C(\\mathbb {R}^{d})$ is called mean periodic if the set $span$ $\\lbrace f(\\mathbf {x}-\\mathbf {b}):\\ \\mathbf {b}\\in \\mathbb {R}^{d}\\rbrace $ is not dense in $C(\\mathbb {R}^{d})$ in the topology of uniform convergence on compacta (see [141]).", "Thus, Schwartz proved that the set $span\\text{\\ }\\lbrace \\sigma (y-\\theta ):\\ \\theta \\in \\mathbb {R}\\rbrace $ is dense in $C(\\mathbb {R)}$ in the topology of uniform convergence.", "We learned about this result from Pinkus [135].", "This density result means that for the given $\\varepsilon $ there exist numbers $c_{ij},\\theta _{ij}\\in \\mathbb {R}$ , $i=1,2,...,k$ , $j=1,...,m_{i}$ such that $\\left|g_{i}(y)-\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (y-\\theta _{ij})\\right|\\,<\\frac{\\varepsilon }{k+1}(5.4)$ for all $y\\in Y_{i},\\ i=1,2,...,k.$ From (5.3) and (5.4) we obtain that $\\left\\Vert f(\\mathbf {x})-\\sum _{i=1}^{k}\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (\\mathbf {a}^{i}{\\cdot }\\mathbf {x}-\\theta _{ij})\\right\\Vert _{C(X)}<\\varepsilon .", "(5.5)$ Hence $\\overline{\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}=C(X).$ Part (2).", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ and the set $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ be dense in $C(X).$ Then for an arbitrary positive real number $\\varepsilon $ , inequality (5.5) holds with some coefficients $c_{ij},\\theta _{ij},\\ i=1,2,\\ j=1,...,m_{i}.$ Since for each $i=1,2,...,k$ , the function $\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (\\mathbf {a}^{i}{\\cdot }\\mathbf {x}-\\theta _{ij})$ is a function of the form $g_{i}(\\mathbf {a}^{i}{\\cdot }\\mathbf {x}),$ the subspace $\\mathcal {R}_{X}\\left(\\mathbf {a}^{1},...,\\mathbf {a}^{k}\\right) $ is dense in $C(X)$ .", "Then by Lemma 5.1, the set $X$ contains no cycles.", "The above theorem still holds if the set of weights $W=\\lbrace \\mathbf {a}^{1},...,\\mathbf {a}^{k}\\rbrace $ is replaced by the set $W_{1}=\\lbrace t_{1}\\mathbf {a}^{1},...,t_{k}\\mathbf {a}^{k}:\\ t_{1},...,t_{k}\\in \\mathbb {R}\\rbrace $ .", "In fact, for $W_{1}$ , the above restrictions on the activation function $\\sigma $ may be weakened.", "Theorem 5.3.", "Assume $\\sigma \\in C(\\mathbb {R})$ is not a polynomial.", "Let $W_{1}=\\lbrace t_{1}\\mathbf {a}^{1},...,t_{k}\\mathbf {a}^{k}:\\ t_{1},...,t_{k}\\in \\mathbb {R}\\rbrace $ be the given set of weights and $X$ be a compact subset of $\\mathbb {R}^{d}$ .", "The following assertions are valid: (1) if $\\cap _{n=1,2,...}\\tau ^{n}(X)=\\emptyset $ , then the set $\\mathcal {M}_{X}(\\sigma ;W_{1},\\mathbb {R})$ is dense in the space of all continuous functions on $X$ .", "(2) if $\\mathcal {M}_{X}(\\sigma ;W_{1},\\mathbb {R})$ is dense in $C(X)$ , then the set $X$ does not contain cycles.", "The proof of this theorem is similar to that of Theorem 5.2 and based on the following result of Leshno, Lin, Pinkus and Schocken [108]: if $\\sigma $ is not a polynomial, then the set $span\\text{\\ }\\lbrace \\sigma (ty-\\theta ):\\ t,\\theta \\in \\mathbb {R}\\rbrace $ is dense in $C(\\mathbb {R)}$ in the topology of uniform convergence on compacta.", "The above example with the set $\\lbrace (0,0,\\frac{1}{2}),(0,0,1),(0,1,0),(1,0,1),(1,1,0),(\\frac{1}{2},\\frac{1}{2},0),(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})\\rbrace $ shows that the sufficient condition in part (1) of Theorem 5.2 is not necessary.", "The necessary condition in part (2), in general, is not sufficient.", "But it is not easily seen.", "Here, is the nontrivial example showing that nonexistence of cycles is not sufficient for the density $\\overline{\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}=C(X).$ For the sake of simplicity, we restrict ourselves to $\\mathbb {R}^{2}.$ Let $\\mathbf {a}^{1}=(1;1),$ $\\mathbf {a}^{2}=(1;-1)$ and the set of weights $W=\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace .$ The set $X$ can be constructed as follows.", "Let $X_{1} $ be the union of the four line segments $[(-3;0),(-1;0)],$ $[(-1;2),(1;2)],$ $[(1;0),(3;0)]$ and $[(-1;-2),(1;-2)].$ Rotate one segment in $X_{1}$ $90^{\\circ }$ about its center and remove the middle one-third from each line segment.", "The obtained set denote by $X_{2}$ .", "By the same way, one can construct $X_{3},X_{4},$ and so on.", "It is clear that the set $X_{i}$ has $2^{i+1}$ line segments.", "Let $X$ be a limit of the sets $X_{i}$ , $i=1,2,...$ .", "Note that there are no cycles.", "By $S_{i}$ , $i=\\overline{1,4},$ denote the closed discs with the unit radius and centered at the points $(-2;0),$ $(0;2),$ $(2;0)$ and $(0;-2)$ respectively.", "Consider a continuous function $f_{0}$ such that $f_{0}(\\mathbf {x})=1$ for $\\mathbf {x}\\in (S_{1}\\cup S_{3})\\cap X$ , $f_{0}(\\mathbf {x})=-1$ for $\\mathbf {x}\\in (S_{2}\\cup S_{4})\\cap X$ , and $-1<f_{0}(\\mathbf {x})<1$ elsewhere on $\\mathbb {R}^{2}$ .", "Let $p=(\\mathbf {y}^{1},\\mathbf {y}^{2},...)$ be any infinite path in $X.$ Note that the points $\\mathbf {y}^{i}, $ $i=1,2,...,$ are alternatively in the sets $(S_{1}\\cup S_{3})\\cap X$ and $(S_{2}\\cup S_{4})\\cap X$ .", "Obviously, $E(f_{0},X)\\overset{def}{=}\\inf _{g\\in \\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) }\\left\\Vert f_{0}-g\\right\\Vert _{C(X)}\\le \\left\\Vert f_{0}\\right\\Vert _{C(X)}=1.", "(5.6)$ For each positive integer $k=1,2,...$ , set $p_{k}=(\\mathbf {y}^{1},...,\\mathbf {y}^{k})$ and consider the path functionals $G_{p_{k}}(f)=\\frac{1}{k}\\sum _{i=1}^{k}(-1)^{i-1}f(\\mathbf {y}^{i}).$ $G_{p_{k}}$ is a continuous linear functional obeying the following obvious properties: $\\left\\Vert G_{p_{k}}\\right\\Vert =G_{p_{k}}(f_{0})=1;$ $G_{p_{k}}(g_{1}+g_{2})\\le \\frac{2}{k}(\\left\\Vert g_{1}\\right\\Vert +\\left\\Vert g_{2}\\right\\Vert )$ for ridge functions $g_{1}={g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) }$ and $g_{2}={g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) .", "}$ $\\left\\Vert G_{p_{k}}\\right\\Vert =G_{p_{k}}(f_{0})=1;$ $G_{p_{k}}(g_{1}+g_{2})\\le \\frac{2}{k}(\\left\\Vert g_{1}\\right\\Vert +\\left\\Vert g_{2}\\right\\Vert )$ for ridge functions $g_{1}={g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) }$ and $g_{2}={g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) .", "}$ By property (1), the sequence $\\lbrace G_{p_{k}}\\rbrace _{k=1}^{\\infty }$ has a weak$^{\\text{*}}$ cluster point.", "This point will be denoted by $G.$ By property (2), $G\\in \\mathcal {R} _{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right)^{\\bot }.$ Therefore, $1=G(f_{0})=G(f_{0}-g)\\le \\left\\Vert f_{0}-g\\right\\Vert _{C(X)}\\text{ \\ forany }g\\in \\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) .$ Taking $\\inf $ over $g$ in the right-hand side of the last inequality, we obtain that $1\\le E(f_{0},X).$ Now it follows from (5.6) that $E(f_{0},X)=1.", "$ Recall that $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})\\subset \\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) .$ Thus $\\inf _{h\\in \\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}\\left\\Vert f-h\\right\\Vert _{C(X)}\\ge 1.$ The last inequality finally shows that $\\overline{\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}\\ne C(X).$ For neural networks with weights consisting of only two vectors (or directions) the problem of density becomes more clear.", "In this case, under some minor restrictions on $X,$ the necessary condition in part (2) of Theorem 5.2 (nonexistence of cycles) is also sufficient for the density of $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ in $C(X)$ .", "These restrictions are imposed on the following equivalent classes of $X$ induced by paths.", "The relation $\\mathbf {x}\\sim \\mathbf {y}$ when $\\mathbf {x}$ and $\\mathbf {y}$ belong to some path in a given compact set $X\\subset \\mathbb {R}^{d}$ defines an equivalence relation.", "Recall that the equivalence classes are called orbits (see Section 1.3.4).", "Theorem 5.4.", "Let $\\sigma \\in C(\\mathbb {R})\\cap L_{p}(\\mathbb {R)}$ , where $1\\le p<\\infty $ , or $\\sigma $ be a continuous, bounded, nonconstant function, which has a limit at infinity (or minus infinity).", "Let $W=\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace \\subset \\mathbb {R}^{d}$ be the given set of weights and $X$ be a compact subset of $\\mathbb {R}^{d}$ with all its orbits closed.", "Then $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R}) $ is dense in the space of all continuous functions on $X$ if and only if $X$ contains no closed paths with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Sufficiency.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ with all its orbits closed.", "Besides, let $X$ contain no closed paths.", "By Theorem 1.6 (see Section 1.3.4), the set $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is dense in $C(X)$ .", "This means that for any positive real number $\\varepsilon $ there exist continuous univariate functions $g_{1}$ and $g_{2}$ such that $\\left|f(\\mathbf {x})-{g_{1}\\left( \\mathbf {a}^{1}{\\cdot }\\mathbf {x}\\right) -g_{2}\\left( \\mathbf {a}^{2}{\\cdot }\\mathbf {x}\\right) }\\right|<\\frac{\\varepsilon }{3}(5.7)$ for all $\\mathbf {x}\\in X$ .", "Since $X$ is compact, the sets $Y_{i}=\\lbrace \\mathbf {a}^{i}{\\cdot }\\mathbf {x:\\ x}\\in X\\rbrace ,\\ i=1,2,$ are also compacts.", "As mentioned above, Schwartz [141] proved that continuous and $p$ -th degree Lebesgue integrable univariate functions or continuous, bounded, nonconstant functions having a limit at infinity (or minus infinity) are not mean-periodic.", "Thus, the set $span\\text{\\ }\\lbrace \\sigma (y-\\theta ):\\ \\theta \\in \\mathbb {R}\\rbrace $ is dense in $C(\\mathbb {R)}$ in the topology of uniform convergence.", "This density result means that for the given $\\varepsilon $ there exist numbers $c_{ij},\\theta _{ij}\\in \\mathbb {R}$ , $i=1,2,$ $j=1,\\dots ,m_{i}$ such that $\\left|g_{i}(y)-\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (y-\\theta _{ij})\\right|\\,<\\frac{\\varepsilon }{3}(5.8)$ for all $y\\in Y_{i},\\ i=1,2.$ From (5.7) and (5.8) we obtain that $\\left\\Vert f(\\mathbf {x})-\\sum _{i=1}^{2}\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (\\mathbf {a}^{i}{\\cdot }\\mathbf {x}-\\theta _{ij})\\right\\Vert _{C(X)}<\\varepsilon .", "(5.9)$ Hence $\\overline{\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}=C(X).$ Necessity.", "Let $X$ be a compact subset of $\\mathbb {R}^{n}$ with all its orbits closed and the set $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ be dense in $C(X).$ Then for an arbitrary positive real number $\\varepsilon $ , inequality (5.9) holds with some coefficients $c_{ij},\\theta _{ij},\\ i=1,2,\\ j=1,\\dots ,m_{i}.$ Since for $i=1,2,$ $\\sum _{j=1}^{m_{i}}c_{ij}\\sigma (\\mathbf {a}^{i}{\\cdot }\\mathbf {x}-\\theta _{ij})$ is a function of the form $g_{i}(\\mathbf {a}^{i}{\\cdot }\\mathbf {x}),$ the subspace $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is dense in $C(X)$ .", "Then by Theorem 1.6, the set $X$ contains no closed paths.", "Remark 5.1.", "It can be shown that the necessity of the theorem is valid without any restriction on orbits of $X$ .", "Indeed if $X$ contains a closed path, then it contains a closed path $p=(\\mathbf {x}^{1},\\dots ,\\mathbf {x}^{2m})$ with different points.", "The functional $G_{p}=\\sum _{i=1}^{2m}(-1)^{i-1}f(\\mathbf {x}^{i})$ belongs to the annihilator of the subspace $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) .$ There exist nontrivial continuous functions $f_{0}$ on $X$ such that $G_{p}(f_{0})\\ne 0$ (take, for example, any continuous function $f_{0}$ taking values $+1$ at $\\lbrace \\mathbf {x}^{1},\\mathbf {x}^{3},\\dots ,\\mathbf {x}^{2m-1}\\rbrace $ , $-1$ at $\\lbrace \\mathbf {x}^{2},\\mathbf {x}^{4},\\dots ,\\mathbf {x}^{2m}\\rbrace $ and $-1<f_{0}(\\mathbf {x})<1$ elsewhere).", "This shows that the subspace $\\mathcal {R}_{X}\\left( \\mathbf {a}^{1},\\mathbf {a}^{2}\\right) $ is not dense in $C(X)$ .", "But in this case, the set $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ cannot be dense in $C(X)$ .", "The obtained contradiction means that our assumption is not true and $X$ contains no closed paths.", "Theorem 5.4 remains valid if the set of weights $\\ W=\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace $ is replaced by the set $W_{1}=\\lbrace t_{1}\\mathbf {a}^{1},t_{2}\\mathbf {a}^{2}:\\ t_{1},t_{2}\\in \\mathbb {R\\rbrace }$ .", "In fact, for the set $W_{1}$ , the required conditions on $\\sigma $ may be weakened.", "As in Theorem 5.3, the activation function $\\sigma $ can be taken only non-polynomial.", "Theorem 5.5.", "Assume $\\sigma \\in C(\\mathbb {R})$ is not a polynomial.", "Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be fixed vectors and $W_{1}=\\lbrace t_{1}\\mathbf {a}^{1},t_{2}\\mathbf {a}^{2}:\\ t_{1},t_{2}\\in \\mathbb {R\\rbrace }$ be the set of weights.", "Let $X$ be a compact subset of $\\mathbb {R}^{d}$ with all its orbits closed.", "Then $\\mathcal {M}_{X}(\\sigma ;W_{1},\\mathbb {R})$ is dense in the space of all continuous functions on $X$ if and only if $X$ contains no closed paths with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "The proof is analogous to that of Theorem 5.4 and based on the above mentioned result of Leshno, Lin, Pinkus and Schocken [108].", "Examples: (a) Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two noncollinear vectors in $\\mathbb {R}^{2}.$ Let $B=B_{1}...B_{k}$ be a broken line with the sides $B_{i}B_{i+1},\\ i=1,...,k-1,$ alternatively perpendicular to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Besides, let $B$ does not contain vertices of any parallelogram with sides perpendicular to these vectors.", "Then the set $\\mathcal {M}_{B}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(B).$ (b) Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two noncollinear vectors in $\\mathbb {R}^{2}.$ If $X$ is the union of two parallel line segments, not perpendicular to any of the vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , then the set $\\mathcal {M}_{X}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(X).$ (c) Let now $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two collinear vectors in $\\mathbb {R}^{2}.$ Note that in this case any path consisting of two points is automatically closed.", "Thus the set $\\mathcal {M}_{X}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(X)$ if and only if $X$ contains no path different from a singleton.", "A simple example is a line segment not perpendicular to the given direction.", "(d) Let $X$ be a compact set with an interior point.", "Then Theorem 5.4 fails, since any such set contains vertices of some parallelogram with sides perpendicular to the given directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , that is a closed path.", "Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two noncollinear vectors in $\\mathbb {R}^{2}.$ Let $B=B_{1}...B_{k}$ be a broken line with the sides $B_{i}B_{i+1},\\ i=1,...,k-1,$ alternatively perpendicular to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Besides, let $B$ does not contain vertices of any parallelogram with sides perpendicular to these vectors.", "Then the set $\\mathcal {M}_{B}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(B).$ Let $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two noncollinear vectors in $\\mathbb {R}^{2}.$ If $X$ is the union of two parallel line segments, not perpendicular to any of the vectors $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , then the set $\\mathcal {M}_{X}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(X).$ Let now $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ be two collinear vectors in $\\mathbb {R}^{2}.$ Note that in this case any path consisting of two points is automatically closed.", "Thus the set $\\mathcal {M}_{X}(\\sigma ;\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace ,\\mathbb {R})$ is dense in $C(X)$ if and only if $X$ contains no path different from a singleton.", "A simple example is a line segment not perpendicular to the given direction.", "Let $X$ be a compact set with an interior point.", "Then Theorem 5.4 fails, since any such set contains vertices of some parallelogram with sides perpendicular to the given directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ , that is a closed path.", "In this subsection we give a necessary condition for the representation of functions by neural networks with weights from a finitely many straight lines.", "Before formulating our result, we introduce new objects, namely semicycles with respect to directions $\\mathbf {a}^{1},...,\\mathbf {a}^{k}\\in \\mathbb {R}^{d}\\backslash \\lbrace \\mathbf {0}\\rbrace $ .", "Definition 5.1.", "A set of points $l=(\\mathbf {x}^{1},\\ldots ,\\mathbf {x}^{n})\\subset \\mathbb {R}^{d}$ is called a semicycle with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{k}$ if there exists a vector $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})\\in \\mathbf {Z}^{n}\\setminus \\lbrace \\mathbf {0}\\rbrace $ such that for any $i=1,\\ldots ,k,$ we have $\\sum _{j=1}^{n}\\lambda _{j}\\delta _{\\mathbf {a}^{i}\\cdot \\mathbf {x}^{j}}=\\sum _{s=1}^{r_{i}}\\lambda _{i_{s}}\\delta _{\\mathbf {a}^{i}\\cdot \\mathbf {x}^{i_{s}}},\\quad where~r_{i}\\le k.(5.10)$ Here $\\delta _{a}$ is the characteristic function of the single point set $\\lbrace a\\rbrace $ .", "Note that for $i=1,\\ldots ,k$ , the set $\\lbrace \\lambda _{i_{s}},~s=1,...,r_{i}\\rbrace $ is a subset of the set $\\lbrace \\lambda _{j},~j=1,...,n\\rbrace $ .", "Thus, Eq.", "(5.10) means that for each $i$ , we actually have at most $k$ terms in the sum $\\sum _{j=1}^{n}\\lambda _{j}\\delta _{\\mathbf {a}^{i}\\cdot \\mathbf {x}^{j}}$ .", "Recall that if in (5.10) for any $i=1,\\ldots ,k$ , we have $\\sum _{j=1}^{n}\\lambda _{j}\\delta _{\\mathbf {a}^{i}\\cdot \\mathbf {x}^{j}}=0,$ then the set $l=(\\mathbf {x}^{1},\\ldots ,\\mathbf {x}^{n})$ is a cycle with respect to the directions $\\mathbf {a}^{1},...,\\mathbf {a}^{k}$ (see Section 1.2).", "Thus a cycle is a special case of a semicycle.", "Let us give a simple example of a semicycle.", "Assume $k=2$ and $\\mathbf {a}^{1}\\cdot \\mathbf {x}^{1}=\\mathbf {a}^{1}\\cdot \\mathbf {x}^{2}$ , $\\mathbf {a}^{2}\\cdot \\mathbf {x}^{2}=\\mathbf {a}^{2}\\cdot \\mathbf {x}^{3}$ , $\\mathbf {a}^{1}\\cdot \\mathbf {x}^{3}=\\mathbf {a}^{1}\\cdot \\mathbf {x}^{4}$ ,..., $\\mathbf {a}^{2}\\cdot \\mathbf {x}^{n-1}=\\mathbf {a}^{2}\\cdot \\mathbf {x}^{n}$ .", "Then it is not difficult to see that for a vector $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ with the components $\\lambda _{j}=(-1)^{j},$ the following equalities hold: $\\sum _{j=1}^{n}\\lambda _{j}\\delta _{\\mathbf {a}^{1}\\cdot \\mathbf {x}^{j}}&=&\\lambda _{n}\\delta _{\\mathbf {a}^{1}\\cdot \\mathbf {x}^{n}}, \\\\\\sum _{j=1}^{n}\\lambda _{j}\\delta _{\\mathbf {a}^{2}\\cdot \\mathbf {x}^{j}}&=&\\lambda _{1}\\delta _{\\mathbf {a}^{2}\\cdot \\mathbf {x}^{1}}.$ Thus, by Definition 5.1, the set $l=\\lbrace \\mathbf {x}^{1},\\ldots ,\\mathbf {x}^{n}\\rbrace $ is a semicycle with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ .", "Note that this set, in the given order of its points, forms a path with respect to the directions $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ (see Section 1.3).", "It is not difficult to see that any path with respect to $\\mathbf {a}^{1}$ and $\\mathbf {a}^{2}$ is a semicycle with respect to these directions.", "But semicycles may also involve some union of paths.", "Note that one can construct many semicycles by adding not more than $k$ arbitrary points to a cycle with respect to the directions $\\mathbf {a}^{1},\\mathbf {a}^{2},...,\\mathbf {a}^{k}$ .", "A cycle (or semicycle) $l$ is called a $q$ -cycle ($q$ -semicycle) if the vector $\\lambda $ associated with $l$ can be chosen so that $\\left|\\lambda _{i}\\right|\\le q,$ $i=1,...,n,$ and $q$ is the minimal number with this property.", "The semicycle considered above is a 1-semicycle.", "If in that example, $\\mathbf {a}^{2}\\cdot \\mathbf {x}^{n-1}=\\mathbf {a}^{2}\\cdot \\mathbf {x}^{1}$ , then the set $\\lbrace x_{1},x_{2},...,x_{n-1}\\rbrace $ is a 1-cycle.", "Let us give a simple example of a 2-cycle with respect to the directions $\\mathbf {a}^{1}=(1,0)$ and $\\mathbf {a}^{2}=(0,1)$ .", "Consider the union $\\lbrace 0,1\\rbrace ^{2}\\cup \\lbrace 0,2\\rbrace ^{2}=\\lbrace (0,0),(1,1),(2,2),(0,1),(1,0),(0,2),(2,0)\\rbrace .$ It is easy to see that this set is a 2-cycle with the associated vector $(2,1,1,-1,-1,-1,-1).$ Similarly, one can construct a $q$ -cycle or $q$ -semicycle for any positive integer $q$ .", "Theorem 5.6.", "Assume $W=\\lbrace t_{1}\\mathbf {a}^{1},...,t_{k}\\mathbf {a}^{k}:\\ t_{1},...,t_{k}\\in \\mathbb {R}\\rbrace $ is the given set of weights.", "If $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})=C(X)$ , then $X$ contains no cycles and the lengths (number of points) of all $q$ -semicycles in $X$ are bounded by some positive integer.", "Let $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})=C(X).$ Then $\\mathcal {R}_{1}+\\mathcal {R}_{2}+...+\\mathcal {R}_{k}=C\\left( X\\right) $ , where $\\mathcal {R}_{i}=\\lbrace g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x):~}g_{i}\\in C(\\mathbb {R)}\\rbrace ,~i=1,2,...,k.$ Consider the linear space $\\mathcal {U}=\\prod _{i=1}^{k}\\mathcal {R}_{i}=\\lbrace (g_{1},\\ldots ,g_{k}):~g_{i}\\in \\mathcal {R}_{i},~i=1,\\ldots ,k\\rbrace $ endowed with the norm $\\Vert (g_{1},\\ldots ,g_{k})\\Vert =\\Vert g_{1}\\Vert +\\cdots +\\Vert g_{k}\\Vert .$ By $\\mathcal {U}^{\\ast }$ denote the dual space of $\\mathcal {U}$ .", "Each functional $F\\in \\mathcal {U}^{\\ast }$ can be written as $F=F_{1}+\\cdots +F_{k},$ where the functionals $F_{i}\\in \\mathcal {R}_{i}^{\\ast }$ and $F_{i}(g_{i})=F[(0,\\ldots ,g_{i},\\ldots ,0)],\\quad i=1,\\ldots ,k.$ We see that the functional $F$ determines the collection $(F_{1},\\ldots ,F_{k})$ .", "Conversely, every collection $(F_{1},\\ldots ,F_{k})$ of continuous linear functionals $F_{i}\\in \\mathcal {R}_{i}^{\\ast }$ , $i=1,\\ldots ,k$ , determines the functional $F_{1}+\\cdots +F_{k},$ on $\\mathcal {U}$ .", "Considering this, in what follows, elements of $\\mathcal {U}^{\\ast }$ will be denoted by $(F_{1},\\ldots ,F_{k})$ .", "It is not difficult to verify that $\\Vert (F_{1},\\ldots ,F_{k})\\Vert =\\max \\lbrace \\Vert F_{1}\\Vert ,\\ldots ,\\Vert F_{k}\\Vert \\rbrace .", "(5.11)$ Let $l=(\\mathbf {x}^{1},\\ldots ,\\mathbf {x}^{n})$ be any $q$ -semicycle (with respect to the directions $\\mathbf {a}^{1}$ ,...,$\\mathbf {a}^{k}$ ) in $X$ and $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ be a vector associated with it.", "Consider the following functional $G_{l,\\lambda }(f)=\\sum _{j=1}^{n}\\lambda _{j}f(\\mathbf {x}^{j}),\\quad f\\in C(X).$ Since $l$ satisfies (5.10), for each function $g_{i}\\in \\mathcal {R}_{i}$ , $i=1,\\ldots ,k$ , we have $G_{l,\\lambda }(g_{i})=\\sum _{j=1}^{n}\\lambda _{j}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}^{j})=\\sum _{s=1}^{r_{i}}\\lambda _{i_{s}}g_{i}(\\mathbf {a}^{i}\\cdot \\mathbf {x}^{i_{s}}),(5.12)$ where $r_{i}\\le k$ .", "That is, for each set $\\mathcal {R}_{i}$ , $G_{l,\\lambda } $ can be reduced to a functional defined with the help of not more than $k$ points of the semicycle $l$ .", "Consider the operator $A:\\mathcal {U}\\rightarrow C(X),\\quad A[(g_{1},\\ldots ,g_{k})]=g_{1}+\\cdots +g_{k}.$ Clearly, $A$ is a linear continuous operator with the norm $\\Vert A\\Vert =1$ .", "Besides, since $\\mathcal {R}_{1}+\\mathcal {R}_{2}+...+\\mathcal {R}_{k}=C(X)$ , $A$ is a surjection.", "Consider also the conjugate operator $A^{\\ast }:C(X)^{\\ast }\\rightarrow \\mathcal {U}^{\\ast },~A^{\\ast }[H]=(F_{1},\\ldots ,F_{k}),$ where $F_{i}(g_{i})=H(g_{i})$ , for any $g_{i}\\in \\mathcal {R}_{i}$ , $i=1,\\ldots ,k$ .", "Set $A^{\\ast }[G_{l,\\lambda }]=(G_{1},\\ldots ,G_{k})$ .", "From (5.12) it follows that $|G_{i}(g_{i})|=|G_{l,\\lambda }(g_{i})|\\le \\Vert g_{i}\\Vert \\sum _{s=1}^{r_{i}}|\\lambda _{i_{s}}|\\le kq\\Vert g_{i}\\Vert ,\\quad i=1,\\ldots ,k,$ Therefore, $\\Vert G_{i}\\Vert \\le kq,\\quad i=1,\\ldots ,k.$ From (5.11) we obtain that $\\Vert A^{\\ast }[G_{l,\\lambda }]\\Vert =\\Vert (G_{1},\\ldots ,G_{k})\\Vert \\le kq.", "(5.13)$ Since $A$ is a surjection, there exists a positive real number $\\delta $ such that $\\Vert A^{\\ast }[H]\\Vert >\\delta \\Vert H\\Vert $ for any functional $H\\in C(X)^{\\ast }$ (see [139]).", "Taking into account that $\\Vert G_{l,\\lambda }\\Vert =\\sum _{j=1}^{n}|\\lambda _{j}|$ , for the functional $G_{l,\\lambda }$ we have $\\Vert A^{\\ast }[G_{l,\\lambda }]\\Vert >\\delta \\sum _{j=1}^{n}|\\lambda _{j}|.", "(5.14)$ It follows from (5.13) and (5.14) that $\\delta <\\frac{kq}{\\sum _{j=1}^{n}|\\lambda _{j}|}.$ The last inequality shows that $n$ (the length of the arbitrarily chosen $q$ -semicycle $l$ ) cannot be as great as possible, otherwise $\\delta =0$ .", "This simply means that there must be some positive integer bounding the lengths of all $q$ -semicycles in $X$ .", "It remains to show that there are no cycles in $X$ .", "Indeed, if $l=(\\mathbf {x}^{1},\\ldots ,\\mathbf {x}^{n})$ is a cycle in $X$ and $\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{n})$ is a vector associated with it, then the above functional $G_{l,\\lambda }$ annihilates all functions from $\\mathcal {R}_{1}+\\mathcal {R}_{2}+...+\\mathcal {R}_{k}$ .", "On the other hand, $G_{l,\\lambda }(f)=\\sum _{j=1}^{n}|\\lambda _{j}|\\ne 0$ for a continuous function $f$ on $X$ satisfying the conditions $f(\\mathbf {x}^{j})=1$ if $\\lambda _{j}>0$ and $f(\\mathbf {x}^{j})=-1$ if $\\lambda _{j}<0$ , $j=1,\\ldots ,n$ .", "This implies that $\\mathcal {R}_{1}+\\mathcal {R}_{2}+...+\\mathcal {R}_{k}\\ne C\\left( X\\right) $ .", "Since $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})\\subseteq \\mathcal {R}_{1}+\\mathcal {R}_{2}+...+\\mathcal {R}_{k}$ , we obtain that $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})\\ne C\\left( X\\right) $ on the contrary to our assumption.", "Remark 5.2.", "Assume $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ is dense in $C(X).$ Is it necessarily closed?", "Theorem 5.6 may describe cases when it is not.", "For example, let $\\mathbf {a}^{1}=(1;-1),\\ \\mathbf {a}^{2}=(1;1),$ $W=\\lbrace \\mathbf {a}^{1},\\mathbf {a}^{2}\\rbrace $ and $\\sigma $ be any continuous, bounded and nonconstant function, which has a limit at infinity.", "Consider the set $X &=&\\lbrace (2;\\frac{2}{3}),(\\frac{2}{3};\\frac{2}{3}),(0;0),(1;1),(1+\\frac{1}{2};1-\\frac{1}{2}),(1+\\frac{1}{2}+\\frac{1}{4};1-\\frac{1}{2}+\\frac{1}{4}), \\\\&&(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8};1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{8}),...\\rbrace .$ It is clear that $X$ is a compact set with all its orbits closed.", "(In fact, there is only one orbit, which coincides with $X$ ).", "Hence, by Theorem 5.4, $\\overline{\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})}=C(X).$ But by Theorem 5.6, $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})\\ne C(X).$ Therefore, the set $\\mathcal {M}_{X}(\\sigma ;W,\\mathbb {R})$ is not closed in $C(X).$ A single hidden layer perceptron is able to approximate a given data with any degree of accuracy.", "But in applications it is necessary to define how many neurons one should take in a hidden layer.", "The more the number of neurons, the more the probability of the network to give precise results.", "Unfortunately, practicality decreases with the increase of the number of neurons in the hidden layer.", "In other words, single hidden layer perceptrons are not always effective if the number of neurons in the hidden layer is prescribed.", "In this section, we show that this phenomenon is no longer true for perceptrons with two hidden layers.", "We prove that a two hidden layer neural network with $d$ inputs, $d$ neurons in the first hidden layer, $2d+2$ neurons in the second hidden layer and with a specifically constructed sigmoidal and infinitely differentiable activation function can approximate any continuous multivariate function with arbitrary accuracy.", "Note that if $r$ is fixed in (5.1), then the set $\\mathcal {M}_{r}(\\sigma )=\\left\\lbrace \\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i}):~c_{i},\\theta _{i}\\in \\mathbb {R},\\mathbf {w\\in }\\mathbb {R}^{d}\\right\\rbrace $ is no longer dense in in the space $C(\\mathbb {R}^{d})$ (in the topology of uniform convergence on compact sets) for any activation function $\\sigma $ .", "The set $\\mathcal {M}_{r}(\\sigma )$ will not be dense even if we variate over all univariate continuous functions $\\sigma $ (see [112]).", "In the following, we will see that this property of single hidden layer neural networks does not carry over to networks with more than one hidden layer.", "A two hidden layer network is defined by iteration of the single hidden layer neural network model.", "The output of two hidden layer perceptron with $r$ units in the first layer, $s$ units in the second layer and the input $x=(x_{1},...,x_{d})$ is $\\sum _{i=1}^{s}d_{i}\\sigma \\left( \\sum _{j=1}^{r}c_{ij}\\sigma (\\mathbf {w}^{ij}\\cdot \\mathbf {x-}\\theta _{ij})-\\gamma _{i}\\right) .$ Here $d_{i},c_{ij},\\theta _{ij},\\gamma _{i}$ are real numbers, $\\mathbf {w}^{ij}$ are vectors of $\\mathbb {R}^{d}$ and $\\sigma $ is a fixed univariate function.", "In many applications, it is convenient to take the activation function $\\sigma $ as a sigmoidal function which is defined as $\\lim _{t\\rightarrow -\\infty }\\sigma (t)=0\\quad \\text{ and }\\quad \\lim _{t\\rightarrow +\\infty }\\sigma (t)=1.$ The literature on neural networks abounds with the use of such functions and their superpositions.", "The following are typical examples of sigmoidal functions: $\\sigma (t)& =\\frac{1}{1+e^{-t}} & & \\text{(the squashing function),} \\\\\\sigma (t)& ={\\left\\lbrace \\begin{array}{ll}0, & t\\le -1, \\\\\\dfrac{t+1}{2}, & -1\\le t\\le 1, \\\\1, & t\\ge 1\\end{array}\\right.", "}& & \\text{(the piecewise linear function),} \\\\\\sigma (t)& =\\frac{1}{\\pi }\\arctan t+\\frac{1}{2} & & \\text{(the arctansigmoid function),} \\\\\\sigma (t)& =\\frac{1}{\\sqrt{2\\pi }}\\int \\limits _{-\\infty }^{t}e^{-x^{2}/2}dx& & \\text{(the Gaussian function).", "}$ In this section, we prove that there exists a two hidden layer neural network model with $d$ units in the first layer and $2d+2$ units in the second layer such that it has the ability to approximate any $d$ -variable continuous function with arbitrary accuracy.", "As an activation function for this model we take a specific sigmoidal function.", "The idea behind the proof of this result is very much connected to the Kolmogorov superposition theorem (see Section 4.1).", "This theorem has been much discussed in neural network literature (see, e.g., [135]).", "In our opinion, the most remarkable application of the Kolmogorov superposition theorem to neural networks was given by Maiorov and Pinkus [119].", "They showed that there exists a sigmoidal, strictly increasing, analytic activation function, for which a fixed number of units in both hidden layers are sufficient to approximate arbitrarily well any continuous multivariate function.", "Namely, the authors of [119] proved the following theorem.", "Theorem 5.7 (Maiorov and Pinkus [119]).", "There exists an activation function $\\sigma $ which is analytic, strictly increasing and sigmoidal and has the following property: For any $f\\in C[0,1]^{d}$ and $\\varepsilon >0,$ there exist constants $d_{i},$ $c_{ij},$ $\\theta _{ij},$ $\\gamma _{i}$ , and vectors $\\mathbf {w}^{ij}\\in \\mathbb {R}^{d}$ for which $\\left|f(\\mathbf {x})-\\sum _{i=1}^{6d+3}d_{i}\\sigma \\left(\\sum _{j=1}^{3d}c_{ij}\\sigma (\\mathbf {w}^{ij}\\cdot \\mathbf {x-}\\theta _{ij})-\\gamma _{i}\\right) \\right|<\\varepsilon (5.15)$ for all $\\mathbf {x}=(x_{1},...,x_{d})\\in [0,1]^{d}.$ This theorem is based on the following version of the Kolmogorov superposition theorem given by Lorentz [114] and Sprecher [145].", "Theorem 5.8 (Kolmogorov's superposition theorem).", "For the unit cube $\\mathbb {I}^{d},~\\mathbb {I}=[0,1],~d\\ge 2,$ there exists constants $\\lambda _{q}>0,$ $q=1,...,d,$ $\\sum _{q=1}^{d}\\lambda _{q}=1,$ and nondecreasing continuous functions $\\phi _{p}:[0,1]\\rightarrow [0,1],$ $p=1,...,2d+1,$ such that every continuous function $f:\\mathbb {I}^{d}\\rightarrow \\mathbb {R}$ admits the representation $f(x_{1},...x_{d})=\\sum _{p=1}^{2d+1}g\\left( \\sum _{q=1}^{d}\\lambda _{q}\\phi _{p}(x_{q})\\right) (5.16)$ for some $g\\in C[0,1]$ depending on $f.$ In the next subsection, using the general ideas developed in [119], we show that the bounds of units in hidden layers in (5.15) may be chosen even equal to the bounds in the Kolmogorov superposition theorem.", "More precisely, these bounds can be taken as $2d+2$ and $d$ instead of $6d+3$ and $3d$ .", "To attain this purpose, we change the “analyticity\" of $\\sigma $ to “infinite differentiability\".", "In addition, near infinity we assume that $\\sigma $ is “$\\lambda $ -strictly increasing\" instead of being “strictly increasing\".", "We begin this subsection with a definition of a $\\lambda $ -monotone function.", "Let $\\lambda $ be any nonnegative number.", "A real function $f$ defined on $(a,b)$ is called $\\lambda $ -increasing ($\\lambda $ -decreasing) if there exists an increasing (decreasing) function $u:(a,b)\\rightarrow \\mathbb {R}$ such that $\\left|f(x)-u(x)\\right|\\le \\lambda ,$ for all $x\\in (a,b)$ .", "If $u$ is strictly increasing (or strictly decreasing), then the above function $f$ is called a $\\lambda $ -strictly increasing (or $\\lambda $ -strictly decreasing) function.", "Clearly, 0-monotonicity coincides with the usual concept of monotonicity and a $\\lambda _{1}$ -monotone function is $\\lambda _{2}$ -monotone if $\\lambda _{1}\\le \\lambda _{2}$ .", "It is also clear from the definition that a $\\lambda $ -monotone function behaves like a usual monotone function as $\\lambda $ gets very small.", "Our purpose is to prove the following theorem.", "Theorem 5.9.", "For any positive numbers $\\alpha $ and $\\lambda $ , there exists a $C^{\\infty }(\\mathbb {R}),$ sigmoidal activation function $\\sigma :$ $\\mathbb {R\\rightarrow R}$ which is strictly increasing on $(-\\infty ,\\alpha )$ , $\\lambda $ -strictly increasing on $[\\alpha ,+\\infty )$ , and satisfies the following property: For any $f\\in C[0,1]^{d}$ and $\\varepsilon >0,$ there exist constants $d_{p}, $ $c_{pq},$ $\\theta _{pq},$ $\\gamma _{p}$ , and vectors $\\mathbf {w}^{pq}\\in \\mathbb {R}^{d}$ for which $\\left|f(\\mathbf {x})-\\sum _{p=1}^{2d+2}d_{p}\\sigma \\left(\\sum _{q=1}^{d}c_{pq}\\sigma (\\mathbf {w}^{pq}\\cdot \\mathbf {x-}\\theta _{pq})-\\gamma _{p}\\right) \\right|<\\varepsilon (5.17)$ for all $\\mathbf {x}=(x_{1},...,x_{d})\\in [0,1]^{d}.$ Let $\\alpha $ be any positive number.", "Divide the interval $[\\alpha ,+\\infty )$ into the segments $[\\alpha ,2\\alpha ],$ $[2\\alpha ,3\\alpha ],...$ .", "Let $h(t)$ be any strictly increasing, infinitely differentiable function on $[\\alpha ,+\\infty )$ with the properties 1) $0<h(t)<1$ for all $t\\in [\\alpha ,+\\infty )$ ; 2) $1-h(\\alpha )\\le \\lambda ;$ 3) $h(t)\\rightarrow 1,$ as $t\\rightarrow +\\infty .$ The existence of a strictly increasing smooth function satisfying these properties is easy to verify.", "Note that from conditions (1)-(3) it follows that any function $f(t)$ satisfying the inequality $h(t)<f(t)<1$ for all $t\\in [\\alpha ,+\\infty ),$ is $\\lambda $ -strictly increasing and $f(t)\\rightarrow 1,$ as $t\\rightarrow +\\infty .$ We are going to construct $\\sigma $ obeying the required properties in stages.", "Let $\\lbrace u_{n}(t)\\rbrace _{n=1}^{\\infty }$ be the sequence of all polynomials with rational coefficients defined on $[0,1].$ First, we define $\\sigma $ on the closed intervals $[(2m-1)\\alpha ,2m\\alpha ],$ $m=1,2,...$ , as the function $\\sigma (t)=a_{m}+b_{m}u_{m}(\\frac{t}{\\alpha }-2m+1),\\text{ }t\\in [(2m-1)\\alpha ,2m\\alpha ],(5.18)$ or equivalently, $\\sigma (\\alpha t+(2m-1)\\alpha )=a_{m}+b_{m}u_{m}(t),\\text{ }t\\in [0,1],(5.19)$ where $a_{m}$ and $b_{m}\\ne 0$ are appropriately chosen constants.", "These constants are determined from the condition $h(t)<\\sigma (t)<1,(5.20)$ for all $t\\in [(2m-1)\\alpha ,2m\\alpha ].$ There is a simple procedure for determining a suitable pair of $a_{m}$ and $b_{m}$ .", "Indeed, let $M=\\max h(t)\\text{, }A_{1}=\\min u_{m}(\\frac{t}{\\alpha }-2m+1)\\text{, }A_{2}=\\max u_{m}(\\frac{t}{\\alpha }-2m+1),$ where in all the above $\\max $ and $\\min $ , the variable $t$ runs over the closed interval $[(2m-1)\\alpha ,2m\\alpha ].$ Note that $M<1$ .", "If $A_{1}=A_{2} $ (that is, if the function $u_{m}$ is constant on $[0,1]$ ), then we can set $\\sigma (t)=(1+M)/2$ and easily find a suitable pair of $a_{m}$ and $b_{m}$ from (5.18).", "Let now $A_{1}\\ne A_{2}$ and $y=a+bx,$ $b\\ne 0,$ be a linear function mapping the segment $[A_{1},A_{2}]$ into $(M,1).$ Then it is enough to take $a_{m}=a$ and $b_{m}=b.$ At the second stage we define $\\sigma $ on the intervals $[2m\\alpha ,(2m+1)\\alpha ],$ $m=1,2,...,$ so that it is in $C^{\\infty }(\\mathbb {R})$ and satisfies the inequality (5.20).", "Finally, in all of $(-\\infty ,\\alpha )$ we define $\\sigma $ while maintaining the $C^{\\infty }$ strict monotonicity property, and also in such a way that $\\lim _{t\\rightarrow -\\infty }\\sigma (t)=0.$ We obtain from the properties of $h$ and the condition (5.20) that $\\sigma (t)$ is a $\\lambda $ -strictly increasing function on the interval $[\\alpha ,+\\infty )$ and $\\sigma (t)\\rightarrow 1$ , as $t\\rightarrow +\\infty .", "$ From the above construction of $\\sigma $ , that is, from (5.19) it follows that for each $m=1,2,...,$ there exists numbers $A_{m}$ ,$\\ B_{m}$ and $r_{m}$ such that $u_{m}(t)=A_{m}\\sigma (\\alpha t-r_{m})-B_{m},(5.21)$ where $A_{m}\\ne 0.$ Let $f$ be any continuous function on the unit cube $[0,1]^{d}.$ By the Kolmogorov superposition theorem the expansion (5.16) is valid for $f.$ For the exterior continuous univariate function $g(t)$ in (5.16) and for any $\\varepsilon >0$ there exists a polynomial $u_{m}(t)$ of the above form such that $\\left|g(t)-u_{m}(t)\\right|<\\frac{\\varepsilon }{2(2d+1)},$ for all $t\\in [0,1].$ This together with (5.21) means that $\\left|g(t)-[a\\sigma (\\alpha t-r)-b]\\right|<\\frac{\\varepsilon }{2(2d+1)},(5.22)$ for some $a,b,r\\in \\mathbb {R}$ and all $t\\in [0,1].$ Substituting (5.22) in (5.16) we obtain that $\\left|f(x_{1},...,x_{d})-\\sum _{p=1}^{2d+1}\\left( a\\sigma \\left( \\alpha \\cdot \\sum _{q=1}^{d}\\lambda _{q}\\phi _{p}(x_{q})-r\\right) -b\\right)\\right|<\\frac{\\varepsilon }{2}(5.23)$ for all $(x_{1},...,x_{d})\\in [0,1]^{d}.$ For each $p\\in \\lbrace 1,2,...,2d+1\\rbrace $ and $\\delta >0$ there exist constants $a_{p},b_{p}$ and $r_{p}$ such that $\\left|\\phi _{p}(x_{q})-[a_{p}\\sigma (\\alpha x_{q}-r_{p})-b_{p}]\\right|<\\delta ,(5.24)$ for all $x_{q}\\in [0,1].$ Since $\\lambda _{q}>0,$ $q=1,...,d,$ $\\sum _{q=1}^{d}\\lambda _{q}=1,$ it follows from (5.24) that $\\left|\\sum _{q=1}^{d}\\lambda _{q}\\phi _{p}(x_{q})-\\left[\\sum _{q=1}^{d}\\lambda _{q}a_{p}\\sigma (\\alpha x_{q}-r_{p})-b_{p}\\right]\\right|<\\delta ,(5.25)$ for all $(x_{1},...,x_{d})\\in [0,1]^{d}.$ Now since the function $a\\sigma (\\alpha t-r)$ is uniformly continuous on every closed interval, we can choose $\\delta $ sufficiently small and obtain from (5.25) that $\\left|\\sum _{p=1}^{2d+1}a\\sigma \\left( \\alpha \\sum _{q=1}^{d}\\lambda _{q}\\phi _{p}(x_{q})-r\\right) \\ -\\sum _{p=1}^{2d+1}a\\sigma \\left( \\alpha \\left[ \\sum _{q=1}^{d}\\lambda _{q}a_{p}\\sigma (\\alpha x_{q}-r_{p})-b_{p}\\right] -r\\right) \\right|$ $< \\frac{\\varepsilon }{2}.$ This inequality may be rewritten as $\\left|\\sum _{p=1}^{2d+1}a\\sigma \\left( \\alpha \\sum _{q=1}^{d}\\lambda _{q}\\phi _{p}(x_{q})-r\\right)-\\sum _{p=1}^{2d+1}d_{p}\\sigma \\left( \\sum _{q=1}^{d}c_{pq}\\sigma (\\mathbf {w}^{pq}\\cdot \\mathbf {x}-\\theta _{pq})-\\gamma _{p}\\right) \\right|<\\frac{\\varepsilon }{2}.", "(5.26)$ From (5.23) and (5.26) it follows that $\\left|f(\\mathbf {x})-\\left[ \\sum _{p=1}^{2d+1}d_{p}\\sigma \\left(\\sum _{q=1}^{d}c_{pq}\\sigma (\\mathbf {w}^{pq}\\cdot \\mathbf {x-}\\theta _{pq})-\\gamma _{p}\\right) -s\\right] \\right|<\\varepsilon ,(5.27)$ where $s=(2d+1)b$ .", "Since the constant $s$ can be written in the form $s=d\\sigma \\left( \\sum _{q=1}^{d}c_{q}\\sigma (\\mathbf {w}^{q}\\cdot \\mathbf {x-}\\theta _{q})-\\gamma \\right) ,$ from (5.27) we finally obtain the validity of (5.17).", "Remark 5.3.", "It is easily seen in the proof of Theorem 5.9 that all the weights $\\mathbf {w}^{ij}$ are fixed (see (5.26)).", "Namely, $\\mathbf {w}^{ij}=\\alpha \\mathbf {e}^{j},$ for all $i=1,...,2d+2,$ $j=1,...,d,$ where $\\mathbf {e}^{j}$ is the $j$ -th coordinate vector of the space $\\mathbb {R}^{d}$ .", "The next theorem follows from Theorem 5.9 easily, since the Kolmogorov superposition theorem is valid for all compact sets of $\\mathbb {R}^{d}$ .", "Theorem 5.10.", "Let $Q$ be a compact set in $\\mathbb {R}^{d}.$ For any numbers $\\alpha \\in \\mathbb {R}$ and $\\lambda >0,$ there exists a $C^{\\infty }(\\mathbb {R}),$ sigmoidal activation function $\\sigma :$ $\\mathbb {R\\rightarrow R}$ which is strictly increasing on $(-\\infty ,\\alpha ) $ , $\\lambda $ -strictly increasing on $[\\alpha ,+\\infty )$ , and satisfies the following property: For any $f\\in C(Q)$ and $\\varepsilon >0$ there exist real numbers $d_{i},$ $c_{ij},$ $\\theta _{ij}$ , $\\gamma _{i},$ and vectors $\\mathbf {w}^{ij}\\in \\mathbb {R}^{d}$ for which $\\left|f(\\mathbf {x})-\\sum _{i=1}^{2d+2}d_{i}\\sigma \\left(\\sum _{j=1}^{d}c_{ij}\\sigma (\\mathbf {w}^{ij}\\cdot \\mathbf {x-}\\theta _{ij})-\\gamma _{i}\\right) \\right|<\\varepsilon $ for all $\\mathbf {x}=(x_{1},...,x_{d})\\in Q.$ Remark 5.4.", "In some literature, a single hidden layer perceptron is defined as the function $\\sum _{i=1}^{r}c_{i}\\sigma (\\mathbf {w}^{i}\\mathbf {\\cdot x}-\\theta _{i})-c_{0}.$ A two hidden layer network then takes the form $\\sum _{i=1}^{s}d_{i}\\sigma \\left( \\sum _{j=1}^{r}c_{ij}\\sigma (\\mathbf {w}^{ij}\\cdot \\mathbf {x-}\\theta _{ij})-\\gamma _{i}\\right) -d_{0}.", "(5.28)$ The proof of Theorem 5.9 shows that for networks of type (5.28) the theorem is valid if we take $2d+1$ neurons in the second hidden layer (instead of $2d+2$ neurons as above).", "That is, there exist networks of type (5.28) having the universal approximation property and for which the number of units in the hidden layers is equal to the number of summands in the Kolmogorov superposition theorem.", "Remark 5.5.", "It is known that the $2d+1$ in the Kolmogorov superposition theorem is minimal (see Sternfeld [151]).", "Thus it is doubtful if the number of neurons in Theorems 5.9 and 5.10 can be reduced.", "Remark 5.6.", "Inequality (5.22) shows that single hidden layer neural networks of the form (5.28) with the activation function $\\sigma $ and with only one neuron in the hidden layer can approximate any continuous function on the interval $[0,1]$ with arbitrary precision.", "Since the number $b$ in (5.22) can always be written as $b=a_1\\sigma (0 \\cdot t-r_1)$ for some $a_1$ and $r_1$ , we see that two neurons in the hidden layer are sufficient for traditional single hidden layer neural networks with the activation function $\\sigma $ to approximate continuous functions on $[0,1]$ .", "Applying the linear transformation $x=a+(b-a)t$ it can be proven that the same argument holds for any interval $[a,b]$ .", "In the preceding section, we considered two theorems (Theorem 5.7 of Maiorov and Pinkus, and Theorem 5.9) on the approximation capabilities of the MLP model of neural networks with a prescribed number of hidden neurons.", "Note that both results are more theoretical than practical, as they indicate only the existence of the corresponding activation functions.", "In this section, we construct algorithmically a smooth, sigmoidal, almost monotone activation function $\\sigma $ providing approximation to an arbitrary continuous function within any degree of accuracy.", "This algorithm is implemented in a computer program, which computes the value of $\\sigma $ at any reasonable point of the real axis.", "In this subsection, we construct algorithmically a sigmoidal function $\\sigma $ which we use in our results in Section 5.3.3.", "To start with the construction of $\\sigma $ , assume that we are given a closed interval $[a, b]$ and a sufficiently small real number $\\lambda $ .", "We construct $\\sigma $ algorithmically, based on two numbers, namely $\\lambda $ and $d := b - a$ .", "The following steps describe the algorithm.", "Step 1.", "Introduce the function $h(x) := 1 - \\frac{\\min \\lbrace 1/2, \\lambda \\rbrace }{1 + \\log (x - d + 1)}.$ Note that this function is strictly increasing on the real line and satisfies the following properties: $0 < h(x) < 1$ for all $x \\in [d, +\\infty )$ ; $1 - h(d) \\le \\lambda $ ; $h(x) \\rightarrow 1$ , as $x \\rightarrow +\\infty $ .", "$0 < h(x) < 1$ for all $x \\in [d, +\\infty )$ ; $1 - h(d) \\le \\lambda $ ; $h(x) \\rightarrow 1$ , as $x \\rightarrow +\\infty $ .", "We want to construct $\\sigma $ satisfying the inequalities $h(x)<\\sigma (x)<1(5.29)$ for $x\\in [d,+\\infty )$ .", "Then our $\\sigma $ will tend to 1 as $x$ tends to $+\\infty $ and obey the inequality $|\\sigma (x)-h(x)|\\le \\lambda ,$ i.e., it will be a $\\lambda $ -increasing function.", "Step 2.", "Before proceeding to the construction of $\\sigma $ , we need to enumerate the monic polynomials with rational coefficients.", "Let $q_n$ be the Calkin–Wilf sequence (see [21]).", "Then we can enumerate all the rational numbers by setting $r_0 := 0, \\quad r_{2n} := q_n, \\quad r_{2n-1} := -q_n, \\ n = 1, 2, \\dots .$ Note that each monic polynomial with rational coefficients can uniquely be written as $r_{k_0} + r_{k_1} x + \\ldots + r_{k_{l-1}} x^{l-1} + x^l$ , and each positive rational number determines a unique finite continued fraction $[m_0; m_1, \\ldots , m_l] := m_0 + \\dfrac{1}{m_1 + \\dfrac{1}{m_2 + \\dfrac{1}{\\ddots + \\dfrac{1}{m_l}}}}$ with $m_0 \\ge 0$ , $m_1, \\ldots , m_{l-1} \\ge 1$ and $m_l \\ge 2$ .", "We now construct a bijection between the set of all monic polynomials with rational coefficients and the set of all positive rational numbers as follows.", "To the only zeroth-degree monic polynomial 1 we associate the rational number 1, to each first-degree monic polynomial of the form $r_{k_0} + x$ we associate the rational number $k_0 + 2$ , to each second-degree monic polynomial of the form $r_{k_0} + r_{k_1} x + x^2$ we associate the rational number $[k_0; k_1+ 2] = k_0 + 1 / (k_1 + 2)$ , and to each monic polynomial $r_{k_0} + r_{k_1} x + \\ldots + r_{k_{l-2}} x^{l-2} + r_{k_{l-1}} x^{l-1} +x^l$ of degree $l \\ge 3$ we associate the rational number $[k_0; k_1 + 1, \\ldots ,k_{l-2} + 1, k_{l-1} + 2]$ .", "In other words, we define $u_1(x) := 1$ , $u_n(x) := r_{q_n-2} + x$ if $q_n \\in \\mathbb {Z}$ , $u_n(x) := r_{m_0} + r_{m_1-2} x + x^2$ if $q_n = [m_0; m_1]$ , and $u_n(x) := r_{m_0} + r_{m_1-1} x + \\ldots + r_{m_{l-2}-1} x^{l-2} +r_{m_{l-1}-2} x^{l-1} + x^l$ if $q_n = [m_0; m_1, \\ldots , m_{l-2}, m_{l-1}]$ with $l \\ge 3$ .", "For example, the first few elements of this sequence are $1, \\quad x^2, \\quad x, \\quad x^2 - x, \\quad x^2 - 1, \\quad x^3, \\quad x - 1,\\quad x^2 + x, \\quad \\ldots .$ Step 3.", "We start with constructing $\\sigma $ on the intervals $[(2n-1)d, 2nd]$ , $n = 1, 2, \\ldots $ .", "For each monic polynomial $u_n(x) =\\alpha _0 + \\alpha _1 x + \\ldots + \\alpha _{l-1} x^{l-1} + x^l$ , set $B_1 := \\alpha _0 + \\frac{\\alpha _1-|\\alpha _1|}{2} + \\ldots + \\frac{\\alpha _{l-1} - |\\alpha _{l-1}|}{2}$ and $B_2 := \\alpha _0 + \\frac{\\alpha _1+|\\alpha _1|}{2} + \\ldots + \\frac{\\alpha _{l-1} + |\\alpha _{l-1}|}{2} + 1.$ Note that the numbers $B_1$ and $B_2$ depend on $n$ .", "To avoid complication of symbols, we do not indicate this in the notation.", "Introduce the sequence $M_n := h((2n+1)d), \\qquad n = 1, 2, \\ldots .$ Clearly, this sequence is strictly increasing and converges to 1.", "Now we define $\\sigma $ as the function $\\sigma (x):=a_{n}+b_{n}u_{n}\\left( \\frac{x}{d}-2n+1\\right) ,\\quad x\\in [(2n-1)d,2nd],(5.30)$ where $a_{1}:=\\frac{1}{2},\\qquad b_{1}:=\\frac{h(3d)}{2},(5.31)$ and $a_{n}:=\\frac{(1+2M_{n})B_{2}-(2+M_{n})B_{1}}{3(B_{2}-B_{1})},\\qquad b_{n}:=\\frac{1-M_{n}}{3(B_{2}-B_{1})},\\qquad n=2,3,\\ldots .", "(5.32)$ It is not difficult to notice that for $n>2$ the numbers $a_{n}$ , $b_{n}$ are the coefficients of the linear function $y=a_{n}+b_{n}x$ mapping the closed interval $[B_{1},B_{2}]$ onto the closed interval $[(1+2M_{n})/3,(2+M_{n})/3]$ .", "Besides, for $n=1$ , i.e.", "on the interval $[d,2d] $ , $\\sigma (x)=\\frac{1+M_{1}}{2}.$ Therefore, we obtain that $h(x)<M_{n}<\\frac{1+2M_{n}}{3}\\le \\sigma (x)\\le \\frac{2+M_{n}}{3}<1,(5.33)$ for all $x\\in [(2n-1)d,2nd]$ , $n=1$ , 2, $\\ldots $ .", "Step 4.", "In this step, we construct $\\sigma $ on the intervals $[2nd,(2n+1)d]$ , $n=1,2,\\ldots $ .", "For this purpose we use the smooth transition function $\\beta _{a,b}(x):=\\frac{\\widehat{\\beta }(b-x)}{\\widehat{\\beta }(b-x)+\\widehat{\\beta }(x-a)},$ where $\\widehat{\\beta }(x):={\\left\\lbrace \\begin{array}{ll}e^{-1/x}, & x>0, \\\\0, & x\\le 0.\\end{array}\\right.", "}$ Obviously, $\\beta _{a,b}(x)=1$ for $x\\le a$ , $\\beta _{a,b}(x)=0$ for $x\\ge b$ , and $0<\\beta _{a,b}(x)<1$ for $a<x<b$ .", "Set $K_n := \\frac{\\sigma (2nd) + \\sigma ((2n+1)d)}{2}, \\qquad n = 1, 2, \\ldots .$ Note that the numbers $\\sigma (2nd)$ and $\\sigma ((2n+1)d)$ have already been defined in the previous step.", "Since both the numbers $\\sigma (2nd)$ and $\\sigma ((2n+1)d)$ lie in the interval $(M_n, 1)$ , it follows that $K_n \\in (M_n, 1)$ .", "First we extend $\\sigma $ smoothly to the interval $[2nd,2nd+d/2]$ .", "Take $\\varepsilon :=(1-M_{n})/6$ and choose $\\delta \\le d/2$ such that $\\left|a_{n}+b_{n}u_{n}\\left( \\frac{x}{d}-2n+1\\right) -\\left(a_{n}+b_{n}u_{n}(1)\\right) \\right|\\le \\varepsilon ,\\quad x\\in [2nd,2nd+\\delta ].", "(5.34)$ One can choose this $\\delta $ as $\\delta :=\\min \\left\\lbrace \\frac{\\varepsilon d}{b_{n}C},\\frac{d}{2}\\right\\rbrace ,$ where $C>0$ is a number satisfying $|u_{n}^{\\prime }(x)|\\le C$ for $x\\in (1,1.5)$ .", "For example, for $n=1$ , $\\delta $ can be chosen as $d/2$ .", "Now define $\\sigma $ on the first half of the interval $[2nd,(2n+1)d]$ as the function $\\sigma (x) :=K_{n}-\\beta _{2nd,2nd+\\delta }(x)$ $\\times \\left(K_{n}-a_{n}-b_{n}u_{n}\\left( \\frac{x}{d}-2n+1\\right) \\right) , x\\in \\left[2nd,2nd+\\frac{d}{2}\\right].", "(5.35)$ Let us prove that $\\sigma (x)$ satisfies the condition (5.29).", "Indeed, if $2nd+\\delta \\le x\\le 2nd+d/2$ , then there is nothing to prove, since $\\sigma (x)=K_{n}\\in (M_{n},1)$ .", "If $2nd\\le x<2nd+\\delta $ , then $0<\\beta _{2nd,2nd+\\delta }(x)\\le 1$ and hence from (5.35) it follows that for each $x\\in [2nd,2nd+\\delta )$ , $\\sigma (x)$ is between the numbers $K_{n}$ and $A_{n}(x):=a_{n}+b_{n}u_{n}\\left( \\frac{x}{d}-2n+1\\right) $ .", "On the other hand, from (5.34) we obtain that $a_{n}+b_{n}u_{n}(1)-\\varepsilon \\le A_{n}(x)\\le a_{n}+b_{n}u_{n}(1)+\\varepsilon ,$ which together with (5.30) and (5.33) yields $A_{n}(x)\\in \\left[ \\frac{1+2M_{n}}{3}-\\varepsilon ,\\frac{2+M_{n}}{3}+\\varepsilon \\right] $ for $x\\in [2nd,2nd+\\delta )$ .", "Since $\\varepsilon =(1-M_{n})/6$ , the inclusion $A_{n}(x)\\in (M_{n},1)$ is valid.", "Now since both $K_{n}$ and $A_{n}(x)$ belong to $(M_{n},1)$ , we finally conclude that $h(x)<M_{n}<\\sigma (x)<1,\\quad \\text{for }x\\in \\left[ 2nd,2nd+\\frac{d}{2}\\right] .$ We define $\\sigma $ on the second half of the interval in a similar way: $\\begin{split}\\sigma (x)& :=K_{n}-(1-\\beta _{(2n+1)d-\\overline{\\delta },(2n+1)d}(x)) \\\\& \\times \\left( K_{n}-a_{n+1}-b_{n+1}u_{n+1}\\left( \\frac{x}{d}-2n-1\\right)\\right) ,\\quad x\\in \\left[ 2nd+\\frac{d}{2},(2n+1)d\\right] ,\\end{split}$ where $\\overline{\\delta }:=\\min \\left\\lbrace \\frac{\\overline{\\varepsilon }d}{b_{n+1}\\overline{C}},\\frac{d}{2}\\right\\rbrace ,\\qquad \\overline{\\varepsilon }:=\\frac{1-M_{n+1}}{6},\\qquad \\overline{C}\\ge \\sup _{[-0.5,0]}|u_{n+1}^{\\prime }(x)|.$ One can easily verify, as above, that the constructed $\\sigma (x)$ satisfies the condition (5.29) on $[2nd+d/2,2nd+d]$ and $\\sigma \\left( 2nd+\\frac{d}{2}\\right) =K_{n},\\qquad \\sigma ^{(i)}\\left( 2nd+\\frac{d}{2}\\right) =0,\\quad i=1,2,\\ldots .$ Steps 3 and 4 construct $\\sigma $ on the interval $[d, +\\infty )$ .", "Step 5.", "On the remaining interval $(-\\infty ,d)$ , we define $\\sigma $ as $\\sigma (x):=\\left( 1-\\widehat{\\beta }(d-x)\\right) \\frac{1+M_{1}}{2},\\quad x\\in (-\\infty ,d).$ It is not difficult to verify that $\\sigma $ is a strictly increasing, smooth function on $(-\\infty ,d)$ .", "Note also that $\\sigma (x)\\rightarrow \\sigma (d)=(1+M_{1})/2$ , as $x$ tends to $d$ from the left and $\\sigma ^{(i)}(d)=0$ for $i=1$ , 2, $\\ldots $ .", "This final step completes the construction of $\\sigma $ on the whole real line.", "It should be noted that the above algorithm allows one to compute the constructed $\\sigma $ at any point of the real axis instantly.", "The code of this algorithm is available at http://sites.google.com/site/njguliyev/papers/monic-sigmoidal.", "As a practical example, we give here the graph of $\\sigma $ (see Figure REF ) and a numerical table (see Table REF ) containing several computed values of this function on the interval $[0, 20]$ .", "Figure REF shows how the graph of $\\lambda $ -increasing function $\\sigma $ changes on the interval $[0,100]$ as the parameter $\\lambda $ decreases.", "The above $\\sigma $ obeys the following properties: $\\sigma $ is sigmoidal; $\\sigma \\in C^{\\infty }(\\mathbb {R})$ ; $\\sigma $ is strictly increasing on $(-\\infty , d)$ and $\\lambda $ -strictly increasing on $[d, +\\infty )$ ; $\\sigma $ is easily computable in practice.", "$\\sigma $ is sigmoidal; $\\sigma \\in C^{\\infty }(\\mathbb {R})$ ; $\\sigma $ is strictly increasing on $(-\\infty , d)$ and $\\lambda $ -strictly increasing on $[d, +\\infty )$ ; $\\sigma $ is easily computable in practice.", "All these properties are easily seen from the above exposition.", "But the essential property of our sigmoidal function is its ability to approximate an arbitrary continuous function using only a fixed number of translations and scalings of $\\sigma $ .", "More precisely, only two translations and scalings are sufficient.", "We formulate this important property as a theorem in the next section.", "Figure: The graph of σ\\protect \\sigma on [0,20][0, 20] (d=2d = 2, λ=1/4\\protect \\lambda = 1/4)Table: Some computed values of σ\\protect \\sigma (d=2d = 2, λ=1/4\\protect \\lambda = 1/4)Figure: The graph of σ\\protect \\sigma on [0,100][0, 100] (d=2d = 2)The following theorems are valid.", "Theorem 5.11.", "Assume that $f$ is a continuous function on a finite segment $[a,b]$ of $\\mathbb {R}$ and $\\sigma $ is the sigmoidal function constructed in Section 5.3.1.", "Then for any sufficiently small $\\varepsilon >0 $ there exist constants $c_{1}$ , $c_{2}$ , $\\theta _{1}$ and $\\theta _{2}$ such that $|f(x)-c_{1}\\sigma (x-\\theta _{1})-c_{2}\\sigma (x-\\theta _{2})|<\\varepsilon $ for all $x\\in [a,b]$ .", "Set $d:=b-a$ and divide the interval $[d,+\\infty )$ into the segments $[d,2d]$ , $[2d,3d]$ , $\\ldots $ .", "It follows from (5.30) that $\\sigma (dx+(2n-1)d)=a_{n}+b_{n}u_{n}(x),\\quad x\\in [0,1](5.36)$ for $n=1$ , 2, $\\ldots $ .", "Here $a_{n}$ and $b_{n}$ are computed by (5.31) and (5.32) for $n=1$ and $n>1$ , respectively.", "From (5.36) it follows that for each $n=1$ , 2, $\\ldots $ , $u_{n}(x)=\\frac{1}{b_{n}}\\sigma (dx+(2n-1)d)-\\frac{a_{n}}{b_{n}}.", "(5.37)$ Let now $g$ be any continuous function on the unit interval $[0,1]$ .", "By the density of polynomials with rational coefficients in the space of continuous functions on any compact subset of $\\mathbb {R}$ , for any $\\varepsilon >0$ there exists a polynomial $p(x)$ of the above form such that $|g(x)-p(x)|<\\varepsilon $ for all $x\\in [0,1]$ .", "Denote by $p_{0}$ the leading coefficient of $p$ .", "If $p_{0}\\ne 0$ (i.e., $p\\lnot \\equiv 0$ ) then we define $u_{n}$ as $u_{n}(x):=p(x)/p_{0}$ , otherwise we just set $u_{n}(x):=1$ .", "In both cases $|g(x)-p_{0}u_{n}(x)|<\\varepsilon ,\\qquad x\\in [0,1].$ This together with (5.37) means that $|g(x)-c_{1}\\sigma (dx-s_{1})-c_{0}|<\\varepsilon (5.38)$ for some $c_{0}$ , $c_{1}$ , $s_{1}\\in \\mathbb {R}$ and all $x\\in [0,1]$ .", "Namely, $c_{1}=p_{0}/b_{n}$ , $s_{1}=d-2nd$ and $c_{0}=p_{0}a_{n}/b_{n}$ .", "On the other hand, we can write $c_{0}=c_{2}\\sigma (dx-s_{2})$ , where $c_{2}:=2c_{0}/(1+h(3d))$ and $s_{2}:=-d$ .", "Hence, $|g(x)-c_{1}\\sigma (dx-s_{1})-c_{2}\\sigma (dx-s_{2})|<\\varepsilon .", "(5.39)$ Note that (5.39) is valid for the unit interval $[0,1]$ .", "Using linear transformation it is not difficult to go from $[0,1]$ to the interval $[a,b]$ .", "Indeed, let $f\\in C[a,b]$ , $\\sigma $ be constructed as above, and $\\varepsilon $ be an arbitrarily small positive number.", "The transformed function $g(x)=f(a+(b-a)x)$ is well defined on $[0,1]$ and we can apply the inequality (5.39).", "Now using the inverse transformation $x=(t-a)/(b-a)$ , we can write $|f(t)-c_{1}\\sigma (t-\\theta _{1})-c_{2}\\sigma (t-\\theta _{2})|<\\varepsilon $ for all $t\\in [a,b]$ , where $\\theta _{1}=a+s_{1}$ and $\\theta _{2}=a+s_{2}$ .", "The last inequality completes the proof.", "Since any compact subset of the real line is contained in a segment $[a,b]$ , the following generalization of Theorem 5.11 holds.", "Theorem 5.12.", "Let $Q$ be a compact subset of the real line and $d$ be its diameter.", "Let $\\lambda $ be any positive number.", "Then one can algorithmically construct a computable sigmoidal activation function $\\sigma \\colon \\mathbb {R}\\rightarrow \\mathbb {R}$ , which is infinitely differentiable, strictly increasing on $(-\\infty ,d)$ , $\\lambda $ -strictly increasing on $[d,+\\infty )$ , and satisfies the following property: For any $f\\in C(Q)$ and $\\varepsilon >0$ there exist numbers $c_{1}$ , $c_{2}$ , $\\theta _{1}$ and $\\theta _{2}$ such that $|f(x)-c_{1}\\sigma (x-\\theta _{1})-c_{2}\\sigma (x-\\theta _{2})|<\\varepsilon $ for all $x\\in Q$ .", "Remark 5.7.", "Theorems 5.11 and 5.12 show that single hidden layer neural networks with the constructed sigmoidal activation function $\\sigma $ and only two neurons in the hidden layer can approximate any continuous univariate function.", "Moreover, in this case, one can fix the weights equal to 1.", "For the approximation of continuous multivariate functions two hidden layer neural networks with $3d+2$ hidden neurons can be taken.", "Namely, Theorem 5.9 (and hence Theorem 5.10) is valid with the constructed in Section 5.3.1 activation function $\\sigma $ .", "Indeed, the proof of this theorem shows that any activation function with the property (5.22) suffices.", "But the activation function constructed in Section 5.3.1 satisfies this property (see (5.38)).", "We prove in Theorem 5.11 that any continuous function on $[a,b]$ can be approximated arbitrarily well by single hidden layer neural networks with the fixed weight 1 and with only two neurons in the hidden layer.", "An activation function $\\sigma $ for such a network is constructed in Section 5.3.1.", "We have seen from the proof that our approach is totally constructive.", "One can evaluate the value of $\\sigma $ at any point of the real axis and draw its graph instantly, using the programming interface at the URL shown at the beginning of Section 5.3.2.", "In the current subsection, we demonstrate our result in various examples.", "For different error bounds we find the parameters $c_{1}$ , $c_{2}$ , $\\theta _{1}$ and $\\theta _{2}$ in Theorem 5.11.", "All computations were done in SageMath [147].", "For computations, we use the following algorithm, which works well for analytic functions.", "Assume $f$ is a function, whose Taylor series around the point $(a+b)/2$ converges uniformly to $f$ on $[a,b]$ , and $\\varepsilon >0$ .", "Consider the function $g(t) := f(a + (b - a) t)$ , which is well-defined on $[0, 1]$ ; Find $k$ such that the $k$ -th Taylor polynomial $T_k(x) := \\sum _{i=0}^k \\frac{g^{(i)}(1/2)}{i!}", "\\left( x - \\frac{1}{2}\\right)^i$ satisfies the inequality $|T_k(x) - g(x)| \\le \\varepsilon / 2$ for all $x\\in [0, 1]$ ; Find a polynomial $p$ with rational coefficients such that $|p(x) - T_k(x)| \\le \\frac{\\varepsilon }{2}, \\qquad x \\in [0, 1],$ and denote by $p_0$ the leading coefficient of this polynomial; If $p_0 \\ne 0$ , then find $n$ such that $u_n(x) = p(x) / p_0$ .", "Otherwise, set $n := 1$ ; For $n=1$ and $n>1$ evaluate $a_{n}$ and $b_{n}$ by (5.31) and (5.32), respectively; Calculate the parameters of the network as $c_1 := \\frac{p_0}{b_n}, \\qquad c_2 := \\frac{2 p_0 a_n}{b_n (1 + h(3d))},\\qquad \\theta _1 := b - 2 n (b - a), \\qquad \\theta _2 := 2 a - b;$ Construct the network $\\mathcal {N}=c_{1}\\sigma (x-\\theta _{1})+c_{2}\\sigma (x-\\theta _{2}).$ Then $\\mathcal {N}$ gives an $\\varepsilon $ -approximation to $f.$ Consider the function $g(t) := f(a + (b - a) t)$ , which is well-defined on $[0, 1]$ ; Find $k$ such that the $k$ -th Taylor polynomial $T_k(x) := \\sum _{i=0}^k \\frac{g^{(i)}(1/2)}{i!}", "\\left( x - \\frac{1}{2}\\right)^i$ satisfies the inequality $|T_k(x) - g(x)| \\le \\varepsilon / 2$ for all $x\\in [0, 1]$ ; Find a polynomial $p$ with rational coefficients such that $|p(x) - T_k(x)| \\le \\frac{\\varepsilon }{2}, \\qquad x \\in [0, 1],$ and denote by $p_0$ the leading coefficient of this polynomial; If $p_0 \\ne 0$ , then find $n$ such that $u_n(x) = p(x) / p_0$ .", "Otherwise, set $n := 1$ ; For $n=1$ and $n>1$ evaluate $a_{n}$ and $b_{n}$ by (5.31) and (5.32), respectively; Calculate the parameters of the network as $c_1 := \\frac{p_0}{b_n}, \\qquad c_2 := \\frac{2 p_0 a_n}{b_n (1 + h(3d))},\\qquad \\theta _1 := b - 2 n (b - a), \\qquad \\theta _2 := 2 a - b;$ Construct the network $\\mathcal {N}=c_{1}\\sigma (x-\\theta _{1})+c_{2}\\sigma (x-\\theta _{2}).$ Then $\\mathcal {N}$ gives an $\\varepsilon $ -approximation to $f.$ In the sequel, we give four practical examples.", "To be able to make comparisons between these examples, all the considered functions are given on the same interval $[-1,1]$ .", "First we select the polynomial function $f(x)=x^{3}+x^{2}-5x+3$ as a target function.", "We investigate the sigmoidal neural network approximation to $f(x)$ .", "This function was considered in [53] as well.", "Note that the authors of [53] chose the sigmoidal function as $\\sigma (x)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if }x\\ge 0, \\\\0, & \\text{if }x<0,\\end{array}\\right.", "}$ and obtained the numerical results (see Table REF ) for single hidden layer neural networks with 8, 32, 128, 532 neurons in the hidden layer (see also [25] for an additional constructive result concerning the error of approximation in this example).", "Table: The Heaviside function as a sigmoidal functionAs it is seen from the table, the number of neurons in the hidden layer increases as the error bound decreases in value.", "This phenomenon is no longer true for our sigmoidal function.", "Using Theorem 5.11, we can construct explicitly a single hidden layer neural network model with only two neurons in the hidden layer, which approximates the above polynomial with arbitrarily given precision.", "Here by explicit construction we mean that all the network parameters can be computed directly.", "Namely, the calculated values of these parameters are as follows: $c_{1}\\approx 2059.373597$ , $c_{2}\\approx -2120.974727$ , $\\theta _{1}=-467$ , and $\\theta _{2}=-3$ .", "It turns out that for the above polynomial we have an exact representation.", "That is, on the interval $[-1,1]$ we have the identity $x^{3}+x^{2}-5x+3\\equiv c_{1}\\sigma (x-\\theta _{1})+c_{2}\\sigma (x-\\theta _{2}).$ Let us now consider the other polynomial function $f(x) = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{6} + \\frac{x^4}{24} + \\frac{x^5}{120} + \\frac{x^6}{720}.$ For this function we do not have an exact representation as above.", "Nevertheless, one can easily construct a $\\varepsilon $ -approximating network with two neurons in the hidden layer for any sufficiently small approximation error $\\varepsilon $ .", "Table REF displays numerical computations of the network parameters for six different approximation errors.", "Table: Several ε\\protect \\varepsilon -approximators of the function 1+x+x 2 /2+x 3 /6+x 4 /24+x 5 /120+x 6 /7201 + x+ x^2 / 2 + x^3 / 6 + x^4 / 24 + x^5 / 120 + x^6 / 720Figure: The graphs of f(x)=1+x+x 2 /2+x 3 /6+x 4 /24+x 5 /120+x 6 /720f(x) = 1 + x + x^2 / 2 + x^3 / 6 + x^4 / 24 + x^5 /120 + x^6 / 720 and some of its approximators (λ=1/4\\protect \\lambda = 1/4)At the end we consider the nonpolynomial functions $f(x) = 4x / (4 + x^2)$ and $f(x) = \\sin x - x \\cos (x + 1)$ .", "Tables REF and REF display all the parameters of the $\\varepsilon $ -approximating neural networks for the above six approximation error bounds.", "As it is seen from the tables, these bounds do not alter the number of hidden neurons.", "Figures REF , REF and REF show how graphs of some constructed networks $\\mathcal {N}$ approximate the corresponding target functions $f$ .", "Table: Several ε\\protect \\varepsilon -approximators of the function 4x/(4+x 2 )4x /(4 + x^2)Table: Several ε\\protect \\varepsilon -approximators of the function sinx-xcos(x+1)\\sin x- x \\cos (x + 1)Figure: The graphs of f(x)=4x/(4+x 2 )f(x) = 4x / (4 + x^2) and some of its approximators(λ=1/4\\protect \\lambda = 1/4)Figure: The graphs of f(x)=sinx-xcos(x+1)f(x)=\\sin x-x\\cos (x+1) and some of itsapproximators (λ=1/4\\protect \\lambda =1/4)tocchapterBibliography" ] ]

[ [ "Role of bridge nodes in epidemic spreading: Different regimes and\n crossovers" ], [ "Abstract Power-law behaviors are common in many disciplines, especially in network science.", "Real-world networks, like disease spreading among people, are more likely to be interconnected communities, and show richer power-law behaviors than isolated networks.", "In this paper, we look at the system of two communities which are connected by bridge links between a fraction $r$ of bridge nodes, and study the effect of bridge nodes to the final state of the Susceptible-Infected-Recovered model, by mapping it to link percolation.", "By keeping a fixed average connectivity, but allowing different transmissibilities along internal and bridge links, we theoretically derive different power-law asymptotic behaviors of the total fraction of the recovered $R$ in the final state as $r$ goes to zero, for different combinations of internal and bridge link transmissibilities.", "We also find crossover points where $R$ follows different power-law behaviors with $r$ on both sides when the internal transmissibility is below but close to its critical value, for different bridge link transmissibilities.", "All of these power-law behaviors can be explained through different mechanisms of how finite clusters in each community are connected into the giant component of the whole system, and enable us to pick effective epidemic strategies and to better predict their impacts." ], [ "Introduction", "Network theory is a powerful tool that can be applied in many disciplines.", "In this framework, real systems such as the power grid, the brain, and societies are represented by a network [1], which is a graph composed of nodes and links that represent the interaction between nodes.", "Many researchers use network theory to study the spreading of an epidemic in order to predict its evolution and to implement strategies to decrease its impact in healthy populations [2].", "Diseases like Ebola [3], H1N1 [4], and the novel coronavirus COVID-19 [5] spread not only domestically, but also from one country to another, mainly through air transportation [6].", "These international airports are bridge nodes, which establish connections between more than one community.", "In this work, we explore how bridge nodes affect the disease spreading.", "The most used model that reproduces the final state of nonrecurrent epidemics is the Susceptible-Infected-Recovered (SIR) model [7], [8], [2].", "In this model a susceptible individual (S) in contact with an infected one (I) gets infected with probability $q$ at each time step.", "An infected individual recovers (R) after $t_r$ time steps since it was infected.", "Once an individual is recovered, it does not play any role in the spreading.", "In this model the fraction of recovered individuals $R$ is the order parameter of a continuous phase transition with a control parameter $T=1-(1-q)^{t_r}$ , where $T$ is the effective probability of infection denoted as the transmissibility.", "It is known that there exists a critical value $T_c$ that separates a nonepidemic phase from an epidemic phase, so that in the thermodynamic limit $R=0$ for $T\\le T_c$ , and $R>0$ for $T>T_c$ [9], [10].", "It was shown [11], [10], [12] that the final state of the SIR model can be mapped into link percolation, due to the fact that infecting through a link in SIR is equivalent to occupying a link in link percolation, and thus the final state of SIR can be solved using percolation tools.", "Each realization of the final state of SIR is one cluster in link percolation, and an epidemic corresponds to the giant component (GC) in link percolation, which is distinguished from outbreaks (corresponding to the finite clusters) by a threshold for the cluster size $s_c$ [13].", "In random complex networks it is worthwhile to find exact solutions for the main magnitudes of the final state of the SIR model using the generating function formalism.", "In this approach two generating functions are used [14], [15].", "One of them is the generating function of the degree distribution $G_0(x)=\\sum _k P(k) x^k$ , where $P(k)$ is the degree distribution with $k_{\\min } \\le k \\le k_{\\max }$ , and $k_{\\min }$ and $k_{\\max }$ are the minimum and maximum degree respectively.", "The other is the generating function of the excess degree distribution $G_1(x)=\\sum _k k P(k)/\\langle k \\rangle x^{k-1}$ , where $\\langle k \\rangle $ is the average degree of the network.", "In the SIR model for isolated networks, the probability $f_{\\infty }$ that a branch of infected nodes reach the infinity for a given transmissibility $T$ satisfies the self-consistent equation $f_{\\infty } =1-G_1(1-T f_{\\infty })$ [16], [10].", "Note that $G_1(1-T f_{\\infty })$ is the probability that following a random chosen link, which leads to a node, the branch of infection does not reach the infinity through its $(k-1)$ outgoing links.", "The fraction of recovered individuals $R$ , which is equivalent to the fraction $P_{\\infty }$ of nodes belonging to the GC in link percolation, is given by $R=1-G_0(1-T f_{\\infty })$ [16], [10], since $G_0(1-T f_{\\infty })$ is the probability that a random chosen node can not reach the infinity with infected nodes through any of its $k$ links.", "The critical value of the transmissibility is $T_c=1/(\\kappa -1)$ , where $\\kappa =\\langle k^2\\rangle /\\langle k \\rangle $ is the branching factor, and $\\langle k^2\\rangle $ is the second moment of the degree distribution [17], [18].", "For Erdös-Rényi (ER) networks [19], the degree follows a Poisson distribution $P(k)=\\langle k \\rangle ^k e^{-\\langle k \\rangle }/k!\\,$ , and thus $T_c=1/\\langle k \\rangle $ .", "Around criticality $T_c$ , many physical quantities behave as power laws, e.g., $P(s) \\sim s^{-\\tau +1}\\exp (-s/s_{\\max })$ , where $P(s)$ is the probability to find a cluster of size $s$ , $s_{\\max }\\sim |T-T_c|^{-1/\\sigma }$ is the largest finite cluster size, and the fraction of recovered $R\\sim |T-T_c|^{\\beta }$ [10], [20].", "Before the last decade, researchers concentrated on studying these processes in isolated networks [17], [21].", "However, real networks are rarely isolated [22], [23].", "For example, each country has its own transportation network, and those networks from different countries are connected into a larger network due to international transportation.", "Also, different communities of people can hold different opinions, but their opinions can exchange through influencers.", "Thus it is more realistic to consider systems composed of many networks, which are called a network of networks (NON) [24], [25], [23], [26], [27], [28].", "A case of NON is a system composed of several communities (or layers), where a fraction of nodes $r$ from each layer are bridge nodes which are connected to bridge nodes from other communities through $k^b$ bridge links.", "Bridge nodes, which can represent airports connecting countries, may have a huge impact on the system because they can influence individuals in other communities.", "As these kinds of nodes are few compared to the number of nodes inside a community, it is reasonable to study these problems in the limit $r\\rightarrow 0$ .", "In Ref.", "[29], the authors studied node percolation in two ER communities with an ER distribution of bridge links, and studied the behavior of $R$ with $r$ , in the limit $r\\rightarrow 0$ , with the constraint $r\\langle k^b \\rangle =\\text{constant}$ .", "They found, using scaling relations, that $R \\propto r^{1/\\epsilon }$ , where $r$ was associated to an external field.They used $\\delta $ instead of $\\epsilon $ since this behavior is analogous to the relation $M \\propto H^{1/\\delta }$ between the magnetization $M$ and the external field $H$ in the Ising model [30].", "In Ref.", "[31], the authors extended this result to an SIR model and also studied the dynamics.", "In the final state for $r\\rightarrow 0$ , they found the same value of the exponent as in Ref.", "[29], and explained it from a geometrical point of view.", "In their interpretation, the GC was formed by finite clusters in both communities connected through bridges links at $T=T_c(r=0)$ .", "Thus the exponent $\\epsilon $ was associated with the exponent $\\tau $ of the finite cluster size distribution, which allowed them to derive this exponent theoretically and obtained $\\epsilon =1/(\\tau -2)$ .", "Note that due to the constraint $r\\langle k^b \\rangle = \\text{constant}$ , the average external connectivity $\\langle k^b \\rangle $ diverges as $r\\rightarrow 0$ .", "In addition, they used the same transmissibility $T$ along intra- and interlinks.", "However, from a realistic point of view, the fraction of bridge nodes and the average external connectivity do not have to be related, and building a large number of connections for one node is practically expensive, so it is unrealistic to study the case when $\\langle k^b \\rangle \\rightarrow \\infty $ .", "On the other hand, the interaction mechanisms are in general different for internal links than those for bridge links, and strategies like cutting international flights can be used to reduce the disease spreading, so the transmissibility along internal links and bridge links can be very different.", "In this paper, we use a more realistic approach in which the fraction of bridge nodes $r$ and the average external connectivity $\\langle k^b\\rangle $ are independent, and the transmissibility along bridges links $T^b$ is different from the internal transmissiblity $T^I$ .", "When $r$ is small, we find very rich behaviors of $R$ with $r$ , many of which are power laws $R\\propto r^{1/\\epsilon }$ , depending on the values of $T^b$ and $T^I$ .", "In these regions the exponent $\\epsilon $ follows different functions of the exponents in the finite cluster size distributions.", "Our theoretical results are in very good agreement with simulations." ], [ "Model", "In our model we consider a system composed of two communities $A$ and $B$ , with degree distributions $P^A(k)$ and $P^B(k)$ .", "The communities are connected through a fraction $r$ of bridges nodes with degree distribution $P^b(k)$ .", "The transmissibility within each community is $T^I$ and the transmissibility along bridge links is $T^b$ .", "To reduce the number of parameters, we will assume that both communities have the same degree distribution, i.e., $P^A(k)=P^B(k) \\equiv P(k)$ .", "Using the generation function formalism, the self-consistent equations of the system are given by $f &=& (1-r)\\left[1-G_1(1-T^I f)\\right] + r\\left[1-G_1(1-T^I f)G_0^b(1-T^bf^b)\\right],\\\\f^b &=& 1-G_1^b(1-T^bf^b)G_0(1-T^I f),$ where $f$ is the probability to expand a branch to the infinity through an internal link, $f^b$ is the probability to expand a branch to the infinity through a bridge link, and $G_0(\\cdot )$ , $G_1(\\cdot )$ , $G_0^b(\\cdot )$ , and $G_1^b(\\cdot )$ are the generating functions of the degree and excess degree distributions for internal and bridge links, respectively.", "The first term on the RHS of Eq.", "(REF ) is the contribution of nonbridge nodes that transmit only internally, while the second term is the contribution of bridge nodes which transmit both internally with $T^I$ and to the other community through bridge links with $T^b$ .", "Thus the fraction of recovered nodes of the system and the fraction of recovered nodes of bridge nodes are given by $ R & = &(1-r)\\left[1-G_0(1-T^I f)\\right] + r\\left[1-G_0(1-T^I f)G_0^b(1-T^bf^b)\\right], \\\\ R^b &= &\\left[1-G_0(1-T^I f)G_0^b(1-T^bf^b)\\right].$ Figure: RR as a function of T b T^b when both internal links and bridge links are ER networks, with 〈k〉=4\\langle k \\rangle =4 and 〈k b 〉=10\\langle k^b \\rangle =10, respectively.", "Theoretical solutions (dark blue solid lines for r=0.1r=0.1 and light orange solid lines for r=0.01r=0.01) are compared with stochastic simulation results of SIR in the final state (dark blue circles for r=0.1r=0.1 and light orange squares for r=0.01r=0.01), for (a) T I =0.25T^I=0.25 and (b) T I =0.2T^I=0.2.", "For the simulations, system sizes N A =N B =10 5 N_A=N_B=10^5, k min =0k_{\\min }=0, k max =100k_{\\max }=100, s c =200s_c=200, and are averaged over 10 3 10^3 realizations.In Fig.", "REF we show the fraction of the recovered $R$ as a function of $T^b$ for different values of $T^I$ and $r$ , for a system where both internal links and bridge links follow an ER degree distribution, with $\\langle k \\rangle =4$ , and $\\langle k^b \\rangle =10$ .", "The solid lines show the numerical solutions obtained from Eqs.", "(REF )-(REF ), and the square and circle symbols are the results from the SIR stochastic simulations.", "We can see that the theory agrees very well with the simulation results, and thus we will mainly use theoretical solutions hereafter.", "Both theoretical solutions and simulation results in Fig.", "REF show a critical value of $T^b$ that depends on $T^I$ and $r$ .", "The system is in a nonepidemic phase, with $R=0$ , when $T^b \\le T^b_c$ , and is in an epidemic phase with a finite positive $R$ when $T^b > T^b_c$ .", "This is due to the fact that the self-consistent Eqs.", "(REF ) and () have only one solution $f=f^b=0$ when $T^b \\le T^b_c$ , and a nontrivial physical solution exists only when $T^b>T^b_c$ .", "The theoretical value of $T^b_c$ can be obtained by solving $|J-I|_{f,f^b=0}=0$ , where $|\\cdot |$ is the determinant, $J$ is the Jacobian matrix, and $I$ is the identity.", "Note that the elements of the Jacobian matrix are given by $J_{i,j}= \\left.\\frac{\\partial f_i}{\\partial f_j}\\right|_{f,f^b=0}$ , where each of $f_i$ and $f_j$ represents $f$ or $f^b$ .", "Thus explicitly $|J-I|_{f,f^b=0}=0$ can be written as $\\begin{vmatrix}T^I(\\kappa -1)-1 & rT^{b}_{c}\\langle k^b\\rangle \\\\[1ex]T^I\\langle k \\rangle & T^{b}_{c}(\\kappa ^b-1)-1\\end{vmatrix}= 0.$ So $T^{b}_{c}$ is given by $T^b_c =\\frac{T^I(\\kappa -1)-1}{(T^I\\left(\\kappa -1\\right)-1)(\\kappa ^b-1)-rT^I\\langle k\\rangle \\langle k^b\\rangle }.$ In the equation above, $T^b_c$ has physical meaning only when $T^I \\le 1/(\\kappa -1)$ (see Appendix for details).", "This implies that any strategy that reduces the transmissibility between communities will prevent a macroscopic number of infected nodes only if the internal transmissibility is below the critical value for an isolated community.", "We can see from Eq.", "(REF ) that, as $r$ approaches 0, the critical value $T^b_c(r \\rightarrow 0)= 1/(\\kappa ^b-1)$ ." ], [ "Different Regimes: Asymptotic Behaviors", "In Fig.", "REF we show the phase diagram for two ER communities connected by ER bridge links with $\\langle k \\rangle =4$ and $\\langle k^b \\rangle =10$ , for different values of $r$ .", "As $r\\rightarrow 0$ , the nonepidemic phase tends to be a rectangle.", "The boundaries of the rectangle are $T^I = T^I_c=1/(\\kappa -1)$ and $T^b=T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , which split the whole space into several regimes, where the relation between $R$ and $r$ follows different behaviors asymptotically.", "In this section, we derive the asymptotic behavior of $R$ versus $r$ , i.e., as $r\\rightarrow 0$ , by first looking at how $R$ depends on $(rR^b)$ , and then how $R^b$ depends on $R$ .", "Figure: An illustration of how finite clusters of recovered nodes in each community (circled in green) are connected due to clusters by bridge links (circled in blue), and thus there exists a GC of recovered nodes in the entire system.", "(a) Only finite clusters exist in each community and for bridge links.", "(b) Only finite clusters exist in each community but a giant component exists for bridge links.", "All recovered nodes are plotted in green, except for patient zero, which is plotted in black, and all nodes that are never infected are plotted in gray.", "Links through which the disease is transmitted are plotted in green, while links that fail to transmit the disease are plotted in gray.", "Squares denote bridge nodes, and circles denote internal nodes.When $T^I\\le T^I_c$ , there are only finite clusters of recovered nodes within each community.", "However, in the epidemic phase, these finite clusters in each community are connected due to bridge links, and thus form a GC of recovered nodes in the entire system, as illustrated in Fig.", "REF .", "Using the mapping between the SIR model and link percolation, as any node in each community has a probability $r$ to be a bridge node, and each bridge node has a probability $R^b$ to be recovered, a finite cluster of size $s$ has a probability $1-(1-r R^b)^s$ to have at least one recovered bridge node, and thus belong to the GC of recovered.", "Thus, the size of the GC as $r \\rightarrow 0$ is given by $R = 1-\\sum _{s=1}^\\infty P(s)(1-r R^b)^s,$ where $P(s) \\sim s^{-\\tau +1}\\exp (-s/s_{\\max })$ is the probability of a finite cluster of size $s$ within a community, $\\tau $ is the Fisher exponent of each community, and the largest finite cluster size $s_{\\max }\\sim |T^I-T^I_c|^{-1/\\sigma }$ .", "Then we can derive the behavior of $R$ with $r$ for $T^I$ below, or equal to the critical internal transmissibility.", "At the critical value $T^I=T^I_c=1/(\\kappa -1)$ so that $s_{\\max }$ diverges and thus $P(s) \\sim s^{-\\tau +1}$ , Eq.", "(REF ) can be simplified into $R\\propto (rR^b)^{\\tau -2}$ [see Eqs.", "(REF ) and (REF ) in Appendix for details].", "This is due to the fact that the average number of infected bridge nodes in each finite cluster of a community depends on the topology of the community, and thus depends on $\\tau $ .", "When $T^I < T^I_c$ , Eq.", "(REF ) can be reduced to $R\\propto rR^b$ , due to the finite $s_{\\max }$ [see Eqs.", "(REF ) and (REF ) in Appendix for details].", "This is intuitive since as $T^I$ is so small that each finite cluster of a community has very few bridge nodes, then the number of nodes in the GC will be proportional to the number of bridge nodes in the GC [as in Fig.", "REF (a)].", "When $\\kappa ^b<\\infty $ , which is always the case in reality, we need to explore the behavior of $R^b$ as well.", "For each cluster connected through bridge links, each bridge node has a probability $1-G_0(1-T^If)$ to be connected to the GC through internal links.", "So as $r \\rightarrow 0$ , a finite cluster of bridge nodes of size $s$ has a probability $1-[G_0(1-T^If)]^s$ to belong to the recovered bridge nodes, and thus $R^b = 1-\\sum _{s}^\\infty P^b(s)[G_0(1-T^If)]^s,$ where $P^b(s) \\sim s^{-\\tau ^b+1}\\exp (-s/s^b_{\\max })$ is the probability of a finite cluster of size $s$ connected by bridge links, $\\tau ^b$ is the Fisher exponent of bridge links, and $s^b_{\\max }\\sim |T^b-T^b_c(r\\rightarrow 0)|^{-1/\\sigma ^b}$ is the largest finite cluster size of bridge links.", "From Eq.", "(REF ) we know that $R\\approx 1-G_0(1-T^If)$ as $r\\rightarrow 0$ , so $R^b \\approx 1-\\sum _{s}^\\infty P^b(s)(1-R)^s.$ At the critical value $T^b=T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , $s^b_{\\max }$ diverges, and thus $R^b\\propto R^{\\tau ^b-2}$ [see Eq.", "(REF ) in Appendix for details].", "When $T^b < T^b_c(r\\rightarrow 0)$ , $R^b\\propto R$ since $s^b_{\\max }<\\infty $ [see Eq.", "(REF ) in Appendix for details].", "When $T^b>T^b_c(r\\rightarrow 0)$ , most bridge nodes are connected into one big cluster through bridge links, so Eqs.", "(REF ) and (REF ) do not apply and $R^b$ is not a power law of $R$ [see Fig.", "REF (b)].", "In summary, $R \\propto {\\left\\lbrace \\begin{array}{ll}rR^b, & \\text{if } T^I<1/(\\kappa -1) \\\\(rR^b)^{\\tau -2}, & \\text{if } T^I=1/(\\kappa -1) \\\\\\text{not a power law of } (rR^b), & \\text{if } T^I>1/(\\kappa -1)\\end{array}\\right.", "},$ $R^b \\propto {\\left\\lbrace \\begin{array}{ll}R, & \\text{if } T^b<1/(\\kappa ^b-1) \\\\R^{\\tau ^b-2}, & \\text{if } T^b=1/(\\kappa ^b-1) \\\\\\text{not a power law of } R, & \\text{if } T^b>1/(\\kappa ^b-1)\\end{array}\\right.", "}.$ Table: Asymptotic power-law behaviors of RR with rr in different regimes.", "The exponent ϵ\\epsilon in R∝r 1/ϵ R\\propto r^{1/\\epsilon } is independent of the specific values of κ\\kappa or κ b \\kappa ^b, but varies with the regimes where the combination of T I T^I and T b T^b falls in.", "⌀\\varnothing means there is no power-law relation in that regime.Combining Eqs.", "(REF ) and (REF ), we obtain the asymptotic power-law behaviors of $R$ with $r$ in many regimes.", "Different values of $\\epsilon $ in the relation $R \\propto r^{1/\\epsilon }$ are summarized in Table REF .", "As an example, when both communities and the bridge links are all ER networks, we have $\\tau = \\tau ^b = 5/2$ , and in the limit $\\kappa ^b \\rightarrow \\infty $ so that $T^b>1/(\\kappa ^b-1)$ all the time, we obtain the same exponents that were found in Refs.", "[29], [31], in which $r \\langle k^b \\rangle = \\text{constant}$ , and $r \\rightarrow 0$ .", "Note that the results in Table REF apply to networks with any degree distributions, i.e., either internal or bridge links or both can be homogeneous or heterogeneous.", "Also, a similar methodology can be applied to a system when the two communities have different degree distributions, i.e., $P^A(k)\\ne P^B(k)$ , and we can still correctly predict the asymptotic power-law relations between $R$ and $r$ for all regimes (see Appendix for details).", "Figure: RR as a function of rr for different regimes where a power law exists: (a) Regime II: T I =0.2T^I=0.2, T b =0.1T^b=0.1; (b) Regime III: T I =0.2T^I=0.2, T b =0.2T^b=0.2; (c) Regime IV: T I =0.25T^I=0.25, T b =0.05T^b=0.05; (d) Regime V: T I =0.25T^I=0.25, T b =0.1T^b=0.1; and (e) Regime VI: T I =0.25T^I=0.25, T b =0.2T^b=0.2.", "Both internal links and bridge links are ER networks, with 〈k〉=4\\langle k \\rangle =4 and 〈k b 〉=10\\langle k^b \\rangle =10, respectively.", "In each regime, numerical solutions of Eqs.", "()-() are plotted in black solid lines, and a dashed line is drawn with the slope predicted by Table .In Fig.", "REF we show the numerical solutions of Eqs.", "(REF )-(REF ) with the log-log plot of $R$ with $r$ for small $r$ in different regimes, for a system of two ER communities connected by ER bridge links, and thus $\\tau =\\tau ^b=5/2$ .", "In each regime, we plot a dashed line with the slope predicted by the theory (see Table REF ).", "We can see that our predictions are in good agreement with the numerical results.", "In Appendix , we also compared simulations using link percolation with numerical solutions.", "We use link percolation mapping instead of SIR to simulate the final state because the former is much less time-consuming for big system sizes.", "The simulation results agree well with theoretical solutions, except for some finite-size effects when $r$ is very small (see Appendix for details).", "When a highly infectious epidemic occurs, one of the first strategies used by many countries is to shut down some international airports.", "Those international airports serve as bridge nodes in the whole system of global transportation, so shutting them down is essentially reducing the percentage of bridge nodes $r$ .", "Meanwhile, international flights are cut for those airports that are still open, which mitigates the disease spreading by reducing the transmissibility along bridge links.", "Also, social distancing strategies like staying at home as long as it is possible or wearing facial masks if having to go outside reduce the chance of face-to-face infection, which is utilized by most countries as another strategy to reduce both $T^I$ and $T^b$ .", "As can easily be seen from our results in this section, strategies like shutting down international airports are not as effective in some regimes as in others.", "In those regimes with a smaller $\\epsilon $ , shutting down international airports to reduce $r$ will significantly reduce the fraction of recovered $R$ , while in regimes with a larger $\\epsilon $ , $R$ will be reduced only slightly.", "This helps us to decide what kind of strategies we are supposed to use to control disease spreading effectively; i.e., shutting down international airports had better be combined with strategies to reduce $T^I$ and $T^b$ , so that it falls in a regime with a small $\\epsilon $ ." ], [ "Crossovers when $T^I\\lesssim T^I_c$", "In Table REF , we can see that the asymptotic values of the exponent $\\epsilon $ change abruptly between regimes.", "However, in this section, we are going to show that, for $T^I \\lesssim T^I_c$ , the system behaves in the same way as $T^I=T^I_c$ for a relatively large value of $r$ , but changes to its asymptotic behavior continuously as $r$ decreases.", "From a percolation point of view, a finite cluster belongs to the GC if it contains recovered bridge nodes (with an overall percentage of $r R^b$ ).", "As $r R^b$ decreases from 1, the GC starts to lose some finite clusters so that $R$ also decreases from 1.", "When $r$ is not too small, finite clusters of smaller sizes are more likely to be detached from the GC.", "Note that the probability of a cluster of size $s$ is $P(s) \\sim s^{-\\tau +1}\\exp (-s/s_{\\max })$ for $T^I<T^I_c$ and $P(s) \\sim s^{-\\tau +1}$ for $T^I=T^I_c$ , which are the same for smaller cluster sizes, so the behaviors of $R$ versus $r$ for different values of $T^I \\lesssim T^I_c$ are the same when $r$ is not small enough.", "The distribution $P(s)$ starts to differ when the GC starts to lose relatively large finite clusters, i.e., when $s$ is comparable to $s_{\\max }$ , and a crossover is going to show up.", "Denote $r^*$ as where the crossover occurs, $R^*$ and $R^{b*}$ as the fraction of recovered nodes and bridge nodes at the crossover respectively, we have $r^* R^{b*}\\sim 1/s_{\\max }$ (see Appendix ).", "Figure: Theoretical solutions of RR as a function of rr when T I ≲T c I T^I\\lesssim T^I_c, i.e., T I =0.248,0.2485,0.249,0.2495,0.2499T^I=0.248,0.2485,0.249,0.2495,0.2499 (from light blue to dark blue solid lines) with (a) T b =0.05T^b=0.05, (b) T b =0.1T^b=0.1, (c) T b =0.2T^b=0.2.", "Both internal links and bridge links are ER networks, with 〈k〉=4\\langle k \\rangle =4 and 〈k b 〉=10\\langle k^b \\rangle =10, respectively.", "Black dashed lines are drawn with the slope predicted by Table  for different regimes.In Fig.", "REF we use different values of $T^I \\lesssim T^I_c=1/(\\kappa -1)$ to show the behavior of $R$ versus $r$ near the critical point, when $T^b$ is below, equal to, and above $T^b_c(r\\rightarrow 0)$ .", "We can see a crossover, where the exponent $\\epsilon $ is the same as the $\\epsilon $ for $T^I = T^I_c$ when $r$ is not small enough (i.e., $r^*\\ll r \\ll 1$ ), and has the same $\\epsilon $ as the asymptotic one for $T^I < T^I_c$ when $r$ is extremely small (i.e., $r\\ll r^*$ ).", "For example, when $T^b<T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , i.e., $T^b=0.05$ , which is shown in Fig.", "REF (a), we can see the power-law behavior $R\\propto r^{1/\\epsilon }$ with $\\epsilon =1$ (as in regime IV) when $r$ is relatively large, but it is in the nonepidemic phase with $R=0$ (as in regime I) as $r\\rightarrow 0$ .", "When $T^b=T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , i.e., $T^b=0.1$ , which is shown in Fig.", "REF (b), the exponent $\\epsilon $ changes from $3/2$ (as in regime V) to $1/2$ (as in regime II) as $r\\rightarrow 0$ .", "When $T^b>T^b_c(r\\rightarrow 0)$ , e.g., $T^b=0.2$ , which is shown in Fig.", "REF (c), the exponent $\\epsilon $ changes from 2 (as in regime VI) to 1 (as in regime III) as $r\\rightarrow 0$ .", "As was mentioned above, since the crossover occurs when $r^* R^{b*}\\sim 1/s_{\\max }$ , and $s_{\\max }$ depends on the internal transmissibility $T^I$ , the values of $r^*$ and $R^*$ also depend on $T^I$ .", "To be more explicit, they follow power laws of the difference between $T^I$ and its critical value, i.e., $r^*\\sim |T^I-T^I_c|^{\\beta _r}$ , and $R^*\\sim |T^I-T^I_c|^{\\beta _R}$ .", "The values of $\\beta _r$ and $\\beta _R$ can be derived as the following.", "Combined with the criteria $1/s_{\\max } \\sim r^*R^{b*}$ , and considering the relation between $R^b$ and $R$ as in Eq.", "(REF ), we will get $1/s_{\\max } \\sim {\\left\\lbrace \\begin{array}{ll}r^*R^*, & \\text{if } T^b<1/(\\kappa ^b-1) \\\\r^*(R^*)^{\\tau ^b-2}, & \\text{if } T^b=1/(\\kappa ^b-1) \\\\r^*, & \\text{if } T^b>1/(\\kappa ^b-1)\\end{array}\\right.", "}.$ Since curves with different values of $T^I$ overlap for a relatively large $r$ (which is also verified in Fig.", "REF ), we also have the relation $R^* \\propto (r^*)^{1/\\epsilon }$ , where $\\epsilon $ is the one for $T^I=1/(\\kappa -1)$ , respectively.", "If we combine $R^* \\propto (r^*)^{1/\\epsilon }$ with Eq.", "(REF ), and knowing that $s_{\\max }\\sim |T^I-T^I_c|^{-1/\\sigma }$ , we obtain that $r^* \\sim {\\left\\lbrace \\begin{array}{ll}|T^I-T^I_c|^{(3-\\tau )/\\sigma } \\sim |T^I-T^I_c|^\\gamma , & \\text{if } T^b<1/(\\kappa ^b-1) \\\\|T^I-T^I_c|^{(1-(\\tau -2)(\\tau ^b-2))/\\sigma } \\sim |T^I-T^I_c|^{(\\tau ^b-2)\\gamma +(3-\\tau ^b)/\\sigma }, & \\text{if } T^b=1/(\\kappa ^b-1) \\\\|T^I-T^I_c|^{1/\\sigma }, & \\text{if } T^b>1/(\\kappa ^b-1)\\end{array}\\right.", "}.$ In any region, the crossover point $r^*$ goes to 0 as $T^I$ approaches $T^I_c$ , so we do not see a crossover unless $T^I$ is below but very close to $T^I_c$ .", "Knowing that the mean finite cluster size $\\langle s \\rangle \\sim |T^I-T^I_c|^{-\\gamma }$ , and the largest finite cluster size $s_{\\max }\\sim |T^I-T^I_c|^{-1/\\sigma }$ , Eq.", "(REF ) can also be written as $1/r^* \\sim {\\left\\lbrace \\begin{array}{ll}\\langle s \\rangle , & \\text{if } T^b<1/(\\kappa ^b-1) \\\\\\langle s \\rangle ^{\\tau ^b-2} {s_{\\max }}^{3-\\tau ^b}, & \\text{if } T^b=1/(\\kappa ^b-1) \\\\s_{\\max }, & \\text{if } T^b>1/(\\kappa ^b-1)\\end{array}\\right.", "},$ and for all three regions of $T^b$ , $R^* \\sim |T^I-T^I_c|^{(\\tau -2)/\\sigma } \\sim |T^I-T^I_c|^\\beta $ , whose exponent is the same as the one in $R\\propto |T-T_c|^\\beta $ for an isolated network.", "The scaling relation between $R$ and $r$ around the critical internal transmissibility ($T^I\\lesssim T^I_c$ ) can then be written as $R = R^*\\,F\\left(\\frac{r}{r^*}\\right),$ where $F(x)$ is given by $F(x)\\sim x^{1/\\epsilon }$ and $\\epsilon ={\\left\\lbrace \\begin{array}{ll}\\epsilon (T^I<T^I_c), & \\text{if } x \\ll 1 \\\\\\epsilon (T^I=T^I_c), & \\text{if } x \\gg 1\\end{array}\\right.", "}.$ (See Table REF for values of $\\epsilon $ for different values of $T^b$ .)", "Figure: Theoretical solutions of R/R * R/R^* as a function of r/r * r/r^*, where R * =|T I -T c I | β R R^*=|T^I-T^I_c|^{\\beta _R} and r * =|T I -T c I | β r r^*=|T^I-T^I_c|^{\\beta _r} when T I ≲T c I T^I\\lesssim T^I_c, i.e., T I =0.248,0.2485,0.249,0.2495,0.2499T^I=0.248,0.2485,0.249,0.2495,0.2499 (from light blue to dark blue solid lines) with (a) T b =0.05T^b=0.05, so that β r =1\\beta _r=1 and β R =1\\beta _R=1, (b) T b =0.1T^b=0.1, so that β r =3/2\\beta _r=3/2 and β R =1\\beta _R=1, and (c) T b =0.2T^b=0.2, so that β r =2\\beta _r=2 and β R =1\\beta _R=1.", "Both internal links and bridge links are ER networks, with 〈k〉=4\\langle k \\rangle =4 and 〈k b 〉=10\\langle k^b \\rangle =10, respectively.", "All curves with different T I ≲T c I T^I\\lesssim T^I_c collapse under the scaling relation.In Fig.", "REF we show the plot of $R$ versus $r$ rescaled by $R^*\\sim |T^I-T^I_c|^{\\beta _R}$ and $r^*\\sim |T^I-T^I_c|^{\\beta _r}$ for three different values of $T^b$ .", "Since both communities and the bridge links are all ER networks, $\\tau =\\tau ^b=5/2$ , $\\gamma =1$ , $\\sigma =1/2$ , and $\\beta =1$ .", "For all values of $T^b$ , $\\beta _R=1$ , which is the same as the exponent $\\beta $ in $R \\propto |T-T_c|^\\beta $ of an isolated network, as mentioned above.", "When $T^b<1/(\\kappa ^b-1)$ , e.g., $T^b=0.05$ , we have $\\beta _r=\\gamma =1$ .", "When $T^b=1/(\\kappa ^b-1)$ , e.g., $T^b=0.1$ , we have $\\beta _r=(\\tau ^b-2)\\gamma +(3-\\tau ^b)/\\sigma =3/2$ .", "When $T^b>1/(\\kappa ^b-1)$ , e.g., $T^b=0.2$ , we have $\\beta _r=1/\\sigma =2$ .", "We can see that the curves of $R/R^*$ versus $r/r^*$ for different values of $T^I \\lesssim T^I_c$ collapse.", "Empirically, there are cases when we can not or do not want to further reduce internal transmissibility, for example due to the shortage of facial masks or to avoid severe economic consequences, so that $T^I$ is below but close to its critical value.", "In those scenarios, depending on the value of $T^I$ , if there are too many open international airports, we may go through a section where shutting them down does not show a huge effect on the total number of infected individuals.", "However, as long as the internal transmissibility is below its critical value so that the disease can not spread massively within a community, if we keep reducing $r$ , then after a point, which is the crossover, the total number of infected individuals is going to drop dramatically.", "A good understanding of this crossover is going to help us better estimate the impact of epidemic strategies." ], [ "Conclusions", "In this paper, we study the effect of bridge nodes to the final state of the SIR model, by mapping it to link percolation.", "We find power-law asymptotic behaviors between $R$ and $r$ in different regimes, depending on how $T^I$ and $T^b$ are compared to their critical values.", "The different exponents are related to the different mechanisms of how finite clusters in each community are connected into the GC of the whole system.", "Additionally, around but below the critical point of internal transmissibility (when $T^I \\lesssim T^I_c$ ), we find the crossover points $r^*$ such that $R$ versus $r$ follows a different power-law behavior when $r^* \\ll r\\ll 1$ compared to its asymptotic one (when $r\\ll r^*$ ).", "The methodology and results in this paper can easily be generalized for NONs with multiple communities.", "The results can provide the authorities with helpful guidance on making decisions about epidemic strategies.", "They enable us to better anticipate the impacts of epidemic strategies before adopting them, and help us understand why strategies like shutting down international airports had better be combined with adequate social distancing strategies to be more effective." ], [ "Acknowledgments", "The authors would like to thank Shlomo Havlin for useful discussions.", "J.M., L.D.V., and L.A.B.", "acknowledge support from NSF Grant No.", "PHY-1505000, DTRA Grant No.", "HDTRA-1-14-1-0017, and DTRA Grant No.", "HDTRA-1-19-1-0016.", "L.A.B.", "and L.D.V.", "thank UNMdP (Grant EXA956/20) and CONICET (Grant No.", "PIP 00443/2014) for financial support." ], [ "Physical and Non-Physical Critical Values", "When $T^I \\le 1/(\\kappa -1)$ , e.g., $T^I=0.2$ , as in Fig.", "REF (a), Eq.", "(REF ) gives the physical critical value $T^b_c$ , from which point the physical solution of $f$ and $f^b$ becomes nontrivial.", "However, when $T^I > 1/(\\kappa -1)$ , e.g., $T^I=0.4$ , as in Fig.", "REF (b), Eq.", "(REF ) gives the value of $T^b$ where more nonphysical solutions show up, while the physical solution stays smooth, so there is no critical phenomenon.", "Figure: Numerical solutions of ff (orange) and f b f^b (blue) of Eqs.", "() and () with r=0.1r=0.1, given (a) T I =0.2T^I=0.2 and (b) T I =0.4T^I=0.4.The only physical solution in each case is plotted in solid lines, and pairs of nonphysical solutions are in dashed lines with different symbols.Both internal links and bridge links are ER networks, with 〈k〉=4\\langle k \\rangle =4 and 〈k b 〉=10\\langle k^b \\rangle =10, respectively.The vertical dashed line represents the value of T c b T^b_c predicted by Eq.", "()." ], [ "Derivations of the Relations between $R$ , {{formula:b76ae811-dc22-4f4d-8f96-7d356dafdf41}} and {{formula:ad1baf8a-5c03-45ed-8759-600793ab7bc0}} in Different Regimes", "To derive how $R$ depends on $r R^b$ , recall that $P(s) \\sim s^{-\\tau +1}\\exp (-s/s_{\\max })$ , so Eq.", "(REF ) becomes $\\begin{split}R & = 1-\\sum _{s=1}^\\infty P(s)(1-r\\,R^b)^s \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }}(1-r\\,R^b)^s ds}{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }} ds} \\\\& = 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }}e^{s\\ln (1-r\\,R^b)} ds}{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }} ds} \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-s(1/s_{\\max }+r\\,R^b)} ds}{\\int _1^\\infty s^{-\\tau +1}\\exp (-s/s_{\\max }) ds}.\\end{split}$ When $T^I=T^I_c=1/(\\kappa -1)$ , so $s_{\\max }$ diverges and thus $P(s) \\sim s^{-\\tau +1}$ , or if $T^I\\lesssim T^I_c=1/(\\kappa -1)$ , but $r$ is not too small, so that $1/s_{\\max }\\ll r R^b \\ll 1$ and thus $1/s_{\\max }$ can be ignored, $\\begin{split}R & \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-r\\,R^b\\,s} ds}{\\int _1^\\infty s^{-\\tau +1} ds} \\\\& = 1-(\\tau -2) \\int _1^\\infty s^{-\\tau +1}e^{-r\\,R^b\\,s} ds \\\\& = 1-(\\tau -2)(r\\,R^b)^{\\tau -2} \\int _{r\\,R^b}^\\infty u^{-\\tau +1}e^{-u} du, \\text{ where }u=r\\,R^bs \\\\& = 1-(\\tau -2)(rR^b)^{\\tau -2} \\left[\\frac{(rR^b)^{-\\tau +2}}{\\tau -2}e^{-rR^b} - \\int _{rR^b}^\\infty \\frac{u^{-\\tau +2}}{\\tau -2}e^{-u} du \\right] \\\\& \\approx (r\\,R^b)^{\\tau -2} \\int _{r\\,R^b}^\\infty u^{-\\tau +2}e^{-u} du \\\\& \\propto (r\\,R^b)^{\\tau -2}.\\end{split}$ When $T^I<T^I_c=1/(\\kappa -1)$ , so $s_{\\max }$ is finite and thus $P(s) \\sim s^{-\\tau +1}\\exp (-s/s_{\\max })$ , or if $T^I\\lesssim T^I_c=1/(\\kappa -1)$ , but $r$ is very small, so that $1/s_{\\max }$ can not be ignored compared with $r R^b$ , $\\begin{split}R & \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }}(1-r\\,R^b)^s ds}{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }} ds} \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }}(1-r\\,R^bs) ds}{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }} ds} \\\\& = r\\,R^b \\frac{\\int _1^\\infty s^{-\\tau +2}e^{-s/s_{\\max }} ds}{\\int _1^\\infty s^{-\\tau +1}e^{-s/s_{\\max }} ds} \\\\& \\propto r\\,R^b.\\end{split}$ When $T^I>T^I_c=1/(\\kappa -1)$ , it is in the epidemic phase and $R$ does not approach 0 as $r\\rightarrow 0$ , so there is no power-law relation between $R$ and $r R^b$ in this regime.", "On the other hand, to derive how $R^b$ depends on $R$ , recall that $P^b(s) \\sim s^{-\\tau ^b+1}\\exp (-s/s^b_{\\max })$ .", "When $T^b=T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , so $s^b_{\\max }$ diverges and $P^b(s) \\sim s^{-\\tau ^b+1}$ , $\\begin{split}R^b & \\approx 1-\\sum _{s=1}^\\infty P^b(s)(1-R)^s \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau ^b+1}(1-R)^s ds}{\\int _1^\\infty s^{-\\tau ^b+1} ds} \\\\& = 1-(\\tau ^b-2) \\int _1^\\infty s^{-\\tau ^b+1}e^{s\\ln (1-R)} ds \\\\& \\approx 1-(\\tau ^b-2) \\int _1^\\infty s^{-\\tau ^b+1}e^{-Rs} ds \\\\& = 1-(\\tau ^b-2)R^{\\tau ^b-2} \\int _R^\\infty u^{-\\tau ^b+1}e^{-u} du, \\text{ where }u=Rs \\\\& = 1-(\\tau ^b-2)R^{\\tau ^b-2} \\left[\\frac{R^{-\\tau ^b+2}}{\\tau ^b-2}e^{-R} - \\int _R^\\infty \\frac{u^{-\\tau ^b+2}}{\\tau ^b-2}e^{-u} du \\right] \\\\& \\approx R^{\\tau ^b-2} \\int _R^\\infty u^{-\\tau ^b+2}e^{-u} du \\\\& \\propto R^{\\tau ^b-2}.\\end{split}$ When $T^b<T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , and thus $s^b_{\\max }<\\infty $ , $\\begin{split}R^b & \\approx 1-\\sum _{s=1}^\\infty P^b(s)(1-R)^s \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau ^b+1}e^{-s/s^b_{\\max }}(1-R)^s ds}{\\int _1^\\infty s^{-\\tau ^b+1}e^{-s/s^b_{\\max }} ds} \\\\& \\approx 1-\\frac{\\int _1^\\infty s^{-\\tau ^b+1}e^{-s/s^b_{\\max }}(1-Rs) ds}{\\int _1^\\infty s^{-\\tau ^b+1}e^{-s/s^b_{\\max }} ds} \\\\& = R \\frac{\\int _1^\\infty s^{-\\tau ^b+2}e^{-s/s^b_{\\max }} ds}{\\int _1^\\infty s^{-\\tau ^b+1}e^{-s/s^b_{\\max }} ds} \\\\& \\propto R.\\end{split}$ When $T^b>T^b_c(r\\rightarrow 0)=1/(\\kappa ^b-1)$ , most bridge nodes are connected into one big cluster through bridge links, so Eqs.", "(REF ) and (REF ) do not hold and $R^b$ is not a power law of $R$ ." ], [ "Extension: Regimes when $P^A(k)\\ne P^B(k)$", "When the two communities of the system have different degree distributions, i.e., $P^A(k)\\ne P^B(k)$ , a similar methodology can be applied to make predictions about the asymptotic power-law relations between $R$ and $r$ in different regimes.", "In this case, instead of Eqs.", "(REF ) and (), we are going to have the following theoretical equations: $f^A &=& (1-r)\\left[1-G_1^A(1-T^I f^A)\\right] + r\\left[1-G_1^A(1-T^I f^A)G_0^b(1-T^bf^{A,b})\\right],\\\\f^B &=& (1-r)\\left[1-G_1^B(1-T^I f^B)\\right] + r\\left[1-G_1^B(1-T^I f^B)G_0^b(1-T^bf^{B,b})\\right],\\\\f^{A,b} &=& 1-G_1^b(1-T^bf^{B,b})G_0^B(1-T^I f^B),\\\\f^{B,b} &=& 1-G_1^b(1-T^bf^{A,b})G_0^A(1-T^I f^A),$ where $f^A$ (or $f^B$ ) is the probability to expand a branch to the infinity through an internal link in community A (or B), $f^{A,b}$ (or $f^{B,b}$ ) is the probability to expand a branch to the infinity through a bridge link, which starts from a bridge node in community A (or B); instead of Eqs.", "(REF ) and (), we are going to have $ R^A & = &(1-r)\\left[1-G_0^A(1-T^I f^A)\\right] + r\\left[1-G_0^A(1-T^I f^A)G_0^b(1-T^bf^{A,b})\\right], \\\\ R^B & = &(1-r)\\left[1-G_0^B(1-T^I f^B)\\right] + r\\left[1-G_0^B(1-T^I f^B)G_0^b(1-T^bf^{B,b})\\right], \\\\R & = & (R^A+R^B)/2, \\\\R^{A,b} &=& 1-G_0^b(1-T^bf^{A,b})G_0^A(1-T^I f^A), \\\\R^{B,b} &=& 1-G_0^b(1-T^bf^{B,b})G_0^B(1-T^I f^B), \\\\R^b & = & (R^{A,b}+R^{B,b})/2.$ It is easy to see that the equations above will be reduced to Eqs.", "(REF )-() if $P^A(k)=P^B(k)=P(k)$ .", "To predict the relation between $R^A$ , $R^B$ , and $R$ as a function of $r$ , we assume $\\kappa ^A<\\kappa ^B$ , without loss of generality.", "Then there will be 15 regimes, namely, the five regimes for $T^I$ ($T^I<\\frac{1}{\\kappa ^B-1}<\\frac{1}{\\kappa ^A-1},T^I=\\frac{1}{\\kappa ^B-1}<\\frac{1}{\\kappa ^A-1},\\frac{1}{\\kappa ^B-1}<T^I<\\frac{1}{\\kappa ^A-1},\\frac{1}{\\kappa ^B-1}<T^I=\\frac{1}{\\kappa ^A-1},\\frac{1}{\\kappa ^B-1}<\\frac{1}{\\kappa ^A-1}<T^I$ ), combined with the three regimes for $T^b$ ($T^b<\\frac{1}{\\kappa ^b-1},T^b=\\frac{1}{\\kappa ^b-1},T^b>\\frac{1}{\\kappa ^b-1}$ ).", "Here, we select two regimes as examples: (a) $T^I<\\frac{1}{\\kappa ^B-1}<\\frac{1}{\\kappa ^A-1},T^b=\\frac{1}{\\kappa ^b-1}$ , and (b) $T^I=\\frac{1}{\\kappa ^B-1}<\\frac{1}{\\kappa ^A-1},T^b=\\frac{1}{\\kappa ^b-1}$ .", "In case (a), where $T^I<\\frac{1}{\\kappa ^A-1}$ and $T^I<\\frac{1}{\\kappa ^B-1}$ , using a similar methodology as in Eq.", "(REF ), we will get $R^A \\propto rR^{A,b}$ and $R^B \\propto rR^{B,b}$ .", "Since $T^b=\\frac{1}{\\kappa ^b-1}$ , using a similar methodology as in Eq.", "(REF ), but substituting $(1-R)$ by $\\sqrt{(1-R^A)(1-R^B)}$ $\\endcsname $For a finite cluster of bridge nodes, especially if it is large enough to belong to the GC, there are approximately the same number of nodes that belong to each community, so instead of $R^b\\approx 1-\\sum _s^\\infty P^b(s)(1-R)^s$ as in Eq.", "(REF ), we will have $R^b\\approx 1-\\sum _s^\\infty P^b(s)(1-R^A)^{s/2}(1-R^B)^{s/2}$ in this case.", ", we will get $R^{A,b}\\sim R^{B,b}\\sim R^b\\propto \\left[1-\\sqrt{(1-R^A)(1-R^B)}\\right]^{\\tau ^b-2}$ .", "In the case of $r\\rightarrow 0$ so that $R^A,R^B\\rightarrow 0$ , as well as $\\kappa ^A<\\kappa ^B$ so that $R^B\\gg R^A$ , we will have $1-\\sqrt{(1-R^A)(1-R^B)}\\approx \\frac{R^A+R^B}{2}=R\\propto R^B$ .", "That is to say, $R$ is dominated by the community with a larger $\\kappa $ .", "As a result, we will have $R^b\\propto (R^B)^{\\tau ^b-2}$ .", "Combining $R^A \\propto rR^b,R^B \\propto rR^b$ with $R^b\\propto (R^B)^{\\tau ^b-2}$ , we will get $R^A\\propto r^{1/\\epsilon ^A}$ , where $\\epsilon ^A=1-(\\tau ^b-2)$ , and $R^B\\propto r^{1/\\epsilon ^B}$ , where $\\epsilon ^B=1-(\\tau ^b-2)$ .", "We have $\\epsilon ^A=\\epsilon ^B$ , as expected, since $T^I$ is below critical in both communities, and the $\\epsilon $ as in $R\\propto r^{1/\\epsilon }$ has the same value $\\epsilon =1-(\\tau ^b-2)$ as well.", "In case (b), where $T^I<\\frac{1}{\\kappa ^A-1}$ and $T^I=\\frac{1}{\\kappa ^B-1}$ , similarly, we will get $R^A \\propto rR^b$ and $R^B \\propto (rR^b)^{\\tau -2}$ .", "Since $T^b=\\frac{1}{\\kappa ^b-1}$ , we still get $R^b\\propto \\left[1-\\sqrt{(1-R^A)(1-R^B)}\\right]^{\\tau ^b-2}\\propto \\left(\\frac{R^A+R^B}{2}\\right)^{\\tau ^b-2}\\propto (R^B)^{\\tau ^b-2}$ .", "Combining them, we will get $R^A\\propto r^{1/\\epsilon ^A}$ , where $\\epsilon ^A=1-(\\tau -2)(\\tau ^b-2)$ , and $R^B\\propto r^{1/\\epsilon ^B}$ , where $\\epsilon ^B=\\frac{1}{\\tau -2}-(\\tau ^b-2)$ .", "The $\\epsilon $ as in $R\\propto r^{1/\\epsilon }$ is dominated by $\\epsilon ^B$ , i.e., $\\epsilon =\\epsilon ^B=\\frac{1}{\\tau -2}-(\\tau ^b-2)$ .", "Figure: R A R^A (blue), R B R^B (orange), and R=(R A +R B )/2R=(R^A+R^B)/2 (black) as a function of rr, for two example regimes when P A (k)≠P B (k)P^A(k)\\ne P^B(k): (a) T I =0.0625T^I=0.0625, T b =0.1T^b=0.1; (b) T I =0.125T^I=0.125, T b =0.1T^b=0.1.", "Both internal links and bridge links are ER networks, with 〈k A 〉=4\\langle k^A\\rangle =4, 〈k B 〉=8\\langle k^B\\rangle =8,and 〈k b 〉=10\\langle k^b\\rangle =10.In each regime, numerical solutions of Eqs.", "()-() are plotted in solid lines, and dashed lines are drawn with predicted slopes.As in Fig.", "REF , we consider a system where both internal and external links are ER networks, with $\\langle k^A\\rangle =4$ , $\\langle k^B\\rangle =8$ , and $\\langle k^b\\rangle =10$ .", "As a result, $\\tau ^I=\\tau ^b=5/2$ , $\\kappa ^A=5$ such that $T^A_c=\\frac{1}{\\kappa ^A-1}=0.25$ , $\\kappa ^B=9$ such that $T^B_c=\\frac{1}{\\kappa ^B-1}=0.125$ , and $\\kappa ^b=11$ such that $T^b_c(r\\rightarrow 0)=\\frac{1}{\\kappa ^b-1}=0.1$ .", "We can see from Fig.", "REF that the numerical solutions of $R^A,R^B$ , and $R$ from Eqs.", "(REF )-() (solid lines) agree well with dashed lines, whose slopes are predicted as above, as $r\\rightarrow 0$ .", "It can be verified that a similar methodology can be used to give correct predictions for all 15 regimes." ], [ "Simulation Results Compared with Numerical Solutions", "In Fig.", "REF , we show the simulation results of the link percolation mapping and numerical solutions of Eqs.", "(REF )-(REF ) when both internal links and bridge links are ER networks, with $\\langle k \\rangle =4$ , and $\\langle k^b \\rangle =10$ .", "When $T^I=1/\\langle k \\rangle =0.25$ , $T^b=1/\\langle k^b \\rangle =0.1$ , the simulation agrees well with theoretical solutions [see Fig.", "REF (a)].", "When $T^I=0.25$ , $T^b=0.05$ , a finite-size effect shows up and a much larger system size is required in order to obtain the theoretical results.", "From Fig.", "REF (b) we can see that as the system size increases, the simulation results converge to the theoretical solution.", "This is further verified in Fig.", "REF , in which we show the box plots of the simulation results of $R$ for different system sizes.", "We can see that as system size increases, the distribution of $R$ narrows and converges to the theoretical solution (horizontal dashed line)." ] ]

[ [ "Emergent gravity through non-linear perturbation" ], [ "Abstract As of now, all analogue gravity models available in the literature deal with the emergence of an acoustic geometry through linear perturbations of transonic fluids only.", "It has never been investigated whether the analogue gravity phenomena is solely a consequence of linear perturbations, or rather a generic property of arbitrary perturbations of inhomogeneous, inviscid and irrotational fluids.", "In the present work, for the first time in the literature, we demonstrate that acoustic spacetimes may be formed through higher order non-linear perturbations, and thus establish that analogue gravity phenomena is rather more general than what was thought before.", "We consider spherically accreting astrophysical systems as a natural classical analogue gravity model, and develop a formalism to investigate non-linear perturbations of such accretion flows to arbitrary order.", "Our iterative approach involves a coupled set of equations for the mass accretion rate and the density of the fluid.", "In particular, we demonstrate that the wave equation for the mass accretion rate involves an acoustic metric which can be perturbatively constructed to all orders.", "We numerically solve the coupled equations about the leading transonic Bondi flow solution.", "This analysis uses boundary conditions set to the original unperturbed values, with the time dependence of the mass accretion rate perturbation taken to be exponentially damped.", "The perturbed solutions indicate that second order and higher perturbations of the metric generically cause the original acoustic horizon to oscillate and change in size.", "We explain this phenomenon in detail and its implications on non-linear perturbations of accretion flows in general." ], [ "Introduction", "Perturbations of irrotational, inhomogeneous, inviscid, transonic fluid flows may lead to the formation of an acoustic metric embedded within the background stationary flow.", "Such metrics are like black holes in that they contain acoustic/sonic horizon(s) from within which phonons cannot escape [1], [2], [3], [4], [5], [6], [7].", "As of now, analogue gravity models have been constructed only through linear perturbation.", "In our present work, we shall generalize the idea of analogue spacetimes by demonstrating their existence within non-linear perturbations up to arbitrary order as well.", "Such generalizations of the emergence of analogue spacetimes has never been presented in the literature before.", "It has been recently proposed that accreting astrophysical systems may be considered as natural examples of classical analogue gravity models [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24].", "In our present work, we shall consider such a model to manifest our formalism.", "Arguably the simplest accretion model comprises the system of a non-relativistic fluid in the presence of a gravitating central mass.", "The potential of the accretor is taken to be spherically symmetric, $\\Phi (r) = -\\frac{GM}{c^2 r}$ , where $M$ denotes the mass of the object, $G$ is Newton's constant, $c$ is the light speed and $r$ is the distance from the center of body placed at the origin.", "The quantity $r_g = \\frac{GM}{c^2}$ is the only relevant length scale in the problem which we hereafter normalize by working in units where $G = M = c =1$ .", "The fluid is assumed to not self accrete and spherical symmetry ensures that the fluid velocity is radial and irrotational.", "Denoting the radial component of the velocity vector, pressure and density of the fluid by $v$ , $P$ and $\\rho $ respectively, the dynamics is governed by the continuity equation $\\dot{\\rho } + \\frac{1}{r^2}\\left(r^2 \\rho v\\right)^{\\prime } = 0\\,, $ and the Euler equation $\\dot{v} + v v^{\\prime } + \\frac{1}{\\rho } P^{\\prime } + \\Phi ^{\\prime }(r) = 0 \\,,$ where dots denote time derivatives and primes are derivatives with respect to $r$ .", "In most cases, it is convenient to assume that the fluid is isentropic and satisfies the adiabatic equation of state $P= \\kappa \\rho ^{\\gamma }$ .", "The adiabatic exponent $\\gamma $ is the ratio of the constant specific heats $c_P$ and $c_V$ of the fluid, while $\\kappa $ is a measure of the fluid's constant specific entropy.", "The local adiabatic sound speed of the fluid can also be determined and is given by $c_s^2 = \\frac{\\partial P}{\\partial \\rho } = \\kappa \\gamma \\rho ^{\\gamma - 1}$ Steady state solutions of Eq.", "(REF ) and Eq.", "(REF ) along with linear perturbations about them have been well investigated in the literature.", "The solution space involves a critical point where $v = c_s$ .", "There exist a family of subsonic flow solutions with $v<c_s$ which never pass the critical point.", "Likewise, a class of supersonic solutions with $v>c_s$ also exist and reside only in the region between the critical point and the boundary of the accretor.", "Bondi demonstrated that there exists a unique transonic flow solution which passes from the subsonic region into the supersonic region through the critical point [25].", "This solution is relevant for accretion flows on black hole spacetimes in the approximation where astrophysical effects close the event horizon are ignored.", "Linear perturbations of the velocity potential and mass accretion rate [26] also demonstrate the stability of the Bondi flow solution.", "Perhaps the most striking consequence arising from a linear perturbation analysis involves the identification of an analogue spacetime.", "Unruh demonstrated that linear perturbations of the velocity potential about the transonic solution can be considered as propagations of a massless scalar field on an acoustic geometry [2].", "This spacetime has a well defined causal structure and a positive definite curvature, with the critical point being the location of the acoustic horizon.", "The existence of stable transonic solutions and an acoustic geometry are important characteristics which define accretion flows in general.", "In this paper we initiate a systematic investigation of non-linear perturbations of spherically symmetric accreting flows.", "We first demonstrate that Eq.", "(REF ) and Eq.", "(REF ) can be cast as a pair of coupled differential equations to be solved for the mass accretion rate and the density.", "In particular, the second-order differential equation for the mass accretion rate is manifestly in the form of a wave equation on an acoustic background.", "We reproduce the known linear order perturbation equation of the mass accretion rate using our approach, before deriving the second, third and $n^{\\text{th}}$ order perturbation equations.", "The acoustic metric to all orders is shown to admit a non-vanishing curvature and causal structure.", "We then proceed to investigate solutions of the perturbation equations.", "We consider solutions of the linear mass accretion rate perturbation that are exponentially damped in time in two regimes, when $\\omega < 1$ (low frequency) and $\\omega \\ge 1$ (high frequency).", "By high frequency, we mean that the corresponding wavelength is at least the order of the accretor boundary or lower, while low frequency means that the corresponding wavelength is at least of the size of the accreting fluid or higher.", "In numerically solving the coupled perturbation equations at linear order, we find that for high frequency damped solutions, the mass accretion rate grows slowly and the density perturbation decays behind the horizon.", "Conversely, up to linear perturbation in the low frequency regime, the density perturbation initially grows behind the horizon.", "In all cases, the sonic horizon of the perturbed acoustic geometry oscillates and either grows to a new stable value (high frequency) or shrinks to a new stable value (low frequency) at late times.", "We find that second-order solutions of the perturbed equations in the high frequency regime possess the qualitative properties of the first-order solutions.", "On the other hand, higher order perturbations in the low frequency regime have the property that high frequency properties resurface.", "This ensures that the acoustic horizon does shrink indefinitely and tends to the configuration of high frequency damped perturbations." ], [ "Algorithm for nonlinear perturbations", "Linear perturbations of Eq.", "(REF ) and Eq.", "(REF ) are usually considered by first expanding the fields around some stationary solution of the equations, such as the Bondi flow.", "The perturbed field of interest is taken to either be the mass accretion rate $f = \\rho v r^2 \\,,$ or in other cases the velocity potential $\\Psi $ , where $\\vec{v} = \\vec{\\nabla } \\Psi $ .", "The equation governing linear perturbations of either field can then be shown to satisfy the Klein-Gordon equation for a massless field on an acoustic spacetime.", "In the following, we base our perturbation scheme on the mass accretion rate.", "We first demonstrate that the unperturbed Eq.", "(REF ) and Eq.", "(REF ) provide a pair of coupled equations - a continuity equation and a wave equation, entirely in terms of the fluid density $\\rho $ and mass accretion rate $f$ .", "By expanding these fields about a stationary solution, the equations can be iteratively solved to any order in perturbation.", "Using the definition in Eq.", "(REF ), we find that Eq.", "(REF ) can be readily expressed as $\\dot{\\rho } + \\frac{1}{r^2}f^{\\prime } = 0$ This is the continuity equation in our perturbation scheme.", "The wave equation follows from taking the time derivative of the Euler equation.", "Using Eq.", "(REF ) we also have $\\frac{1}{\\rho } P^{\\prime } = \\kappa \\gamma \\rho ^{\\gamma - 2} \\rho ^{\\prime }$ Substituting this expression in Eq.", "(REF ) and taking the time derivative gives $\\ddot{v} + \\left( v \\dot{v} + \\kappa \\gamma \\rho ^{\\gamma -2} \\dot{\\rho } \\right) ^{\\prime } = 0$ By taking the time derivative of Eq.", "(REF ) and using the expression for $\\dot{\\rho }$ in Eq.", "(REF ), we find the following expression for $\\dot{v}$ in terms of $f$ and its time derivative $\\dot{v} = \\frac{1}{\\rho r^2}\\left( \\dot{f} + \\frac{f \\partial _r f}{\\rho r^2} \\right)$ Substituting Eq.", "(REF ) and Eq.", "(REF ) in Eq.", "(REF ) then gives $\\partial _t \\left(\\frac{1}{\\rho r^2} \\partial _t f \\right) + \\partial _t \\left(\\frac{f}{\\rho ^2 r^4} \\partial _t f\\right) + \\partial _r \\left(\\frac{f}{\\rho ^2 r^4} \\partial _r f \\right) + \\partial _r \\left(\\left(\\frac{f^2}{\\rho ^3 r^6} - \\frac{\\gamma \\kappa }{r^2} \\rho ^{\\gamma -2}\\right)\\partial _r f\\right) =0$ If we now define the inverse acoustic metric components $g^{tt} = \\frac{1}{\\rho r^2} \\,, \\qquad g^{rt} = \\frac{f}{\\rho ^2 r^4} = g^{tr} \\,, \\qquad g^{rr} = \\frac{f^2}{\\rho ^3 r^6} - \\frac{\\gamma \\kappa }{r^2} \\rho ^{\\gamma -2}\\,,$ we find that Eq.", "(REF ) has the expression $\\partial _{\\mu }\\left(g^{\\mu \\nu } \\partial _{\\nu } f\\right) = 0$ Eq.", "(REF ) is the wave equation that will feature in our perturbative approach.", "While we will continue to work with $f$ and $\\rho $ treated as the independent variables in this section, it will be useful here and later on in our paper to consider the inverse metric components in terms of the velocity $v$ and the speed of sound $c_s$ .", "Using the expressions in Eq.", "(REF ) and Eq.", "(REF ), we find that the inverse metric components in Eq.", "(REF ) can be expressed as $g^{tt} = \\frac{v}{f} \\,, \\qquad g^{tr} = \\frac{v^2}{f} \\,, \\qquad g^{rr} = \\frac{v}{f}\\left(v^2 - c_s^2 \\right) \\,.", "$ We will be considering perturbations about stationary solutions which always involve a positive definite (inflowing) velocity and a constant positive mass accretion rate.", "Taking these properties to hold in the perturbative solution, we see that $g^{tt}$ and $g^{tr}$ are always positive.", "On the other hand, $g^{rr}$ is negative in the subsonic region where $v<c_s$ , positive in the supersonic region where $v>c_s$ and vanishes at the critical point $v=c_s$ .", "The critical point provides the location of the acoustic horizon.", "The horizon can change/fluctuate significantly in considering perturbations, which is a central topic of our paper.", "We consider a perturbative expansion of $f$ and $\\rho $ about a known stationary solution up to $n^{\\text{th}}$ order to be of the form $f(r,t) &= f_0 + \\sum _{k=1}^{n}\\epsilon ^k f_k(r,t) = f_0\\left(1 + \\frac{1}{f_0} \\sum _{k=1}^{n}\\epsilon ^k f_k(r,t)\\right) \\\\\\rho (r,t) & = \\rho _0(r) + \\sum _{k=1}^{n} \\epsilon ^k \\rho _k(r,t) = \\rho _0(r)\\left( 1 + \\frac{1}{\\rho _0(r)}\\sum _{k=1}^{n} \\epsilon ^k \\rho _k(r,t)\\right) \\,, $ where $f_0$ and $\\rho _0$ denote the stationary solution values.", "As mentioned, the stationary solution has the property that $f_0$ is a positive constant while $\\rho _0 = \\rho _0(r)$ is independent of time.", "The dimensionless counting parameter $\\epsilon $ measures the strength of the perturbation.", "From the expansions in Eq.", "(REF ) and Eq.", "(), we can also determine all other dependent quantities, such as components of the inverse acoustic metric, the velocity or the speed of sound, to any order in perturbation.", "If $A(r,t)$ with the $n^{\\text{th}}$ order expansion $A(r,t) = A_0(r) + \\sum _{k=1}^{n}\\epsilon ^k A_k(r,t) = A_0\\left(1 + \\frac{1}{A_0(r)} \\sum _{k=1}^{n}\\epsilon ^k A_k(r,t)\\right) \\,,$ denotes any of the fields (dependent or independent) then the expansion is perturbative provided $\\epsilon \\frac{\\vert A_{l+1} \\vert }{\\vert A_{l} \\vert } < 1 \\,, \\qquad l = 0 \\,, \\cdots n-1$ To determine the $n^{\\text{th}}$ order solutions of $f$ and $\\rho $ , we first substitute the expansions of Eq.", "() in Eq.", "(REF ) and Eq.", "(REF ).", "We then collect the expressions in powers of $\\epsilon $ and solve for the coefficients.", "The coefficient of $\\epsilon ^0$ is manifestly satisfied by the stationary solution.", "The coefficient of $\\epsilon $ resulting from Eq.", "(REF ) is $\\dot{\\rho _1} + \\frac{\\partial _r f_1}{r^2} = 0 \\,, $ while from Eq.", "(REF ) we find $\\partial _{\\mu }\\left(g_{(0)}^{\\mu \\nu } \\partial _{\\nu } f_1\\right)+\\partial _{\\mu }\\left(g_{(1)}^{\\mu \\nu } \\partial _{\\nu } f_0\\right) = 0 \\,,$ where $g^{tt}_{(0)} = \\frac{1}{r^2 \\rho _0} \\,, \\qquad g^{tr}_{(0)} = \\frac{f_0}{r^4 \\rho _0^2} = g^{rt}_{(0)} \\,, \\qquad g^{rr}_{(0)} = \\frac{f_0^2}{r^6 \\rho _0^3}-\\frac{\\gamma \\kappa \\rho _0^{\\gamma -2}}{r^2}$ $g^{tt}_{(1)} = \\frac{1}{r^2 \\rho _0}\\left(-\\frac{\\rho _1}{\\rho _0}\\right)& \\,, \\qquad g^{tr}_{(1)} = \\frac{f_0}{r^4 \\rho _0^2}\\left(\\frac{f_1}{f_0} - 2 \\frac{\\rho _1}{\\rho _0}\\right) = g^{rt}_{(1)} \\,, \\\\g_{(1)}^{rr} =&\\frac{f_0^2}{r^6 \\rho _0^3} \\left(2 \\frac{f_1}{f_0} - 3 \\frac{\\rho _1}{\\rho _0}\\right)-\\frac{\\gamma \\kappa \\rho _0^{\\gamma -2}}{r^2}\\left(\\left(\\gamma -2\\right)\\frac{\\rho _1}{\\rho _0}\\right)$ Since $f_0$ is a constant, Eq.", "(REF ) simplifies to $\\partial _{\\mu }\\left(g_{(0)}^{\\mu \\nu } \\partial _{\\nu } f_1\\right) = 0 \\,,$ The inverse metric $g^{\\mu \\nu }_{(0)}$ is completely determined by the given stationary solution.", "We can thus solve Eq.", "(REF ) for the unknown first order pertubation $f_1$ , which is then used to determine $\\rho _1$ from Eq.", "(REF ).", "We note that the acoustic metric at this order is completely time independent and that Eq.", "(REF ) is the known linear order equation describing the propagation of the perturbed field on the background.", "The second order equations are coefficients of $\\epsilon ^2$ resulting from substituting the expansions in Eq.", "(REF ) and Eq.", "(), which are $\\dot{\\rho _2} + \\frac{\\partial _r f_2}{r^2} = 0 \\,, $ from Eq.", "(REF ) and $\\partial _{\\mu }\\left(g_{(0)}^{\\mu \\nu } \\partial _{\\nu } f_2\\right) &+ \\partial _{\\mu }\\left(g_{(1)}^{\\mu \\nu } \\partial _{\\nu } f_1\\right) + \\partial _{\\mu }\\left(g_{(2)}^{\\mu \\nu } \\partial _{\\nu } f_0\\right)= 0$ from Eq.", "(REF ), where the coefficients of $g_{(0)}^{\\mu \\nu }$ and $g_{(1)}^{\\mu \\nu }$ are those in Eq.", "(REF ) and Eq.", "(REF ) respectively, while the coefficients of $g_{(2)}^{\\mu \\nu }$ are $g^{tt}_{(2)} &= \\frac{1}{r^2 \\rho _0}\\left(\\left(\\frac{\\rho _1}{\\rho _0}\\right)^2 -\\frac{\\rho _2}{\\rho _0}\\right) \\,, \\qquad g_{(2)}^{tr} = \\frac{f_0}{r^4 \\rho _0^2}\\left(\\frac{f_2}{f_0} - 2 \\left(\\frac{\\rho _2}{\\rho _0} + \\frac{f_1}{f_0}\\frac{\\rho _1}{\\rho _0}\\right) + 3 \\left(\\frac{\\rho _1}{\\rho _0}\\right)^2\\right) = g^{rt}_{(2)} \\,,\\\\g^{rr}_{(2)} &= \\frac{f_0^2}{r^6 \\rho _0^3} \\left(\\left(\\frac{f_1}{f_0}\\right)^2 + 2 \\frac{f_2}{f_0} - 3 \\frac{\\rho _2}{\\rho _0} + 6\\left(\\left(\\frac{\\rho _1}{\\rho _0}\\right)^2 - \\frac{f_1}{f_0}\\frac{\\rho _1}{\\rho _0}\\right)\\right) \\\\&\\qquad \\qquad - \\frac{\\gamma \\kappa \\rho _0^{\\gamma -2}}{r^2}\\left(\\frac{1}{2}\\left(\\gamma -2\\right)\\left(\\gamma -3\\right)\\left(\\frac{\\rho _1}{\\rho _0}\\right)^2 + \\left(\\gamma -2\\right)\\frac{\\rho _2}{\\rho _0}\\right)$ As in the linear perturbation analysis, the constant $f_0$ implies that Eq.", "(REF ) is simply $\\partial _{\\mu }\\left(g_{(0)}^{\\mu \\nu } \\partial _{\\nu } f_2\\right) = - \\partial _{\\mu }\\left(g_{(1)}^{\\mu \\nu } \\partial _{\\nu } f_1\\right) $ Unlike the first order equation, we now see that the equation for the second order perturbed field $f_2$ involves a source term constructed from the solutions of the preceding order in perturbation.", "This is a generic trait which distinguishes the first-order perturbation equation from those at higher orders.", "We may interpret Eq.", "(REF ) simply as the perturbed mass accretion rate $f_2$ propagating on the lowest order acoustic background in the presence of a first order source.", "However, from Eq.", "(REF ) we know that we always have an equation for a massless scalar field on an acoustic background.", "Hence a more accurate description is that the combined field at this order $f_0 + \\epsilon f_1 + \\epsilon ^2 f_2$ propagates on a background expanded up to linear order $g_{(0)}^{\\mu } + \\epsilon g_{(1)}^{\\mu \\nu }$ .", "This property also continues to hold at higher orders, namely, the perturbed mass accretion rate up to order $n$ propagates on the $n-1$ acoustic spacetime.", "The perturbation solutions at second order follow from first solving Eq.", "(REF ) for $f_2$ and then substituting it in Eq.", "(REF ) to determine $\\rho _2$ .", "The iterative procedure continues up to order $n$ .", "Assuming that we have solved all equations up to order $n-1$ , we now collect the $\\epsilon ^n$ coefficient resulting from substituting Eq.", "(REF ) and Eq.", "() in Eq.", "(REF ) and Eq.", "(REF ).", "We find the expressions $\\dot{\\rho _n} &= - \\frac{\\partial _r f_n}{r^2} \\\\\\partial _{\\mu }\\left(g^{\\mu \\nu }_{(0)} \\partial _{\\nu } f_n\\right) &= - \\sum _{k=1}^{n-1} \\partial _{\\mu }\\left(g^{\\mu \\nu }_{(k)} \\partial _{\\nu } f_{n-k}\\right) $ The total mass accretion rate $f$ up to order $n$ propagates on an acoustic background metric $g_{\\text{eff}(n-1)}^{\\mu \\nu }$ constructed from the preceding $n-1$ orders in perturbation $g_{\\text{eff}(n-1)}^{\\mu \\nu } = \\sum _{k=0}^{n-1} \\epsilon ^k g^{\\mu \\nu }_{(k)}$" ], [ "Curvature and causal structure of the acoustic metric", "In the previous section, we determined that an effective inverse acoustic metric can be described to every order in perturbation theory.", "The properties of the inverse metric to all orders is most conveniently investigated by using the form in either Eq.", "(REF ) or Eq.", "(REF ), with the understanding that the expressions at a particular perturbation order results from using the expansions in Eq.", "(REF ) and Eq. ().", "In this section, we will be interested in the causal structure and curvature of the acoustic metric.", "We will thus use the components of the inverse metric given in Eq.", "(REF ) $g^{\\mu \\nu } =\\begin{pmatrix}g^{tt} & g^{tr} \\\\g^{rt} & g^{rr}\\end{pmatrix}=\\begin{pmatrix}\\frac{v}{f} & \\frac{v^2}{f}\\\\\\frac{v^2}{f} & \\frac{v}{f}\\left(v^2 - c_s^2 \\right)\\end{pmatrix}$ The determinant of Eq.", "(REF ) is the inverse of the determinant of the metric, which we denote as $g^{-1}$ , with the expression $g^{-1} = - \\frac{v^2 c_s^2}{f^2}$ The negative value indicates the Lorentzian signature of the spacetime.", "Using Eq.", "(REF ), we find the metric $g_{\\mu \\nu } =\\begin{pmatrix}g_{tt} & g_{tr} \\\\g_{rt} & g_{rr}\\end{pmatrix}=\\begin{pmatrix}g g^{rr} & - g g^{rt} \\\\- g g^{tr} & g g^{tt}\\end{pmatrix}=\\begin{pmatrix}\\frac{f}{v} \\left(1 - \\frac{v^2}{c^2_s}\\right) & \\frac{f}{c_s^2}\\\\\\frac{f}{c_s^2} & -\\frac{f}{v c_s^2}\\end{pmatrix}$ We can now define the line element in the usual way $ds^2 = g_{\\mu \\nu } dx^{\\mu } dx^{\\nu } = g_{tt}dt^2 + 2 g_{tr} dt dr + g_{rr}dr^2$ The causal structure on the acoustic background follows from considering the analogue of the light cone in spacetime, here also characterized by $ds^2 = 0$ in Eq.", "(REF ).", "This leads to a quadratic equation in $dt$ admitting two roots $dt = \\left(\\frac{g_{tr}}{g_{tt}} \\pm \\sqrt{\\left(\\frac{g_{rt}}{g_{tt}}\\right)^2 - \\frac{g_{rr}}{g_{tt}}}\\right) dr$ Using the metric components in Eq.", "(REF ) we find the solutions $dt &= \\frac{1}{c+v}dr \\,, \\\\dt &= -\\frac{1}{c-v} dr $ Eq.", "(REF ) and Eq.", "() describe motion with and against the fluid respectively, which characterizes the causal properties of this spacetime with respect to the flow.", "These solutions further motivate the definition of ingoing and outgoing null coordinates $du = \\frac{1}{\\sqrt{2}}\\left(dt - \\frac{1}{c+v}dr\\right) \\,, \\qquad dv = \\frac{1}{\\sqrt{2}}\\left(dt + \\frac{1}{c-v}dr\\right) \\,,$ with which the sonic line element in Eq.", "(REF ) can be expressed as $ds^2 = 2 du dv$ This metric describes the causal structure in the subsonic region.", "From the acoustic horizon up to the accretor boundary we only have ingoing flows.", "In particular, in the supersonic region where $v>c$ , the expression for $dv$ in Eq.", "(REF ) also describes an ingoing flow.", "We can also demonstrate that there exists a non-trivial curvature of the spacetime.", "A 2 dimensional Riemmanian spacetime has only one independent curvature term - the Ricci scalar $R$ .", "The Riemann tensor is expressed in terms of $R$ as $R_{t r t r} = \\frac{1}{2}\\left( g_{tt} g_{rr} - g_{tr}^2\\right) R$ To compute the curvature, we will need to determine the connection components $\\Gamma ^{\\alpha }_{\\mu \\nu } = \\frac{1}{2}g^{\\alpha \\beta } \\left(g_{\\mu \\beta \\,, \\nu } + g_{\\nu \\beta \\,, \\mu } - g_{\\mu \\nu \\,, \\beta }\\right) $ and subsequently the Ricci tensor and Ricci scalar $R_{\\mu \\nu } &= \\Gamma ^{\\alpha }_{\\mu \\nu \\,, \\alpha } - \\Gamma ^{\\alpha }_{\\mu \\alpha \\,, \\nu } + \\Gamma ^{\\alpha }_{\\beta \\alpha }\\Gamma ^{\\beta }_{\\mu \\nu } - \\Gamma ^{\\alpha }_{\\beta \\nu }\\Gamma ^{\\beta }_{\\mu \\alpha } \\,, \\\\R &= g^{\\mu \\nu }R_{\\mu \\nu } = g^{tt}R_{tt} + 2g^{tr}R_{tr} + g^{rr}R_{rr} \\,.", "$ Explicitly, one can derive the connection components $\\Gamma ^t_{tt} = \\frac{1}{2}\\frac{g_{rr}\\dot{g}_{tt}+ g_{tr}\\left(g^{\\prime }_{tt} - 2 \\dot{g}_{tr}\\right)}{g} &\\,, \\qquad \\Gamma ^t_{tr} = \\frac{1}{2}\\frac{g_{rr}\\dot{g}^{\\prime }_{tt} - g_{tr}g^{\\prime }_{rr}}{g} \\\\\\Gamma ^t_{rr} = -\\frac{1}{2}\\frac{g_{tr}g^{\\prime }_{rr} + g_{rr}\\left(\\dot{g}_{rr} - 2 g^{\\prime }_{tr}\\right)}{g} &\\,, \\qquad \\Gamma ^r_{tt} = -\\frac{1}{2}\\frac{g_{tr}\\dot{g}_{tt}+ g_{tt}\\left(g^{\\prime }_{tt} - 2 \\dot{g}_{tr}\\right)}{g}\\\\\\Gamma ^r_{rt} = -\\frac{1}{2}\\frac{g_{tr}g^{\\prime }_{tt} - g_{tt}\\dot{g}_{rr}}{g} &\\,,\\qquad \\Gamma ^r_{rr} = -\\frac{1}{2}\\frac{g_{tt}g^{\\prime }_{rr}+ g_{tr}\\left(\\dot{g}_{rr} - 2 g^{\\prime }_{tr}\\right)}{g}$ Using Eq.", "(REF ) and Eq.", "() we find the following Ricci scalar $R & = \\frac{1}{2\\left(g_{tt}g_{rr} - g_{rt}\\right)^2}\\Bigg [g_{tr}\\left(g^{\\prime }_{rr}\\dot{g}_{tt} - 2g_{tr}\\left(g^{\\prime }_{tt} - 2 \\dot{g}_{tr}\\right) - \\dot{g}_{rr}\\left(g^{\\prime }_{tt} + 2 \\dot{g}_{tr}\\right)\\right) + g_{tt}\\left(g^{\\prime }_{rr}\\left(g^{\\prime }_{tt} - 2 \\dot{g}_{tr}\\right) + \\dot{g}^2_{rr}\\right)\\\\&\\qquad \\qquad + 2 g^2_{tr}\\left(g^{\\prime \\prime }_{tt} - 2 \\dot{g}^{\\prime }_{tr} + \\dot{g}_{rr}\\right) + g_{rr}\\left(g^{\\prime 2}_{tt} + \\dot{g}_{tt}\\left(\\dot{g}_{rr} - 2g^{\\prime }_{tr}\\right) -2 g_{tt}\\left(g^{\\prime \\prime }_{tt} - 2 \\dot{g}^{\\prime }_{tr} + \\ddot{g}_{rr}\\right)\\right)\\Bigg ]$ The metric at linear order in perturbation is constructed from the given stationary solution and is time independent.", "However, in considering second order and higher perturbations the curvature acquires a time dependent profile.", "It can be noted that if the perturbative solutions of $f$ and $\\rho $ are well behaved, the acoustic spacetime curvature is well defined to all orders in perturbation." ], [ "Numerical solutions", "In this section, we will numerically investigate the solutions of $f$ , $\\rho $ and inverse acoustic metric components up to second order in perturbation.", "We will first need to describe the unperturbed transonic Bondi solution and its properties.", "In our analysis, we set $\\gamma =1.35$ and $\\kappa =1$ throughout the flow.", "The solution is determined about the critical point where $v = c_s$ using the Runge-Kutta 4th order method with the Bernoulli constant chosen to be $E = 1.001$ , where $E = \\frac{v^2}{2} + \\frac{\\gamma \\kappa }{\\gamma -1}\\rho ^{\\gamma -1} + \\Phi (r)$ Given our choice of units, the boundary of the accretor is located at $r=1$ and we assume a maximum radius of $R_{\\infty }=100$ to denote the outer boundary of the accreting fluid.", "With this setup, we determine $v_0$ and $c_{s0}$ throughout the flow whose plots are given in Figure 1.", "Figure: Plot of v 0 (r)v_0(r) (left) and c s0 (r)c_{s0 }(r) (right).", "The origin is at (1,0)(1,0) in both plots.", "We note that v 0 (r)=0.96v_0(r) = 0.96 at r=1r=1 and falls rapidly, while c s0 =0.48c_{s0} = 0.48 at r=1r=1 and changes slowly past the horizon.From these solutions for the fluid velocity and sound speed, we can uniquely determine the expressions of $f_0$ and $\\rho _0$ using Eq.", "(REF ) and Eq.", "(REF ).", "We find that $f_0 = 0.0129$ and the plot for $\\rho _0(r)$ is given in Figure 2.", "Figure: Plot of ρ 0 (r)\\rho _0(r).", "The origin is at (1,0)(1,0).The transonic solution also determines the lowest order metric components using Eq.", "(REF ).", "This is the acoustic metric on which first order perturbations of the mass accretion rate propagate.", "At this order, the inverse metric components are spatial and the graphs of its components are provided in Figure 3.", "We find that $g^{rr}_{(0)}$ vanishes at $r_0=2.362$ , which is the location of the acoustic horizon.", "We also see that $g^{rr}_{(0)}>0$ for $r<r_0$ and $g^{rr}_{(0)}<0$ for $r>r_0$ .", "On the other hand, $g^{tt}_{(0)}$ and $g^{tr}_{(0)}$ remain positive definite throughout the flow.", "Figure: Plots of inverse metric components g (0) tt (r)g^{tt}_{(0)}(r) (upper), g (0) tr (r)g^{tr}_{(0)}(r) (middle) and g (0) rr (r)g^{rr}_{(0)}(r) (lower).", "The origin is at (1,0)(1,0) in all plots.", "The location where g 0 rr =0g^{rr}_{0} = 0 is at r=2.362r=2.362.With the stationary solution characterized, we will now consider perturbations which are exponentially damped in time.", "There are two aspects which will factor into our numerical analysis - the range of time and the frequency.", "Since we are considering exponentially damped perturbations, we need to choose a time range which is neither too short, such that the effect of the damping is irrelevant, nor too long, such that the damping is overwhelming rendering the perturbation irrelvant.", "Since we expect that damped waves are accompanied by a superposition of exponentially growing and decaying modes in space, we choose the time domain to be an order larger than the spatial range and thus $t \\in \\lbrace 0, 10^3\\rbrace $ .", "The upper limit in the time domain will be taken as representing `late times'.", "The other aspect we will keep track of is the frequency.", "Here we will characterize the perturbations as `high frequency' and `low frequency'.", "By high frequency, we will mean the corresponding wavelengths are the size of the accretor boundary or lower.", "Accordingly, $\\omega \\ge 1$ (since $r_g = 1$ in our units) is the high frequency domain.", "Conversely, by low frequency we mean that the corresponding wavelength is the size of the accretion fluid or higher.", "Hence $\\omega \\le 10^{-2}$ will represent the low frequency domain.", "In the following subsections, we will in fact be concerned with sufficiently high and low frequencies.", "We will take the high frequency value to be $10^{3}$ and the low frequency value to be $10^{-3}$ ." ], [ "High frequency perturbations", "As mentioned, we set $\\omega = 10^3$ throughout our high frequency analysis with $t \\in \\lbrace 0, 10^3\\rbrace $ .", "In solving Eq.", "(REF ), we set the temporal boundary condition $f_1(r,0) = f_0$ and the condition $e^{-\\omega t}$ at initial time from $r=1$ to $r=100$ .", "This results in the solution plotted in figure 4.", "We see the resulting interference of exponentially growing and decaying waves in space, over time.", "We expect that at high frequencies, the growing mode in space dominates the decaying mode in the supersonic region.", "As a consequence, on average $\\partial _r f_1 < 0$ as we move away from $r=1$ .", "We also see from Eq.", "(REF ) that $\\rho _1$ is suppressed by $r^{-2}$ and is thus largely sensitive to the behaviour of $\\partial _r f_1$ in the supersonic region near the accretor.", "These aspects of $f_1$ and $\\rho _1$ provide a significant conclusion.", "A negative change in the mass accretion rate moving away from the accretor implies a negative change in the density perturbation.", "In solving Eq.", "(REF ) numerically, we find that this is precisely the case and the result is plotted in figure 5.", "Figure: Solution of the first order pertubed mass accretion rate f 1 (r,t)f_1(r,t).Figure: Solution of the first order perturbed density ρ 1 (r,t)\\rho _1(r,t).", "The perturbation is most relevant near the accretor boundary.", "The negative value reflects the increasing mass accretion towards the accretor boundary in the supersonic region.Since $\\rho _1$ is negative near the accretor, the density up to linear order perturbations will be less than that of the stationary solution.", "In addition, we have also noted that the mass accretion rate on average and up to linear order in perturbation is slightly greater than $f_0$ .", "These results are consistent in the presence of a decreasing density accompanied by an increasing fluid velocity.", "We thus expect the first order corrected metric to have an acoustic horizon greater than the stationary solution, with a time dependent profile.", "To explore this, we substitute the stationary and first order perturbation solutions of $f$ and $\\rho $ in the metric component expressions given in Eq.", "(REF ).", "These components are plotted in figure 6.", "In order to construct the perturbed metric, we now need to adopt an appropriate choice for $\\epsilon $ such that the condition in Eq.", "(REF ) is satisfied.", "The magnitudes are largest near the accretor boundary, with the highest magnitude in all the plots appearing for $g^{rr}_{(1)}(r,t)$ at approximately $t=620$ , where $g^{rr}_{(1)}(1,620) = 2.29 \\times 10^6$ .", "We thus set $\\epsilon = \\frac{1}{7} \\times 10^{-6}$ .", "This ensures $\\epsilon \\frac{\\vert g_{(1)} \\vert }{\\vert g_{(0)} \\vert }$ to be roughly $0.3$ for all metric components near the accretor boundary and about $0.03$ near the acoustic horizon.", "Figure: Perturbed metric expressions for g (1) tt (r,t)g^{tt}_{(1)}(r,t) (upper), g (1) tr (r,t)g^{tr}_{(1)}(r,t) (middle) and g (1) rr (r,t)g^{rr}_{(1)}(r,t) (lower).By using Eq.", "(REF ) and the metric components in Eq.", "(REF ), we can construct the effective metric that will enter in the second order perturbation analysis $g_{\\text{eff}(1)}^{\\mu \\nu } = g^{\\mu \\nu }_{(0)} + \\epsilon g^{\\mu \\nu }_{(1)}$ The corresponding plots of the effective metric Eq.", "(REF ) are given in figure 7.", "While there are small time dependent perturbations in the region near the horizon and into the supersonic region, these are barely noticeable in these plots of the full effective metric.", "This is a desired result, since it confirms that our choice in $\\epsilon $ had the effect of constituting small perturbations.", "Nevertheless, in close up considerations of the near horizon region, we can see a pronounced effect of the perturbation.", "A close up overhead view of the horizon region, from $r=2.362$ to $r=2.37$ has been provided in figure 8.", "The blue shaded plane is that of $g^{rr}_{\\text{eff}(1)} = 0$ .", "We see that the horizon fluctuates and grows from the original value $r=2.362$ at $t=0$ to $r=2.365$ at $t=10^3$ .", "At no point after $t=0$ does the horizon ever return to its original value.", "The fluctuations are expected from considering the effect of a damped exponential wave about a local minima, which appears near the original acoustic horizon.", "Figure: Perturbed metric expressions for g eff(1) tt (r,t)g^{tt}_{\\text{eff}(1)}(r,t) (upper), g eff(1) tr (r,t)g^{tr}_{\\text{eff}(1)}(r,t) (middle) and g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) (lower).Figure: Overhead view of g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) from r=2.362r=2.362 to r=2.37r=2.37 over all time.", "The blue plane demarcates g eff(1) rr (r,t)=0g^{rr}_{\\text{eff}(1)}(r,t) = 0 and the intersection with the orange curve of g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) traces out the horizon location in time.", "We note that the horizon fluctuates and grows to its new value of r=2.365r=2.365 at t=10 3 t= 10^3.We now proceed to solve the second order perturbation equations of the mass accretion rate and the density given in Eq.", "(REF ) and Eq.", "(REF ) respectively.", "In solving for the mass accretion rate, we provide the initial boundary conditions $f_2(r,0) = f_1(r,0)$ , $f_2(1,t) = f_1(1,t)$ and $f_2(100,t) = f_1(100,t)$ , which ensures consistency with the first order perturbation solution and the exponentially damped in time behaviour.", "Likewise for the density perturbation, we require only one condition at initial time $\\rho _2(r,0) = \\rho _1(r,0)$ .", "The plots of the resulting solution has been provided in figure 9 and figure 10.", "We see that the second order perturbation are more pronounced versions of the first order solutions and maintain its properties.", "The density perturbation decays even more at this order as can be seen from comparing figure 10 with figure 5.", "Figure: Solution of the second order pertubed mass accretion rate f 1 (r,t)f_1(r,t).Figure: Solution of the second order perturbed density ρ 1 (r,t)\\rho _1(r,t).", "The solution is more negative near the accretor boundary than the corresponding solution at first order given in figure 5.With the second order solutions in hand, we can construct the second order perturbed inverse metric and determine the effective metric up to second order.", "These components are actually relevant for third order perturbations of the mass accretion rate and density, which will not be considered here.", "Nevertheless, it is important to consider these solutions since this is the next order where time dependence is involved in the metric solutions.", "As such, we can compare the metric at this order with those in the preceding order to identify time evolution trends appearing in higher orders of perturbation.", "The third order inverse metric components are calculated using the expressions in Eq.", "(REF ) with the corresponding plots given in figure 11.", "Comparison of these plots with those of the first order solutions in figure 6 demonstrate that the trends are simply more pronounced at this order, with some small perturbations developing in the subsonic region.", "The largest magnitude increase is in $g^{rr}_{(2)}(r,t)$ at $r=1\\,, t=620$ where we now have $g^{rr}_{(2)}(1,620) = 10^7$ , which is an order of magnitude larger than the peak in the first order solution.", "However, we note that $\\epsilon ^2 \\sim 10^{-12}$ which is several orders of magnitudes smaller than $\\epsilon $ at the first order.", "From this we conclude that while solutions for $f$ and $\\rho $ at higher orders in perturbation are more pronounced versions of the first order solutions, they are highly suppressed in our perturbative analysis.", "Figure: Perturbed metric expressions for g (2) tt (r,t)g^{tt}_{(2)}(r,t) (upper), g (2) tr (r,t)g^{tr}_{(2)}(r,t) (middle) and g (2) rr (r,t)g^{rr}_{(2)}(r,t) (lower).We can construct the effective metric up to this order $g_{\\text{eff}(2)}^{\\mu \\nu } = g^{\\mu \\nu }_{(0)} + \\epsilon g^{\\mu \\nu }_{(1)} + \\epsilon ^2 g^{\\mu \\nu }_{(2)} \\,.$ We have not plotted the components of the effective metric as they appear almost identical to those in figure 7.", "It is important to consider the effect close to the horizon at this order.", "In figure 12 we have plotted the overhead view of $g^{rr}_{\\text{eff}(2)}(r,t)$ from $r=2.362$ to $r=2.37$ over all time, just as in figure 8 for $g^{rr}_{\\text{eff}(1)}(r,t)$ .", "In comparing with $g^{rr}_{\\text{eff}(1)}(r,t)$ , we see that while the horizon of $g^{rr}_{\\text{eff}(2)}(r,t)$ has grown at late times, the changes are very small and do not affect the qualitative result at first order.", "Figure: Overhead view of g eff(2) rr (r,t)g^{rr}_{\\text{eff}(2)}(r,t) from r=2.362r=2.362 to r=2.37r=2.37 over all time.", "While the horizon has grown slightly more than those of g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) in figure 8, they are very small corrections which do not affect the overall qualitative behaviour of the metric at first order.Our analysis demonstrates that higher order perturbation solutions in the high frequency regime are amplified versions of the first order solutions.", "For weak perturbations, higher order solutions provide minor corrections to first order results.", "It is possible to set $\\epsilon $ to be closer to 1 to represent strong perturbations.", "In this case, we should expect the corrections to be more significant and in accordance with the properties of the first order solutions." ], [ "Low frequency perturbations", "We will now consider low frequency damped perturbations by setting $\\omega = 10^{-3}$ in an interval $t \\in \\lbrace 0, 10^3\\rbrace $ .", "We solve Eq.", "(REF ) with the initial boundary conditions $f_1(r,0) = f_0$ and $f_1(1,t) = e^{-\\omega t} = f_1(100,t)$ .", "We now find solutions which have been plotted in figure 13.", "We one again find a solution resulting from the interference of exponentially growing and decaying waves in space, over time.", "We now expect that as a consequence of considering very low frequencies, the supersonic region involves the decaying mode to slightly dominate the growing mode in space.", "As a consequence $\\partial _r f_1 > 0$ on average as we move away from the accretor boundary at $r=1$ , and we expect to find a net positive change in the density perturbation.", "This is confirmed through the solution plot of Eq.", "(REF ) in figure 14.", "Figure: Solution of the first order pertubed mass accretion rate f 1 (r,t)f_1(r,t).Figure: Solution of the first order perturbed density ρ 1 (r,t)\\rho _1(r,t).", "The perturbation is once again most relevant near the accretor boundary and is positive unlike in the high frequency analysis.Since $\\rho _1$ is positive near the accretor boundary for low frequency damped perturbations, we now expect the density up to linear order to be greater than that of the stationary solution.", "This increases the first order corrected sound speed solution relative to the stationary solution value.", "The net decrease in the mass accretion rate at linear order, given an increase in the density perturbation, can only mean that the first order corrected velocity is less than that of the given stationary solution.", "We thus expect that the first order corrected metric has an acoustic horizon which is less than that of the stationary solution at late times.", "Let us first consider the effect of the first order perturbation solutions of $f$ and $\\rho $ on the inverse metric components in Eq.", "(REF ).", "The plots of these components are given in figure 15.", "Like the high frequency solutions, the corrections are dominant near the accretor boundary.", "However, unlike the corresponding solutions in the high frequency analysis (plotted in figure 6), the corrections are negative and are an indication of the shrinking of the horizon we expect to observe.", "We now choose $\\epsilon $ with respect to the highest value present in $g^{rr}_{(1)}$ in such a way that $\\epsilon \\frac{\\vert g_{(1)} \\vert }{\\vert g_{(0)} \\vert }$ is roughly $0.3$ for all metric components near the accretor boundary.", "By making this choice, we one again find that $\\epsilon \\frac{\\vert g_{(1)} \\vert }{\\vert g_{(0)} \\vert }$ near the acoustic horizon is approximately $0.03$ .", "Hence the results to follow are considered at the same relative strength as those we assumed for the high frequency analysis in the previous subsection.", "Figure: Perturbed metric expressions for g (1) tt (r,t)g^{tt}_{(1)}(r,t) (upper), g (1) tr (r,t)g^{tr}_{(1)}(r,t) (middle) and g (1) rr (r,t)g^{rr}_{(1)}(r,t) (lower).", "Unlike the corresponding plots given in figure 6 for the high frequency case, we now find that the corrections are negative near the accretor boundary at r=1r=1We can now use Eq.", "(REF ) to construct the effective inverse metric up to first order in perturbation.", "Since the strength of the perturbation in this case is also very small, we end up with full range plots which appear as in figure 7.", "As noted in the high frequency analysis, the interesting effects are present in close up plots from the near horizon region.", "In figure 16, we have provided a close up overhead view of the horizon region, now from $r=2.348$ to $r=2.362$ .", "The blue shaded plane as before demarcates $g^{rr}_{\\text{eff}(1)} = 0$ and intercepts with the horizon over time.", "We see that the horizon fluctuates and shrinks from the original value $r=2.362$ at $t=0$ to $r=2.3595$ at $t=10^3$ .", "The horizon never returns to its original value at $t=0$ at any time in its evolution.", "Since the perturbation strengths are comparable with those in the high frequency analysis, with the frequencies being inverses of one another, we may compare the results to draw inferences.", "We note that the amount the acoustic horizon shrinks in the low frequency case is marginally less than the horizon grows in the case of high frequencies.", "This suggests a tendency for the horizon to always grow under perturbations.", "We will see evidence to this effect moving forward.", "Figure: Overhead view of g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) from r=2.348r=2.348 to r=2.3632r=2.3632 over all time.", "The horizon fluctuates and shrinks to its new value of r=2.3595r=2.3595 at t=10 3 t= 10^3.We now solve the second order perturbation equations of the mass accretion rate and the density given in Eq.", "(REF ) and Eq.", "(REF ) respectively.", "The initial boundary conditions for the mass accretion rate are chosen to be $f_2(r,0) = f_1(r,0)$ , $f_2(1,t) = f_1(1,t)$ and $f_2(100,t) = f_1(100,t)$ , while for the density perturbation we choose $\\rho _2(r,0) = \\rho _1(r,0)$ .", "The plots of the resulting solutions has been provided in figure 17 and figure 18.", "We can make an interesting observation on comparing with the first order and second order solutions.", "In first considering the high frequency regime, the comparison of figure 4 and figure 9 reveals that the second order mass accretion rate perturbation is an amplified version of the first order perturbation, with an increase in amplitude by two orders in magnitude.", "In comparing figure 5 and figure 10, we likewise see that the density perturbation increases by a factor of 3 in going from first order to second order perturbations in the supersonic region.", "In now considering the low frequency results, we find on comparing figure 13 and figure 17 that the magnitude of mass accretion perturbations remains in the same order and does not necessarily increase on average.", "The comparison of figure 14 and figure 18 shows that the density perturbation decreases by an order in magnitude going from first order perturbation solutions to those at second order.", "This implies that the exponentially growing mode in the mass accretion rate perturbation tends to dominate at higher orders in the supersonic region.", "The density and mass accretion rate perturbation results demonstrate a tendency to return to the behaviour present in the high frequency analysis at higher orders in perturbation.", "Figure: Solution of the second order pertubed mass accretion rate f 1 (r,t)f_1(r,t).Figure: Solution of the second order perturbed density ρ 1 (r,t)\\rho _1(r,t).", "The solution is more negative relative to the corresponding first order solution in figure 15.We can now compute the second order perturbed inverse metric using the expressions in Eq.", "(REF ).", "These components are plotted in figure 19.", "We see that at the second order itself, the inverse metric components have the qualitative form present in the case of high frequency perturbations, whose plots are given in figure 6 and figure 11.", "We would thus expect to see an increase in the horizon size relative to the first order solution.", "However, due to fact that $\\epsilon ^2$ falls faster than the any change in the metric components, this effect is highly suppresed in considering weak perturbations (as in this paper).", "In figure 20, we have provided a close up overhead plot of the horizon region from $r=2.348$ to $r=2.362$ for the second order effective metric component $g^{rr}_{(2)}(r,t)$ .", "While there is an increase in the horizon size relative to the first order effective metric component, this increase is marginal.", "The horizon grows from $r=2.3595$ to $r=2.3598$ at $t=10^3$ .", "Figure: Perturbed metric expressions for g (2) tt (r,t)g^{tt}_{(2)}(r,t) (upper), g (2) tr (r,t)g^{tr}_{(2)}(r,t) (middle) and g (2) rr (r,t)g^{rr}_{(2)}(r,t) (lower).", "Unlike the first order solutions in figure 16, we now find growing perturbations near the accretor boundary as in the high frequency plots in figure 6 and figure 11.Figure: Overhead view of g eff(2) rr (r,t)g^{rr}_{\\text{eff}(2)}(r,t) from r=2.348r=2.348 to r=2.362r=2.362 over all time.", "The horizon has grown slightly more than than in g eff(1) rr (r,t)g^{rr}_{\\text{eff}(1)}(r,t) at t=10 3 t=10^3 given in figure 17, from r=2.3595r=2.3595 to r=2.3598r=2.3598 .", "The correction at this and higher orders are suppressed by growing powers of ϵ\\epsilon .", "Thus for very weak perturbations, the original horizon size may not be recovered.", "This might not be the case with stronger perturbations in general.In conclusion, we see that in the case of very low frequency damped perturbations, there is the interesting effect that the acoustic horizon fluctuates and reduces in size.", "This effect however is a low order perturbation artefact, as higher order perturbations tend to increase the mass accretion rate and decrease the density in the supersonic region.", "This leads to metric corrections which increase after the first order perturbation as we go to higher orders.", "In the conservative case where the strength of the perturbation is assumed weak, $\\epsilon \\ll 1$ , the higher order corrections at the acoustic horizon are suppresed by powers of $\\epsilon $ .", "However, in generically considering stronger perturbations, an initially low frequency damped perturbation will be expected to have a horizon which increases in size.", "We may alternatively state that initial low frequency damped perturbations tends to the configuration of high frequency damped perturbations at higher orders." ] ]

[ [ "The Impacts of Convex Piecewise Linear Cost Formulations on AC Optimal\n Power Flow" ], [ "Abstract Despite strong connections through shared application areas, research efforts on power market optimization (e.g., unit commitment) and power network optimization (e.g., optimal power flow) remain largely independent.", "A notable illustration of this is the treatment of power generation cost functions, where nonlinear network optimization has largely used polynomial representations and market optimization has adopted piecewise linear encodings.", "This work combines state-of-the-art results from both lines of research to understand the best mathematical formulations of the nonlinear AC optimal power flow problem with piecewise linear generation cost functions.", "An extensive numerical analysis of non-convex models, linear approximations, and convex relaxations across fifty-four realistic test cases illustrates that nonlinear optimization methods are surprisingly sensitive to the mathematical formulation of piecewise linear functions.", "The results indicate that a poor formulation choice can slow down algorithm performance by a factor of ten, increasing the runtime from seconds to minutes.", "These results provide valuable insights into the best formulations of nonlinear optimal power flow problems with piecewise linear cost functions, a important step towards building a new generation of energy markets that incorporate the nonlinear AC power flow model." ], [ "Nomenclature", "tocsectionNomenclature [$Y^s = g^s-$ ] $N$ - The set of nodes $E$ , $E^R$ - The set of from and to branches $G$ - The set of generators $G_i$ - The subset of generators at bus $i$ $p_k$ - The number of points in piecewise linear cost for generator $k$ $C_k$ - The cost points $[1,2,\\dots ,p_k]$ for generator $k$ $C^{\\prime }_k$ - The cost points $[2,3,\\dots ,p_k]$ for generator $k$ $C^{\\prime \\prime }_k$ - The cost points $[3,4,\\dots ,p_k]$ for generator $k$ $cg_{kl}$ - The cost of generator $k$ at point $l \\in C_k$ $pg_{kl}$ - The power of generator $k$ at point $l \\in C_k$ ${\\Delta cg}_{kl}$ - The incremental cost of generator $k$ between points $l \\in C^{\\prime }_k$ and $l-1$ $bcg_{kl}$ - The cost offset of generator $k$ between points $l \\in C^{\\prime }_k$ and $l-1$ $a_k,b_k,c_k$ - Polynomial cost coefficients of generator $k$ $\\mathbf {i}$ - Imaginary number constant $S = p+ \\mathbf {i}q$ - AC power $V = v \\angle \\theta $ - AC voltage $Y = g + \\mathbf {i}b$ - Branch admittance $W$ - Product of two AC voltages $s^u$ - Branch apparent power thermal limit $\\theta ^{\\Delta l}, \\theta ^{\\Delta u}$ - Voltage angle difference limits $S^g$ , $S^d$ - AC power generation and demand $\\Re (\\cdot ), \\Im (\\cdot )$ - Real and imaginary parts of a complex num.", "$(\\cdot )^*$ , $|\\cdot |$ - Conjugate and magnitude of a complex num.", "$x^l, x^u$ - Lower and upper bounds of $x$ , respectively $\\mathbf {x}$ - A constant value Introduction Over the last several decades, competitive energy markets have proven to be an effective mechanism to generate power at minimal cost.", "In such markets potential energy generation units provide offers in the form of piecewise convex functions of generation production cost, and an independent system operator (ISO) or regional transmission organization (RTO) solves a mathematical optimization problem to determine the cheapest dispatch of those generating units, considering a wide variety of network reliability criteria, that is market clearing [1], [2], [3].", "Given the uncertainty of future energy demands, the market clearing process is repeated at different time scales ranging from day-head to real-time, at approximately 1 hour and 15 minute intervals, respectively.", "The significant size of real-world market clearing problems, which feature 100s to 1000s of generating units, and tight run-time requirements, just a few minutes, present a significant computational challenge to optimization algorithms.", "Modern energy markets currently solve this challenging optimization problem by utilizing commercial mixed-integer linear optimization software, such as CPLEX [4], Gurobi [5], and Xpress [6], all of which provide high reliability and state-of-the-art computational performance.", "However, a key limitation of these commercial tools is a focus on linear equations, limiting market clearing optimization to linear approximations of power flow physics, such as the seminal DC Power Flow [7].", "This approximation of the true nonlinear physics of AC power networks results in out-of-market corrections by network operators to adjust for inaccuracies in the market's physics model.", "Recent advances in nonlinear optimization [8] and convex nonlinear relaxations [9] have spurred aspirations for a new generation of market clearing optimization software that considers the full AC power flow physics.", "An AC market design has the promise of both reducing out-of-market corrections and incorporating prices for valuable ancillary services such as voltage support capabilities, which are currently priced by ad-hoc methods.", "From an algorithmic standpoint, the realization of an AC market in practice requires the fast and reliable solution of challenging mixed-integer non-convex nonlinear optimization problems, which is an active area of research in the optimization community.", "Recognizing the near-term potential for AC power flow markets, in 2019 ARPA-e conducted a Grid Optimization Competition [10], to identify the most promising algorithmic approaches for building the next generation of power network optimization software.", "Pursuing a future AC market design, this work explores how to best model piecewise convex functions of generation production costs in nonlinear optimization algorithms.", "Specifically, it considers different mathematically-equivalent formulations of piecewise linear cost functions and evaluates their computational implications on nonlinear power network optimization.", "The core observations of this work are threefold: (1) the mathematical modeling lessons learned from linear active-power-only markets do not necessarily carry over to forthcoming AC power markets; (2) a poor choice of the piecewise linear cost function representation can result in a solution time slowdown of as much as 10 times on realistic test systems; (3) ultimately, the “$\\lambda $ ” and “$\\Delta $ ” formulations of the piecewise cost functions prove to be the most suitable for current nonlinear optimization software.", "To the best of our knowledge, this paper provides the first synthesis of state-of-the-art models for piecewise linear cost functions coming from the market operations literature [11] with state-of-the-art models for nonlinear AC optimal power flow literature [12], [13].", "The rest of the paper is organized as follows: Section provides a brief introduction to optimal power flow and Section reviews mathematical models for piecewise linear cost functions.", "Section conducts computational experiments of 12 optimization models across 54 test cases providing the core contributions of the paper, and Section finishes with closing remarks.", "Optimal Power Flow The most prevalent power network optimization task is arguably the optimal power flow problem (OPF).", "At a high level, the OPF problem is a single-time period optimization task that consists of finding the cheapest way to generate sufficient power to meet a specified demand.", "The challenge of OPF is that an AC power network with a variety of operating constraints is used to transmit the power from the generators to the demands.", "Capturing both these operational constraints and the AC physics of a power network gives rise to a challenging non-convex nonlinear optimization task and a thriving body of algorithmic research [9].", "It is important to highlight, although OPF forms a foundational sub-problem of an energy market, that a real-world market requires many extensions including co-optimization of multiple-time periods, commitment of generation units, and contingency constraints, to name a few.", "This work focuses on OPF as a first necessary step to building a comprehensive and reliable AC energy market optimization.", "This work begins with the canonical academic AC-OPF formulation [14] presented in Model REF .", "At a high level, the network is defined by a set of buses $N$ , lines $E$ , and generators $G$ .", "A notable feature of this formulation is that the active power generator costs are provided as convex quadratic functions, that is, $\\mbox{Cost}_k(x) = \\mathbf {c_k} x^2 + \\mathbf {b_k} x + \\mathbf {a_k} \\;\\; \\forall k \\in G $ The objective function (REF ) seeks to minimize total of the generation costs.", "The constraints (),() capture the bus voltage requirements and generator output limits, respectively.", "The constraints (),() model the AC power flow physics via power balance and Ohm's law, respectively.", "Finally, the constraints (),() enforce the thermal and angle stability limits across the power lines in the network.", "For additional details, a first-principles derivation of this model is presented in [12].", "OPF Solution Methods The mathematical optimization problem presented by Model REF is a non-convex NonLinear Program (NLP).", "This problem is known to feature local minima [15] and to be NP-Hard in the general case [16], [17].", "The standard solution for challenging non-convex NLP models is use of global optimization solvers, such as Baron [18], Couenne [19], and Alpine [20], which provide solution quality bounds and optimality proofs.", "However, these approaches can only solve AC-OPF problems with a few hundred buses, which is an order of magnitude less than real-world applications that require thousands of buses.", "It has been observed that interior point methods, such as Ipopt [8] and Knitro [21], can quickly find high-quality solutions to real-world AC-OPF problems [12].", "However, these methods do not provide global guarantees of convergence and optimality of the solutions that they find.", "This performance gap between off-the-self interior point and global optimization solvers has yielded a wide variety of bespoke solutions for solving AC-OPF with quality guarantees, including problem-specific convex relaxations [22], [12], polynomial optimization [23], and bound tightening [24].", "See [9] for a comprehensive review of different approaches.", "In addition to the AC-OPF formulation presented in Model REF , this work also considers two canonical alternatives: the seminal DC Power Flow approximation and a simple convex relaxation of the OPF problem.", "These alternatives serve to position the results of this work in the ongoing transition of power network optimization from linear active-power-only approximations to nonlinear active-and-reactive optimization.", "Convex Relaxation Convex relaxations of the non-convex AC model have drawn significant interest in recent years [9], in large part due to their ability to provide tight bounds on the AC-OPF solution quality.", "Following those lines, this work considers the Second-Order Cone (SOC) relaxation of the AC power flow equations [22].", "Although some relaxations are stronger than SOC [12], [25], [26] and others are faster than SOC [27], the SOC relaxation is selected because it provides an appealing tradeoff between bounding strength and runtime performance.", "The first insight of the SOC relaxation is that the voltage product terms $V_i^* V_j$ can be lifted into a higher dimensional $W$ -space as follows: $|V_i|^2 &\\Rightarrow W_{ii} \\;\\; \\forall i \\in N \\\\V_i V^*_j &\\Rightarrow W_{ij} \\;\\; \\forall (i,j) \\in E $ Note that lifting Model REF into the $W$ -space makes all of the non-convex constraints linear.", "The second insight of the SOC relaxation is that the $W$ -space relaxation can be strengthened by adding the valid inequality, $& |W_{ij}|^2 \\le W_{ii}W_{jj} \\;\\; \\forall (i,j) \\in E $ which is a convex rotated SOC constraint that is supported by a wide variety of commercial optimization tools.", "Utilizing these two insights, the SOC-OPF relation of the AC-OPF problem is presented in Model REF .", "Many of the constraints remain the same; the core differences are as follows.", "Constraints (REF ),(),(),() capture the bus voltage requirements, Ohm's law, and voltage angle stability limits in the lifted $W$ -space.", "Constraint () is a new constraint that strengthens the relaxation.", "The virtues of this model are that it is convex (i.e., global optimality is achieved by local solvers like Ipopt and Knitro) and it provides a lower bound to the objective function value of the non-convex AC-OPF model.", "The principal weakness of this model is that its voltage solution is non-physical and usually does not provide useful insights into the solution of the non-convex AC-OPF model.", "Linear Approximation The most widely used approach to solving OPF problems is to approximate the power flow physics with the DC Power Flow approximation [7].", "This model is achieved by taking a first-order Taylor expansion around the nominal voltage operating point of $V_i \\approx 1 \\; \\angle \\; 0$ .", "This expansion yields the following voltage product approximation, $V_i V^*_j &\\Rightarrow 1 \\; \\angle \\; (\\theta _i - \\theta _j)$ and results in omitting the reactive power variables and constraints from the model as they are constant values in the first-order Taylor expansion.", "A more detailed derivation of this model from first principles is available in [7], [28].", "Applying this transformation to the AC-OPF problem yields the DC-OPF approximation in Model REF .", "The constraints are similar to the AC-OPF model but with the reactive power components omitted.", "The objective function (REF ) is the same but applied to the active-power variables directly.", "The constraint () captures the generator output limits.", "The constraints (),() approximate the power flow physics of power balance and Ohm's law, respectively.", "Finally, the constraints (),() enforce the thermal and angle stability limits across the power lines in the network.", "The virtue of this model is that it is a linear optimization problem, which benefits from decades of research and mature commercial optimization software.", "The principal weakness of this model is that it cannot consider bus voltage and reactive power requirements.", "Piecewise Linear Costs Mathematical formulations for piecewise linear cost functions are a cornerstone of linear and mixed-integer linear optimization tools [29], [30].", "The first linear programming formulation for convex piecewise linear costs appeared shortly after the simplex method was popularized for a practical reason – to approximate separable convex functions [31].", "This was quickly followed by several alternative formulations for separable convex piecewise linear costs [32], [33], [34].", "Formulations for non-convex piecewise linear cost functions arrived shortly after the development of Mixed-integer linear programming (MILP), a much richer modeling framework, that can capture many discrete mathematical structures [35].", "Both the convex and non-convex formulations of piecewise linear cost have been a fruitful research topic over the years.", "See [36] for a recent comprehensive review of different approaches.", "In the context of power systems, the majority of the piecewise linear cost literature has focused on the formulation of convex piecewise linear cost functions in the context of the unit commitment (UC) problem, which focuses on the temporal constraints of assigning generators to deliver power for several hours or days.", "The use of convex piecewise linear cost in the UC context dates back to Garver [37], who proposed them in a MILP formulation for UC.", "Over the years research demonstrated that the formulation of the piecewise linear production cost can have a profound impact on modern MILP solver performance [38], [3], [11].", "While several papers have studied quadratic and polynomial convex production costs for UC [39], [40], [41], these formulations have not been adopted by industry in the United States, mainly due to the increased complexity of the resulting nonlinear mixed-integer optimization problem.", "At this time, it is still the standard practice in the United States for ISOs and RTOs to require generators to submit convex piecewise linear offers for generation production costs.", "As previously discussed, there is an increasing interest for market operators to consider AC physics directly in market clearing problems, like OPF and UC [10].", "However, there is currently an inconsistency between the existing market structures, which model generator costs as convex piecewise linear functions, and the AC-OPF literature that has standardized around convex quadratic cost functions [14].", "The principal objective of this work is to consider the breadth of convex piecewise linear cost formulations developed in the UC literature in the context of AC-OPF, to understand their performance implications on forthcoming nonlinear optimization problems.", "Figure: An example of typical piecewise linear data processing.", "In this case, the points 1 and 2 can be removed due to the generation lower bound, ℜ(S gl )\\Re (S^{gl}).", "Point 5 can be removed due to no change in the slope of the adjacent segments, and point 7 needs to be extended to include the generation upper bound, ℜ(S gu )\\Re (S^{gu}).Piecewise Linear Data Model and Assumptions In this work piecewise linear functions are defined by a sequence of points representing line segments for each generator $k \\in G$ , $(pg_{kl}, cg_{kl}) \\;\\; \\forall l \\in C_k $ where $cg_{kl}$ is the cost of generating $pg_{kl}$ megawatts of power and the set $C_k$ determines how many points each generator cost function has.", "In some cases it is convenient to consider the piecewise linear function as a collection of lines defined by slope-intercept pairs as follows, ${\\Delta cg}_{kl} = \\frac{cg_{k,l} - cg_{k,l-1}}{pg_{k,l} - pg_{k,l-1}} \\;\\; \\forall l \\in C^{\\prime }_k \\\\bcg_{kl} = cg_{kl} - {\\Delta cg}_{kl}pg_{kl} \\;\\; \\forall l \\in C^{\\prime }_k $ where ${\\Delta cg}_{kl}$ and $bcg_{kl}$ are the slope and intercept respectively.", "Note that the set $C^{\\prime }_k$ omits the first index of the standard point set $C_k$ so that $l-1$ is well defined.", "Figure REF provides an illustration of a prototypical generation cost function encoded as a sequence of points.", "Assumptions: A core challenge of working with piecewise linear functions is their generality.", "Hence, it is important to provide a detailed specification of the inputs that are permitted.", "Throughout this work it is assumed that the optimization problem of interest is a minimization problem and the points of each generator encode a convex piecewise linear cost function.", "Specifically, this work requires three key properties of these functions: (1) the generator's lower bound, $S^{gl}_k$ , occurs in the first segment of function; (2) the generator's upper bound, $S^{gu}_k$ , occurs in the last segment of function; (3) the slope of each linear segment is strictly increasing.", "These properties are summarized as follows, $pg_{k,1} & \\le \\Re (S^{gl}_k) < pg_{k,2} & \\;\\; \\forall k \\in G \\\\pg_{k,{p_k}-1} & < \\Re (S^{gu}_k) \\le pg_{k,p_k} & \\;\\; \\forall k \\in G \\\\{\\Delta cg}_{k,l-1} & < {\\Delta cg}_{k,l} & \\forall l \\in C^{\\prime \\prime }_k \\;\\; \\forall k \\in G $ If the given piecewise linear function is convex, these properties can be ensured by the following data processing procedure.", "Data Processing: For a variety of reasons, real-world piecewise linear generation cost functions often benefit from data cleaning before encoding them in an optimization model.", "In this work we conduct the following data processing procedures: (1) if the generator bounds are outside of the first and last segments, that is $\\Re (S^{gl}_k) < pg_{k,1}$ or $pg_{k,p_k} < \\Re (S^{gu}_k)$ , then the segments are extended to include the generator bounds; (2) if the piecewise linear function includes segments beyond the bounds of the generator, that is $pg_{k,2} \\le \\Re (S^{gl}_k)$ or $\\Re (S^{gu}_k) \\le pg_{k,{p_k}-1}$ , then the extra out-of-bounds segments are removed; (3) if there is little change in the slope of two adjacent segments, that is ${\\Delta cg}_{k,l-1} \\approx {\\Delta cg}_{k,l}$ , then they are combined into one segment.", "These simple data processing steps increase the performance of optimization algorithms by removing redundant constraints and serve to enforce the mathematical requirements of the formulations considered by this work.", "Figure REF provides an illustration of generation cost function that requires these data processing procedures.", "Formulations of OPF with Piecewise Linear Costs The interest of this work is variants of the OPF problems from Section where the polynomial active power generation cost functions are replaced with piecewise linear functions, that is, $\\mbox{Cost}_k(x) = \\max _{l \\in C^{\\prime }_k} \\lbrace \\mathbf {\\Delta cg}_{kl} x + \\mathbf {bcg}_{kl}\\rbrace \\;\\; \\forall k \\in G $ Unlike the previously considered polynomial functions, there are a wide variety of equivalent mathematical encodings of piecewise linear functions.", "Following the seminal works on piecewise linear formulations [34], [42], this work considers the four “standard” formulations for convex piecewise linear functions, referred to as the $\\Psi $ , $\\lambda $ , $\\Delta $ and $\\Phi $ models by [42].", "All four formulations are mathematically equivalent but can have significant performance implications for the numerical methods used in optimization algorithms.", "While other formulations for piecewise linear costs exist, three of these standard formulations are commonly used in power systems problems (i.e., $\\Psi $ , $\\lambda $ , $\\Delta $ ) [11].", "This work includes a fourth formulation, $\\Phi $ , for completeness and because it shares an interesting mathematical connection as the dual of the $\\Delta $ formulation.", "At a high level, these formulations represent two distinct perspectives on modeling piecewise linear functions, focusing on either the function evaluation points (i.e., $\\Psi $ ,$\\lambda $ ) or integration of the cost function's derivative (i.e., $\\Delta $ ,$\\Phi $ ).", "The $\\Psi $ Formulation: This formulation is arguably the most popular and intuitive.", "The $\\Psi $ formulation explicitly models the epigraph of (REF ), that is the region on and above the objective function on the graph ($x$ , $f(x)$ ).", "However, because mathematical programming solvers do not usually have explicit support for the $\\max $ function, an auxiliary cost variable $c^g_k \\in [\\mathbf {cg}_{k,1}, \\mathbf {cg}_{k,p_k}]$ is introduced for each generator, which combined with inequality constraints, captures the semantics of the $\\max $ function.", "The complete AC-OPF problem using the $\\Psi $ cost formulation is presented in Model REF .", "The $\\lambda $ Formulation: This formulation reflects the most natural encoding of a convex hull from the collection of points (REF ), which is a popular modeling approach in the linear programming literature [43].", "The core idea is to introduce an interpolation variable $\\lambda ^{cg}_{kl} \\in [0, 1]$ for each point in the piecewise linear function and link all of these interpolation variables together with the constraint $\\sum _{l \\in C_k} \\lambda ^{cg}_{kl} = 1$ .", "The power and cost of the interpolated point can be recovered with the expressions $\\sum _{l \\in C_k} \\mathbf {pg}_{kl} \\lambda ^{cg}_{kl}$ and $\\sum _{l \\in C_k} \\mathbf {cg}_{kl} \\lambda ^{cg}_{kl}$ , respectively.", "The complete AC-OPF problem using the $\\lambda $ cost formulation is presented in Model REF .", "The $\\lambda $ formulation is also interesting because it represents the mathematical dual of the $\\Psi $ piecewise linear formulation [42].", "The $\\Delta $ Formulation: This formulation breaks the cost function into a collection of generation bins, ${\\Delta pg}_{kl}$ , based on consecutive points in the piecewise linear function, that is, ${\\Delta pg}_{kl} \\in [0, \\mathbf {pg}_{kl} - \\mathbf {pg}_{k,l-1}]$ .", "The key observation is that each bin can be associated with a linear cost based on the slope of that line segment, i.e., $\\mathbf {\\Delta cg}_{kl} {\\Delta pg}_{kl}$ .", "An interpretation of this formulation is that it computes the integral of the cost's derivative along the power-axis of the piecewise linear function.", "The complete AC-OPF problem using the $\\Delta $ cost formulation is presented in Model REF .", "Both the $\\Delta $ formulation and the $\\lambda $ formulation are particularly interesting as they have proven the most effective formulations in practical UC problems [11].", "The $\\Phi $ Formulation: This formulation is the most challenging to interpret and combines elements from both the $\\Psi $ and $\\Delta $ models.", "It begins by extracting a linear cost function from the first segment of the piecewise linear function, i.e., $\\mathbf {\\Delta cg}_{k,2}\\Re (S^g_k) + \\mathbf {bcg}_{k,2}$ .", "It then defines bins for how much power is supplied by each segment after the first, i.e., $\\Phi _{kl} \\in [0, \\Re (\\mathbf {S^{gu}}_k) - \\mathbf {pg}_{k,l-1}]$ .", "The extra incremental cost of each segment over the previous segments is then captured by $(\\mathbf {\\Delta cg}_{k,l} - \\mathbf {\\Delta cg}_{k,l-1}) \\Phi _{kl}$ .", "An interpretation of this formulation is that it computes the integral of the cost's derivative along the cost-axis of the piecewise linear function.", "The complete AC-OPF problem using the $\\Phi $ cost formulation is presented in Model REF .", "Although this $\\Phi $ formulation is uncommon in the literature, it is interesting in this context as the mathematical dual of the $\\Delta $ piecewise linear formulation [42].", "Relaxation and Approximation Variants Models REF through REF present variants of the AC-OPF model with different formulations of piecewise linear cost functions.", "As these formulations only changed the objective function in that model, it is clear how similar modifications could also be applied to the DC approximation and SOC relaxation models that were presented in Section .", "Computational Evaluation Combining both the power flow formulations from Section with the piecewise linear formulations from Section , this section conducts a detailed computational evaluation of 12 different OPF formulations, i.e., $\\lbrace \\mbox{AC}, \\mbox{SOC}, \\mbox{DC}\\rbrace \\times \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace $ .", "The overarching observation is that the choice of piecewise linear formulation has a much more dramatic impact on state-of-the-art nonlinear optimization algorithms than it does for linear optimization algorithms.", "This result is demonstrated by three computational studies, the first focusing on a solution quality comparison, the second highlighting the runtime trends of different piecewise linear formulations, and the third comparing different algorithms for solving convex nonlinear optimization problems, which is particularly relevant when considering convex relaxations of OPF problems.", "Test Cases and Computational Setting The traditional AC-OPF benchmark problems, such as PGLib-OPF [14], have standardized around convex quadratic cost functions, which precludes their use in this work.", "To the best of our knowledge, ARPA-e's Grid Optimization Competition Challenge 1 datasets [10] represent the first comprehensive source of AC-OPF test cases that feature piecewise linear cost functions and hence these were leveraged for building a test suite in this work.", "The publicly available Challenge 1 Final Event data archive consists of 340 AC-OPF cases spanning 18 distinct networks, which range from 500 to 30,000 buses.", "This work down selected that collection to a set of 54 representative AC-OPF cases, three representatives from each distinct network.", "Finally, the cases were converted from the Grid Optimization Competition data format into the Matpower data format [44] to be compatible with established state-of-the-art OPF evaluation tools [13].", "The specific details of which scenarios were selected are available in Table REF .", "The proposed OPF formulations were implemented in Julia v1.5 as an extension to the PowerModels v0.17 [45], [13] framework, which utilizes JuMP v0.21 [46] as a general-purpose mathematical optimization modeling layer.", "The NLP formulations were primarily solved with Ipopt [8] using the HSL MA27 linear algebra solver [47] and a cross validation is conducted with Knitro v12.2.", "The LP and QCQP formulations were solved with Gurobi v9.0 [5].", "All of the solvers were configured to terminate once the optimality gap was less than $10^{-6}$ without an explicit time limit.", "The evaluation was conducted on HPE ProLiant XL170r servers with two Intel 2.10 GHz CPUs and 128 GB of memory; however, for consistency the algorithms were configured to only utilize one thread.", "Solution Quality Validation The first observation of this experiment is that all of the formulations considered found solutions of identical quality, up to the numerical tolerances of the optimization algorithms.", "For the convex DC-OPF and SOC-OPF formulations this serves as an important validation of the implementation's correctness.", "Since these problems are convex and all of the piecewise linear formulation are proven to be mathematically equivalent, all models should converge to a consistent globally optimal solution, up to the accuracy of floating point arithmetic.", "Detailed results for these models are omitted in the interest of brevity.", "A more surprising result is that all four variants of the non-convex AC-OPF formulation also converge to solutions of identical quality, up to the numerical tolerance of the optimization algorithm.", "This result is demonstrated in the detailed results presented in Table REF .", "The first AC-OPF-$\\lambda $ column shows the locally optimal objective value and the following three columns indicate the absolute difference in the objective value from the other three formulations, all of which are well below the optimality tolerance of $10^{-6}$ .", "These are encouraging results as they suggest the choice of piecewise linear formulation in nonlinear optimization algorithms can be taken solely on the criteria of runtime performance without concern for solution quality degradation.", "Figure: Runtime comparison of all power flow models across all piecewise linear formulations.", "The horizontal lines indicate the mean runtime of each piecewise linear formulation.Table: Quality and Runtime Results of AC Power Flow Formulations Linear and Nonlinear Runtime Trends Given the stability of solution quality across the piecewise linear formulations, runtime performance becomes the most important criteria for comparison.", "In this analysis the metric of interest is the relative runtime increase of a given piecewise linear formulation over the best runtime across all of the formulations.", "Specifically, we define the runtime-ratio as, $\\mbox{runtime-ratio}_m = \\frac{\\mbox{runtime}_m}{\\displaystyle \\min _{n \\in \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace } \\mbox{runtime}_n} \\;\\; \\forall m \\in \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace $ Figure REF presents the runtime-ratio of each formulation broken down by power flow formulation.", "In these figures the y-axis presents the runtime-ratio in a log scale and the x-axis orders the 54 OPF problems from smallest to largest.", "DC-OPF Results: With a variety of outliers occurring in each formulation, there is no clear winner in this formulation.", "However, on average the $\\lambda $ and $\\Delta $ formulations perform best, which is consistent with similar benchmarking studies from the UC literature [11].", "It is important to highlight that the range of the y-axis in this case goes up to 1.7, which indicates that a poor selection of piecewise linear formulation can result in a performance reduction of no more than 70%.", "These results provide further validation of the experiment design as they replicate well-known results from the literature.", "AC-OPF Results: These results provide a stark contrast to the DC-OPF study.", "The first significant difference is that the $\\Psi $ model is significantly worse than all other formulations.", "It is 5 times slower on average and can be 10 times slower in the worst case; that is 500% and 1000% slower, respectively.", "The second significant difference is that although the remaining three models have similar performance, they all have notable outliers that are more than 2 times slower than the best formulation considered.", "On average, the $\\Delta $ formulation appears to be the best, but still suffers from notable outliers.", "A case-by-case runtime breakdown for the AC-OPF model is presented in Table REF , for further inspection of specific cases.", "Overall, these results highlight a drastically increased sensitivity of the AC-OPF problem to piecewise linear formulations.", "SOC-OPF Results: By and large these results are similar to the AC-OPF results, which suggests that the increased sensitivity of the AC-OPF problem to piecewise linear formulations is a feature of the interior point algorithm, rather than an issue of non-convexity.", "Two notable differences in these results are the overall reduction of outliers and the $\\lambda $ formulation becoming a clear winner in terms of performance.", "Overall, these results present novel insights into the impact of piecewise linear formulations on nonlinear optimization and suggest that a detailed analysis of nonlinear optimization algorithm behavior is required to find a consistent-best formulation for piecewise linear generation costs.", "Convex Nonlinear Algorithm Comparison The results from the previous section highlight a stark distinction between the behavior of linear programming algorithms (i.e., DC-OPF) and general purpose nonlinear interior point algorithms (i.e., AC-OPF and SOC-OPF).", "However, the SOC-OPF model is a convex quadratic nonlinear model, which can be solved by specialized nonlinear optimization algorithms, such as Quadratic Constrained Quadratic Programming (QCQP) solvers.", "This experiment briefly explores the possible benefit of convex quadratic optimization algorithms by comparing the solution of the SOC-OPF model via general purpose NLP (i.e., Ipopt) and a more specialized QCQP solver (i.e., Gurobi).", "The first observation is that there is a significant difference in each solver's reliability, which is highlighted by Table REF .", "In at least 8 of the 54 networks considered, the QCQP solver reports a numerical error.", "Interestingly, there is significant variability in the QCQP solver's reliability in different piecewise linear formulations.", "Putting these reliability issues aside, the remaining analysis focuses on the subset of OPF cases where both solvers report convergence to an optimal solution.", "In those cases, both solvers find solutions of identical quality, up to the numerical tolerances of the optimization algorithms, which suggests a correct implementation of both approaches.", "Table: Solver Reliability on SOC-OPF ModelsThe second observation is a significant difference in performance of the two algorithms.", "Figure REF presents a side-by-side comparison of the nonlinear solver runtimes for the subset of cases that both can solve.", "In the figure, points below the diagonal line indicate a performance increase for the NLP solver and points above the line indicate a performance increase for the QCQP solver.", "These results indicate the QCQP algorithm brings increased performance to the $\\Psi $ model, which has very poor performance in the NLP solver.", "However, for the other piecewise linear formulations, the NLP solver has a consistent performance advantage.", "Overall, these results suggest that general purpose nonlinear interior point algorithms remain the most reliable and performant solution to large-scale convex nonlinear OPF problems.", "Figure: A runtime comparison of solving the SOC-OPF problem with NLP and QCQP algorithms.", "Non-convex Nonlinear Solver Comparison The results from the previous sections highlight distinctions between the behavior of specialized convex optimization algorithms (e.g., linear programming and second order cone programming) and the general purpose nonlinear interior point algorithms, as implemented by Ipopt.", "However, it is possible that the poor performance of the $\\Psi $ model in the previous experiments is a consequence of a deficiency in Ipopt's implementation and is not inherent to interior point algorithms more broadly.", "To better understand the consistency of the results presented in this work, the following experiment compares the performance of Ipopt to the commercial nonlinear optimization solver Knitro.", "Similar to the previous analysis, the first observation is that there is a difference in each solver's reliability, which is highlighted by Table REF .", "There appears to be a consistent convergence issue with two of the models considered, however this can likely be overcome by careful tuning of Knitro's convergence tolerance parameters.", "Interestingly, the most significant variability in Knitro's reliability occurs in the $\\Psi $ model, which provides further evidence that this formulation is particularly problematic for interior point algorithms.", "Putting these reliability issues aside, the remaining analysis focuses on the subset of OPF cases where both solvers converge to a locally optimal solution.", "In these cases, both solvers find solutions of identical quality, up to the numerical tolerances of the optimization algorithms, which suggests a correct implementation of both approaches.", "Table: Solver Reliability on AC-OPF ModelsThe second observation are the trends in runtime performance of the two algorithms.", "Figure REF presents a side-by-side comparison of the solver runtimes for the subset of cases that both can solve.", "In the figure, points below the diagonal line indicate a performance increase for Ipopt and points above the line indicate a performance increase for Knitro.", "These results indicate the two algorithms have similar performance on all of the models considered, which indicates that the performance challenges of the $\\Psi $ model may persists in a variety of interior point algorithms.", "It is worth noting that Knitro tends to show slightly better performance on the $\\Psi $ model while Ipopt has slightly better performance on the other models.", "Knitro's implementation reduces the typical runtime of the $\\Psi $ model by about half.", "This is a notable improvement but it is not sufficient to change the overall conclusion of this work that the $\\Delta $ and $\\Lambda $ formulations a preferable for solving large-scale non-convex OPF problems.", "Figure: A runtime comparison of solving the AC-OPF problem with Ipopt and Knitro solvers.", "Conclusion Pursuing future nonlinear AC market clearing optimization algorithms, this work considered how to best formulate AC optimal power flow problems with piecewise linear generation cost functions.", "To that end, core insights from the unit commitment literature in piecewise linear cost formulations [11] were combined with insights from the optimal power flow literature [12], [13], resulting in 12 variants of the optimal power flow problem.", "A comprehensive numerical evaluation of these models on 54 realistic power network cases indicates that the “$\\lambda $ ” and “$\\Delta $ ” formulations of the piecewise cost functions prove to be the most suitable for current nonlinear optimization software, with the “$\\lambda $ ” formulation being particularly suitable for convex relaxations of the power flow equations.", "However, notable outliers in both models suggest ongoing research is required to ensure performance reliability of nonlinear optimization software, in preparation for real-world deployments with strict runtime requirements.", "Acknowledgment The authors would like to thank Richard O'Neill for helpful feedback on a preliminary draft of this work.", "LA-UR-20-23777" ], [ "Introduction", "Over the last several decades, competitive energy markets have proven to be an effective mechanism to generate power at minimal cost.", "In such markets potential energy generation units provide offers in the form of piecewise convex functions of generation production cost, and an independent system operator (ISO) or regional transmission organization (RTO) solves a mathematical optimization problem to determine the cheapest dispatch of those generating units, considering a wide variety of network reliability criteria, that is market clearing [1], [2], [3].", "Given the uncertainty of future energy demands, the market clearing process is repeated at different time scales ranging from day-head to real-time, at approximately 1 hour and 15 minute intervals, respectively.", "The significant size of real-world market clearing problems, which feature 100s to 1000s of generating units, and tight run-time requirements, just a few minutes, present a significant computational challenge to optimization algorithms.", "Modern energy markets currently solve this challenging optimization problem by utilizing commercial mixed-integer linear optimization software, such as CPLEX [4], Gurobi [5], and Xpress [6], all of which provide high reliability and state-of-the-art computational performance.", "However, a key limitation of these commercial tools is a focus on linear equations, limiting market clearing optimization to linear approximations of power flow physics, such as the seminal DC Power Flow [7].", "This approximation of the true nonlinear physics of AC power networks results in out-of-market corrections by network operators to adjust for inaccuracies in the market's physics model.", "Recent advances in nonlinear optimization [8] and convex nonlinear relaxations [9] have spurred aspirations for a new generation of market clearing optimization software that considers the full AC power flow physics.", "An AC market design has the promise of both reducing out-of-market corrections and incorporating prices for valuable ancillary services such as voltage support capabilities, which are currently priced by ad-hoc methods.", "From an algorithmic standpoint, the realization of an AC market in practice requires the fast and reliable solution of challenging mixed-integer non-convex nonlinear optimization problems, which is an active area of research in the optimization community.", "Recognizing the near-term potential for AC power flow markets, in 2019 ARPA-e conducted a Grid Optimization Competition [10], to identify the most promising algorithmic approaches for building the next generation of power network optimization software.", "Pursuing a future AC market design, this work explores how to best model piecewise convex functions of generation production costs in nonlinear optimization algorithms.", "Specifically, it considers different mathematically-equivalent formulations of piecewise linear cost functions and evaluates their computational implications on nonlinear power network optimization.", "The core observations of this work are threefold: (1) the mathematical modeling lessons learned from linear active-power-only markets do not necessarily carry over to forthcoming AC power markets; (2) a poor choice of the piecewise linear cost function representation can result in a solution time slowdown of as much as 10 times on realistic test systems; (3) ultimately, the “$\\lambda $ ” and “$\\Delta $ ” formulations of the piecewise cost functions prove to be the most suitable for current nonlinear optimization software.", "To the best of our knowledge, this paper provides the first synthesis of state-of-the-art models for piecewise linear cost functions coming from the market operations literature [11] with state-of-the-art models for nonlinear AC optimal power flow literature [12], [13].", "The rest of the paper is organized as follows: Section provides a brief introduction to optimal power flow and Section reviews mathematical models for piecewise linear cost functions.", "Section conducts computational experiments of 12 optimization models across 54 test cases providing the core contributions of the paper, and Section finishes with closing remarks." ], [ "Optimal Power Flow", "The most prevalent power network optimization task is arguably the optimal power flow problem (OPF).", "At a high level, the OPF problem is a single-time period optimization task that consists of finding the cheapest way to generate sufficient power to meet a specified demand.", "The challenge of OPF is that an AC power network with a variety of operating constraints is used to transmit the power from the generators to the demands.", "Capturing both these operational constraints and the AC physics of a power network gives rise to a challenging non-convex nonlinear optimization task and a thriving body of algorithmic research [9].", "It is important to highlight, although OPF forms a foundational sub-problem of an energy market, that a real-world market requires many extensions including co-optimization of multiple-time periods, commitment of generation units, and contingency constraints, to name a few.", "This work focuses on OPF as a first necessary step to building a comprehensive and reliable AC energy market optimization.", "This work begins with the canonical academic AC-OPF formulation [14] presented in Model REF .", "At a high level, the network is defined by a set of buses $N$ , lines $E$ , and generators $G$ .", "A notable feature of this formulation is that the active power generator costs are provided as convex quadratic functions, that is, $\\mbox{Cost}_k(x) = \\mathbf {c_k} x^2 + \\mathbf {b_k} x + \\mathbf {a_k} \\;\\; \\forall k \\in G $ The objective function (REF ) seeks to minimize total of the generation costs.", "The constraints (),() capture the bus voltage requirements and generator output limits, respectively.", "The constraints (),() model the AC power flow physics via power balance and Ohm's law, respectively.", "Finally, the constraints (),() enforce the thermal and angle stability limits across the power lines in the network.", "For additional details, a first-principles derivation of this model is presented in [12]." ], [ "OPF Solution Methods", "The mathematical optimization problem presented by Model REF is a non-convex NonLinear Program (NLP).", "This problem is known to feature local minima [15] and to be NP-Hard in the general case [16], [17].", "The standard solution for challenging non-convex NLP models is use of global optimization solvers, such as Baron [18], Couenne [19], and Alpine [20], which provide solution quality bounds and optimality proofs.", "However, these approaches can only solve AC-OPF problems with a few hundred buses, which is an order of magnitude less than real-world applications that require thousands of buses.", "It has been observed that interior point methods, such as Ipopt [8] and Knitro [21], can quickly find high-quality solutions to real-world AC-OPF problems [12].", "However, these methods do not provide global guarantees of convergence and optimality of the solutions that they find.", "This performance gap between off-the-self interior point and global optimization solvers has yielded a wide variety of bespoke solutions for solving AC-OPF with quality guarantees, including problem-specific convex relaxations [22], [12], polynomial optimization [23], and bound tightening [24].", "See [9] for a comprehensive review of different approaches.", "In addition to the AC-OPF formulation presented in Model REF , this work also considers two canonical alternatives: the seminal DC Power Flow approximation and a simple convex relaxation of the OPF problem.", "These alternatives serve to position the results of this work in the ongoing transition of power network optimization from linear active-power-only approximations to nonlinear active-and-reactive optimization." ], [ "Convex Relaxation", "Convex relaxations of the non-convex AC model have drawn significant interest in recent years [9], in large part due to their ability to provide tight bounds on the AC-OPF solution quality.", "Following those lines, this work considers the Second-Order Cone (SOC) relaxation of the AC power flow equations [22].", "Although some relaxations are stronger than SOC [12], [25], [26] and others are faster than SOC [27], the SOC relaxation is selected because it provides an appealing tradeoff between bounding strength and runtime performance.", "The first insight of the SOC relaxation is that the voltage product terms $V_i^* V_j$ can be lifted into a higher dimensional $W$ -space as follows: $|V_i|^2 &\\Rightarrow W_{ii} \\;\\; \\forall i \\in N \\\\V_i V^*_j &\\Rightarrow W_{ij} \\;\\; \\forall (i,j) \\in E $ Note that lifting Model REF into the $W$ -space makes all of the non-convex constraints linear.", "The second insight of the SOC relaxation is that the $W$ -space relaxation can be strengthened by adding the valid inequality, $& |W_{ij}|^2 \\le W_{ii}W_{jj} \\;\\; \\forall (i,j) \\in E $ which is a convex rotated SOC constraint that is supported by a wide variety of commercial optimization tools.", "Utilizing these two insights, the SOC-OPF relation of the AC-OPF problem is presented in Model REF .", "Many of the constraints remain the same; the core differences are as follows.", "Constraints (REF ),(),(),() capture the bus voltage requirements, Ohm's law, and voltage angle stability limits in the lifted $W$ -space.", "Constraint () is a new constraint that strengthens the relaxation.", "The virtues of this model are that it is convex (i.e., global optimality is achieved by local solvers like Ipopt and Knitro) and it provides a lower bound to the objective function value of the non-convex AC-OPF model.", "The principal weakness of this model is that its voltage solution is non-physical and usually does not provide useful insights into the solution of the non-convex AC-OPF model." ], [ "Linear Approximation", "The most widely used approach to solving OPF problems is to approximate the power flow physics with the DC Power Flow approximation [7].", "This model is achieved by taking a first-order Taylor expansion around the nominal voltage operating point of $V_i \\approx 1 \\; \\angle \\; 0$ .", "This expansion yields the following voltage product approximation, $V_i V^*_j &\\Rightarrow 1 \\; \\angle \\; (\\theta _i - \\theta _j)$ and results in omitting the reactive power variables and constraints from the model as they are constant values in the first-order Taylor expansion.", "A more detailed derivation of this model from first principles is available in [7], [28].", "Applying this transformation to the AC-OPF problem yields the DC-OPF approximation in Model REF .", "The constraints are similar to the AC-OPF model but with the reactive power components omitted.", "The objective function (REF ) is the same but applied to the active-power variables directly.", "The constraint () captures the generator output limits.", "The constraints (),() approximate the power flow physics of power balance and Ohm's law, respectively.", "Finally, the constraints (),() enforce the thermal and angle stability limits across the power lines in the network.", "The virtue of this model is that it is a linear optimization problem, which benefits from decades of research and mature commercial optimization software.", "The principal weakness of this model is that it cannot consider bus voltage and reactive power requirements.", "Mathematical formulations for piecewise linear cost functions are a cornerstone of linear and mixed-integer linear optimization tools [29], [30].", "The first linear programming formulation for convex piecewise linear costs appeared shortly after the simplex method was popularized for a practical reason – to approximate separable convex functions [31].", "This was quickly followed by several alternative formulations for separable convex piecewise linear costs [32], [33], [34].", "Formulations for non-convex piecewise linear cost functions arrived shortly after the development of Mixed-integer linear programming (MILP), a much richer modeling framework, that can capture many discrete mathematical structures [35].", "Both the convex and non-convex formulations of piecewise linear cost have been a fruitful research topic over the years.", "See [36] for a recent comprehensive review of different approaches.", "In the context of power systems, the majority of the piecewise linear cost literature has focused on the formulation of convex piecewise linear cost functions in the context of the unit commitment (UC) problem, which focuses on the temporal constraints of assigning generators to deliver power for several hours or days.", "The use of convex piecewise linear cost in the UC context dates back to Garver [37], who proposed them in a MILP formulation for UC.", "Over the years research demonstrated that the formulation of the piecewise linear production cost can have a profound impact on modern MILP solver performance [38], [3], [11].", "While several papers have studied quadratic and polynomial convex production costs for UC [39], [40], [41], these formulations have not been adopted by industry in the United States, mainly due to the increased complexity of the resulting nonlinear mixed-integer optimization problem.", "At this time, it is still the standard practice in the United States for ISOs and RTOs to require generators to submit convex piecewise linear offers for generation production costs.", "As previously discussed, there is an increasing interest for market operators to consider AC physics directly in market clearing problems, like OPF and UC [10].", "However, there is currently an inconsistency between the existing market structures, which model generator costs as convex piecewise linear functions, and the AC-OPF literature that has standardized around convex quadratic cost functions [14].", "The principal objective of this work is to consider the breadth of convex piecewise linear cost formulations developed in the UC literature in the context of AC-OPF, to understand their performance implications on forthcoming nonlinear optimization problems.", "Figure: An example of typical piecewise linear data processing.", "In this case, the points 1 and 2 can be removed due to the generation lower bound, ℜ(S gl )\\Re (S^{gl}).", "Point 5 can be removed due to no change in the slope of the adjacent segments, and point 7 needs to be extended to include the generation upper bound, ℜ(S gu )\\Re (S^{gu})." ], [ "Piecewise Linear Data Model and Assumptions", "In this work piecewise linear functions are defined by a sequence of points representing line segments for each generator $k \\in G$ , $(pg_{kl}, cg_{kl}) \\;\\; \\forall l \\in C_k $ where $cg_{kl}$ is the cost of generating $pg_{kl}$ megawatts of power and the set $C_k$ determines how many points each generator cost function has.", "In some cases it is convenient to consider the piecewise linear function as a collection of lines defined by slope-intercept pairs as follows, ${\\Delta cg}_{kl} = \\frac{cg_{k,l} - cg_{k,l-1}}{pg_{k,l} - pg_{k,l-1}} \\;\\; \\forall l \\in C^{\\prime }_k \\\\bcg_{kl} = cg_{kl} - {\\Delta cg}_{kl}pg_{kl} \\;\\; \\forall l \\in C^{\\prime }_k $ where ${\\Delta cg}_{kl}$ and $bcg_{kl}$ are the slope and intercept respectively.", "Note that the set $C^{\\prime }_k$ omits the first index of the standard point set $C_k$ so that $l-1$ is well defined.", "Figure REF provides an illustration of a prototypical generation cost function encoded as a sequence of points.", "Assumptions: A core challenge of working with piecewise linear functions is their generality.", "Hence, it is important to provide a detailed specification of the inputs that are permitted.", "Throughout this work it is assumed that the optimization problem of interest is a minimization problem and the points of each generator encode a convex piecewise linear cost function.", "Specifically, this work requires three key properties of these functions: (1) the generator's lower bound, $S^{gl}_k$ , occurs in the first segment of function; (2) the generator's upper bound, $S^{gu}_k$ , occurs in the last segment of function; (3) the slope of each linear segment is strictly increasing.", "These properties are summarized as follows, $pg_{k,1} & \\le \\Re (S^{gl}_k) < pg_{k,2} & \\;\\; \\forall k \\in G \\\\pg_{k,{p_k}-1} & < \\Re (S^{gu}_k) \\le pg_{k,p_k} & \\;\\; \\forall k \\in G \\\\{\\Delta cg}_{k,l-1} & < {\\Delta cg}_{k,l} & \\forall l \\in C^{\\prime \\prime }_k \\;\\; \\forall k \\in G $ If the given piecewise linear function is convex, these properties can be ensured by the following data processing procedure.", "Data Processing: For a variety of reasons, real-world piecewise linear generation cost functions often benefit from data cleaning before encoding them in an optimization model.", "In this work we conduct the following data processing procedures: (1) if the generator bounds are outside of the first and last segments, that is $\\Re (S^{gl}_k) < pg_{k,1}$ or $pg_{k,p_k} < \\Re (S^{gu}_k)$ , then the segments are extended to include the generator bounds; (2) if the piecewise linear function includes segments beyond the bounds of the generator, that is $pg_{k,2} \\le \\Re (S^{gl}_k)$ or $\\Re (S^{gu}_k) \\le pg_{k,{p_k}-1}$ , then the extra out-of-bounds segments are removed; (3) if there is little change in the slope of two adjacent segments, that is ${\\Delta cg}_{k,l-1} \\approx {\\Delta cg}_{k,l}$ , then they are combined into one segment.", "These simple data processing steps increase the performance of optimization algorithms by removing redundant constraints and serve to enforce the mathematical requirements of the formulations considered by this work.", "Figure REF provides an illustration of generation cost function that requires these data processing procedures." ], [ "Formulations of OPF with Piecewise Linear Costs", "The interest of this work is variants of the OPF problems from Section where the polynomial active power generation cost functions are replaced with piecewise linear functions, that is, $\\mbox{Cost}_k(x) = \\max _{l \\in C^{\\prime }_k} \\lbrace \\mathbf {\\Delta cg}_{kl} x + \\mathbf {bcg}_{kl}\\rbrace \\;\\; \\forall k \\in G $ Unlike the previously considered polynomial functions, there are a wide variety of equivalent mathematical encodings of piecewise linear functions.", "Following the seminal works on piecewise linear formulations [34], [42], this work considers the four “standard” formulations for convex piecewise linear functions, referred to as the $\\Psi $ , $\\lambda $ , $\\Delta $ and $\\Phi $ models by [42].", "All four formulations are mathematically equivalent but can have significant performance implications for the numerical methods used in optimization algorithms.", "While other formulations for piecewise linear costs exist, three of these standard formulations are commonly used in power systems problems (i.e., $\\Psi $ , $\\lambda $ , $\\Delta $ ) [11].", "This work includes a fourth formulation, $\\Phi $ , for completeness and because it shares an interesting mathematical connection as the dual of the $\\Delta $ formulation.", "At a high level, these formulations represent two distinct perspectives on modeling piecewise linear functions, focusing on either the function evaluation points (i.e., $\\Psi $ ,$\\lambda $ ) or integration of the cost function's derivative (i.e., $\\Delta $ ,$\\Phi $ ).", "The $\\Psi $ Formulation: This formulation is arguably the most popular and intuitive.", "The $\\Psi $ formulation explicitly models the epigraph of (REF ), that is the region on and above the objective function on the graph ($x$ , $f(x)$ ).", "However, because mathematical programming solvers do not usually have explicit support for the $\\max $ function, an auxiliary cost variable $c^g_k \\in [\\mathbf {cg}_{k,1}, \\mathbf {cg}_{k,p_k}]$ is introduced for each generator, which combined with inequality constraints, captures the semantics of the $\\max $ function.", "The complete AC-OPF problem using the $\\Psi $ cost formulation is presented in Model REF .", "The $\\lambda $ Formulation: This formulation reflects the most natural encoding of a convex hull from the collection of points (REF ), which is a popular modeling approach in the linear programming literature [43].", "The core idea is to introduce an interpolation variable $\\lambda ^{cg}_{kl} \\in [0, 1]$ for each point in the piecewise linear function and link all of these interpolation variables together with the constraint $\\sum _{l \\in C_k} \\lambda ^{cg}_{kl} = 1$ .", "The power and cost of the interpolated point can be recovered with the expressions $\\sum _{l \\in C_k} \\mathbf {pg}_{kl} \\lambda ^{cg}_{kl}$ and $\\sum _{l \\in C_k} \\mathbf {cg}_{kl} \\lambda ^{cg}_{kl}$ , respectively.", "The complete AC-OPF problem using the $\\lambda $ cost formulation is presented in Model REF .", "The $\\lambda $ formulation is also interesting because it represents the mathematical dual of the $\\Psi $ piecewise linear formulation [42].", "The $\\Delta $ Formulation: This formulation breaks the cost function into a collection of generation bins, ${\\Delta pg}_{kl}$ , based on consecutive points in the piecewise linear function, that is, ${\\Delta pg}_{kl} \\in [0, \\mathbf {pg}_{kl} - \\mathbf {pg}_{k,l-1}]$ .", "The key observation is that each bin can be associated with a linear cost based on the slope of that line segment, i.e., $\\mathbf {\\Delta cg}_{kl} {\\Delta pg}_{kl}$ .", "An interpretation of this formulation is that it computes the integral of the cost's derivative along the power-axis of the piecewise linear function.", "The complete AC-OPF problem using the $\\Delta $ cost formulation is presented in Model REF .", "Both the $\\Delta $ formulation and the $\\lambda $ formulation are particularly interesting as they have proven the most effective formulations in practical UC problems [11].", "The $\\Phi $ Formulation: This formulation is the most challenging to interpret and combines elements from both the $\\Psi $ and $\\Delta $ models.", "It begins by extracting a linear cost function from the first segment of the piecewise linear function, i.e., $\\mathbf {\\Delta cg}_{k,2}\\Re (S^g_k) + \\mathbf {bcg}_{k,2}$ .", "It then defines bins for how much power is supplied by each segment after the first, i.e., $\\Phi _{kl} \\in [0, \\Re (\\mathbf {S^{gu}}_k) - \\mathbf {pg}_{k,l-1}]$ .", "The extra incremental cost of each segment over the previous segments is then captured by $(\\mathbf {\\Delta cg}_{k,l} - \\mathbf {\\Delta cg}_{k,l-1}) \\Phi _{kl}$ .", "An interpretation of this formulation is that it computes the integral of the cost's derivative along the cost-axis of the piecewise linear function.", "The complete AC-OPF problem using the $\\Phi $ cost formulation is presented in Model REF .", "Although this $\\Phi $ formulation is uncommon in the literature, it is interesting in this context as the mathematical dual of the $\\Delta $ piecewise linear formulation [42]." ], [ "Relaxation and Approximation Variants", "Models REF through REF present variants of the AC-OPF model with different formulations of piecewise linear cost functions.", "As these formulations only changed the objective function in that model, it is clear how similar modifications could also be applied to the DC approximation and SOC relaxation models that were presented in Section ." ], [ "Computational Evaluation", "Combining both the power flow formulations from Section with the piecewise linear formulations from Section , this section conducts a detailed computational evaluation of 12 different OPF formulations, i.e., $\\lbrace \\mbox{AC}, \\mbox{SOC}, \\mbox{DC}\\rbrace \\times \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace $ .", "The overarching observation is that the choice of piecewise linear formulation has a much more dramatic impact on state-of-the-art nonlinear optimization algorithms than it does for linear optimization algorithms.", "This result is demonstrated by three computational studies, the first focusing on a solution quality comparison, the second highlighting the runtime trends of different piecewise linear formulations, and the third comparing different algorithms for solving convex nonlinear optimization problems, which is particularly relevant when considering convex relaxations of OPF problems." ], [ "Test Cases and Computational Setting", "The traditional AC-OPF benchmark problems, such as PGLib-OPF [14], have standardized around convex quadratic cost functions, which precludes their use in this work.", "To the best of our knowledge, ARPA-e's Grid Optimization Competition Challenge 1 datasets [10] represent the first comprehensive source of AC-OPF test cases that feature piecewise linear cost functions and hence these were leveraged for building a test suite in this work.", "The publicly available Challenge 1 Final Event data archive consists of 340 AC-OPF cases spanning 18 distinct networks, which range from 500 to 30,000 buses.", "This work down selected that collection to a set of 54 representative AC-OPF cases, three representatives from each distinct network.", "Finally, the cases were converted from the Grid Optimization Competition data format into the Matpower data format [44] to be compatible with established state-of-the-art OPF evaluation tools [13].", "The specific details of which scenarios were selected are available in Table REF .", "The proposed OPF formulations were implemented in Julia v1.5 as an extension to the PowerModels v0.17 [45], [13] framework, which utilizes JuMP v0.21 [46] as a general-purpose mathematical optimization modeling layer.", "The NLP formulations were primarily solved with Ipopt [8] using the HSL MA27 linear algebra solver [47] and a cross validation is conducted with Knitro v12.2.", "The LP and QCQP formulations were solved with Gurobi v9.0 [5].", "All of the solvers were configured to terminate once the optimality gap was less than $10^{-6}$ without an explicit time limit.", "The evaluation was conducted on HPE ProLiant XL170r servers with two Intel 2.10 GHz CPUs and 128 GB of memory; however, for consistency the algorithms were configured to only utilize one thread." ], [ "Solution Quality Validation", "The first observation of this experiment is that all of the formulations considered found solutions of identical quality, up to the numerical tolerances of the optimization algorithms.", "For the convex DC-OPF and SOC-OPF formulations this serves as an important validation of the implementation's correctness.", "Since these problems are convex and all of the piecewise linear formulation are proven to be mathematically equivalent, all models should converge to a consistent globally optimal solution, up to the accuracy of floating point arithmetic.", "Detailed results for these models are omitted in the interest of brevity.", "A more surprising result is that all four variants of the non-convex AC-OPF formulation also converge to solutions of identical quality, up to the numerical tolerance of the optimization algorithm.", "This result is demonstrated in the detailed results presented in Table REF .", "The first AC-OPF-$\\lambda $ column shows the locally optimal objective value and the following three columns indicate the absolute difference in the objective value from the other three formulations, all of which are well below the optimality tolerance of $10^{-6}$ .", "These are encouraging results as they suggest the choice of piecewise linear formulation in nonlinear optimization algorithms can be taken solely on the criteria of runtime performance without concern for solution quality degradation.", "Figure: Runtime comparison of all power flow models across all piecewise linear formulations.", "The horizontal lines indicate the mean runtime of each piecewise linear formulation.Table: Quality and Runtime Results of AC Power Flow Formulations" ], [ "Linear and Nonlinear Runtime Trends", "Given the stability of solution quality across the piecewise linear formulations, runtime performance becomes the most important criteria for comparison.", "In this analysis the metric of interest is the relative runtime increase of a given piecewise linear formulation over the best runtime across all of the formulations.", "Specifically, we define the runtime-ratio as, $\\mbox{runtime-ratio}_m = \\frac{\\mbox{runtime}_m}{\\displaystyle \\min _{n \\in \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace } \\mbox{runtime}_n} \\;\\; \\forall m \\in \\lbrace \\Psi , \\lambda , \\Delta , \\Phi \\rbrace $ Figure REF presents the runtime-ratio of each formulation broken down by power flow formulation.", "In these figures the y-axis presents the runtime-ratio in a log scale and the x-axis orders the 54 OPF problems from smallest to largest.", "DC-OPF Results: With a variety of outliers occurring in each formulation, there is no clear winner in this formulation.", "However, on average the $\\lambda $ and $\\Delta $ formulations perform best, which is consistent with similar benchmarking studies from the UC literature [11].", "It is important to highlight that the range of the y-axis in this case goes up to 1.7, which indicates that a poor selection of piecewise linear formulation can result in a performance reduction of no more than 70%.", "These results provide further validation of the experiment design as they replicate well-known results from the literature.", "AC-OPF Results: These results provide a stark contrast to the DC-OPF study.", "The first significant difference is that the $\\Psi $ model is significantly worse than all other formulations.", "It is 5 times slower on average and can be 10 times slower in the worst case; that is 500% and 1000% slower, respectively.", "The second significant difference is that although the remaining three models have similar performance, they all have notable outliers that are more than 2 times slower than the best formulation considered.", "On average, the $\\Delta $ formulation appears to be the best, but still suffers from notable outliers.", "A case-by-case runtime breakdown for the AC-OPF model is presented in Table REF , for further inspection of specific cases.", "Overall, these results highlight a drastically increased sensitivity of the AC-OPF problem to piecewise linear formulations.", "SOC-OPF Results: By and large these results are similar to the AC-OPF results, which suggests that the increased sensitivity of the AC-OPF problem to piecewise linear formulations is a feature of the interior point algorithm, rather than an issue of non-convexity.", "Two notable differences in these results are the overall reduction of outliers and the $\\lambda $ formulation becoming a clear winner in terms of performance.", "Overall, these results present novel insights into the impact of piecewise linear formulations on nonlinear optimization and suggest that a detailed analysis of nonlinear optimization algorithm behavior is required to find a consistent-best formulation for piecewise linear generation costs." ], [ "Convex Nonlinear Algorithm Comparison", "The results from the previous section highlight a stark distinction between the behavior of linear programming algorithms (i.e., DC-OPF) and general purpose nonlinear interior point algorithms (i.e., AC-OPF and SOC-OPF).", "However, the SOC-OPF model is a convex quadratic nonlinear model, which can be solved by specialized nonlinear optimization algorithms, such as Quadratic Constrained Quadratic Programming (QCQP) solvers.", "This experiment briefly explores the possible benefit of convex quadratic optimization algorithms by comparing the solution of the SOC-OPF model via general purpose NLP (i.e., Ipopt) and a more specialized QCQP solver (i.e., Gurobi).", "The first observation is that there is a significant difference in each solver's reliability, which is highlighted by Table REF .", "In at least 8 of the 54 networks considered, the QCQP solver reports a numerical error.", "Interestingly, there is significant variability in the QCQP solver's reliability in different piecewise linear formulations.", "Putting these reliability issues aside, the remaining analysis focuses on the subset of OPF cases where both solvers report convergence to an optimal solution.", "In those cases, both solvers find solutions of identical quality, up to the numerical tolerances of the optimization algorithms, which suggests a correct implementation of both approaches.", "Table: Solver Reliability on SOC-OPF ModelsThe second observation is a significant difference in performance of the two algorithms.", "Figure REF presents a side-by-side comparison of the nonlinear solver runtimes for the subset of cases that both can solve.", "In the figure, points below the diagonal line indicate a performance increase for the NLP solver and points above the line indicate a performance increase for the QCQP solver.", "These results indicate the QCQP algorithm brings increased performance to the $\\Psi $ model, which has very poor performance in the NLP solver.", "However, for the other piecewise linear formulations, the NLP solver has a consistent performance advantage.", "Overall, these results suggest that general purpose nonlinear interior point algorithms remain the most reliable and performant solution to large-scale convex nonlinear OPF problems.", "Figure: A runtime comparison of solving the SOC-OPF problem with NLP and QCQP algorithms." ], [ "Non-convex Nonlinear Solver Comparison", "The results from the previous sections highlight distinctions between the behavior of specialized convex optimization algorithms (e.g., linear programming and second order cone programming) and the general purpose nonlinear interior point algorithms, as implemented by Ipopt.", "However, it is possible that the poor performance of the $\\Psi $ model in the previous experiments is a consequence of a deficiency in Ipopt's implementation and is not inherent to interior point algorithms more broadly.", "To better understand the consistency of the results presented in this work, the following experiment compares the performance of Ipopt to the commercial nonlinear optimization solver Knitro.", "Similar to the previous analysis, the first observation is that there is a difference in each solver's reliability, which is highlighted by Table REF .", "There appears to be a consistent convergence issue with two of the models considered, however this can likely be overcome by careful tuning of Knitro's convergence tolerance parameters.", "Interestingly, the most significant variability in Knitro's reliability occurs in the $\\Psi $ model, which provides further evidence that this formulation is particularly problematic for interior point algorithms.", "Putting these reliability issues aside, the remaining analysis focuses on the subset of OPF cases where both solvers converge to a locally optimal solution.", "In these cases, both solvers find solutions of identical quality, up to the numerical tolerances of the optimization algorithms, which suggests a correct implementation of both approaches.", "Table: Solver Reliability on AC-OPF ModelsThe second observation are the trends in runtime performance of the two algorithms.", "Figure REF presents a side-by-side comparison of the solver runtimes for the subset of cases that both can solve.", "In the figure, points below the diagonal line indicate a performance increase for Ipopt and points above the line indicate a performance increase for Knitro.", "These results indicate the two algorithms have similar performance on all of the models considered, which indicates that the performance challenges of the $\\Psi $ model may persists in a variety of interior point algorithms.", "It is worth noting that Knitro tends to show slightly better performance on the $\\Psi $ model while Ipopt has slightly better performance on the other models.", "Knitro's implementation reduces the typical runtime of the $\\Psi $ model by about half.", "This is a notable improvement but it is not sufficient to change the overall conclusion of this work that the $\\Delta $ and $\\Lambda $ formulations a preferable for solving large-scale non-convex OPF problems.", "Figure: A runtime comparison of solving the AC-OPF problem with Ipopt and Knitro solvers.Pursuing future nonlinear AC market clearing optimization algorithms, this work considered how to best formulate AC optimal power flow problems with piecewise linear generation cost functions.", "To that end, core insights from the unit commitment literature in piecewise linear cost formulations [11] were combined with insights from the optimal power flow literature [12], [13], resulting in 12 variants of the optimal power flow problem.", "A comprehensive numerical evaluation of these models on 54 realistic power network cases indicates that the “$\\lambda $ ” and “$\\Delta $ ” formulations of the piecewise cost functions prove to be the most suitable for current nonlinear optimization software, with the “$\\lambda $ ” formulation being particularly suitable for convex relaxations of the power flow equations.", "However, notable outliers in both models suggest ongoing research is required to ensure performance reliability of nonlinear optimization software, in preparation for real-world deployments with strict runtime requirements." ], [ "Acknowledgment", "The authors would like to thank Richard O'Neill for helpful feedback on a preliminary draft of this work.", "LA-UR-20-23777" ] ]

[ [ "First-principles calculations of thermal electron emission from H$^-$ in\n silicon" ], [ "Abstract Thermal electron emission process of a hydrogen impurity is an important topic of fundamental semiconductor physics.", "Despite of decades-long study, theory is not established yet.", "Here, we study the process of $\\mathrm{H}^{-}$ in silicon, $\\mathrm{H^{-}} \\to \\mathrm{H^{0}} + e^{-}$, using a first-principles calculation.", "Our calculation indicates that the process consists of two steps: slow diffusion of H$^{-}$ from a tetrahedral site to a bond-center site, which is the rate-limiting step, and faster nonradiative transition from $\\mathrm{H}^{-}$ to $\\mathrm{H}^{0} + e^{-}$ that occurs subsequently at the body-center site.", "The calculated rate is consistent with a deep level transient spectroscopy experiment" ], [ "Introduction", "Due to its amphoteric nature and fast diffusivity, hydrogen impurities passivate various defects and help increase the efficiency of various semiconductor devices.", "Their nonradiative carrier capture/emission capability [1] significantly affects diffusivity and carrier lifetimes.", "Due to its technological importance, this phenomenon has been studied over the past decades.", "In silicon, hydrogen is known to exist as H$^{+}$ or H$^{-}$ depending on doping condition[2], [3], while H$^{0}$ is metastable[4] and can exist only at very low temperatures[5] or under illumination[6].", "These properties can be consistently explained by first-principles total energy calculations[7], [8], which indicate that H$^{+}$ or H$^{-}$ is thermodynamically stable while H$^{0}$ is always unstable against the reaction $2\\mathrm {H}^{0} \\rightarrow \\mathrm {H}^{+} + \\mathrm {H}^{-}$ .", "The transition between different charge states was experimentally studied with a deep level transient spectroscopy (DLTS)[8].", "The DLTS experiment showed that thermal electron emission from H$^{-}$ , i.e., $\\mathrm {H}^{-} \\rightarrow \\mathrm {H}^{0} + e^{-}$ , has a slow reaction rate, $r = 2.8 \\times 10^{-1} \\, \\mathrm {s}^{-1}$ at room temperature with activation energy $E_{\\mathrm {a}} = 0.84 \\, \\mathrm {eV}$ .", "The activation energy is usually thought to correspond to the energy difference between H$^{-}$ and H$^{0} + e^{-}$ .", "However, density functional theory (DFT) calculation of this quantity only amounts to 0.50 eV.", "The nature of the observed activation energy in the experiment therefore remains to be investigated.", "Here, we show our theoretical study on the rate for thermal electron emission by H$^{-}$ in silicon.", "Although there are many studies treating hydrogen impurity in silicon, as far as we know, there has been no first-principles study on the rate because it requires accurate description of the band gap, the defect thermodynamics, and electron-phonon couplings.", "Recently, however, Kim et al.", "demonstrated that first-principles calculation can reproduce the rate for nonradiative carrier capture from the $DX$ center in GaAs despite large anhnarmonicity of the potential energy surface (PES)[9].", "Because the PES for hydrogen impurity in silicon is also known to be anharmonic[8], their approach should help us to elucidate the reason for the high activation energy observed in the DLTS experiment on H$^{-}$ .", "In this context, we calculated thermodynamic transition level with screened-hybrid DFT and confirmed that the calculated result is different from the experimental activation energy.", "Then, we constructed a configuration coordinate diagram for the transition between $\\mathrm {H}^{-}$ and $\\mathrm {H}^{0} + e^{-}$ .", "The calculated diagram for H$^{-}$ was found to have local minimum not only at its most stable tetrahedral site but also at a bond center site, which is the global minimum for $\\mathrm {H}^{0} + e^{-}$ .", "The existence of local minimum at the bond-center site is a key to explain the fast thermal electron emission because nuclear wavefunctions overlap considerably at the bond-center site.", "The calculated rate shows that, with the assumption of thermal equilibrium, thermal electron emission is much faster than the experiment, indicating another step should rate-determine the whole process.", "Finally, we calculated the migration barrier of H$^{-}$ between tetrahedral and bond-center site, and conclude that the migration is the rate-limiting step for thermal electron emission from H$^{-}$ ." ], [ "Methods", "We performed first-principles calculations based on DFT [10], [11] with Vienna Ab initio Simulation Package (vasp)[12], [13], [14], [15], where the Kohn-Sham orbitals are expressed by the plane-wave basis.", "We used the screened hybrid functional formulated as Heyd-Scuseria-Ernzerhof exchange correlation functional (HSE06 functional)[16].", "Projector augmented wave (PAW) method was used for the interactions between valence electrons and ions.", "The cutoff energy for the plane-wave basis was $400 \\, \\mathrm {eV}$ , which was found to give sufficient accuracy for defect thermodynamics and electron-phonon coupling in our system.", "We optimized the lattice constant of silicon with conventional cell calculations using $\\Gamma $ -centered $6 \\times 6 \\times 6$ k-point mesh for Brillouin-zone integration.", "The calculated lattice constant is 5.433 Å, in good agreement with experimental one extrapolated to $0\\, \\mathrm {K}$ , 5.430 Å[17].", "For defect calculations, we use $3 \\times 3\\times 3$ super cell of conventional cell, including 216 Si atoms, in order to suppress spurious interactions between defects.", "To reduce the computational cost, $\\Gamma $ -point calculation was performed.", "The calculated band gap of silicon is 1.31 eV which is larger than the experimental one, 1.17 eV, because the conduction band bottom is not correctly sampled with $\\Gamma $ -point in $3\\times 3\\times 3$ super cell.", "To obtain correct energetics between H$^{-}$ and H$^{0}$ + $e^{-}$ , we adjust the Kohn-Sham energies of the conduction bands by shifting them by $-0.14 \\, \\mathrm {eV}$ in the calculated configuration coordinate diagram.", "Finite-size correction for charged defect is calculated with the Markov-Payne correction[18] by extrapolating the calculated energies with $2\\times 2\\times 3$ , $2\\times 3\\times 3$ and $3\\times 3\\times 3$ supercells to infinite cell size.", "For the extrapolation, we used $2 \\times 2 \\times 2$ k-point for all supercells: this k-point mesh is found to be necessary for the extrapolation to work well.", "The correction was calculated for H$^{-}$ at a tetrahedral site, and the same correction was used for all other configurations.", "The finite-size correction changes the calculated thermodynamic transition level, $\\varepsilon (0/-)$ , by +0.06 eV.", "The rate for thermal electron emission was calculated with the formalism proposed by Alkauskas et al[19].", "In the formalism, with the aid of the static approximation for electronic states and one-dimensional approximation for nuclear degrees of freedom, the rate for nonadiabatic transition from electronic state $i$ to $j$ with multiphonon absorption is calculated with the Fermi's golden rule: $r_{ij} = \\frac{2\\pi }{\\hbar } |W_{ij}|^2 \\sum _{n} \\omega _n \\sum _{m}|\\langle \\chi _{jm} |Q-Q_0| \\chi _{in} \\rangle |^2 \\delta (E_{in} - E_{jm}).$ $W_{ij}$ is the electron matrix element defined as $W_{ij} = \\langle \\psi _j | \\frac{\\partial H}{\\partial Q}| \\psi _i \\rangle $ , where $|\\psi _{i(j)} \\rangle $ is the Kohn-Sham orbital for state $i(j)$ .", "With first-order perturbation theory, $W_{ij}$ can be expressed as $W_{ij} = (\\varepsilon _j - \\varepsilon _i) \\langle \\psi _j | \\frac{\\partial \\psi _i }{\\partial Q}\\rangle $ , where $\\varepsilon _{i}$ is the Kohn-Sham energy for a state $i$ .", "The Kohn-Sham orbitals overlaps, $\\langle \\psi _j | \\frac{\\partial \\psi _i }{\\partial Q}\\rangle $ , is calculated with all-electron wavefunctions including PAW augmented core contributions reproduced by pawpyseed code[20].", "$\\omega _n$ is the thermal weight for the n-th eigenstate of nuclear wavefunction, $|\\chi _{in}\\rangle $ .", "The nuclear wave functions and eigenenergies are calculated by solving the one-dimensional Schrödinger equation for configuration coordinate diagram to account for large anharmonicity of the potential energy surface[9].", "$Q$ is the configuration coordinate defined as $Q = \\sqrt{M} R$ , where M is a diagonal matrix with masses in its diagonal element and R is the $3N$ -dimensional coordinate for lattice.", "We assume a linear reaction pathway between the most stable configurations of two charge states, H$^{0}$ and H$^{-}$ , following the scheme proposed by Alkauskas et al.", "[19] although this treatment is known to underestimate the rate[21] because the linear pathway does not necessarily give a major contribution.", "The electron emission rate from H$^{-}$ is calculated by summing up the contributions of the transition from a defect state to conduction band states.", "The electron emission rate is thus calculated as $r_{i} = \\sum _{j=\\mathrm {CB}} r_{ij}$ , where $i$ is the defect state, and j is the conduction band state.", "As the summation for the conduction bands, we take the bands within $0.15 \\, \\mathrm {eV}$ above the conduction band minimum.", "We have confirmed that conduction bands above them give only negligible effects to the calculated rate.", "The delta function in Eq.", "(REF ) is approximated as Gaussian with finite width, $\\sigma $ , which corresponds to a lifetime of the vibrational mode considered in the configuration coordinate.", "Although lifetimes of local vibrational modes at low temperatures are known[22], we cannot use those values as $\\sigma $ because the local vibrational modes are different from our reaction pathway.", "Thus, we tried several values of $\\sigma $ to investigate its effect on the calculated reaction rate.", "The value of $\\sigma $ is, indeed, found to affect the calculated rate.", "However, our conclusion is robust against the choice of $\\sigma $ value, and we used $\\sigma = 0.026 \\, \\mathrm {eV}$ throughout this study." ], [ "Thermodynamics", "Firstly, we investigate the thermodynamics of hydrogen in silicon.", "The calculated energies suggest that the most stable site is the bond-center (BC) site for H$^{0}$ while the tetrahedral (Td) site for H$^{-}$ , and the calculated thermodynamic transition level is $\\varepsilon (0/-) = 0.67 \\, \\mathrm {eV}$ from the valence band maximum.", "These results are in good agreement with previous theoretical study using local density functional[8].", "For the electron emission process, the energy difference between $\\mathrm {H}^{0} + e^{-}$ and $\\mathrm {H}^{-}$ is calculated to be $E_{g} - \\varepsilon (0/-) = 0.50 \\, \\mathrm {eV}$ , where $E_{g}$ is the band gap of silicon.", "Thus, the activation energy for thermal electron emission is expected to be $0.50 \\, \\mathrm {eV}$ from first-principles calculations.", "This value is, however, inconsistent with an experimental activation energy, $0.84 \\, \\mathrm {eV}$ , obtained by a DLTS experiment[8].", "Considering the inconsistency, we investigated possible origins of the discrepancy from computational point of views.", "We calculated the effects of thermal expansion of silicon at room temperature, zero-point energy of hydrogen and the type of pseudopotentials (hard/standard).", "These effects change the calculated $\\varepsilon (0/-)$ as $-0.01 \\, \\mathrm {eV}$ , $+0.06 \\, \\mathrm {eV}$ , and $-0.02 \\, \\mathrm {eV}$ , respectively.", "None of these effects is thus confirmed to fill the gap between theoretical and experimental values.", "In conclusion, our calculation indicates that thermodynamics could not explain the activation energy observed in the DLTS experiment." ], [ "Configuration Coordinate Diagrams", "Figure REF shows the configuration coordinate diagram for the transition between $\\mathrm {H}^{-}$ and $\\mathrm {H}^{0}+e^{-}$ .", "The configuration coordinate is taken as the linear interpolation between their most stable Td site for H$^{-}$ and BC site for H$^{0}$ .", "The diagram for $\\mathrm {H}^{0} + e^{-}$ is calculated as the summation of total energy of $\\mathrm {H}^{0}$ and the Kohn-Sham energy of the conduction band minimum of pristine silicon.", "The calculated configuration coordinate diagrams have two unique features: firstly, the H$^{0}$ state shows large anharmonicity around the configurations of the Td site.", "At the configuration of the Td site, the hydrogen occupies the interstitial void of silicon.", "Because the radius of H$^{0}$ is smaller than that of H$^{-}$ , the H$^{0}$ interacts less strongly with Si.", "Thus, the flat PES appears around the Td site for H$^{0}$ while the PES of H$^{-}$ has a deep minimum at the Td site.", "The calculated energy difference of $\\mathrm {H}^{0} + e^{-}$ and $\\mathrm {H}^{-}$ around Td site is large, $1.0 \\, \\mathrm {eV}$ .", "The large energy gap indicates that the activation energy of 1.0 eV is required for H$^{-}$ to emit an electron around Td site.", "Secondly, H$^{-}$ state has a local minimum at the BC site configuration.", "The existence of local minimum has been confirmed by additional structural optimization calculation.", "Because both $\\mathrm {H}^{-}$ and $\\mathrm {H}^{0} + e^{-}$ have local minimum at the same configuration, the nuclear wavefunctions should overlap significantly.", "The electron emission rate is, therefore, expected to be large at the BC site.", "We should note that the local minimum at the BC site for H$^{-}$ was not observed in the work by Herring et al[8].", "The reason for the discrepancy is likely to be the difference in the configuration coordinates or exchange-correlation functional used to calculate the PES, i.e., HSE06 functional in our calculation and local density functional in their work.", "Figure: (Color online) Configuration coordinate diagrams of H 0 +e - \\mathrm {H}^{0} + e^{-} and H - ^{-}.For H 0 +e - \\mathrm {H}^{0} + e^{-}, electron is assumed to occupy the conduction band bottom.The dot symbols show the calculated energies from first principles,and the lines show the ones calculated with the spline interpolation between the calculated ones.Q=0Q=0 amu 1/2 ^{1/2} Å  is the most stable configuration for H 0 ^{0} (BC site)while the configuration for Q=4.49Q=4.49 amu 1/2 ^{1/2} Å  is the most stable one for H - ^{-} (Td site).Figure REF shows the Kohn-Sham energies of valence bands, a defect state, and conduction bands along the configuration coordinate.", "The hydrogen insertion into the BC site makes the anti-bonding state of Si-Si bonding stable due to expansion of Si-Si bond and attractive interaction by proton.", "Thus, the anti-bonding state appears in the band gap as a defect state.", "As the configuration changes from that of BC site to Td site, the anti-bonding state disappears while another in-gap state appears.", "The new in-gap state corresponds to the defect state at Td site, which is the electronic state localized around the hydrogen in the interstitial void of Si.", "The defect levels of H$^{-}$ , however, do not exist in the band gap at the Td site configuration as shown in the Figure REF .", "This apparently causes a problem in applying the Fermi's golden rule formalism because the formalism requires to identify the initial defect state but the state is hybridized with the valence band.", "It does not, however, hamper the calculation of nonradiative electron emission from H$^{-}$ .", "Because, as discussed in the previous paragraph, the emission is likely to be dominated at the BC site where the energy required to activate the emission is much smaller than that at the Td site.", "Therefore, we can use the Kohn-Sham orbitals around BC site to calculate $W_{ij}$ in Eq.", "(REF ) to discuss the electron emission at room temperature.", "It should be noted that we neglect relative efficiency of the Td-to-BC migration of H$^{-}$ and the thermal electron emission at the BC site.", "Hence the migration is assumed to occur quickly toward thermal equilibrium so that the thermal weight, $\\omega _n$ , is allowed to be used in Eq.", "REF .", "Later, we will discuss the validity of this assumption based on the calculated rate from first principles.", "Figure: (Color online)The calculated Kohn-Sham energies of H - ^{-} for the bands around defect level along the configuration coordinate.Left and right inset figures show the partial charge densities of defect statewith H at the BC site and near the Td site, respectively.In the insets, large blue spheres correspond to Si atoms and small pink spheres to H atom.Filled and open circles correspond to occupied and unoccupied states, respectively.Configuration coordinate is same as Figure ." ], [ "Nonradiative Electron Emission", "The electronic matrix element, $W_{ij}$ , is calculated with the finite-displacement method at the configuration of the BC site as discussed in the previous section.", "In table REF , we tabulated the values of $W_{ij}$ which give major contributions to the calculated rates.", "Table: The calculated values of W ij W_{ij} for H - ^{-}between a defect state, ii, and conduction band states, {j}\\lbrace j \\rbrace , at the BC site configuration.ε j \\varepsilon _j is the Kohn-Sham energy for state jj.Only the W ij W_{ij}'s that have major contributions to the electron emission rateare tabulated.As shown in the table, not only the CBM but also other bands within $0.15 \\ \\mathrm {eV}$ from CBM have large values of $W_{ij}$ , indicating the importance to include those contributions into the calculations.", "Only eleven conduction bands are included in the calculations because the other conduction bands are well-seperated by $0.5 \\ \\mathrm {eV}$ and are thus negligible.", "The nuclear wavefunctions in Eq.", "(REF ) are obtained by solving the one-body Schrödinger equation with PES obtained with spline interpolation of the calculated PES.", "We note that the configuration coordinate diagrams around the BC site is in the Marcus inverted region[23]: because positions of the local minima are virtually the same within our configuration coordinate, the overlaps of nuclear wavefunctions are expected to be large compared to that in the normal region, thus possibly giving large rate for thermal electron emission.", "Figure REF shows the calculated rate for the thermal electron emission from H$^{-}$ to conduction bands.", "The calculated rate at room temperature is $5.2 \\times 10^{7} \\, \\mathrm {s}^{-1}$ and the activation energy is estimated to be $0.46 \\, \\mathrm {eV}$ from the Arrhenius plot.", "The calculated activation energy is consistent with the results expected with thermodynamics calculations, $E_g - \\varepsilon (0/-) = 0.50 \\, \\mathrm {eV}$ .", "Although the calculated rates depend on the smearing factor $\\sigma $ to describe the $\\delta $ function, the calculated rates are much larger than the experimental value, $r = 2.8 \\times 10^{-1} \\, \\mathrm {s}^{-1}$ at room temperature[8].", "Figure: The calculated rate for electron emission from H - ^{-}.The black circles are the calculated values and the black line is guide for eyes.Activation energy, E a E_{a}, is estimated to be 0.46 eV 0.46 \\, \\mathrm {eV}by fitting the calculated rates to the Arrhenius plot, r=r 0 exp(-E a /kT)r = r_{0}\\exp (-E_{a}/kT)The argument so far assumes thermal equilibrium: i.e.", "the diffusion of $\\mathrm {H}^{-}$ from the Td site to the BC site is assumed to be much faster than the thermal electron emission at BC site, thus enabling the use of thermal weight, $\\omega _n$ in Eq.", "(REF ) as a population of initial state.", "To test this assumption, we calculate the activation energy for the migration of H$^{-}$ with the climbing image nudged elastic band method[24], [25].", "Figure REF shows the calculated PES for H$^{-}$ migration between Td site and BC site.", "Here, the configuration of the BC site is optimized for H$^{-}$ , thus slightly different from the one used in the configuration coordinate diagram in Figure REF , which is the configuration optimized for H$^{0}$ .", "The calculated activation energy is $0.96 \\, \\mathrm {eV}$ , which is comparable to the one from the DLTS experiment, $0.84 \\, \\mathrm {eV}$ .", "The calculated activation energy is used to estimate the rate for the migration with a simple formula, $r = \\frac{kT}{\\hbar }\\exp (-E_{\\mathrm {a}}/kT)$ .", "Then, the calculated rate at $T = 300 \\, \\mathrm {K}$ is $4.3 \\times 10^{-3} \\, \\mathrm {s}^{-1}$ , which is much smaller than the calculated rate for thermal electron emission under thermal equilibrium.", "Thus, the rate-determining step is suggested to be the migration of H$^{-}$ from Td site to BC site.", "If the experimental activation energy, $E_{\\mathrm {a}} = 0.84 \\, \\mathrm {eV}$ , is used, the calculated migration rate is $3.0 \\times 10^{-1} \\, \\mathrm {s}^{-1}$ , showing good agreement with the experimental rate, $2.8 \\times 10^{-1} \\, \\mathrm {s}^{-1}$ .", "This fact strongly suggests that the activation energy observed in the DLTS experiment corresponds to the one for the migration process of H$^{-}$ , and the migration is the rate-limiting step for the thermal electron emission from H$^{-}$ .", "It should be noted here that the rate for the migration is estimated with treating hydrogen as a classical particle, thus neglecting nuclear quantum effect such as quantum tunneling.", "Although the quantum tunneling can be dominant as migration mechanism at room temperature, we expect that its effect is small because the distance between the Td site and the BC site is large as shown in Fig.", "REF .", "Figure: The minimum energy path for H - ^{-} migration calculated with the climbing image nudged elastic band method.The calculated activation energy is 0.96 eV from the Td site." ], [ "Conclusion", "Here, we study the thermal electron emission process of H$^{-}$ in silicon, $\\mathrm {H}^{-} \\rightarrow \\mathrm {H}^{0} + e^{-}$ , with density functional calculations.", "Our calculation indicates that the electron emission process from H$^{-}$ in Si consists of two steps.", "The first step is H$^{-}$ migration from its most stable Td site to a metastable BC site.", "This migration is the rate-limiting step for the whole reaction with calculated activation energy of 0.96 eV.", "The second step is the electron emission from H$^{-}$ in the BC site to a conduction band.", "This step is much faster than the first step, having the calculated activation energy of 0.46 eV.", "The activation energy observed in a DLTS experiment, 0.84 eV, is likely to correspond to the one in the first step.", "Our study reveals that modern first-principles calculation can elucidate nonadiabatic process of hydrogen impurity, which is the most ubiquitous and mysterious impurity in semiconductors, and thus shows the possibility to clarify the degradation mechanism of semiconductor devices by hydrogen impurity from first principles.", "The calculations were performed on the supercomputers at the Institute for Solid State Physics, the University of Tokyo.", "This research was supported by MEXT as “Priority Issue on Post-K computer” (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage and use) and by JSPS KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Hydrogenomics”, No.", "JP18H05519.", "We receive further support from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO)." ] ]

[ [ "On the number of intersection points of lines and circles in $\\mathbb\n R^3$" ], [ "Abstract We consider the following question: Given $n$ lines and $n$ circles in $\\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families.", "We prove that if there are no $n^{1/2}$ curves (lines or circles) lying on an algebraic surface of degree at most two, then the number of these intersection points is $O(n^{3/2})$." ], [ "Introduction", "It is easy to see that $n$ distinct lines in $\\mathbb {R}^3$ can have a quadratic number of intersection points.", "For instance, consider $n$ lines in general position in the plane.", "Another example is the following.", "Consider $n$ lines lying on the surface of a hyperboloid with one sheet: $\\lfloor n/2 \\rfloor $ of these lines belong to one family of generators, $\\lceil n/2 \\rceil $ of these lines belong to another.", "Note that in this example hyperboloid with one sheet can be replaced by any regulus, that is, the surface spanned by all lines that meet three pairwise skew lines in $\\mathbb {R}^3$ .", "A non-trivial upper bound on the number of intersection points of $n$ lines is proven in the following theorem provided that there is no surface of small degree containing many lines.", "Theorem 1 (Guth–Katz [2]) Let $\\mathcal {L}$ be a collection of $n$ lines in $\\mathbb {R}^3$ .", "Let $A \\ge 100 n^{1/2}$ and suppose that there are at least $100An$ points incident to at least two lines of $\\mathcal {L}$ .", "Then there exists a plane or regulus $Z \\subset \\mathbb {R}^3$ that contains at least $A$ lines from $\\mathcal {L}$ .", "In [2] Theorem REF was applied to prove the lower bound in old difficult Erdős' problem about the number of distinct distances between points on the plane.", "A more general result was proven for curves of arbitrary degree in [3].", "Theorem 2 (Guth–Zahl [3]) Let $D > 0$ .", "Then there are constants $c_1, C_1, C_2>0$ so that the following holds.", "Let $k$ be a field and let $\\mathcal {L}$ be a collection of $n$ irreducible curves in $k^3$ of degree at most $D$ .", "Suppose that $\\operatorname{char}(k) = 0$ or $n \\le c_1 (\\operatorname{char}(k))^2$ .", "Then for each $A \\ge C_1n^{1/2}$ , either there are at most $C_2An$ points in $k^3$ incident to two or more curves from $\\mathcal {L}$ , or there is an irreducible surface $Z$ of degree at most $100D^2$ that contains at least $A$ curves from $\\mathcal {L}$ .", "In the current paper we prove an analogous result about the number of intersection points between lines and circles in $\\mathbb {R}^3$ claiming that an irreducible surface $Z$ in this case is either a plane or a hyperboloid with one sheet.", "For a collection $\\mathcal {L}$ of lines in $\\mathbb {R}^3$ and a collection $\\mathcal {C}$ of circles in $\\mathbb {R}^3$ , denote by $P(\\mathcal {L}, \\mathcal {C})$ the set of points lying on at least one line of $\\mathcal {L}$ and at least one circle of $\\mathcal {C}$ .", "Theorem 3 Let $\\mathcal {L}$ be a collection of $n$ lines and $\\mathcal {C}$ be a collection of $m$ circles in $\\mathbb {R}^3$ .", "Then for $A \\ge 10^5 \\min (n, m)^{1/2}$ , either $P(\\mathcal {L}, \\mathcal {P}) \\le 1000A (n + m)$ or there is a plane or a hyperboloid with one sheet containing at least $A$ curves of $\\mathcal {L} \\cup \\mathcal {C}$ .", "Remark It is known that a hyperboloid with one sheet in $\\mathbb {R}^3$ can be defined in some Cartesian coordinate system by an equation $ a x^2 + b y^2 - c z^2 = r $ , where $a, b, c, r > 0$ .", "If $c = 0$ and $a, b, r > 0$ , then this equation defines elliptic cylinder, and if $r = 0$ and $a, b, c > 0$ , then it defines elliptic cone.", "In the current paper, we consider elliptic cones and cylinders as hyperboloids with one sheet.", "Each of these surfaces contains infinite families of circles and lines in $\\mathbb {R}^3$ .", "In $ \\mathbb {C}^3 $ there are surfaces of degree four containing families of generating lines and circles; we refer a reader to the definition of complex circle given after Corollary REF .", "One of the examples is the surface defined by the equation $(x^2 + y^2 + z^2)^2 + (x + iy)^2 - z^2 = 0$ (see [5]).", "It can be parametrised in $ \\mathbb {CP}^3 $ as $ t^2 - 1 : i (t^2 - 1 - 2st) : s(t^2 + 1) : s(t^2 - 1) + 4t $ .", "This surface contains the family of lines $t = \\mbox{const} $ and the family of circles $ s = \\mbox{const} $ .", "Any $n$ lines and $m$ circles on this surface have $nm$ intersections.", "Thus Theorem REF is incorrect in $\\mathbb {C}^3$ .", "Our proof is based on two ideas.", "The first one is to find a surface of small degree containing all curves of our collections.", "This stage is similar to the idea developed by Guth and Katz in [2] to prove Theorem REF .", "The second one is to show that each irreducible component of the surface constructed in the previous step is ruled and has many circles on it lying in parallel planes.", "These properties of the irreducible components makes it possible to claim that each of them has degree at most two.", "In the similar way this idea is presented in the work [5] of Nilov and Skopenkov, where the authors proved that if through each point of a surface in $\\mathbb {R}^3$ one can draw both a straight line segment and a circular arc, then this surface is a part of algebraic surface of degree at most two.", "The paper is organised as follows.", "In Section we state auxiliary facts from algebraic geometry.", "Next, in Section we show Theorem REF using Proposition REF , which is proved in Section ." ], [ "Preliminaries", "Let $\\mathbb {CP}^3$ be the three-dimensional complex projective space with homogeneous coordinates $x : y : z : w$ .", "The infinitly distant plane is the plane defined by the equation $w = 0$ .", "We consider only algebraic surfaces in $\\mathbb {R}^3$ or $\\mathbb {CP}^3$ , and for this reason we usually omit word “algebraic”.", "From now on in this section, by $k^3$ we denote $\\mathbb {R}^3$ or $\\mathbb {CP}^3$ .", "A surface $Z \\in k^3$ is called ruled if every point $p \\in Z$ is incident to a line $\\ell \\in Z$ .", "A surface is doubly ruled if every its point of it is incident to two distinct lines contained in $ Z $ .", "It is well-known that if an algebraic surface in $ k^3 $ is doubly ruled, then it is a plane or a regulus.", "Ruled surfaces distinct from planes or reguli are called singly ruled.", "Our main tool to prove that some surface is ruled is the following theorem.", "Theorem 4 (Cayley–Salmon [7], Monge [4]) If $S \\subset \\mathbb {CP}^3$ is a surface of degree $d$ , which does not contain irreducible ruled components, then there is a surface $T$ of degree at most $11d - 24$ such that $S$ and $T$ do not have common irreducible components and each line of $S$ is contained in $T$ .", "Under the conditions of Theorem REF , the degree of the intersection curve of surfaces $S$ and $T$ does not exceed $d(11d - 24)$ .", "Therefore, we obtain the following corollary.", "Corollary 5 Let $S \\subset \\mathbb {CP}^3$ be a surface of degree $d$ , which does not contain irreducible ruled components.", "Then $S$ contains at most $d(11d - 24)$ lines.", "The absolute conic in $\\mathbb {CP}^3$ is given by the equations $x^2 + y^2 + z^2 = 0, w = 0$ .", "A complex circle is an irreducible conic in $\\mathbb {CP}^3$ having two distinct common points with the absolute conic.", "Clearly, a circle in $\\mathbb {R}^3$ is a subset of a complex circle.", "A line in $\\mathbb {CP}^3$ can be naturally defined in the Plücker coordinates $\\mathbb {CP}^5$ : The line passing through points $x_1 : y_1 : z_1 : w_1$ and $x_2 : y_2 : z_2 : w_2$ is identified with the point $x_1y_2 - x_2y_1 : x_1z_2 - x_2z_1 : x_1w_2 - x_2w_1 : y_1z_2 - y_2z_1 : y_1w_2 - y_2w_1 : z_1w_2 - z_2w_1\\in \\mathbb {CP}^5.$ Let $Z \\subset k^3$ be an irreducible singly ruled surface.", "A line $\\ell \\subset Z$ is called an exceptional line of $Z$ if every point of $\\ell $ is incident to another line contained in $Z$ .", "A point $p \\in Z$ is called an exceptional point of $Z$ if it is incident to infinitely many lines contained in $Z$ .", "An upper bound on the number of exceptional lines and points on a irreducible ruled surface in $\\mathbb {CP}^3$ is proved in the following proposition (see [2]).", "Proposition 6 If $Z \\subset \\mathbb {CP}^3$ is an irreducible ruled surface different from a plane and a regulus, then $Z$ contains at most two exceptional lines and at most one exceptional point.", "Lines that are not exceptional we call generators or generating lines.", "To describe all generating lines contained in ruled surface, we use the following theorem (see [1]).", "Theorem 7 If $Z \\subset \\mathbb {CP}^3$ be an irreducible ruled surface different from a plane and a regulus, then all generating lines of $Z$ can be parameterized by an irreducible algebraic curve in the Plücker space.", "We also use the following property of generating lines (see [6]).", "Lemma 8 If $Z \\subset \\mathbb {CP}^3$ be an irreducible ruled surface, then each algebraic curve on $Z$ , which is not a generating line, intersects all generating lines of $Z$ .", "Our main tool to show Theorem REF is the following proposition, which we prove in Section .", "Proposition 9 Let $\\mathcal {L}$ be a collection of $n$ lines and $\\mathcal {C}$ be a collection of $m$ circles in $\\mathbb {R}^3$ .", "Suppose that for some $A \\ge 10^5 \\min (n, m)^{1/2}$ , each curve of $\\mathcal {L} \\cup \\mathcal {C}$ contains at least $A$ points of $P(\\mathcal {L}, \\mathcal {C})$ .", "Then there are at most $500 \\min (n, m) / A$ planes and hyperboloids with one sheet containing all curves of $\\mathcal {L} \\cup \\mathcal {C}$ ." ], [ "Bounding $P(\\mathcal {L}, \\mathcal {C})$", "[Proof of Theorem REF ] Suppose that there are no $A$ curves of $\\mathcal {L} \\cup \\mathcal {C}$ lying on a plane or on a hyperboloid with one sheet.", "Let us show that $|P(\\mathcal {L}, \\mathcal {C})| \\le 1000 (n + m)A$ .", "If a curve $\\omega \\in \\mathcal {L} \\cup \\mathcal {C}$ contains at most $A$ points of $P(\\mathcal {L}, \\mathcal {C})$ , then $|P(\\mathcal {L}\\setminus \\lbrace \\omega \\rbrace , \\mathcal {C}\\setminus \\lbrace \\omega \\rbrace )| \\ge |P(\\mathcal {L}, \\mathcal {C})|- A$ .", "Removing one by one such curves $ \\omega _1, \\ldots , \\omega _k $ , one achieves the following scenario: Both collections $\\mathcal {L}^{\\prime } := \\mathcal {L} \\setminus \\lbrace \\omega _1, \\ldots , \\omega _k \\rbrace $ and $\\mathcal {C}^{\\prime } := \\mathcal {C} \\setminus \\lbrace \\omega _1, \\ldots , \\omega _k \\rbrace $ contain only curves with at least $A$ points of $P(\\mathcal {L}^{\\prime }, \\mathcal {C}^{\\prime })$ .", "It is enough to show that the theorem holds for these collections.", "Indeed, $P(\\mathcal {L}, \\mathcal {C}) \\le P(\\mathcal {L}^{\\prime }, \\mathcal {C}^{\\prime }) + kA \\le 1000A(|\\mathcal {L}^{\\prime }| + |\\mathcal {C}^{\\prime }|) + A (|\\mathcal {L}| + |\\mathcal {C}| - |\\mathcal {L}^{\\prime }| - |\\mathcal {C}^{\\prime }|) \\le 1000A(|\\mathcal {L}| + |\\mathcal {C}|).$ Thus without loss of generality, we can assume that each curve of $\\mathcal {L}\\cup \\mathcal {C}$ contains at least $A$ points of $P(\\mathcal {L}, \\mathcal {C})$ .", "By Proposition REF , there exist $k \\le 500 \\min (n, m) / A \\le A / 10$ planes and hyperboloids with one sheet such that their union contains all curves of $\\mathcal {L} \\cup \\mathcal {C}$ .", "Denote these surfaces by $Z_1, \\ldots , Z_k$ .", "Let $\\mathcal {L}_i$ and $\\mathcal {C}_i$ be the subcollections of $\\mathcal {L}$ and $\\mathcal {C}$ lying on $Z_i$ but not lying on $Z_j$ for all $j > i$ .", "Clearly $\\bigcup \\mathcal {L}_i = \\mathcal {L}$ and $\\bigcup \\mathcal {C}_i = \\mathcal {C}$ .", "Let $\\omega $ be any curve of $\\mathcal {L}_i\\cup \\mathcal {C}_i$ .", "By Bézout's theorem, $\\omega $ contains at most 4 points of intersection with $Z_j$ for $j > i$ .", "Thus for all $i < j$ we have $|P(\\mathcal {L}_i, \\mathcal {C}_j)| + |P(\\mathcal {L}_j, \\mathcal {C}_i)| \\le 4(|\\mathcal {L}_i| + |\\mathcal {C}_i|) \\le 4A$ .", "Since $|\\mathcal {L}_i| + |\\mathcal {C}_i| \\le A $ , we have $|P(\\mathcal {L}_i, \\mathcal {C}_i)| \\le 2 |\\mathcal {L}_i| \\cdot |\\mathcal {C}_i| \\le A^2$ .", "Hence $|P(\\mathcal {L}_i, \\mathcal {C}_i)| + \\sum _{j > i} (|P(\\mathcal {L}_i, \\mathcal {C}_j)| + |P(\\mathcal {L}_j, \\mathcal {C}_i)|) < A^2 + 4k(|\\mathcal {L}_i| + |\\mathcal {C}_i|) \\le A^2 + 4kA \\le 2A^2$ for all $i$ .", "Finally, we obtain $|P(\\mathcal {L}, \\mathcal {C})| \\le \\sum _i \\left(|P(\\mathcal {L}_i, \\mathcal {C}_i)| + \\sum _{j > i} \\left( |P(\\mathcal {L}_i, \\mathcal {C}_j)| + |P(\\mathcal {L}_j, \\mathcal {C}_i)| \\right) \\right) < 2kA^2 \\le 1000 \\min (n, m) A,$ which finishes the proof." ], [ "Covering $\\mathcal {L}$ and {{formula:055e2681-3426-4592-a3d7-0f618b59607e}} by a small number of quadrics", "First, we prove a weak version of Proposition REF .", "Proposition 10 Let $\\mathcal {L}$ be a collection of $n$ lines and $\\mathcal {C}$ be a collection of $m$ circles in $\\mathbb {R}^3$ .", "Suppose that for some $A \\ge 100 \\min (n, m)^{1/2}$ , each curve of $\\mathcal {L}$ and $\\mathcal {C}$ contains at least $A$ points of $P(\\mathcal {L}, \\mathcal {C})$ .", "Then there is an algebraic surface of degree at most $500 \\min (n, m) / A$ , which contains all curves of $\\mathcal {L}$ and $\\mathcal {C}$ ." ], [ "Constructing a surface", "First, we state and prove several simple observations that we need to show Proposition REF .", "Lemma 11 (Multiplicative Chernoff bound) Let $X_1, \\ldots , X_n$ be independent Bernoulli random variables with parameter $p$ .", "Then the following inequalities hold $\\mathbb {P}\\left(\\frac{1}{n}\\sum _{i=1}^n X_i \\ge (1 + \\delta )p\\right) \\le \\left( \\frac{e^\\delta }{(1 + \\delta )^{1 + \\delta }} \\right)^{np};$ $\\mathbb {P}\\left(\\frac{1}{n}\\sum _{i=1}^n X_i \\le (1 - \\delta )p\\right) \\le \\left( \\frac{e^{-\\delta }}{(1 - \\delta )^{1 - \\delta }} \\right)^{np}.$ Substituting $\\delta = 1$ in the first inequality and $\\delta = \\frac{1}{2}$ in the second one, we obtain the following corollary.", "Corollary 12 Let $X_1, \\ldots X_n$ be independent Bernoulli random variables with parameter $p$ .", "Then the following inequalities hold $\\mathbb {P}\\left(\\frac{1}{n}\\sum _{i=1}^n X_i \\ge 2p\\right) \\le \\exp \\left( - \\frac{1}{4} np \\right);$ $\\mathbb {P}\\left(\\frac{1}{n}\\sum _{i=1}^n X_i \\le \\frac{1}{2} p\\right) \\le \\exp \\left( -\\frac{1}{8} np \\right).$ A trivial upper bound on the minimal degree of the surface containing all curves of $ \\mathcal {L} \\cup \\mathcal {C} $ is shown in the following lemma.", "Lemma 13 If $\\mathcal {L}$ is a collection of $n$ lines in $ \\mathbb {R}^3 $ and $\\mathcal {C}$ be a collection of $m$ circles in $\\mathbb {R}^3$ , then there is a surface of degree at most $(12(n + m))^{1/2}$ containing all curves of $ \\mathcal {L} \\cup \\mathcal {C} $ .", "Set $d = \\left\\lfloor (12(n + m))^{1/2} \\right\\rfloor $ .", "Choose $2d + 1$ pairwise distinct points on each curve of $\\mathcal {L} \\cup \\mathcal {C}$ .", "In total there are $k = (n + m)(2d + 1)$ chosen points.", "Since $k < 2(n + m)(d + 1) = \\frac{1}{6} \\cdot 12(n + m) (d + 1) \\le \\frac{(d + 1)^3}{6} < \\binom{d + 3}{3},$ there is a surface $Z$ of degree $d$ containing all chosen points.", "By Bézout's theorem and the fact that each curve $ \\omega \\in \\mathcal {L} \\cup \\mathcal {C} $ intersects $Z$ at $2d + 1$ points, we obtain that $\\omega $ lies on $Z$ .", "Finally, we need the following trivial combinatorial lemma.", "Lemma 14 If $G$ is a graph without isolated vertices, then there is a matching $M$ between $V(G)$ and $E(G)$ of size at least $|V(G)| / 2$ such that for each pair $(v, e) \\in M$ vertex $v$ is endpoint of edge $e$ .", "Consider any connected component of $G$ and choose arbitrary vertex $v$ in this component.", "Let $d(u)$ be the distance from $v$ to $u$ in $G$ , that is, the minimal number of edges in the path between $u$ and $v$ .", "Let us match vertex $u \\ne v$ to an edge $uw \\in E(G)$ with property $d(u) = d(w) + 1$ .", "Obviously, all matched edges are different.", "In each component of size $n$ , there are at least $n - 1 \\ge n / 2$ matched vertices.", "[Proof of Proposition REF ] The proof of Proposition REF is by induction on $n + m$ .", "The base of induction $ n + m = 0 $ is trivial.", "The step of the induction can be done using the following lemma.", "Lemma 15 Let the conditions of Proposition REF hold.", "If $n \\le m$ , then there is a surface of degree at most $ D = \\lceil 200n / A \\rceil $ containing at least $ \\lceil \\frac{3n}{4} \\rceil $ lines; If $n \\ge m$ , then there is a surface of degree at most $ D = \\lceil 200m / A \\rceil $ containing at least $ \\lceil \\frac{3m}{4} \\rceil $ circles.", "Lemma REF is proved at the end of Subsection REF .", "By Lemma REF , there is a surface $ Z_0 $ of degree at most $ D = \\lceil 200\\min (n, m) / A \\rceil $ containing at least three quarters of curves of a smaller collection among $\\mathcal {L}$ and $\\mathcal {C}$ .", "Let $n_1 = |\\mathcal {L}_1|$ and $m_1 = |\\mathcal {C}_1|$ , where $\\mathcal {L}_1 \\subset \\mathcal {L}$ and $\\mathcal {C}_1 \\subset \\mathcal {C}$ are the collections of lines and circles respectively that are not contained in $Z_0$ .", "If $n < m$ , then $n_1 \\le n/4$ and $4\\min (n_1, m_1) \\le 4n_1 \\le n = \\min (n, m)$ .", "Similarly, the inequality $4\\min (n_1, m_1) \\le \\min (n, m)$ holds in the case $m \\ge n$ .", "Every line of $ \\mathcal {L}_1 $ (every circle of $ \\mathcal {C}_1 $ ) intersects at least $A - 2\\deg Z_0 \\ge A - 2D > 0$ circles of $ \\mathcal {C}_1 $ (lines of $ \\mathcal {L}_1 $ ).", "Thus there are two possible cases: $n_1 = m_1 = 0$ or $n_1 > 0$ and $m_1 > 0$ .", "In the first case, we find the surface of degree at most $\\lceil 200 \\min (n, m) /A \\rceil $ containing all curves of $\\mathcal {L} \\cup \\mathcal {C}$ .", "Suppose $n_1 > 0$ and $m_1 > 0$ .", "Each curve of $\\mathcal {L}_1 \\cup \\mathcal {C}_1$ has at least $A_1 = A - 2D$ points of $P(\\mathcal {L}_1, \\mathcal {C}_1)$ .", "Hence we get $|A_1| \\ge \\frac{1}{2}|A| \\ge 50 \\min (n, m)^{1/2} \\ge 50 (4 \\min (n_1, m_1))^{1/2} = 100 \\min (n_1, m_1)^{1/2}$ By the induction hypothesis, there is a surface of degree $500 \\min (n_1, m_1) /A_1$ containing all curves of $\\mathcal {L}_1$ and $\\mathcal {C}_1$ .", "The union of this surface and $Z_0$ is a surface of degree at most $\\left\\lceil 200 \\min (n, m) / A \\right\\rceil + 500 \\min (n_1, m_1) / A_1 \\le 500n/A$ containing all curves of $\\mathcal {L}$ and $\\mathcal {C}$ .", "[Proof of Lemma REF ] Suppose $n \\le m$ .", "The opposite case $n \\ge m$ is obtained from proof below by changing lines to circles and vice versa.", "Let $p = \\frac{D^2}{25n}$ .", "Clearly, $D \\ge 10$ and $p = \\frac{D^2}{25n} \\le \\frac{(200n + A)^2}{25n \\cdot A^2} \\le \\frac{(200n + 2n)^2}{25n \\cdot A^2} < 1$ .", "Let $\\mathcal {L}_0 \\subset \\mathcal {L}$ be a random subset chosen by picking every line independently with probability $p$ .", "Note that $\\mathbb {E}|\\mathcal {L}| = pn = \\frac{D^2}{25}$ .", "By Corollary REF , we obtain $\\mathbb {P}\\left(|\\mathcal {L}_0| > \\frac{2D^2}{25}\\right) = \\mathbb {P}\\left(|\\mathcal {L}_0| > 2pn\\right) \\le \\exp \\left( -\\frac{1}{4}pn \\right) = \\exp \\left( -\\frac{D^2}{100} \\right) < \\frac{1}{2}\\text{.", "}$ For each $q \\in P(\\mathcal {L}, \\mathcal {C})$ , let $X_q$ be the event $\\lbrace \\exists \\ell \\in \\mathcal {L}_0 : q \\in \\ell \\rbrace $ .", "Consider an arbitrary circle $\\gamma $ in $\\mathcal {C}$ .", "Set $Q_\\gamma = P(\\mathcal {L}, \\lbrace \\gamma \\rbrace )$ .", "By the premises of the proposition, $|Q_\\gamma | \\ge A$ .", "Let us match some points of $Q_\\gamma $ to distinct lines passing through these points in the following way.", "If a point $q \\in Q_\\gamma $ belongs to a line $\\ell \\in \\mathcal {L}$ with $|\\ell \\cap \\gamma | = 1$ , then $q$ is matched to $\\ell $ .", "Let $Q_\\gamma ^1$ be the set of points that are matched after previous operation and $Q_\\gamma ^2 = Q_\\gamma \\setminus Q_\\gamma ^1$ .", "Let $G$ be the graph such that $V(G) = Q_\\gamma ^2$ and $E(G)$ is a set of pairs of points of $Q_\\gamma ^2$ belonging to one line of $\\mathcal {L}$ .", "Since there are no isolated vertices in $G$ , by Lemma REF , distinct lines can be matched to at least half of the points of $Q_\\gamma ^2$ .", "Let $Q_\\gamma ^{\\prime }$ be the set of matched points of $Q_\\gamma $ .", "Clearly, $|Q_\\gamma ^{\\prime }| \\ge |Q_\\gamma ^1| + |Q_\\gamma ^2| / 2 \\ge A / 2$ .", "For each $q \\in Q_\\gamma ^{\\prime }$ , let $X^\\gamma _q$ be the event that the line corresponding to $q$ is contained in $\\mathcal {L}_0$ .", "Clearly, for each $q \\in Q_\\gamma ^{\\prime }$ we have $X^\\gamma _q \\subset X_q$ , and thus $\\mathbb {P}(X_q) \\ge \\mathbb {P}(X^\\gamma _q) = p$ .", "If $S_\\gamma = |P(\\mathcal {L}_0, \\lbrace \\gamma \\rbrace )|$ and $S_\\gamma ^{\\prime } = \\sum _{q \\in Q_\\gamma ^{\\prime }} \\mathbf {I}_{X^\\gamma _q}$ , then $S_\\gamma = \\sum _{q \\in Q_\\gamma } \\mathbf {I}_{X_q} \\ge \\sum _{q \\in Q_\\gamma ^{\\prime }} \\mathbf {I}_{X^\\gamma _q} = S_\\gamma ^{\\prime }$ and $\\mathbb {E}S_\\gamma ^{\\prime } =\\sum _{q \\in Q_\\gamma ^{\\prime }} \\mathbb {P}(X_q) \\ge \\sum _{q \\in Q_\\gamma ^{\\prime }} \\mathbb {P}(X^\\gamma _q) \\ge |Q_\\gamma ^{\\prime }| \\cdot p \\ge Ap / 2 \\text{.", "}$ Since events $X^\\gamma _q$ are independent, by Corollary REF , we obtain $\\mathbb {P}\\left(S_\\gamma \\le 2D\\right) \\le \\mathbb {P}\\left(S_\\gamma ^{\\prime } \\le 2D\\right) \\le \\mathbb {P}\\left(S_\\gamma ^{\\prime } \\le \\frac{AD}{100n} D \\right) =\\mathbb {P}\\left(S_\\gamma ^{\\prime } \\le \\frac{1}{4}Ap\\right) \\le \\exp \\left(-\\frac{1}{8} Ap \\right) =$ $= \\exp \\left(-\\frac{AD^2}{200n} \\right) \\le \\exp \\left( -\\frac{200n}{A} \\right) \\le e^{-100} \\le \\frac{1}{16}\\text{.", "}$ Consider any line $\\ell \\in \\mathcal {L}$ .", "Let $\\mathcal {C}_\\ell $ be a maximum size family of circles of $\\mathcal {C}$ such that it is possible to match all circles of $\\mathcal {C}_{\\ell }$ to different points of $P(\\lbrace \\ell \\rbrace , \\mathcal {C})$ .", "Similarly, using Lemma REF , we obtain $|\\mathcal {C}_l| \\ge A/2$ .", "Let $T_{\\ell }$ be the number of circles of $\\mathcal {C}_{\\ell }$ such that each of them contains at most $2D$ intersection points with lines of $\\mathcal {L}_0$ .", "We have $\\mathbb {E} T_{\\ell } = \\sum _{\\gamma \\in \\mathcal {C}_{\\ell }} \\mathbb {P}\\left(S_\\gamma \\le 2D\\right) \\le \\frac{|\\mathcal {C}_{\\ell }|}{16}\\text{.", "}$ By Markov's inequality and the inequality $|\\mathcal {C}_\\ell | \\ge A / 2 \\ge 1000n / A \\ge 4D$ , we get $\\mathbb {P}(T_{\\ell } \\ge |\\mathcal {C}_{\\ell }| - 2D) \\le \\frac{\\mathbb {E}T_{\\ell }}{|\\mathcal {C}_{\\ell }| - 2D} \\le \\frac{|\\mathcal {C}_{\\ell }|}{16(|\\mathcal {C}_{\\ell }| - 2D)} \\le \\frac{1}{8}\\text{.", "}$ Let $\\mathcal {L}^{\\prime }$ be the collection of lines $\\ell \\in \\mathcal {L}$ such that $T_{\\ell } \\ge |\\mathcal {C}_{\\ell }| - 2D$ .", "By Markov's inequality, we have $\\mathbb {P} \\left( |\\mathcal {L}^{\\prime }| > \\frac{n}{4} \\right) \\le \\frac{4\\mathbb {E}|\\mathcal {L}^{\\prime }|}{n} = \\frac{4}{n} \\sum \\limits _{\\ell \\in \\mathcal {L}} \\mathbb {P}(T_{\\ell } \\ge |\\mathcal {C}_{\\ell }| - 2D) \\le \\frac{1}{2}\\text{.", "}$ Therefore, there is $\\mathcal {L}_0 \\subset \\mathcal {L}$ such that $|\\mathcal {L}_0| \\le \\frac{2D^2}{25}$ and $|\\mathcal {L}^{\\prime }| \\le n / 4$ .", "Thus, by Lemma REF , there is a surface of degree at most $D$ containing all lines of $\\mathcal {L}_0$ .", "Let $Z_0$ be a surface of minimal degree containing every line of $\\mathcal {L}_0$ .", "By Bézout's theorem, every line or circle intersecting $Z_0$ in at least $2D + 1$ points lies on $Z_0$ .", "Since each line $\\ell $ of $\\mathcal {L} \\setminus \\mathcal {L}^{\\prime }$ has at least $2D + 1$ intersection points with circles that has at least $2D + 1$ intersections with $Z_0$ , the line $\\ell $ lies on $Z_0$ ." ], [ "Decomposition into quadrics", "In this section we show Proposition REF .", "From now on we assume that all polynomials have coefficients in $\\mathbb {R}$ and all surfaces are considered in $\\mathbb {CP}^3$ except otherwise is explicitly indicated.", "For a polynomial $Q \\in \\mathbb {R}[x, y, z]$ , let $Z(Q)$ be the surface in $\\mathbb {CP}^3$ defined by the equation $Q = 0$ .", "[Proof of Proposition REF ] Let $Q$ be the polynomial of minimum degree such that $Z(Q)$ contains all curves of $\\mathcal {L} \\cup \\mathcal {C}$ .", "Among all polynomials of minimum degree, we choose one with maximum number of irreducible over $\\mathbb {R}$ components.", "By Proposition REF , we have $d \\le 500 \\min (n, m) / A \\le A / 100$ .", "Let $Q_1, \\ldots , Q_k$ be the irreducible over $\\mathbb {R}$ components of $Q$ .", "Suppose that $Q_i$ is reducible over $\\mathbb {C}$ , that is, $Q_i = Q_{i, 1} \\cdot Q_{i, 2}$ , where $Q_{i, j} \\in \\mathbb {C}[x, y, z] $ .", "If we replace all coefficients in $Q_{i, j}$ be their real part, then we obtain polynomial $Q_{i, j}^{\\prime } \\in \\mathbb {R}[x, y, z]$ such that $\\deg Q_{i, j}^{\\prime } \\leqslant \\deg Q_{i, j}$ and $Q_{i, j}^{\\prime }$ contains all curves of $\\mathcal {L} \\cup \\mathcal {C}$ lying in $Q_{i, j}$ .", "Thus replacement $Q_i$ with $Q_{i, 1}^{\\prime } \\cdot Q_{i, 2}^{\\prime }$ decreases degree of the polynomial or increases the number of irreducible over $\\mathbb {R}$ components, a contradiction.", "Let $\\mathcal {L}_i \\subseteq \\mathcal {L}$ and $\\mathcal {C}_i \\subseteq \\mathcal {C}$ be subcollections of curves contained in $Z(Q_i)$ , but not contained in $Z(Q_j)$ for all $j \\ne i$ .", "Since $Q$ is the polynomial of minimum degree, $\\mathcal {L}_i \\cup \\mathcal {C}_i \\ne \\varnothing $ .", "Let $A_i$ be the maximum number $k$ such that every curve of $\\mathcal {L}_i \\cup \\mathcal {C}_i$ contains at least $k$ points of $P(\\mathcal {L}_i, \\mathcal {C}_i)$ .", "By Bézout's theorem, each curve of $\\mathcal {L}_i \\cup \\mathcal {C}_i$ has at most $2d$ intersection points with $\\bigcup _{j \\ne i} Z(Q_j)$ .", "Hence each curve of $\\mathcal {L}_i \\cup \\mathcal {C}_i$ contains at most $2d$ points of $P(\\mathcal {L}, \\mathcal {C}) \\setminus P(\\mathcal {L}_i, \\mathcal {C}_i)$ and $A_i \\ge A - 2d \\ge \\frac{9}{10}A$ .", "Denoting $n_i = |\\mathcal {L}_i|$ , $m_i = |\\mathcal {C}_i|$ and $d_i = \\deg Q_i$ , by Propositions REF , we have $d_i \\le \\frac{500 \\min (n_i, m_i)}{A_i} \\le \\frac{1000 \\min (n_i, m_i)}{A} \\le \\frac{1000 \\min (n_i, m_i)}{5000 (\\min (n, m))^{1/2}} \\le \\frac{(\\min (n_i, m_i))^{1/2}}{5}.$ Using inequality $11d_i^2 \\le \\frac{11}{25} \\min (n_i, m_i) < n_i$ and Corollary REF , we get that $Z(Q_i)$ is ruled in $\\mathbb {CP}^3$ .", "Suppose that $Z(Q_i)$ contains the absolute conic.", "By Lemma REF , each generating line passes through the absolute conic.", "Since the absolute conic does not contain real points and every line of $\\mathcal {L}_i$ is real, these lines do not intersect the absolute conic and thus are exceptional.", "By Proposition REF , the surface $Z(Q_i)$ contains at most two exceptional lines, but $n_i \\ge (5d_i)^{1/2} > 2$ .", "Hence $Z(Q_i)$ does not contain the absolute conic.", "Let $S_i$ be the set of intersection points of $Z(Q_i)$ and the absolute conic.", "By Bézout's theorem, the inequality $|S_i| \\le 2d_i$ holds.", "Each circle of $\\mathcal {C}_i$ is a subset of some complex circle in $\\mathbb {CP}^3$ .", "Each of these complex circle intersects the absolute conic in exactly two points, that is, contains two points of $S_i$ .", "Thus there are at least $\\frac{|\\mathcal {C}_i|}{\\binom{|S_i|}{2}} \\ge \\frac{2|\\mathcal {C}_i|}{|S_i|^2} \\ge \\frac{m_i}{2d_i^2} \\ge 5$ complex circles passing through the same pair of points of $S_j$ .", "Let $\\gamma _1, \\ldots , \\gamma _5 \\in \\mathcal {C}_i$ be the circles in $\\mathbb {R}^3$ corresponding to these complex circles.", "Since for any $i, j \\in \\lbrace 1, \\ldots , 5\\rbrace $ complex circles corresponding to $\\gamma _i$ and $\\gamma _j$ have the same intersection points with the infinite plane, $\\gamma _i$ and $\\gamma _j$ lie in the same plane or in parallel planes.", "Lemma REF applied to the irreducible ruled surface $Z(Q_i)$ implies that each generating line passes through each $\\gamma _i$ .", "Suppose that there are two circles lying in one plane $ P $ .", "For each point $ p $ of these circles, but, probably, the points of intersection, there is a line passing $ p $ and contained in $ P $ and in $ Z(Q_i) $ .", "Since infinitely many lines lies in the intersection of $ P $ and $ Z(Q_i) $ , plane $ P $ is contained in $ Z(Q_i) $ and hence $ Q_i $ is linear.", "From now on, we assume that $\\gamma _i$ lies in distinct real planes.", "Let $\\mathcal {X}$ be the family of polynomials of the form $h(x, y, z) = (x^2 + y^2) \\cdot p_1(z) + x \\cdot p_2(z) + y \\cdot p_3(z) + p_3(z) \\in \\mathbb {R}[x, y, z]$ , where $\\deg h \\le 4$ .", "This family is a vector space of dimension 16 and thus there is a polynomial of this family passing through any 15 points in the space.", "Assume that the planes containing circles $\\gamma _i$ are parallel to $Oxy$ .", "Choose arbitrary three points on each circle $\\gamma _i$ .", "There is a polynomial $Q_i^{\\prime } \\in \\mathcal {X}$ such that $Z(Q_i^{\\prime })$ passes through chosen points.", "Since the intersection of $Z(Q_i^{\\prime })$ with a plane parallel to $Oxy$ is a circle, a line or the empty set, this surface contains each of $\\gamma _i$ .", "Moreover, each generating line of $Z(Q_i)$ intersects $Z(Q_i^{\\prime })$ in at least five points and thus is contained in it.", "Therefore, $Q_i$ divides $Q_i^{\\prime }$ .", "If $Q_i^{\\prime }$ is irreducible over $\\mathbb {R}$ , then $Q_i \\in \\mathcal {X}$ .", "Suppose to the contrary that $Q_i^{\\prime } = f \\cdot g$ , where $f, g \\in \\mathbb {R}[x, y, z]$ and $\\deg f > 0, \\deg g > 0$ .", "Suppose that both $f$ and $g$ depends on variables $x$ or $y$ .", "In this case, these polynomials can be expressed in form $f(x, y, z) = x \\cdot u_1(z) + y \\cdot u_2(z) + u_3(z)$ and $g(x, y, z) = x \\cdot v_1(z) + y \\cdot v_2(z) + v_3(z)$ .", "From these representations follows that $u_1 v_1 = u_2 v_2$ and $u_1 v_2 + u_2 v_1 = 0$ .", "It means that $u_1 = u_2 = 0$ or $v_1 = v_2 = 0$ , a contradiction.", "If $Q_i^{\\prime }$ cannot be expressed as a product of polynomials, which depends on $x$ or $y$ , then each factor of $Q_i^{\\prime }$ lies in $\\mathcal {X}$ .", "Hence $Q_i \\in \\mathcal {X}$ .", "If $Q_i$ has degree at most two, then it is a plane or a hyperboloid with one sheet.", "Otherwise, the line $\\ell \\in \\mathbb {CP}^3$ defined by $z = w = 0$ lies on $Z(Q_i)$ , because $\\deg Q_i \\ge 3$ and degree on variables $ x $ and $ y $ in $ Q_i $ does not exceed two.", "Consider the following two cases.", "Case 1: $\\ell $ is exceptional.", "Since planes containing $\\ell $ are parallel to $Oxy$ , every line intersecting $\\ell $ is parallel to $Oxy$ .", "A line that is parallel to $Oxy$ cannot intersect all circles $\\gamma _i$ and thus it cannot be a generating line.", "But by Lemma REF , for each point of $\\ell $ there is a generating line passing through it, a contradiction.", "Case 2: $\\ell $ is generating.", "Let $I$ be the intersection of $\\ell $ and the absolute conic.", "By Theorem REF , the generating lines form an algebraic curve in Plücker space, and, therefore, there is a sequence of generating lines $\\ell _i$ converging to $\\ell $ .", "As in the Case 1, there are no generating lines passing through $\\ell $ , thus $\\ell _i \\cap I = \\varnothing $ .", "For each $j \\in \\lbrace 1, \\ldots , 5\\rbrace $ , the line $\\ell _i$ intersects $\\gamma _j$ at some point $P_i^j$ .", "Each of the five points $P_i^j$ converges to one of the two points of the set $I$ .", "We may assume that $P_i^1$ and $P_i^2$ converge to the same point $P \\in I$ .", "For $ j \\in \\lbrace 1, 2\\rbrace $ , the sequence of lines $PP^j_i$ converges to tangent line to $\\gamma _i$ at $P$ .", "Since $\\gamma _1$ and $\\gamma _2$ are not coplanar, these tangent lines are distinct and the plane $PP_i^1P_i^2$ converges to the projective plane $\\Omega $ containing them.", "The projective plane $\\Omega $ has a unique common point with $\\gamma _1$ , while $\\ell \\subset \\Omega $ intersects $\\gamma _1$ by the two-point set $I$ , a contradiction.", "Therefore, for each $i$ , the degree of $Q_i$ does not exceed two." ], [ "The author acknowledge the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926.", "The author is grateful to Alexandr Polyanskii for fruitful and inspirational discussions, to Alexey Balitskiy for numerous valuable comments that helped to significantly improve the presentation of the paper and to Mikhail Skopenkov for advices on references." ] ]

[ [ "MACER: A Modular Framework for Accelerated Compilation Error Repair" ], [ "Abstract Automated compilation error repair, the problem of suggesting fixes to buggy programs that fail to compile, has generated significant interest in recent years.", "Apart from being a tool of general convenience, automated code repair has significant pedagogical applications for novice programmers who find compiler error messages cryptic and unhelpful.", "Existing approaches largely solve this problem using a blackbox-application of a heavy-duty generative learning technique, such as sequence-to-sequence prediction (TRACER) or reinforcement learning (RLAssist).", "Although convenient, such black-box application of learning techniques makes existing approaches bulky in terms of training time, as well as inefficient at targeting specific error types.", "We present MACER, a novel technique for accelerated error repair based on a modular segregation of the repair process into repair identification and repair application.", "MACER uses powerful yet inexpensive discriminative learning techniques such as multi-label classifiers and rankers to first identify the type of repair required and then apply the suggested repair.", "Experiments indicate that the fine-grained approach adopted by MACER offers not only superior error correction, but also much faster training and prediction.", "On a benchmark dataset of 4K buggy programs collected from actual student submissions, MACER outperforms existing methods by 20% at suggesting fixes for popular errors that exactly match the fix desired by the student.", "MACER is also competitive or better than existing methods at all error types -- whether popular or rare.", "MACER offers a training time speedup of 2x over TRACER and 800x over RLAssist, and a test time speedup of 2-4x over both." ], [ "Introduction", "The ability to code is a staple requirement in science and engineering and programmers rely heavily on feedback from the programming environment, such as the compiler, linting tools, etc., to correct their programs.", "However, given the formal nature of these tools, it is difficult to master their effective use without extended periods of exposure.", "Thus, and especially for beginners, these tools can pose a pedagogical hurdle.", "This is particularly true of compiler error messages which, although always formally correct, can often be unhelpful in guiding the novice programmer on how to correct their error [14].", "This is sometimes due to the terse language used in error messages.", "For example, see Fig REF where the error message uses terms such as “specifier” and ”statement” which may be unfamiliar to a novice.", "At other times this is due to the compiler being unable to comprehend the intent of the user.", "For example, statements such as 0 = i; (where i is an integer variable) in the C programming language generate an error informing the programmer that the “expression is not assignable” (such as with the popular LLVM compiler [13]).", "The issue here is merely the direction of assignment but the compiler brings in concepts of expressions and assignability which may confuse a beginner.", "At still other times, there are compiler quirks, such as reporting a missing terminal semicolon as begin present on the next line.", "Several of these issues can be irritating, although not debilitating, to more experienced programmers as well.", "However, for novices, this frequently means seeking guidance from a human mentor who can then explain the program repair steps in more accessible terms.", "Given that educational institutions struggle to keep up with increasing student strengths, human mentorship is not a scalable solution [5].", "Consequently, automated program repair has generated a lot of interest in recent years due to its promising applications to programming education and training.", "A tool that can automatically take a program with compilation errors and suggest repairs to remove those errors can greatly facilitate programming instructors, apart from being a source of convenience for even seasoned programmers.", "In this work we report MACER, a tool for accelerated repair of programs that fail to compile.", "Figure: Two examples of actual repairs carried out by MACER.", "The erroneous line in the first example requires multiple replacements to repair the error.", "Specifically, two occurrences of ',' need to be replaced with ';' as indicated by the repair class description.", "The erroneous line the second example uses incorrect syntax to check for equality and requires replacing the '=' symbol with the '==' symbol." ], [ "Related Works", "The area of automated program repair has seen much interest in recent years.", "The DeepFix method [9] was one of the first major efforts at using deep learning techniques such as sequence-to-sequence models to locate as well as repair errors.", "The TRACER method [1] proposed segregating this pipeline into repair line localization and repair prediction and reported better repair performance.", "This work also introduced the use of the $\\mathsf {Pred} @\\mathsf {k} $ metric to compilation error repair which demands not just elimination of compilation errors but actually an exact match with the fix desired by the student.", "This was a much more stringent metric than the prevailing repair accuracy metric which simply counted reductions in compilation errors.", "Recent works have focused on several other aspects of this problem.", "The RLAssist method [8] introduced self learning techniques using reinforcement learning to eliminate the need for training data.", "However, the technique offers slow training times.", "The work of [10] proposes to use generative techniques using variational auto-encoders to introduce diversity in the fixes suggested by the technique.", "The work of [19] focuses on locating and repairing a special class of errors called variable-misuse errors which are logical errors where programmers use an inappropriate identifier, possibly due to confusion in identifier names.", "The TEGCER method [2] focuses not on error repair but rather repair demonstration by showing students, fixes made by other students on similar errors which can be argued to have greater pedagogical utility.", "The works DeepFix, RLAssist and TRACER most directly relate to our work and we will be comparing to all of them experimentally.", "MACER outperforms all these methods in terms of repair accuracy, exact match ($\\mathsf {Pred} @\\mathsf {k} $ ) accuracy, training and prediction time, or all of the above." ], [ "Our Contributions", "MACER makes the following key contributions to compilation error repair MACER sets up a modular pipeline that, in addition to locating lines that need repair, further segregates the repair pipeline by identifying what is the type of repair needed on each line (the repair-class of that line), and where in that line to apply that repair (the repair-profile of that line).", "This presents a significant departure from previous works like TRACER and DeepFix that rely on a heavy-duty generative mechanism to perform the last two operations (repair type identification and application) in a single step to directly generate repair suggestions.", "Although convenient, these generative mechanisms used in previous works come at a cost – not only are they expensive at training and prediction, but their one-step approach also makes it challenging to fine tune their method to focus more on certain types of errors than others.", "We show that MACER on the other hand, is able to specifically target certain error types.", "Specifically, MACER is able to pay individual attention to each repair class to offer superior error repair.", "MACER introduces techniques used in large-scale multi-class and multi-label learning tasks, such as hierarchical classification and reranking techniques, to the problem of program repair.", "To the best of our knowledge, these highly efficient and scalable techniques have hitherto not been applied to the problem of compilation error repair.", "MACER accurately predicts the repair class (see Tab REF ).", "Thus, instructors can manually rewrite helpful feedback (to accompany MACER's suggested repair) for popular repair classes which may offer greater pedagogical value.", "We present a highly optimized implementation of an end-to-end tool-chainThe MACER tool-chain is available at https://github.com/purushottamkar/macer/ for compilation error repair that effectively uses these scalable techniques.", "MACER's repair pipeline is end-to-end and entirely automated i.e.", "steps such as creation of repair classes can be replicated for any programming language for which static type inference is possible.", "The resulting implementation of MACER not only outperforms existing techniques on various metrics, but also offers training and prediction times that are several times to orders of magnitude faster than those of existing techniques." ], [ "Problem Setting and Data Preprocessing", "MACER learns error repair strategies given training data in the form of several pairs of programs, with one program in the pair failing to compile (called the source program) and the other program in the pair being free of compilation errors (called the target program).", "Such a supervised setting is standard in previous works in compilation error repair [1], [8].", "Similar to [1], we train only on those pairs where the two programs differ in a single line (e.g.", "in Fig REF , the programs differ only in line 3).", "However, we stress that MACER is able to perform repair on programs where multiple lines may require repairs as well, and we do include such datasets in our experiments.", "The differing line in the source (resp.", "target) program is called the source line (resp.", "target line).", "With every such program pair, we also receive the errorID and message generated by the Clang compiler [13] when compiling the source program.", "Tab REF lists a few errorIDs and error messages.", "It is clear from the table data that some error types are extremely rarely encountered whereas others are very common.", "Table: Some examples of the 148 compiler errorIDs listed in decreasing order of their frequency of occurrence in the data (reported in the Count column).", "It is clear that some error types are extremely frequent whereas other error types rarely occur in data.", "The symbol □\\Box is a placeholder for program specific tokens such as identifiers, reserved keywords, punctuation marks etc.", "For example, a specific instance of errorID E6 is shown in Figure .", "A specific instance of errorID E1 could be “Expected ; after expression”." ], [ "Notation", "We use angular brackets to represent n-grams.", "For example, the statement $a = b + c;$ contains the unigrams $\\langle $a$\\rangle $ , $\\langle $=$\\rangle $ , $\\langle $b$\\rangle $ , $\\langle $+$\\rangle $ , $\\langle $c$\\rangle $ and $\\langle $;$\\rangle $ , as well as contains the bigrams $\\langle $a =$\\rangle $ , $\\langle $= b$\\rangle $ , $\\langle $b +$\\rangle $ , $\\langle $+ c$\\rangle $ , $\\langle $c ;$\\rangle $ and $\\langle $; EOL$\\rangle $ .", "When encoding bigrams, we include an end-of-line character EOL as well.", "This helps MACER distinguish this location since several repairs (such as insertion of expression termination symbols) require edits at the end of the line." ], [ "Feature Encoding", "The source lines contain several user-defined literals and identifiers (variable names) which can be diverse but are not informative for error repair.", "To avoid overwhelming the machine learning method with these uninformative tokens, it is common in literature to reduce the input vocabulary size.", "MACER does this by replacing literals and identifiers with their corresponding abstract LLVM token type while retaining keywords and symbols.", "An exception is string literals where format-specifiers (such as For example, the raw or concrete statement int abc = 0; is converted into the abstract statement int VARIABLE_INT = LITERAL_INT ;.", "An attempt is made to infer the datatypes of identifiers (which is possible even though compilation fails on these programs since the compiler is often nevertheless able to generate a partial symbol table while attempting compilation).", "Undeclared/unrecognized identifiers are replaced with a generic token INVALID.", "Such abstraction is common in literature.", "DeepFix [9] replaces each program variable name with a generic identifier ID and removes the contents of string literals.", "We use the abstraction module described in TRACER [1] that retains type information, which is helpful in fixing type errors.", "For our data, this abstraction process yielded a vocabulary size of 161 tokens and 1930 unique bigrams.", "Both uni and bigrams were included in the representation since this feature representation will be used to predict the repair class of the line which involves predicting which tokens (i.e.", "unigrams) need to be replaced/deleted/inserted, as well as be used to predict the repair profile of the line which involves predicting bigrams as locations.", "Thus, having uni and bigrams natively in the representation eased the task of these classifiers.", "Including trigrams in the representation did not offer significant improvements but increased training and prediction times.", "MACER represents each source line as a 2239 dimensional binary vector.", "The first 148 dimensions in this representation store a one-hot encoding of the compiler errorID generated on that source line (see Tab REF for examples).", "The next 161 dimensions store a one-hot unigram feature encoding of the source line and the remaining 1930 dimensions store a one-hot bigram feature encoding of the abstracted source line.", "We used one-hot encodings rather than TF-IDF encodings since the additional frequency information for uni and bigrams did not offer any predictive advantage.", "We also found the use of trigrams to not significantly increase performance but make the method slower.", "It is important to note that the feature creation step does not use the target line in any manner.", "This is crucial to allow feature creation for test examples as well.", "Table: Some examples of the 1016 repair classes used by MACER listed in decreasing order of their frequency of occurrence (frequencies reported in the column Count).", "For example, Class C2 concerns the use of undeclared identifiers and the solution is to replace the undeclared identifier (INVALID token) with an integer variable or literal.", "A ∅\\emptyset indicates that no token need be inserted/deleted for that class.", "For example, no token need be inserted to perform repair for repair class C22 whereas no token need be deleted to perform repair for repair class C115.", "See the text for a description of the notation used in the second column.Figure: Repair classes generated by MACER arranged in descending order of the number of training programs associated with them.", "Only the 500 most popular classes are shown.", "The classes exhibit heavy-tailed behavior: less than 400 of the 1016 classes have 3 or more training data points associated with them.", "On the other hand, the top 10 classes have more than 200 training points each." ], [ "Repair Class Creation", "The repair class of a source line encodes what repair to apply to that line.", "As noted in Table REF , the Clang compiler offers 148 distinct errorIDs.", "However, diverse repair strategies may be required to handle all instances of any given errorID.", "For example, errorID E6 in Fig REF can of course signal missing semicolons within the header of a for loop as the example indicates, but it can also be used by the compiler to signal missing semicolons ; at the end of a do-while block, as well as missing colons : in a switch case block.", "To consider the above possibilities, similar to TEGCER [2], we first expand the set of 148 compiler-generated errorIDs into a much bigger set of 1016 repair classes.", "It is notable that these repair classes are generated automatically from training data and do not require any manual supervision.", "For each training instance, token abstraction (see Sec REF ) is done on both the source and target lines and a diff is taken between the two.", "This gives us the set of tokens that must be deleted from the (abstracted) source line, as well as those that must be inserted into the source line, in order to obtain the (abstracted) target line.", "A tuple is then created consisting of the compiler errorID for that source line, followed by an enumeration of tokens that must be deleted (in order of their occurrence in the source line from left to right), followed by an enumeration of tokens that must be inserted (in order of their insertion point in the source line from left to right).", "Such a tuple of the form [ErrID [redTOK$^-_1$ TOK$^-_2$ ...] [mygreenTOK$^+_1$ TOK$^+_2$ ...]] is called a repair class.", "We identified 1016 such classes.", "A repair class requiring no insertions (resp.", "deletions) is called a Delete (resp.", "Insert) repair class.", "A repair class requiring as many insertions as deletions with insertions at exactly the locations of the deletions is called a Replace repair class.", "Tab REF illustrates a few repair classes.", "Repair classes exhibit a heavy tail (see Fig REF ) with popular classes having hundreds of training points whereas the vast majority of (rare) repair classes have merely single digit training instances." ], [ "Repair Profile Creation", "The repair profile of a source line encodes where in that line to apply the repair encoded in its repair class.", "For every source line, taking the diff of the abstracted source and abstracted target lines (as done in Sec REF ) also tells us which bigrams in the abstracted source line require some edit operation (insert/delete/replace) in order to obtain the abstracted target line.", "The repair profile for a training pair stores the identity of these bigrams which require modification for that source line.", "A one-hot representation of the set of these bigrams i.e.", "a binary vector $\\in {0,1}^{1930}$ is taken to be the repair profile of that source line.", "We note that the repair profile is a sparse fixed-dimensional binary vector (that does not depend on the number of tokens in the source line) and ignores repetition information.", "Thus, even if a bigram requires multiple edit operations, or even if a bigram appears several times in the source line and only one of those occurrences requires an edit, we record a 1 in the repair profile corresponding to that bigram.", "This was done in order to simplify prediction of the repair profile for erroneous programs at testing time." ], [ "Working Dataset", "After the steps in Sec REF , REF , and REF have been carried out, we have with us, corresponding to every source-target pair in the training dataset, a class-label $y^i \\in [1016]$ telling us the repair class for that source line, a feature representation $^i \\in {0,1}^{2239}$ that tells us the errorID, and the uni/bigram representation of the source line, and a sparse Boolean vector $^i \\in {0,1}^{1930}$ that tells us the repair profile.", "Altogether, this constitutes a dataset of the form ${(^i,y^i,^i)}_{i=1}^n$ ." ], [ "MACER (Modular Accelerated Compilation Error Repair) segregates the repair process into six distinct steps Repair Lines: Locate within the source code, which line(s) are erroneous and require repair.", "Feature Encoding: For each of the identified lines, perform code abstraction and obtain a 2239-dimensional feature vector (see Sec REF ).", "Repair Class Prediction: Use the feature vector to predict which of the 1016 repair classes is applicable i.e.", "which type of repair is required.", "Repair Localization: Use the feature vector to predict locations within the source lines at which repairs should be applied.", "Repair Application: Apply the predicted repairs at the predicted locations Repair Concretization: Undo code abstraction and compile.", "Although previous works do incorporate some of the above steps, e.g., TRACER incorporates code abstraction and locating repair lines within the source code, MACER departs most notably from previous approaches in segregating the subsequent repair process into repair class prediction, localization, and application steps.", "Among other things such as greater training and prediction speed, this allows MACER to learn a customized repair location and repair application strategy for different repair classes which can be beneficial.", "For instance, if it is known that the repair required is the insertion of a semi-colon, then the location where the repair must be performed is narrowed down significantly.", "In contrast, existing methods expect a generative mechanism such as sequence-to-sequence prediction or reinforcement learning, to jointly perform all these tasks.", "This precludes any opportunity to exploit the type of repair required to perform better on specific repair types, apart from making these techniques slow at training and prediction.", "Below we detail the working of each of the above steps." ], [ "Repair Lines", "One of the key tasks of a compiler is to report line numbers where an error was encountered.", "However, this does not necessarily correspond to the location where the repair must be performed.", "In our training data set where errors are localized to a single line, the repair line location was the same as the compiler reported line-number in only about 80% of the cases.", "Existing works have used different techniques for repair line localization.", "RLAssist [8] use reinforcement learning to perform localization by navigating the program using movement based actions to maximize a predefined reward.", "DeepFix [9] trains a dedicated neural-network to identify suspicious tokens (and hence their location) to achieve around 86% repair line localization accuracy.", "TRACER [1] relies on compiler reported line numbers and considers two additional lines, one above and one below the compiler error line, and obtains a localization accuracy of around 87%.", "MACER uses this same technique which gave a repair line localization recall of around 90% on our training dataset.", "Figure: The prediction hierarchy used by MACER to predict the repair class." ], [ "Repair Class Prediction", "As outlined in Sec , MACER considers 1016 repair classes which is quite large.", "In order to make fast and accurate predictions for the correct repair class that apply to a given source line, MACER uses hierarchical classification techniques that are popular in the domain of large-scale multi-class and multi-label classification problems [12], [17].", "As there exists a natural hierarchy in our problem setting, we found it suitable (as suggested by [12]) to use a fixed hierarchy rather than a learnt hierarchy.", "Given that a significant fraction of repair classes (around 40%) involve replacement repairs, we found it advantageous to first segregate source lines that require replacement repairs from others.", "The classification hierarchy used by MACER is shown in Fig REF .", "The root node decides whether a source line requires a replacement or some other form of repair using a feed-forward network with 2 hidden layers with 128 nodes each and trained on cross entropy loss.", "All other internal nodes use a linear one-vs-rest classifier trained on cross entropy loss to perform their respective multi-way splits.", "It is well-known [11] that discriminative classifiers struggle to do well on rare classes due to paucity of data.", "Our repair classes do exhibit significant heavy-tailed behavior (see Fig REF ) with most classes occurring infrequently and only a few being popular.", "To improve MACER's performance, we first augment the classification tree into a ranking tree that ranks classes instead of just predicting one class, and then introduce a reranking step which modifies the ranking given by the ranking tree." ], [ "Repair Class Ranking", "We converted the classification tree into a probabilistic ranking tree that could assign a likelihood score to each repair class.", "More specifically, given the feature representation of a source line $\\in {0,1}^{2239}$ , the tree is used to assign, for every repair class $c \\in [1016]$ , a likelihood score $s^\\mathsf {tree} _c() := ¶{y = c }$ .", "We followed a process similar to ([12], [17]) to obtain these likelihood scores from the tree.", "This construction is routine and detailed in the appendix Sec .", "Although these scores $s^\\mathsf {tree} _c()$ can themselves be used to rank the classes, doing so does not yield the best results.", "This is due to the large number of extremely rare repair classes (Tab REF shows that only $\\approx $ 150 of the 1016 repair classes have more than 10 training examples).", "Table: Performance benefits of reranking.", "The table shows the performance accuracy (in terms of various ranking metrics) achieved by MACER in predicting the correct repair class.", "The first three columns report 𝖳𝗈𝗉@k\\mathsf {Top} @k i.e.", "the fraction of test examples on which the correct errorID or correct repair tokens were predicted within the top kk locations of the ranking.", "The last column reports the mean-average precision i.e.", "the average reciprocal rank at which the correct repair class was predicted in terms of tokens to be inserted or deleted.", "Note that in all cases, reranking significantly boosts the performance.", "In particular, the last column indicates that reranking ensures that the correct tokens to be inserted/deleted were almost always predicted within the first two ranks." ], [ "Repair Class Reranking", "To improve classification performance on rare repair classes, MACER uses prototype classifiers [11], [16] that have been found to be effective and scalable when dealing with a large number of rare classes.", "Suppose a repair class $c \\in [1016]$ is associated with $n_c$ training points.", "The k-means algorithm is used to obtain $k_c = {\\frac{n_c}{25}}$ clusters out of these $n_c$ training points and the centroids of these clusters, say $\\tilde{}_c^1,\\ldots ,\\tilde{}_c^{k_c}$ , are taken as prototypes for this repair class.", "This is repeated for all repair classes.", "At test time, given a source line $\\in {0,1}^{2239}$ , these prototypes are used to assign a new score to each repair class as follows $s^\\mathsf {prot} _c() := \\max _{k \\in [k_c]}\\ \\exp {-\\frac{1}{2}{- \\tilde{}_c^k}_2^2}$ Thus, for each repair class, the source line searches for the closest prototype of that repair class and uses it to generate a score.", "MACER uses the scores assigned by the probabilistic ranking tree and those assigned by the prototypes to get a combined score as $s_c() = 0.8\\cdot s^\\mathsf {tree} _c() + 0.2\\cdot s^\\mathsf {prot} _c()$ (the constants 0.8, 0.2 are standard in literature and we did not tune them).", "MACER uses this combined score $s_c()$ to rank repair classes in decreasing order of their applicability.", "Tab REF outlines how the reranking step significantly boosts MACER's ability to accurately predict the relevant compiler errorID and the repair class.", "Sec  will present additional ablation studies that demonstrate how the reranking step boosts not just the repair class prediction accuracy, but MACER's error repair performance as well." ], [ "Repair Localization", "Having predicted the repair class, MACER proceeds to locate regions within the source line where those repairs must be made.", "MACER reformulates this a problem of predicting which bigram(s) within the source line require edits (multiple locations may require edits on the same line).", "Note that this is exactly the same as predicting the repair profile vector of that source line (apart from any ambiguity due to the same bigram appearing multiple times in the line which is handled during repair application).", "This observation turns repair localization into a multi-label learning problem, that of predicting the sparse Boolean repair profile vector $$ using the source feature representation $$ and the predicted repair class $\\hat{y}$ as inputs.", "Each of the 1930 bigrams in our vocabulary, now turns into a potential “label” which if turned on, indicates that repair is required at bigrams of that type.", "Fig REF explains the process pictorially.", "MACER adopts the “one-vs-rest” (OVR) approach that is a state-of-the-art in large-scale multi-label classification [4].", "Thus, 1930 binary classifiers are trained (using standard implementations of decision trees), each one predicting whether that particular bigram needs repair or not.", "At test time, classifiers corresponding to all bigrams present in the source line are queried as to whether the corresponding bigrams require repair or not.", "MACER trains a separate OVR multi-label classifier per repair class that is trained only on training points of that class (as opposed to having a single OVR classifier handle examples of all repair classes).", "Training repair-class specific localizers improved performance since the kind of bigrams that require edits for replacement repairs e.g.", "substituting = with ==, are very different from the kind of bigrams that require edits for insertion repairs e.g.", "inserting a semicolon ;.", "At test time, after predicting the repair class for a source line, MACER invokes the OVR classifier of the predicted repair class to perform repair localization.", "During repair localization, we only invoke OVR classifiers corresponding to bigrams that are actually present in the source line.", "Thus, we are guaranteed that any bigrams predicted to require edits will always be present in the source line.", "In our experiments, we found this strategy to work well with MACER offering a Hamming loss of just 1.43 in terms of predicting the repair profile as a Boolean vector.", "Thus, on an average, only about one bigram was either predicted to require repair when it did not, or not predicted to require repair when it actually did." ], [ "Repair Application", "The above two steps provide MACER with information on what repairs need to be applied as well as where they need to be applied.", "Frugal but effective techniques are then used to apply the repairs which we discuss in this subsection.", "Let $$ denote the ordered set of all bigrams (and their locations, ordered from left to right) in the source line which were flagged by the repair localizer as requiring edits.", "For example, if the repair localizer predicts the bigram $\\langle $, VARIABLE_INT$\\rangle $ to require edits and this bigram appears twice in the source line (note that this would indeed happen in the example in Fig REF ), then both those bigrams would be included in $$ .", "This is repeated for all bigrams flagged by the repair localizer.", "Below we discuss the repair application strategy for various repair class types." ], [ "Insertion Repairs", "Recall that these are repairs where no token needs to be deleted from the source line but one or more tokens need to be inserted.", "We observed that in an overwhelmingly large number of situations that require multiple tokens to be inserted, all tokens need to be inserted at the same location, for instance the repair for(i=0;i<5) $\\rightarrow $ for(i=0;i<5;i++) has the repair class [E6 [red$\\emptyset $] [mygreen; VARIABLE_INT ++]] and requires three tokens, a semicolon ;, an integer variable identifier, and the increment operator ++ to be inserted, all at the same location i.e.", "within the bigram $\\langle $5 )$\\rangle $ which is abstracted as $\\langle $LITERAL_INT )$\\rangle $ .", "Thus, for insert repairs, MACER concatenates all tokens marked for insertion in the predicted repair class and attempts insertion of this ensemble into all bigrams in the set $$ .", "Attempting insertion into a single bigram itself requires 3 attempts since each bigram offers 3 positions for insertion within itself.", "After each attempt, MACER concretizes the resulting program (see below) and attempts to compile it.", "MACER stops at a successful compilation and keeps trying otherwise." ], [ "Deletion Repairs", "Recall that these are repairs where no token needs to be inserted into the source line but one or more tokens need to be deleted.", "In this case, MACER scans the list of tokens marked for deletion in the predicted repair class from right to left.", "For every such token, the first bigram in the ordered set $$ (also scanned from right to left) that has that token, gets edited by deleting that token.", "Once all tokens in the repair class are exhausted, MACER concretizes the resulting program (see below) and attempts to compile it." ], [ "Replace Repairs", "Recall that these are repairs where an insertion and a deletion, both happen at the same location and this process may be required multiple times.", "In such cases, MACER scans the list of tokens marked for deletion in the predicted repair class from right to left and also considers the corresponding token marked for insertion.", "Let this pair be (TOK$^-$ , TOK$^+$ ).", "As in the deletion repair case, the first bigram in the ordered set $$ (also scanned from right to left) that contains TOK$^-$ , gets edited by deleting TOK$^-$ from that bigram and inserting TOK$^+$ in its place.", "Once all tokens in the repair class are exhausted, MACER concretizes the resulting program (see below) and attempts to compile it." ], [ "Miscellaneous Repairs", "In the most general case, an unequal number of tokens may need to be inserted and deleted from the source line, that too possibly at varied locations.", "Handling all cases in this situation separately is unwieldy and thus, MACER adopts a generic approach which works well in a large number of cases.", "First, MACER ignores the insertion tokens in the repair class and performs edits as if the repair class were a deletion type class.", "Subsequently, it considers the insertion tokens (all deletion tokens having been considered by now) and processes the resulting edited line as if it were an insertion type class." ], [ "Repair Concretization", "The above repair process generates a line that still contains abstract LLVM tokens such as LITERAL_INT.", "In order to make the program compilable, these abstract tokens are replaced with concrete program tokens such as literals and identifiers through an approximate process reverses the abstraction.", "To do this, we replace each abstract token with the most recently used concrete variable/literal of the same type, that already exists in the current scope.", "This is an approximate process since the repair application could suggest the insertion of a particular type of variable, which does not exist in the current scope.", "For example, if the repair application stage suggests the insertion of a variable of type Variable_Float, then at least one floating point variable should be declared in the same scope as the erroneous line.", "Nevertheless, we observe that this concretization strategy of MACER is able to recover the correct replacement in 90+% of the instances in our datasets.", "MACER considers each candidate repair line reported by the repair line localizer (recall that these include compiler-reported lines as well as lines immediately above and below those lines).", "For each such candidate repair line, MACER applies its predicted repair (including concretization) and then compiles the resulting program.", "If the number of compilation errors in the program reduce, then this repair is accepted and the process is repeated for the remaining candidate repair lines.", "Table: Comparison between TRACER and MACER on the single-line and multi-line test datasets.", "MACER achieves similar 𝖯𝗋𝖾𝖽@𝗄\\mathsf {Pred} @\\mathsf {k} and repair accuracy as TRACER on the single-line dataset.", "On multi-line dataset, where programs require repairs on multiple different lines, MACER achieves 14% improvement over TRACER.Table: Comparison of all methods on the DeepFix dataset.", "Values take from * ^{*} and † ^{\\dagger }.", "MACER offers the highest repair accuracy on this dataset.", "The nearest competitor is TRACER that is 12.5% behind.", "MACER offers a prediction time that is 4×4\\times faster than TRACER and 2×2\\times faster than the rest, and a train time that is 2×2\\times faster than TRACER and more than 800×800\\times faster than RLAssist.Figure: The two figures compare MACER and TRACER on the head repair classes (top 60 in terms of popularity with 35+ training points in each class) and torso repair classes (top 60-120 with 15+ training points in each class).", "To avoid clutter, only 30 classes from each category are shown.", "MACER has a substantial lead over TRACER on head classes with an average of 20% higher prediction hit rate (i.e.", "predicting the exact repair as desired by the student).", "On torso classes, MACER continues to dominate albeit with a smaller margin.", "On rare classes, the two methods are competitive." ], [ "Experiments", "We compared MACER's performance against previous works, as well as performed ablation studies to study the relative contribution of its components.", "All MACER implementationsThe MACER tool-chain is available at https://github.com/purushottamkar/macer/ were done using standard machine learning libraries such as sklearn [15] and keras [6].", "Experiments were performed on a system with Intel(R) Core(TM) i7-4770 CPU @ 3.40GHz $\\times $ 8 CPU having 32 GB RAM." ], [ "Datasets", "We report MACER accuracy on three different datasets.", "All these datasets were curated from the same 2015-2016 fall semester course offering of CS-1 course at IIT-Kanpur (a large public university) where 400+ students attempted more than 40 different programming assignments.", "The dataset was recorded using Prutor [7], an online IDE.", "The DeepFix datasethttps://www.cse.iitk.ac.in/users/karkare/prutor/prutor-deepfix-09-12-2017.zip contains 6,971 programs that fail to compile, each containing between 75 and 450 tokens [9].", "The single-line (17,669 train program pairs + 4,578 test program pairs) and multi-line (17,451 test program pairs) datasetshttps://github.com/umairzahmed/tracer released by [1] contain program pairs where error-repair is required, respectively, on a single line or multiple lines ." ], [ "Metrics", "We report our results on two metrics i) repair-accuracy, the popular metric widely adopted by repair tools, and ii) $\\mathsf {Pred} @\\mathsf {k} $ , a metric introduced by TRACER  [1].", "Repair accuracy denotes the the fraction of test programs that were successfully repaired by a tool i.e.", "all compilation errors were removed, thereby producing a correct program devoid of any compilation errors.", "On the other hand, $\\mathsf {Pred} @\\mathsf {k} $ metric captures the fraction of test programs where at least one of the top $\\mathsf {k} $ abstract repair suggestions (since MACER and other competing algorithms are capable of offering multiple suggestions for repair in a ranked list) exactly matched the student's own abstract repair.", "The choice of the $\\mathsf {Pred} @\\mathsf {k} $ metric is motivated by the fact that the goal of program repair is not to generate any program that merely compiles.", "This is especially true of repair tools designed for pedagogical settings.", "rated by the tool is exactly same as the student generated one, at the abstraction level.", "The purpose of this metric is further motivated in the Sec REF ." ], [ "Training Details", "Our training is divided into two parts, learning models to perform i) Repair Class prediction, and ii) Repair Location prediction.", "For repair class prediction we followed the prediction hierarchy shown in Figure  REF .", "The root node uses a feed forward neural net with two hidden layers of 128 nodes each.", "We tried {1,2,3} hidden layers with each layer containing nodes varying in {128,256,512}, and the structure currently used by us (with 2 hidden layers of 128 nodes each) was found to be the best.", "For re-ranking the repair classes, we created prototype(s) of each class depending on the size of class using KMeans clustering.", "The second part of training is Repair Location Prediction.", "We followed one-vs-rest (OVR) approach and performed binary classification for each of the 1930 bigrams, each binary classification telling us whether the corresponding bigram is worthy of edits or not.", "We recall that these OVR classifiers were trained separately for all repair classes to allow greater flexibility.", "The binary classification was performed using standard implementations of decision trees that use Gini impurity to ensure node purity.", "Decision trees were chosen due to their speed of prediction and relatively high accuracy." ], [ "A Naive Baseline and Importance of $\\mathsf {Pred} @\\mathsf {k} $", "We consider a naive method Kali' that simply deletes all lines where the compiler reported an error.", "This is inspired by Kali [18], an erstwhile state-of-art semantic-repair tool that repaired programs by functionality deletion alone.", "This naive baseline Kali' gets 48% repair accuracy on the DeepFix dataset whereas DeepFix  [9], TRACER  [1] and MACER get respectively 27%, 44% and 56% (Tab REF ).", "Although Kali' seems to offer better repair accuracy than TRACER, its $\\mathsf {Pred} @1$ accuracy on the single-line dataset is just 4%, compared to 59.6% and 59.7% by TRACER and MACER respectively (Tab REF ).", "This demonstrates the weakness of reporting on repair accuracy metric in isolation, and motivates the usage of additional complex metrics such as $\\mathsf {Pred} @\\mathsf {k} $ , to better capture the efficacy of repair tools.", "Table: Performance of MACER on several sample test instances.", "Pred?", "= Yes if MACER's top suggestion exactly matched the student's abstracted fix, else Pred?", "= No is recorded.", "Rep?", "= Yes if MACER's top suggestion removed all compilation errors else Rep?", "= Yes is recorded.", "ZS?", "records whether the example was a “zero-shot” test example where MACER had never seen the corresponding repair class in training data.", "On the first three examples, MACER not only offers successful compilation, but offers a repair that exactly matches that desired by the student.", "Note that the second example involves an undeclared identifier.", "In the next two examples, although MACER does not offer the exact match desired by the student, it nevertheless offers sane fixes that eliminate all compilation errors.", "In the fifth example, MACER errs on the side of caution and inserts a matching parenthesis rather than risk eliminating an unmatched parenthesis.", "The last two are zero-shot examples.", "Although MACER could handle one of the zero-shot cases gracefully, it could not handle the other case.", "Obtaining better performance on zero shot repair classes is a valuable piece of future work for MACER.Figure: A graph comparing the prediction and repair hit rate of MACER on the top 120 most popular repair classes.", "To avoid clutter, every 4 class is shown.", "For these popular classes – which still may have as low as 15 training data points – MACER frequently achieves perfect or near perfect score in terms of prediction accuracy or repair accuracy or both.Figure: A study similar to the one presented in Fig  but with respect to prediction (exact match) accuracy instead of repair accuracy.", "A study of the prediction accuracy offered by MACER on repair classes with at least 3 training data points – a total of 391 such classes were there.", "On a majority of these classes 221/391 = 56%, MACER offers greater than 90% prediction accuracy.", "On a much bigger majority 287/391 = 73% of these classes, MACER offers more than 50% prediction accuracy.", "The second graph indicates that MACER's prediction accuracy drops below 50% only on classes which have less than around 30 points.Figure: A study of the repair accuracy offered by MACER on repair classes with at least 3 training data points – a total of 391 such classes were there.", "On a majority of these classes 267/391 = 68%, MACER offers greater than 90% repair accuracy.", "On a much bigger majority 327/391 = 84% of these classes, MACER offers more than 50% repair accuracy.", "The second graph indicates that MACER's repair accuracy drops below 50% only on classes which have less than around 30 points.", "This indicates that MACER is very effective at utilizing even small amounts of training data.Table: An ablation study on the differential contributions of MACER's components.", "ZS stands for “zero-shot”.", "For the “ZS included” column all test points are considered while reporting accuracies.", "For the “ZS excluded” column, only those test points are considered whose repair class was observed at least once in the training data.", "RR stands for Reranking.", "RCP stands for Repair Class Prediction, RLP stands for Repair Location Prediction.", "RCP = P (resp.", "RLP = P) implies that we used the repair class (resp.", "repair location) predicted by MACER.", "RCP = G (resp.", "RLP = G) implies that we used the true repair class (resp.", "true repair profile vector).", "It is evident from the difference in the results of the first two rows that (whether we include ZS or not), reranking gives 10-12% boost in both prediction accuracy.", "This highlights the importance of reranking in the presence of rare classes.", "Similarly, it can be seen that predicting the repair class (resp location) correctly accounts for 5-12% (resp.", "6%) of the performance.", "The final row shows that MACER loses 6-8% performance owing to improper repair application/concretization.", "In the last two rows, 𝖯𝗋𝖾𝖽@1\\mathsf {Pred} @1 is higher than 𝖱𝖾𝗉@1\\mathsf {Rep} @1 (1-2% cases) owing to concretization failures – even though the predicted repair matched the student's repair in abstracted form, the program failed to compile after abstraction was removed." ], [ "Breakup of Training Time", "Of the total 7 minute train time (see Tab REF ), MACER took less than 5 seconds to create repair classes and repair profiles from the raw dataset.", "The rest of the training time was taken up more or less evenly by repair class prediction training (tree ranking + reranking) and repair profile prediction training." ], [ "Comparisons with other methods", "The values for $\\mathsf {Pred} @k$ (resp.", "$\\mathsf {Rep} @k$ ) were obtained by considering the top $k$ repairs suggested by a method and declaring success if any one of them matched the student repair (resp.", "removed compilation errors).", "For $\\mathsf {Pred} @\\mathsf {k} $ computations, all methods were given the true repair line and did not have to perform repair line localization.", "For $\\mathsf {Rep} @\\mathsf {k} $ computations, all methods had to localize then repair.", "Tabs REF and REF compare MACER with competitor methods.", "MACER offers superior repair performance at much lesser training and prediction costs.", "Fig REF shows that MACER outperforms TRACER by $\\approx $ 20% on popular classes while being competitive or better on others." ], [ "Ablation studies with ", "To better understand the strengths and limitations of MACER, we report on further experiments.", "Fig REF shows that for top 120 most popular repair classes (which still may have as low as 15 training data points), MACER frequently achieves perfect or near perfect score in terms of prediction accuracy or repair accuracy or both.", "Figs REF and REF shows that MACER is effective at utilizing even small amounts of training data and that its prediction accuracy drops below 50% only on repair classes which have less than 30 examples in the training set.", "Tab REF offers examples of actual repairs by MACER.", "Although it performs favorably on repair classes seen during training, it often fails on zero-shot repair classes which were never seen during training.", "Tab REF presents an explicit ablation study analyzing the differential contributions of MACER's individual components on the single-line dataset.", "Re-ranking gives 10-12% boost to both $\\mathsf {Pred} @\\mathsf {k} $ and repair accuracy.", "Predicting the repair class (resp.", "profile) correctly accounts for 5-12% (resp.", "6%) of the performance.", "MACER loses a mere 6% accuracy on account of improper repair application.", "For all figures and tables, details are provided in the captions." ], [ "Conclusion", "In this paper we presented MACER, a novel technique to accelerated compilation error-repair.", "A key contribution of MACER is a fine-grained segregation of the error repair process into efficiently solvable ranking and labelling problems.", "These reductions are novel in this problem area where most existing techniques prefer to directly apply a single powerful generative learning technique instead.", "MACER offers significant advantages over existing techniques namely superior error repair accuracy on various error classes and increased training and prediction speed.", "Targeting rare error classes and “zero-shot” cases (Tab REF ) is an important area of future improvement.", "A recent large scale user-study [3] demonstrated that students who received automated repair feedback from TRACER  [1] resolved their compilation errors faster on average, as opposed to human tutored students; with the performance gain increasing with error complexity.", "We plan to conduct a similar systematic user study in the future, to better understand the correlation between the $\\mathsf {Pred} @\\mathsf {k} $ metric scores and error resolution efficiency (performance) of students." ], [ "Acknowledgments", "The authors thank the reviewers for helpful comments and are grateful to Pawan Kumar for support with benchmarking experiments.", "P. K. thanks Microsoft Research India and Tower Research for research grants." ], [ "Hierarchical Repair Class Ranking", "Let $T$ denote the tree in Fig REF with root $r(T)$ .", "The leaves of $T$ correspond to individual repair classes.", "The set of leaf nodes of a subtree rooted at any node $t$ will be denoted by $L(t)$ .", "The set of children of a node $t$ will be denoted by $C(t)$ and the parent of node $t$ will be denoted by $P(t)$ .", "The set of nodes on the path from the root $r(T)$ to any leaf $l \\in L(r(T))$ is denoted by $W(l)$ .", "Note that each leaf $l$ corresponds to a repair class $c_l \\in [1016]$ .", "For any node $t$ , let the indicator random variable $V_t$ indicate if we visited node $t$ .", "Then using the chain rule, we can express $¶{y = c } = ¶{V_{l_c} = 1 } = \\prod _{t \\in W(l_c)}¶{V_t = 1 , V_{P(t)} = 1}$ The correctness of the above can be deduced from the fact that $V_t = 1$ implies $V_{P(t)} = 1$ .", "Given that all nodes in our tree train their classifiers probabilistically using the cross entropy loss, we are readily able to, for every internal node $tn$ and its, say $k$ children $c_1,\\ldots ,c_k$ , assign the probability $¶{V_{c_i} = 1 , V_t = 1}$ using the sigmoidal activation (for binary split at the root), or the softmax activation (for all other multi-way splits).", "This allows us to compute the score $\\hat{s}_c() := ¶{y = c }$ for any repair class $c$ by just traversing the tree from the root node to the leaf node corresponding to the repair class $c$ ." ] ]

[ [ "Ultrafast Electron Cooling in an Expanding Ultracold Plasma" ], [ "Abstract Plasma dynamics critically depends on density and temperature, thus well-controlled experimental realizations are essential benchmarks for theoretical models.", "The formation of an ultracold plasma can be triggered by ionizing a tunable number of atoms in a micrometer-sized volume of a Bose-Einstein condensate (BEC) by a single femtosecond laser pulse.", "The large density combined with the extremely low temperature of the BEC give rise to an initially strongly coupled plasma in a so far unexplored regime bridging ultracold neutral plasma and ionized nanoclusters.", "Here, we report on ultrafast cooling of electrons, trapped on orbital trajectories in the long-range Coulomb potential of the dense ionic core, with a cooling rate of 400 K/ps.", "Furthermore, our experimental setup grants direct access to the electron temperature that relaxes from 5250 K to below 10 K in less than 500 ns." ], [ "Introduction", "Ultrashort laser pulses provide pathways for manipulating and controlling atomic quantum gases on femtosecond time-scales.", "In particular the strong light-field of a femtosecond laser pulse is able to instantaneously ionize a controlled number of atoms in a Bose-Einstein condensate (BEC).", "Above a critical number of charged particles, the attractive ionic Coulomb potential is large enough to trap a fraction of the photoelectrons, thus forming an ultracold plasma [1].", "Well-controlled ultracold plasmas in the laboratory provide benchmarks for multi-scale theories and can shed light on extreme conditions present in inertial confinement fusion [2], the core of Jovian planets and white dwarfs [3].", "As depicted in Fig.", "REF , the plasma density inherited from the BEC surpasses the densities achieved so far in supersonic expansion [4] and magneto-optical traps (MOTs) [5], [1], [6], [7] by orders of magnitude.", "Strongly-coupled plasmas where the Coulomb energy exceeds the thermal energy are of particular interest because the charge carriers develop spatial correlations [8] and self-assembled ordered structures.", "A recent work [9] approaches this regime by laser cooling of the ions.", "In small clusters ionized by ultrashort laser pulses, strongly coupled plasmas can be realized as well.", "In such systems, the interplay between interparticle Coulomb energies and molecular bonds is essential to understand energy transfer between electrons and ions [10], [11], [12] (see Fig.", "REF ).", "Recent experiments have studied charged particle dynamics at solid-state densities in finite-size nanoclusters [13], [14], [15] and observed the emergence of low-energy electrons [16].", "Photoionization of a BEC with a femtosecond laser pulse enables access to an unexplored plasma regime with high charge carrier densities above $10^{20}$  m$^{-3}$ , cold ion temperatures below 40 mK and hot electron temperatures above 5000 K. Density $\\rho $ and temperature $T$ entirely determine not only the coupling parameter but also the dominating length- and time-scales in a plasma.", "Compared to macroscopic ultracold neutral plasmas (UNP) at MOT densities, these initial parameters allow for creating a micrometer-sized plasma with large charge imbalance and high plasma frequencies where the Coulomb energies initially exceed the ionic thermal energies by three orders of magnitude.", "Such an initially strongly coupled microplasma with a few hundred to thousands of particles bridges the dynamics and energy transfer studied in photoionized nanoclusters and ultracold neutral plasma.", "Moreover, this plasma regime allows neglecting three-body-recombination and interatomic binding energies in the theoretical description, which are relevant in UNP or ionized nanoclusters, respectively.", "This considerably simplifies theoretical models of the dynamics for benchmark comparisons.", "Here, we report on the dynamics of ultracold microplasmas triggered in a $^{87}$ Rb BEC by a femtosecond laser pulse.", "Our experimental setup grants access to the electronic kinetic energy distribution with meV resolution by combining state-of-the-art techniques of ultrashort laser pulses and ultracold atomic gases.", "So far, the electron temperature of ultracold plasmas has only been inferred indirectly by comparing the fraction of spilled electrons in an extraction field [17], the free plasma expansion [18] or the three-body recombination rate [19] to theoretical models.", "We directly observe an electron cooling from 5250 K to below 10 K in less than 500 ns, which is in excellent agreement with charged particle tracing (CPT) simulations that we have performed in parallel.", "The small number of particles involved in our microplasma is a key feature that allows for an accurate comparison between experimental results and simulations.", "In addition, the dynamics investigated here reveals striking effects that cannot be captured by an hydrodynamic description such as the ultrafast electron cooling and the increasing electron coupling parameter approaching unity [6].", "Such a laboratory experiment, which grants access to additional, microscopic observables, allows testing the validity of macroscopic models, thus leading to a better understanding of similar systems in nature.", "Figure: Number density and temperature diagram of plasmas.", "Plasmas occurring in nature or prepared in the laboratory span several orders of magnitude in size, temperature and number density.", "The majority of naturally occurring plasmas are weakly coupled (Γ i <1\\Gamma _\\mathrm {i} < 1), however, the intriguing regime of strongly coupled plasmas (Γ i >1\\Gamma _\\mathrm {i} > 1) is realized in astronomical objects like Jupiter's core or white dwarfs.", "The dynamics in this challenging region can experimentally be approached by ultracold neutral plasmas, ionized nanoclusters and ultracold microplasmas, where the latter investigated in this work (highlighted by a white circle) bridges the length- and time-scales of the former two." ], [ "Ultracold Microplasma", "We experimentally investigate the dynamics of an ultracold microplasma by combining ultracold quantum gases with the ultrashort timescales of femtosecond laser pulses.", "As shown in Fig.", "REF a, a $^{87}$ Rb BEC is locally ionized by a single laser pulse at 511 nm wavelength with a full width at half maximum (FWHM) duration of 215$^{+20} _{-15}$  fs.", "Whereas earlier photoionization studies in $^{87}$ Rb BECs applied nanosecond laser pulses [20], here, the pulse duration is significantly shorter than the timescale for the electron dynamics given by the inverse electron plasma frequency $\\omega _\\mathrm {p,e}^{-1} = \\sqrt{\\frac{m_{\\mathrm {e}} \\epsilon _0}{\\rho _{\\mathrm {e}} e^2}} = 1.3\\text{ ps},$ where $m_\\mathrm {e}$ is the electron mass and $\\rho _\\mathrm {e}$ is the initial electronic density.", "Therefore, the initial plasma dynamics is not perturbed by the laser pulse and the creation of the charged particles can be considered as instantaneous.", "A high-resolution objective with a numerical aperture of 0.5 focuses the femtosecond laser pulse down to a waist of $w_0$ $\\approx $ 1 µm leading to peak intensities up to $2 \\times 10^{13}$  $\\mathrm {W}\\,\\text{cm}^{-2}$ (see Methods).", "The number of ionized atoms can be tuned from a few hundred to $N_\\mathrm {e,i}$ $\\approx $  4000 in a controlled manner via the pulse intensity.", "At the highest intensity, the ionization probability reaches unity within the center region of a cylindrical volume depicted in Fig.", "REF b [21].", "The radius of 1.35 µm is determined by the laser focus, while the height of 5 µm is limited by the atomic target, thus providing a locally ionized volume within the atomic cloud (see supplementary note 1).", "As depicted in Fig.", "REF c, the photoionization of $^{87}$ Rb at 511 nm can be described as a non-resonant two-photon-process.", "The excess energy of 0.68 eV is almost entirely transferred to the lighter photoelectrons.", "Due to the low initial ionic temperature of $T_\\mathrm {i}$ $\\approx $ 33 mK dominated by the photoionization recoil, we attain a remarkably high initial ionic coupling parameter of $\\Gamma _\\mathrm {i} = 4800$ , which compares the Coulomb energy to the thermal energy per particle: $\\Gamma _\\mathrm {e,i} = \\frac{e^2}{4 \\pi \\epsilon _0 a_\\mathrm {e,i} k_\\mathrm {B} T_\\mathrm {e,i}}.$ Here, $T_\\mathrm {e,i}$ describes the electron/ion temperature determined by the mean kinetic energy per particle and $a_\\mathrm {e,i} = \\left( \\frac{3}{4 \\pi \\rho _\\mathrm {e,i}} \\right)^{1/3}$ denotes the Wigner-Seitz radius at the electron/ion density $\\rho _\\mathrm {e,i}$ .", "As both the interparticle distance and the kinetic energy of the ions increase during plasma evolution, $\\Gamma _\\mathrm {i}$ decreases rapidly (see supplementary note 7).", "As a key feature, this experimental setup grants access to the atomic density via absorption imaging as well as the energy distribution of the photoionization products (Fig.", "REF a).", "A tunable electric field separates electrons and ions and directs them onto opposite imaging microchannel plates (MCP) (see Methods).", "Using CPT simulations, the spatial distribution on the detector can be assigned to an electronic kinetic energy distribution in a quantitative manner (see Methods).", "The simulated detector images for different kinetic energy at $\\pm U_\\mathrm {ext} = 300$  V are depicted in Fig.", "REF d: photoelectrons resulting from two-photon ionization (0.68 eV) or from above-threshold ionization (ATI) processes (3.1 eV) [22], [23] can be clearly identified.", "Figure REF a-c shows the averaged electron signals measured for increasing laser pulse peak intensities.", "For low intensities (a) the dominant structure on the detector is the spatial distribution of the electrons emerging from the non-resonant two-photon ionization process with a kinetic energy of 0.68 eV, corresponding to an initial electron temperature of $T_\\mathrm {e}\\approx 5250$  K, in excellent agreement with the trajectory simulation results (Fig.", "REF d).", "For the highest intensity shown (c), a second class of electrons appears stemming from the three-photon ATI (compare to Fig.", "REF d).", "As a central result, in (b) and (c) an narrow peak appears, corresponding to electrons having a very small kinetic energy.", "At these intensities, the number of photoionized atoms exceeds the critical number of ions $N^* \\approx 960$ required for plasma formation at our excess energies and a fraction of the photoelectrons is trapped and cooled in the resulting space charge potential generated by the unpaired ions.", "Experimentally, the threshold intensity depends on the density of the atomic target, which gives a clear evidence of a critical number of ions required for plasma formation (see supplementary note 6).", "This rules out low energetic electrons directly created in the strong-field ionization process as reported at high Keldysh parameters [24] or speculated in alkali atoms at high intensities [22], [25].", "Figure REF d shows radially averaged electron distribution in the depicted circular sector in Fig.", "REF a.", "The vertical lines mark the limit of the distributions obtained from the trajectory simulations for 0.01 eV and 0.68 eV depicted in Fig.", "REF d. At the lowest laser intensity, the kinetic energy distribution is flat up to the energy corresponding to two-photon ionization.", "At the higher intensity, a large fraction of cold plasma electrons is concentrated below the 10 meV line, which corresponds to a temperature lower than 77 K. An increasing number of generated ions deepens the space charge potential, which not only enables trapping more electrons in the plasma, but also significantly decelerates the escaping electrons.", "This can be clearly seen in the averaged spectrum (Fig.", "REF d) but also as a decrease of the area of the kinetic energy distribution in Fig.", "REF a-c.", "The plasma dynamics is reproduced by CPT plasma simulations including the mutual Coulomb-interactions between all charged particles (see Methods).", "Figure REF e-f shows the simulated results with electron/ion numbers of $N_\\mathrm {i,e} =$ 500 (e) and 4000 (f) with an initial electron energy of 0.68 eV, which are in excellent agreement with the measured kinetic energy distributions in Fig.", "REF a-b.", "In the simulations slow plasma electrons emerge only above the critical charge carrier density required for plasma formation.", "The extraction field sets the expansion time towards the detectors and, thus, the velocity resolution, which can be tuned from 10 meV at $\\pm U_\\mathrm {ext} = 300$  V to the 1 meV level at $\\pm U_\\mathrm {ext} = 5$  V (corresponding to static electric fields of 162 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ and 4.6 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ in the center, respectively).", "Figure REF g shows the spatial extent of the low-energy plasma electrons as characteristic elliptical structure for an extraction field of 5 V. The trajectory simulation results for different initial energies are depicted in Fig.", "REF i.", "Comparison of Fig.", "REF g and Fig.", "REF i yields a measured kinetic energy of the plasma electrons of approximately 1 meV, which corresponds to a final electron temperature below 10 K. Figure REF h shows the plasma simulation result for $N_\\mathrm {i,e}$ = 4000.", "Even in this experimentally challenging regime, the measurements and the plasma simulation almost perfectly agree.", "Beyond the excellent agreement with the measured kinetic distributions, these CPT simulations grant access to the dynamics of each particle.", "They reveal two cooling mechanisms occurring on distinct timescales: an ultrafast cooling during the plasma formation (picosecond timescale) and a subsequent process driven by the Coulomb expansion of the ionic cloud (nanosecond timescale).", "Figure REF a-e shows snapshots of the CPT simulations for an ultracold microplasma consisting of a few thousand charged particles.", "While UNP are realized at low excess energies (typically below 0.13 eV) [5], we are able to create an ultracold plasma at high excess energies (0.68 eV), corresponding to an initial electron temperature of $T_\\mathrm {e}\\approx 5250$  K. Therefore, the majority of photoelectrons leaves the ionization volume within a few picoseconds, while the ions can be regarded as static (Fig.", "REF a).", "The decrease of the electronic density in the plasma reduces the amount of shielding between the ions, which thus gain potential energy.", "In addition, this charge separation process gives rise to a space charge potential that strongly decelerates the escaping electrons (Fig.", "REF b).", "As a result, the kinetic energy of the electronic component is converted to potential energy of the ions.", "Whereas half of the electrons entirely escape the ionic core (escaping electrons), the other half is trapped within the evoked space charge potential (plasma electrons, see supplementary note 4 for definition).", "Figure REF f displays the evolution of the mean total electron (light blue line) and ion energy (red line) per particle determined by the sum of kinetic and potential energy of each component.", "Additionally, the kinetic energy of the plasma electrons (dark blue line) is shown.", "Within the electronic expansion, 50% of the plasma electrons' kinetic energy is transferred onto the potential energy of the ionic component within the first 7 ps.", "On this timescale, the trapped electrons are cooled down from 5250 K to about 2500 K, yielding an ultrafast cooling rate for electrons of approximately 400 $\\mathrm {K}\\,\\mathrm {ps}^{-1}$ .", "Disorder-induced heating of the electrons is negligible as the associated temperature of $T_\\mathrm {DIH} \\approx 70~K$ in this density regime is exceeded by orders of magnitude by the initial electron temperature [9].", "The large charge imbalance of our plasma strongly influences the many-body dynamics.", "As depicted in Fig.", "REF c, the electrons are trapped in orbital trajectories within the Coulomb potential of a quasi-static ionic core.", "This leads to an oscillatory exchange of energy between the captured electrons and the ions (see Fig.", "REF f).", "The period of $2 \\pi / \\omega _\\mathrm {p,e} \\approx 8$  ps of these oscillations is consistent with the inverse of the initial electron plasma frequency given in Eq.", "REF .", "This energy transfer between the individual electrons and the ions is predominantly in phase during the initial electron expansion but it dephases over time leading to a damping behavior (see supplementary note 4).", "In contrast to UNP, here, the ionic plasma period always exceeds the evolution time, preventing ionic thermalization.", "On a nanosecond timescale, the potential energy stored in the ions gradually translates into kinetic energy leading to a Coulomb explosion of the plasma (Fig.", "REF d,e).", "Whereas UNP typically exhibit hydrodynamic expansion after equilibration due to the electrons thermal pressure [26], in this work, the positively charged plasma expansion is dominated by the Coulomb pressure of the charge imbalance, yielding an asymptotic expansion velocity of the root mean square (rms) ion radius of 418 $\\mathrm {m}\\,\\mathrm {s}^{-1}$ (see supplementary note 4).", "This is in reasonable agreement with the expected hydrodynamic expansion velocity $v_\\mathrm {hyd} = \\sqrt{k_\\mathrm {B} (T_\\mathrm {e,0} + T_\\mathrm {i,0})/m_\\mathrm {i}} = 710$  $\\mathrm {m}\\,\\mathrm {s}^{-1}$ for the initial electron/ion temperature $T_\\mathrm {e/i,0}$  [6].", "The ionic expansion leads to a further reduction of the electronic temperature down to $T_\\mathrm {e}$ $\\approx $ 100 K within tens of nanoseconds.", "The simulations reveal an increasing electron coupling parameter towards $\\Gamma _\\mathrm {e} = 0.3$ approaching significant coupling (see supplementary note 7).", "The evolution of the mean kinetic energy of the plasma electrons as well as the depth of the effective space charge potential during the plasma expansion are shown in Fig.", "REF g for CPT simulations for different extraction fields (see supplementary note 5).", "In addition, the expected electron kinetic energy progression given by adiabatic cooling during the plasma expansion $E_\\mathrm {kin,e}(t) = E_\\mathrm {kin,e}(0) \\left(1+ t^2/\\tau _\\mathrm {exp}^2\\right)$ is shown (dotted blue line) [6].", "Here, $\\tau _\\mathrm {exp} = \\sqrt{m_\\mathrm {i} \\sigma ^2 / \\left[ k_\\mathrm {B}\\left(T_\\mathrm {e,0} + T_\\mathrm {i,0}\\right) \\right]}$ denotes the plasma expansion time and $\\sigma $ is given by the initial rms ion radius.", "The observed electron cooling rates during the plasma expansion largely follow the prediction by the hydrodynamic model.", "However, the initial ultrafast electron cooling is not captured by this model.", "The decrease of the ionic density over time lowers the binding Coulomb potential to the point where its gradient is exceeded by the extraction field [27].", "At this point, the plasma electrons are escaping the space charge potential and are drawn to the detector.", "Hence, the final electron temperatures can be controlled by the extraction field, which determines the plasma lifetime and thus the duration of the electron cooling process.", "Without extraction field, electrons can be cooled down to sub meV energies in less than 1 µs.", "We determine the lifetime of the microplasma experimentally by implementing a gated detection scheme.", "In order to separate the slow plasma electrons from the fast escaping electrons, a short repulsive voltage pulse is applied onto the electron extraction electrode after a certain time $t_\\mathrm {delay}$ to repel electrons that have not yet passed the electrode (see supplementary note 9).", "The resulting accumulated electron signals for different time delays are depicted in Fig.", "REF a.", "The escaping electrons arrive on the detector within the first 200 ns and display a homogeneous signal as expected for photoelectrons at 0.68 eV kinetic energies at these low extraction fields.", "Up to 600 ns, only a small fraction of cold plasma electrons is detected leaking out of the plasma during expansion.", "In the last 400 ns the fraction of plasma electrons significantly increases as the expanded plasma is torn apart by the extraction field.", "For a quantitative analysis, Fig.", "REF b shows the corresponding spatially-integrated electron counts for different extraction voltages.", "One can clearly distinguish two plateaus that can be associated to the escaping electrons arriving first and the plasma electrons delayed by several hundreds of nanosecond [5].", "The arrival time distributions obtained from plasma simulations show excellent agreement with respect to their temporal profile (see Fig.", "REF b - solid lines, vertically scaled with one free parameter).", "Figure REF c shows the electron arrival rates obtained from the time derivative of a double-sigmoid fit to the data in Fig.", "REF b (see supplementary note 9).", "The plasma lifetime corresponds to the time delay between the arrival of escaping and plasma electrons.", "The measured lifetimes are plotted for different extraction fields in Fig.", "REF d. For the lowest extraction field we achieve a lifetime of 498(46) ns, which is in excellent agreement with the calculated vanishing time of the space charge potential in Fig.", "REF g." ], [ "Discussion", "In summary, we have experimentally realized an ultracold plasma in a charge carrier density regime unexplored so far, which supports an initial ion coupling parameter of $\\Gamma _\\mathrm {i}$  = 4800.", "We have directly measured electron cooling from 5250 K to below 10 K within 500 ns, arising from the unique combination of a high charge carrier density and a micrometer-sized ionization volume.", "The CPT simulation results are in excellent agreement with the measurements and reveal an ultrafast energy transfer of 50% of the initial electron energy onto the ionic component within the first plasma period yielding an ultrafast initial electron cooling rate of $\\approx $  400 $\\mathrm {K}\\,\\mathrm {ps}^{-1}$ .", "The high degree of experimental control over the initial state, together with the almost perfect agreement of the CPT simulations, provides a unique model system to investigate the validity of statistical plasma models.", "Besides the fundamental interest in the plasma dynamics, the low electron temperatures and enormous cooling rates may be used in plasma-based ultracold electron sources [28] (see supplementary note 11).", "The generated low-emittance electron bunches can be utilized for seeding high-brilliance particle accelerators [29] and coherent imaging of biological systems [30].", "Our system links contemporary source designs utilizing MOTs [31], [32] where the electron-cathode interactions are negligible and traditional state-of-the-art electron sources [33], [34], [35] where the emittance is fundamentally limited by such interactions.", "Our density regime indeed promotes an electron cooling mechanism based on their interaction with the ionic core acting as photoemission cathode.", "By exploiting the toolbox of ultrafast dynamics further, our experimental setup allows investigating more advanced dynamical schemes.", "The impact of the plasma geometry can be studied by taking advantage of the non-linearity of the strong-field ionization process in order to shape the ionization volume beyond Gaussian distributions.", "The interaction between several microplasmas, launched simultaneously within a BEC, can also be explored.", "Moreover, pump-probe schemes utilizing a second synchronized terahertz pulse for controlling the plasma evolution can offer direct experimental access to the ultrafast dynamics of the microplasma.", "The photoionization of a BEC instead of a magneto-optically trapped gas paves the way towards plasmas supporting strong ion as well as electron coupling, mimicking conditions in gas planets [3], confinement fusion [2] or even more exotic systems like quark-gluon-plasma [7].", "Indeed, strong coupling can be reached by tuning the wavelength of the femtosecond laser close to the ionization threshold, thus reducing the initial kinetic energy of the electrons by orders of magnitude.", "Below the ionization threshold, the large spectral bandwidth of the femtosecond laser pulses shall prevent Rydberg blockade effects and enable an efficient excitation of a dense ultracold Rydberg gas, which can form strongly-coupled plasma by avalanche ionization [36], [4], [37].", "Finally, disorder-induced heating as limiting process for Coulomb coupling, can be suppressed by loading the condensate into a 3D optical lattice that establishes an initial spatial correlation [38]." ], [ "$^{87}$ Rb Bose-Einstein Condensates", "The $^{87}$ Rb atoms, evaporated from dispensers, are captured in a 2D MOT used to efficiently load a 3D MOT.", "Bright molasses cooling allows reaching sub-Doppler temperatures ($T_\\text{D} = 148$  µK).", "The atomic cloud is then loaded into a hybrid trap [39] combining a magnetic quadrupole field and a far-detuned dipole trap at 1064 nm and further cooled by forced radio-frequency evaporation.", "The dipole trap beam transports the ultracold ensemble into the center of the imaging MCP detectors.", "After further evaporative cooling in a crossed dipole trap, quantum degeneracy can be reached.", "A BEC with $N = 4.2 (3) \\times 10^{4}$ atoms is held in the far-detuned optical dipole trap at trap frequencies of $\\omega _{x,y} = 2 \\pi \\times 113(3)$  Hz and $\\omega _{\\mathrm {z}} = 2 \\pi \\times 128(1)$  Hz providing a peak atomic density of $\\rho = 2 \\times 10^{14}$  cm$^{-3}$ .", "In this work, we use laser pulses with a central wavelength of 511.4 nm and a bandwidth of 1.75 nm (FWHM).", "The duration of the Gaussian temporal profile is 215$^{+20} _{-15}$  fs (FWHM), measured by autocorrelation.", "A detailed description of the pulse generation can be found in [21].", "The pulses are focused by a high resolution microscope objective with a numerical aperture of 0.5 onto the optical dipole trap.", "The focal waist is measured with an additional, identical objective to $w_1 = 0.99(3)$  µm and $w_2 = 1.00(5)$  µm.", "The pulse energies $E_\\mathrm {p}$ are inferred from the measured averaged laser power $P$ at a pulse repetition rate of 100 kHz, taking the mirror reflectivities as well as the calibrated transmission of the objective and the shielding mesh into account (see supplementary note 10).", "With the measured quantities, the applied peak intensities $I_0$ are determined by $I_0 = \\frac{2 P_0}{\\pi w_1 w_2}$ with the peak power $P_0 = E_\\mathrm {p}/\\left(\\sqrt{2 \\pi } \\tau \\right)$ including the rms pulse duration $\\tau = \\tau _{\\mathrm {FWHM}}/\\left(2\\sqrt{2 \\ln (2)}\\right)$ .", "The resulting pulse intensity distributions as well as the atomic density distribution allow determining the initial electron/ion distributions within the ionization volume (see supplementary note 1).", "The experimental setup enables direct detection of electrons and ions on opposite spatially resolving detectors.", "For this purpose, an extraction field accelerates the charged particles onto the detectors.", "In this work, we analyze the recorded photoelectron kinetic energy distribution.", "The static extraction field is created by two opposing mesh electrodes at $\\pm U_\\mathrm {ext}$ , which consist of gold plated etched copper meshes with about 70-80% permeability.", "After passing the meshes, the electrons are post-accelerated towards two MCPs in Chevron configuration with a channel diameter of 12 µm and a channel pitch of 15 µm attached to a P46 (Y$_3$ Al$_5$ O$_{12}$ :Ce) phosphor screen.", "The emitted light from the phosphor screen is recorded by a highspeed camera, which is operated at a frame rate of 1000 fps.", "The detection efficiency for electrons $\\eta \\approx 0.4$ is given by the product of the transparency of the extraction meshes and the open area ratio as well as the quantum efficiency of the MCP.", "In order to ensure a constant, optimal quantum efficiency for all extraction voltages, the electrons are post-accelerated onto the same front potential $U_\\mathrm {front} = 268$  V of the MCP.", "Absolute electron numbers are challenging to obtain.", "Indeed, when decreasing the extraction field, the absolute detection efficiency decreases as electrons with high kinetic energies cannot be detected efficiently.", "Moreover, the high flux of plasma electrons on a small part of the MCP area as seen for $\\pm U_\\mathrm {ext} = 300$  V in Fig.", "REF c leads to electron depletion in the microchannel material and thus a systematic underestimation of the number of incident electrons in this part of the detector.", "Furthermore, detection of low kinetic energy electrons is notoriously challenging and typically limited by residual electric and magnetic fields.", "Therefore, it requires high experimental control over such stray fields (see supplementary note 3).", "In order to obtain a full simulation of the experiment (including the charged particle detectors), trajectory simulations are performed using the Electrostatics as well as the Particle Tracing Module within the COMSOL Multiphysics® software [40].", "For this reason, we include the 3D computer-aided design (CAD) geometry of our setup into the simulation.", "The use of finite element methods (FEM) allows calculating the electric potential landscape produced by the different electrodes (see supplementary note 2).", "In addition, a global magnetic offset field of 370 mG perpendicular to the detection axis and the ionization beam axis is included, which is used in the experiment to center the electron signal onto the detector.", "The trajectory simulation results in Fig.", "REF d are obtained for monoenergetic ensembles of electrons having randomized positions within the ionization volume and velocity directions.", "Due to the expansion during the time-of-flight, the spatial extent of the electron signal grants access to the underlying velocity distribution.", "The CPT simulations of the plasma dynamics are carried out with the Particle Tracing Module within the COMSOL Multiphysics® software [40].", "For these simulations $N_\\mathrm {i,e}$ electrons and ions are randomly distributed in a cylindrical ionization volume.", "The particles are created monoenergetically according to the two-photon excess energies whereas the distribution of velocity directions is randomized.", "For $t > 0$ , the 3D differential equation of motion is solved numerically for all particles including Coulomb interaction.", "Typical calculations for $N_\\mathrm {i,e} = 4000$ and few microseconds of time-evolution last between 5-22 days while being paralleled on 35 processing units (2.2 GHz) corresponding to a CPU time of a few hundred days.", "The divergence of the interparticle Coulomb potential $U_{\\mathrm {C}}(r)$ is circumvented by introducing a bounded interaction potential $\\tilde{U}_\\mathrm {C}(r) ={\\left\\lbrace \\begin{array}{ll}U_\\mathrm {C}(r), & r > r_0 \\\\U_\\mathrm {C}(r_0), & r \\le r_0\\end{array}\\right.", "}$ where $r$ denotes the interparticle distance and $r_0 = 20$  nm the cut-off radius, which is chosen well below the mean initial interparticle distance.", "The simulations furthermore neglect interactions with the remaining neutral atoms and radiative energy losses of the charged particles.", "Beyond macroscopic quantities such as the mean kinetic energies of the electron/ion ensembles, the plasma simulations offer detailed access to the dynamics at a single-particle level (see supplementary note 4) [41].", "The authors would like to thank Thomas C. Killian, Nikolay M. Kabachnik and Andrey K. Kazansky for fruitful discussions.", "We thank Bernhard Ruff, Jakob Butlewski, Julian Fiedler, Donika Imeri and Jette Heyer for contributions during an early stage of the experiment.", "This work is funded by the Cluster of Excellence 'CUI: Advanced Imaging of Matter' of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 - project ID 390715994 as well as by the Cluster of Excellence 'The Hamburg Centre for Ultrafast Imaging' of the DFG - EXC 1074 - project ID 194651731.", "Supplementary Material: Ultrafast Electron Cooling in an Expanding Ultracold Plasma In order to calculate the initial electron/ion distributions $\\rho _\\mathrm {e/i}(x,y,z) = \\rho _\\mathrm {a}(x,y,z) \\times P(x,y,z),$ the full 3D distribution of the atomic density $\\rho _\\mathrm {a}(x,y,z)$ is modeled and multiplied with the non-linear 3D ionization probability distribution $P(x,y,z)$ given by the intensity distribution of the laser pulse.", "Here, $z$ denotes the pulse propagation direction.", "The ionization probabilities are obtained by solving the time-dependent Schrödinger equation.", "We have demonstrated in a previous work that this theoretical description is in perfect agreement with the measured ionization probabilities [21].", "Figure: Ionization volume.", "a.", "2D projection of the simulated 3D electron/ion density distribution ρ e /i(x,y,z)\\rho _\\mathrm {e/i}(x,y,z) after strong-field ionization by a single pulse with I 0 =1.9×10 13 I_0 = 1.9 \\times 10^{13} Wcm -2 \\mathrm {W}\\,\\text{cm}^{-2} The dashed white lines mark the cylindrical volume, which is used for the CPT plasma simulations.", "b. Ionization probability in xx-direction (solid blue line), in zz-direction (red dotted line) and normalized atomic density distribution in xx-direction (dashed yellow line).Figure REF a shows the 2D-projection of the obtained electron/ion density distribution for a pulse of $I_0 = 1.9 \\times 10^{13}$  $\\mathrm {W}\\,\\text{cm}^{-2}$ together with the cylindrical volume used for the plasma simulations.", "Fig.", "REF b depicts the ionization probability in $x$ -direction at $(y,z) = (0,0)$ (solid blue line), in $z$ -direction at $(x,y) = (0,0)$ (red dotted line) as well as the normalized atomic density distribution in $x$ -direction at $(y,z) = (0,0)$ (dashed yellow line) as a Thomas-Fermi profile of the BEC for the experimental trap frequencies.", "The atomic cloud is almost spherical as the trapping frequencies are similar in all three dimensions.", "Whereas the photoionization can be regarded as local in in $x$ - and $y$ -direction, we fully ionize the atomic ensemble in the direction of the pulse propagation $z$ .", "We use the Electrostatics Module within the COMSOL Multiphysics® software [40] to calculate the electrostatic field configuration for each extraction voltage.", "For this purpose, we include the 3D computer-aided design (CAD) geometry of our setup into the simulation.", "The use of finite element methods (FEM) allows for the calculation of the electric potential landscape produced by the different electrodes.", "We used the physics-controlled mesh option 'finer' with a minimum/maximum element size of 0.88/12 mm.", "Figure REF shows a sectional view into the vacuum chamber together with the equipotential lines obtained for $\\pm U_\\mathrm {ext} = 300$  V. Since the high-resolution objective needs to be close to the ionization volume, the reentrant window has to be shielded by a grounded high-transparency copper mesh (identical to the ones used for the electrodes) to avoid accumulation of static charge.", "This leads to a non-linear extraction field strongly increasing towards the detector, as well as a top-down asymmetry, which explains the non-spherical distributions observed experimentally (compare Fig. 3).", "The extraction field in the interaction region can be precisely controlled by the voltages $U_\\mathrm {ext}$ .", "However, its amplitude is not perfectly proportional to the applied voltage for low extraction fields ($\\pm U_\\mathrm {ext} = 5~$ V).", "This is due to the voltages on the electron MCP, which give rise to an electric field on the order of 2 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ in the center.", "In addition, the shielding by the vacuum chamber is included in the simulations.", "The FEM simulations of the electromagnetic fields provide valuable insight into the level of control over the electric and magnetic stray fields.", "At an extraction voltage of $\\pm U_\\mathrm {ext} = 5$  V, which corresponds to an electric field of 4.6 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ at the center, the extraction field still clearly dominates over the electrical stray fields.", "For smaller extraction fields however, our measurements deviate from the theoretical predictions due to electric fields.", "In our experimental setup, these electric stray fields are passively shielded by the grounded vacuum chamber and electric gradients are controlled down to the $\\mathrm {V}\\,\\mathrm {m}^{-1}$ level.", "Helmholtz coils are used to compensate magnetic fields in all three spatial dimensions with an accuracy of 10 mG. A homogeneous compensation over the extent of the detection units is enabled by meter-sized coils.", "We have been working here with a magnetic field offset of 370 mG along the $y$ -axis, which increases our energy resolution and centers the electron signal onto the detectors.", "The plasma simulations are based on CPT simulations and include Coulomb interaction between the charged particles (see Methods).", "Besides quantities such as the mean kinetic energies of the electron/ion ensembles, the CPT plasma simulations provide detailed access to the dynamics of each charged particle.", "Figure REF a shows the time-evolution of the distance from the ionization center of single plasma electrons (blue lines) for the simulation depicted in Fig.", "4f-g without extraction field.", "For clarity, the graph only depicts a random selection of 31 plasma electrons.", "One clearly identifies an oscillatory motion for the different particles exhibiting different frequencies ranging from hundreds of gigahertz to a few megahertz.", "The frequency is decreasing with increasing amplitude of the oscillations due to screening by the more closely bound electrons.", "The maximal ion radius (dashed red line) given by the maximum of all ion distances from the ionization center is used to distinguish between plasma and escaping electrons.", "In this simulation electrons are regarded as plasma electrons, if their distance at t = 2500 ns is less than twice the maximum ion radius.", "In addition, the rms electron/ion radius is given for each time step (bold solid blue line / solid light red line).", "The asymptotic expansion velocity of the rms ion radius of $v_\\mathrm {i,rms} = 418~\\mathrm {m}\\,\\mathrm {s}^{-1}$ is in reasonable agreement with the expected plasma expansion velocity $v_\\mathrm {hyd} = \\sqrt{k_\\mathrm {B} \\left( T_\\mathrm {e,0} + T_\\mathrm {i,0} \\right) / m_\\mathrm {i}} \\approx 710~\\mathrm {m}\\,\\mathrm {s}^{-1}$ for hydrodynamic expansion (dark red dotted line) at an initial electron temperature of $T_\\mathrm {e,0} = 5250$  K and negligible initial ion temperature $T_\\mathrm {i,0}$  [6].", "Figure REF b depicts the corresponding kinetic energies of the plasma electrons for the first 100 ps (blue lines) as well as the mean kinetic energy of all 1961 plasma electrons (bold white line).", "The observed oscillations in the mean kinetic energy are caused by the oscillatory motion of the electrons with similar frequencies and the apparent damping is evoked by their dephasing.", "For the simulations including an extraction field, the differentiation between plasma and escaping electrons is more challenging, since electrons enter large orbits, where the non-linear extraction field become dominant, and escape from the plasma.", "Thus, the extraction fields reduce the number of plasma electrons by about 80% over time.", "For the simulations shown in Fig.", "4g at $\\pm U_\\mathrm {ext} = 300$  V and $\\pm U_\\mathrm {ext} = 5$  V electrons are regarded as plasma electrons, if their distance from the ionization center after 40 ns / 300 ns is less than 1 mm.", "The CPT plasma simulations furthermore provide access to the space charge potential well created by the unpaired ions during the plasma expansion.", "The extraction field $E_\\mathrm {ext}$ adds up to the 1D space charge potential along the detection axis lowering the effective trapping potential $U_\\mathrm {eff}(r) = U(r) - E_\\mathrm {ext} \\cdot r,$ where $r$ denotes the distance to the ionization center in the direction of the extraction field.", "Figure 4g (red lines) shows the evolution of the effective space charge potential depth for the plasma simulations without extraction field as well as for extraction voltages of $\\pm U_\\mathrm {ext} = 5$  V and $\\pm U_\\mathrm {ext} = 300$  V (corresponding to $E_\\mathrm {ext}$ = 4.6 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ and $E_\\mathrm {ext}$ = 162 $\\mathrm {V}\\,\\mathrm {m}^{-1}$ ).", "Here, for each time-step $U(r)$ is approximated by the Coulomb potential of a homogeneously charged sphere $U(r) ={\\left\\lbrace \\begin{array}{ll}\\frac{Q}{8 \\pi \\epsilon _0 R} \\left(3 - \\frac{{r}^2}{R^2}\\right), & r \\le R\\\\\\frac{Q}{4 \\pi \\epsilon _0 r}, & r > R\\end{array}\\right.", "}$ where the radius $R$ is given by the maximal ion radius and the charge $Q = e \\cdot N_{\\mathrm {diff}}$ is given by the difference $N_\\mathrm {diff}$ of the number of ions and electrons within the maximum ion radius.", "The potential depth is determined by the difference of the local maximum $U_\\mathrm {eff}(r_\\mathrm {max})$ and the local minimum $U_\\mathrm {eff}(r_\\mathrm {min})$ of the effective potential at $r_\\mathrm {min} = \\frac{4 \\pi \\epsilon _0 E_\\mathrm {ext} R^3}{\\text{$Q$}} \\text{ and } r_\\mathrm {max} = \\sqrt{\\frac{Q}{4 \\pi \\epsilon _0 E_\\mathrm {ext}}}.$ For plasma formation, the depth of the space charge potential has to exceed the electronic excess energy $E_\\mathrm {kin,e}$ .", "Thus, for a given excess energy and a Gaussian spatial distribution of charge carriers, the creation of a minimum ion number $N^* = E_\\mathrm {kin,e}/U_0$ is required, where $U_0 = \\sqrt{\\frac{2}{\\pi }} \\frac{e^2}{4 \\pi \\epsilon _0 \\sigma }$ and $\\sigma $ denotes the rms radius of the ionic distribution [5].", "The experimentally realized ionic distribution is approximated by a Gaussian distribution with the arithmetic mean radius of $\\sigma = (2\\times 1.35~\\mu \\mathrm {m} + 5~\\mu \\mathrm {m})/3$ leading to a critical ion number of $N^* =960$ .", "Figure REF shows the measured brightness at $\\pm U_\\mathrm {ext} = 300$  V in an elliptical area around the plasma electrons on the detector.", "The number of plasma electrons, as signature of the plasma formation, displays a strong dependency on the critical charge carrier density.", "This density has been varied either by the pulse intensity (from $0.1 \\times 10^{13}$  $\\mathrm {W}\\,\\text{cm}^{-2}$ to $1.7 \\times 10^{13}$  $\\mathrm {W}\\,\\text{cm}^{-2}$ ) or the atomic density (from $6 \\times 10^{17}$ m$^{-3}$ to $\\rho $ = $1.3 \\times 10^{20}$ m$^{-3}$ ).", "In order to vary the atomic density, we modify the evaporation efficiency, which leads to a reduced number of atoms in the final optical dipole trap.", "The density is scanned from an ultracold thermal cloud with $\\rho $ = $6 \\times 10^{17}$ m$^{-3}$ to an almost pure condensate with $\\rho $ = $1.3 \\times 10^{20}$ m$^{-3}$ .", "The densities are determined by the analysis of the optical density distributions in a time-of-flight measurement recorded by absorption imaging for different timesteps after switching off the optical dipole trap.", "Whereas the calculation of in-situ atomic densities is reliable for the limiting cases of a fully condensed or thermal atomic sample, it is known to be difficult for partly condensed samples as the determination requires a bimodal fit using a Gaussian as well as the Thomas-Fermi density model.", "However, the critical number of ionized atoms can be extracted from the numbers of detected electrons for each intensity density combination.", "As the ionization volume slightly increases with increasing peak intensity, the critical number increases as well.", "While for $I_\\mathrm {0} = 1.7 \\times 10^{12} $  $\\mathrm {W}\\,\\text{cm}^{-2}$ around 500 ions need to be created, for $I_\\mathrm {0} = 1.9 \\times 10^{13} $  $\\mathrm {W}\\,\\text{cm}^{-2}$ approximately 1000 electrons are required, which agrees well with the expected value of $N^*$ = 960.", "Figure: Critical charge carrier density for plasma formation.", "Measured brightness of the electron signal in an elliptical area enclosing low kinetic energy electrons.", "The atomic target density has been varied over three orders of magnitude and the pulse intensity from 0.1×10 13 0.1 \\times 10^{13} Wcm -2 \\mathrm {W}\\,\\text{cm}^{-2} to 1.7×10 13 1.7 \\times 10^{13} Wcm -2 \\mathrm {W}\\,\\text{cm}^{-2}.Figure: Time-evolution of the simulated plasma parameters.", "a. Density of electrons (blue line) and ions (red line) in a spherical volume around the ionization center.", "b. Electron/ion plasma period τ p ,e/i=2πω p ,e/i -1 \\tau _\\mathrm {p,e/i} = 2 \\pi \\omega _\\mathrm {p,e/i}^{-1} (dashed blue/red line) and frequencies ω p ,e/i\\omega _\\mathrm {p,e/i} (solid blue/red lines).", "c. Coulomb coupling parameter Γ e ,i\\Gamma _\\mathrm {e,i} of the electron/ion component (blue/red line) determined by the particle densities and the plasma electron/ion mean kinetic energy.", "d. Electron/ion de Broglie wavelength λ dB ,e/i\\lambda _\\mathrm {dB,e/i} (solid blue/red line) and Wigner-Seitz radius a WS ,e/ia_\\mathrm {WS,e/i} (dashed blue/red line).During the expansion, the densities and kinetic energies of the plasma vary over several orders of magnitude.", "Figure REF displays the time evolution of central parameters extracted from the CPT plasma simulation.", "Figure REF a shows the evolution of the electron and ion density in the center of the plasma.", "The electron density drops within the initial expansion and stays constant for the first nanosecond.", "As the ionic component expands, the electron and ion density both decrease and even fall below typical initial UNP densities.", "The density evolution determines the plasma frequency as well as the plasma period given in Fig.", "REF b.", "The decrease of the electron density explains the deceleration of the electronic orbital oscillations during the plasma expansion.", "In addition, the ionic plasma frequency significantly decreases within the first plasma period (given by the initial ionic density).", "In contrast to UNP, where plasma expansion can be regarded as slow in relation to the inverse ionic plasma frequency, here, the ionic plasma period exceeds the plasma expansion duration, thus preventing ionic thermalization.", "In Fig.", "REF c the ion and electron coupling parameters are depicted.", "The ionic coupling parameter decreases after the first electron plasma period when charge imbalance is established due to ionic acceleration and increasing interparticle distance.", "On the contrary, the electronic coupling parameter increases during the plasma expansion since the electron temperature decreases over orders of magnitude.", "The simulations reveal a maximum coupling parameter of $\\Gamma _\\mathrm {e} = 0.3$ approaching significant electron coupling.", "Electron temperatures in the Kelvin domain raise the question of quantum degeneracy for the electronic ensemble.", "Fig.", "REF d illustrates the electron/ion de Broglie wavelengths $\\lambda _\\mathrm {dB,e/i}$ at different expansion times.", "Whereas the ionic wavelength quickly decreases, the electrons reach a maximum de Broglie wavelength on the order of $\\lambda _\\mathrm {dB,e} \\approx 100~$ nm at the end of the plasma expansion.", "However, the ratio of mean interparticle distance given by the Wigner-Seitz radius $a_\\mathrm {WS,e/i}$ (dashed blue/red line) and the de Broglie wavelength never exceeds 1.3 %, which yields $E_\\mathrm {kin,e}/E_\\mathrm {F} > 6000$ , where $E_\\mathrm {F} = \\frac{\\hbar ^2}{2 m_\\mathrm {e}} \\left(3 \\pi ^2 \\rho _\\mathrm {e}\\right)^{2/3}$ denotes the electron Fermi energy with the reduced Planck constant $\\hbar $ .", "Thus, a quantum mechanical description required for a fermionic ensemble close to degeneracy can be safely disregarded.", "For ultracold plasma in the density and temperature regime described in this manuscript, three-body recombination (TBR) is expected to be the dominant process of electron-ion recombination.", "The TBR rate $K_\\mathrm {TBR}$ per ion according to classical TBR theory is given by $K_\\mathrm {TBR} \\approx 3.8 \\times 10^{-9}~T_\\mathrm {e}^{-9/2} \\rho _\\mathrm {e}$  s$^{-1}$ , where $T_\\mathrm {e}$ is the electron temperature in K and the electron density $\\rho _\\mathrm {e}$ is given in cm$^{-3}$  [6].", "Figure REF shows the calculated TBR rate per ion (solid line) as well as the time-integrated TBR probability per ion (dashed line).", "After 2.5 µs of plasma expansion, a cumulated TBR probability of approximately 1% is reached.", "As a result, the plasma lifetime is expected to be on the order of 100 µs before a significant fraction of Rydberg excitations are created.", "However, on the ten microsecond timescale, when the plasma is dilute and collisions barely occur, radiative and dielectronic recombination might further limit the plasma lifetime.", "The experimental setup gives access to the distribution of arrival times of the detected electrons by a gated detection scheme (Fig.", "REF ).", "For this purpose, a repulsive voltage pulse is applied to the electron extraction mesh after a variable delay $t_\\mathrm {delay}$ after the femtosecond laser pulse.", "The rapidly switched potential prevents electrons from passing the extraction mesh for time-of-flight durations $\\tau _\\mathrm {ToF} > t_\\mathrm {delay}$ .", "The applied pulse has a duration of 2 µs and is capacitively coupled onto the meshes.", "This enables to measure the accumulated electron signal up to $t_\\mathrm {delay}$ , while maintaining the spatial resolution.", "The estimated timing uncertainty of 30 ns is caused by the temporal jitter and the pulse rise time.", "In order to analyze the obtained temporal profiles (see Fig.", "5b) quantitatively, a double-sigmoid function $f(t) = \\frac{a_1}{1+e^{-b_1\\cdot (t-c_1)}} + \\frac{a_2}{1+e^{-b_2\\cdot (t-c_2)}}$ is fitted to the measured data.", "Figure REF shows the fit functions for $\\pm U_\\mathrm {ext} = 100$  V, $\\pm U_\\mathrm {ext} = 50$  V, $\\pm U_\\mathrm {ext} = 25$  V, $\\pm U_\\mathrm {ext} = 15$  V, $\\pm U_\\mathrm {ext} = 10$  V and $\\pm U_\\mathrm {ext} = 5$  V (solid lines, from light blue to dark blue).", "The two inflection points for the double-sigmoid functions are given by $c_1$ and $c_2$ (vertical dashed lines).", "The spectra in Fig.", "5c are the time derivatives of the fitted functions.", "The arrival time difference $c_2 - c_1$ determines the plasma lifetime shown in Fig. 5d.", "The error bars are given by the 95% confidence interval for the inflection points.", "Figure: Electron arrival time.", "a.-f.", "Accumulated electron counts measured at a peak intensity of 1.2×10 13 1.2 \\times 10^{13} Wcm -2 \\mathrm {W}\\,\\text{cm}^{-2} for different extraction voltages (see Fig.", "5b) (black data points).", "The vertical error bars of the binned data (blue data points) are given by the standard deviation over all realizations and the horizontal ones indicate the time uncertainty of the repulsive voltage pulse.", "The solid lines show the double-sigmoid functions according to Eq.", "() obtained by a fit to the data points.", "The dashed lines depict the inflection points c 1 c_1 and c 2 c_2 of each sigmoid function.The experimental setup only provides access to the femtosecond laser power before passing the high resolution microscope objective.", "As the transmittance $\\alpha _\\mathrm {T}$ critically depends on pointing and angle of the incident laser beam, the actual peak intensities inside the vacuum chamber have to be calibrated.", "The averaged laser power used for the calculation of the applied peak intensity (see Methods) is given by $P = \\alpha _\\mathrm {T}P_\\mathrm {front}$ .", "Here, $P_\\mathrm {front}$ denotes the power in front of the objective, which is measured through a circular aperture with the same diameter as the objective aperture (4 mm) at a pulse repetition rate of 100 kHz.", "Figure REF shows the measured number of electrons for different powers $P_\\mathrm {front}$ at an extraction voltage of $\\pm U_\\mathrm {ext} = 200$  V (data points).", "The solid lines depict the expected numbers of electrons assuming a transmittance of 0.2, 0.1 and 0.05, which are calculated by use of the absolute ionization probabilities reported in [21] as well as the beam waist measured with an identical objective (see Methods).", "The best agreement is obtained for $\\alpha _\\mathrm {T} = 0.1$ .", "The electron cooling mechanisms in ultracold plasma can be exploited for plasma-based ultracold electron sources [28] producing low-emittance electron bunches.", "These bunches can be used to seed high-brilliance particle accelerators [29] and for coherent imaging of biological systems [30].", "With the final electron temperature of $T_\\mathrm {e}$  $\\approx $  10 K and an rms electron bunch radius $\\sigma _\\mathrm {r}$ = 0.52 mm, we achieve a normalized rms emittance of $\\epsilon _\\mathrm {r} = \\sigma _\\mathrm {r}\\cdot \\sqrt{k_\\mathrm {B} T_\\mathrm {e} / m_\\mathrm {e} c^2} = 21$ nm rad and a relative transverse coherence length $C_{\\perp } = \\hbar /(m_\\mathrm {e} c \\epsilon _\\mathrm {r}) = 2 \\times 10^{-5}$ .", "In our experimental setup, the electron excess energy can be reduced further by working closer to the ionization threshold, allowing to approach the value of $C_{\\perp } = 10^{-3}$ required for single-shot electron diffraction [31]." ] ]

[ [ "Trajectorial dissipation and gradient flow for the relative entropy in\n Markov chains" ], [ "Abstract We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates.", "These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance.", "Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time.", "Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions.", "Extension are developed to general \"convex divergences\" and to countable state-spaces.", "The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context." ], [ "Introduction and Summary", "We present a trajectorial approach to the temporal dissipation of variance and relative entropy, in the context of ergodic Markov Chains in continuous time.", "We follow the methodology of the recent work by Karatzas, Schachermayer & Tschiderer (2019), which is based on stochastic calculus and uses time-reversal in a critical fashion.", "By aggregating the trajectorial results, i.e., by averaging them with respect to the invariant measure, we obtain a very crisp, geometric picture of the steepest descent property for the curve of time-marginals, relative to local perturbations.", "This holds for an appropriate, locally flat metric on configuration space, defined in terms of a suitable discrete Sobolev norm.", "We adopt then a more global approach, and establish also the gradient flow property—to the effect that the temporal evolution for the curve of the Chain's time-marginals is prescribed by an appropriate Riemannian metric on the manifold of probability measures on configuration space, and by the differential of the relative entropy functional along this curve; cf.", "Maas (2011), Mielke (2011), Erbar & Maas (2012, 2014).", "Both steepest descent and gradient flow are manifestations of the seminal Jordan, Kinderlehrer & Otto (1998) results and of their outgrowth, the so-called “Otto Calculus\" initiated in Otto (2001).", "Preview: For a finite state-space, we set up the probabilistic framework in Section and the functional-analytic one in Section .", "The appropriate stochastic-analytic machinery and results appear in Sections and .", "Temporal dissipation and steepest descent are developed in increasing generality: First in Section for the variance and its associated, globally determined and flat, metric; then in Section for the Boltzmann-Gibbs-Shannon relative entropy; and finally in Section for general entropies induced by convex functions.", "Gradient flows and their associated geometries are taken up in Section , culminating with a very direct proof of a discrete version of the celebrated HWI inequality of Otto & Villani (2000).", "Some extensions to state-spaces with a countable infinity of elements are developed in Section ." ], [ "The Setting", "On a probability space $ ( \\Omega , {\\cal F}, \\mathbb {P}) , $ we start with an irreducible, positive recurrent, discrete-time Markov Chain $\\, {\\cal Z}= (Z_n )_{n \\in \\mathbb {N}_0}\\,$ with state-space $\\,{\\cal S} ,$ transition probability matrix $ \\,\\Pi = ( \\pi _{xy} )_{(x,y) \\in {\\cal S}^2}\\, $ with entries $ \\pi _{xy} = \\mathbb {P}(Z_{n+1} =y \\,|\\, Z_n =x)\\, $ for $\\,n \\in \\mathbb {N}_0 ,$ and initial distribution $\\,P(0) = ( p (0, x) )_{x \\in {\\cal S}}\\,$ which is a column vector with components $\\,p (0,x) : = \\mathbb {P}(Z_0 =x) >0\\,$ for all $\\,x \\in {\\cal S}.$ Throughout Sections –, the state-space $\\,{\\cal S}\\,$ is assumed to be finite; extensions to countable state-spaces are taken up in Section .", "It is straightforward to check that the sequence of random variables $\\,\\big (M^f_n \\big )_{n \\in \\mathbb {N}_0}\\,$ with $\\, M^f_0 :=f(Z_0)\\,,$ $M^f_n \\,:=\\, f(Z_n) - \\sum _{k=0}^{n-1} \\big ( \\Pi f - f \\big ) (Z_k)\\,, \\qquad n \\in \\mathbb {N}\\,,$ is a martingale of the filtration generated by the Markov Chain $ {\\cal Z}$ , for any given function $\\, f : {\\cal S} \\rightarrow \\mathbb {R}\\,.$  Here and in what follows, we denote $\\, ( \\Pi f ) (z) := \\sum _{y \\in {\\cal S}} \\, \\pi _{zy} \\, f(y) , ~ z \\in {\\cal S}$ .", "It is well known that such a Chain has a unique invariant distribution: that is, a column vector $\\, Q = \\big ( q (y) \\big )_{y \\in {\\cal S}}\\,$ of positive numbers adding up to 1 and satisfying $\\, \\Pi ^\\prime Q = Q\\,$ or, more explicitly, $q (y) \\,=\\, \\sum _{z \\in {\\cal S}} \\, q(z) \\, \\pi _{zy} \\,, \\qquad \\forall ~~y \\in {\\cal S}\\,.$ Here and throughout this paper, prime $\\,^\\prime \\,$ denotes transposition of a matrix or vector.", "A major result of discrete-time Markov Chain theory states that, when $ {\\cal Z}$ is also aperiodic, the $k-$ step transition probabilities $\\pi ^{(0)}_{xy} := \\mathbf { 1}_{x=y}\\,, \\qquad \\pi ^{(k)}_{xy} := \\mathbb {P}\\big ( Z_k =y \\, \\big |\\, Z_0 =x\\big )\\,, \\quad k \\in \\mathbb {N}$ converge as $k$ tends to infinity to $ \\,q(y) $ , for every pair of states $\\, (x,y) \\in {\\cal S}^2$ .", "We refer to Chapter 1 in Norris (1997), in particular Theorems 1.7.7 and 1.8.3, for an excellent account of the relevant theory." ], [ "From Discrete- to Continuous-Time ", "Consider now on the same probability space a Poisson process $\\, {\\cal N}= \\big (N(t) \\big )_{0 \\le t < \\infty }\\,$ with parameter $\\, \\lambda =1\\,$ and independent of the discrete-time Markov Chain $\\, {\\cal Z}.$ We construct via time-change the continuous-time process $X(t) \\,:=\\, Z_{N(t)}\\,, \\qquad 0 \\le t < \\infty \\,,$ as well as the filtration $\\,\\mathbb {F}^X = \\big \\lbrace {\\cal F}^X (t) \\big \\rbrace _{0 \\le t < \\infty }\\,$ this process generates via $\\, {\\cal F}^X (t) := \\sigma \\big ( X(s), \\, 0 \\le s \\le t \\big ) .$ Straightforward computation shows that this new, continuous-time process $\\, {\\cal X}= \\big (X(t) \\big )_{0 \\le t < \\infty }\\,$ has the Markov property, and time-homogeneous transition probabilities $\\varrho _h (x,y) \\, :=\\, \\mathbb {P}\\big ( X(t+h) =y \\, \\big |\\, X(t )=x\\big ) = e^{-h} \\sum _{k \\in \\mathbb {N}_0} \\frac{\\, h^k\\,}{k!", "}\\, \\pi ^{(k)}_{xy}\\,, \\qquad t \\ge 0, ~ h > 0$ with the notation of (REF ); we set $\\, \\varrho _0 (x,y) := \\mathbf { 1}_{x=y}\\,$ .", "The functions $\\, h \\mapsto \\varrho _h (x,y) \\,$ in (REF ) are uniformly continuous and continuously differentiable; cf.", "Theorems 2.13, 2.14 in Liggett (2010).", "More generally, for arbitrary $\\, n \\in \\mathbb {N}$ , $\\, 0 < \\theta _1 < \\cdots < \\theta _n = \\theta < t < \\infty \\,$ , $(x, y_1, \\cdots , y_n, z) \\in {\\cal S}^{n+2}\\,$ with $\\, y = y_n\\,,$ the finite-dimensional distributions of this process are $ \\mathbb {P}\\big ( X(0 )=x , X(\\theta _1) = y_1, \\cdots , X(\\theta _n)= y_n, X(t) =z \\big ) \\,=~~~~~~~~~~~~~~~~~~~~$ $~~~~~~~~~~~~~~~~~~~~~~~~\\,= \\,p (0,x)\\, \\varrho _{\\theta _1} (x,y_1)\\, \\varrho _{\\theta _2 - \\theta _1} (y_1,y_2)\\cdots \\varrho _{\\theta _n - \\theta _{n-1}} (y_{n-1},y_n) \\cdot \\varrho _{t - \\theta } (y ,z)$ and we deduce the time-homogeneous Markov property $ \\mathbb {P}\\big ( X(t) =z \\, \\big |\\, {\\cal F}^X (\\theta ) \\big )=\\varrho _{t - \\theta } \\big (X(\\theta ) ,z \\big )= \\mathbb {P}\\big ( X(t) =z \\, \\big |\\, X (\\theta ) \\big ) .$ Finally, from the Chapman-Kolmogorov equations $\\,\\pi ^{(m+n)}_{xy}=\\sum _{z \\in {\\cal S}} \\, \\pi ^{(m)}_{xz} \\pi ^{(n)}_{zy} \\,$ for the $k-$ step transition probabilities of $ \\,{\\cal Z} \\, $ in (REF ), we deduce these same equations for the quantities in (REF ): $\\varrho _{t+\\theta } (x,y) \\, =\\, \\sum _{z \\in {\\cal S}} \\, \\varrho _{ \\theta } (x,z) \\, \\varrho _{t } (z,y) \\, , \\qquad (\\theta , t) \\in [0, \\infty )^2, ~~ (x,y) \\in {\\cal S}^2$ Here we think of the temporal argument $\\theta $ as the “backward variable\", and of $t$ as the “forward variable\"." ], [ "Infinitesimal Generators and Martingales", "We introduce now the matrix ${\\cal K} := \\,\\Pi - \\mathrm {I}\\, = \\big \\lbrace \\kappa (x,y) \\big \\rbrace _{(x,y) \\in {\\cal S}^2} \\qquad \\text{with elements} \\qquad \\kappa (x,y) := \\pi _{xy}- \\mathbf { 1}_{x=y}\\,:$ non-negative off the diagonal, adding up to zero across each row.", "From (REF ) and with the help of time-homogeneity, we obtain for $\\, t \\ge 0\\,,$ $\\, h >0\\,$ the infinitesimal “transition rates\" $ \\mathbb {P}\\big ( X(t+h) =y \\, \\big |\\, X (t) =x \\big )\\,=\\, h \\cdot \\kappa (x,y) + o (h)\\,, \\qquad x \\ne y\\,,$ $ \\mathbb {P}\\big ( X(t+h) =x \\, \\big |\\, X (t) =x \\big )\\,=\\, 1+ h \\cdot \\kappa (x,x) + o (h)$ with the standard convention $\\, \\lim _{h \\downarrow 0 } \\big ( o (h) / h \\big ) =0,$ valid uniformly over $t \\in [0, \\infty ).$ In particular, (REF ) and (REF ) give the infinitesimals $\\, \\varrho _{h} (x,y) - \\varrho _{0} (x,y) = h \\cdot \\kappa (x,y) + o (h)\\,$ for all $\\,(x,y) \\in {\\cal S}^2\\,$ , and thus $\\partial \\varrho _h (x,y) \\, \\big |_{h=0} \\,=\\, \\kappa (x,y)\\,.$ Here and throughout the paper, $\\, \\partial g \\,$ denotes partial differentiation of a function $g$ with respect to its temporal argument.", "A bit more generally, for any $ f : {\\cal S} \\rightarrow \\mathbb {R}$ we have from (REF ), (REF ) the semigroup computation $\\big ( T_h f \\big ) (x) \\,:=\\, \\mathbb {E}\\big [ f \\big ( X(t+h)) \\, \\big |\\, X(t)=x \\big ] \\,=\\, f(x)+ h \\cdot \\big ( {\\cal K} f \\big ) (x) + o (h) \\,.$ We deploy, here and in what follows, the infinitesimal generator of the Chain, i.e., the linear operator $\\big ( {\\cal K} f \\big ) (x) := \\big ( \\Pi f \\big ) (x)- f(x) = \\sum _{y \\in {\\cal S}} \\, \\kappa (x,y)\\, f(y)= \\sum _{y \\in {\\cal S}} \\, \\kappa (x,y)\\, \\big [ f(y) - f (x) \\big ] \\,, \\quad x \\in {\\cal S} \\,.$ Using the computation (REF ), it is shown fairly easily that the exact analogue of the random sequence (REF ) in our present setting, namely, the process $f \\big ( X(t) \\big ) - \\int _0^t \\big ( {\\cal K} f \\big ) \\big ( X (\\theta ) \\big ) \\, \\mathrm {d}\\theta \\,, \\qquad 0 \\le t < \\infty \\,,$ is an $\\,\\mathbb {F}^X-$ martingale; cf.", "Theorem 3.32 in Liggett (2010).", "As a slight generalization, we obtain also the following result (Lemma IV.20.12 in Rogers & Williams (1987)).", "Proposition 2.1 Given any function $\\, g : [0, \\infty ) \\times {\\cal S} \\rightarrow \\mathbb {R}\\,$ whose temporal derivative $ \\, t \\mapsto \\partial g (t, x)\\, $ is continuous for every state $\\, x \\in {\\cal S} ,$ the process below is a local $\\,\\mathbb {F}^X-$ martingale: $M^g (t) \\,:= \\, g \\big ( t,X(t) \\big ) - \\int _0^t \\big ( \\partial g + {\\cal K} g \\big ) \\big (\\theta , X (\\theta ) \\big )\\, \\mathrm {d}\\theta \\,, \\qquad 0 \\le t < \\infty \\,.$ Remark 2.1 The General Case: Instead of starting with transition probabilities $\\pi _{x y}$ and defining $ \\kappa (x,y) = \\pi _{xy}- \\mathbf { 1}_{x=y}\\,$ as in (REF ), one can work instead with any transition rates $\\kappa (x,y)$ satisfying: $(i)$ $\\kappa (x,y) \\ge 0\\, $ for $ \\,x \\ne y\\,$ ; and $(i)$ $\\sum _{\\,y \\in {\\cal S}} \\kappa (x,y) = 0\\,$ for every $x \\in {\\cal S}$ .", "In this manner, arbitrary irreducible continuous-time Markov chains on finite state spaces can be constructed, and studied with little extra effort.", "We have opted here for the somewhat less general, but very concrete and intuitive, approach of the present Section." ], [ "Forward and Backward ", "Let us differentiate both sides of the equations in (REF ) with respect to the backward variable $\\theta $ , then set $\\theta =0$ .", "We obtain on account of (REF ) the Backward Kolmogorov differential equations $\\partial \\varrho _t (x,y) \\,=\\, \\sum _{z \\in {\\cal S}} \\, \\kappa (x, z) \\, \\varrho _t (z,y).$ We can write this system of equations, for the matrix-valued function $\\, t \\mapsto {\\cal P}_t = \\big ( \\varrho _t (x,y) \\big )_{(x,y) \\in {\\cal S}^2}\\,$ of the forward variable $\\, t \\in [0, \\infty )\\,,$ in the form $\\,\\partial \\, {\\cal P}_t\\,=\\, {\\cal K } \\,{\\cal P}_t\\,, \\,~{\\cal P}_0 = \\mathrm {I}\\,.$ In a similar manner, differentiating formally the equations (REF ) with respect to the forward variable $t$ , then evaluating at $t=0$ and recalling the transpose ${\\cal K}^\\prime \\,:=\\, \\big ( \\kappa ^\\prime ( y,z) \\big )_{( y,z) \\in {\\cal S}^2}\\,, \\qquad \\kappa ^\\prime ( y,z) \\,:=\\, \\kappa (z,y)$ of the $\\,{\\cal K}-$ matrix, we obtain the Forward Kolmogorov equations $\\partial \\varrho _\\theta (x,y) \\,=\\, \\sum _{z \\in {\\cal S}} \\, \\varrho _\\theta (x,z)\\, \\kappa ( z, y) \\,=\\, \\sum _{z \\in {\\cal S}} \\, \\kappa ^\\prime ( y,z) \\, \\varrho _\\theta (x,z) \\, ,\\qquad \\text{or} \\qquad \\partial \\,{\\cal P}_\\theta \\,=\\, {\\cal K }^\\prime {\\cal P}_\\theta \\,, \\quad {\\cal P}_0 = \\mathrm {I} .$" ], [ "A Curve of Probability Vectors", "For every $\\,t > 0$ , let us consider the column vector $\\,P(t) = \\big ( p (t, y) \\big )_{y \\in {\\cal S}}\\,$ of probabilities for the $ \\mathbb {P}-$ distribution $p(t,y) := \\mathbb {P}\\big ( X(t) = y\\big ) = e^{-t} \\sum _{x \\in {\\cal S}} \\,p(0,x) \\sum _{k \\in \\mathbb {N}_0} \\frac{\\, t^k\\,}{k!", "}\\, \\pi ^{(k)}_{xy}\\,>\\,0$ of the random variable $X(t)$ .", "The forward Kolmogorov equations of (REF ), the law of total probability, and the Markov property, show that these satisfy their own forward Kolmogorov equations, namely $\\partial p (t,y) \\,=\\, \\sum _{z \\in {\\cal S}}\\, p (t,z) \\, \\kappa ( z,y) \\,=\\, \\sum _{z \\in {\\cal S}} \\, \\kappa ^\\prime ( y,z)\\, p (t,z) \\,=: \\, \\big ( {\\cal K}^\\prime p \\big ) (t,y)\\,;$ or, more compactly and in matrix form, $\\, \\partial P(t)= {\\cal K}^\\prime P(t) \\,,~ ~0 \\le t < \\infty \\, $ in the notation of (REF ).", "We shall think of $ ( P (t) )_{0 \\le t < \\infty }\\,$ as a curve on the manifold $\\, {\\cal M } = {\\cal P}_+ ({\\cal S})\\,,$ of vectors $\\, P = ( p (x) )_{x \\in {\\cal S}}\\,$ with strictly positive elements and total mass $\\, \\sum _{x \\in {\\cal S}} p (x)=1, $ viewed as probability measures and governed by (REF ).", "Suppose that the initial distribution $P(0)$ of the discrete-time Markov Chain $\\,{\\cal Z}$ coincides with the column vector $ \\,Q = \\big ( q (y) \\big )_{y \\in {\\cal S}}\\,$ of (REF ) satisfying $ \\,\\Pi ^\\prime Q = Q\\,,$ or equivalently $\\, {\\cal K}^\\prime Q=0 $ on account of (REF ).", "It follows that $\\, P(t) \\equiv Q\\,,$ $\\, \\forall \\, t \\in [0, \\infty )\\,$ provides now the solution of (REF ): the distribution $Q$ is invariant also for the continuous-time Markov Chain $\\,{\\cal X}$ in (REF ).", "A bit more generally, $\\,Q\\,$ is the equilibrium distribution of $\\,{\\cal X}$ , in the sense that for every initial distribution $\\,P(0)= \\big ( p (0, x) \\big )_{x \\in {\\cal S}} \\, $ and function $\\, f : {\\cal S} \\rightarrow \\mathbb {R}\\,$ we have the limiting behavior $\\lim _{t \\rightarrow \\infty } p(t,y)\\, = \\,q(y)\\,,\\qquad \\forall ~~ y \\in {\\cal S} ,$ $\\lim _{T \\rightarrow \\infty } \\frac{1}{T} \\int _0^T f \\big ( X(t) \\big ) \\, \\mathrm {d}t \\,=\\, \\sum _{y \\in {\\cal S}} \\, q(y)\\, f(y)\\,, \\quad \\mathbb {P}-\\text{a.e.", ";}$ see Sections 3.6 –3.8 in Norris (1997) for an account of these results.", "In the present, continuous-time context, aperiodicity plays no role." ], [ "A Curve of Likelihood Ratios", "Let us compare now the components of the probability vector $P(t)$ in (REF ), with those of the invariant probability vector $\\,Q\\,$ in (REF ).", "One way to do this, very fruitful in the present context, is by considering the likelihood ratio column vector ${\\mathbf {\\ell }}_t \\equiv {\\mathbf {\\ell }} (t)= \\big ( \\ell (t, y) \\big )_{y \\in {\\cal S}} \\qquad \\text{with components} \\qquad \\ell (t,y)\\, := \\frac{\\,p(t,y)\\,}{q(y)} \\,.$ Substituting the product $\\, p (t,y) = \\ell (t,y) \\,q (y)\\,$ into the forward Kolmogorov equation (REF ), we obtain for the likelihood ratios of (REF ) the Backward Equation $\\partial \\ell (t,y) \\,=\\, \\sum _{z \\in {\\cal S}}\\, \\widehat{\\kappa } ( y,z) \\, \\ell (t,z) \\,=\\, \\sum _{z \\in {\\cal S}}\\, \\widehat{\\kappa } ( y,z) \\, \\big [ \\ell (t,z) - \\ell (t,y) \\big ] \\,=: \\, \\big ( \\widehat{{\\cal K}} \\, {\\mathbf {\\ell }} \\big ) (t,y)\\,,$ or equivalently $\\, \\partial {\\mathbf {\\ell }} (t ) = \\widehat{{\\cal K}} \\, {\\mathbf {\\ell }} (t)\\, $ in matrix form, with the new transition rates $\\widehat{{\\cal K}} \\,:=\\, \\Big ( \\widehat{\\kappa } ( y,z) \\Big )_{( y,z) \\in {\\cal S}^2}\\,, \\qquad \\widehat{\\kappa } ( y,z) \\,:=\\, \\frac{\\, q(z)}{q(y)} \\, \\kappa (z,y)\\,.$ The entries of this matrix $ \\widehat{{\\cal K}} $ are non-negative off the diagonal, and add up to zero $\\, \\sum _{z \\in {\\cal S}} \\widehat{\\kappa } ( y,z)=0\\,$ across every row $y \\in {\\cal S} ,$ on account of (REF ), (REF ).", "We shall think of $\\, ( {\\mathbf {\\ell }} (t) )_{0 \\le t < \\infty }\\,$ as a curve, now in the space ${\\cal L = L_+ (S)}$ of vectors $\\, \\Lambda = ( \\lambda (x) )_{x \\in {\\cal S}}\\,$ with strictly positive elements and $\\, \\sum _{x \\in {\\cal S}} q (x)\\,\\lambda (x)=1,$ viewed as likelihood ratios with respect to the invariant distribution and evolving in time via (REF ).", "Presently, we shall identify $\\, \\widehat{{\\cal K}}\\,$ of (REF ) with the infinitesimal generator of a suitable continuous-time Markov Chain, run backwards in time.", "A special case, however, is worth mentioning already.", "Definition 3.1 Detailed Balance: The invariant distribution $\\,Q\\,$ in (REF ) is said to satisfy the detailed-balance conditions, if $q(y) \\, \\kappa (y,z) \\,=\\, q(z) \\, \\kappa ( z,y) \\, , \\quad \\forall ~ (y,z) \\in {\\cal S}^2.$ This requirement turns out to be equivalent to the identity $\\,q(y) \\, \\varrho _t (y,z)= q(z) \\, \\varrho _t ( z,y)$ for all $ t \\in (0, \\infty ), ~(y,z) \\in {\\cal S}^2 \\,;$ one leg of the equivalence is immediate, courtesy of (REF ).", "When (REF ) prevails, $\\, \\widehat{{\\cal K}} \\equiv {\\cal K}\\,$ holds in (REF ); and the backward equation (REF ) for the likelihood ratios $\\, ( \\ell _t (x) )_{x \\in {\\cal S}}\\,$ of (REF ), is then exactly the same as the backward equation (REF ) for $\\, ( \\varrho _t ( x, y) )_{x \\in {\\cal S}}\\,$ .", "We stress that, whenever the detailed-balance conditions (REF ) are needed in the sequel, they will be invoked explicitly." ], [ "Discrete Gradient and Divergence; ", "It is apt at this point to introduce some necessary notation and functional-analytic notions.", "For a given function $\\,f : {\\cal S} \\rightarrow \\mathbb {R}\\,$ we consider the discrete gradient $\\,\\nabla f: {\\cal S}^2 \\rightarrow \\mathbb {R}\\,$ given by $\\nabla f (x,y): = f(y) - f(x)\\,.$ In a similar spirit, we consider the discrete divergence $\\big (\\nabla \\cdot F\\big ) (x ) \\,: =\\, \\frac{\\,1\\,}{2} \\sum _{y \\in {\\cal S}, \\, y \\ne x} \\kappa (x,y)\\,\\big [ F(x,y) - F(y,x) \\big ]$ of a function $\\, F : {\\cal S} \\times {\\cal S} \\rightarrow \\mathbb {R}\\,,$ and note the familiar concatenation formula ${\\cal K} f = \\nabla \\cdot \\big ( \\nabla f \\big ) \\,.$ This allows us to think of the operator ${\\cal K}$ in (REF ) also as a “discrete Laplacian\".", "We introduce also the set $\\, \\mathcal {Z}:= \\lbrace (x,y) \\in \\mathcal {S}\\times \\mathcal {S}\\ : \\ \\kappa (x,y) > 0\\rbrace \\,$ consisting of all edges in the incidence graph associated with the Markov chain, and the measure $\\,C$ on $\\,\\mathcal {Z}$ defined by the “conductances\" $C\\lbrace (x,y)\\rbrace \\equiv c(x,y) := \\frac{1}{2} \\,\\kappa (x,y)\\,q(x) \\,, \\qquad (x,y) \\in \\mathcal {Z}.$ With these ingredients, we consider the bilinear forms $\\big \\langle { f , g }\\big \\rangle _{L^2(\\mathcal {S}, Q)} \\,:=\\, \\sum _{x \\in {\\cal S}} \\, q(x)\\,f(x) \\,g(x)\\,,\\qquad \\big \\langle { F , G }\\big \\rangle _{L^2(\\mathcal {Z}, C)} \\,:= \\,\\sum _{(x,y) \\in {\\mathcal {Z}} } \\, c(x,y) \\,F(x,y)\\, G(x,y)$ for real-valued functions defined on ${\\cal S}$ (lowercase $f,\\,g$ ) and on ${\\cal S} \\times {\\cal S} $ (uppercase $F,\\,G$ ), respectively.", "They induce the $\\mathbb {L}^2-$ norms $ \\big \\Vert f \\big \\Vert _{L^2(\\mathcal {S}, Q)} $ (relative to the probability measure $Q$ ) and $ \\big \\Vert F \\big \\Vert _{L^2(\\mathcal {Z}, C)} $ (relative to the unnormalized measure $C$ on $\\mathcal {Z}$ in (REF )), given respectively via $\\begin{aligned}\\big \\Vert f \\big \\Vert _{\\mathbb {L}^2(\\mathcal {S}, Q)}^2\\,&:= \\big \\langle { f , f }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S}, Q)}= \\sum _{x \\in {\\cal S}} \\, q(x)\\,f^2(x)\\,, \\\\\\big \\Vert F \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, C)}^2\\,&:= \\big \\langle { F , F }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, C)}= \\sum _{(x,y) \\in {\\mathcal {Z}} } \\,c(x,y) \\,F^2(x,y)\\,.\\end{aligned}$ Remark 4.1 We note from (REF )-(REF ) the adjoint relationship $\\big \\langle { f, \\widehat{{\\cal K}} g }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S}, Q)} \\,=\\, \\big \\langle { {\\cal K} f, g }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S}, Q)}\\,.$ Thus (REF ) holds if, and only if, the operator ${\\cal K}$ in (REF ) is self-adjoint on $ \\mathbb {L}^2 ({\\cal S}, Q).$ Finally, we introduce the bilinear Dirichlet form associated with the Markov Chain: ${\\cal E} (f,g)\\, := \\,- \\big \\langle { f, {\\cal K} g }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S}, Q)}\\,= \\,- \\sum _{y \\in {\\cal S}} \\, q(y)\\, f(y) \\, \\big ( {\\cal K} g \\big ) (y)\\,=\\,- \\, \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\, q(y) \\, \\kappa (y,x) \\, f ( y) \\, g ( x).$ This form is not symmetric, in general; but satisfies $\\,{\\cal E} (f,f)\\ge 0\\,,$ as follows from Lemma REF below.", "Lemma 4.1 The Dirichlet form (REF ) can be cast equivalently as ${\\cal E } (f,g) = \\frac{1}{\\,2\\,} \\, \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) \\, \\big ( f ( y) -g ( x) \\big )^2\\,.$ Proof: We have clearly $\\,\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) f^2 ( y)=0\\,$ on account of $\\, \\sum _{x \\in {\\cal S}} \\, \\kappa ( y,x )=0\\,$ for every $y \\in {\\cal S} \\, $ ; as well as $\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) \\, g^2 ( x)\\,=\\, \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x,y) \\, q(x) \\, g^2 ( x) \\,=\\,0\\,,$ from the adjoint rates of (REF ) and their property $\\, \\sum _{y \\in {\\cal S}} \\,\\widehat{\\kappa } ( x, y )=0\\,$ for every $x \\in {\\cal S} .$ It follows from (REF ) that $~~~~~~~~~~ ~~\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) \\, \\big ( f ( y) -g ( x) \\big )^2\\,= \\, - 2\\,\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) \\, f ( y) \\, g ( x)\\,= \\,2\\, {\\cal E } (f,g) \\,.", "\\qquad \\Box $" ], [ "Consequences of Detailed Balance ", "The detailed-balance conditions (REF ) can be thought of as positing that “the conductances of (REF ) do not depend on the direction of the current's flow\".", "Under these conditions, we have for functions $\\,f : {\\cal S} \\rightarrow \\mathbb {R}\\,$ and $\\, F : {\\cal S} \\times {\\cal S} \\rightarrow \\mathbb {R}\\, $ the discrete integration-by-parts formula $\\big \\langle { \\, \\nabla f , F }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\,=\\,- \\,\\big \\langle { f , \\nabla \\cdot F }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)} ,$ in addition to the concatenation property (REF ).", "As a result, the bilinear Dirichlet form of (REF ) is now symmetric, and induces the Hilbert $\\, \\mathbb {H}^{ 1}-$ inner product and norm $\\big \\langle { f , g}\\big \\rangle _{\\mathbb {H}^1(\\mathcal {S},Q)} \\,:=\\, {\\cal E} (f,g) \\,=\\, \\Big \\langle {\\nabla f , \\nabla g}\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\,,$ $\\big \\Vert f \\big \\Vert ^2_{\\mathbb {H}^1(\\mathcal {S},Q)}\\,:= \\,{\\cal E} (f,f)\\,= \\,\\sum _{(x,y) \\in {\\cal Z} } \\, c(x,y)\\, \\big ( f ( y) - f ( x) \\big )^2\\,=\\,- \\big \\langle f, {\\cal K} f \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}\\,=\\, \\big \\Vert \\nabla f \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)}^2\\,,$ respectively.", "We introduce also the dual of this norm, the Hilbert $\\, \\mathbb {H}^{-1}-$ norm $\\,\\big | \\big | f \\big | \\big |_{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\,,$ via $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\,:= \\,\\big \\Vert \\nabla \\big ({\\cal K}^{-1} f \\big ) \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)}\\,, ~~~~ \\text{if} ~ f \\in \\text{Range} ({\\cal K})\\,; \\qquad \\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)} \\,:= \\, + \\infty \\,,~~ ~~ \\text{otherwise;}$ and note the variational characterizations $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\,=\\, \\sup _{g : {\\cal S} \\rightarrow \\mathbb {R}} \\frac{\\,\\big \\langle f, g \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}\\,}{\\, \\big \\Vert g \\big \\Vert _{\\mathbb {H}^1(\\mathcal {S},Q)}\\,} \\,,$ $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\,=\\,\\inf _{F : {\\cal Z} \\rightarrow \\mathbb {R}} \\big \\lbrace \\big \\Vert F \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)}: f = \\nabla \\cdot F \\big \\rbrace \\,=\\, \\inf _{g : {\\cal S} \\rightarrow \\mathbb {R}} \\big \\lbrace \\big \\Vert \\nabla g\\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)}: f = {\\cal K} g \\big \\rbrace \\,.$ Basic Hilbert space theory shows that these two infima are attained.", "Lemma 4.2 Under the conditions of (REF ), the expression (REF ) for the Dirichlet form becomes ${\\cal E } (f,g)= \\frac{1}{\\,2\\,} \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\kappa (y,x) \\big [ f ( y) -f ( x) \\big ] \\,\\big [ g ( y) - g ( x) \\big ] =\\Big \\langle { \\nabla f , \\nabla g }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\langle f , g \\big \\rangle _{\\mathbb {H}^1(\\mathcal {S},Q)} .~~$ Proof: Let us write the double summation in the above display as $\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) q(y) \\, \\Big [ \\, f ( y) \\, g( y)- f ( y) \\, g ( x) - f( x) \\, g ( y)+ f ( x) \\, g ( x)\\, \\Big ]=$ $=\\,- \\,\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) q(y) \\, \\Big [ \\, f ( y) \\, g ( x) + f ( x) \\, g ( y)\\, \\Big ]\\,=\\, - \\,2\\,\\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) q(y) \\, f ( y) \\, g ( x) \\,=\\, - \\, 2\\, {\\cal E } (f,g)\\,.$ Here, the first equality uses (REF ), as well as the properties $\\, \\sum _{x \\in {\\cal S}} \\, \\kappa ( y,x )=0\\,$ for every $y \\in {\\cal S} ,$ and $\\, \\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x ,y )=0\\,$ for every $x \\in {\\cal S} ; $ whereas, the second equality uses the conditions (REF ), and the third equality is just (REF ).", "This proves the first equality in (REF ).", "The second and third are just restatements of (REF ).", "$\\Box $ Remark 4.2 Additional Consequences: It follows from (REF )–(REF ) that, under the detailed-balance conditions (REF ), the mapping $\\, \\nabla \\,: ~{\\mathbb {H}^1 (\\mathcal {S}, Q)} \\rightarrow {\\mathbb {L}^2(\\mathcal {Z},C)}$ is an isometric embedding.", "Whereas, the discrete divergence mapping $\\, \\nabla \\cdot \\,$ in (REF ) is, up to a minus sign, the adjoint of the mapping $\\, \\nabla \\,: ~\\mathbb {L}^2(\\mathcal {S},Q) \\rightarrow \\mathbb {L}^2(\\mathcal {Z},C)$ .", "Remark 4.3 A Counterexample.", "In the absence of detailed balance, the Dirichlet form $\\,{\\cal E } (f,g) $ is not an inner product; indeed, Remark REF shows that there exist functions $f :{\\cal S} \\rightarrow \\mathbb {R}\\,,$ $\\,g: {\\cal S} \\rightarrow \\mathbb {R}$ with $\\, {\\cal E}(f,g)=- \\big \\langle f, {\\cal K} g \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)} \\ne - \\big \\langle g, {\\cal K} f \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= {\\cal E}(g,f)$ .", "An explicit example of this situation is provided by the matrix ${\\cal K}\\,=\\,\\begin{pmatrix}\\,-1 & 1 & 0~ \\\\\\,0 & -1 & 1~ \\\\\\,1 & 0 & -1 ~\\end{pmatrix},$ whose invariant distribution $\\,Q =(1/3, 1/3, 1/3)$   is uniform on the state space $\\,{\\cal S} = \\lbrace 1, 2, 3\\rbrace \\, $ and for which detailed balance fails.", "Whereas, with $\\,f = \\mathfrak {e}_1 = (1, 0, 0)\\,$ and $\\,g = \\mathfrak {e}_2 = (0, 1, 0)\\,$ the first and second unit row vectors, respectively, and noting $\\, 3\\, {\\cal E} (\\varphi , \\gamma ) = \\varphi \\, {\\cal K}^{\\prime } \\gamma ^{\\prime }\\,$ from (REF ), we observe $3\\, {\\cal E} (f, g) = \\big (1, 0, 0\\big ) \\begin{pmatrix}\\, 1 ~ \\\\\\, -1 ~ \\\\\\, 0 ~\\end{pmatrix} = \\, -1\\,, \\qquad 3\\, {\\cal E} ( g, f) = \\big ( 0,1, 0\\big ) \\begin{pmatrix}\\, -1 ~ \\\\\\, 0 ~ \\\\\\, 1 ~\\end{pmatrix} = \\, 0\\,.$ Nevertheless, $\\,\\big \\Vert f \\big \\Vert _{\\mathbb {H}^1(\\mathcal {S},Q)} = \\sqrt{{\\cal E} (f,f)\\,}\\,$ is always a Hilbert norm, with associated inner product given by the Dirichlet form $\\, {\\cal E}_{\\text{sym}} (f,g)\\,$ of the reversible Markov Chain, with symmetrized rates $\\, \\kappa _{\\text{sym}} (x,y) := ( \\kappa (x,y) + \\widehat{\\kappa } (x,y) ) /2\\,$ in the manner of (REF ), (REF ); namely, $\\, {\\cal E}_{\\text{sym}} (f,f) \\equiv {\\cal E} (f,f)\\, $ and $\\big \\langle f,g \\big \\rangle _{\\mathbb {H}^1(\\mathcal {S},Q)} = -\\frac{1}{2} \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\big [ q(y) \\kappa (y,x) + q(x) \\kappa (x,y) \\big ] f(x) g (y) =\\,- \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\, q(y) \\, \\kappa _{\\text{sym}} (y,x) \\, f ( y) \\, g ( x) .$" ], [ "Time Reversal and Associated Martingales", "It is well known that the Markov property is invariant under reversal of time (interchanging the roles of “past\" and “future\", keeping the “present\" as is).", "This means, in particular, that the time-reversed process $\\widehat{X} (s) : = X (T-s)\\,, ~~ ~~ 0 \\le s \\le T\\,$ is a Markov Chain, for any given $T \\in (0, \\infty )$ .", "But how about the transition probabilities of this time-reversed process?", "These are fairly easy to compute, namely, $ \\mathbb {P}\\big ( \\widehat{X}(s_2) =z \\, \\big |\\, \\widehat{{\\cal G}} (s_1 ) \\big )=\\, \\mathbb {P}\\big ( \\widehat{X}(s_2) =z \\, \\big |\\, \\widehat{X}(s_1) \\big ) = \\,\\rho ^* \\big (s_1, \\widehat{X}(s_1) ; s_2, z \\big )$ for $\\, 0 \\le s_1 \\le s_2 \\le T $ , $\\, z \\in {\\cal S} $ , where $\\rho ^* \\big (s_1, y ; s_2, z \\big ) \\,:=\\, \\frac{\\, p (T-s_2, z)\\,}{p (T-s_1, y)} \\,\\, \\varrho _{s_2 - s_1} \\big ( z, y\\big )\\,;$ but need not be time-homogeneous in general.", "However: Let us compute these same transition probabilities when the Chain starts at its invariant distribution $Q$ .", "We introduce at this point another probability measure $\\,\\mathbb {Q}\\,$ on the underlying measurable space $\\, ( \\Omega , {\\cal F}),$ under which the Markov Chain $\\, {\\cal X}\\,$ has exactly the same dynamics as before, but its initial distribution is the invariant probability vector $ Q = \\big ( q (y) \\big )_{y \\in {\\cal S}}\\,$ in (REF ).", "Then, in lieu of (REF ), the finite-dimensional distributions of the Chain are $\\mathbb {Q} \\big ( X(0 )=x , X(\\theta _1) = y_1, \\cdots , X(\\theta _n)= y_n, X(t) =z ) \\,=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \\,q ( x)\\, \\varrho _{\\theta _1} (x,y_1)\\, \\varrho _{\\theta _2 - \\theta _1} (y_1,y_2)\\cdots \\varrho _{\\theta _n - \\theta _{n-1}} (y_{n-1},y_n) \\cdot \\varrho _{t - \\theta } (y ,z)\\,.$ On each $\\sigma -$ algebra $\\,{\\cal F}^X (t),$ $\\, 0 \\le t < \\infty ,$ the two probability measures $ \\mathbb {P}$ and $\\mathbb {Q}$ are equivalent; in fact, on the smaller $\\sigma -$ algebra $\\, \\sigma (X(t)) \\,,$ we single out in the notation of (REF ) the so-called likelihood process $L (t)\\,:=\\,\\frac{\\,\\mathrm {d} \\mathbb {P}\\,}{\\mathrm {d}\\mathbb {Q}}\\bigg |_{\\sigma (X(t)) } \\,=\\, \\ell \\big ( t, X(t) \\big ) \\,, \\qquad 0 \\le t < \\infty \\,.$ Under this dispensation, the transition probabilities are $\\mathbb {Q} \\big ( \\widehat{X}(s_2) =z \\, \\big |\\, \\widehat{{\\cal G}} (s_1 ) \\big )=\\, \\mathbb {Q}\\big ( \\widehat{X}(s_2) =z \\, \\big |\\, \\widehat{X}(s_1) \\big ) = \\, \\widehat{\\varrho }_{s_2-s_1} \\big ( \\widehat{X}(s_1) , z \\big ) ,$ i.e., time-homogeneous, with $\\widehat{\\varrho }_{h} (y,z) \\,:=\\, \\frac{\\, q ( z)\\,}{q (y)} \\,\\, \\varrho _{h} \\big ( z, y\\big )\\,.$ Invoking (REF ) and (REF ), we see that the $\\, \\mathbb {Q}-$ infinitesimal-generator of this time-reversed Markov Chain $ \\widehat{X} (s) = X (T-s), \\,~ 0 \\le s \\le T$  in (REF ), is given precisely by $\\, \\widehat{{\\cal K}} = ( \\widehat{\\kappa } ( y,z ) )_{(y,z) \\in {\\cal S}^2}\\,$ as in (REF ).", "(We note parenthetically that, when the detailed-balance conditions (REF ) hold, the initial distributions and transition probabilities of the continuous-time Markov Chain $ X(t),\\, 0 \\le t \\le T ,$ and of its time-reversal (REF ), are exactly the same under the probability measure $\\mathbb {Q}$ .)", "Remark 5.1 The standing assumption $P(0) \\in {\\cal M},$ i.e., that all entries of the initial distribution are strictly positive, is made for economy of exposition.", "For even the probability vector $P(0)$ belongs to the closure $ \\overline{{\\cal M}} $ of this manifold, i.e., some of its entries are allowed to vanish, there is at least one $x \\in {\\cal S} $ with $p(0,x)>0$ ; then (REF ) and irreducibility imply $p(t,y) >0$ for all $t>0,\\, y \\in {\\cal S}.$ Thus, even if the curve $ ( P (t) )_{0 \\le t < \\infty }\\,$ starts out on the boundary $ \\, \\overline{{\\cal M}} \\setminus {\\cal M},$ it enters $ {\\cal M} $ immediately and stays there for all times $\\,t \\in (0, \\infty ).$ By complete analogy with Proposition REF , we formulate now the following result.", "Proposition 5.1 For any given function $\\, g : [0, T] \\times {\\cal S} \\rightarrow \\mathbb {R}\\,$ whose temporal derivative $ \\, s \\mapsto \\partial g (s, x) \\, $ is continuous for every state $\\, x \\in {\\cal S} ,$  the process below is a $\\,\\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-\\,$ local martingale: $\\widehat{M}^{\\,g} (s) \\, := \\, g \\big ( s,\\widehat{X}(s) \\big ) - \\int _0^s \\big ( \\partial g + \\widehat{{\\cal K}} g \\big ) \\big (u, \\widehat{X} (u) \\big )\\, \\mathrm {d}u \\,, \\qquad 0 \\le s \\le T\\,.$ The following important result is due to Fontbona & Jourdain (2016) in the context of diffusions.", "Its proof (cf.", "Theorem 4.2 in Karatzas, Schachermayer & Tschiderer (2019)) uses only the Markov property and the definition of conditional expectation, and carries over verbatim to our present context.", "An alternative argument, specific to the Markov Chain context, uses Proposition REF and is given right below.", "Proposition 5.2 Time-Reversed Likelihood Process as Martingale: Fix $\\,T \\in (0, \\infty )\\,$ and consider the time-reversed Chain (REF ), as well as the filtration $\\, \\widehat{\\mathbb {G}} = \\big \\lbrace \\widehat{{\\cal G}} (s) \\big \\rbrace _{0 \\le s \\le T}\\,$ this process generates via $\\, \\widehat{{\\cal G}} (s) := \\sigma \\big ( \\widehat{X} (u), ~ 0 \\le u \\le s \\big ).$ Then, the time-reversed likelihood process $L(T-s)\\,=\\, \\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\,, \\quad 0 \\le s \\le T~~~~~~~\\text{is a}~\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )- \\text{martingale.", "}$ Proof: We consider in (REF ) the function $\\,g(s, x) = \\ell (T-s, x)\\,, ~\\, 0 \\le s \\le T, ~~ x \\in {\\cal S}\\,$ and note that $ \\partial g(s, x) = - \\partial \\ell (T-s, x) = - \\big ( \\widehat{{\\cal K}} \\, \\ell \\big ) (T-s, x) \\, $ holds on account of (REF ).", "It follows from (REF ), whose integrand now vanishes, that the time-reversed likelihood ratio process $ \\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\,, ~ 0 \\le s \\le T \\,$ is a $\\, \\mathbb {Q}-$ local-martingale of the time-reversed filtration $\\,\\widehat{\\mathbb {G}}\\,$ .", "But this process is positive, thus also a $ \\mathbb {Q}-$ supermartingale, and its expectation $\\mathbb {E^Q} \\big [ \\ell \\big (T-s, X(T-s) \\big ) \\big ] \\,=\\, \\sum _{y \\in {\\cal S}} \\, q(y)\\, \\frac{\\,p (T-s, y)\\,}{q(y)}\\,=\\,1\\,, \\qquad 0 \\le s \\le T$ is constant.", "Therefore $ \\,\\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\,, ~ 0 \\le s \\le T \\,$ is a true $\\, \\mathbb {Q}-$ martingale, exactly as stated in (REF ).", "$\\Box $" ], [ "The Variance Process", "For a probability vector $\\, P = ( p(y) )_{y \\in {\\cal S}}$ with positive components, we introduce its likelihood vector ${\\mathbf {\\ell }} = ( \\ell (y) )_{y \\in {\\cal S}} \\in {\\cal L}\\,$ with $\\, \\ell (y) = p(y) / q(y)$ as in (REF ), relative to the invariant distribution $Q$ of the Chain.", "We define then in the manner of (REF ) the Variance of $P$ relative to $Q,$ also known as $\\chi ^2-$ divergence, as $V \\big (P\\,|\\,Q\\big ) \\, \\equiv \\, \\text{Var}^\\mathbb {Q} \\big ( {\\mathbf {\\ell }} \\big )\\, := \\, \\sum _{y \\in {\\cal S}} \\, q (y) \\,\\ell ^2 ( y) -1 \\,= \\,\\big | \\big | {\\mathbf {\\ell }} \\big | \\big |_{\\mathbb {L}^2(\\mathcal {S},Q)}^2 -1 \\,.$ Let us recall now from (REF ) the curve $ \\big ( P(t) \\big )_{0 \\le t < \\infty } $ of time-marginal distributions for our continuous-time Markov Chain, and the corresponding curve of likelihoods $\\,{\\mathbf {\\ell }}_t = ( \\ell (t,y) )_{y \\in {\\cal S}}\\,,~ 0 \\le t < \\infty \\,$ in the space $ {\\cal L}\\,,$ with $\\, \\ell (t,y) = p(t,y) / q(y)$ .", "We will show in Proposition REF that the variance just defined in (REF ) plays the role of Lyapunov function for the convergence to equilibrium along this curve.", "To see this, we summon the likelihood process $\\, L(t) = \\ell ( t, X(t)), ~ 0 \\le t < \\infty \\,$ from (REF ) and consider its square $\\, L^2(t) , ~ 0 \\le t < \\infty \\,,$ the so-called Variance Process, under time-reversal.", "Proposition 6.1 For any given $T\\in (0, \\infty ),$ we have the Doob-Meyer decomposition $\\ell ^2 \\big ( T-s, \\widehat{X} (s) \\big ) \\,=\\,\\widehat{M} (s) + \\int _0^s \\sum _{ y \\ne x } \\bigg ( \\widehat{\\kappa } (x,y) \\, \\Big ( \\ell (t, y) - \\ell (t, x ) \\Big )^2 \\bigg )\\bigg |_{t = T-u \\atop x = \\widehat{X} (u)}\\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T~~~~$ of the time-reversed variance process $\\, \\ell ^2 \\big ( T-s, \\widehat{X} (s) \\big ) ,$ $\\,0 \\le s \\le T\\,,$ where $\\widehat{M}$ is a $ \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ martingale.", "Proof: The first claim follows from Proposition REF and the Jensen inequality.", "For the second claim we deploy Proposition REF with $\\,g(s, x) := \\ell ^2 (T-s, x)\\,, ~\\, 0 \\le s \\le T, ~~ x \\in {\\cal S}\\,,$ to conclude via the calculation $\\big ( \\partial g + \\widehat{{\\cal K}} g \\big ) (T-s, x) \\,= \\,\\sum _{y \\in {\\cal S} } \\, \\,\\widehat{\\kappa } (x,y) \\Big ( \\ell (T-s, y) - \\ell (T-s, x) \\Big )^2$ that $\\widehat{M}$ is a local $\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ martingale.", "The uniform continuity of $ [0,T] \\ni t \\mapsto p_t (x,y) \\in [0,1] $ and the finiteness of the state-space imply that this process is actually bounded, thus a true $\\, \\mathbb {Q}-$ martingale.", "Let us now justify the claim (REF ).", "From the Backwards Equation (REF ), we have $\\partial g (T-s, x) \\,= - 2 \\ell (T-s, x)\\, \\partial \\ell (T-s, x) = - 2 \\ell (T-s, x) \\sum _{y \\in {\\cal S}} \\widehat{\\kappa } (x,y) \\ell (T-s, y),$ $\\big ( \\widehat{{\\cal K}} g \\big ) (T-s, x) \\,= \\,\\sum _{y \\in {\\cal S} } \\, \\widehat{\\kappa } (x,y) \\, \\ell ^2 (T-s, y)\\,= \\,\\sum _{y \\in {\\cal S} } \\, \\widehat{\\kappa } (x,y) \\, \\Big [ \\ell ^2 (T-s, y) + \\ell ^2 (T-s, x) \\Big ]$ on account of the property $\\, \\sum _{y \\in {\\cal S}} \\widehat{\\kappa } ( x, y )=0\\,$ for every $x \\in {\\cal S} ;$ now (REF ) follows readily.", "$\\Box $ Proposition REF deals with the trajectorial behavior of the variance process; and for this, it is crucial to let time run backwards.", "Now, we want to adopt also an “aggregate\" point of view, and take $\\mathbb {Q}-$ expectations in (REF ).", "When doing this, it does not matter any more whether time runs forwards or backwards, so we state the following result “forwards in time\".", "Recalling (REF ), we obtain thus the dissipation of the variance.", "Proposition 6.2 Along the curve $ \\big ( P(t) \\big )_{0 \\le t < \\infty } $ of time-marginal distributions in (REF ), the variance $V \\big (P(t)\\,|\\,Q\\big )= {\\rm Var}^\\mathbb {Q} \\big ( {\\mathbf {\\ell }}_t \\big )= \\sum _{y \\in {\\cal S}} \\, q (y) \\,\\ell ^2 (t, y) -1 \\,= \\, \\big \\Vert {\\mathbf {\\ell }}_t \\big \\Vert _{\\mathbb {L}^2(\\mathcal {S},Q)}^2 -1\\,, \\quad 0 \\le t < \\infty \\,\\,$ is decreasing with $\\, \\lim _{t \\rightarrow \\infty } \\downarrow V \\big (P(t)\\,|\\,Q\\big )=0 ,$ and the rate of its decrease is given by $\\partial \\, \\big \\Vert {\\mathbf {\\ell }}_t \\big \\Vert _{\\mathbb {L}^2(\\mathcal {S},Q)}^2 \\,=\\, \\partial \\, V \\big (P(t)\\,|\\,Q\\big ) \\,=\\, - 2 \\, {\\cal E} \\big ( {\\mathbf {\\ell }}_t\\, , {\\mathbf {\\ell }}_t \\,\\big )$ (thus by $\\, - 2\\, \\big \\Vert {\\mathbf {\\ell }}_t \\big \\Vert ^2_{\\mathbb {H}^1(\\mathcal {S},Q)} \\,$ under the detailed-balance conditions (REF )).", "More precisely, $V \\big (P(T)\\,|\\,Q\\big ) \\,= \\,V \\big (P(0)\\,|\\,Q\\big ) - \\int _0^T \\sum _{(x,y) \\in {\\cal Z} } \\, q (y) \\, \\kappa (y,x)\\, \\Big ( \\ell (t,y) - \\ell (t,x) \\Big )^2 \\, \\mathrm {d}t$ $= \\int _T^\\infty \\sum _{(x,y) \\in {\\cal Z} } \\, q (y) \\, \\kappa (y,x) \\, \\Big ( \\ell (t,y) - \\ell (t,x) \\Big )^2 \\, \\mathrm {d}t \\,.$ The decomposition (REF ) is a trajectorial version of this variance dissipation, at the level of the individual particle viewed under the probability measure $\\mathbb {Q}$ and under time-reversal.", "As a consequence of (REF ) and of the Bayes rule, we deduce from (REF ) the Doob-Meyer decomposition $\\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\,=\\,\\widehat{N} (s) + \\int _0^s \\sum _{ y \\ne x } \\bigg ( \\frac{\\, \\widehat{\\kappa } (x,y)\\,}{\\, \\ell \\big ( t, x \\big )\\,} \\, \\Big ( \\ell (t, y) - \\ell (t, x ) \\Big )^2 \\bigg )\\bigg |_{t = T-u \\atop x = \\widehat{X} (u)} \\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T~~~~$ of the time-reversed likelihood process, where $\\widehat{N}$ is a $\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {P}\\big )-$ martingale." ], [ "Steepest Descent of the Variance, under Detailed Balance ", "We will establish now the following Theorem REF .", "As pointed out in Jordan, Kinderlehrer & Otto (1998), results of this type go as far back as the paper by Courant, Friedrichs & Lewy (1928) in the Brownian motion context.", "We deploy the notation of (REF ) for the likelihood ratios relative to the invariant distribution, as well as the following notion.", "Definition 6.1 We say that a smooth curve of probability measures $ ( P (t) )_{t_0 \\le t < \\infty } \\subset \\mathcal {M}= \\mathcal {P}_+(\\mathcal {S})$ is of steepest descent locally at $\\,t = t_0\\,,$ for a given smooth functional $F : \\mathcal {M}\\rightarrow \\mathbb {R}$ and relative to a given metric $\\varrho $ on $\\mathcal {M},$ if it minimizes, among all curves $ \\, ( \\widetilde{P} (t) )_{t_0 \\le t < \\infty } \\subset \\mathcal {M}\\,$ satisfying $\\widetilde{P} (t_0) = P (t_0),$ the infinitesimal rate of change of $F$ as measured on $\\mathcal {M}$ in terms of $\\varrho \\,,$ namely, $\\lim _{h \\downarrow 0} \\,\\frac{\\, F \\big (\\widetilde{P} (t_0 +h) \\big ) - F \\big (P (t_0) \\big )}{\\,\\varrho \\big ( \\widetilde{P} (t_0+h), P (t_0) \\big )\\, } \\,.$ Theorem 6.3 Steepest Descent for the Variance: Under the conditions (REF ) of detailed balance, the curve $\\, ( P (t) )_{0 \\le t < \\infty }\\,$ of time-marginal distributions in (REF ) has the property of steepest decent for the variance of (REF ) with respect to the metric distance bequeathed by the norm of (REF ), i.e., $\\varrho \\big ( P_1, P_2 \\big )\\,:= \\, \\big \\Vert \\, {\\mathbf {\\ell }}_1 - {\\mathbf {\\ell }}_2 \\,\\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S}, Q)} \\qquad \\text{ for \\, $P_1 = {\\mathbf {\\ell }}_1 Q$ \\, and $~P_2 = {\\mathbf {\\ell }}_2\\, Q\\,.$}$ The proof of this result needs Proposition REF below.", "We pave the way towards it by formulating first a variational version of Propositions REF , REF .", "For this purpose, we fix an arbitrary time-point $t_0 \\in (0, \\infty )$ and let $ \\psi (\\cdot ) = ( \\psi (t) )_{t_0 \\le t < t_0 + \\varepsilon }$ be a continuous curve of real-valued functions on the state-space ${\\cal S}$ .", "With these ingredients, we define a new curve $\\, \\ell ^\\psi (\\cdot ) = ( \\ell ^\\psi (t) )_{t_0 \\le t < t_0 + \\varepsilon }$ of such functions, for a suitable $\\varepsilon >0,$ by specifying in the space $\\, {\\cal L = L_+ (S)}\\,$ of subsection REF the initial condition $\\ell ^\\psi (t_0) = {\\mathbf {\\ell }} (t_0) \\in {\\cal L}$ and the dynamics $\\, \\partial \\ell ^\\psi (t) = ( \\widehat{{\\cal K}} \\psi ) (t)$ for $\\, t \\in [t_0, t_0 + \\varepsilon );$ in the manner of (REF ) and a bit more explicitly, $\\partial \\ell ^\\psi (t,x) \\,=\\, \\sum _{y \\in {\\cal S}}\\, \\widehat{\\kappa } ( x,y) \\, \\psi (t,y) \\,,\\qquad x \\in {\\cal S}.$ The curve $\\, \\ell ^\\psi (\\cdot ) = ( \\ell ^\\psi (t) )_{t_0 \\le t < t_0 + \\varepsilon }\\,,$ the “output\" of the system (REF ) corresponding to the “input\" $ \\psi (\\cdot )$ , is only defined on an interval $\\, [ t_0 , t_0 + \\varepsilon )\\,$ and lives in the space $\\, {\\cal L} ,$ since $\\partial \\, \\sum _{x \\in {\\cal S}} \\,q(x) \\,\\ell ^\\psi (t,x)\\,=\\, \\sum _{x \\in {\\cal S}} \\,q(x) \\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } (x,y)\\, \\psi (t,y)\\,=\\, \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\,q(y) \\, \\kappa ( y,x)\\, \\psi (t,y)\\,=\\,0$ implies $ \\sum _{x \\in {\\cal S}} \\,q(x) \\,\\ell ^\\psi (t,x) = \\sum _{x \\in {\\cal S}} \\,q(x) \\, {\\mathbf {\\ell }} (t_0,x) =1$ for all $ t \\in [t_0 + \\varepsilon )$ .", "Thus, the recipe $p^\\psi (t,x):= q(x) \\,\\ell ^\\psi (t,x), \\qquad (t,x) \\in [t_0, t_0 + \\varepsilon ) \\times {\\cal S}$ procures a curve $\\, ( P^\\psi (t))_{0 \\le t < t_0 + \\varepsilon }\\,, $ on the manifold $\\,{\\cal M = P_+ (S)}\\,$ in subsection REF consisting of vectors $\\, P = \\big ( p (x) \\big )_{x \\in {\\cal S}}\\,$ with strictly positive elements and total mass $\\, \\sum _{x \\in {\\cal S}} p (x)=1\\,.$ Conversely: By irreducibility, the “input curve\" $\\, \\psi (\\cdot )\\,$ is determined by the “output curve\" $\\, \\ell ^\\psi (\\cdot )\\,$ up to an additive time-dependent constant.", "In particular, every smooth curve $\\, \\ell ^* (\\cdot ) = ( \\ell ^* (t) )_{t_0 \\le t < t_0 + \\varepsilon }$ in $\\, {\\cal L}\\,$ with $\\, \\ell ^* (t_0) = \\ell (t_0)\\,$ is representable as $\\, \\ell ^\\psi (\\cdot )\\,$ for a suitable continuous $\\, \\psi (\\cdot )\\,$ as above.", "For instance, $ \\, {\\mathbf {\\ell }} (\\cdot ) \\in {\\cal L}\\,$ of (REF ) is the “output\" that corresponds in this manner to the “input\" $ \\, \\psi (\\cdot ) \\equiv {\\mathbf {\\ell }} (\\cdot ) $ in (REF ), via (REF ).", "We have the following generalization of Proposition REF , to which it reduces when $ \\, \\psi (\\cdot ) \\equiv {\\mathbf {\\ell }} (\\cdot ) .$ Proposition 6.4 In the above context, we have for $\\,t \\in [t_0 + \\varepsilon )\\,$ the properties $\\partial \\,V \\big (P^\\psi (t)\\,|\\,Q\\big )= \\partial \\, \\mathbb {E^Q} \\Big [ \\big ( \\ell ^\\psi \\big )^2 \\big (t, X (t) \\big ) \\Big ]= 2 \\,\\big \\langle \\psi _t, {\\cal K} \\ell ^\\psi _t \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)} = -2 \\, {\\cal E} \\big ( \\psi _t, \\ell ^\\psi _t \\big ) .$ Whereas, under the detailed balance conditions (REF ), this expression becomes $\\partial \\,V \\big (P^\\psi (t)\\,|\\,Q\\big )\\,=\\,-2 \\, {\\cal E} \\big ( \\ell ^\\psi _t , \\psi _t \\big )\\,=\\, -2\\, \\Big \\langle { \\, \\nabla \\ell ^\\psi _t , \\nabla \\psi _t }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)} \\,=\\, -2\\, \\Big \\langle \\ell ^\\psi _t , \\psi _t \\Big \\rangle _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,.$ Proof: A reasoning similar to that in Propositions REF and REF , and carried out once again in the backwards direction of time, can be deployed by applying Proposition REF to $\\,g(s, x) := \\big ( \\ell ^\\psi \\big )^2 (T-s, x)\\,, ~~\\, 0 \\le s \\le T,$ $ x \\in {\\cal S}\\, $ for arbitrary but fixed $\\, T \\in (0, t_0 + \\varepsilon )\\,.$ But here is a simpler argument : $\\partial \\,V \\big (P^\\psi (t)\\,|\\,Q\\big )= \\,\\partial \\big \\Vert \\ell ^\\psi _t \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {S},Q)} = \\, 2 \\, \\big \\langle \\ell ^\\psi _t , \\widehat{{\\cal K}} \\psi _t \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)} = \\,2 \\,\\big \\langle \\psi _t , {\\cal K} \\ell ^\\psi _t \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)} = -2 \\, {\\cal E} \\big ( \\psi _t, \\ell ^\\psi _t \\big )$ on account of (REF ), (REF ) and (REF ).", "This reasoning proves Proposition REF as well.", "$\\Box $ We compute now the “infinitesimal cost of moving the curve\" $\\, \\big ( \\ell ^\\psi (t) \\big )_{t_0 \\le t < t_0 + \\varepsilon }\\,.", "$ Proposition 6.5 Under the conditions (REF ) of detailed balance, we have $\\lim _{h \\downarrow 0}\\, \\frac{1}{h}\\, \\big \\Vert \\,{\\mathbf {\\ell }}_{t+h} - {\\mathbf {\\ell }}_{t} \\,\\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\, = \\,\\big \\Vert \\, {\\cal K} \\,{\\mathbf {\\ell }}_{t} \\, \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)} \\, = \\,\\big \\Vert \\, {\\mathbf {\\ell }}_{t} \\, \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)}$ for every $t \\in [t_0, t_0 + \\varepsilon ); $ and a bit more generally, in the notation just developed, $\\lim _{h \\downarrow 0}\\, \\frac{1}{h}\\, \\big \\Vert \\, \\ell _{t+h}^\\psi - \\ell _{t}^{ \\psi } \\,\\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)}\\, = \\,\\big \\Vert \\, {\\cal K} \\, \\psi _{t} \\, \\big \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S},Q)} \\, = \\,\\big \\Vert \\, \\psi _{t} \\, \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,.$ Proof: From (REF ), (REF ) it follows that for every $\\, x \\in {\\cal S}\\, $ we have $\\lim _{h \\downarrow 0}\\,\\frac{1}{\\,h\\,} \\Big [ \\ell ^{ \\psi }_{t+h} (x) - \\ell ^{ \\psi }_{t } ( x) \\Big ] \\,=\\, \\big ( {\\cal K} \\, \\psi _t \\big ) ( x) ,$ so the first equality in (REF ) is evident.", "For the second equality in (REF ) it suffices to recall (REF )–(REF ), observe that $\\, \\nabla \\psi _t\\,$ is the unique element $\\, F \\in \\mathbb {L}^2(\\mathcal {Z},C)\\,$ with the property $\\, \\nabla \\cdot F = {\\cal K} \\,\\psi _t\\,,$ and note from Remark REF the isometry $\\,\\big \\Vert F \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)} = \\big \\Vert \\psi _t \\big \\Vert _{\\mathbb {H}^1(\\mathcal {S},Q)}\\,$ from the space $\\mathbb {L}^2(\\mathcal {Z},C)$ to $\\mathbb {H}^1(\\mathcal {S}, Q)$ .", "Now, (REF ) is just a special case of (REF ) with $ \\, \\psi (\\cdot ) \\equiv {\\mathbf {\\ell }} (\\cdot ) ,$ as discussed above.", "$\\Box $" ], [ "The Proof of Theorem ", "We are ready to tackle the proof of Theorem REF .", "Along any smooth curve of the form $\\, ( P^\\psi (t))_{t_0 \\le t < t_0 + \\varepsilon }\\,$ created as in (REF ), (REF ) on the manifold of probability vectors $\\,{\\cal M = P_+(S)} $ and with $\\ell ^\\psi (t_0) = {\\mathbf {\\ell }} (t_0) \\in {\\cal L}$ , we have from Propositions REF , REF the respective rates for the variance and the metric distance $\\lim _{h \\downarrow 0} \\frac{\\, V \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big ) - V \\big (P (t_0)\\,|\\,Q\\big )}{h} \\,=\\, -2\\, \\Big \\langle \\, {\\mathbf {\\ell }}_{t_0} , \\psi _{t_0} \\Big \\rangle _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,,$ $\\lim _{h \\downarrow 0} \\, \\frac{\\,\\varrho \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\,}{h} \\,=\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,.$ Thus, the rate of change for the variance along the perturbed curve $\\, ( P^\\psi (t) )_{t_0 \\le t < t_0 + \\varepsilon }\\,,$ when measured on the manifold ${\\cal M}$ by the metric distance in (REF ), is $\\lim _{h \\downarrow 0} \\frac{\\, V \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big ) - V \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\, } \\,=\\, -2\\, \\,\\bigg \\langle \\, {\\mathbf {\\ell }}_{t_0} , \\frac{ \\psi _{t_0} }{\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,.$ On the other hand, along the original curve $\\, ( P (t) )_{0 \\le t < \\infty }\\,$ of time-marginal distributions for the Chain (that is, with $ \\psi (\\cdot ) \\equiv {\\mathbf {\\ell }} (\\cdot ) $ modulo an affine transformation, as noted above), the rate of variance dissipation measured in terms of the metric distance traveled on the manifold ${\\cal M}\\,$ is $\\lim _{h \\downarrow 0} \\frac{\\, V \\big (P (t_0 +h)\\,|\\,Q\\big ) - V \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho \\big ( P (t_0+h), P (t_0) \\big )\\, } \\,=\\, -2\\, \\,\\big \\Vert \\, {\\mathbf {\\ell }}_{t_0} \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,<\\,0\\,.$ A simple comparison of the last two displays, via Cauchy-Schwarz, gives the steepest descent property of the variance with respect to the metric distance in (REF ), i.e., $\\lim _{h \\downarrow 0} \\frac{\\, V \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big ) - V \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\, } \\,- \\,\\lim _{h \\downarrow 0} \\frac{\\, V \\big (P (t_0 +h)\\,|\\,Q\\big ) - V \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho \\big ( P (t_0+h), P (t_0) \\big )\\, }$ $= \\, 2 \\left( \\,\\big \\Vert {\\mathbf {\\ell }}_{t_0} \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} - \\bigg \\langle \\, {\\mathbf {\\ell }}_{t_0} , \\frac{ \\psi _{t_0} }{\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^{1}(\\mathcal {S}, Q)} \\right) \\,\\ge \\, 0\\,,$ along the original curve $\\, ( P(t))_{0 \\le t < \\infty }\\,$ of time-marginals for the Markov Chain.", "Equality holds here if, and only if, $\\,c+\\psi _{t_0}$ is a positive constant multiple of $\\,{\\mathbf {\\ell }}_{t_0}$ for some $\\,c \\in \\mathbb {R}\\,.$ $\\Box $ We will revisit this theme in Sections and ." ], [ "The Relative Entropy Process", "For an arbitrary probability vector $\\, P = ( p (x) )_{x \\in {\\cal S}}\\,$ with strictly positive elements, let us recall the definition of its relative entropy, or Kullback–Leibler divergence, $H (P\\,|\\,Q) \\,:= \\,\\sum _{x \\in {\\cal S}}\\, p(x) \\log \\Big ( \\frac{\\,p (x) \\,}{ q (x)} \\Big )$ with respect to the invariant distribution $\\, Q = ( q (x) )_{x \\in {\\cal S}}\\,$ of (REF ).", "In terms of the likelihood function in (REF ), the relative entropy of the probability vector $P(t)$ in (REF ) with respect to $\\,Q,$ is $H \\big ( P(t) \\, \\big | \\, Q \\big ) \\, =\\, \\mathbb {E^P} \\Big [ \\log \\ell \\big ( t, X(t) \\big ) \\Big ] , \\qquad 0 \\le t < \\infty \\,,$ the $ \\mathbb {P}-$ expectation of the log-likelihood at time $t$ .", "We shall see presently that this function $t \\, \\longmapsto \\,H \\big ( P(t) \\, \\big | \\, Q \\big ) ~~\\text{is non-negative, and satisfies} ~\\, \\lim _{t \\rightarrow \\infty } \\downarrow H \\big ( P(t) \\, \\big | \\, Q \\big ) =0\\,.$ In other words, the relative entropy functional of (REF ) is a Lyapunov function for the curve $ ( P(t) )_{0 \\le t < \\infty } $ of time-marginal distributions for our continuous-time Markov Chain.", "We shall compute in subsection REF the rate of temporal decrease for the function in (REF ).", "Of course, all this is in accordance with general thermodynamic principles governing the approach to equilibrium in physical systems (e.g., Chapter 2 in Cover & Thomas (1991) in the discrete-time Markov Chain context of our Section ).", "Let us note also, that the relative entropy in (REF ) can be cast equivalently as the $\\mathbb {Q}-$ expectation $H \\big ( P(t) \\, \\big | \\, Q \\big ) \\,=\\, \\sum _{y \\in {\\cal S}} \\, q ( y) \\,\\frac{\\,p(t,y)\\,}{q(y)}\\, \\log \\left( \\frac{\\,p(t,y)\\,}{q(y)} \\right)\\,=\\, \\mathbb {E^Q} \\Big [ \\ell \\big ( t, X(t) \\big )\\log \\ell \\big ( t, X(t) \\big ) \\Big ]$ of the relative entropy process $\\,\\ell \\big ( t, X(t) \\big ) \\cdot \\log \\ell \\big ( t, X(t) \\big )\\,, ~\\, 0 \\le t < \\infty \\,.$ This allows us to justify the first claim in (REF ), regarding non-negativity.", "Indeed, the convexity of the function $\\, (0, \\infty ) \\ni \\ell \\mapsto \\Phi (\\ell ) := \\ell \\, \\log \\ell \\,\\,$ gives, on the strength of the Jensen inequality, $H \\big ( P(t) \\, \\big | \\, Q \\big ) \\,=\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( \\ell \\big ( t, X(t) \\big ) \\big ) \\big ]\\, \\ge \\, \\Phi \\Big ( \\mathbb {E^Q} \\big [ \\ell \\big ( t, X(t) \\big ) \\big ] \\Big )\\,=\\, f (1) \\,=\\, 0\\,.$ Alternatively, this follows from $\\,H \\big ( P(t) \\, \\big | \\, Q \\big ) = \\mathbb {E^Q} [ \\Psi \\big ( \\ell ( t, X(t) ) ) ] ,$ with $\\Psi \\ge 0$ as in (REF ) below.", "Proposition 7.1 In the context of Proposition REF , the time-reversed relative entropy process $\\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\cdot \\log \\ell \\big ( T-s, \\widehat{X} (s) \\big ) , ~~~~0 \\le s \\le T\\,~~~~~~~\\text{is a}~\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )- \\text{submartingale;}$ the properties in (REF ) hold; and the time-reversed log-likelihood process $\\log \\ell \\big ( T-s, \\widehat{X} (s) \\big ) , ~~~~0 \\le s \\le T ~~~~~~~\\text{is a}~\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {P}\\big )- \\text{submartingale.", "}$ Proof: The first claim follows from (REF ) and the convexity of the function $\\, \\Phi (\\ell ) = \\ell \\, \\log \\ell \\,\\,$ appearing inside the expectation in (REF ), along with the Jensen inequality.", "The $\\,\\mathbb {Q}-$ expectation $H \\big ( P (T-s) \\,| \\,Q)\\,=\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( \\ell \\big ( T-s, \\widehat{X} (s) \\big ) \\big ) \\big ]\\,,~~~~~~ ~ 0 \\le s \\le T$ of the process in (REF ) is thus increasing.", "This is precisely the monotonicity in (REF ); the remaining claim $\\, \\lim _{t \\rightarrow \\infty } \\downarrow H \\big ( P(t) \\, \\big | \\, Q \\big ) =0\\,$ there, follows now from (REF ), (REF ), and the finiteness of $\\, {\\cal S}.$ The claim of (REF ) is a consequence of (REF ), (REF ), and the familiar Bayes rule (Lemma 3.5.3 in Karatzas & Shreve (1988)).", "$\\Box $" ], [ "Trajectorial Relative Entropy Dissipation", "We read now Proposition REF with $\\,\\Phi (\\ell ) = \\ell \\, \\log \\ell \\,$ and the function $h (s,x) \\,=\\, \\Phi \\big ( \\ell (T-s, x) \\big )\\,, \\qquad 0 \\le s \\le T, ~ ~x \\in {\\cal S}\\,.$ As argued in the discussion following Proposition REF , and Proposition REF , the “time-reversed relative entropy\" $ H \\big ( P (T-s) \\,| \\,Q) = \\mathbb {E^Q} \\big [ h \\big ( s, \\widehat{X}(s) \\big ) \\big ] , ~~ 0 \\le s \\le T\\,\\,$ is increasing; and $\\widehat{M}^{ h} (s) \\, := \\, h \\big ( s,\\widehat{X}(s) \\big ) - \\int _0^s \\big ( \\partial h + \\widehat{{\\cal K}} h \\big ) \\big (u, \\widehat{X} (u) \\big )\\, \\mathrm {d}u \\,, \\qquad 0 \\le s \\le T$ is a $\\mathbb {Q}-$ local-martingale of the time-reversed filtration $\\,\\widehat{\\mathbb {G}}\\,.$ The integrand in (REF ) is straightforward to compute: from (REF ), (REF ), and with $t = T-s$ for notational convenience, we get $\\partial h (s,x) = - \\big ( 1 + \\log \\ell (t, x) \\big ) \\, \\big ( \\widehat{{\\cal K}} \\, \\ell \\big ) (t,x) = -\\big ( 1 + \\log \\ell (t, x) \\big ) \\,\\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x, y) \\, \\ell (t, y),\\quad \\text{thus}$ $\\big ( \\partial h + \\widehat{{\\cal K}} h \\big ) (s,x) \\,=\\, \\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x, y)\\, \\ell (t, y) \\Big [ \\log \\frac{\\, \\ell (t, y)\\,}{\\ell (t, x)} - 1 \\Big ] = \\,\\ell (t,x) \\sum _{y \\in {\\cal S} \\atop y \\ne x } \\, \\widehat{\\kappa } ( x, y) \\, \\Psi \\Big ( \\frac{\\, \\ell (t, y)\\,}{\\ell (t, x)} \\Big ) \\ge 0\\,.$ Here the function $\\Psi (r)\\, :=\\, r \\log r -r +1\\,,~~~~~~~r>0$ is nonnegative, convex, and attains its minimum $\\Psi (1)=0$ at $r=1$ .", "We have used in the last equality of (REF ) the property $\\, \\sum _{y \\in {\\cal S}} \\,\\widehat{\\kappa } ( x, y )=0\\,$ for every $x \\in {\\cal S} .$ Proposition 7.2 The submartingales of (REF ), (REF ) admit the respective Doob-Meyer decompositions $\\ell \\big ( T-s, \\widehat{X} ( s) \\big ) \\log \\big ( \\ell \\big ( T-s, \\widehat{X} ( s) \\big ) \\big )\\,=\\, \\,\\widehat{M}^{ \\,h} (s) + \\int _0^s \\lambda ^{\\mathbb {Q}} (u) \\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T \\,,$ $\\log \\big ( \\ell \\big ( T-s, \\widehat{X} ( s) \\big ) \\big )\\,=\\, \\,\\widehat{N}^{\\, h} (s) + \\int _0^s \\lambda ^{\\mathbb {P}} (u) \\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T\\,,$ in the notation of (REF ), (REF ), with $\\,\\lambda ^{\\mathbb {Q}} (s) =\\Lambda ^{\\mathbb {Q}} \\big ( T-s, \\widehat{X}(s)\\big ) \\,,$ $\\,\\lambda ^{\\mathbb {P}} (s) =\\Lambda ^{\\mathbb {P}} \\big ( T-s, \\widehat{X}(s)\\big ) $ and $\\Lambda ^{\\mathbb {Q}} (t,x) \\,:=\\, \\ell (t,x) \\,\\Lambda ^{\\mathbb {P}} (t,x) \\ge 0\\,, \\qquad \\Lambda ^{\\mathbb {P}} (t,x) \\,:=\\, \\sum _{y \\in {\\cal S}, \\, y \\ne x } \\, \\widehat{\\kappa } ( x, y)\\, \\Psi \\Big ( \\frac{\\, \\ell (t, y)\\,}{\\ell (t, x)} \\Big ) \\ge 0\\,.$ Here $\\,\\widehat{M}^{ \\,h} \\,$ is the process of (REF ) and a $\\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ martingale, whereas $\\,\\widehat{N}^{ \\,h} \\,$ is a $\\big ( \\widehat{\\mathbb {G}}, \\mathbb {P}\\big )-$ martingale.", "Proof: Let us take a look at the expressions of (REF )–(REF ).", "We have already noted that each function $ [0,T] \\ni t \\mapsto p (t,x) \\in (0,1)\\,$ is uniformly continuous.", "This fact, along with the finiteness of the state space $ {\\cal S} ,$ implies that the quantities in (REF ), (REF ) are actually uniformly bounded.", "This implies a similar boundedness for the $\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ local martingale in (REF ), which is thus seen to be a true $\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ martingale.", "The remaining claims follow from the Bayes rule.", "$\\Box $ The decomposition (REF ) is a trajectorial version of relative entropy dissipation.", "This manifests itself at the level of the individual particles that undergo the Markov Chain motion viewed under the lens of the probability measure $\\mathbb {Q}$ and under time-reversal, rather than only at the level of their ensembles.", "We note that the quantity of (REF ) provides the exact rate of relative entropy dissipation, in the sense that for every $\\,0 < t < T < \\infty \\,$ we have the following convergence, a.e.", "and in $\\, \\mathbb {L}^1 (\\mathbb {P})$ : $\\lim _{s \\uparrow T-t} \\frac{1}{\\, T - t -s\\,} \\, \\bigg ( \\mathbb {E^P}\\Big [ \\log \\ell \\big ( t , X(t) \\big ) \\, \\Big | \\,\\widehat{\\mathcal {G}} (s) \\Big ] - \\log \\big ( \\ell \\big ( T-s, \\widehat{X} ( s) \\big ) \\big ) \\bigg ) =\\,\\Lambda ^{\\mathbb {P}} \\big ( t, X ( t) \\big ) .$ The decomposition (REF ) and the trajectorial rate (REF ) are exact analogues of those in Theorem 3.6 and Proposition 3.12 of Karatzas, Schachermayer & Tschiderer (2019).", "They constitute trajectorial versions of relative entropy dissipation, viewed now under the original probability measure $\\mathbb {P}$ — and again under time-reversal." ], [ "With this preparation, we are now in a position to recover the precise rate of decay for the relative entropy function in (REF ); cf.", "Diaconis & Saloff-Coste (1996), Lemma 2.5.", "All it takes, is to “aggregate\" (take $\\mathbb {Q}-$ expectations) in (REF ).", "This leads to an analogue of equation (3.31) in Karatzas, Schachermayer & Tschiderer (2019) as we describe now.", "The seminal paper Stam (1959), from the early days of Information Theory, establishes the first identity of this type, and in a context where the invariant measure $Q$ is standard Gaussian.", "A.J.", "Stam gives credit for this result to his teacher, the analyst, number theorist, combinatorialist and logician Nicolaas de Bruijn.", "Theorem 7.3 de Bruijn-type identity for the Dissipation of Relative Entropy: The relative entropy of (REF ) is a decreasing function of time, and satisfies $H \\big ( P (T ) \\,| \\,Q)= H \\big ( P (0) \\,| \\,Q) - \\int _0^T I ( t) \\, \\mathrm {d}t = \\int _T^\\infty I ( t) \\, \\mathrm {d}t\\,, \\qquad I(t) \\,=\\, {\\cal E} \\big ( {\\mathbf {\\ell }}_t, \\log {\\mathbf {\\ell }}_t \\big ) \\ge 0$ for all $ \\, T \\in [0, \\infty )$ , in the notation of (REF ), (REF ).", "Proof: The first claim is simply a restatement of (REF ); and by taking $\\mathbb {Q}-$ expectations in (REF ) we obtain in conjunction with (REF ) the first equality of (REF ), with $I(t) \\, := \\, \\mathbb {E^Q} \\big [ \\big ( \\partial h + \\widehat{{\\cal K}} h \\big ) \\big (T-t, X (t) \\big ) \\big ] .$ From (REF ) and (REF ), this quantity coincides with the last expression in (REF ): to wit, $I(t) \\, =\\, \\sum _{x \\in {\\cal S}} \\, q(x) \\, \\big ( \\partial h + \\widehat{{\\cal K}} h \\big ) (T-t,x) \\,=\\, \\sum _{(x,y) \\in {\\cal Z} } \\,q(x)\\, \\widehat{\\kappa } ( x, y)\\, \\ell (t, y) \\Big [ \\log \\frac{\\, \\ell (t, y)\\,}{\\ell (t, x)} - 1 \\Big ]$ $=\\,\\sum _{x \\in {\\cal S}} \\, q(x)\\, \\ell (t,x) \\sum _{y \\in {\\cal S} \\atop y \\ne x } \\,\\widehat{\\kappa } ( x, y) \\, \\Psi \\Big ( \\frac{\\, \\ell (t, y)\\,}{\\ell (t, x)} \\Big ) = \\sum _{x \\in {\\cal S}} \\sum _{y \\in {\\cal S}} \\, \\kappa (y,x) \\, q(y) \\, \\ell (t, y) \\, \\log \\ell (t,x)\\,=\\, {\\cal E} \\big ( {\\mathbf {\\ell }}_t, \\log {\\mathbf {\\ell }}_t \\big ).$ It is non-negative on account of the non-negativity of the last expression in (REF ), and uniformly continuous as a function of time.", "In the display (REF ), the second equality follows from the first equality in (REF ); the third from the last equality in (REF ); the fourth from (REF ) and the property $\\, \\sum _{y \\in {\\cal S}} \\, \\kappa ( y,x )=0\\,$ for every $y \\in {\\cal S} $ ; and the fifth from the definition (REF ).", "We deduce $\\,H \\big ( P (0) \\,| \\,Q) = \\int _0^\\infty I ( t) \\, \\mathrm {d}t \\,$ by letting $\\, T \\rightarrow \\infty \\,$ in (REF ) and recalling (REF ); then the second identity in (REF ) follows.", "$\\Box $ Remark 7.1 Whenever there exists a positive real constant $\\, \\alpha \\,$ (respectively, $\\, \\beta \\,$ ) such that the Poincaré (resp., the modified log-Sobolev) inequality $\\alpha \\, \\le \\, \\frac{\\, 2\\, {\\cal E} (f, f) \\,}{\\,\\sum _{y \\in {\\cal S}} \\, q(y) f^2 (y) -1\\,} \\qquad \\bigg (\\text{resp.,} \\quad \\beta \\, \\le \\, \\frac{\\, {\\cal E} (f, \\log f)\\,}{\\,\\sum _{y \\in {\\cal S}} \\, q(y) f (y) \\log f(y)\\,}\\bigg )$ holds for every function $ f : {\\cal S} \\rightarrow (0, \\infty ) $ with $\\,\\sum _{y \\in {\\cal S}} \\, q(y) f (y) =1,$ it is clear from (REF ), (REF ) and (REF ), (REF ) that the variance (resp., the relative entropy) decays exponentially: $\\text{Var}^\\mathbb {Q} \\big ( L (t) \\big ) \\,\\le \\, \\text{Var}^\\mathbb {Q}\\big ( L (0) \\big ) \\,e^{\\, - \\alpha \\, t} \\qquad \\Big (\\text{resp.,} ~~~H \\big ( P (t) \\,| \\,Q) \\,\\le \\, H \\big ( P (0) \\,| \\,Q) \\,e^{\\, - \\beta \\, t} \\Big )\\,.$ Remark 7.2 To the best of our knowledge, the identities (REF ), (REF ) appear in the Markov Chain context first in Lemma 2.5 of Diaconis & Saloff-Coste (1996); see also Bobkov & Tetali (2006), Montenegro & Tetali (2006), Caputo et al.", "(2009) and Conforti (2020).", "These authors use slightly different arguments, based on semigroups.", "One advantage of the more probabilistic approach we follow here, is that it provides a very sharp picture for the dissipation of relative entropy along trajectories, as exemplified in subsection REF ." ], [ "The following is now a direct consequence of Lemma REF .", "Proposition 7.4 Under the detailed-balance condition (REF ), the rate of relative entropy dissipation in (REF ) can be cast as $I (t) \\,=\\, {\\cal E} \\big ( {\\mathbf {\\ell }}_t, \\log {\\mathbf {\\ell }}_t \\big ) \\,=\\, \\frac{1}{\\,2\\,} \\sum _{(x,y) \\in {\\cal Z} } \\, \\Big ( \\log \\ell \\big ( t,y \\big ) - \\log \\ell \\big ( t,x \\big ) \\Big )^2 \\,\\, \\Theta \\big ( \\ell (t,y), \\ell (t,x)\\big )\\, \\kappa (y,x) \\, q(y)$ $~~~~~~~~~~~~~~~~~~~~~~~~\\,=\\, \\frac{1}{\\,2\\,} \\sum _{(x,y) \\in {\\cal Z} } \\, \\frac{\\,\\big ( \\ell (t,y) - \\ell (t,x) \\big )^2 \\,}{\\Theta \\big ( \\ell (t,y), \\ell (t,x)\\big )}\\, \\kappa (y,x) \\, q(y) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ in terms of the “logarithmic mean\" function $\\Theta ( q, p) \\,:=\\, \\frac{ q-p}{\\, \\log q- \\log p \\, } \\,=\\, \\int _0^1 q^r \\,p^{1-r}\\,\\mathrm {d}r\\,, \\qquad ( q,p) \\in (0, \\infty )^2.$ Remark 7.3 The expression in (REF ) is reminiscent of the familiar Fisher Information in Statistics and Information Theory; cf.", "Bobkov & Tetali (2006).", "Always under the detailed-balance condition (REF ), the expression of (REF ) can be expressed in terms of a “score function\", the discrete logarithmic-gradient of the likelihood ratio, as $\\, \\big \\langle { \\nabla {\\mathbf {\\ell }}_t , \\nabla \\log {\\mathbf {\\ell }}_t }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)} \\,$ in the notation of (REF )-(REF ).", "As shown in Bobkov & Tetali (2006), the inequality $\\, 2 (a-b)^2 \\le (a^2 - b^2) \\, \\log (a / b)\\,$ for $0 < a, b < \\infty \\,$ leads under detailed-balance (REF ) to the Diaconis and Saloff-Coste (1996) estimate $\\,{\\cal E} \\big ( e^g, g \\big ) \\ge \\,4 \\, \\,{\\cal E} \\big ( e^{g/2}, \\,e^{g/2} \\big )\\,,~~~~~~~~\\text{and thus to}~~~~~~~~\\, I(t) = {\\cal E} \\big ( {\\mathbf {\\ell }}_t, \\log {\\mathbf {\\ell }}_t \\big ) \\ge \\,4 \\, {\\cal E} \\big ( \\sqrt{ {\\mathbf {\\ell }}_t\\,} , \\sqrt{{\\mathbf {\\ell }}_t\\,} \\,\\big ).$" ], [ "The $\\Phi -$ Relative Entropy Process", "In order to reveal the common thread running through the examples of the last two Sections, let us consider now a continuously differentiable and convex function $\\, \\Phi : (0, \\infty ) \\rightarrow \\mathbb {R}$ with $\\Phi (1)=0$ with continuous, strictly positive second derivative.", "We denote by $\\, \\varphi : (0, \\infty ) \\rightarrow \\mathbb {R}\\, $ its derivative $\\, \\Phi ^{\\prime } = \\varphi \\,.$ For each $\\eta >0$ , $\\xi >0$ we define the Bregman $\\, \\Phi -$divergence $\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big ) \\,:=\\, \\Phi (\\eta ) - \\Phi (\\xi )- (\\eta - \\xi ) \\, \\varphi (\\xi ) \\,,$ a quantity which is non-negative on account of the convexity of $\\Phi $ (and has nothing to do with the “discrete divergence\" we introduced in (REF )).", "For instance, $\\,\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big )= (\\xi - \\eta )^2\\,$ for $\\, \\Phi (\\xi ) = \\xi ^2-1\\,;$ whereas, for $\\, \\Phi (\\xi ) = \\xi \\, \\log \\xi \\,$ and in the notation of (REF ), we have $\\,\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big )\\,=\\, \\text{div}^\\Psi \\big ( \\eta \\, | \\, \\xi \\big )\\,=\\, \\xi \\, \\Psi (\\eta / \\xi )\\,.$ Let us consider now, for a general convex $\\Phi $ as above, the so-called $\\, \\Phi -$relative entropy $H^\\Phi \\big ( P(t) \\big | Q \\big ) \\,:=\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( \\ell (t, X(t) \\big ) \\big ]\\,=\\, \\sum _{y \\in {\\cal S}} \\, q(y) \\, \\Phi \\Big ( \\frac{p (t,y)}{q(y)} \\Big )\\,, \\qquad 0 \\le t < \\infty \\,;$ see Chafaï (2004) for a general study of such functions.", "The convexity of $\\Phi $ and the Jensen inequality imply that this function is nonnegative, since $\\Phi (1)=0$ ; and from Proposition REF , that the time-reversed $\\, \\Phi -$ relative entropy process $\\, \\Phi \\big ( \\ell (T-s, \\widehat{X} ( s) \\big ), ~~~~~~~\\, 0 \\le s \\le T\\,$ is a $\\, (\\mathbb {\\widehat{G} , Q})-$ submartingale, for every fixed $T \\in (0, \\infty )$ .", "As a consequence the function in (REF ) decreases, in fact satisfies $\\, \\lim _{t \\rightarrow \\infty } \\downarrow H^\\Phi \\big ( P(t) \\, \\big | \\, Q \\big ) =0\\,$ by virtue of (REF ) and the finiteness of the state space." ], [ "Trajectorial Dissipation of the $\\, \\Phi -$ Relative Entropy", "The Doob-Meyer decomposition of this submartingale is obtained from Proposition REF as follows: Consider the function $g(s, x) = \\Phi \\big ( \\ell (T-s, x)\\big )$ and compute, in the manner of (REF ), the quantities $\\partial g (s,x) = - \\varphi \\big ( \\ell (t, x) \\big )\\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x, y) \\big [ \\ell (t, y)- \\ell (t,x) \\big ],~~~~~\\big ( \\widehat{{\\cal K}} g \\big ) (s,x) =\\sum _{y \\in {\\cal S}} \\, \\widehat{\\kappa } ( x, y) \\big [ \\Phi \\big ( \\ell (t, y) \\big ) - \\Phi \\big ( \\ell (t,x) \\big ) \\big ]$ with $t=T-s$ , on account of (REF ).", "Putting these expressions together with (REF ), we deduce $\\big ( \\partial g+ \\widehat{{\\cal K}} g \\big ) (s,x) \\,=\\sum _{y \\in {\\cal S}, \\, y \\ne x} \\, \\widehat{\\kappa } ( x, y) \\, \\,\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big ) \\bigg |_{\\eta = \\ell (t, y) \\atop \\xi = \\ell (t, x)} \\, =: \\, \\Lambda ^{\\Phi , \\mathbb {Q} } (t,x) \\, \\ge \\, 0\\,.$ The following result is now a direct consequence of Proposition REF and the Bayes rule.", "Once again, the finiteness of the state-space and the continuity of the functions involved, turn local into true martingales.", "Proposition 8.1 For any given $T \\in (0, \\infty )$ , the process below is a $\\, (\\mathbb {\\widehat{G} , Q})-$ martingale: $\\Phi \\big ( \\ell (T-s, \\widehat{X}(s))\\big )- \\int _0^s \\Lambda ^{\\Phi , \\mathbb {Q} } \\big ( T-u, \\widehat{X}(u) \\big )\\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T\\,.$ Whereas, with $\\,\\Lambda ^{\\Phi , \\mathbb {P} } (t,x) := \\Lambda ^{\\Phi , \\mathbb {Q} } (t,x)/ \\ell (t,x),$ the process below is a $\\, (\\mathbb {\\widehat{G} , P})-$ martingale: $\\frac{\\Phi \\big ( \\ell (T-s, \\widehat{X}(s))\\big )}{\\ell (T-s, \\widehat{X}(s))} - \\int _0^s \\Lambda ^{\\Phi , \\mathbb {P} } \\big ( T-u, \\widehat{X}(u) \\big )\\, \\mathrm {d}u\\,, \\qquad 0 \\le s \\le T.$" ], [ "Generalized ", "In view of these considerations, it is now straightforward to “aggregate\" (i.e., take $\\mathbb {Q}-$ expectations of) the $ (\\mathbb {\\widehat{G} , Q})-$ martingale of (REF ).", "We obtain in the manner of (REF ) the following result, stated again in the forward direction of time; cf.", "Chafaï (2004), Proposition 1.1.", "Proposition 8.2 Generalized de Bruijn-type identity: The temporal dissipation of the $\\, \\Phi -$ relative entropy of (REF ) is given for $\\,0 \\le T < \\infty \\,$ as $H^\\Phi \\big ( P (T ) \\,| \\,Q)=H^\\Phi \\big ( P (0 ) \\,| \\,Q)- \\int _0^T I^\\Phi ( t) \\, \\mathrm {d}t= \\int ^\\infty _T I^\\Phi ( t) \\, \\mathrm {d}t \\,, \\qquad I^\\Phi (t) \\,:=\\, \\mathbb {E^Q} \\big [ \\Lambda ^{\\Phi , \\mathbb {Q} }\\big ( t, X(t) \\big ) \\big ] \\ge 0 \\,.$ On the strength of (REF ), the dissipation rate in (REF ) is given by the $\\, \\Phi -$Fisher Information $I^\\Phi (t) = \\sum _{(x,y) \\in {\\cal Z} }q(x)\\, \\widehat{\\kappa } ( x, y) \\, \\,\\mathrm {div}^\\Phi \\big ( \\ell (t, y) \\big | \\ell (t, x) \\big ) = \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S} } \\,q(x)\\, \\widehat{\\kappa } ( x, y) \\,\\mathrm {div}^\\Phi \\big ( \\ell (t, y) \\big | \\ell (t, x) \\big )$ $= - \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S}} q(y)\\, \\kappa ( y,x) \\, \\ell (t, y)\\, \\varphi \\big ( \\ell (t,x) \\big ) = {\\cal E} \\big ( {\\mathbf {\\ell }}_t, \\varphi ({\\mathbf {\\ell }}_t)\\big ) .$ Proof: The third equality in (REF ) is a consequence of the properties $\\, \\sum _{y \\in {\\cal S}} \\,\\widehat{\\kappa } ( x, y )=0\\,$ for every $x \\in {\\cal S} ,$ and $\\, \\sum _{x \\in {\\cal S}} \\, \\kappa ( y,x )=0\\,$ for every $y \\in {\\cal S} ,$ as well as of (REF ).", "It underscores the fact that, when passing from the trajectorial to the “aggregate\" point of view (that is, when taking $\\mathbb {Q}-$ expectations), the term $\\, \\xi \\varphi (\\xi ) -\\Phi (\\xi )\\,$ that depends only on the variable $\\xi = \\ell (t,x)$ , as well as the term $\\,\\Phi (\\eta )\\,$ that depends only on the variable $\\eta = \\ell (t,y)$ , can be ignored in (REF ); only the “mixed term\" $\\, -\\eta \\, \\varphi (\\xi )\\,$ remains relevant.", "We note that similar reasoning was deployed in the proof of Lemma REF .", "$\\Box $ Remark 8.1 Some Special Cases:  (i)   For the convex function $\\, \\Phi (\\xi ) = \\xi \\log \\xi \\, , $   and recalling (REF ), (REF ), the quantity $I^\\Phi (t)$ of (REF ) is seen to coincide with $I(t)$ in (REF ), (REF ).", "(ii)  On the other hand, when $\\, \\Phi (\\xi ) = \\xi ^2-1\\, $ we have $\\,\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big ) = (\\eta - \\xi )^2\\,$ in (REF ) and $H^\\Phi \\big ( P(t) \\big | Q \\big ) =\\, \\mathbb {E^Q} \\big ( \\ell ^2 (t, X(t) \\big ) - 1\\,= \\,\\big | {\\mathbf {\\ell }}_t \\big |_{\\mathbb {L}^2(\\mathcal {S},Q)}^2 -1\\,=\\, \\text{Var}^{\\mathbb {Q}} (L (t))\\,=\\,V \\big (P(t)\\,|\\,Q\\big )\\,, \\qquad 0 \\le t < \\infty $ as in (REF ), and the rate of temporal dissipation for this function is precisely the integrand in (REF ): $I^\\Phi (t) \\, = \\,-2\\, \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S}} \\, q(y)\\, \\kappa ( y,x) \\, \\ell (t, x)\\, \\ell (t,y) \\,=\\, 2\\, {\\cal E} \\big ( {\\mathbf {\\ell }}_t, {\\mathbf {\\ell }}_t \\big )\\,.$ (iii)   A bit more generally, the choice of convex function $\\, \\Phi (\\xi ) = (\\xi ^m-1) / (m-1)\\, $ with $\\, m >1 ,$ leads to the so-called “Rényi relative entropy\" $H^\\Phi \\big ( P(t) \\big | Q \\big ) \\,= \\,\\frac{\\,\\mathbb {E^Q} \\big ( \\ell ^m (t, X(t) \\big ) - 1\\,}{m-1} \\,, \\qquad 0 \\le t < \\infty $ whose rate of temporal dissipation is a generalized version of (REF ): $I^\\Phi (t) \\, = \\,-\\, \\frac{m}{m-1}\\, \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S}} \\, q(y)\\, \\kappa ( y,x)\\, \\ell (t,y) \\, \\big ( \\ell (t, x) \\big )^{m-1} \\,=\\, \\frac{m}{m-1} \\, \\,{\\cal E} \\big ( {\\mathbf {\\ell }}_t, {\\mathbf {\\ell }}_t^{\\,m-1} \\big )\\,.$ The variance $\\text{Var}^{\\mathbb {Q}} (L (t))\\,$ is thus a special case of the Rényi relative entropy, corresponding to $\\,m=2\\,$ ; whereas the relative entropy in (REF ) corresponds to the limit of (REF ) as $\\, m \\downarrow 1.$ We stress that nowhere in this subsection, or in the one preceding it, did we invoke the detailed-balance conditions of (REF )." ], [ "Locally Steepest Descent for the $\\, \\Phi -$ Relative Entropy Under Detailed Balance", "We formulate now a variational version of Proposition REF under the conditions (REF ) of detailed balance.", "These will be in force throughout the current subsection.", "Remark 8.2 First, let us take a look at the expression of (REF ).", "From the consequence $\\,q(x)\\, \\widehat{\\kappa } ( x, y)= q(y)\\, \\kappa ( y,x) = q(y)\\, \\widehat{\\kappa } ( y,x)\\,$ of the detailed balance conditions (REF ), as well as from the consequence $\\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big ) + \\text{div}^\\Phi \\big ( \\xi \\, | \\, \\eta \\big ) \\,=\\, \\big ( \\eta - \\xi \\big ) \\big ( \\varphi (\\eta ) - \\varphi (\\xi ) \\big )$ of (REF ), we see that the $\\, \\Phi -$Fisher Information of (REF ) can be cast in this case as $\\begin{aligned}I^\\Phi (t)& = \\,\\frac{1}{2}\\, \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S}} q(x)\\, \\widehat{\\kappa } ( x, y) \\, \\Big ( \\text{div}^\\Phi \\big ( \\eta \\, | \\, \\xi \\big ) + \\text{div}^\\Phi \\big ( \\xi \\, | \\, \\eta \\big ) \\Big )\\bigg |_{\\eta = \\ell (t, y) \\atop \\xi = \\ell (t, x)}\\\\ & = \\,\\frac{1}{2}\\, \\sum _{x \\in {\\cal S} }\\sum _{y \\in {\\cal S}} q(y)\\, \\kappa ( y,x) \\, \\Big ( \\big ( \\eta - \\xi \\big ) \\big ( \\varphi (\\eta ) - \\varphi (\\xi ) \\big ) \\Big )\\bigg |_{\\eta = \\ell (t, y) \\atop \\xi = \\ell (t, x)} = {\\cal E} \\big ( \\varphi ({\\mathbf {\\ell }}_t), {\\mathbf {\\ell }}_t \\big )\\\\& = \\,\\frac{1}{2} \\sum _{(x,y) \\in {\\cal Z} }q(x)\\, \\kappa ( x, y) \\,\\Theta ^\\Phi ( \\xi , \\eta ) \\big (\\varphi (\\xi ) - \\varphi (\\eta )\\big )^2 \\, \\bigg |_{\\xi = \\ell (t, x) \\atop \\eta = \\ell (t, y) } \\,\\,\\end{aligned}$ in the manner of (REF ); we recall the notation $\\, \\varphi = \\Phi ^{\\prime } .$ Here, the function $\\Theta ^\\Phi ( q, p) \\,:=\\, \\frac{\\, q- p \\, }{\\, \\varphi (q)- \\varphi (p) \\,} \\,, \\quad 0 < q \\ne p < \\infty \\,, \\qquad ~~\\Theta ^\\Phi ( p, p) \\,:=\\,\\frac{1}{\\, \\Phi ^{^{\\prime \\prime }} (p)\\,}\\,, \\quad 0 < p < \\infty \\,,$ extends the “logarithmic mean\" of (REF ), to which it reduces when $\\Phi (\\xi ) = \\xi \\log \\xi .$ With $\\, \\Phi (\\xi ) = \\xi ^2 -1$ we get $\\,\\Theta ^\\Phi \\equiv 1/2,\\,$ and the last expression in (REF ) reduces to $\\,\\,\\sum _{(x,y) \\in {\\cal Z} } \\,q(x)\\, \\kappa ( x, y) \\cdot \\big (\\ell (t, x) - \\ell (t, y) \\big )^2 \\,$ as in (REF ).", "We shall comment further on this choice of (REF ), in subsection 9.2 below.", "We set out now to find a metric on the manifold $\\, {\\cal M = P_+ (S)}$ of probability vectors on the state-space, relative to which the time-marginals for the Markov Chain $ ( P (t) )_{0 \\le t < \\infty } $ constitute a curve of steepest descent for the $\\Phi -$ relative entropy.", "In other words, we look for a metric on $\\, {\\cal M}\\,$ that can play — in the current general context — a role similar to that played by the Hilbert norm $\\, \\Vert \\cdot \\Vert _{\\mathbb {H}^{-1} (\\mathcal {S}, Q)}\\, $ in Section .", "This norm defines the metric distance of (REF ) that works for the variance $V (P(t)|Q)$ , i.e., in the special case $\\, \\Phi (\\xi ) = \\xi ^2-1.$ But except for such very special cases, the Riemannian metric on the manifold ${\\cal M}$ will not be flat; i.e., not induced by such a simple norm as in Proposition REF .", "For this reason we are forced to consider the machinery of Riemannian geometry, which we take up in the next Section .", "In this Section we avoid Riemannian terminology, and present the steepest descent property of the curve $\\, ( P (t) )_{0 \\le t < \\infty } \\, $ in terms of appropriate Hilbert norms that capture the local behavior of the Riemannian metric." ], [ " Locally Weighted ", "We start this effort by recalling from (REF ) the norm $ \\Vert F \\Vert _Q^2 $ for functions $F: {\\cal Z} \\rightarrow \\mathbb {R}\\,.$ This is defined on the “off-diagonal Cartesian product\" $ {\\cal Z}$ by assigning to its elements $\\,(x,y), \\, x \\ne y\\,$ the weights $\\, q(x)\\, \\kappa ( x,y) / 2\\,$ and taking the usual $\\,\\mathbb {L}^2-$ norm relative to the positive measure with these weights.", "For a fixed likelihood ratio $\\ell $ in the space ${\\cal L = L_+ (S)}\\,$ of subsection REF we consider now, in place of $\\, c(x,y) \\equiv q(x)\\, \\kappa ( x,y) / 2\\,$ and with the notation of (REF ), the new weights $c(x,y) \\cdot \\vartheta _\\ell (x,y)\\,,\\qquad \\text{ where\\, }\\quad \\vartheta _\\ell (x,y) := \\Theta ^\\Phi \\big ( \\ell (x), \\ell (y) \\big ) \\,=\\, \\frac{\\nabla \\ell (x,y)}{\\nabla ( \\varphi \\circ \\ell ) (x,y)}\\,.$ The resulting weighted inner product and norm, extensions of the respective quantities for real-valued functions on $\\, {\\cal S} \\times {\\cal S}\\,$ in (REF ), (REF ) (to which they reduce when $\\Phi (\\xi ) = \\xi ^2 / 2\\,$ ), are respectively $\\begin{aligned}\\big \\langle { F , G }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\,& := \\,\\sum _{(x,y) \\in {\\cal Z} }\\, c(x,y)\\, \\vartheta _\\ell (x,y)F(x,y)\\, G(x,y)\\,\\,= \\,\\big \\langle { \\, \\vartheta _\\ell F,G\\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\,,\\\\\\big \\Vert F \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}^2&\\, := \\,\\big \\langle { F , F }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\,.\\end{aligned}$ We define now for $ f: {\\cal S} \\rightarrow \\mathbb {R}$ the Weighted Sobolev Norm $ \\Vert \\cdot \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} ,$ by replacing on the right-hand sides of (REF )–(REF ) the norm $\\, \\Vert \\cdot \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, C)}\\,$ by the new norm $\\, \\Vert \\cdot \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\,$ in (REF ): $\\big \\langle f, g \\big \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,:=\\,\\big \\langle { \\nabla f , \\nabla g }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\,,\\quad \\big \\Vert f \\big \\Vert ^2_{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}\\,:= \\, \\big \\langle f, f\\big \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}= \\big \\Vert \\nabla f \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}^2\\,.$ Remark 8.3 It is interesting to note at this point, and will become quite important down the road, that the $ \\Phi -$Fisher Information of (REF ), (REF ) can be expressed in terms of the square of this new, weighted Sobolev norm.", "Indeed, for any $\\ell \\in {\\cal L_+ (S)}$ we have ${\\cal E} \\big ({ \\ell }, \\varphi ( { \\ell } ) \\big )= \\big \\langle { \\nabla { \\ell } ,\\nabla \\varphi ( { \\ell } ) }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\langle { \\vartheta _\\ell \\nabla \\varphi ( { \\ell } ) ,\\nabla \\varphi ( { \\ell } ) }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\Vert \\nabla \\varphi ( { \\ell }) \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}= \\big \\Vert \\varphi ( { \\ell } ) \\big \\Vert ^2_{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}.$ Thus, the $\\, \\Phi -$Fisher Information of (REF ) takes the form $\\,I^\\Phi (t) = \\big \\Vert \\varphi ( {\\mathbf {\\ell }}_t ) \\big \\Vert ^2_{\\mathbb {H}^1_\\Theta (\\mathcal {S}, { \\mathbf {\\ell }}_t Q)}.$ Finally, we introduce in the manner of (REF ), (REF ), the dual of this weighted Sobolev norm $\\begin{aligned}\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}\\,:=\\,\\sup _{g : {\\cal S} \\rightarrow \\mathbb {R}}\\frac{\\,\\big \\langle {f, g}\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}\\,}{\\, \\big \\Vert g \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}\\,} .\\end{aligned}$ This admits a variational characterization, analogous to (REF ), that will be crucial in what follows.", "Proposition 8.3 Variational Interpretation: For any function $\\,f : \\mathcal {S}\\rightarrow \\mathbb {R}\\,$ we have $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)} & = \\inf _{G : \\mathcal {Z}\\rightarrow \\mathbb {R}}\\bigg \\lbrace \\Vert G \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\ : \\ f + \\nabla \\cdot \\big (\\vartheta _\\ell G\\big )=0 \\bigg \\rbrace \\, .$ Moreover, $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}$ is finite if, and only if, $\\,\\sum _{x \\in \\mathcal {S}} q(x) f(x) = 0\\,;$ in this case the infimum is attained, and uniquely, by the unique discrete gradient that is admissible.", "Consider a function $f : \\mathcal {S}\\rightarrow \\mathbb {R}$ such that $\\sum _{x \\in \\mathcal {S}} q(x) f(x) = 0;$ if this is not the case, it is straightforward to verify that both sides in (REF ) are infinite.", "We note that the set of admissible $G$ on the right-hand side of (REF ) is non-empty (indeed, $G_0 := - \\frac{1}{\\vartheta _\\ell }\\nabla \\mathcal {K}^{-1} f$ is admissible) and that a minimizer exists.", "Let $G : \\mathcal {Z}\\rightarrow \\mathbb {R}$ be such a minimizer.", "We show first that $G$ is a discrete gradient, by a projection argument in the Hilbert space $\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)$ .", "To this end, let us denote by $\\nabla h$ the orthogonal projection of $G$ onto the subspace of discrete gradients in $\\mathcal {H}_\\ell $ .", "We claim that $\\nabla h$ is admissible on the right-hand side of (REF ).", "Indeed, $ G - \\nabla h$ is orthogonal in $\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)$ to $\\nabla g$ for any $g : \\mathcal {S}\\rightarrow \\mathbb {R}$ .", "This implies $- \\big \\langle {\\, g , \\nabla \\cdot \\big ( \\vartheta _\\ell (G - \\nabla h)\\big ) \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}=\\big \\langle {\\, \\nabla g , \\vartheta _\\ell (G - \\nabla h) \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\langle {\\,\\nabla g , G - \\nabla h}\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}= 0$ and yields $\\nabla \\cdot \\big ( \\vartheta _\\ell G \\big ) = \\nabla \\cdot \\big ( \\vartheta _\\ell \\nabla h \\big ),$ proving the claim.", "By orthogonality, we have $ \\Vert G \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}^2 = \\Vert \\nabla h\\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}^2 + \\Vert G - \\nabla h\\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}^2$ .", "Since $G$ is a minimizer, we infer $\\Vert G - \\nabla h\\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)} = 0$ , which implies that $G \\equiv \\nabla h$ .This shows that $\\nabla h$ is a minimizer, and that the right-hand side of (REF ) is equal to $\\Vert h \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}$ .", "It is shown in Maas (2011) that $\\nabla h$ is actually the unique discrete gradient satisfying the constraint in (REF ).", "To prove the equality in (REF ), we note for any $g : \\mathcal {S}\\rightarrow \\mathbb {R}$ the identities $\\langle {f,g}\\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= - \\big \\langle {\\nabla \\cdot \\big ( \\vartheta _\\ell \\nabla h \\big ),g}\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= \\big \\langle {\\vartheta _\\ell \\nabla h ,\\nabla g}\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\langle h , g \\big \\rangle _{{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,\\,.", "}$ Writing the dual norm as a Legendre transform, we obtain $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2& = \\sup _{g : \\mathcal {S}\\rightarrow \\mathbb {R}}\\bigg \\lbrace 2\\langle {f,g}\\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}-\\big \\Vert g \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}^2\\bigg \\rbrace \\\\& = \\sup _{g : \\mathcal {S}\\rightarrow \\mathbb {R}}\\bigg \\lbrace 2 \\, \\big \\langle h , g \\big \\rangle _{{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}} -\\big \\Vert g \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}^2\\bigg \\rbrace = \\Vert h \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}^2 \\,\\,,$ which establishes the equality in (REF ).", "Let us consider now as in Section , for some $\\varepsilon >0$ an arbitrary smooth curve $\\, \\ell ^\\psi (\\cdot ) = ( \\ell ^\\psi (t) )_{t_0 \\le t < t_0 + \\varepsilon }$ with initial position $\\ell ^\\psi (t_0) = {\\mathbf {\\ell }} \\equiv {\\mathbf {\\ell }} (t_0)$ in ${\\cal L = \\cal L_+ (S)}$ .", "In order to compute ${\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}$ -norms, it is natural in view of Proposition REF to write the time-evolution in the manner of a “discrete continuity equation” $\\,\\partial \\ell ^\\psi _t + \\nabla \\cdot ( \\vartheta _{\\ell _t} \\nabla \\psi _t ) = 0$ as in subsection REF , where $\\psi _t : \\mathcal {S}\\rightarrow \\mathbb {R}$ is unique up to an additive constant.", "We regard $ \\psi (\\cdot )$ as an input, whose gradient is the velocity vector field that yields the infinitesimal change $\\partial \\ell ^\\psi _t$ of the likelihood ratio flow.", "In light of (REF ), (REF ) and detailed balance, the original backward equation $ \\,\\partial {\\mathbf {\\ell }}_t = \\widehat{{\\cal K}} {\\mathbf {\\ell }}_t= {\\cal K} {\\mathbf {\\ell }}_t = \\nabla \\cdot ( \\nabla {\\mathbf {\\ell }}_t ) $ corresponds to $\\, \\psi _t = - \\varphi ({\\mathbf {\\ell }}_t)\\,$ in this scheme of things.", "We define as in (REF ) the corresponding curve $ P^\\psi (\\cdot ) = \\big ( P^\\psi (t) \\big )_{t_0 \\le t < t_0 + \\varepsilon } $ on the manifold $ {\\cal M = P_+ (S)}$ of probability vectors on the state-space.", "We obtain the following generalization of Proposition REF .", "Proposition 8.4 In the above context, we have $\\begin{aligned}\\partial H^\\Phi \\big (P^\\psi (t)\\,|\\,Q\\big )= \\big \\langle { \\, \\varphi (\\ell ^\\psi _t ) , \\psi _t \\,}\\big \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell _t Q)} \\,.\\end{aligned}$ Using the discrete continuity equation, a discrete integration by parts, and the definitions of the scalar products, we deduce $\\partial H^\\Phi \\big (P^\\psi (t)\\,|\\,Q\\big )& = \\,\\partial \\,\\mathbb {E^Q} \\Big [ \\Phi \\big ( \\ell ^\\psi \\big (t, X (t) \\big ) \\big ) \\Big ]\\,= - \\big \\langle { \\, \\varphi (\\ell ^\\psi _t ),\\nabla \\cdot ( \\vartheta _{\\ell _t} \\nabla \\psi _t ) \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}\\\\&= \\big \\langle { \\, \\nabla \\varphi (\\ell ^\\psi _t ), \\vartheta _{\\ell _t} \\nabla \\psi _t \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\langle { \\, \\nabla \\varphi (\\ell ^\\psi _t ), \\nabla \\psi _t \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _{\\ell _t} C)}= \\big \\langle { \\, \\varphi (\\ell ^\\psi _t ) , \\psi _t \\,}\\big \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell _t Q)}\\,,$ as desired.", "With the context and notation just established, and always for $\\,{\\mathbf {\\ell }} \\equiv {\\mathbf {\\ell }} (t_0)= ( \\ell (t_0, x) )_{x \\in {\\cal S}}\\,,$ we can formulate the following analogue of Proposition REF .", "This result uses the characterizations of the weighted $\\mathbb {H}^{-1}-$ norm in (REF ), along with the identity $\\mathcal {K}{\\mathbf {\\ell }} = \\nabla \\cdot \\big ( \\vartheta _{\\mathbf {\\ell }} \\nabla \\varphi ({\\mathbf {\\ell }})\\big )$ .", "Proposition 8.5 Under the conditions (REF ) of detailed balance, we have $\\lim _{h \\downarrow 0}\\, \\frac{1}{h}\\, \\big \\Vert \\,{\\mathbf {\\ell }}_{t_0 +h} - {\\mathbf {\\ell }}_{t_0} \\,\\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)}\\, = \\,\\big \\Vert \\, {\\cal K}\\, {\\mathbf {\\ell }}_{t_0} \\, \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)} \\,= \\,\\big \\Vert \\, \\varphi ( {\\mathbf {\\ell }}_{t_0} )\\, \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)} \\,;$ and a bit more generally, $\\lim _{h \\downarrow 0}\\, \\frac{1}{h}\\, \\big \\Vert \\, \\ell _{t_0+h}^\\psi - \\ell _{t_0}^{ \\psi } \\,\\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)}\\, = \\,\\big \\Vert \\,\\nabla \\cdot ( \\vartheta _{{\\mathbf {\\ell }}_{t_0}} \\nabla \\psi _{t_0}) \\, \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)}\\, = \\,\\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,.$ We pass now to the principal result of the Section.", "This generalizes Theorem REF , to which it reduces when $\\Phi (\\xi ) = \\xi ^2-1.$ It is also a direct analogue of Theorem 3.4 in Karatzas, Schachermayer & Tschiderer (2019), where a similar steepest-descent for the relative entropy is established for Langevin diffusions and with distance on the ambient space measured by the quadratic Wasserstein metric.", "The role of that metric is played now by the locally flat metric defined in (REF ) below.", "Theorem 8.6 Steepest Descent for the $\\Phi -$ Relative Entropy: Under the detailed-balance conditions (REF ), the curve $\\, ( P (t) )_{t_0 \\le t < \\infty }\\,$ of time-marginal distributions in (REF ) has the property of steepest descent in Definition REF for the $\\Phi -$ Relative Entropy of (REF ), locally at $t=t_0\\,,$ and with respect to the distance induced by the “flat metric\" $\\varrho _\\star \\big ( P_1, P_2 \\big )\\,:= \\, \\big \\Vert \\, {\\mathbf {\\ell }}_1 - {\\mathbf {\\ell }}_2 \\,\\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)} \\qquad \\text{ for \\, $P_1 = {\\mathbf {\\ell }}_1 Q$ \\, and $~P_2 = {\\mathbf {\\ell }}_2 \\,Q.$}$ Proof: This is proved exactly as in subsection REF , with the caveat that the distance-inducing flat metric is now determined “locally\", that is, depends on $(t_0, {\\mathbf {\\ell }}) \\equiv (t_0,{\\mathbf {\\ell }} (t_0))$ in the weighted norms of (REF )–(REF ).", "We go through the argument again, however, in order to highlight the role that these weighted norms play in the present, more general context.", "From (REF ), and recalling the initial position $\\,\\ell ^\\psi (t_0) = {\\mathbf {\\ell }} (t_0) \\in {\\cal L}\\,,$ we obtain $\\lim _{h \\downarrow 0} \\frac{\\,H^\\Phi \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big ) - H^\\Phi \\big (P (t_0)\\,|\\,Q\\big )}{h}\\,=\\, \\,\\Big \\langle \\, \\varphi ({\\mathbf {\\ell }}_{t_0}) , \\psi _{t_0} \\Big \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,;$ whereas (REF ) gives $\\lim _{h \\downarrow 0} \\, \\frac{\\,\\varrho _\\star \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\,}{h} \\,=\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,,$ thus $\\lim _{h \\downarrow 0} \\frac{\\, H^\\Phi \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big )- H^\\Phi \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho _\\star \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\, } \\,=\\, \\,\\bigg \\langle \\, \\varphi ( {\\mathbf {\\ell }}_{t_0}) , \\frac{ \\psi _{t_0} }{\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,.$ This is the rate of change for the $\\Phi -$ relative entropy along the perturbed curve $\\, \\big ( P^\\psi (t) \\big )_{t_0 \\le t < t_0 + \\varepsilon }\\,,$ as measured on the manifold ${\\cal M}$ with respect to the distance in (REF ).", "On the other hand, we have from (REF ), (REF ) and (REF ), the following observation: Along the original curve of time-marginal distributions $\\, ( P (t) )_{t_0 \\le t < \\infty }\\,$ for the Chain, which corresponds to taking $ \\psi (\\cdot ) \\equiv {\\mathbf {\\ell }} (\\cdot ) $ above, the rate of $\\Phi -$ relative entropy dissipation measured in terms of the “flat metric\" distance traveled on the manifold ${\\cal M}, $ is given as $\\lim _{h \\downarrow 0} \\frac{\\, H^\\Phi \\big (P (t_0 +h)\\,|\\,Q\\big ) - H^\\Phi \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho _\\star \\big ( P (t_0+h), P (t_0) \\big )\\, } \\,=\\, - \\,\\big \\Vert \\varphi ( {\\mathbf {\\ell }}_{t_0} ) \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,<\\,0\\,.$ A simple comparison of the last two displays, via Cauchy-Schwarz, gives the steepest descent property $\\lim _{h \\downarrow 0} \\frac{\\, H^\\Phi \\big (P^\\psi (t_0 +h)\\,|\\,Q\\big ) - H^\\Phi \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho _\\star \\big ( P^\\psi (t_0+h), P (t_0) \\big )\\, } \\,- \\,\\lim _{h \\downarrow 0} \\frac{\\, H^\\Phi \\big (P (t_0 +h)\\,|\\,Q\\big ) - H^\\Phi \\big (P (t_0)\\,|\\,Q\\big )}{\\,\\varrho _\\star \\big ( P (t_0+h), P (t_0) \\big )\\, }$ $= \\, \\big \\Vert \\varphi ( {\\mathbf {\\ell }}_{t_0} ) \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} + \\bigg \\langle \\, \\varphi ( {\\mathbf {\\ell }}_{t_0} ) , \\frac{ \\psi _{t_0} }{\\, \\big \\Vert \\psi _{t_0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q) } \\,\\ge \\, 0$ of the $\\Phi -$ relative entropy with respect to the distance in (REF ), along the original curve of Markov Chain time-marginals.", "Equality holds if, and only if, $\\nabla \\psi _{t_0}$ is a negative constant multiple of $\\nabla \\varphi ({\\mathbf {\\ell }}_{t_0})$ .", "$\\Box $" ], [ "Non-uniqueness of the Flat Metric", "There exist norms other than $\\,\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)$ of (REF ), for which Theorem REF remains valid; see Dietert (2015) and Proposition REF below.", "Here we exhibit an explicit example.", "Fix $\\ell \\in \\mathcal {L}_+(\\mathcal {S})$ and consider the “modified weighted $\\,\\mathbb {H}^{-1}-$ norm” given by $\\big \\Vert f \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2\\, := \\,\\Big \\langle { \\frac{1}{\\vartheta _\\ell } \\nabla \\big (\\mathcal {K}^{-1} f\\big ), \\nabla \\big ( \\mathcal {K}^{-1} f \\big ) }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}$ for functions $f : \\mathcal {S}\\rightarrow \\mathbb {R}$ with $\\,\\sum _{x \\in \\mathcal {S}} f(x) q(x) = 0\\,$ .", "This norm is never smaller than the original $\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)-$ norm; namely, $\\big \\Vert f \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)} \\, \\ge \\, \\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)} \\, .$ And equality holds when $f = \\mathcal {K}\\ell \\,;$ to wit, $\\big \\Vert \\mathcal {K}\\ell \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)} \\,= \\, \\big \\Vert \\mathcal {K}\\ell \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}.$ These two facts imply that the curve $\\, ( P (t) )_{t_0 \\le t < \\infty }$ from Theorem REF , which corresponds to the original backward equation $ \\,\\partial {\\mathbf {\\ell }}_t = \\nabla \\cdot ( \\nabla {\\mathbf {\\ell }}_t ) = {\\cal K} {\\mathbf {\\ell }}_t $ of (REF ), is a curve of steepest descent also with respect to the modified norms ${\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)$ in (REF ).", "To prove the inequality (REF ), we use Proposition REF and the identity $\\mathcal {K}f = \\nabla \\cdot ( \\nabla f)$ to obtain $\\begin{aligned}\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2&= \\inf _{G : \\mathcal {Z}\\rightarrow \\mathbb {R}}\\bigg \\lbrace \\big \\langle { G, \\vartheta _\\ell G }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}\\ : \\ f + \\nabla \\cdot \\big (\\vartheta _\\ell G\\big )=0 \\bigg \\rbrace \\\\& \\le \\Big \\langle {\\, \\frac{1}{\\vartheta _\\ell } \\nabla \\big ( \\mathcal {K}^{-1} f \\big ), \\, \\vartheta _\\ell \\Big ( \\frac{1}{\\vartheta _\\ell } \\nabla \\big ( \\mathcal {K}^{-1} f \\big ) \\Big ) }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}= \\big \\Vert f \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2 \\, .\\end{aligned}$ The equality (REF ) holds since, on the one hand, $\\big \\Vert \\mathcal {K}\\ell \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2= \\Big \\langle { \\frac{1}{\\vartheta _\\ell } \\nabla \\ell , \\nabla \\ell }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}& = \\Big \\langle { \\nabla \\varphi (\\ell ) , \\nabla \\ell }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\\\& = \\Big \\langle { \\nabla \\varphi (\\ell ) , \\vartheta _\\ell \\nabla \\varphi (\\ell ) }\\Big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}= \\big \\Vert \\varphi (\\ell ) \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}^2\\,\\, ;$ while, on the other hand, Proposition REF yields $\\big \\Vert \\mathcal {K}\\ell \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2= \\big \\Vert \\nabla \\cdot ( \\nabla \\ell ) \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2= \\big \\Vert \\nabla \\cdot \\big (\\vartheta _\\ell \\nabla \\varphi (\\ell )\\big ) \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}^2= \\big \\Vert \\varphi (\\ell ) \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)}^2 \\,\\, .$ Remark 8.4 In general, the norms $\\big \\Vert f \\big \\Vert _{{\\widetilde{\\mathbb {H}}}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}$ and $\\big \\Vert f \\big \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\ell Q)}$ are different.", "Indeed, it follows from Proposition REF and (REF ) that equality of norms holds if, and only if, $\\frac{1}{\\vartheta _\\ell } \\nabla \\big ( \\mathcal {K}^{-1} f \\big )$ is a discrete gradient.", "This is in general false, but it is true in the following very special cases: At equilibrium, i.e., with $\\,\\ell \\equiv 1,$ we have $\\,\\vartheta _\\ell \\equiv 1$ , so that $\\,\\frac{1}{\\vartheta _\\ell } \\nabla \\big ( \\mathcal {K}^{-1} f \\big ) = \\nabla \\big ( \\mathcal {K}^{-1} f \\big )$ ; For the multiple $\\Phi (\\xi ) = \\frac{1}{2}( \\xi ^2 - 1)$ of the variance in Section , $\\vartheta _\\ell \\equiv 1$ for every likelihood ratio $\\ell \\,$ ; If the state space $\\mathcal {S}$ consists of only two points, $\\frac{1}{\\vartheta _\\ell } \\nabla \\big ( \\mathcal {K}^{-1} f \\big ) $ is a discrete gradient, since this holds for every anti-symmetric function on $\\,\\mathcal {S}\\times \\mathcal {S}$ ." ], [ "Gradient Flows", "Let us reconsider now, under conditions of detailed balance, the results of Sections – from a different, “Riemannian” point of view.", "We shall see here that, under the conditions (REF ), the curve $ ( P(t) )_{0 \\le t < \\infty }$ of time-marginal distributions for the Chain evolves as a gradient flow of the relative $\\Phi -$ entropy.", "This takes place in a suitable geometry on the space of probability measures, in the spirit of the pioneering work by Jordan, Kinderlehrer & Otto (1998).", "We refer to Erbar & Maas (2012, 2014), Mielke (2011, 2013) and to the expository paper Maas (2017), for an in-depth study of such issues in discrete spaces.", "We summon from subsection REF the manifold $\\, {\\cal M = P_+ (S)} \\,$ of probability vectors $ P = \\big ( p (x) \\big )_{x \\in {\\cal S}}$ with strictly positive entries; i.e., $ \\, {\\cal M }\\, $ is the interior of the lateral face of the unit simplex in $\\mathbb {R}^{n},$ with $\\,n=|{\\cal S}| $ the cardinality of the state-space.", "We denote by $ {\\cal M}_0 ({\\cal S}) $ the collection of vectors $\\, W = \\big ( w (x) \\big )_{x \\in {\\cal S}}\\,$ with total mass $\\, \\sum _{x \\in {\\cal S}} w (x)=0\\,,$ viewed as “signed measures\", and observe that $\\, {\\cal M}\\,$ is a relatively open subset of the $(n-1)-$ dimensional affine space $\\, P + {\\cal M}_0 ({\\cal S})= \\lbrace P+W : W \\in {\\cal M}_0 ({\\cal S}) \\rbrace ,$ for an arbitrary $\\, P \\in {\\cal M}\\,$ .", "This observation allows us to identify the tangent space at each $\\, P \\in {\\cal M}\\, $ with $\\, {\\cal M}_0 ({\\cal S})\\,$ ." ], [ "Gradient Flow for the Variance\n", "As a warmup, let us start as in Section with a derivation for the gradient flow property for the variance functional $\\, {\\cal M} \\ni P \\mapsto V(P|Q)\\in \\mathbb {R}_+ $ of (6.1).", "Following de Giorgi's approach to curves of maximal slope (cf.", "Ambrosio, Gigli & Savaré (2008)), we compute the dissipation of this functional along an arbitrary smooth curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } $ on ${\\cal M} $ ; or equivalently, along the curve $( \\widetilde{\\ell }_t)_{0 \\le t < \\infty } $ induced on the space $ {\\cal L} $ by the likelihood ratios $\\widetilde{\\ell }_t (y) = \\widetilde{p}_t ( y)/ q(y)$ .", "As in Section , we express the time-evolution of this likelihood ratio curve as $\\,\\partial \\widetilde{\\ell }_t = {\\cal K} f_t= \\nabla \\cdot \\big ( \\nabla f_t \\big )\\,$ in the manner of (REF ), for a suitable curve $\\, ( f_t )_{0 \\le t < \\infty }\\,$ with $f_t : {\\cal S} \\rightarrow \\mathbb {R}\\,.$ This is uniquely determined up to an additive constant on account of the Chain's irreducibility, and its discrete gradient provides the “momentum vector field\" of the motion.", "Recalling the consequences $\\widehat{{\\cal K}}f = {\\cal K}f = \\nabla \\cdot ( \\nabla f )$ of detailed balance and of (REF ), (REF ), as well as the fact that $\\,\\nabla \\cdot \\,$ is the adjoint of $- \\nabla $ from (REF ), we obtain $\\begin{aligned}\\partial V \\big ( \\widetilde{P}_t \\big | Q \\big )& = \\partial \\big \\Vert \\widetilde{\\ell }_t \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {S},Q)}= 2 \\big \\langle { \\widetilde{\\ell }_t , \\partial \\widetilde{\\ell }_t}\\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= 2 \\big \\langle \\widetilde{\\ell }_t , {\\cal K} f_t \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= -2 \\big \\langle { \\nabla \\widetilde{\\ell }_t , \\nabla f_t }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\\\ & \\ge \\, -2\\, \\big \\Vert \\nabla \\widetilde{\\ell }_t \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)} \\, \\big \\Vert \\nabla f_t \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)}\\,\\ge \\, - \\, \\big \\Vert \\nabla \\widetilde{\\ell }_t \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z},C)} \\,-\\, \\big \\Vert \\nabla f_t \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z},C)}\\,.\\end{aligned}$ Equality holds in the first (resp., the second) of these inequalities if, and only if, $\\nabla f_t $ and $\\nabla \\widetilde{\\ell }_t$ are positively collinear (resp., have the same norm).", "In other words, both these last two inequalities hold as equalities if and only if $\\,\\nabla f_t =\\nabla \\widetilde{\\ell }_t$ , and this leads to the backwards equation (REF ) on account of detailed balance: $\\partial \\widetilde{\\ell }_t = {\\cal K} f_t = \\nabla \\cdot \\big ( \\nabla f_t \\big ) =\\nabla \\cdot \\big ( \\nabla \\widetilde{\\ell }_t \\big ) = {\\cal K} \\,\\widetilde{\\ell }_t = \\widehat{{\\cal K}} \\,\\widetilde{\\ell }_t\\,.$ But the last two norms in (REF ) are $ \\,\\Vert \\nabla f_t \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)} = \\Vert \\nabla ({\\cal K}^{-1} (\\partial \\widetilde{\\ell }_t)) \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)} = \\Vert \\partial \\widetilde{\\ell }_t \\Vert _{\\mathbb {H}^{-1}(\\mathcal {S}, Q)}\\,$ as well as $\\, \\Vert \\nabla \\widetilde{\\ell }_t \\Vert _{\\mathbb {L}^2(\\mathcal {Z},C)} = \\Vert \\widetilde{\\ell }_t \\Vert _{\\mathbb {H}^{1}(\\mathcal {S},Q)} \\,.$ In this manner we obtain from (REF ) the following classical result.", "This provides another proof for Theorem REF by identifying the solutions of $\\, \\partial P_t = {\\cal K}^\\prime P_t\\,$ in (REF ) as curves in the direction of steepest descent for the variance, relative to the distance induced by the $\\mathbb {H}^{-1}(\\mathcal {S},Q)$ norm.", "But it also strengthens Theorem REF , by identifying also the correct velocity with which the gradient flow moves into this direction.", "Theorem 9.1 For any given probability vector $P \\in {\\cal M} $ and with $\\,{\\mathbf {\\ell }} \\in {\\cal L}\\,$ the likelihood ratio vector corresponding to $P$ , we have along any smooth curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } $ on ${\\cal M} $ with $\\widetilde{P}_0=P $ the inequality $\\bigg ( \\partial V \\big ( \\widetilde{P}_t \\big | Q \\big )+ \\big \\Vert \\partial \\widetilde{\\ell }_t \\big \\Vert ^2_{\\mathbb {H}^{-1}(\\mathcal {S},Q)} \\bigg )\\bigg |_{t=0} \\,\\ge \\, - \\, \\big \\Vert {\\mathbf {\\ell }} \\big \\Vert ^2_{\\mathbb {H}^{1}(\\mathcal {S},Q)} \\, .$ Equality holds if, and only if, the curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } \\subset {\\cal M} $ satisfies the forward equation $ \\partial \\widetilde{P}_t = {\\cal K}^\\prime \\widetilde{P}_t$ $($ equi- valently, the induced likelihood ratio curve $( \\widetilde{\\ell }_t)_{0 \\le t < \\infty } \\subset {\\cal L} $ satisfies the backward equation $ \\partial \\widetilde{\\ell }_t = {\\cal K} \\widetilde{\\ell }_t).$" ], [ "Gradient Flow for the $\\Phi -$ Relative Entropy\n", "Let us examine now, how these ideas might work in the context of the generalized relative entropy functional ${\\cal M} \\ni P \\, \\longmapsto \\, H^\\Phi \\big ( P \\big | Q \\big ) \\, :=\\, \\sum _{y \\in {\\cal S}} \\, q(y) \\, \\Phi \\Big ( \\frac{p ( y)}{q(y)} \\Big ) \\, \\in \\, [0, \\infty )$ corresponding to a convex function $\\Phi ,$ as in Section .", "We fix a smooth curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } $ on ${\\cal M} $ emanating from a given $\\widetilde{P}_0= P \\in {\\cal M};$ and consider the induced curve $( \\widetilde{\\ell }_t)_{0 \\le t < \\infty } \\subset {\\cal L} $ of likelihood ratios $\\widetilde{\\ell }_t (y) = \\widetilde{p}_t ( y)/ q(y), ~ y \\in {\\cal S}\\,$ emanating from $\\,{\\mathbf {\\ell }} = \\ell _0$ .", "As in subsection REF , we cast the time-evolution of the likelihood ratio curve as a continuity equation $\\partial \\widetilde{\\ell }_t \\,+\\, \\nabla \\cdot \\big ( \\widetilde{\\vartheta }_t \\nabla f_t \\big ) = 0$ where the “velocity vector field” is the discrete gradient of a suitable function $\\,f_t : {\\cal S} \\rightarrow \\mathbb {R}\\,$ , and $\\widetilde{\\vartheta }_t$ is a shorthand for $\\vartheta _{\\widetilde{\\ell }_t}$ from (REF ).", "In the manner of (REF ), this expresses the time-evolution of the $\\Phi -$ relative entropy functional $\\,H^\\Phi \\big ( \\widetilde{P}_t \\big | Q \\big ) = \\sum _{y \\in {\\cal S}} \\, q(y) \\,\\Phi \\big (\\widetilde{\\ell }_t (y) \\big ) $ in (REF ) along the curve $\\,\\big ( \\widetilde{P}_t\\big )_{0 \\le t < \\infty } \\, $ as $\\begin{aligned}\\partial H^\\Phi \\big ( \\widetilde{P}_t \\big | Q \\big )& = \\big \\langle \\varphi ( \\widetilde{\\ell }_t ), \\partial \\widetilde{\\ell }_t \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= - \\big \\langle \\varphi ( \\widetilde{\\ell }_t ), \\nabla \\cdot ( \\widetilde{\\vartheta }_t \\nabla f_t ) \\big \\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}= \\big \\langle { \\,\\nabla \\varphi ( \\widetilde{\\ell }_t ) , \\widetilde{\\vartheta }_t\\nabla f_t \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z},C)}\\\\ &= \\big \\langle { \\,\\nabla \\varphi ( \\widetilde{\\ell }_t ) , \\nabla f_t \\, }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)}\\ge \\, - \\big \\Vert \\nabla \\varphi ( \\widetilde{\\ell }_t ) \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)} \\, \\big \\Vert \\nabla f_t \\big \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)}\\\\&\\ge - \\, \\Big ( \\big \\Vert \\nabla \\varphi ( \\widetilde{\\ell }_t ) \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)} + \\big \\Vert \\nabla f_t \\big \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)}\\Big )\\Big /2\\,.\\end{aligned}$ Once again, equality holds if and only if $\\,\\nabla f_t =\\nabla \\big ( \\varphi (\\widetilde{\\ell }_t) \\big ),$ and this leads by detailed balance to the backwards equation $\\partial \\widetilde{\\ell }_t= - \\nabla \\cdot \\big ( \\widetilde{\\vartheta }_t \\,\\nabla f_t \\big )= \\nabla \\cdot \\big (\\widetilde{\\vartheta }_t \\,\\nabla (\\varphi ( \\widetilde{\\ell }_t ))\\big )= \\nabla \\cdot \\big ( \\nabla \\widetilde{\\ell }_t \\big )= {\\cal K} \\,\\widetilde{\\ell }_t= \\widehat{{\\cal K}} \\,\\widetilde{\\ell }_t$ of (REF ).", "We have used here the elementary but crucial consequence $\\,\\widetilde{\\vartheta }_t \\,\\nabla (\\varphi ( \\widetilde{\\ell }_t ))= \\nabla \\widetilde{\\ell }_t\\,$ of (REF ), a “discrete chain rule\" that sheds light on our choice of weight-function $\\Theta ^\\Phi $ in (REF ).", "But the last two norms displayed in (REF ) are $ \\,\\Vert \\nabla f_t \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)} = \\Vert \\partial \\widetilde{\\ell }_t \\Vert _{_{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\widetilde{\\ell }_t Q)}}\\,$ and $ \\, \\Vert \\nabla \\varphi ( \\widetilde{\\ell }_t ) \\Vert _{\\mathbb {L}^2(\\mathcal {Z}, \\widetilde{\\vartheta }_t C)}=\\Vert \\varphi ( \\widetilde{\\ell }_t ) \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\widetilde{\\ell }_t Q)} \\,.$ We summarize the situation in Theorem REF below; this corresponds to Theorem REF , in the same manner as Theorem REF corresponds to Theorem REF .", "Again, the de Giorgi argument (REF ) gives not only the “direction of steepest descent\" into which the gradient flow travels, but also the velocity of this flow.", "Theorem 9.2 For any given probability vector $P \\in {\\cal M},$ and with $\\,{\\mathbf {\\ell }} \\in {\\cal L}\\,$ the likelihood ratio vector corresponding to $P$ , we have along any smooth curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } $ on ${\\cal M} $ with $\\widetilde{P}_0=P $ the inequality $\\bigg ( 2\\, \\partial H^\\Phi \\big ( \\widetilde{P}_t \\big | Q \\big )+ \\big \\Vert \\partial \\widetilde{\\ell }_t \\big \\Vert ^2_{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\mathbf {\\ell }Q)} \\bigg )\\bigg |_{t=0} \\,\\ge \\, - \\, \\big \\Vert \\varphi ( {\\mathbf {\\ell }} ) \\big \\Vert ^2_{\\mathbb {H}^{1}_\\Theta (\\mathcal {S}, \\mathbf {\\ell }Q)} \\, .$ Equality holds here if, and only if, the curve $( \\widetilde{P}_t)_{0 \\le t < \\infty } \\subset {\\cal M} $ satisfies the forward equation $\\,\\partial \\widetilde{P}_t = {\\cal K}^\\prime \\widetilde{P}_t\\,$ $($ equivalently, the likelihood ratio curve $\\,( \\widetilde{\\ell }_t)_{0 \\le t < \\infty } \\subset {\\cal L} \\, $ satisfies the backward equation $\\,\\partial \\widetilde{\\ell }_t = {\\cal K} \\, \\widetilde{\\ell }_t \\,,$ and the corresponding “driver\" in (REF ) is $\\, f_t = - \\varphi (\\widetilde{\\ell }_t)\\, .", ")$" ], [ "A Riemannian Framework", "Let us take up these same ideas again, but now in a Riemannian-geometric framework as for instance in Maas (2011), Mielke (2011).", "For any given probability vector $\\, P \\in {\\cal M}\\,, $ we define the “likelihood ratio\" vector $\\, {\\mathbf {\\ell }} = \\big ( \\ell (x) \\big )_{x \\in {\\cal S}} \\in \\mathcal {L}$ with strictly positive elements $\\, \\ell (x) := p(x) / q(x)$ .", "We consider then the Riemannian metric $(g_{\\mathbf {\\ell }})_{{\\mathbf {\\ell }} \\in \\mathcal {L}}$ on $\\mathcal {L}$ induced by the scalar products $\\, \\,g_{\\mathbf {\\ell }} (\\partial \\ell _1, \\partial \\ell _2 ) \\,:= \\,\\big \\langle { \\nabla \\psi _1, \\nabla \\psi _2 }\\big \\rangle _{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)} \\, ,$ where $\\nabla \\psi _i$ is the unique discrete gradient satisfying the continuity equation $\\partial \\ell _i = \\nabla \\cdot ( \\vartheta _\\ell \\nabla \\psi _i ) $ for $i=1, 2$ .", "In particular, $g_{\\mathbf {\\ell }}(\\partial \\ell , \\partial \\ell ) = \\Vert \\nabla \\psi \\Vert ^2_{\\mathbb {L}^2(\\mathcal {Z}, \\vartheta _\\ell C)}= \\Vert \\partial \\ell \\Vert _{\\mathbb {H}^{-1}_\\Theta (\\mathcal {S}, \\mathbf {\\ell }Q)}^2$ on account of (REF ).", "The Riemannian gradient $\\operatornamewithlimits{grad}F$ of a smooth functional $F : \\mathcal {L}\\rightarrow \\mathbb {R}$ is then given by $\\operatornamewithlimits{grad}F = - \\nabla \\cdot \\Big (\\vartheta _\\ell \\,\\,\\nabla D_\\ell F \\Big ) \\,, \\qquad \\text{where} \\quad D_\\ell F \\, \\equiv \\, \\frac{\\delta F}{\\delta \\ell }$ is the $\\mathbb {L}^2(\\mathcal {S},Q)$ -derivative defined by $\\lim _{\\varepsilon \\rightarrow 0} \\varepsilon ^{-1}\\big ( F(\\ell + \\varepsilon \\eta ) - F(\\ell ) \\big ) = \\langle { D_\\ell F, \\eta }\\rangle _{\\mathbb {L}^2(\\mathcal {S},Q)}$ for $\\eta : \\mathcal {S}\\rightarrow \\mathbb {R}$ with $\\sum _{x \\in \\mathcal {X}} \\eta (x) q(x) = 0$ .", "In particular, the gradient flow equation $\\,\\partial \\ell = - \\operatornamewithlimits{grad}F(\\ell )\\,$ reads $\\partial \\ell = \\nabla \\cdot \\Big (\\vartheta _\\ell \\,\\, \\nabla D_\\ell F \\Big ) \\ .$ The Riemannian metric $g$ on $\\mathcal {L}$ can be turned into a Riemannian metric $G$ on the manifold of probability measures $\\mathcal {M}$ , via $\\,G_P(\\partial P_1, \\partial P_2) := g_{\\mathbf {\\ell }} (\\partial \\ell _1, \\partial \\ell _2)$ , where $P = {\\mathbf {\\ell }}\\, Q$ and $\\,\\partial P_i = \\partial \\ell _i \\, Q\\,$ for $i = 1,2$ .", "Theorem 9.3 (Maas (2011), Mielke (2011)): Under the detailed balance conditions (REF ), and with $ \\Theta $ the function of (REF ), the Forward Kolmogorov equation $\\,\\partial P(t)= {\\cal K}^\\prime P(t)\\,$ in (REF ) is the gradient flow of the $\\Phi -$ relative entropy in (REF ) with respect to the Riemannian metric $G$ induced on the manifold $ {\\cal M} .$ Proof: Let $\\,( P(t))_{0 \\le t < \\infty }$ solve the Forward Kolmogorov equation $\\,\\partial P(t)= {\\cal K}^\\prime P(t)\\,$ .", "By detailed balance, the associated likelihood ratio curve $\\,\\big ({\\mathbf {\\ell }} (t)\\big )_{0 \\le t < \\infty } \\subset {\\cal L} \\, $ satisfies the backward equation $\\,\\partial {\\mathbf {\\ell }} (t) = {\\cal K} \\,{\\mathbf {\\ell }} (t)$ .", "In view of (REF ), we thus need to verify the identity ${\\cal K} \\,{\\ell } = \\nabla \\cdot \\Big (\\vartheta _\\ell \\, \\nabla D_\\ell h^\\Phi \\Big ) \\ ,$ where $h^\\Phi : \\mathcal {L}\\rightarrow \\mathbb {R}$ is defined by $h^\\Phi (\\ell ) = H^\\Phi (\\ell Q|Q)$ .", "For $\\,{\\mathbf {\\ell }} \\in \\mathcal {L}\\,$ and $\\,\\eta : \\mathcal {S}\\rightarrow \\mathbb {R}\\,$ with $\\,\\sum _{x \\in \\mathcal {S}} \\eta (x) q(x) = 0\\,,$ we have the directional derivative computation $\\frac{\\mathrm {d}~}{\\mathrm {d}\\varepsilon } \\, h^\\Phi \\big ( {\\mathbf {\\ell }} + \\varepsilon \\eta \\, \\big ) \\, \\bigg |_{\\varepsilon = 0}\\,=\\,\\sum _{x \\in {\\cal S}} \\, \\eta (x) \\,\\varphi \\big ( \\ell (x) \\big );\\quad ~~~~\\text{thus}~ ~~~~~ D_\\ell h^\\Phi \\equiv \\frac{\\delta h^\\Phi }{\\delta \\ell }= \\varphi ( {\\mathbf {\\ell }} ) : = \\Big ( \\varphi ( \\ell (x) ) \\Big )_{x \\in {\\cal S}} .$ Invoking the “discrete chain-rule” $\\,\\vartheta _\\ell \\,\\nabla (\\varphi ( {{\\mathbf {\\ell }}} ))= \\nabla {\\ell }$ we obtain the desired identity $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nabla \\cdot \\Big (\\vartheta _\\ell \\, \\nabla D_\\ell h^\\Phi \\Big )= \\nabla \\cdot \\Big (\\vartheta _\\ell \\, \\nabla (\\varphi ({\\mathbf {\\ell }}))\\Big )= \\nabla \\cdot \\big ( \\nabla \\ell \\big )= \\mathcal {K}\\ell .\\qquad \\qquad \\qquad \\qquad \\quad \\Box $ Theorem REF has a converse, developed in Dietert (2015) as follows.", "Proposition 9.4 Suppose that there exists a $\\,{\\cal C}^1$ Riemannian metric on the manifold of probability vectors ${\\cal M} ,\\,$ under which the Forward Kolmogorov equation $\\,\\partial P(t)= {\\cal K}^\\prime P(t)\\,$ of (REF ) is the gradient flow for the relative entropy in (REF ).", "Then the Markov Chain satisfies the detailed balance conditions (REF )." ], [ "The HWI Inequality", "In the Riemannian framework of this Section, we present now a version of the celebrated HWI inequality of Otto & Villani (2000).", "The basic ingredient is the notion of Ricci curvature in the present context, as in Definition 1.3 of Maas (2011).", "We recast this definition using the more general notion of $\\Phi $ -entropy in Section – rather than the classical entropy which is, of course, a special case.", "We recall also from subsection 3.1, Remark REF the manifold ${\\cal M}$ of probability vectors on ${\\cal S}$ with strictly positive entries, its closure $\\overline{{\\cal M}}$ of probability vectors with nonnegative entries, and the corresponding manifolds ${\\cal L}\\,,$ $\\overline{{\\cal L}}$ of likelihood ratios.", "Definition 9.1 Ricci$^\\Phi $ -curvature: We say that our finite-state Markov Chain with generator $\\mathcal {K}$ has non-local Ricci curvature bounded from below by $\\kappa \\in \\mathbb {R}$ relative to $\\Phi $ as above, and write  Ricci$^\\Phi (\\cal K)\\ge \\kappa ,\\,$ if for every constant-speed geodesic $(P_t)_{0 \\le t \\le 1}$ on the closed manifold $\\overline{\\cal M}$ , the inequality $H^\\Phi \\big (P_t \\big | Q\\big ) \\le (1-t) H^\\Phi \\big (P_0 \\big | Q\\big ) + tH^\\Phi \\big (P_1 \\big | Q\\big ) - \\frac{\\,\\kappa \\,}{2} \\,t (1-t)\\, \\mathcal {W}^2 (P_0,P_1) ,\\quad 0 \\le t \\le 1$ holds.", "Here $\\cal {W} (\\cdot , \\cdot )$ is the geodesic distance with respect to the Riemannian metric of subsection 9.3.", "We shall apply the above inequality in the form of the following result about functions of a real variable.", "Proposition 9.5 Let $\\big ( f(t) \\big )_{0 \\le t \\le 1}$ be a continuous, real-valued function such that $f(t+h) - 2f(t) + f(t-h) \\ge \\kappa h^2$ holds for some $\\kappa \\in \\mathbb {R}$ and every pair $(t,h) \\in \\mathbb {R}_+^2$ with $\\, t \\le 1 \\pm h .", "$ Suppose also that f is right-differentiable at $t=0$ with derivative $f^{\\prime }(0)$ .", "Then $f(1) \\ge f(0) + f^{\\prime }(0) + \\frac{\\kappa }{2}.$ Proof: If $f$ is twice differentiable, the condition (REF ) amounts to $f^{\\prime \\prime } \\ge \\kappa $ .", "For general $f$ and supposing $\\kappa = 0,$ condition (REF ) is tantamount to the convexity of $f$ , so the inequality (REF ) becomes obvious.", "The case of general $\\kappa $ follows by subtracting from $f(t)$ the quadratic $ \\,\\kappa \\, t^2 /2$ .", "$\\Box $ Under the assumption Ricci$^\\Phi (\\cal K)\\ge \\kappa $ , for a constant-speed geodesic $(P_t)_{0 \\le t \\le 1}$ joining $ P_0 \\in \\cal M$ with $P_1 \\in \\overline{\\cal M}$ and such that $\\mathcal {W} (P_0,P_1) =1$ , the function $f(t)=H^{\\Phi }(P_t | Q)$ satisfies the assumptions of Proposition 9.5.", "Indeed, $(P_u)_{t-h \\le u \\le t+h}$ is then a constant-speed geodesic which joins $P_{t-h} $ with $P_{t+h}$ and satisfies $\\mathcal {W} (P_{t-h},P_{t+h}) =2h$ , so (REF ) applies with $t= 1/2 $ .", "The existence of constant-speed geodesics and of $f^{\\prime }(0)$ , follows respectively from Theorem 3.2 and Proposition 3.4 in Erbar & Maas (2012).", "We formulate now a version of the HWI inequality in the present context.", "This sharpens slightly Theorem 7.3 of Erbar & Maas (2012), where $P_1$ in the following theorem is the invariant measure $\\,Q\\,$ ; and its proof does not rely on the “evolution variational inequality\" (the EVI of Theorem 4.5 in Erbar & Maas (2012)), but rather on the elementary estimate of Proposition REF .", "Theorem 9.6 HWI Inequality of Otto-Villani: Under the assumptions of subsection 9.2, suppose that $ \\, \\textnormal {Ricci}^\\Phi (\\cal K) \\ge \\kappa \\, $ holds for some $\\kappa \\in \\mathbb {R}$ .", "With $P_0,$ $P_1$ any probability measures in $\\cal M,$ $\\overline{\\cal M},$ respectively, denote by $\\mathcal {W} (P_0,P_1)$ their geodesic distance and by $I^\\Phi (P_0|Q)$ the $\\Phi $ -Fisher information of (REF ) with $t=0$ .", "We have then $H^\\Phi (P_0|Q) - H^\\Phi (P_1|Q)\\,\\le \\, \\big ( I^\\Phi (P_0|Q)\\big )^{1/2} \\ \\mathcal {W} (P_0,P_1) - \\frac{\\kappa }{2} \\, \\mathcal {W}^2 (P_0,P_1) .$ Proof: We follow the argument in the proof of Proposition 3.21 of Karatzas, Schachermayer & Tschiderer (2019), where the HWI inequality is established for diffusions in $\\mathbb {R}^n$ .", "We let $(P_t)_{0 \\le t \\le 1} \\subset \\overline{{\\cal M}}$ be a constant-speed geodesic of probability measures joining $P_0$ with $P_1$ (which we know exists, by Theorem 3.2 of Erbar & Maas (2012)), denote by $(\\ell _t)_{0\\le t\\le 1} \\subset \\overline{{\\cal L}}\\,$ the corresponding likelihood-ratio curve, consider the function $ \\,f(t):=H^{\\Phi }(P_t | Q)\\, , ~~ 0 \\le t \\le 1 ,$ and pass to the parametrization $u=u(t)=\\frac{w}{i^{1/2}}t\\, , \\qquad 0 \\le u \\le \\frac{w}{i^{1/2}},$ where $\\,i=I^\\Phi (P_0)=\\mathcal {E} (\\ell _0,\\phi (\\ell _0))\\,$ and $w= \\mathcal {W} (P_0,P_1)$ .", "We set $g(u) = g(u(t)) = f(t)$ .", "Recalling the likelihood ratio $\\ell _t$ corresponding to $P_t$ , consider the continuous curve of likelihood ratios $\\widetilde{\\ell }(u) = \\ell _{t}\\, , \\qquad 0 \\le u \\le \\frac{w}{i^{1/2}}$ so that $\\widetilde{\\ell }(0) = \\ell _0$ and $\\widetilde{\\ell }(w\\,i^{-1/2}) = \\ell _1$ , as well as the corresponding curve $\\widetilde{P} (u), ~ 0 \\le u \\le w\\, i^{- 1/2}$ of probabilities.", "Since $(P_t)_{0 \\le t \\le 1}$ is a geodesic of constant speed $w$ , we have $\\big \\Vert \\partial \\ell _0 \\big \\Vert _{H^{-1}_\\Theta (\\mathcal {S} , \\ell Q)} = \\mathcal {W} (P_0 , P_1 ) = w , \\qquad \\text{thus} \\qquad \\big \\Vert \\partial \\widetilde{\\ell }(0) \\big \\Vert ^{2}_{H^{-1}_\\Theta (\\cal S , \\ell Q)} = i$ with $\\ell = \\ell _0;$ this last display gives the second term in (REF ), Theorem REF .", "As for the term $ \\, \\big \\Vert \\varphi ( {\\mathbf {\\ell }} ) \\big \\Vert ^2_{\\mathbb {H}^{1}_\\Theta (\\mathcal {S}, \\mathbf {\\ell }Q)}$ in (REF ), the expression (REF ) and Remark REF give $\\, \\big \\Vert \\varphi (\\widetilde{\\ell }(0))\\big \\Vert ^2_{H^{1}_\\Theta (\\cal S , \\ell Q)} = \\mathcal {E} (\\ell _0, \\varphi (\\ell _0)) = i.", "$ In this manner, (REF ) leads to the inequality $g^{\\prime }(0)=\\partial H^\\Phi (\\widetilde{P}_u | Q)\\Big |_{u=0} \\ge -i,$ where the existence of the right-derivative $g^{\\prime }(0)$ is assured by Proposition 3.4 of Erbar & Maas (2012).", "Going back to the original parametrization, we obtain $ f^{\\prime }(0) \\ge - w i^{1/2}.$ The assumption Ricci$^\\Phi ({\\cal K})\\ge \\kappa $ implies that $f$ satisfies (REF ), with $\\kappa $ replaced by $\\kappa w^2$ .", "In conclusion, (REF ) gives $H^\\Phi (P_1 | Q) \\ge H^\\Phi (P_0 | Q) - i^{1/2} w + \\frac{\\kappa }{2} w^2,$ which is tantamount to the HWI inequality (REF ).", "$\\Box $ Remark 9.1 As is well known (e.g., Erbar & Maas (2012)), the HWI inequality leads directly to the corresponding versions of the Modified Log-Sobolev and Talagrand inequalities, by taking $ \\Phi (\\cdot ) = \\Psi (\\cdot ) $ as in (7.13) and $P_1 = Q$ .", "Poincaré-type inequalities also follow this way, by linearizing the Modified Log-Sobolev inequality.", "The HWI inequality (REF ) can be sharpened.", "In the above proof, we estimated the slope of the function $H^\\Phi (P_t | Q)$ at $t= 0$ in terms of the worst case, i.e., the steepest descent; this led to the square root $\\big ( I^\\Phi (P_0|Q)\\big )^{1/2}$ of the Fisher information, by Theorem REF .", "But Propositions REF , REF allow us to calculate the slope of this function, with respect to the norm ${H^{-1}_\\Theta (\\cal S , \\ell Q)}$ which induces the local Riemannian metric at $\\ell = \\ell _{0}$ .", "We obtain in this manner the following more precise result, in the spirit of Otto & Villani (2000), Cordero-Erausquin (2005) or Karatzas, Schachermayer & Tschiderer (2019).", "Proposition 9.7 Under the assumptions of Theorem REF , suppose in addition that the curve $(P_t)_{0 \\le t \\le 1}$ is driven by a continuous function $(\\psi _t)_{0 \\le t \\le 1}$ via the “discrete continuity equation\" $\\,\\partial \\ell ^\\psi _t + \\nabla \\cdot ( \\vartheta _{\\ell _t} \\nabla \\psi _t ) = 0.$ Then we have the inequality $H^\\Phi (P_0|Q) - H^\\Phi (P_1|Q)\\,\\le \\, \\mathcal {W} (P_0,P_1)\\, \\bigg \\langle \\, \\varphi ( {\\mathbf {\\ell }}_{0}) , \\frac{ \\psi _{0} }{\\, \\big \\Vert \\psi _{0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, \\ell Q)} - \\frac{\\kappa }{2} \\, \\mathcal {W}^2 (P_0,P_1) .$ From (REF ), the slope of the function $H^\\Phi (P_t | Q)$ with respect to the norm ${H^{-1}_\\Theta (\\cal S , \\ell Q)}$ on $\\mathcal {M}$ , which induces the local Riemannian metric at $(t, {\\mathbf {\\ell }} )=(0, \\ell _{0})$ , is given by the bracket term on the right hand side of (REF ).", "Hence, we may replace the inequality (REF ) by the more precise equality $g^{\\prime }(0) = - \\bigg \\langle \\, \\varphi ( {\\ell }_{0}) , \\frac{ \\psi _{0} }{\\, \\big \\Vert \\psi _{0} \\big \\Vert _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)} \\,} \\bigg \\rangle _{\\mathbb {H}^1_\\Theta (\\mathcal {S}, {\\mathbf {\\ell }} Q)}.$ The rest of the proof of Theorem REF can be repeated verbatim, to obtain (REF ) instead of (REF ).", "Remark 9.2 What happens when $P_0$ is on the boundary of $\\mathcal {M}$ , as in Remark REF ?", "that is, when the set $ \\,\\mathcal {N}_0 = \\lbrace x \\in {\\cal S}: P_0(x)=0\\rbrace $ is non-empty?", "To be specific, let us concentrate on the classical entropy $\\,\\Phi (\\ell ) = \\ell \\, \\log \\ell \\,$ .", "Then the Fisher information $ I^\\Phi (P_0|Q)$ is infinite, and the HWI inequality (REF ) holds trivially.", "On the other hand, the refined version (REF ) may deliver some nontrivial information.", "Indeed, suppose that $(P_t)_{0 \\le t \\le 1}$ is driven by a continuous function $(\\psi _t)_{0 \\le t \\le 1}$ via the “discrete continuity equation\" (REF ).", "If $\\psi _0$ also vanishes on $\\mathcal {N}_0\\,,$ the bracket term in (REF ) is finite (via the rule $ 0 \\cdot \\infty = 0$ ).", "As we assume that $\\psi (\\cdot )$ is continuous (actually, we only need this continuity at $t=0$ ), we can still apply the above argument and conclude that (REF ) holds, yielding a nontrivial result.", "The geometric interpretation of $\\psi _0$ vanishing on $\\mathcal {N}_0\\,, $ is that the curve $(P_t)_{0 \\le t \\le 1}$ starts “tangentially to the boundary of $\\mathcal {M}$ \", when departing from $P_0$ at this boundary." ], [ "Countable State-Space", "It is well known that the results of Sections and hold also for countably infinite state-spaces ${\\cal S}\\,;$ see Chapters 2, 3 in Norris (1997) and Liggett (2010).", "In particular, the ergodic property (REF ) holds at least for bounded functions $\\,f : {\\cal S} \\rightarrow \\mathbb {R}\\,.$ The crucial Proposition REF also remains valid.", "Propositions REF , REF carry over to countable state-spaces under the assumption $ V \\big (P(0)\\,|\\,Q\\big ) < \\infty $ .", "To see this, we start by observing that we can guarantee now prima facie only the local martingale property of the processes $\\,\\widehat{M}\\,$ in (REF ).", "Still, we can localize $\\,\\widehat{M} \\, $ by an increasing sequence $ \\big \\lbrace \\sigma _n \\big \\rbrace _{n \\in \\mathbb {N}} \\,$ of $\\,\\widehat{\\mathbb {G}}-$ stopping-times with values in $[0,T]$ and $ \\, \\lim _{n \\rightarrow \\infty } \\uparrow \\sigma _n = T,$ and create the bounded $(\\widehat{\\mathbb {G}}, \\mathbb {Q})-$ martingales $\\,\\widehat{M} (s \\wedge \\sigma _n)\\,, ~ 0 \\le s \\le T.$ Taking expectations in (REF ) with $\\,s=\\sigma _n\\,$ ,  then letting $n \\rightarrow \\infty $ and using monotone convergence, the $ \\mathbb {Q}-$ submartingale property of $\\,\\ell ^2 \\big ( T-s, \\widehat{X} (s) \\big ) \\,, ~~ 0 \\le s \\le T$ from Proposition REF , and optional sampling, we obtain from (REF ) the inequality $\\mathbb {E^Q} \\big [\\ell ^2 \\big ( T , X (T) \\big ) \\big ]+ \\int _0^T2 \\, {\\cal E} \\big ( \\ell _t\\, , \\ell _t \\,\\big )\\,\\mathrm {d}t \\,= \\,\\lim _{n \\rightarrow \\infty } \\uparrow \\mathbb {E^Q} \\big [\\ell ^2 \\big ( T-\\sigma _n , \\widehat{X} (\\sigma _n) \\big ) \\big ] \\,\\le \\, \\mathbb {E^Q} \\big [\\ell ^2 \\big ( 0 , X (0) \\big ) \\big ].$ But the reverse of this last inequality also holds, on account of Fatou's Lemma; thus (REF ) follows for countable state-spaces as well, and $\\,\\widehat{M} \\, $ is seen to be a true $(\\widehat{\\mathbb {G}}, \\mathbb {Q})-$ martingale.", "Then $\\, \\lim _{t \\rightarrow \\infty } V \\big (P(t)\\,|\\,Q\\big )=0$ , and with it (REF ), are proved for a countable state-space in the manner of Proposition REF below." ], [ "Relative Entropy Dissipates all the way down to Zero", "When the state-spaces ${\\cal S}$ is countably infinite, the results of Section pertaining to the relative entropy need the additional assumption $H \\big (P(0)\\big | Q \\big ) = \\sum _{y \\in {\\cal S}} \\, p (0,y) \\, \\log \\left( \\frac{\\,p(0,y)\\,}{q(y)} \\right)< \\infty \\,.$ Then everything goes through as before, including the non-negativity and decrease claims in (REF ) – except for the argument establishing (REF ), which uses the finiteness of the state-space in a crucial manner.", "Here is a proof for this result in the countable case.", "Proposition 10.1 The dissipation of relative entropy all the way down to zero, as in (REF ), holds for a countable state-space under the condition (REF ).", "Proof:   Let us recall the likelihood ratio process $\\, L (t) := \\ell \\big (t, X(t)\\big ),$ $ 0 \\le t < \\infty \\,$ of (REF ), and from (REF ) that its time-reversal $\\, L (T-s) , ~ 0 \\le s \\le T\\,$ is a $\\, \\big ( \\widehat{\\mathbb {G}}, \\mathbb {Q}\\big )-$ martingale.", "Fix $\\, 0 \\le t_1 < t_2 < \\infty \\,$ .", "For any $\\, T \\in (t_2 , \\infty )\\,$ , this means $\\, \\mathbb {E^Q} \\big [ L (T-s_1) \\, \\big | \\, {\\cal G} (T-s_2 )\\big ] = L (T-s_2)\\,$ for $s_1 = T - t_1,$ $~s_2 = T - t_2,$ or equivalently: $\\,\\mathbb {E^Q} \\big [ L (t_1) \\, \\big | \\, \\sigma \\big ( X (\\theta ) ,\\, \\, t_2 \\le \\theta \\le T \\big ) \\big ] \\,= \\,L (t_2)\\,.\\,$ But this last identity holds for any $\\, T \\in (t_2 , \\infty )$ , so it leads — on the strength of the P. Lévy martingale convergence Theorem 9.4.8 in Chung (1974) — to $\\mathbb {E^Q} \\big [ L (t_1) \\, \\big | \\, {\\cal H } (t_2) \\big ] = L (t_2)\\,, \\qquad ~~~{\\cal H } (t) := \\sigma \\big ( X (\\theta ) ,\\, \\, t \\le \\theta < \\infty \\big ).$ To wit, the likelihood ratio process $\\, \\big ( L(t) \\big )_{0 \\le t < \\infty }\\,$ is a martingale of the backwards filtration $\\, \\big ( {\\cal H}(t) \\big )_{0 \\le t < \\infty }\\,,$ whose “tail\" sigma-algebra is trivial on account of the ergodicity property (REF ) of the Markov Chain (Blackwell & Freedman (1964)): ${\\cal H}(\\infty ) \\,:=\\, \\bigcap _{0 \\le t < \\infty } {\\cal H}(t) \\,=\\, \\big \\lbrace \\emptyset , \\Omega \\big \\rbrace \\,, \\quad \\text{mod.}", "~ \\mathbb {Q}\\,.$ We invoke now the martingale version of the backward submartingale convergence Theorem 9.4.7 in Chung (1974).", "It follows from this result that $\\, \\big ( L(t) \\big )_{0 \\le t < \\infty }\\,$ is a $\\,\\mathbb {Q}-$ uniformly integrable family; that the limit $\\,L(\\infty ) := \\lim _{t \\rightarrow \\infty } L(t)\\,$ exists, both a.e.", "and in $\\,\\mathbb {L}^1$ under $\\mathbb {Q}\\,;$ and that the backward martingale property (REF ) extends all the way to infinity, namely $\\mathbb {E^Q} \\big [ L (t_1 ) \\, \\big | \\, {\\cal H } (\\infty ) \\big ] = L (\\infty )\\,.$ But the triviality under $\\,\\mathbb {Q} $ of the tail sigma-algebra, implies that $\\,L(\\infty ) \\,$ is $\\,\\mathbb {Q}-$ a.e. constant.", "Then the extended martingale property (REF ) identifies this constant as $\\,L (\\infty ) = \\mathbb {E^Q} \\big [L (\\infty ) \\big ] = \\mathbb {E^Q} \\big [L (t_1) \\big ] = 1\\,.$ We recall the relative entropy from (REF ).", "The convexity of the function $\\, \\Phi (\\ell ) = \\ell \\, \\log \\ell \\,$ shows, in conjunction with (REF ) and the Jensen inequality, that $\\Big ( \\Phi \\big ( L (t) \\big ), {\\cal H } (t) \\Big )_{0 \\le t < \\infty } \\qquad \\text{is a backward} ~~ \\mathbb {Q}-\\text{submartingale,}$ with decreasing expectation $\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( L( t ) \\big ]= H \\big ( P(t) \\, \\big | \\, Q \\big ) \\ge 0.\\,$ Because this expectation is bounded from below, we can appeal once again to the backward submartingale convergence Theorem 9.4.7 in Chung (1974).", "We deduce that the process in (REF ) is a $\\,\\mathbb {Q}-$ uniformly integrable family which converges, again both a.e.", "and in $\\,\\mathbb {L}^1$ under $\\mathbb {Q}\\,,$ to $\\,\\lim _{t \\rightarrow \\infty } \\Phi \\big ( L(t) \\big ) =\\Phi \\big ( L (\\infty ) \\big ) = \\Phi (1) = 0\\,.$ Furthermore, the aforementioned uniform integrability gives $\\lim _{t \\rightarrow \\infty } \\downarrow H \\big (P(t) \\, \\big | \\,Q\\big ) \\,=\\,\\lim _{t \\rightarrow \\infty } \\, \\mathbb {E^Q} \\big [ \\Phi \\big ( L( t ) \\big ) \\big ]\\,=\\, \\mathbb {E^Q} \\Big ( \\lim _{t \\rightarrow \\infty } \\Phi \\big ( L( t ) \\big ) \\Big ) \\,=\\, 0\\,;$ that is, (REF ) is also valid in this general case with countable state-space.", "$\\Box $" ], [ "Relative Entropy is Continuous at the Origin", "We discuss now the validity of the de Bruijn identities of (REF ) when the state-space is countable.", "Proposition 10.2 The de Bruijn identities of (REF ) for the dissipation of relative entropy are valid for a countable state-space, under the condition (REF ).", "To justify this claim, we would like to use the argument already deployed; but there is now no obvious, general way to turn the local martingale $\\,\\widehat{M}^{\\, h} $ of (REF ) into a true $\\mathbb {Q}-$ martingale.", "Thus, we localize $\\,\\widehat{M}^{\\, h} $ by an increasing sequence $ \\big \\lbrace \\sigma _n \\big \\rbrace _{n \\in \\mathbb {N}} \\,$ of $\\,\\widehat{\\mathbb {G}}-$ stopping-times with values in $[0,T]$ and $ \\, \\lim _{n \\rightarrow \\infty } \\uparrow \\sigma _n = T.$ In this manner we create the bounded $(\\widehat{\\mathbb {G}}, \\mathbb {Q})-$ martingales $\\,\\widehat{M}^{\\, h} (s \\wedge \\sigma _n)\\,, ~ 0 \\le s \\le T,$ which then give $\\mathbb {E^Q}\\int _0^{\\sigma _n} \\big ( \\partial h + \\widehat{{\\cal K}} h \\big ) \\big (u, \\widehat{X} (u) \\big )\\, \\mathrm {d}u\\,=\\, \\mathbb {E^Q} \\big [ h \\big ( \\sigma _n, \\widehat{X}(\\sigma _n) \\big ) \\big ] - \\,\\mathbb {E^Q} \\big [ h \\big ( 0, \\widehat{X}(0) \\big ) \\big ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $~~~~~~~~~~~~=\\, H \\big ( P(T-\\sigma _n) \\, \\big | \\, Q \\big ) - H \\big ( P(T) \\, \\big | \\, Q \\big ) \\, \\le \\, H \\big ( P(0 ) \\, \\big | \\, Q \\big ) - H \\big ( P(T) \\, \\big | \\, Q \\big ) \\, \\le \\, H \\big ( P(0 ) \\, \\big | \\, Q \\big ) \\, < \\, \\infty $ for every $\\, n \\in \\mathbb {N},$ on account of (REF ); see also the argument straddling (REF ) below.", "In particular, the sequence of real numbers in (REF ) takes values in the compact interval $\\, [ - H ( P(0 ) \\, | \\, Q ), H ( P(0 ) \\, | \\, Q ) ] .$ We would like now to let $ n \\rightarrow \\infty $ in (REF ), and establish the de Bruijn identity (REF ) in this case.", "The issue once again is continuity of the relative entropy — though now at the origin (rather than at infinity, as in (REF )); and not along fixed times, but rather along an appropriate sequence of stopping times, i.e., $\\lim _{n \\rightarrow \\infty } \\uparrow H \\big ( P(T- \\sigma _n) \\, \\big | \\, Q \\big )\\,=\\, H \\big ( P(0) \\, \\big | \\, Q \\big )\\,.$ Accepting this for a moment, and letting $\\, n \\rightarrow \\infty \\,$ in (REF ), we obtain the de Bruijn identity (REF ), i.e., $\\int _0^T I (t) \\, \\mathrm {d}t\\,=\\,\\mathbb {E^Q} \\int _0^{T} \\Big ( \\partial h + \\widehat{{\\cal K}} h \\Big ) \\big (u, \\widehat{X} (u) \\big ) \\, \\mathrm {d}u \\,=\\, H \\big ( P(0) \\, \\big | \\, Q \\big ) -H \\big ( P(T) \\, \\big | \\, Q \\big )$ by monotone convergence.", "We let now $ T \\rightarrow \\infty \\,$ in (REF ) and arrive at the second identity in (REF ), thanks to the property (REF ) already established in Proposition REF .", "Proof of (REF ): By analogy with (REF ), and invoking now additionally the optional sampling theorem for the bounded stopping times $ \\big \\lbrace \\sigma _n \\big \\rbrace _{n \\in \\mathbb {N}} \\,$ of $\\,\\widehat{\\mathbb {G}}\\,$ with values in $[0,T]$ , we deduce that the sequence of non-negative real numbers $H \\big ( P (T-\\sigma _n) \\,| \\,Q)\\,=\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( \\ell \\big ( T-\\sigma _n, \\widehat{X} (\\sigma _n) \\big ) \\big ) \\big ]\\,,~~~~~~ ~n \\in \\mathbb {N}$ is increasing; in particular, $\\,\\lim _{n \\rightarrow \\infty } H \\big ( P(T-\\sigma _n) \\, \\big | \\, Q \\big ) \\le H \\big ( P(0) \\, \\big | \\, Q \\big ) .$ On the other hand, the boundedness-from-below of the function $\\,\\Phi (\\ell ) = \\ell \\,\\log \\ell \\, $ gives $\\lim _{n \\rightarrow \\infty } H \\big ( P(T-\\sigma _n) \\, \\big | \\, Q \\big )\\, \\ge \\, \\mathbb {E^Q} \\Big [ \\lim _{n \\rightarrow \\infty } \\, \\Phi \\big ( \\ell \\big ( T-\\sigma _n, \\widehat{X} (\\sigma _n) \\big ) \\big ) \\Big ]\\,=\\, \\mathbb {E^Q} \\big [ \\Phi \\big ( \\ell \\big ( 0, X (0) \\big ) \\big ) \\big ]\\,=\\, H \\big ( P(0) \\, \\big | \\, Q \\big )$ with the help of Fatou's Lemma, and (REF ) follows.", "$\\Box $ Remark 10.1 The General Case: Exacly the same methods show that the results of Propositions REF and REF , pertaining to a general convex function $\\, \\Phi : (0, \\infty ) \\rightarrow \\mathbb {R}\\,$ with the properties imposed there, continue to hold for the generalized relative entropy functional of 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[ [ "Finite generation of cohomology for Drinfeld doubles of finite group\n schemes" ], [ "Abstract We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology.", "That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra.", "As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e.", "have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions.", "This paper completes earlier work of E. M. Friedlander and the author." ], [ "Introduction", "Fix $k$ an arbitrary field of finite characteristic.", "Let us recall some terminology [21]: A finite $k$ -linear tensor category ${C}$ is said to be of finite type (over $k$ ) if the self-extensions of the unit object $\\operatorname{Ext}^\\ast _{C}(,)$ are a finitely generated $k$ -algebra, and for any object $V$ in ${C}$ the extensions $\\operatorname{Ext}^\\ast _{C}(,V)$ are a finitely generated module over this algebra.", "In this case, the Krull dimension $\\operatorname{Kdim}{C}$ of ${C}$ is the Krull dimension of the extension algebra of the unit.", "One is free to think of ${C}$ here as the representation category $\\operatorname{rep}(A)$ of a finite-dimensional Hopf algebra $A$ , with monoidal structure induced by the comultiplication, and unit $=k$ provided by the trivial representation.", "It has been conjectured [10] [14] that any finite tensor category, over an arbitrary base field, is of finite type.", "Here we consider the category of representations for the Drinfeld double $D(G)$ of a finite group scheme $G$ , which is identified with the so-called Drinfeld center $Z(\\operatorname{rep}(G))$ of the category of finite $G$ -representations [18], [9].", "The Drinfeld double $D(G)$ is the smash product ${O}(G)\\rtimes kG$ of the algebra of global functions on $G$ with the group ring $kG$ , under the adjoint action.", "So, one can think of $Z(\\operatorname{rep}(G))$ , alternatively, as the category of coherent $G$ -equivariant sheaves on $G$ under the adjoint action $Z(\\operatorname{rep}(G))=\\operatorname{rep}(D(G))=\\operatorname{Coh}(G)^G.$ In the present work we prove the following.", "Theorem (REF ) For any finite group scheme $G$ , the Drinfeld center $Z(\\operatorname{rep}(G))$ is of finite type and of Krull dimension $\\operatorname{Kdim}Z(\\operatorname{rep}(G))\\le \\operatorname{Kdim}\\operatorname{rep}(G)+\\operatorname{embed.dim}(G).$ Here $\\operatorname{embed.dim}(G)$ denotes the minimal dimension of a smooth (affine) algebraic group in which $G$ embeds as a closed subgroup.", "The above theorem was proved for $G=\\mathbb {G}_{(r)}$ a Frobenius kernel in a smooth algebraic groups $\\mathbb {G}$ in work of E. M. Friedlander and the author [11].", "Thus Theorem REF completes, in a sense, the project of [11].", "One can apply Theorem REF , and results of J. Plavnik and the author [21], to obtain an additional finite generation result for all dual tensor categories $\\operatorname{rep}(G)^\\ast _{M}(:=\\operatorname{End}_{\\operatorname{rep}(G)}({M}))$ , calculated relative to an exact $\\operatorname{rep}(G)$ -module category ${M}$ [10].", "Corollary 1.1 Let $G$ be a finite group scheme, and ${M}$ be an arbitrary exact $\\operatorname{rep}(G)$ -module category.", "Then the dual category $\\operatorname{rep}(G)^\\ast _{M}$ is of finite type and of uniformly bounded Krull dimension $\\operatorname{Kdim}\\operatorname{rep}(G)^\\ast _{M}\\le \\operatorname{Kdim}\\operatorname{rep}(G)+\\operatorname{embed.dim}(G).$ Immediate from Theorem REF and [21].", "We view Theorem REF , and Corollary REF , as occurring in a continuum of now very rich studies of cohomology for finite group schemes, e.g.", "[12], [14], [23], [13], [26], [7], [3].", "Remark 1.2 Exact $\\operatorname{rep}(G)$ -module categories have been classified by Gelaki [15], and correspond to pairs $(H,\\psi )$ of a subgroup $H\\subset G$ and certain 3-cocycle $\\psi $ which introduces an associativity constraint for the action of $\\operatorname{rep}(G)$ on $\\operatorname{rep}(H)$ .", "Remark 1.3 For an analysis of support theory for Drinfeld doubles of some solvable height 1 group schemes, one can see [20], [19].", "The problem of understanding support for general doubles $D(G)$ is, at this point, completely open." ], [ "Approach via equivariant deformation theory", "In [11], where the Frobenius kernel $\\mathbb {G}_{(r)}$ in a smooth algebraic group $\\mathbb {G}$ is considered, we basically use the fact that ambient group $\\mathbb {G}$ provides a smooth, equivariant, deformation of $\\mathbb {G}_{(r)}$ parametrized by the quotient $\\mathbb {G}/\\mathbb {G}_{(r)}\\cong \\mathbb {G}^{(r)}$ in order to gain a foothold in our analysis of cohomology.", "In particular, the adjoint action of $\\mathbb {G}_{(r)}$ on $\\mathbb {G}$ descends to a trivial action on the twist $\\mathbb {G}^{(r)}$ , so that the Frobenius map $\\mathbb {G}\\rightarrow \\mathbb {G}^{(r)}$ can be viewed as smoothly varying family of $\\mathbb {G}_{(r)}$ -algebras which deforms the algebra of functions ${O}(\\mathbb {G}_{(r)})$ .", "Such a deformation situation provides “deformation classes\" in degree 2, $\\lbrace \\text{deformation classes}\\rbrace =T_1\\mathbb {G}^{(r)}\\subset \\operatorname{Ext}^2_{\\operatorname{Coh}(\\mathbb {G}_{(r)})^{\\mathbb {G}_{(r)}}}(,)=\\operatorname{Ext}^2_{D(\\mathbb {G}_{(r)})}(,).$ One uses these deformation classes, in conjunction with work of Friedlander and Suslin [14], to find a finite set of generators for extensions.", "For a general finite group scheme $G$ , we can try to pursue a similar deformation approach, where we embed $G$ into a smooth algebraic group $\\mathcal {H}$ , and consider $\\mathcal {H}$ as a deformation of $G$ parametrized by the quotient $\\mathcal {H}/G$ .", "However, a general finite group scheme may not admit any normal embedding into a smooth algebraic group.", "(This is the case for certain non-connected finite group schemes, and should also be the case for restricted enveloping algebras $kG=u^{\\rm res}(\\mathfrak {g})$ of Cartan type simple Lie algebras, for example).", "So, in general, one accepts that $G$ acts nontrivially on the parametrization space $\\mathcal {H}/G$ , and that the fibers in the family $\\mathcal {H}$ are permuted by the action of $G$ here.", "Thus we do not obtain a smoothly varying family of $G$ -algebras deforming ${O}(G)$ in this manner.", "One can, however, consider a type of equivariant deformation theory where the group $G$ is allowed to act nontrivially on the parametrization space, and attempt to obtain higher deformation classes in this instance $\\lbrace \\text{higher deformation classes}\\rbrace \\subset \\operatorname{Ext}^{\\ge 2}_{\\operatorname{Coh}(G)^G}(,)=\\operatorname{Ext}_{D(G)}^{\\ge 2}(,).$ We show in Sections  and  that this equivariant deformation picture can indeed be formalized, and that–when considered in conjunction with work of Touzé and Van der Kallen [26]–it can be used to obtain the desired finite generation results for cohomology (see in particular Theorems REF and REF ).", "Remark 1.4 From a geometric perspective, one can interpret our main theorem as a finite generation result for the cohomology of non-tame stacky local complete intersections.", "(Formally speaking, we only deal with the maximal codimension case here, but the general situation is similar.)", "One can compare with works of Gulliksen [16], Eisenbud [8], and many others regarding the homological algebra of complete intersections." ], [ "Acknowledgements", "Thanks to Ben Briggs, Christopher Drupieski, Eric Friedlander, Julia Pevtsova, Antoine Touzé, and Sarah Witherspoon for helpful conversations.", "The proofs of Lemmas REF and REF are due to Ben Briggs and Ragnar Buchweitz (with any errors in their reproduction due to myself).", "This material is based upon work supported by the National Science Foundation under Grant No.", "DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester." ], [ "Differential generalities", "Throughout $k$ is a field of finite characteristic, which is not necessarily algebraically closed.", "Schemes and algebras are $k$ -schemes and $k$ -algebras, and $\\otimes =\\otimes _k$ .", "All group schemes are affine group schemes which are of finite type over $k$ , and throughout $G$ denotes an (affine) group scheme." ], [ "Commutative algebras and modules", "A finite type commutative algebra over a field $k$ is a finitely generated $k$ -algebra.", "A coherent module over a commutative Noetherian algebra is a finitely generated module.", "We adopt this language, at times, to distinguish clearly between these two notions of finite generation." ], [ "$G$ -equivariant dg algebras", "Consider $G$ an affine group scheme.", "We let $\\operatorname{rep}(G)$ denote the category of finite-dimensional $G$ -representations, $\\operatorname{Rep}(G)$ denote the category of integrable, i.e.", "locally finite, representations, and $\\operatorname{Ch}(\\operatorname{Rep}(G))$ denote the category of cochain complexes over $\\operatorname{Rep}(G)$ .", "Each of these categories is considered along with its standard monoidal structure.", "By a $G$ -algebra we mean an algebra object in $\\operatorname{Rep}(G)$ , and by a dg $G$ -algebra we mean an algebra object in $\\operatorname{Ch}(\\operatorname{Rep}(G))$ .", "For $T$ any commutative $G$ -algebra, by a $G$ -equivariant dg $T$ -algebra $S$ we mean a $T$ -algebra in $\\operatorname{Ch}(\\operatorname{Rep}(G))$ .", "Note that, for such a dg algebra $S$ , the associated sheaf $S^\\sim $ on $\\operatorname{Spec}(T)$ is an equivariant sheaf of dg algebras, and vice versa.", "Note also that a dg $G$ -algebra is the same thing as an equivariant dg algebra over $T=k$ ." ], [ "DG modules and resolutions", "For $S$ a dg $G$ -algebra, we let $S\\text{-dgmod}^G$ and $D(S)^G$ denote the category of $G$ -equivariant dg modules over $S$ and its corresponding derived category $D(S)^G=(S\\text{-dgmod}^G)[{\\rm quis}^{-1}]$ .", "(Of course, by an equivariant dg module we mean an $S$ -module in the category of cochains over $G$ .)", "If we specify some commutative Noetherian graded $G$ -algebra $T$ , and equivariant $T$ -algebra structure on cohomology $T\\rightarrow H^\\ast (S)$ , then we take $D_{coh}(S)^G:=\\left\\lbrace \\begin{array}{c}\\text{The full subcategory in $D(S)^G$ consisting of dg modules}\\\\\\text{$M$ with finitely generated cohomology over $T$}\\end{array}\\right\\rbrace .$ When $T=k$ we take $D_{fin}(S)^G=D_{coh}(S)^G$ .", "A (non-equivariant) free dg $S$ -module is an $S$ -module of the form $\\oplus _{j\\in J}\\Sigma ^{n_j}S$ , where $J$ is some indexing set.", "A semi-projective resolution of a (non-equivariant) dg $S$ -module $M$ is a quasi-isomorphism $F\\rightarrow M$ from a dg module $F$ equipped with a filtration $F=\\cup _{i\\ge 0} F(i)$ by dg submodules such that each subquotient $F(i)/F(i-1)$ is a summand of a free $S$ -module.", "An equivariant semi-projective resolution of an equivariant dg module $M$ is a $G$ -linear quasi-isomorphism $F\\rightarrow M$ from an equivariant dg module $F$ which is non-equivariantly semi-projective.", "The usual shenanigans, e.g.", "[6], shows that equivariant semi-projective resolutions always exist." ], [ "Homotopy isomorphisms", "Consider $S$ and $A$ dg $G$ -algebras, over some given group scheme $G$ .", "By an (equivariant) homotopy isomorphism $f:S\\rightarrow A$ we mean a zig-zag of $G$ -linear dg algebra quasi-isomorphism $S\\overset{\\sim }{\\leftarrow }S_1\\overset{\\sim }{\\rightarrow }S_2\\cdots \\overset{\\sim }{\\leftarrow }S_{N-1}\\overset{\\sim }{\\rightarrow }A.$ We note that we use the term homotopy informally here, as we do not propose any particular model structure on the category of dg $G$ -algebras (cf.", "[24], [25]).", "Throughout the text, when we speak of homotopy isomorphisms between dg $G$ -algebras we always mean equivariant homotopy isomorphisms.", "A homotopy isomorphism $f:S\\rightarrow A$ as in (REF ) specifies a triangulated equivalence between the corresponding derived categories of dg modules $f_\\ast :D(S)^G\\overset{\\sim }{\\rightarrow }D(A)^G,$ via successive application of base change and restriction along the maps to/from the $S_i$ .", "To elaborate, an equivariant quasi-isomorphism $f:S_1\\rightarrow S_2$ specifies mutually inverse equivalences $S_2\\otimes _{S_1}^{\\rm L}-:D(S_1)^G\\rightarrow D(S_2)^G$ and $\\operatorname{res}_f:D(S_2)^G\\rightarrow D(S_1)$ .", "So for a homotopy isomorphism $f:S\\rightarrow A$ , compositions of restriction and base change produce the equivalence (REF ).", "Note that, on cohomology, such a homotopy isomorphism $f:S\\rightarrow A$ induces an actual isomorphism of algebras $H^\\ast (f):H^\\ast (S)\\rightarrow H^\\ast (A)$ , and one can check that for a dg module $M$ over $S$ we have $H^\\ast (f_\\ast M)\\cong H^\\ast (A)\\otimes _{H^\\ast (S)}H^\\ast (M)\\cong \\operatorname{res}_{H^\\ast (f)^{-1}}H^\\ast (M).$ So, in particular, if $H^\\ast (S)$ and $H^\\ast (A)$ are $T$ -algebras, for some commutative Noetherian $T$ , and $H^\\ast (f)$ is $T$ -linear, then the equivalence (REF ) restricts to an equivalence $f_\\ast :D_{coh}(S)^G\\overset{\\sim }{\\rightarrow }D_{coh}(A)^G$ between the corresponding equivariant, coherent, derived categories.", "Definition 2.1 We say a dg $G$ -algebra $S$ is equivariantly formal if $S$ is equivariantly homotopy isomorphic to its cohomology $H^\\ast (S)$ ." ], [ "Derived maps and derived endomorphisms", "Fix $S$ a dg $G$ -algebra, over a group scheme $G$ .", "For such $S$ , the dg $\\operatorname{Hom}$ functor $\\operatorname{Hom}_S$ on $S\\text{-dgmod}^G$ naturally takes values in $\\operatorname{Ch}(\\operatorname{Rep}(G))$ .", "Namely, for $x$ in the group ring $kG={O}(G)^\\ast $ , we act on functions $f\\in \\operatorname{Hom}_S(M,N)$ according to the formula $(x\\cdot f)(m):=x_1 f(S(x_2)m).$ With these actions each $\\operatorname{Hom}_S(M,N)$ is a dg $G$ -representation, and composition $\\operatorname{Hom}_S(N,L)\\otimes \\operatorname{Hom}_S(M,N)\\rightarrow \\operatorname{Hom}_S(M,L)$ is a map of dg $G$ -representations.", "In particular, $\\operatorname{End}_S(M)$ is a dg $G$ -algebra for any equivariant dg module $M$ over $S$ .", "Remark 2.2 One needs to use cocommutativity of $kG$ here to see that $x\\cdot f$ is in fact $S$ -linear for $S$ -linear $f$ .", "We derive the functor $\\operatorname{Hom}_S$ to $\\operatorname{Ch}(\\operatorname{Rep}(G))$ by taking $\\operatorname{RHom}_S(M,N):=\\operatorname{Hom}_S(M^{\\prime },N),$ where $M^{\\prime }\\rightarrow M$ is any equivariant semi-projective resolution of $M$ .", "One can apply their favorite arguments to see that $\\operatorname{RHom}_S(M,N)$ is well-defined as an object in $D(\\operatorname{Rep}(G))$ , or refer to the following lemma.", "Lemma 2.3 For any two equivariant resolutions $M_1\\rightarrow M$ and $M_2\\rightarrow M$ there is an equivariant semi-projective dg module $F$ which admits two surjective, equivariant, quasi-isomorphisms $F\\rightarrow M_1$ and $F\\rightarrow M_2$ .", "By adding on acyclic semi-projective summands we may assume that the given maps $f_i:M_i\\rightarrow M$ are surjective.", "For example, one can take a surjective resolution $N\\rightarrow M$ , consider the mapping cone $\\operatorname{cone}(id_N)$ , then replaces the $M_i$ with $(\\Sigma ^{-1}\\operatorname{cone}(id_N))\\oplus M_i$ .", "So, let us assume that the $f_i$ here are surjective.", "We consider now the fiber product $F_0$ of the maps $f_1$ and $f_2$ to $M$ .", "Note that the structure maps $F_0\\rightarrow M_i$ are surjective, since the $f_i$ are surjective.", "We have the exact sequence $0\\rightarrow F_0\\rightarrow M_1\\oplus M_2\\overset{[f_1\\ -f_2]^T}{\\rightarrow }M\\rightarrow 0$ and by considering the long exact sequence on cohomology find that we have also an exact sequence $0\\rightarrow H^\\ast (F_0)\\rightarrow H^\\ast (M_1)\\oplus H^\\ast (M_2)\\rightarrow H^\\ast (M)\\rightarrow 0,$ with the map from $H^\\ast (M_1)\\oplus H^\\ast (M_2)$ the sum of isomorphisms $\\pm H^\\ast (f_i)$ .", "It follows that the composites $H^\\ast (F_0)\\rightarrow H^\\ast (M_1)\\oplus H^\\ast (M_2)\\rightarrow H^\\ast (M_i)$ are both isomorphisms, and hence that the maps $F_0\\rightarrow M_1$ and $F_0\\rightarrow M_2$ are quasi-isomorphisms.", "One considers $F\\rightarrow F_0$ any surjective, equivariant, semi-projective resolution to obtain the claimed result.", "For $M$ in $D(S)^G$ we take $\\operatorname{REnd}_S(M)=\\operatorname{End}_S(M^{\\prime })$ , for $M^{\\prime }\\rightarrow M$ any equivariant semi-projective resolution.", "The following result should be known to experts.", "The proof we offer is due to Benjamin Briggs and Ragnar Buchweitz.", "I thank Briggs for communicating the proof to me, and allowing me to repeat it here.", "Lemma 2.4 $\\operatorname{REnd}_S(M)$ is well-defined, as a dg $G$ -algebra, up to homotopy isomorphism.", "Furthermore, if $M$ and $N$ are isomorphic in $D(S)^G$ , then $\\operatorname{REnd}_S(M)$ and $\\operatorname{REnd}_S(N)$ are homotopy isomorphic as well.", "Given an explicit isomorphism $\\xi :M\\rightarrow N$ in $D(S)^G$ , the homotopy isomorphism $\\operatorname{RHom}_S(M)\\rightarrow \\operatorname{RHom}_S(N)$ can in particular be chosen to lift the canonical isomorphism $\\operatorname{Ad}_\\xi :\\operatorname{Ext}^\\ast _S(M,M)\\rightarrow \\operatorname{Ext}^\\ast _S(N,N)$ on cohomology.", "Consider two equivariant semi-projective resolutions $M_1\\rightarrow M$ and $M_2\\rightarrow M$ .", "By Lemma REF we may assume that the map $M_1\\rightarrow M$ lifts to a surjective, equivariant, quasi-isomorphism $f:M_1\\rightarrow M_2$ .", "In this case we have the two quasi-isomorphisms $f_\\ast $ and $f^\\ast $ of $\\operatorname{Hom}$ complexes, and consider the fiber product ${3mm}{& B@{-->}[dl]@{-->}[dr] & \\\\\\operatorname{End}_S(M_1)[dr]_{f_\\ast } & & \\operatorname{End}_S(M_2)[dl]^{f^\\ast }\\\\&\\operatorname{Hom}_S(M_1,M_2)}$ As $f_\\ast $ and $f^\\ast $ are maps of dg $G$ -representations, $B$ is a dg $G$ -representation.", "Furthermore, one checks directly that $B$ is a dg algebra, or more precisely a dg subalgebra in the product $\\operatorname{End}(M_1)\\times \\operatorname{End}(M_2)$ .", "So the top portion of (REF ) is a diagram of maps of dg $G$ -algebras.", "As $M_1$ is projective, as a non-dg module, the map $f_\\ast $ is a surjective quasi-isomorphism.", "One can therefore argue as in the proof of Lemma REF to see that the structure maps from $B$ to the $\\operatorname{End}_S(M_i)$ are quasi-isomorphisms.", "So we have the explicit homotopy isomorphism $\\operatorname{End}_S(M_1)\\overset{\\sim }{\\leftarrow }B\\overset{\\sim }{\\rightarrow }\\operatorname{End}_S(M_2).$ Now, if $M$ is isomorphic to $N$ in $D(S)^G$ , then there is a third equivariant dg module $\\Omega $ with quasi-isomorphisms $M\\overset{\\sim }{\\leftarrow }\\Omega \\overset{\\sim }{\\rightarrow }N$ .", "Any resolution $F\\overset{\\sim }{\\rightarrow }\\Omega $ therefore provides a simultaneous resolution of $M$ and $N$ , and we may take $\\operatorname{REnd}_S(M)=\\operatorname{End}_S(F)=\\operatorname{REnd}_S(N)$ ." ], [ "Equivariant deformations and Koszul resolutions", "In Sections  and  we develop the basic homological algebra associated with equivariant deformations.", "Our main aim here is to provide equivariant versions of results of Bezrukavnikov and Ginzburg [4], and Pevtsova and the author [20] (cf.", "[8], [1])." ], [ "Equivariant deformations", "We recall that a deformation of an algebra $R$ , parametrized by an augmented commutative algebra $Z$ , is a choice of flat $Z$ -algebra $Q$ along with an algebra map $Q\\rightarrow R$ which reduces to an isomorphism $k\\otimes _Z Q\\cong R$ .", "We call such a deformation $Q\\rightarrow R$ an equivariant deformation if all of the algebras present are $G$ -algebras, and all of the structure maps $Z\\rightarrow Q$ , $Z\\rightarrow k$ , and $Q\\rightarrow R$ are maps of $G$ -algebras.", "The interesting point here, and the point of deviation with other interpretations of equivariant deformation theory, is that we allow $G$ to act nontrivially on the parametrization space $\\operatorname{Spec}(Z)$ (or $\\operatorname{Spf}(Z)$ in the formal setting)." ], [ "An equivariant Koszul resolution", "We fix a group scheme $G$ , and equivariant deformation $Q\\rightarrow R$ of a $G$ -algebra $R$ with formally smooth parametrization space space $\\operatorname{Spf}(Z)$ .", "We require specifically that $Z$ is isomorphic to a power series $k[\\!\\hspace{0.28453pt}[{x_1,\\dots ,x_n}]\\!\\hspace{0.28453pt}]$ in finitely many variables.", "As the distinguished point $1\\in \\operatorname{Spf}(Z)$ is a fixed point for the $G$ -action, the cotangent space $T_1\\operatorname{Spf}(Z)=m_Z/m_Z^2$ admits a natural $G$ -action, and so does the graded algebra $\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)=\\wedge ^\\ast (m_Z/m_Z^2),$ which we view as a dg $G$ -algebra with vanishing differential.", "Lemma 3.1 (cf.", "[1]) One can associate to the parametrization algebra $Z$ a commutative equivariant dg $Z$ -algebra $\\mathcal {K}_Z$ such that $\\mathcal {K}_Z$ is finite and flat over $Z$ , and $\\mathcal {K}_Z$ admits quasi-isomorphisms $\\mathcal {K}_Z\\overset{\\sim }{\\rightarrow }k$ and $k\\otimes _Z \\mathcal {K}_Z\\overset{\\sim }{\\rightarrow }\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)$ of equivariant dg algebras.", "We first construct an unbounded dg resolution $\\mathcal {K}^{\\prime }$ of $k$ , as in [5], then truncate to obtain $\\mathcal {K}$ .", "We construct $\\mathcal {K}^{\\prime }$ as a union $\\mathcal {K}^{\\prime }=\\varinjlim _{i\\ge 0} \\mathcal {K}(i)$ of dg subalgebras $\\mathcal {K}(i)$ over $Z$ .", "We define the $\\mathcal {K}(i)$ inductively as follows: Take $\\mathcal {K}(0)=Z$ and, for $V_1$ a finite-dimensional $G$ -subspace generating the maximal ideal $m_Z$ in $Z$ , we take $\\mathcal {K}(1)=\\operatorname{Sym}_Z(Z\\otimes \\Sigma V_1)$ with differential $d(\\Sigma v)=v$ , $v\\in V_1$ .", "Suppose now that we have $\\mathcal {K}(i)$ an equivariant dg algebra which is finite and flat over $Z$ in each degree, and has (unique) augmentation $\\mathcal {K}(i)\\rightarrow k$ which is a quasi-isomorphism in degrees $> -i$ .", "Let $V_i$ be an equivariant subspace of cocycles in $\\mathcal {K}(i)^{-i}$ which generates $H^{-i}(\\mathcal {K}(i))$ , as a $Z$ -module.", "Define $\\mathcal {K}(i+1)=\\operatorname{Sym}_Z(Z\\otimes \\Sigma V_i)\\otimes _Z \\mathcal {K}(i),\\ \\text{with extended differential $d(\\Sigma v)=v$ for }v\\in V_i.$ We then have the directed system of dg algebras $\\mathcal {K}(0)\\rightarrow \\mathcal {K}(1)\\rightarrow \\dots $ with colimit $\\mathcal {K}^{\\prime }=\\varinjlim _i \\mathcal {K}(i)$ .", "By construction $\\mathcal {K}^{\\prime }$ is finite and flat over $Z$ in each degree, and has cohomology $H^\\ast (\\mathcal {K}^{\\prime })=k$ .", "Since $Z$ is of finite flat dimension, say $n$ , the quotient $(\\mathcal {K}_Z:=)\\mathcal {K}=\\mathcal {K}^{\\prime }/((\\mathcal {K}^{\\prime })^{<-n}+B^{-n}(\\mathcal {K}^{\\prime }))$ is finite and flat over $Z$ in all degrees.", "Furthermore, $\\mathcal {K}$ inherits a $G$ -action so that the quotient map $\\mathcal {K}^{\\prime }\\rightarrow \\mathcal {K}$ is an equivariant quasi-isomorphism.", "So we have produced a finite flat dg $Z$ -algebra $\\mathcal {K}$ with equivariant quasi-isomorphism $\\mathcal {K}\\overset{\\sim }{\\rightarrow }k$ .", "We consider a section $m_Z/m_Z^2\\rightarrow V_1$ of the projection $V_1\\rightarrow m_Z/m_Z^2$ , and let $\\bar{S}_1\\subset V_1$ denote the image of this section.", "Take $S=\\operatorname{Sym}_Z(Z\\otimes \\Sigma \\bar{S}_1)$ with differential specified by $d(\\Sigma v)=v$ for $v\\in \\bar{S}_1$ .", "Then $S$ the the standard Koszul resolution for $k$ , and the inclusion $S\\rightarrow \\mathcal {K}$ is a (non-equivariant) dg algebra quasi-isomorphism.", "Since $\\mathcal {K}$ and $S$ are bounded above and flat over $Z$ in each degree, the reduction $k\\otimes _Z S\\rightarrow k\\otimes _Z \\mathcal {K}$ remains a quasi-isomorphism and we have an isomorphism of algebras $\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)\\cong H^\\ast (k\\otimes _ZS)\\overset{\\cong }{\\rightarrow }H^\\ast (k\\otimes _Z \\mathcal {K}).$ Note that the dg subalgebra $\\operatorname{Sym}(\\Sigma V_1)\\subset k\\otimes _Z \\mathcal {K}$ consists entirely of cocycles, and furthermore $Z^{-1}(k\\otimes _Z \\mathcal {K})=\\Sigma V_1$ .", "We see also that the intersection $V_1\\cap m_Z^2$ consists entirely of coboundaries, as such vectors $v$ lift to cocycles in the acyclic complex $\\mathcal {K}$ which are of the form $v+m_Z\\otimes V_1$ .", "A dimension count now implies that the projection $V_1=Z^{-1}(k\\otimes _Z \\mathcal {K})\\rightarrow H^{-1}(k\\otimes _Z \\mathcal {K})$ reduces to an isomorphism $V_1/(m_Z^2\\cap V_1)=H^1(k\\otimes _Z K)$ .", "Hence, for the degree $-1$ coboundaries in $k\\otimes _Z\\mathcal {K}$ , we have $B^{-1}=V_1\\cap m_Z^2$ .", "One now consults the diagram ${\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)[d]_\\cong [r]^{\\rm incl} & \\operatorname{Sym}(\\Sigma V_1)[d][r]^(.25){\\rm proj} & \\operatorname{Sym}(\\Sigma V_1)/(B^{-1})\\cong \\operatorname{Sym}(\\Sigma m_Z/m_Z^2)[dl]\\\\H^\\ast (k\\otimes _Z S)[r]^\\cong & H^\\ast (k\\otimes _Z \\mathcal {K}),}$ to see that the intersection $B^\\ast (k\\otimes _Z\\mathcal {K})\\cap \\operatorname{Sym}(\\Sigma V_1)$ is necessarily the ideal $(B^1)$ generated by the degree $-1$ coboundaries.", "So we find that the projection $f:k\\otimes _Z \\mathcal {K}\\rightarrow \\operatorname{Sym}(\\Sigma V_1)/(B^1)\\cong \\operatorname{Sym}(\\Sigma m_Z/m_Z^2)$ which annihilates (the images of) all cells $\\Sigma V_i$ with $i>1$ is an equivariant dg algebra map, and furthermore an equivariant dg algebra quasi-isomorphism.", "In the following $Z$ a commutative $G$ -algebra which is isomorphic to a power series in finitely many variables, as above.", "Definition 3.2 An equivariant Koszul resolution of $k$ over $Z$ is a $G$ -equivariant dg $Z$ -algebra $\\mathcal {K}_Z$ which is finite and flat over $Z$ , comes equipped with an equivariant dg algebra quasi-isomorphism $\\epsilon :\\mathcal {K}_Z\\overset{\\sim }{\\rightarrow }k$ , and also comes equipped with an equivariant dg map $\\pi :\\mathcal {K}_Z\\rightarrow \\operatorname{Sym}(\\Sigma m_Z/m_Z^2)$ which reduces to a quasi-isomorphism $k\\otimes _Z\\mathcal {K}_Z\\overset{\\sim }{\\rightarrow }\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)$ along the augmentation $Z\\rightarrow k$ .", "Lemma REF says that equivariant Koszul resolutions of $k$ , over such $Z$ , always exists." ], [ "The Koszul resolution associated to an equivariant deformation", "Consider $Q\\rightarrow R$ an equivariant deformation, parameterized by a formally smooth space $\\operatorname{Spf}(Z)$ , as in Section REF .", "For any equivariant Koszul resolution $\\mathcal {K}_Z\\overset{\\sim }{\\rightarrow }k$ over $Z$ , the product $\\mathcal {K}_Q:=Q\\otimes _Z \\mathcal {K}_Z$ is naturally a dg $G$ -algebra which is a finite and flat extension of $Q$ .", "Since finite flat modules over $Z$ are in fact free, $\\mathcal {K}_Q$ is more specifically free over $Q$ in each degree.", "Flatness of $Q$ over $Z$ implies that the projection $id_Q\\otimes _Z\\epsilon :\\mathcal {K}_Q\\overset{\\sim }{\\rightarrow }Q\\otimes _Z k=R$ is a quasi-isomorphism of dg $G$ -algebras (cf.", "[1], [4], [2]).", "We call the dg algebra (REF ), deduced from a particular choice of equivariant Koszul resolution for $Z$ , the (or a) Koszul resolution of $R$ associated to the equivariant deformation $Q\\rightarrow R$ ." ], [ "Deformations associated to group embeddings", "Consider now $G$ a finite group scheme, and a closed embedding of $G$ into a smooth affine algebraic group $\\mathcal {H}$ .", "(We mean specifically a map of group schemes $G\\rightarrow \\mathcal {H}$ which is, in addition, a closed embedding.)", "We explain in this section how such an embedding $G\\rightarrow \\mathcal {H}$ determines an equivariant deformation ${O}\\rightarrow {O}(G)$ which fits into the general framework of Section .", "Note that such closed embeddings $G\\rightarrow \\mathcal {H}$ always exists for finite $G$ .", "For example, if we choose a faithful $G$ -representation $V$ then the corresponding action map $G\\rightarrow \\operatorname{GL}(V)$ is a closed embedding of $G$ into the associated general linear group." ], [ "The quotient space", "For any embedding $G\\rightarrow \\mathcal {H}$ of $G$ into smooth $\\mathcal {H}$ we consider the quotient space $\\mathcal {H}/G$ .", "The associated quotient map $\\mathcal {H}\\rightarrow \\mathcal {H}/G$ is $G$ -equivariant, where we act on $\\mathcal {H}$ via the adjoint action and on $\\mathcal {H}/G$ via translation.", "This is all clear geometrically, but let us consider this situation algebraically.", "Functions on the quotient ${O}(\\mathcal {H}/G)$ are the right $G$ -invariants ${O}(\\mathcal {H})^G$ in ${O}(\\mathcal {H})$ , or rather the left ${O}(G)$ -coinvariants.", "Then ${O}(\\mathcal {H}/G)$ is a right ${O}(\\mathcal {H})$ -coideal subalgebra in ${O}(\\mathcal {H})$ , in the sense that the comultiplication on ${O}(\\mathcal {H})$ restricts to a coaction $\\rho :{O}(\\mathcal {H}/G)\\rightarrow {O}(\\mathcal {H}/G)\\otimes {O}(\\mathcal {H}).$ We project along ${O}(\\mathcal {H})\\rightarrow {O}(G)$ to obtain the translation coaction of ${O}(G)$ on ${O}(\\mathcal {H}/G)$ .", "The left translation coaction of ${O}(G)$ on ${O}(\\mathcal {H})$ restricts to a trivial coaction on ${O}(\\mathcal {H}/G)$ .", "So, ${O}(\\mathcal {H}/G)$ is a sub ${O}(G)$ -bicomodule in ${O}(\\mathcal {H})$ .", "We consider the dual action of the group ring $kG={O}(G)^\\ast $ on ${O}(\\mathcal {H})$ , and find that the inclusion ${O}(\\mathcal {H}/G)\\rightarrow {O}(\\mathcal {H})$ is an inclusion of $G$ -algebras, where we act on ${O}(\\mathcal {H})$ via the adjoint action and on ${O}(\\mathcal {H}/G)$ by translation.", "We have the following classical result, which can be found in [17].", "Theorem 4.1 Consider a closed embedding $G\\rightarrow \\mathcal {H}$ of a finite group scheme into a smooth algebraic group $\\mathcal {H}$ .", "The algebra of functions ${O}(\\mathcal {H})$ is finite and flat over ${O}(\\mathcal {H}/G)$ , and ${O}(\\mathcal {H}/G)$ is a smooth $k$ -algebra." ], [ "The associated equivariant deformation sequence", "Consider $G\\rightarrow \\mathcal {H}$ as above and let $1\\in \\mathcal {H}/G$ denote the image of the identity in $\\mathcal {H}$ , by abuse of notation.", "We complete the inclusion ${O}(\\mathcal {H}/G)\\rightarrow {O}(\\mathcal {H})$ at the ideal of definition for $G$ to get a finite flat extension $\\widehat{{O}}_{\\mathcal {H}/G}\\rightarrow \\widehat{{O}}_\\mathcal {H}$ .", "Take $Z=\\widehat{{O}}_{\\mathcal {H}/G}\\ \\ \\text{and}\\ \\ {O}=\\widehat{{O}}_\\mathcal {H}.$ So we have the deformation ${O}\\rightarrow {O}(G)$ , with formally smooth parametrizing algebra $Z$ .", "A proof of the following Lemma can be found at [20].", "Lemma 4.2 The completion ${O}=\\widehat{{O}}_\\mathcal {H}$ is Noetherian and of finite global dimension.", "Note that the ideal of definition for $G$ is the ideal $\\mathfrak {m}{O}(G)$ , where $\\mathfrak {m}\\subset {O}(\\mathcal {H}/G)$ is associated to the closed point $1\\in \\mathcal {H}/G$ .", "Proposition 4.3 Consider a closed embedding $G\\rightarrow \\mathcal {H}$ of a finite group scheme into a smooth algebraic group $\\mathcal {H}$ .", "Take ${O}=\\widehat{{O}}_{\\mathcal {H}}$ and $Z=\\widehat{{O}}_{\\mathcal {H}/G}$ , where we complete at the augmentation ideal $\\mathfrak {m}$ in ${O}(\\mathcal {H}/G)$ .", "Then the quotients ${O}(\\mathcal {H}/G)/\\mathfrak {m}^n$ and ${O}(\\mathcal {H})/\\mathfrak {m}^n{O}(\\mathcal {H})$ inherit $G$ -algebra structures from ${O}(\\mathcal {H}/G)$ and ${O}(\\mathcal {H})$ respectively.", "The completions $Z$ and ${O}$ inherit unique continuous $G$ -actions so that the inclusions ${O}(\\mathcal {H}/G)\\rightarrow Z$ and ${O}(\\mathcal {H})\\rightarrow {O}$ are $G$ -linear.", "Under the actions of (b), the projection ${O}\\rightarrow {O}(G)$ is an equivariant deformation of ${O}(G)$ parametrized by $\\operatorname{Spf}(Z)=(\\mathcal {H}/G)^{\\wedge }_1$ .", "All of (a)–(c) will follow if we can simply show that $\\mathfrak {m}\\subset {O}(\\mathcal {H}/G)$ is stable under the translation action of $kG$ .", "This is clear geometrically, and certainly well-known, but let us provide an argument for completeness.", "If we let $\\ker (\\epsilon )\\subset {O}(\\mathcal {H})$ denote the augmentation ideal, we have $\\mathfrak {m}=\\ker (\\epsilon )\\cap {O}(\\mathcal {H}/G)$ .", "For the adjoint coaction $\\rho _{\\rm ad}:f\\mapsto f_2\\otimes S(f_1)f_3$ of ${O}(\\mathcal {H})$ on itself, and $f\\in \\ker (\\epsilon )$ , we have $\\begin{array}{l}(\\epsilon \\otimes 1)\\circ \\rho _{\\rm ad}(f)=\\epsilon (f_2)S(f_1)f_3\\\\\\hspace{56.9055pt}=S(f_1)(\\epsilon (f_2)f_3)=S(f_1)f_2=\\epsilon (f)=0.\\end{array}$ So we see that under the adjoint coaction $\\rho _{\\rm ad}(\\ker (\\epsilon ))\\subset \\ker (\\epsilon )\\otimes {O}(\\mathcal {H})$ .", "It follows that $\\ker (\\epsilon )$ is preserved under the adjoint coaction of ${O}(G)$ , and hence the adjoint action of $kG$ , as well.", "So, the intersection $\\mathfrak {m}={O}(\\mathcal {H}/G)\\cap \\ker (\\epsilon )$ is an intersection of $G$ -subrepresentations in ${O}(\\mathcal {H})$ , and hence $\\mathfrak {m}$ is stable under the action of $kG$ ." ], [ "Equivariant formality results and deformation classes", "We observe cohomological implications of the existence of a (smooth) equivariant deformation, for a given finite-dimensional $G$ -algebra $R$ .", "The main results of this section can been seen as particular equivariantizations of [4] and [20], as well as of classical results of Gulliksen [16]." ], [ "We fix an equivariant deformation", "We fix a $G$ -equivariant deformation $Z\\rightarrow Q\\rightarrow R$ , with $Z$ isomorphic to a power series in finitely many variables.", "Fix also a choice of equivariant Koszul resolution $\\mathcal {K}:=\\mathcal {K}_Z,\\ \\ \\text{with}\\ \\epsilon :\\mathcal {K}\\overset{\\sim }{\\rightarrow }k\\ \\text{and}\\ \\pi :\\mathcal {K}\\rightarrow \\operatorname{Sym}(\\Sigma m_Z/m_Z^2).$ Recall the associated dg resolution $K_Q\\overset{\\sim }{\\rightarrow }R$ , with $\\mathcal {K}_Q=Q\\otimes _Z\\mathcal {K}$ .", "Via general phenomena (Section REF ) we observe Lemma 5.1 Restriction provides a derived equivalence $D_{fin}(R)^G\\overset{\\sim }{\\rightarrow }D_{coh}(\\mathcal {K}_Q)^G$ .", "Following the notation of [20], we fix $A_Z:=\\operatorname{Sym}(\\Sigma ^{-2}T_1\\operatorname{Spf}(Z))=\\operatorname{Sym}(\\Sigma ^{-2}(m_Z/m_Z^2)^\\ast ).$" ], [ "Equivariant formality and deformation classes", "Lemma 5.2 Consider $\\mathcal {K}$ the regular dg $\\mathcal {K}$ -bimodule.", "There is a ($G$ -)equivariant homotopy isomorphism $\\operatorname{REnd}_{\\mathcal {K}\\otimes _Z \\mathcal {K}}(\\mathcal {K})\\overset{\\sim }{\\rightarrow }A_Z.$ In particular, $\\operatorname{REnd}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}(\\mathcal {K})$ is equivariantly formal.", "Consider our algebra $A=A_Z$ from (REF ) and take $B=\\operatorname{Sym}(\\Sigma m_Z/m_Z^2)$ .", "Let $F\\rightarrow k$ be the standard resolution of the trivial module over $B$ .", "The resolution $F$ is of the form $B\\otimes A^\\ast $ , as a graded space, with differential given by right multiplication by the identity element $\\sum _i x_i\\otimes x^i$ in $B^{-1}\\otimes A^2$ , and so $F$ admits a natural dg $(B,A)$ -bimodule structure.", "The action map for $A$ now provides an equivariant quasi-isomorphism $A\\overset{\\sim }{\\rightarrow }\\operatorname{End}_{B}(F)=\\operatorname{REnd}_{B}(k)$ .", "For the Koszul resolution $\\mathcal {K}$ over $Z$ , we have the equivariant quasi-isomorphism $\\pi \\otimes _Z\\epsilon :\\mathcal {K}\\otimes _Z \\mathcal {K}\\overset{\\sim }{\\rightarrow }B$ and corresponding restriction and base change equivalences $D(\\mathcal {K}\\otimes _Z\\mathcal {K})^G\\leftrightarrows D(B)^G$ , which are mutually inverse.", "Restriction sends the trivial representation $k$ over $B$ to the regular $\\mathcal {K}$ -bimodule $k\\cong \\mathcal {K}$ .", "Hence the base change $B\\otimes ^{\\rm L}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}\\mathcal {K}$ is isomorphic to $k$ .", "We then get then an equivariant quasi-isomorphism $B\\otimes ^{\\rm L}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}-:\\operatorname{REnd}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}(\\mathcal {K})\\overset{\\sim }{\\rightarrow }\\operatorname{REnd}_{B}(B\\otimes ^{\\rm L}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}\\mathcal {K}),$ with the latter algebra homotopy isomorphic to $\\operatorname{REnd}_{B}(k)\\cong A$ by Lemma REF .", "Remark 5.3 In odd characteristic, one can replace the quasi-isomorphism $\\pi \\otimes _Z\\epsilon :\\mathcal {K}\\otimes _Z\\mathcal {K}\\rightarrow B$ with the more symmetric map $mult(\\frac{1}{2}\\pi \\otimes _Z\\frac{-1}{2}\\pi ):\\mathcal {K}\\otimes _Z\\mathcal {K}\\rightarrow B.$ The point is to provide an equivariant quasi-isomorphism which is a retract of the non-equivariant quasi-isomorphism $B\\rightarrow \\mathcal {K}\\otimes _Z\\mathcal {K}$ implicit in [4].", "Recall that we are considering an equivariant deformation $Q\\rightarrow R$ , with associated dg resolution $K_Q\\overset{\\sim }{\\rightarrow }R$ , as in Section REF .", "We have the natural action of $A_Z$ on $D_{coh}(K_Q)$ [20], which is expressed via the algebra map $A_Z=\\operatorname{End}^\\ast _{D(\\mathcal {K}\\otimes _Z\\mathcal {K})}(\\mathcal {K})\\rightarrow Z(D_{coh}(\\mathcal {K}_Q))$ to the center of the derived category $Z(D_{coh}(\\mathcal {K}_Q))=\\oplus _i\\operatorname{Hom}_{\\text{Fun}}(id,\\Sigma ^i)$ .", "Specifically, for any endomorphism $f:\\mathcal {K}\\rightarrow \\Sigma ^n\\mathcal {K}$ in the derived category of $Z$ -central bimodules, and $M$ in $D_{coh}(K_Q)$ , we have the induced endomorphism $f\\otimes _{\\mathcal {K}}^{\\rm L}M:M\\rightarrow \\Sigma ^n M.$ Suppose, for convenience, that $Q$ is of finite global dimension.", "We lift the maps $-\\otimes ^{\\rm L}_\\mathcal {K}M:\\operatorname{End}^\\ast _{D(\\mathcal {K}\\otimes _Z\\mathcal {K})}(\\mathcal {K})\\rightarrow \\operatorname{End}^\\ast _{D(\\mathcal {K}_Q)}(M)$ to a dg level, for equivariant $M$ , as follows [4]: Fix an equivariant semi-projective resolution $F\\rightarrow \\mathcal {K}$ over $\\mathcal {K}\\otimes _Z\\mathcal {K}$ and, at each $M$ , chose an equivariant quasi-isomorphism $M^{\\prime }\\rightarrow M$ from a dg $\\mathcal {K}_Q$ -module which is bounded and projective over $Q$ in each degree.", "(Such a resolution exists since $Q$ is of finite global dimension.)", "Then $F\\otimes _{\\mathcal {K}}M^{\\prime }\\rightarrow M$ is an equivariant semi-projective resolution of $M$ over $\\mathcal {K}_Q$ [20].", "We now have the lift $-\\otimes _\\mathcal {K}M^{\\prime }:\\operatorname{End}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}(F)\\rightarrow \\operatorname{End}_{\\mathcal {K}_Q}(F\\otimes _\\mathcal {K}M^{\\prime })$ of (REF ), and we write this lift simply as $\\mathfrak {def}^G_M:\\operatorname{REnd}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}(\\mathcal {K})\\rightarrow \\operatorname{REnd}_{\\mathcal {K}_Q}(M).$ Direct calculation verifies that $\\mathfrak {def}^G_M$ , constructed in this manner, is in fact $G$ -linear.", "The following result is an equivariantization of [20].", "Theorem 5.4 Consider a $G$ -equivariant deformation $Q\\rightarrow R$ , with $R$ finite-dimensional, $Q$ of finite global dimension, and parametrization algebra $Z$ isomorphic to a power series in finitely many variables.", "Let ${R}$ denote the formal dg algebra $\\operatorname{REnd}_{\\mathcal {K}\\otimes _Z\\mathcal {K}}(\\mathcal {K})$ (Lemma REF ).", "For any $M$ in $D_{coh}(\\mathcal {K}_Q)^G$ , the equivariant dg algebra map $\\mathfrak {def}^G_M:{R}\\rightarrow \\operatorname{REnd}_{\\mathcal {K}_Q}(M)$ defined above has the following properties: The induced map on cohomology $H^\\ast (\\mathfrak {def}^G_M):A_Z\\rightarrow \\operatorname{End}^\\ast _{D(\\mathcal {K}_Q)}(M)$ is a finite morphism of graded $G$ -algebras.", "For any $N$ in $D_{coh}(\\mathcal {K}_Q)^G$ , the induced action of ${R}$ on $\\operatorname{RHom}_{\\mathcal {K}_Q}(M,N)$ is such that $\\operatorname{RHom}_{\\mathcal {K}_Q}(M,N)\\in D_{coh}({R})^G.$ By $D_{coh}({R})^G$ we mean the category of $G$ -equivariant dg modules over ${R}$ with finitely generated cohomology over $A_Z=H^\\ast ({R})$ .", "The map $\\mathfrak {def}^G_M$ was already constructed above.", "We just need to verify the implications for cohomology, which actually have nothing to do with the $G$ -action.", "We note that the cohomology $H^\\ast (\\mathfrak {def}^G_M)$ is, by construction, obtained by evaluating the functor $-\\otimes ^{\\rm L}_{\\mathcal {K}}M:D(\\mathcal {K}\\otimes _Z\\mathcal {K})\\rightarrow D(\\mathcal {K}_Q)$ at the object $\\mathcal {K}$ .", "(Again, we forget about equivariance here.)", "We can factor this functor through the category of $\\mathcal {K}_Q$ -bimodules $D(\\mathcal {K}\\otimes _Z\\mathcal {K})\\overset{-\\otimes _Z^{\\rm L}Q}{\\longrightarrow }D(K_Q\\otimes K_Q)\\overset{-\\otimes ^{\\rm {L}}_{\\mathcal {K}_Q}M}{\\longrightarrow }D(\\mathcal {K}_Q)$ to see that the corresponding map to the center (REF ) agrees with that of [4] [20].", "So the finiteness claims of (1) and (2) follow from [20].", "Via Lemma REF we may replace $D({R})^G$ with $D(A_Z)^G$ , and view $\\operatorname{RHom}_{K_Q}$ , or equivalently $\\operatorname{RHom}_R$ , as a functor to $D(A_Z)^G$ .", "Alternatively, we could work with the dg scheme (shifted affine space) $\\mathcal {T}^\\ast =T_1^\\ast \\operatorname{Spf}(Z)=\\operatorname{Spec}(A_Z)$ , and view $\\operatorname{RHom}_R$ as a functor taking values in the derived category of equivariant dg sheaves on $\\mathcal {T}^\\ast $ .", "From this perspective, Theorem REF tells us that $\\operatorname{RHom}_R$ has image in the subcategory of dg sheaves on $\\mathcal {T}^\\ast $ with coherent cohomology, $\\operatorname{RHom}_R:(D_{fin}(R)^G)^{op}\\times D_{fin}(R)^G\\rightarrow D_{coh}(A_Z)^G\\cong D_{coh}(\\mathcal {T}^\\ast )^G.$ Remark 5.5 We only use the finiteness claims of Theorem REF in the case in which all of $Z$ , $Q$ , and $R$ are commutative.", "In this case in particular, claims (1) and (2) of Theorem REF should be obtainable directly from Gulliksen [16].", "Remark 5.6 One may compare the above analyses with the formality arguments of [1]." ], [ "Touzé-Van der Kallen and derived invariants", "We recall some results of Touzé and Van der Kallen [26].", "Our aim is to take derived invariants of Theorem REF to obtain a finite generation result for equivariant extensions $\\operatorname{Hom}_{D(R)^G}^\\ast $ .", "We successfully realize this aim via an invocation of [26].", "Throughout this section $G$ is a finite group scheme." ], [ "Basics and notations", "For $V$ any $G$ -representation we have the standard group cohomology $H^\\ast (G,V)=\\operatorname{Ext}^\\ast _G(,V)$ .", "For more general objects in $D(\\operatorname{Rep}(G))$ we adopt a hypercohomological notation.", "Notation 6.1 We let $(-)^{\\mathrm {R}G}:D(\\operatorname{Rep}(G))\\rightarrow D(Vect)$ denote the derived invariants functor, $(-)^{\\mathrm {R}G}=\\operatorname{RHom}_G(,-)$ .", "For $M$ in $D(\\operatorname{Rep}(G))$ we take $\\mathbb {H}^\\ast (G,M):=H^\\ast (M^{\\mathrm {R}G}).$ We note that the hypercohomology $\\mathbb {H}^\\ast (G,M)$ is still identified with morphisms $\\operatorname{Hom}^\\ast _{D(\\operatorname{Rep}(G))}(,M)$ in the derived category.", "Since $G$ is assumed to be finite, we are free to employ an explicit identification $(-)^{\\mathrm {R}G}=\\operatorname{Hom}_G(Bar_G,-),$ where $Bar_G$ is the standard Bar resolution.", "For any dg $G$ -algebra $S$ the derived invariants $S^{\\mathrm {R}G}$ are naturally a dg algebra in $Vect$ , and for any equivariant dg $S$ -module $M$ , $M^{\\mathrm {R}G}$ is a dg module over $S^{\\mathrm {R}G}$ .", "(Under our explicit expression of derived invariants in terms of the bar resolution, these multiplicative structures are induced by a dg coalgebra structure on $Bar_G$ , see e.g.", "[22].)", "We therefore obtain at any dg $G$ -algebra a functor $(-)^{\\mathrm {R}G}:D(S)^G\\rightarrow D(S^{\\mathrm {R}G}).$ The following well-known fact can be proved by considering the hypercohomology $\\mathbb {H}^\\ast (G,S)$ as maps $\\rightarrow \\Sigma ^nS$ in the derived category.", "Lemma 6.2 If $A$ is a commutative dg $G$ -algebra, then the hypercohomology $\\mathbb {H}^\\ast (G,A)$ is a also commutative." ], [ "Derived invariants and coherence of dg modules", "We have the following result of Touzé and Van der Kalen.", "Theorem 6.3 ([26]) Consider $G$ a finite group scheme, and $A$ a commutative $G$ -algebra which is of finite type over $k$ .", "Then the cohomology $H^\\ast (G,A)$ is also of finite type, and for any finitely generated equivariant $A$ -module $M$ , the cohomology $H^\\ast (G,M)$ is a finite module over $H^\\ast (G,A)$ .", "One can derive this results to obtain Theorem 6.4 Consider $G$ a finite group scheme, and $S$ a dg $G$ -algebra which is equivariantly formal and has commutative, finite type, cohomology.", "Suppose additionally that the cohomology of $S$ is bounded below.", "Then the derived invariants functor (REF ) restricts to a functor $(-)^{\\mathrm {R}G}:D_{coh}(S)^G\\rightarrow D_{coh}(S^{\\mathrm {R}G}).$ Equivalently, for any equivariant dg $S$ -module $M$ with finitely generated cohomology over $H^\\ast (S)$ , the hypercohomology $\\mathbb {H}^\\ast (G,M)$ is finite over $\\mathbb {H}^\\ast (G,S)$ .", "Take $A=H^\\ast (S)$ .", "We are free to view, momentarily, $A$ as a non-dg object.", "We have that $A$ is finite over its even subalgebra $A^{ev}$ , which is a commutative algebra in the classical sense, so that Theorem REF implies that cohomology $H^\\ast (G,-)$ sends $A$ to a finite extension of $H^\\ast (G,A^{ev})$ , and any finitely generated $A$ -module to a finitely generated $H^\\ast (G,A^{ev})$ -module.", "Hence $H^\\ast (G,A)$ is of finite type over $k$ , and $H^\\ast (G,N)$ is finite over $H^\\ast (G,A)$ for any finitely generated, equivariant, non-dg, $A$ -module $N$ .", "Since $G$ is a finite group scheme, $A$ is also a finite module over its (usual) invariant subalgebra $A^G$ , and any $A$ -module is finitely generated over $A$ if and only if it is finitely generated over $A^G$ .", "Theorem REF then tells us that, for any finitely generated $A$ -module $N$ , the cohomology $H^\\ast (G,N)$ is finitely generated over $H^\\ast (G,A^G)=H^\\ast (G,)\\otimes A^G$ , where $H^\\ast (G,A^G)$ acts through the algebra map $H^\\ast (G,{\\rm incl}):H^\\ast (G,A^G)\\rightarrow H^\\ast (G,A).$ Consider now any dg module $M$ in $D_{coh}(S)^G$ .", "Formality implies an algebra isomorphism $S\\cong A$ in $D(\\operatorname{Rep}(G))$ and so identifies $\\mathbb {H}^\\ast (G,S)$ with $\\mathbb {H}^\\ast (G,A)=H^\\ast (G,A)$ .", "We want to show that, for such a dg module $M$ , the hypercohomology $\\mathbb {H}^\\ast (G,M)$ is a finitely generated module over $\\mathbb {H}^\\ast (G,S)\\cong H^\\ast (G,A)$ .", "It suffices to show that $\\mathbb {H}^\\ast (G,M)$ is finite over $H^\\ast (G,A^G)=H^\\ast (G,)\\otimes A^G$ .", "We have the first quadrant spectral sequence (via our bounded below assumption) $E_2^{\\ast ,\\ast }=H^\\ast (G,H^\\ast (M))\\ \\Rightarrow \\ \\mathbb {H}^\\ast (G,M),$ and the $E_2$ -page is finite over $H^\\ast (G,A^G)$ by the arguments given above.", "Since $H^\\ast (G,A^G)$ is Noetherian, it follows that the associated graded module $E_\\infty ^{\\ast ,\\ast }=\\operatorname{gr}\\mathbb {H}^\\ast (G,M)$ is finite over $H^\\ast (G,A^G)$ , and since the filtration on $\\mathbb {H}^\\ast (G,M)$ is bounded in each cohomological degree it follows that the hypercohomology $\\mathbb {H}^\\ast (G,M)$ is indeed finite over $H^\\ast (G,A^G)\\subset \\mathbb {H}^\\ast (G,S)$  [14]." ], [ "Finite generation of cohomology for Drinfeld doubles", "Consider $G$ a finite group scheme.", "Fix a closed embedding $G\\rightarrow \\mathcal {H}$ into a smooth algebraic group $\\mathcal {H}$ , and fix also the associated $G$ -equivariant deformation $Z\\rightarrow {O}\\rightarrow {O}(G),\\ \\ Z=\\widehat{{O}}_{\\mathcal {H}/G},\\ {O}=\\widehat{{O}}_{\\mathcal {H}},$ as in Section REF .", "Here $kG$ acts on ${O}(G)$ and ${O}$ via the adjoint action, and this adjoint action restricts to a translation action on $Z$ .", "We recall that the embedding dimension of $G$ is the minimal dimension of such smooth $\\mathcal {H}$ admitting a closed embedding $G\\rightarrow \\mathcal {H}$ .", "We consider the tensor category $Z(\\operatorname{rep}(G))\\cong \\operatorname{rep}(D(G))\\cong \\operatorname{Coh}(G)^G$ of representations over the Drinfeld double of $G$ , aka the Drinfeld center of $\\operatorname{rep}(G)$ .", "We prove the following below.", "Theorem 7.1 For any finite group scheme $G$ , the Drinfeld center $Z(\\operatorname{rep}(G))$ is of finite type and of bounded Krull dimension $\\operatorname{Kdim}Z(\\operatorname{rep}(G))\\le \\operatorname{Kdim}\\operatorname{rep}(G)+\\operatorname{embed.dim}(G).$ One can recall our definition of a finite type tensor category, and of the Krull dimension of such a category, from the introduction.", "For $\\mathcal {T}^\\ast $ the cotangent space $T^\\ast _1\\operatorname{Spf}(Z)$ , considered as a variety with a linear $G$ -action, we show in particular that there is a finite map of schemes $\\operatorname{Spec}\\operatorname{Ext}^\\ast _{Z(\\operatorname{rep}(G))}(,)\\rightarrow (G\\setminus \\mathcal {T}^\\ast )\\times \\operatorname{Spec}H^\\ast (G,)$ ." ], [ "Preliminaries for Theorem ", "We let $G$ act on itself via the adjoint action, and have $\\operatorname{Coh}(G)^G=\\operatorname{rep}({O}(G))^G$ .", "The unit object $\\in \\operatorname{Coh}(G)^G$ is the residue field of the fixed point $1:\\operatorname{Spec}(k)\\rightarrow G$ .", "We have $\\operatorname{REnd}_{\\operatorname{Coh}(G)^G}()=\\operatorname{REnd}_{\\operatorname{Coh}(G)}()^{\\mathrm {R}G},$ as an algebra, and for any $V$ in $\\operatorname{Coh}(G)^G$ we have $\\operatorname{RHom}_{\\operatorname{Coh}(G)^G}(,V)=\\operatorname{RHom}_{\\operatorname{Coh}(G)}(,V)^{\\mathrm {R}G},$ as a dg $\\operatorname{REnd}_{\\operatorname{Coh}(G)^G}()$ -module.", "One can observe these identifications essentially directly, by noting that for the projective generator ${O}(G)\\rtimes kG$ we have an identification of $G$ -representations $\\operatorname{Hom}_{\\operatorname{Coh}(G)}({O}(G)\\rtimes kG,V)=\\operatorname{Hom}_k(kG,V)={O}(G)\\otimes V,$ and ${O}(G)\\otimes V$ is an injective over $kG$ for any $V$ .", "Hence the functor $\\operatorname{Hom}_{\\operatorname{Coh}(G)}(-,V)$ sends projectives objects in $\\operatorname{Coh}(G)^G$ to injectives in $\\operatorname{Rep}(G)$ , and for a projective resolution $F\\rightarrow $ we have identifications in the derived category of vector spaces $\\begin{array}{rl}\\operatorname{RHom}_{\\operatorname{Coh}(G)^G}(,V)&=\\operatorname{Hom}_{\\operatorname{Coh}(G)^G}(F,V)\\\\&=\\operatorname{Hom}_{\\operatorname{Coh}(G)}(F,V)^G\\\\&\\cong \\operatorname{Hom}_{\\operatorname{Coh}(G)}(F,V)^{\\mathrm {R}G}=\\operatorname{RHom}_{\\operatorname{Coh}(G)}(,V)^{\\mathrm {R}G}\\end{array}$ and $\\operatorname{REnd}_{\\operatorname{Coh}(G)^G}(,)=\\operatorname{End}_{\\operatorname{Coh}(G)}(F)^G\\cong \\operatorname{End}_{\\operatorname{Coh}(G)}(F)^{\\mathrm {R}G}=\\operatorname{REnd}_{\\operatorname{Coh}(G)}()^{\\mathrm {R}G}.$ The middle identification for derived endomorphisms comes from the diagram ${\\operatorname{End}(F)^G[r][d]_\\sim & \\operatorname{End}(F)^{\\mathrm {R}G}[d]^\\sim \\\\\\operatorname{Hom}(F,)^G[r]_\\sim & \\operatorname{Hom}(F,)^{\\mathrm {R}G}.", "}$" ], [ "Proof of Theorem ", "Fix an embedding $G\\rightarrow \\mathcal {H}$ and associated equivariant deformation ${O}\\rightarrow {O}(G)$ as above, and take $A=A_Z=\\operatorname{Sym}(\\Sigma ^{-2}(m_Z/m_Z^2)^\\ast )$ , as in (REF ).", "Take also ${R}$ the dg $G$ -algebra $\\operatorname{REnd}_{\\mathcal {K}_Z\\otimes _Z\\mathcal {K}_Z}(\\mathcal {K}_Z)$ .", "We recall from Lemma REF that ${R}$ is equivariantly formal, and so homotopy isomorphic to $A$ .", "We adopt the abbreviated notations $\\operatorname{RHom}=\\operatorname{RHom}_{\\operatorname{Coh}(G)}$ and $\\operatorname{REnd}=\\operatorname{REnd}_{\\operatorname{Coh}(G)}$ when convenient.", "We consider the equivariant dg algebra map $\\mathfrak {def}^G_:{R}\\rightarrow \\operatorname{REnd}_{\\operatorname{Coh}(G)}()$ of Theorem REF , and the action of ${R}$ on each $\\operatorname{REnd}_{\\operatorname{Coh}(G)}(,V)$ through $\\mathfrak {def}^G_$ .", "By Theorems REF and REF , the hypercohomology $\\mathbb {H}^\\ast (G,\\operatorname{REnd}())$ is a finite algebra extension of $\\mathbb {H}^\\ast (G,{R})$ , and $\\mathbb {H}^\\ast (G,\\operatorname{RHom}(,V))$ is a finitely generated module over $\\mathbb {H}^\\ast (G,{R})$ for any $V$ in $\\operatorname{Coh}(G)^G$ .", "In particular, $\\mathbb {H}^\\ast (G,\\operatorname{RHom}(,V))$ is finite over $\\mathbb {H}^\\ast (G,\\operatorname{REnd}())$ .", "Since $\\mathbb {H}^\\ast (G,{R})\\cong \\mathbb {H}^\\ast (G,A)$ is of finite type over $k$ , by Touzé-Van der Kallen (Theorem REF ), the above arguments imply that $\\mathbb {H}^\\ast (G,\\operatorname{REnd}_{\\operatorname{Coh}(G)}())=\\operatorname{Ext}^\\ast _{\\operatorname{Coh}(G)^G}(,)$ is a finite type $k$ -algebra, and that each $\\mathbb {H}^\\ast (G,\\operatorname{RHom}_{\\operatorname{Coh}(G)}(,V))=\\operatorname{Ext}^\\ast _{\\operatorname{Coh}(G)^G}(,V)$ is a finitely generated module over this algebra, for $V$ in $\\operatorname{Coh}(G)^G$ .", "That is to say, the tensor category $Z(\\operatorname{rep}(G))\\cong \\operatorname{Coh}(G)^G$ is of finite type over $k$ .", "As for the Krull dimension, $\\mathbb {H}^\\ast (G,A)$ is finite over $H^\\ast (G,A^G)=H^\\ast (G,)\\otimes A^G$ , by Touzé-Van der Kallen, so that $\\begin{array}{rl}\\operatorname{Kdim}Z(\\operatorname{rep}(G))&=\\operatorname{Kdim}\\operatorname{Ext}^\\ast _{Z(\\operatorname{rep}(G))}(,)\\\\&\\le \\operatorname{Kdim}H^\\ast (G,k)\\otimes A^G\\\\&\\hspace{14.22636pt}=\\operatorname{Kdim}H^\\ast (G,k)\\otimes A\\\\&\\hspace{14.22636pt}=\\operatorname{Kdim}\\operatorname{rep}(G)+\\dim \\mathcal {H}/G=\\operatorname{Kdim}\\operatorname{rep}(G)+\\dim \\mathcal {H}.\\end{array}$ When $\\mathcal {H}$ is taken to be of minimal possible dimension we find the proposed bound, $\\operatorname{Kdim}(Z(\\operatorname{rep}(G)))\\le \\operatorname{Kdim}\\operatorname{rep}(G)+\\operatorname{embed.dim}(G).$" ] ]

[ [ "Reactive molecular dynamics at constant pressure via non-reactive force\n fields: extending the Empirical Valence Bond method to the\n isothermal-isobaric ensemble" ], [ "Abstract The Empirical Valence Bond (EVB) method offers a suitable framework to obtain reactive potentials through the coupling of non-reactive force fields.", "However, most of the implemented functional forms for the coupling terms depend on complex spatial coordinates, which precludes the computation of the stress tensor for condensed phase systems and prevents the possibility to carry out EVB molecular dynamics in the isothermal-isobaric (NPT) ensemble.", "In this work, we make use of coupling terms that depend on the energy gaps, defined as the energy differences between the participating non-reactive force fields, and derive an expression for the EVB stress tensor suitable for computations.", "Implementation of this new methodology is tested for a model of a single reactive malonaldehyde solvated in non-reactive water.", "Computed densities and classical probability distributions in the NPT ensemble reveals a negligible role of the reactive potential in the limit of low concentrated solutions, thus corroborating the validity of standard approximations customarily adopted for EVB simulations." ], [ "Introduction", "Molecular dynamics (MD) simulations offer a powerful computational tool to derive atomistic insight of complex phenomena from organic chemistry and biochemistry to heterogeneous catalysis [1], [2], [3].", "The interatomic interactions in classical MD simulations are based on force field (FF) descriptorsA force field is a mathematical construction to model the interactions between atoms without having to compute the electronic Schrödinger equation.", "This construction reduces dramatically the computational effort to obtain energies and forces (and stress tensors for extended systems), allowing the sampling of the phase space up to nano-seconds, depending on the system and the computational resources available.", "which allows for very fast computation of the interactions and access to simulate very large systems.", "Commonly, these FF descriptors have simple functional forms, with parameters either fitted to experimental data or derived from quantum mechanical calculations [4], [5].", "In most of the available FF libraries, functional forms and fitted parameters remain unchanged during the course of the MD simulation.", "In reactive processes, however, the interactions inevitably change due to the breaking and/or formation of chemical species.", "Thus, standard FFs are not suitable to simulate chemical reactions and they are referred to as non-reactive.", "An alternative to simulate chemical reactions with MD is given by Reactive FFs (RFFs) [6], [7], [8], [9], [10], [11], that are designed to model interatomic interactions of multiple states representing different chemical species.", "The task of designing RFFs, however, is very challengingIndeed, designing RFFs requires a high level of expertise to tackle a multi-dimensional problem [11], where the modelled interactions are often expressed by complicated functional forms with many strongly coupled parameters that are optimized via the use of sophisticated tools [12], [13], [14], [15], [16], [17], [18].", "and, despite the enormous progress over the last years [7], [19], a general parameterization is not yet available.", "The Empirical Valence Bond (EVB) method [20], [21], [22], [23], [24], [25] offers, instead, a simple general framework to model reactive processes through the coupling of multiple non-reactive FFs, where each FF corresponds to a different chemical state for the system.", "In this method, a suitable EVB matrix is built using the computed energies of the involved chemical states as well as appropriate coupling terms.", "Matrix diagonalization at each time step allows computation of reactive energy landscapes that account for the change in chemistry when sampling conformations between the participating, chemically different, states.", "In contrast to RFFs, the advantage of the EVB method lies in the large availability of standard non-reactive FFs libraries, which has offered an appealing strategy for computational implementation and development over the past four decades [26], [27], [28], [29].", "Moreover, despite the tedious initial task to calibrate the coupling terms against reference data, research has demonstrated that these couplings are invariant to the surrounding electrostatics, making it possible to simulate the same reactive unit in different environments [30].", "This convenient feature of the EVB method has widely increased its recognition as a powerful tool within the computational chemistry community [24].", "For condensed phase systems, reported MD simulations with the EVB method (herein, MD-EVB simulations) are conducted either in the microcanonical (NVE) or the canonical (NVT) ensemble.", "However, MD-EVB simulations at constant pressure and temperature, i.e.", "using the isothermal-isobaric (NPT) ensemble have not been addressed in the literature.", "In fact, the standard protocol for EVB simulations in condensed phase is to first consider only one of the possible chemical states of the system (preferably the state with lowest free energy) together with the surrounding, non-reactive, environment and carry out a standard NPT simulation (without EVB) at the target pressure and temperature [31].", "The converged volume is then fixed and the MD-EVB simulation is performed using the NVT ensemble.", "This procedure appears to be a sensible strategy for very large, homogeneous soft-matter systems.", "However, for smaller systems or higher concentration of solutes, the validity of this standard protocol to approximate real experimental conditions at constant pressure and temperature has never been corroborated to date.", "In this work, we demonstrate that the use of the standard formulation to compute the stress tensor cannot be directly applied to derive the components of the EVB stress tensor.", "We argue that this limitation explains the absence of MD-EVB simulations in the NPT ensemble.", "In contrast to using complex spatial variables to fit the coupling terms of the EVB matrix, we propose to make use of energy gaps [32], defined as the energy difference between the non-reactive FFs.", "With this choice, we derive an expression for the EVB stress tensor suitable for computational implementation, not only offering a solution to an overlooked limitation of EVB but also extending the applicability of MD-EVB simulations to NPT ensembles for the first time.", "The computational implementation of this new formalism is tested using a model of a solvated reactive malonaldehyde molecule in water.", "MD-EVB simulations at 300 K and 1 atm are used to quantify the role of the reactive potential in the computed density and classical probability distributions of the energy gaps obtained from the sampling of the configurational space.", "Results also allow to evaluate the validity of the standard protocol for MD-EVB simulations, while the derived method offers an opportunity to explore new strategies for future implementation and development of the EVB method.", "The structure of this paper is as follows.", "The fundamentals of the EVB method developed over the years are presented in a convenient notation in section .", "In section , we discuss the limitation of the standard formulation to calculate the EVB stress tensor, and propose a new alternative method.", "A brief overview of the computational implementation is given in section .", "Section discusses general aspects of the coupling terms within the framework of the present paper.", "Details of the model and MD computational setting are provided in section , which is followed by section with the results and discussion.", "Concluding remarks are finally addressed in section ." ], [ "The EVB method", "In this section we present the fundamentals of the EVB formalism in a convenient notation.", "Let us assume an atomic system composed of $N_{p}$ particles with positions described by the set of vectors $\\lbrace \\bf R\\rbrace $ .", "The non-reactive force field (FF) for the chemical state $(m)$ is described by the configurational energy $E_{c}^{(m)}(\\lbrace \\bf R\\rbrace )$ and the set of forces $\\vec{F}_{J}^{(m)}(\\lbrace \\bf R\\rbrace )$ , where the index $J$ runs over the total number of particles.", "The configurational energy function $E_{c}^{(m)}(\\lbrace {\\bf R} \\rbrace )$ can be generally written as a sum of different terms as follows [33] $E_{c}^{(m)}(\\lbrace {\\bf R} \\rbrace )&=& [ E^{(m)}_{shell}+E^{(m)}_{teth}+E^{(m)}_{bond}+E^{(m)}_{ang}+E^{(m)}_{dih}+ \\nonumber \\\\&+& E^{(m)}_{inv}+E^{(m)}_{3body}+E^{(m)}_{4body}+E^{(m)}_{ters}+ \\nonumber \\\\&+& E^{(m)}_{metal}+ E^{(m)}_{vdw}+ E^{(m)}_{coul}] (\\lbrace {\\bf R} \\rbrace )$ where $E^{(m)}_{shell}$ , $E^{(m)}_{teth}$ , $E^{(m)}_{bond}$ , $E^{(m)}_{ang}$ , $E^{(m)}_{dih}$ , $E^{(m)}_{inv}$ , $E^{(m)}_{3body}$ , $E^{(m)}_{4body}$ , $E^{(m)}_{ters}$ , $E^{(m)}_{metal}$ , $E^{(m)}_{vdw}$ and , $E^{(m)}_{coul}$ are the interactions representing core-shell polarization, tethered particles, chemical bonds, valence angles, dihedrals, inversion angles, three-body, four-body, Tersoff, metallic, van der Waals and coulombic contributions, respectively.", "Following Eq.", "(REF ), the forces can be expressed using a similar decomposition.", "In the current notation, we shall use indexes $m$ and $k$ for the chemical states (and FFs), $I$ and $J$ for atoms and Greek letters for Cartesian coordinates.", "Indexes in parenthesis are used to emphasize the particular chemical state.", "The purpose of the EVB method is to couple $N_F$ non-reactive force fields to obtain a reactive potential.", "These FFs are coupled through the Hamiltonian $\\hat{H}_{\\text{EVB}}$ with a matrix representation $H_{\\text{EVB}} \\in \\mathcal {R}^{N_F \\times N_F}$ that has the following components $H^{mk}_{\\text{EVB}}(\\lbrace \\bf {R}\\rbrace )={\\left\\lbrace \\begin{array}{ll} E_{c}^{(m)}(\\lbrace {\\bf R}\\rbrace ) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m=k \\\\C_{mk}(\\epsilon _{mk}) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m \\ne k\\end{array}\\right.", "}$ where each diagonal element corresponds to the configurational energy $E_{c}^{(m)}(\\lbrace {\\bf R} \\rbrace )$ of the non-reactive FF that models the interactions as if the system was in the chemical state $(m)$ , whereas the off-diagonal terms C$_{mk}$ are the couplings between states $m$ and $k$ .", "For convenience in the notation, we shall omit hereinafter the dependence on the set of coordinates $\\lbrace {\\bf R}\\rbrace $ for the particles.", "Even though there are different possible choices for the coupling terms, in the above definition we have set $C_{mk}$ to depend on $\\epsilon _{mk}=E_{c}^{(m)}-E_{c}^{(k)}=-[E_{c}^{(k)}-E_{c}^{(m)}]=-\\epsilon _{km}$ , where $\\epsilon _{mk}$ is commonly referred to as energy gap and defines a possible reaction coordinate for the process [22], [24], [32], [34].", "Since the $H_{\\text{EVB}}$ matrix is Hermitian by construction and the $C_{mk}$ terms are real, the condition of $C_{mk}=C_{km}$ must be imposed to the off-diagonal elements.", "Diagonalization of $H_{\\text{EVB}}$ leads to $N_F$ possible eigenvalues $\\lbrace \\lambda _1,...,\\lambda _{N_{F}}\\rbrace $ with $H_{\\text{EVB}}\\Psi _{\\lambda _m}=\\lambda _m \\Psi _{\\lambda _m}, \\,\\,\\,\\,\\,\\,\\,\\,\\, m=1,...,N_F.$ The EVB energy, $E_{\\text{EVB}}$ , is defined as the lowest eigenvalue $E_{\\text{EVB}}=min(\\lambda _1,...,\\lambda _{N_F})$ with the corresponding normalized EVB eigenvector $\\Psi _{\\text{EVB}}=\\Psi _{min(\\lambda _1,...,\\lambda _{N_F})}.$ and $E_{\\text{EVB}}=\\big \\langle \\Psi _{\\text{EVB}}\\big |\\hat{H}_{\\text{EVB}}\\big | \\Psi _{\\text{EVB}}\\big \\rangle .$ Since the eigenvector $\\Psi _{\\text{EVB}}$ is real and normalized we have $\\sum _{k=1}^{N_F} \\big |\\Psi ^{(k)}_{\\text{EVB}}\\big |^{2}=1$ from which we can interpret $|\\Psi ^{(k)}_{\\text{EVB}}\\big |^{2}$ as the fraction of the chemical state $(k)$ being part of the EVB state.", "The eigenvector $\\Psi _{\\text{EVB}}$ can also be represented as a column vector $\\in \\mathcal {R}^{N_F \\times 1}$ where $\\Psi ^{(k)}_{\\text{EVB}}$ is the element of the $k$ -row.", "Thus, Eq.", "(REF ) is expressed as a matrix multiplication $E_{\\text{EVB}}=\\sum _{m,k=1}^{N_F} \\tilde{\\Psi }^{(m)}_{\\text{EVB}} H^{mk}_{\\text{EVB}}\\Psi ^{(k)}_{\\text{EVB}}$ where $\\tilde{\\Psi }_{\\text{EVB}}$ is the transpose of ${\\Psi }_{\\text{EVB}}$ .", "In section S1 of the Supporting Information we demonstrate that the decomposition of $E_{\\text{EVB}}$ into different types of interactions (bonds, angles, etc) as for $E_{c}^{(m)}$ in Eq.", "(REF ) is not well defined.", "The resulting EVB force over the particle $J$ , $\\vec{F}_{J}^{\\text{EVB}}$ , follows from the Hellman-Feynman theorem [35] $&&\\vec{F}_{J}^{\\text{EVB}}=-\\nabla _{\\vec{R}_J}E_{\\text{EVB}}=-\\big \\langle \\Psi _{\\text{EVB}}\\big | \\nabla _{\\vec{R}_J} \\hat{H}_{\\text{EVB}} \\big | \\Psi _{\\text{EVB}}\\big \\rangle \\nonumber \\\\&&= \\sum _{\\alpha =x,yz} F_{J\\alpha }^{\\text{EVB}} \\,\\, \\check{\\alpha }$ where $\\check{\\alpha }$ corresponds to each of the orthonormal Cartesian vectors and $F_{J\\alpha }^{\\text{EVB}}=-\\big \\langle \\Psi _{\\text{EVB}}\\big | \\frac{\\partial \\hat{H}_{\\text{EVB}}}{\\partial _{R_{J\\alpha }}}\\big | \\Psi _{\\text{EVB}}\\big \\rangle .$ From Eq.", "(REF ) the matrix components of the operator $\\frac{\\partial \\hat{H}_{\\text{EVB}}}{\\partial _{R_{J\\alpha }}}$ are given as follows $\\frac{\\partial H^{mk}_{\\text{EVB}}}{\\partial R_{J\\alpha }}={\\left\\lbrace \\begin{array}{ll}\\frac{\\partial E_{c}^{(m)}}{\\partial R_{J\\alpha }}=-F^{(m)}_{J\\alpha } \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m=k \\\\\\\\\\begin{aligned}\\frac{d C_{mk}}{\\partial R_{J\\alpha }} &=\\frac{d C_{mk}(\\epsilon _{mk})}{d\\epsilon _{mk}}\\frac{\\partial \\epsilon _{mk}}{\\partial R_{J\\alpha }}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m \\ne k\\\\&=\\frac{d C_{mk}(\\epsilon _{mk})}{d\\epsilon _{mk}} \\left[\\frac{\\partial E_{c}^{(m)}}{\\partial J\\alpha }-\\frac{\\partial E_{c}^{(k)}}{\\partial J\\alpha }\\right]\\\\&=C^{\\prime }_{mk}[F^{(k)}_{J\\alpha }-F^{(m)}_{J\\alpha }]\\end{aligned}\\end{array}\\right.", "}$ where $C^{\\prime }_{mk}=\\frac{d C_{mk}(\\epsilon _{mk})}{d\\epsilon _{mk}}$ and $F^{(k,m)}_{J\\alpha }$ is the $\\alpha $ component of the total configurational force over particle $J$ in the chemical state $(k,m)$ .", "Similarly to Eq.", "(REF ), Eq.", "(REF ) can be expressed as a matrix multiplication $F_{J\\alpha }^{\\text{EVB}}=-\\sum _{m,k=1}^{N_F} \\tilde{\\Psi }^{(m)}_{\\text{EVB}} \\left(\\frac{\\partial H^{mk}_{\\text{EVB}}}{\\partial R_{J\\alpha }}\\right) \\Psi ^{(k)}_{\\text{EVB}}.$ The above equations define the standard EVB force field (EVB-FF).", "Even though the EVB formalism was first developed to compute molecular systems, EVB is also applicable to extended systems, customarily modelled using the supercell approximation and periodic boundary conditions (PBCs).", "Nevertheless, MD-EVB simulations have only been conducted for the NVE and NVT ensembles, to the best of our knowledge, as there is no evidence of a previously reported method to compute the EVB stress tensor.", "In the next section, we discuss the intricacies related to computing the stress tensor using the standard formulation and propose a new method that allows extending the applicability of MD-EVB to NPT ensembles the first time." ], [ "The EVB stress tensor", "The key requirement for a NPT simulation with the EVB method is to being able to compute the EVB stress tensor $\\sigma ^{\\text{EVB}}$ .", "Similarly to the energy and forces, the configurational stress tensor for the force field $m$ , ${\\bf \\sigma }^{c(m)}$ , can be decomposed in a general expression equivalent to Eq.", "(REF ), where each contribution is computed separately using well-known functional forms [36], [33].", "For $bonded$ interactions, for example, the $\\alpha \\beta $ contribution to the stress tensor from particle $J$ due to the bonded interactions with the surrounding particles, $\\sigma _{J,\\alpha \\beta }^{\\text{\\tiny {bond}(m)}}$ , is given by $\\sigma _{J,\\alpha \\beta }^{\\text{\\tiny {bond}}(m)}=\\sum _{I}{R}_{JI,\\alpha }\\,\\,{f}^{\\text{\\tiny {bond}}(m)}_{IJ,\\beta }$ where ${R}_{JI,\\alpha }$ is the $\\alpha $ component of the vector separation $\\vec{R}_{JI}=\\vec{R}_J-\\vec{R}_I$ between particles $I$ and $J$ , and $\\vec{f}^{\\text{\\tiny { bond}}(m)}_{IJ}$ the bond force over particle $J$ from its bonded interaction with particle $I$ .", "In Eq.", "(REF ) the sum runs over all particles $I$ interacting with particle $J$ via bonds.", "Analogously, we could in principle propose an expression for the $\\alpha \\beta $ component of the EVB stress tensor resulting from the EVB bonded forces, ${f}^{\\text{EVB}}_{IJ,\\beta }$ , as follows, $\\sigma _{J,\\alpha \\beta }^{\\text{EVB}}={R}_{JI,\\alpha }\\,\\,{f}^{\\text{EVB}}_{IJ,\\beta }.$ In the present case of bonded interactions, the evaluation of Eq.", "(REF ) requires of each individual EVB-bonded force over particle $J$ from interaction with particles $I$ , given by $\\vec{f}^{\\text{ EVB}}_{IJ}$ .", "Nevertheless, the EVB force given in Eq.", "(REF ) represents the total force, $\\vec{F}_{J}^{\\text{ EVB}}$ , resulting from the interaction of particle $J$ with all the neighboring particles, which generally include other type of interactions apart from bonding interactions.", "As far as we can discern, each individual contribution to the force $\\vec{f}^{\\text{ EVB}}_{IJ}$ cannot be computed from the EVB formalism presented in last section and, consequently, the evaluation of the stress tensor via Eq.", "(REF ) is not possible.", "The same reasoning applies to other type of interactions.", "This limitation precludes the computation of the stress tensor within the EVB formalism via standard formulae and, consequently, MD simulations using the NPT ensemble.", "Surprisingly, this inherent limitation of the EVB method has not been previously discussed in the literature, to the best of our knowledge.", "To circumvent this problem, we propose to use the well-known relation between the configurational energy and the configurational stress tensor [37] $\\frac{\\partial E^{(k)}_{c}}{\\partial h_{\\alpha \\beta }}=-V\\sum _{\\gamma =x,y,z}\\sigma _{\\alpha \\gamma }^{c(k)}h^{-1}_{\\beta \\gamma }$ where $h$ is the set of lattice vectors of the supercell with volume $V$ =det($h$ ).", "Multiplying to the left by $h_{\\nu \\beta }$ and summing over $\\beta $ we obtain the inverse relation to Eq.", "(REF ) $\\sigma _{\\alpha \\beta }^{c(k)}=-\\frac{1}{V}\\sum _{\\gamma =x,y,z}h_{\\beta \\gamma }\\frac{\\partial E^{(k)}_{c}}{\\partial h_{\\alpha \\gamma }}$ which can be used to define the EVB stress tensor $\\sigma _{\\alpha \\beta }^{\\text{EVB}}=-\\frac{1}{V}\\sum _{\\gamma =x,y,z}h_{\\beta \\gamma }\\frac{\\partial E_{\\text{EVB}}}{\\partial h_{\\alpha \\gamma }}.$ Similar to the definition of the EVB force, we evaluate $\\partial E_{\\text{EVB}}/\\partial h_{\\alpha \\gamma }$ using the Eq.", "(REF ) and the Hellman-Feynman theorem [35] $\\frac{\\partial E_{\\text{EVB}}}{\\partial h_{\\alpha \\beta }}=\\big \\langle \\Psi _{\\text{EVB}}\\big | \\frac{\\partial \\hat{H}_{\\text{EVB}}}{\\partial h_{\\alpha \\beta }}\\big | \\Psi _{\\text{EVB}}\\big \\rangle .$ The matrix components of the operator $\\frac{\\partial \\hat{H}_{\\text{EVB}}}{\\partial _{h_{\\alpha \\beta }}}$ follow from the definition of the EVB matrix (REF ) and the use of relation (REF ) $\\frac{\\partial H^{mk}_{\\text{EVB}}}{\\partial h_{\\alpha \\beta }}={\\left\\lbrace \\begin{array}{ll}\\frac{\\partial E_{c}^{(m)}}{\\partial h_{\\alpha \\beta }}=-V\\sum _{\\gamma }\\sigma _{\\alpha \\gamma }^{c(m)}h^{-1}_{\\beta \\gamma } \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m=k \\\\\\\\\\begin{aligned}\\frac{d C_{mk}}{\\partial h_{\\alpha \\beta }}&= \\frac{d C_{mk}(\\epsilon _{mk})}{d \\epsilon _{mk}}\\frac{\\partial \\epsilon _{mk}}{\\partial h_{\\alpha \\beta }}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\, m \\ne k\\\\&= \\frac{d C_{mk}(\\epsilon _{mk})}{d\\epsilon _{mk}}\\left[\\frac{\\partial E_{c}^{(m)}}{\\partial h_{\\alpha \\beta }}-\\frac{\\partial E_{c}^{(k)}}{\\partial h_{\\alpha \\beta }} \\right]\\ \\\\&=-VC^{\\prime }_{mk}\\sum _{\\gamma }[\\sigma _{\\alpha \\gamma }^{c(m)}-\\sigma _{\\alpha \\gamma }^{c(k)}] h^{-1}_{\\beta \\gamma }.\\\\\\end{aligned}\\end{array}\\right.}", "\\nonumber $ Finally, the EVB stress tensor of Eq.", "(REF ) can be expressed as a matrix multiplication $\\sigma _{\\alpha \\beta }^{\\text{EVB}}=-\\frac{1}{V}\\sum _{\\gamma =x,y,z}h_{\\beta \\gamma }\\sum _{m,k=1}^{N_F} \\tilde{\\Psi }^{(m)}_{\\text{EVB}} \\left(\\frac{\\partial H^{mk}_{\\text{EVB}}}{\\partial h_{\\alpha \\beta }}\\right) \\Psi ^{(k)}_{\\text{EVB}}.$ These expressions provide an alternative to compute the stress tensor $\\sigma ^{\\text{EVB}}$ from the configurational stress tensors of each non-reactive FF, $\\sigma _{\\alpha \\gamma }^{c(k)}$ .", "It is important to note that this new scheme to compute $\\sigma ^{\\text{EVB}}$ can only be derived if one uses functional forms for $C_{mk}$ that depend on the energy differences $\\epsilon _{mk}$ , for which one can evaluate $\\frac{\\partial E_{c}^{(m)}}{\\partial h_{\\alpha \\beta }}-\\frac{\\partial E_{c}^{(m)}}{\\partial h_{\\alpha \\beta }}$ and use relation (REF ) with the computed configurational stress tensor for each chemical state.", "In contrast, if the choice was to use coupling terms that do not depend on $\\epsilon _{mk}$ but other degrees of freedom such as spatial coordinates (see refs.", "chang1990,truhlar2000,schlegel2006,sonnenberg2007,sonnenberg2009,steffen2017,steffen2019), we cannot discern a clear logic to derive an expression for $\\sigma ^{\\text{EVB}}$ , which might explain the fact there is no evidence of any previous reported method to compute the stress tensor using EVB.", "So far we have presented an alternative to compute the stress tensor $\\sigma _{\\alpha \\beta }^{\\text{EVB}}$ but have not discussed the total virial $\\mathcal {V}_{\\text{EVB}}$ .", "Similarly to the stress tensor, the inability to compute individual contributions of the EVB force prevents the evaluation of the virial using the standard formulation [33], and the usual decomposition of the virial depending of the type of interaction under consideration.", "Within the presented formalism, we compute the virial $\\mathcal {V}_{\\text{EVB}}$ from $\\sigma _{\\alpha \\beta }^{\\text{EVB}}$ as follows $\\mathcal {V}_{\\text{EVB}}=-\\sum _{\\alpha =x,y,z} \\sigma _{\\alpha \\alpha }^{\\text{EVB}}.$ In contrast to the EVB energy, it is possible to decomposed the virial into different type of interactions, as we discuss in section S2 of the Supporting Information.", "The total stress tensor, $\\sigma ^{T}$ , is given by the following general expression $\\sigma ^{T}=\\sigma ^{\\text{kin}}+\\sigma ^{\\text{EVB}}+\\sigma ^{\\text{RB}}+\\sigma ^{\\text{bc}}$ where $\\sigma ^{\\text{kin}}$ , $\\sigma ^{\\text{RB}}$ and $\\sigma ^{\\text{bc}}$ are the contributions to the stress tensor from the kinetic energy, rigid bodies (RB) and bond constraints (bc), respectively.", "The EVB method only accounts for the configurational interactions, as described.", "The kinetic stress tensor is computed as usual from the instantaneous velocities of the particles [33].", "For a particle that is part of a rigid body, the only possible interactions are intermolecular non-bonded interactions (such as coulombic and van der Waals interactions) with other neighboring particles that are not part of the same rigid body.", "Following the computation of the EVB forces via Eq.", "(REF ), the contribution to the stress from the rigid bodies follows from refs.", "smith1987 and essmann1995 $\\sigma _{\\alpha \\beta }^{\\text{RB}}=\\sum _{\\mathcal {B}=1}^{N_{\\text{RB}}}\\sum _{I=1}^{\\eta _{\\mathcal {B}}} {F}_{I_{\\mathcal {B}},\\alpha }^{\\text{EVB}} d_{I_{\\mathcal {B}},\\beta } $ where $\\vec{F}_{I_{\\mathcal {B}}}$ is the total force over particle $I$ of rigid body $\\mathcal {B}$ and $\\vec{d}_{I_{\\mathcal {B}}}$ the vector distance from atom $I_{\\mathcal {B}}$ to the center of mass of the rigid body $\\mathcal {B}$ .", "In the above expression, index $\\mathcal {B}$ runs over all the rigid bodies.", "Each rigid body is composed of $\\eta _{\\mathcal {B}}$ particles.", "Since, by definition, the topology of rigid bodies remain unaltered during the simulation, the use of RBs within in the present framework is meaningful only to model the environment interacting reactive EVB site.", "A common example is the use of rigidly constrained water molecules to model a solution.", "Contributions to the stress tensor from bond constraints, $\\sigma _{\\alpha \\beta }^{\\text{bc}}$ , are obtained using the SHAKE/RATTLE algorithm [45], [46] during the course of the simulation.", "This algorithm is independent of the EVB formalism, and corrects for the dynamics of the constrained particles.", "Finally, frozen particles do not contributed to the stress tensor and are not considered in the formalism.", "It is important to note that the topology defined via the setting of RBs, frozen atoms and bond constraints must be the consistent for all the coupled FFs, as they impose well defined conditions for the dynamics.", "For example, if a group of atoms form a rigid body, they must remain a rigid body independently of chemical state under consideration." ], [ "Overview of the computational implementation", "The EVB method described in section and its extension for the computation of the stress tensor (section ) were implemented within the DL_POLY_4 code [47], [48].", "In the standard format, DL_POLY_4 reads the initial coordinates, velocities and forces from the CONFIG file.", "Each particle is labelled according to its specification in the FIELD file, which contains the information of the FF type and parameters for the interactions between the particles.", "Settings for the MD simulation are specified in the CONTROL file.", "Initially, the code was modified to allow i) reading multiple ($N_F$ ) CONFIG and FIELD files, ii) allocating arrays of dimension $N_F$ for the relevant quantities, iii) checking consistency of specification between all force fields and initial coordinates (including any possible constraint such as rigid bodies), iv) reading EVB settings such as coupling terms and v) preventing the execution if there are MD or FF options that are not consistent with a EVB simulation.", "With regards to this last point, not all type of interactions in the energy decomposition of Eq.", "(REF ) are suitable to describe reactive interactions.", "For example, three-body, four-body, Tersoff and metallic interactions are, by construction, not designed to account for different chemical states.", "Thus, such interactions should only be used to model the surrounding atomic environment interacting with the EVB site.", "Regarding the EVB method in itself, modifications to the code required to allow for the computations of energies, forces, stress tensor and virials for each of the $N_F$ force-fields separately.", "From the computed configurational energy of each FF and the choice of the functional forms for the coupling terms, the EVB matrix (REF ) is built and diagonalized, and the lowest eigenvalue and the corresponding vector are assigned to $E_{EVB}$ and $\\Psi _{EVB}$ , respectively.", "Matrix (REF ) is computed for each particle's Cartesian components and the resulting EVB force is obtained via the matrix multiplication of Eq.", "(REF ).", "From the stress tensors computed for each FF, matrix (REF ) is built for all the $\\alpha \\beta $ terms and the $\\alpha \\beta $ component of the EVB stress tensor obtained via Eq.", "(REF ), and the total virial from Eq.", "(REF ).", "Such EVB calculations are conducted for each time step taking advantage of the domain decomposition as implemented in DL_POLY_4 [47], [48].", "In this implementation, all the $N_F$ force fields are computed in a loop architecture, i.e.", "one after the other, before being coupled via the EVB method.", "This means that all the available processors are used to compute each force-field, in contrast to the alternative strategy of dividing processors for each force field.", "For extended systems, this choice is convenient given the relative high computational cost of the long range Coulombic part in comparison with all the other contributions to the configurational energy.", "This loop structure increases the computational time by a multiplicative factor of approximately $N_F$ with respect to the required time to compute only a single force field." ], [ "Coupling terms", "The quality of EVB method depends on the choice for the coupling terms $C_{mk}$ , particularly to reproduce accurate interactions at the intermediate region between chemical states $m$ and $k$ where the change of chemistry occurs.", "Several sophisticated EVB coupling recipes have been proposed over the years [38], [39], [40], [41], [42], [43], [44].", "Despite their proven success, these recipes use complex internal (spatial) coordinates to couple the force fields.", "Here, however, we aim to use functional forms $C_{mk}$ that depend on the energy gaps $\\epsilon _{mk}=E^{(m)}_{c}-E^{(k)}_{c}$ , because these variables not only constitute a possible generalized reaction coordinate [24], [34], [22], [32] but also allow to compute the EVB stress tensor as described in Sec.", ", which is the main purpose of the present work.", "The dependence of coupling terms $C_{mk}$ on energy gaps has been previously investigated by B. Hartke et al.. [22].", "In their work, Density Functional Theory (DFT)[49], [50] was first used to compute the minimum energy path (MEP) between reactant and product states for the bond breaking-formation of several molecules in gas phase.", "Via highly accurate DFT-derived FFs [51], [52] for the involved chemical states $m$ and $k$ , the authors computed the coupling terms $C_{mk}$ for selected configurations along the corresponding MEP from the individual energies $E^{(m)}_{c}$ and $E^{(k)}_{c}$ and the reference DFT energy.", "By plotting $C_{mk}$ as a function of $\\epsilon _{mk}$ , the data was fitted to constants and Gaussian-type of functions.", "The implementation of such procedure necessarily requires the use of force-fields i) consistent with the level of theory that is used to compute the explicit electronic problem for the reaction and ii) accurate enough far from the reference geometry for which they were fitted.", "Ultimately, meeting these requirements is a non-trivial challenge, generally impossible in many cases, particularly for large systems.", "In addition, previous research claimed that for several reactions the resulting EVB energy leads to large errors along the MEP, especially in the transition region where, artificial minima are created in the worst cases.", "To overcome these limitations, a combination of Gaussian functions were proposed to model the coupling terms [43], thus offering a promising route for future calibration and development of EVB potentials." ], [ "Model study and MD settings", "To test the implementation of EVB method and its extension to the NPT ensemble, we considered a single malonaldehyde molecule in water solution as a our model system.", "Malonaldehyde (MA) is an archetypal example of intramolecular proton transfer between two oxygen atoms.", "Each conformation corresponds to a different chemical configurations for the same molecule, as shown in Fig.", "REF a and c. In a classical description, the system swaps between both configurations only when vibrations promote the proton to overcome the energy barrier via the transition state (TS), as depicted in Fig.", "REF b.", "A reactive FF for MA would aim to model the interatomic interactions for the whole domain with the forming and breaking of the O-H bonds.", "An example of such a FF was proposed by Y. Yang et al.. [53] based on an extension of the molecular mechanics with proton transfer method [54] to non-linear hydrogen bonds.", "More recently, reactive force fields for MA have been derived using machine learning [55] and neural networks [56].", "Here, nevertheless, we use different non-reactive FFs to describe interactions in the vicinity of each conformation, as schematically shown in Fig REF .", "The two FFs were generated with the DL_FIELD program [57] in a format suitable to DL_POLY_4 using the OPLS-2005 FF library [58], [59], which is not only specially designed for liquid simulations but also constitutes an example of non-reactive FF available in the literature.", "Atoms are labelled differently depending on the FF.", "For example, for the conformation of Fig REF a (FF$_1$ from now on), the proton H$_\\text{O}$ is chemically bonded to the O$_\\text{HE}$ site and only interacts with oxygen O via van der Waals and coulombic interactions.", "For this FF$_1$ topology, the conformational energy would be rather large for geometries where the proton H$_\\text{O}$ is at the vicinity of the O site and the realistic chemistry would be better represented by interactions according to the topology of the FF of Fig REF b (FF$_2$ from now on).", "For this reason, if one only used FF$_1$ to described the interactions, atom H$_\\text{O}$ would unlikely explore the vicinity of the O site during the course of a MD simulation.", "The same reasoning can be applied to the complementary non-reactive FF$_2$ .", "Each water molecule of the solvent was simulated with the TIP4P scheme [60], which uses a four-site water model with an off-center point charge for oxygen.", "To maximize the effect of the EVB reactive potential on the solution, the number of the non-reactive water molecules has to be minimized.", "In MD simulations with FFs, this choice is restricted by the van der Waals cutoff radius, which is routinely set to 12 Å.", "Thus, for a cubic supercell with periodic boundary conditions, the minimum size of the box length should be 24 Å.", "To this purpose, models were built to contains 599 rigid water molecules arranged around the MA molecule within a cubic box of 27 Å, while using an initial separation criteria of 1.9 Å between the molecules.", "This amount of water prevented box length values below the limit of 24 Å in all the simulations.", "Such a model already represents an aqueous solution with a rather low concentration of $9.19\\times 10^{-2}$ molality [mol(MA)/kg(H$_2$ O)].", "Using the initial arrangement of atoms, the system was initially computed in the NVT ensemble at 300K using only FF$_1$ for MA and a Nose-Hoover [61], [62] thermostat with a relaxation time of 0.5 ps.", "Equilibration was conducted for 5 ps, scaling the system temperature every 5 fs and resampling instantaneous system momenta distribution every 9 fs.", "Production MD followed for 30 ps.", "The last snapshot with positions, velocities and forces served as the starting point of a NPT simulation at 1 atm, this time using a Nose-Hoover thermostat and barostat with relaxation times of 3.0 and 1.0 ps for the thermostat and barostat, respectively.", "Equilibration was conducted for 2 ps while allowing for a variation of 10% in the system density.", "This was followed by a 200 ps of MD production run.", "The average supercell dimension was used for all the EVB-NVT simulations and as starting point for the EVB-NPT runs.", "All the MD simulations used a timestep of 1 fs, while the electrostatic interactions are computed through the smooth particle mesh Ewald method [37], [48].", "Details of the EVB simulations are given in the next section." ], [ "Results and discussion", "Proton transfer is a quantum mechanical process [63], [64], [65].", "Accounting for the full quantum problem of the nuclei, however, is computationally prohibited for sufficiently large systems, and several approximations have been developed over the years (See ref.", "yamada2014 and references therein).", "In particular, the EVB method has been extended to its Multistate version (MS-EVB) to successfully capture the essential physics and chemistry in different protonated systems, both in the classical and quantum regime [67], [68], [69], [70], [71], [72], [73], [74], [75].", "Nevertheless, MS-EVB inherits the limitation of EVB with respect to the stress tensor and its application has only been been restricted to NVE and NVT ensembles.", "With regards to the model of a single MA in water, A. Yamada et al.. has previously used the quantum-classical molecular dynamics method [76] to compute the quantum reaction dynamics[66].", "One the other hand, Y. Yang et al.. [53] assumed the whole solution as classical, and used MD to compute proton transfer rates in an effective potential for MA that included zero point energy effects.", "The purpose of this work, however, is to compute the solvated MA using the EVB method in the NPT ensemble, and compare the results with the standard protocol.", "Thus, nuclear quantum effects are neglected in the following simulations as well as zero point energy corrections, in line with a previous density-functional tight-binding QM/MM study [77].", "Even though this represents an over simplification of the problem, the assumption of classical mechanics for the whole system has demonstrated to provide a reasonable framework to compute the lower limits for proton transfer [53].", "We started our study by considering the explicit quantum electronic problem of a single MA in vacuum at zero temperature, and used the computed quantities as a reference to calibrate the EVB potential.", "By means of the Nudge Elastic Band (NEB) method [78], [79] combined with DFT calculations, we computed the minimum energy path (MEP) to transfer the proton between the two conformations.", "Details for these calculations are provided in section Settings for DFT simulations.", "Following the geometry relaxation of each MA conformation, the converged structures were used as fixed end-points of the MEP, which was built by using 17 intermediate images.", "Fig.", "REF shows the computed energy profiles along the converged MEP for the B3LYP [80] and the PBE [81] electronic exchange and correlation (XC) functionals.", "Dispersive van der Waals interaction are included via the Grimme's DFT-D3 formalism [82].", "We compute energy barriers of 2.86 and 0.79 kcal/mol for B3LYP and PBE, respectively, which is in agreement with previous work [83] and corroborates the crucial dependence of the XC functional on the energy barrier and length of the MEP.", "These values underestimate previous coupled-cluster calculations, which predicted energy barriers between 4.0-4.3 kcal/mol [84], [85], [86], [53].", "Despite this underestimation, we shall proceed with the computed DFT values, as achieving chemical accuracy is not the main purpose of the present work.", "Even though this choice represents a departure from a more realistic chemistry, lower energy barriers are more convenient to MD, as less computational time is needed to switch between conformations, thus allowing a better sampling of the configurational space.", "Figure: EVB barrier for proton transfer for a MA molecule in vacuum as a function of the coupling term C 12 C_{12}.", "Computed energy barriers obtained from the DFT calculations of Fig.", "are shown as a reference.On the other hand, energy barriers computed with the EVB method depend on the term $C_{12}$ .", "Here, we have assumed $C_{12}$ to be a constant.", "Note that the choice of a constant for the coupling term, even trivial, complies with the functional form requirement for the coupling terms to compute the EVB stress tensor.", "Figure REF shows that the computed EVB barrier for MA decreases as the value of $C_{12}$ increases.", "For the adopted OPLS-2005-FFs and to the purpose of comparison with the DFT energy barriers of Fig.", "REF , we have only considered values of $C_{12}$ in the range between 46.5 and 50.0 kcal/mol.", "These results constitute an example of how coupling terms can be used to calibrate EVB potentials against a reference value, in this case obtained from DFT.", "The advantage of the EVB formalism lies in the assumption that the coupling terms calibrated for reactive molecules in gas phase do not change significantly when transferring the reactive system from one phase to the another [24].", "This approximation has been rigorously validated via Constrained-DFT calculation [30].", "Moreover, the use of constants for the coupling terms is the most common choice in the execution of EVB simulations for solvated reactive sites, as a constant can be finely adjusted until a calculated property (usually free energy) agrees with the experimental value [24], [31].", "Here, we are not interested in comparing with experiments but evaluating how the results are affected by the use of different ensembles.", "Clearly, the reactive EVB potential is different from any of the individual non-reactive FFs, particularly in the TS region.", "Thus, it is natural to argue as to which extent this reactive EVB potential affects the stress tensor and the converged volume (and density), and how results compare with the density resulting from a NPT simulation using one of the involved FF, as in the standard protocol.", "If the energy barrier is sufficiently large, the system will only sample the vicinity of one of the possible configurations.", "Even though the system might occasionally swap conformation, the TS region will be hardly sampled.", "Thus, for cases where the conformations are chemically equivalent and the barrier is large enough, the standard protocol appears to be a sensible approximation.", "In contrast, if the barrier is sufficiently low, the TS region will be better sampled during the course of a MD simulation and the average potential will depart from any of the individual FFs.", "Figure: Computed density (green filled circles) of the model solution composed of one MA and 599 rigid water molecules.", "Different choices of the coupling C 12 C_{12} lead to different energy barriers for proton transfer of a single MA in vacuum (see Fig.", ").", "Each energy barrier (values in the x-axis) can be considered as a different reactive model for MA.", "The region between horizontal red dashed lines corresponds to the range of possible densities from a standard NPT simulation using only one of the non-reactive FFs.", "The horizontal blue line refers to the experimental density of pure water.", "Reference pressure and temperature are 1 atm and 300K, respectively.Figure: Normalized distribution for the ϵ 12 \\epsilon _{12} energy gap following NVT and NPT simulations for the model of MA with an energy barrier of 1.94 kcal/mol a) after MD runs b) upon symmetrization.", "Note values in the y-axis are scaled for the sake of visualization.", "The region within the brown vertical dashed lines is arbitrarily assigned to the transition state region (see text).Following a NPT simulation of the solvated MA using only one of the two FFs at 1 atm and 300 K, the computed density of the system is predicted to be in the region between the horizontal red dashed lines, as shown in Fig.", "REF .", "From the set of $C_{12}$ values considered in Fig.", "REF the classical barrier for proton transfer can be artificially changed.", "Thus, the EVB potentials generated using these different $C_{12}$ values can be considered as different reactive models for MA.", "For each of these reactive models, we run a full EVB-NPT simulation and compute the density and its uncertainty, indicated by the green filled circles with error bars.", "Results demonstrate that the converged density for the solvated MA is statistically independent of the energy barrier for intramolecular proton transfer.", "In addition, the fact that computed EVB-NPT values statistically fall within the boundary of the red dashed lines supports the validity of the standard protocol in determining the size of the system, as least for the present test case.", "To further investigate on the role of the ensemble, we use the converged volume from the first NPT simulation (with only one FF) and run EVB-NVT simulations for each reactive field.", "To compare EVB simulation for both ensembles, here we propose to use the energy gap $\\epsilon _{12}$ , obtained from the energy difference between the FF$_1$ and FF$_2$ at each time step.", "The range of computed values for $\\epsilon _{12}$ are grouped using a total of 150 bins, each bin with an energy window of 20 kcal/mol.", "To remove the dependence on the simulation time (i.e.", "number of configurations sampled), histograms are normalized such as the total area is equal to one.", "Such normalized distributions can be interpreted as the classical probability of finding the system at a given value of $\\epsilon _{12}$ .", "Figure REF a shows the computed distribution following EVB-NVT and EVB-NPT simulations for the solvated model with an energy barrier for MA of 1.94 kcal/mol.", "Distributions exhibit two broad peaks centered at approximately -350 and 350 kcal/mol, which indicates the system mainly samples configurations in the vicinity of the conformations of Figure REF and resembles the well-known probability distribution of a proton in a double well.", "The observed asymmetry in the distributions is attributed to the finite time of MDs run and the lack of control for both configurations to be equally sampled.", "In fact, the reactive potential for MA in gas phase at zero temperature is symmetric along the MEP (Fig.", "REF ), and the same is expected for MA in solution at 300 K despite the electrostatic field created by the surrounding water.", "Symmetric distributions, however, can only be achieved by increasing the sampling of the conformational phase space and, hence, the computational time for the MD runs, which is beyond the purpose of this work.", "Alternatively, we use the computed data and make these distributions to be symmetric around $\\epsilon _{12}=0$ , as shown in Fig REF b.", "To the best of our knowledge, such probability distributions have not been reported before within the framework of EVB.", "Moreover, these computed distributions support previous research that suggest the convenience of $\\epsilon $ as an alternative reaction coordinate [22], [24], [32], [34].", "In fact, in contrast to the MEP, $\\epsilon $ represents a coordinate that implicitly accounts not only for the reactive site but also includes the effect of the surrounding solvent.", "The symmetric distributions obtained for the NVT and NPT ensembles indicate a good level of agreement.", "Nevertheless, it is convenient to quantify this agreement for a better comparison.", "To this purpose, we first estimate the width for both peaks, which is of the order of 320 kcal/mol.", "This range for $\\epsilon _{12}$ was used to define the TS domain, indicated by the region between the dashed lines of Fig.", "REF , located at $\\epsilon ^{TS}_{12}= \\pm $ 160 kcal/mol.", "This TS region is also adopted to be independent of the reactive model for MA.", "We define the probability for the the system to be in the TS region as follows $\\text{TS Probability}(\\epsilon ^{TS}_{12})=\\int _{-\\epsilon ^{TS}_{12}}^{\\epsilon ^{TS}_{12}} \\mathcal {P}(\\epsilon _{12}) \\,\\, d\\epsilon _{12}$ where $\\mathcal {P}(\\epsilon _{12})$ is the normalized distribution obtained from the histograms, as shown in Fig.", "REF .", "Clearly, the computed TS probability depends on the choice for the extension of the TS region, which is completely arbitrary.", "However, Eq.", "(REF ) provides a method to quantify the relevance of the TS region and compare the probability distributions for different ensembles.", "Figure REF shows the computed values for the TS Probabilities for the different reactive models of MA.", "As expected, the classical probability for the system to sample the TS region increases as the barrier reduces.", "Additionally, the computed probability distributions do not depend on the assumed ensemble for the present model.", "Figure: Probability to sample the TS for solvated MA in the NVT and NPT ensembles.", "Results are plotted as a function of the classical energy barrier for proton transfer in MA.Based on these results, we conclude that the standard protocol to compute single reactive sites in water solution with EVB using the NVT ensemble is already a remarkably good approximation to the self-consistent EVB-NPT simulation, at least for concentrations of reactive sites lower than $9.19\\times 10^{-2}$ molality.", "In the present case study of solvated MA, the flexible-reactive MA is composed of nine atoms and contributes with 27 degrees of freedom (d.o.f), whereas the 599 rigid water molecules can only undergo translation and rotation, contributing another 3594 d.o.f.", "Thus, the rather small ratio 27:3594 and the rather low compressibility of water might explain why the reactive degrees of freedom play a negligible role in the dynamics of the whole system.", "It would be interesting to investigate if the dynamics will differ on increasing the concentration of the reactive MA.", "As discussed in section , due to restriction on the size of the simulation cell, higher concentration of MA could only be achieved here by replacing water molecules with MA molecules.", "However, the EVB method is designed for only one reactive component.", "Thus, increasing the concentration of reactive components would require an extension of the EVB method to account for multiple reactive sites in a single simulation.", "This new capability would be ideal not only to the prospect of computing higher concentrations in solutions, but also to perform EVB simulations of molecular crystals composed by reactive units, as for the family of Ketohydrazone-Azoenol systems [87], [88].", "Such a development is part of current research in our group." ], [ "Concluding remarks", "In this work we propose a new formalism to derive the stress tensor within the EVB method, thus allowing EVB simulations of condensed phase systems at constant pressure and temperature for the first time.", "This formalism is based on the use of energy gaps as reactive coordinates to parameterize the coupling terms of the EVB matrix.", "As a test case, we considered the intramolecular proton transfer in MA molecule solvated in water by means of molecular dynamics, while neglecting the role of quantum nuclei and zero point energy corrections.", "In comparison to the standard protocol of converging the system volume using only one of the non-reactive FFs, results from EVB-NPT simulations for different reactive models of MA (i.e.", "different energy barriers) demonstrate a negligible effect of the EVB potential in the computed density of the solution.", "In addition, we performed EVB-NVT simulations using the converged volume from the standard protocol.", "To compare the sampling of the configurational space with respect to EVB-NPT simulations, we use the energy gap as a variable to compute probability distributions of the reactive system.", "Detailed analysis of these distributions also demonstrate a negligible difference between both ensembles.", "We attribute these findings to the relative low concentration for the model of MA in water, where the non-reactive dynamics of the rigid water molecules dominates over the reactive dynamics of the single MA.", "Therefore, future EVB simulations of solvated reactive molecules with higher concentrations would be beneficial to quantify the role of the reactive dynamics on the whole system.", "To this purpose, developments to extend the EVB method are required to accommodate multiple reactive sites at the same time." ], [ "Settings for DFT simulations", "Geometry optimization and NEB calculations for the two conformations of MA were conducted using the library DL_FIND [89] of the ChemShell program [90] via its interface [91] with the ORCA package [92] for the DFT computation of energies and forces.", "Both PBE and B3LYP functionals were used together with DFT-D3 dispersion correction [82].", "The def2-TZVP basis set [93] was used for all the atoms.", "I.S.", "and A. M. E. designed the implementation of the EVB in DL_POLY_4; I.T has lead DL_POLY_4 and participated in initial stages of the EVB project; A. M. E. led the refactoring of DL_POLY_4; I.S.", "developed the mathematical formalism for the EVB stress tensor, implemented the required changes within DL_POLY_4, built the models and performed the MD simulations; A. M. E. provided technical support for the EVB implementation; K. S. prepared the input files for the ChemShell simulations; I.S.", "wrote the manuscript; A. M. E., K. S. and I. T. revised the manuscript.", "This work made use of computational support by CoSeC, the Computational Science Centre for Research Communities, through CCP5: The Computer Simulation of Condensed Phases, EPSRC grant no EP/M022617/1.", "I. S. acknowledges i) fruitful discussions with Vlad Sokhan, Silvia Chiacchiera, Fausto Martelli, Alfredo Caro and Gilberto Teobaldi, ii) Chin Yong for his help with the DL_FIELD code, iii) John Purton for his initial contributions to the EVB funding project iv) Jim Madge for his heroic contribution to the refactoring of DL_POLY_4.", "A. M. E. acknowledges support of EPSRC via grant EP/P022308/1.", "Similarly to Eq.", "(1) of the main manuscript, it is interesting to investigate if $E_{\\text{EVB}}$ can be decomposed in different energy contribution.", "To this purpose we express the diagonal terms of $H^{mk}_{\\text{EVB}}$ as a sum of the individual contributions $H^{mk}_{\\text{EVB}}={\\left\\lbrace \\begin{array}{ll} \\sum _{\\text{\\tiny {type}}} U_{\\text{\\tiny {type}}}^{(m)} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m=k \\\\C_{mk}(\\epsilon _{mk}) \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, m \\ne k\\end{array}\\right.", "}\\qquad \\mathrm {(S1)}$ where the index $\\text{type}$ runs over all type of possible interactions (bonds, angles, coulombic, etc).", "In contrast, $C_{mk}(\\epsilon _{mk})$ cannot be decomposed in terms of the individual contributions $U_{\\text{\\tiny {type}}}^{(m)}$ .", "Consequently, matrix $H_{\\text{EVB}}$ cannot be decomposed for each type of interaction.", "One might consider the particular case of constant coupling terms $C_{mk}(\\epsilon _{mk})=\\mathcal {C}_{mk}$ , $\\forall m,k=1,\\cdots , N_F$ , with $m\\ne k$ to check if a separation into individual terms is possible.", "For the sake of simplicity, let us consider the case of two FFs with $\\mathcal {C}_{12}=\\mathcal {C}_{21}$ .", "Without loss of generality, we can write $\\mathcal {C}_{12}$ as a sum of a set of constants $\\mathcal {C}_{12}=\\sum _{\\text{\\tiny {type}}} \\mathcal {C}^{\\text{\\tiny {type}}}_{12}\\qquad \\mathrm {(S2)}$ and $H_{\\text{EVB}}=\\sum _{\\text{\\tiny {type}}} H_{EVB}^{\\text{\\tiny {type}}},\\,\\,\\,\\, \\text{with} \\,\\,\\,\\, H_{EVB}^{\\text{\\tiny {type}}}=\\begin{pmatrix} U_{\\text{\\tiny {type}}}^{(1)} & \\mathcal {C}^{\\text{\\tiny {type}}}_{12} \\\\\\mathcal {C}^{\\text{\\tiny {type}}}_{12} & U_{\\text{\\tiny {type}}}^{(2)}\\end{pmatrix}\\qquad \\mathrm {(S3)}$ Using the computed EVB eigenvector, $\\Psi _{\\text{EVB}}$ , from diagonalization of the $H_{\\text{EVB}}$ matrix we have EEVB=type EEVBtype,     where EEVBtype=EVB|HEVBtype| EVBS4 which, in principle, offers a possible way to decompose the EVB energy in terms of individual types of interactions.", "However, such a decomposition is not unequivocally defined, as there are infinite ways of writing the sum for $\\mathcal {C}_{12}$ in Eq.", "(REF ).", "This demonstrates that an EVB energy decomposition in individual terms as in Eq.", "(1) is not well defined.", "In fact, only $E_{\\text{EVB}}$ is well defined.", "The purpose of this section is to demonstrate that the EVB stress tensor $\\sigma _{\\alpha \\beta }^{\\text{EVB}}$ and virial $\\mathcal {V}_{\\text{EVB}}$ can be decomposed in different components according to the type of the interaction.", "We note that $\\sigma _{\\alpha \\gamma }^{c(m)}$ in Eq.", "(19) can be decomposed in different contributions, namely $\\sigma _{\\alpha \\gamma }^{c(m)}=\\sum _{\\text{\\tiny {type}}} \\sigma _{\\alpha \\gamma }^{\\text{\\tiny {type}}(m)}$ .", "Thus, HmkEVBh=type Hmktypeh          whereS5 Hmktypeh={ll -Vtype(m)h-1                                  m=k -VCmk[type(m)-type(k)] h-1          m k .", "from which, similarly to Eq.", "(20), we have type=-1V=x,y,zhm,k=1NF (m)EVB (Hmktypeh) (k)EVBS6 and EVB=typetype S7 with the following decomposition for the virial VEVB=typeVtypeEVB= -type=x,y,z type.", "S8" ] ]

[ [ "Rocklines as Cradles for Refractory Solids in the Protosolar Nebula" ], [ "Abstract In our solar system, terrestrial planets and meteoritical matter exhibit various bulk compositions.", "To understand this variety of compositions, formation mechanisms of meteorites are usually investigated via a thermodynamic approach that neglect the processes of transport throughout the protosolar nebula.", "Here, we investigate the role played by rocklines (condensation/sublimation lines of refractory materials) in the innermost regions of the protosolar nebula to compute the composition of particles migrating inward the disk as a function of time.", "To do so, we utilize a one-dimensional accretion disk model with a prescription for dust and vapor transport, sublimation and recondensation of refractory materials (ferrosilite, enstatite, fayalite, forsterite, iron sulfide, metal iron and nickel).", "We find that the diversity of the bulk composition of cosmic spherules, chondrules and chondrites can be explained by their formation close to rocklines, suggesting that solid matter is concentrated in the vicinity of these sublimation/condensation fronts.", "Although our model relies a lot on the number of considered species and the availability of thermodynamic data governing state changes, it suggests that rocklines played a major role in the formation of small and large bodies in the innermost regions of the protosolar nebula.", "Our model gives insights on the mechanisms that might have contributed to the formation of Mercury's large core." ], [ "Introduction", "Meteorites and terrestrial planets show varying proportions of silicates and metallic iron, with Fe being distributed between Fe alloys and silicates [47], [22].", "Chondrites are the most common meteoritical bodies found on Earth, whose unaltered structure gives valuable information on the formation conditions of the building blocks of the solar system [23], [67].", "Chondrules, which are round grains composed primarily of the silicate minerals olivine and pyroxene and found in the three families of chondrites (ordinary, carbonaceous, and enstatite chondrites), are then important to assess the thermodynamic evolution and the initial composition of the protosolar nebula (PSN) [14], [8].", "Another source of meteoritical matter are cosmic spherules (CS), which are micrometeorites formed by the melting of interplanetary dust particles during atmospheric entry.", "CS are believed to sample a broader range of material than the collections of meteorites [62].", "They are sorted by families and types, each one presenting its own structure and composition [58], [62], [2], [15].", "Neither chondrules, chondrites, or CS represent the earliest condensates of refractory phases in the PSN, most of which have been thermally processed before accretion on asteroids and planetary bodies.", "Their composition, however, can reasonably be used as a proxy of the composition of refractory solids in the PSN.", "The observed variations of composition of chondrites, chondrules, and CS suggest a compositional and redox gradient of refractory matter as a function of radial distance in the PSN and/or time.", "Recent studies explored the link between CS and chondrites, showing that it is possible to associate CS to their chondritic precursors, based on compositional analysis [54], [65], potentially providing additional constraints to the thermodynamic conditions of the PSN.", "The question of the origin of the composition differences among chondrites and chondrules is often answered by invoking the cooling of the inner disk where high temperature materials condense first [23], [63], [64].", "However, the influence of the temperature gradient within an evolving PSN, where more refractory materials form at closer distances from the Sun, has never been investigated to address this issue.", "Abundances of materials both in solid and gaseous phases are ruled by the chemical status of the disk and by the location of their condensation/sublimation fronts [19], [48], [49].", "Significant increases of the abundances of solid materials can be generated at the location of these transition lines, due to the dynamics of vapors and grains [17], [16], [9], [3], [18], [43].", "These processes potentially explain the metallicity of Jupiter through its formation near the water snowline [43], as well as the high density of Mercury whose building blocks could have formed in regions where the abundances of Fe-bearing species are prominent [64], [66].", "Here, we use a time-dependent coupled disk/transport model to investigate the role held by rocklines (the concept of snowlines extended to more refractory solids) of the most abundant solids into the shaping of Mg, Fe, and Si abundances profiles in the inner part of the PSN.", "The radial transport of solid grains through the different rocklines, coupled to the diffusion of vapors, leads to local enrichments or depletions in minerals and imply variations of the Mg-Fe-Si composition of dust grains in the inner regions of the PSN.", "We discuss our results in light of the relative abundances and compositions of minerals observed in meteoritic matter and planetary bulk compositions.", "Our approach can be used to derive the formation conditions of the primitive matter in the PSN, and to give insights on the origin of Mercury as well as Super-Mercuries [55], [11].", "Section summarizes the disk/transport model used to compute the composition and the thermodynamic properties of the PSN.", "Our results are presented in Section .", "Section is devoted to discussion and conclusions.", "Our time-dependent PSN model is ruled by the following second-order differential equation [37]: $\\frac{\\partial \\Sigma _{\\mathrm {g}}}{\\partial t} = \\frac{3}{r} \\frac{\\partial }{\\partial r} \\left[ r^{1/2} \\frac{\\partial }{\\partial r} \\left( r^{1/2} \\Sigma _{\\mathrm {g}} \\nu \\right)\\right].", "$ This equation describes the evolution of a viscous accretion disk of surface density $\\Sigma _{\\mathrm {g}}$ of dynamical viscosity $\\nu $ , assuming hydrostatic equilibrium in the $z$ direction.", "This equation can be rewritten as a set of two first-order differential equations coupling the gas surface density $\\Sigma _{\\mathrm {g}}$ field and mass accretion rate $\\dot{M}$ : $\\frac{\\partial \\Sigma _{\\mathrm {g}}}{\\partial t} &= \\frac{1}{2\\pi r} \\frac{\\partial \\dot{M}}{\\partial r}, \\\\\\dot{M} &= 3\\pi \\Sigma _{\\mathrm {g}}\\nu \\left(1+2Q\\right), $ where $Q=\\mathrm {d} \\ln (\\Sigma _{\\mathrm {g}}\\nu )/\\mathrm {d} \\ln (r)$ .", "The first equation is a mass conservation law, and the second one is a diffusion equation.", "The mass accretion rate can be expressed in terms of the gaz velocity field $v_g$ as $\\dot{M}=-2\\pi r \\Sigma _{\\mathrm {g}} v_{\\mathrm {g}}$ .", "The viscosity $\\nu $ is computed using the prescription of [57] for $\\alpha $ -turbulent disks: $\\nu = \\alpha \\frac{c_{\\mathrm {s}}^2}{\\Omega _{\\mathrm {K}}}, $ where $\\Omega _{\\mathrm {K}}=\\sqrt{GM_{\\odot }/r^3}$ is the keplerian frequency with $G$ the gravitational constant, and $c_{\\mathrm {s}}$ is the isothermal sound speed given by $c_{\\mathrm {s}} = \\sqrt{\\frac{RT}{\\mu _{\\mathrm {g}}}}.", "$ In this expression, $R$ is the ideal gas constant and $\\mu _{\\mathrm {g}} = 2.31$ g.mol$^{-1}$ is the gas mean molar mass [39].", "In Eq.", "REF , $\\alpha $ is a non-dimensional parameter measuring the turbulence strength, which also determines the efficiency of viscous heating, hence the temperature of the disk.", "The value of $\\alpha $ typically lies in the range 10$^{-4}$ –10$^{-2}$ , from models calibrated on disk observations [24], [28], [18].", "The midplane temperature $T$ of the disk is computed via the addition of all heating sources, giving the expression [28]: $T^4 = &&\\frac{1}{2\\sigma _{\\mathrm {sb}} } \\left(\\frac{3}{8} \\tau _{\\mathrm {R}}+\\frac{1}{2 \\tau _{\\mathrm {P}}}\\right) \\Sigma _{\\mathrm {g}} \\nu \\Omega _{\\mathrm {K}}^2 \\nonumber \\\\&+& T_{\\odot }^4 \\left[ \\frac{2}{3\\pi } \\left(\\frac{R_{\\odot }}{r}\\right)^3 + \\frac{1}{2} \\left(\\frac{R_{\\odot }}{r}\\right)^2 \\left(\\frac{H}{r}\\right) \\left( \\frac{\\mathrm {d}\\ln H }{\\mathrm {d}\\ln r} - 1\\right) \\right] \\nonumber \\\\&+& T_{\\mathrm {amb}}^4~.", "$ The first term corresponds to the viscous heating [46], where $\\sigma _{\\mathrm {sb}}$ is the Stefan-Boltzmann constant, and $\\tau _{\\mathrm {R}}$ and $\\tau _{\\mathrm {P}}$ are the Rosseland and Planck mean optical depths, respectively.", "For dust grains, we assume $\\tau _{\\mathrm {P}}=2.4 \\tau _{\\mathrm {R}}$ [46].", "$\\tau _{\\mathrm {R}}$ is derived from [28]: $\\tau _{\\mathrm {R}} = \\frac{\\kappa _{\\mathrm {R}} \\Sigma _{\\mathrm {g}}}{2}, $ where $\\kappa _{\\mathrm {R}}$ is the Rosseland mean opacity, computed as a sequence of power laws of the form $\\kappa _{\\mathrm {R}}=\\kappa _0 \\rho ^a T^b$ , where parameters $\\kappa _0$ , $a$ and $b$ are fitted to experimental data in different opacity regimes [4] and $\\rho $ denotes the gas density at the midplane.", "The second term corresponds to the irradiation of the disk by the central star of radius $R_\\odot $ and surface temperature $T_\\odot $ .", "It considers both direct irradiation at the midplane level and irradiation at the surface at a scale height $H=c_{\\mathrm {s}}/\\Omega _{\\mathrm {K}}$ .", "The last term accounts for background radiation of temperature $T_{\\mathrm {amb}} = 10$ K. At each time step, $\\Sigma _{\\mathrm {g}}$ is evolved with respect to Eq.", "(REF ).", "Then Eq.", "(REF ) is solved iteratively with Eqs.", "(REF ), (REF ), (REF ) to produce the new thermodynamic properties of the disk.", "The new velocity field is then computed following Eq.", "().", "The time-step is computed using the diffusion timescale in each bin $\\Delta t = 0.5 \\min (\\Delta r^2/\\nu )\\simeq 0.1$ yr, where $\\Delta r$ is the spatial grid size, and the factor $0.5$ is taken for safety.", "The spatial grid is formed from $N=500$ bins scaled as $r_i = (r_\\mathrm {min}-r_\\mathrm {off})\\times i^\\beta +r_\\mathrm {off}$ , where $\\beta = \\log \\left(\\frac{r_\\mathrm {max}-r_\\mathrm {off}}{r_\\mathrm {min}-r_\\mathrm {off}}\\right)/\\log (N)$ , giving a non-uniform grid, with $r_\\mathrm {min}$ and $r_\\mathrm {max}$ the limits of the grid, and $r_\\mathrm {off}$ is a parameter that allows to control the spatial distribution of points in the simulation box.", "We compute the sum of the mass lost from both limits of the simulation box ($\\int \\dot{M}(r_\\mathrm {min})dt$ and $\\int \\dot{M}(r_\\mathrm {max})dt$ ) along with the mass of the disk itself at each time.", "This quantity remains constant within the precision of the machine.", "The initial condition is the self-similar solution $\\Sigma _{\\mathrm {g}} \\nu \\propto \\exp \\left(-(r/r_{\\mathrm {c}})^{2-p}\\right)$ derived by [37].", "Choosing $p=\\frac{3}{2}$ for an early disk, we derive the initial surface density and mass accretion rate as follows: $\\left\\lbrace \\begin{array}{ll}\\Sigma _{\\mathrm {g},0} = \\frac{\\dot{M}_{\\mathrm {acc},0}}{3\\pi \\nu } \\exp \\left[-\\left(\\frac{r}{r_{\\mathrm {c}}}\\right)^{0.5}\\right], \\\\\\dot{M}_0 = \\dot{M}_{\\mathrm {acc},0} \\left(1- \\left(\\frac{r}{r_{\\mathrm {c}}}\\right)^{0.5} \\right)\\exp \\left[-\\left(\\frac{r}{r_{\\mathrm {c}}}\\right)^{0.5}\\right].\\end{array}\\right.", "$ To compute the numerical value of $\\Sigma _{\\mathrm {g},0}$ , we solve iteratively equation (REF ) with the imposed density profile in the first member of equation (REF ).", "$r_{\\mathrm {c}}$ regulates the size of the disk, and from the second member of equation (REF ) we see that it matches, at $t=0$ to the centrifugal radius.", "The value of $r_{\\mathrm {c}}$ is computed by dichotomy, assuming a disk mass $M_d$ of 0.1 M$_\\odot $ .", "For $\\alpha =10^{-3}$ , $r_{\\mathrm {c}}$ is equal to 1.83 AU, and 99% of the disk's mass is encapsulated within $\\sim 100$ AU.", "We assume the initial mass accretion rate onto the central star $\\dot{M}_{\\mathrm {acc},0}$ to be $10^{-7.6}$ M$_\\odot \\cdot $ yr$^{-1}$ [24].", "The resulting density and temperature profiles are shown in Figure REF at different epochs for $\\alpha =10^{-3}$ .", "Figure: From top to bottom: disk's surface density and temperature profiles at different epochs of its evolution, assuming α=10 -3 \\alpha =10^{-3}." ], [ "Size of dust particles", "The size of dust particles used in our model is determined by a two-populations algorithm derived from [5].", "This algorithm computes the representative size of particles through the estimate of the limiting Stokes number in various dynamical regimes.", "We assume that dust is initially present in the form of particles of sizes $a_0 = 10^{-7}$ m, and grow through mutual collisions on a timescale [35]: $\\tau _{\\mathrm {growth}} = \\frac{a}{\\dot{a}} = \\frac{4\\Sigma _{\\mathrm {g}}}{\\sqrt{3}\\epsilon _{\\mathrm {g}} \\Sigma _{\\mathrm {d}} \\Omega _{\\mathrm {K}}}, $ where $a$ is the size of dust grains and $\\Sigma _{\\mathrm {d}}$ is the dust surface density.", "We set the growth efficiency parameter $\\epsilon _{\\mathrm {g}}$ equal to 0.5 [35].", "The size of particles that grow through sticking is thus given by $a_{\\mathrm {stick}}=a_0 \\exp (t/\\tau _{\\mathrm {growth}})$ .", "However, this growth is limited by several mechanisms preventing particles from reaching sizes greater than $\\sim 1$ cm.", "The first limit arises from fragmentation, when the relative speed between two grains due to their relative turbulent motion exceeds the velocity threshold $u_{\\mathrm {f}}$ .", "This sets a first upper limit for the Stokes number of dust grains, which is [5]: $\\mathrm {St}_{\\mathrm {frag}}= 0.37 \\frac{1}{3 \\alpha } \\frac{u_{\\mathrm {f}}^2}{c_{\\mathrm {s}}^2}, $ where we set $u_{\\mathrm {f}}=10$ m.s$^{-1}$ [5], [43].", "A second limitation for dust growth is due to the drift velocity of grains, i.e.", "when grains drift faster than they grow, setting an other limit for the Stokes number [5]: $\\mathrm {St}_{\\mathrm {drift}} = 0.55 \\frac{\\Sigma _{\\mathrm {d}}}{\\Sigma _{\\mathrm {g}}} \\frac{v_{\\mathrm {K}}^2}{c_{\\mathrm {s}}^2} \\left| \\frac{\\mathrm {d}\\ln P}{\\mathrm {d} \\ln r}\\right|^{-1}, $ where $v_{\\mathrm {K}}$ is the keplerian velocity, and $P$ is the disk midplane pressure.", "Equation (REF ) only considers the relative turbulent motion between grains that are at the same location, but the fragmentation threshold $u_\\mathrm {f}$ can also be reached when dust grains drift at great velocities, and in the process collide with dust grains that are on their path.", "In that case, we obtain a third limitation for grains' size [5]: $\\mathrm {St}_{\\mathrm {df}} = \\frac{1}{(1-N)}\\frac{u_{\\mathrm {f}} v_{\\mathrm {K}}}{c_{\\mathrm {s}}^2} \\left( \\frac{\\mathrm {d}\\ln P}{\\mathrm {d}\\ln r}\\right)^{-1} ,$ where the factor $N=0.5$ accounts for the fact that only particles of bigger size fragment during collisions.", "The relation between Stokes number and dust grains size depends on the flow regime in the disk [31]: $\\mathrm {St} = \\left\\lbrace \\begin{array}{ll}\\sqrt{2\\pi } \\frac{a \\rho _{\\mathrm {b}}}{\\Sigma _{\\mathrm {g}}} & \\text{ if } a \\le \\frac{9}{4}\\lambda \\\\\\frac{8}{9} \\frac{a^2 \\rho _{\\mathrm {b}} c_\\mathrm {s}}{\\Sigma _{\\mathrm {g}} \\nu } & \\text{ if } a \\ge \\frac{9}{4}\\lambda ,\\end{array}\\right.", "$ where $\\rho _{\\mathrm {b}}$ is the bulk density of grains.", "The first case correspond to the Epstein's regime, occuring in the outermost region of the disk, and the second case corresponds to the Stokes regime.", "The limit between the two regimes is set by the mean free path $\\lambda =\\sqrt{\\pi /2}~\\nu /c_\\mathrm {s}$ (computed by equating both terms of Eq.", "(REF )) in the midplane of the disk.", "Combining Eqs.", "(REF -REF ), the limiting Stokes number and representative particle size are computed for both regimes independently, giving $a_\\mathrm {E}$ and $\\mathrm {St}_\\mathrm {E}$ in Epstein regime, and $a_\\mathrm {S}$ and $\\mathrm {St}_\\mathrm {S}$ in Stokes regime.", "Then, as shown by Figure REF , the representative size and Stokes number is given by $a &= \\min \\left(a_\\mathrm {E},a_\\mathrm {S}\\right), \\\\\\mathrm {St} &= \\max \\left(\\mathrm {St}_\\mathrm {E},\\mathrm {St}_\\mathrm {S}\\right).", "$ In the following, two end-cases are considered.", "In case A, we assume that all trace species are entirely independent, i.e.", "a run with several species is equivalent to several runs with a single traces species at a time.", "In case B, we assume that at each orbital distance, dust grains are a mixture of all available solid matter at that distance.", "For this case, the considered dust surface density is the sum over all surface densities $\\Sigma _{\\mathrm {d}} = \\sum _{i} \\Sigma _{\\mathrm {d},i}$ .", "The bulk density $\\rho _\\mathrm {b}$ of resulting grains is also the mass-average of the bulk densities of its constituents (given in Table REF ): $\\rho _\\mathrm {b} = \\frac{\\sum _{i} \\Sigma _{\\mathrm {d},i} \\rho _{\\mathrm {b},i}}{\\sum _{i} \\Sigma _{\\mathrm {d},i}}.$ Case B is favored from a dynamical point of view, as dust grains of different composition mix over long timescales, whereas case A is favored from a thermodynamic point of view, since sublimation and condensation tend to separate species into their pure forms.", "Since we focus on rocklines, whose positions are all at distances $\\le 1$ AU, we consider our disk volatile-free.", "In our model, the closest iceline would be that of H$_2$ O, which is located at $\\sim $ 4 AU.", "This iceline is far enough to ignore its impact on processes at play around rocklines.", "Figure: Visual sketch showing the Stokes number St dependency with respect to the particle size aa, in both considered flow regimes.", "When a<9λ/4a<9\\lambda /4, particles follow Epstein regime.", "When a>9λ/4a>9\\lambda /4, particles follow Stokes regime.", "In both cases, the relevant size is the smallest among the two, and the relevant Stokes number is the greatest, which are depicted as the red dot-dashed line." ], [ "Evolution of vapors and dust", "We follow the approaches of [18] and [19] for the dynamics of trace species in term of motion and thermodynamics, respectively.", "The disk is uniformly filled with seven refractory species considered dominant (see Table REF ), assuming protosolar abundances for Fe, Mg, Ni, Si and S [39], similar Fe/Mg ratios in olivine and pyroxene, and that half of Ni is in pure metallic form while the remaining half is in kamacite.", "Sublimation of grains occurs during their inward drift when partial pressures of trace species become lower than the corresponding vapor pressures.", "Once released, vapors diffuse both inward and outward.", "Because of the outward diffusion, vapors can recondense back in solid form, and condensation occurs either until thermodynamic equilibrium is reached or until no more gas is available to condense.", "Over one integration time step $\\Delta t$ , the amount of sublimated or condensed matter is [19] $\\Delta \\Sigma _{\\mathrm {subl},i} &= \\dot{Q}_{\\mathrm {subl},i} \\Delta t \\nonumber \\\\&= \\mathrm {min}\\left(\\frac{6\\sqrt{2\\pi }}{\\pi \\rho _{\\mathrm {b}} a} \\sqrt{\\frac{\\mu _i}{RT}} P_{\\mathrm {sat},i} \\Sigma _{\\mathrm {d},i}\\Delta t~;~\\Sigma _{\\mathrm {d},i} \\right),$ $\\Delta \\Sigma _{\\mathrm {cond},} &= \\dot{Q}_{\\mathrm {cond},i} \\Delta t \\nonumber \\\\&= \\min \\left( \\frac{2 H \\mu _g}{R T} \\cdot \\left(P_{\\mathrm {v},i}-P_{\\mathrm {sat},i}\\right)~;~ \\Sigma _{\\mathrm {v},i}\\right),$ where $\\mu _i$ is the molar mass of a given trace species, $P_{\\mathrm {sat},i}$ its saturation pressure and $P_{\\mathrm {v},i}$ its partial pressure at a given time and place in the PSN.", "The second term of the min function ensures that the amount of sublimated (resp.", "condensed) matter is at most the amount of available matter to sublimate (resp.", "condense) i.e.", "the dust (resp.", "vapor) surface density $\\Sigma _{\\mathrm {d},i}$ (resp.", "$\\Sigma _{\\mathrm {v},i}$ ).", "We define the rockline as the location at which the surface density $\\Sigma _{\\mathrm {d},i}$ of solid and $\\Sigma _{\\mathrm {v},i}$ and vapor of a given species $i$ are equal.", "Species exist mostly in solid forms at greater heliocentric distances than their rocklines while they essentially form vapors at distances closer to the central star.", "Figure REF shows the locations of the considered rocklines as a function of time in the PSN for both cases.", "No gas phase chemistry is assumed in the disk.", "Figure: Time evolution of the locations of rocklines in the PSN.", "Solid and dashed lines correspond to cases A and B, respectively (see text).", "Only minor differences between the two cases are observed, resulting from changes in radial drift velocities of particles.The motion of dust and vapor, who coexist as separate surface densities $\\Sigma _{\\mathrm {d},i}$ and $\\Sigma _{\\mathrm {v},i}$ , is computed by integrating the 1D radial advection-diffusion equation [5], [18]: $\\frac{\\partial \\Sigma _{i}}{\\partial t}+\\frac{1}{r} \\frac{\\partial }{\\partial r}\\left[r\\left(\\Sigma _{i} v_{i}-D_{i} \\Sigma _{\\mathrm {g}} \\frac{\\partial }{\\partial r}\\left(\\frac{\\Sigma _{i}}{\\Sigma _{\\mathrm {g}}}\\right)\\right)\\right]+\\dot{Q}_{i}=0.$ This equation holds for both vapor and solid phases since the motion is determined by the radial velocity $v_i$ and the radial diffusion coefficient $D_i$ of species $i$ , $\\dot{Q}$ being the source/sink term.", "When a species $i$ is in vapor form, we assume $v_i \\simeq v_g$ and $D_i \\simeq \\nu $ .", "When this species is in solid form, the dust radial velocity is the sum of the gas drag induced velocity and the drift velocity [5]: $v_{\\mathrm {d}} = \\frac{1}{1+ \\mathrm {St}^2}v_{\\mathrm {g}} + \\frac{ 2\\mathrm {St}}{1+ \\mathrm {St}^2} v_{\\mathrm {drift}}, $ where the drift velocity is given by [68], [45]: $v_{\\mathrm {drift}} = \\frac{c_{\\mathrm {s}}^2}{v_{\\mathrm {K}}} \\frac{\\mathrm {d}\\ln P}{\\mathrm {d}\\ln r}.$ This expression usually holds for a population of particles sharing the same size.", "However, we work here with the two-population algorithm from [5], i.e.", "the dust is composed of a mass fraction $f_\\mathrm {m}$ of particles of size $a$ , and a mass fraction $1-f_\\mathrm {m}$ of particles of size $a_0$ .", "The dust radial velocity can be then approximated by a mass-weighted velocity: $v_i = f_\\mathrm {m} v_{d,\\text{size }a} + (1-f_\\mathrm {m}) v_{d,\\text{size }a_0},$ where $v_{d,\\text{size }a/a_0}$ is calculated with respect to Eq.", "(REF ), St is computed for both populations at each heliocentric distance, and $f_{\\mathrm {m}}$ depends on the size limiting mechanism [5]: $f_{\\mathrm {m}} = \\left\\lbrace \\begin{array}{ll}0.97 & \\text{ if } \\mathrm {St}_{\\mathrm {drift}} =\\text{min}\\left(\\mathrm {St}_{\\mathrm {frag}}, \\mathrm {St}_{\\mathrm {drift}}, \\mathrm {St}_{\\mathrm {df}}\\right)\\\\0.75 & \\text{ otherwise.}\\end{array}\\right.", "\\nonumber $ Due to the low Stokes number of dust grains ($\\mathrm {St}<1$ ), we make the approximation $D_i = \\frac{\\nu }{1+\\mathrm {St}^2_i} \\simeq \\nu $ .", "Table: Main refractory phases present in the disk with corresponding initial abundances and references for saturation pressures P sat P_{\\mathrm {sat}}.Figures REF and REF represent the time evolution of the composition of refractory particles evolving throughout the PSN in a Mg-Fe-Si ternary diagram and the time evolution of the Fe abundance profile (in wt%) as a function of heliocentric distance, and in case A and case B, respectively.", "Ternary diagrams display the composition of cosmic spherules and chondrules.", "Solid particles start with a protosolar composition which changes during their drift throughout the innermost regions of the PSN, due to the successive sublimation of minerals.", "Because alloys, which contain only Fe, S and Ni, are the first to sublimate (see Fig.", "REF ), solid particles loose a substantial amount of iron (in the $\\sim $ 20–50$\\%$ range), but not Mg and Si.", "This is illustrated by the composition profiles that shift toward the Mg-Si axis with an unchanged Mg/Si ratio on the ternary diagrams.", "Closer to the Sun, ferrosilite and fayalite begin to sublimate until no Fe remains in solid form.", "In this case, the composition profiles in ternary diagrams are located on the Mg-Si axis, and because some silicon is vaporized too, the Mg/Si ratio increases.", "Finally, if the temperature is high enough to sublimate forsterite, only enstatite remains in solid particles, with an atomic ratio $(\\mathrm {Mg/Si})_{\\mathrm {at}} = 1$ corresponding to 46 wt% of Mg.", "In case A, lighter grains are more subject to the pressure gradient and drift inward faster than dense Fe-rich grains.", "As a result, the Fe wt% increases by $\\sim 10\\%$ in the whole disk.", "This leads to a slightly wider range of possible compositions in the PSN than for case B, but the differences between the two runs are minor from a compositional point of view.", "Figure: Left panel: composition of refractory matter in a Mg-Fe-Si ternary diagram expressed in mass fraction between t=10 4 t~=~10^4 and 2×10 6 2~\\times ~10^6 yr of the PSN evolution with α=10 -3 \\alpha =10^{-3} and for case A (all trace species independent).", "Purple triangles correspond to glass cosmic spherules (S-V type) , suggesting they were formed by condensation in the vicinities of Fe oxides rocklines.", "Green circles correspond to barred olivine spherules (S-BO type) that potentially formed via mixing in the region of iron alloys rocklines.", "Green squares represent porphyritic spherules (S-P type) from the same collection.", "Yellow triangles correspond to a random selection of chondrules from various carbonaceous chondrites studied in .", "Sun and Earth symbols correspond to protosolar and Earth bulk compositions, respectively .", "The red circle represents Mercury's bulk composition .", "Right panel: evolution of the iron abundance (in wt%) in solids as a function of heliocentric distance.", "The different colorboxes correspond to the iron content in chondrules (0–10%), glass cosmic spherules (10–30%), and porphyritic and barred olivine cosmic spherules (30–60%).Figure: Same as figure , but for case B.Figure REF represents the composition of PSN refractory particles in case A (as in Fig.", "REF ) superimposed with the mean bulk compositions of chondrite groups.", "The figure also shows the composition of a random selection of chondrules and matrix [26].", "The mean bulk composition of chondrites is close to the protosolar value, and its spread seems to follow the profiles derived from our model.", "The same behaviour can be observed for the matrix.", "On the other hand, chondrules exhibit a very low amount of bulk Fe, and the average composition seems to be close to the one computed by our model.", "Figure: Ternary diagram (expressed in mass fraction) representing the PSN composition profiles from Fig.", "(case A), with compositions of chondrules, matrixes and mean chondritic types.", "Compositions of chondrules (yellow triangles) and matrix (grey squares) are taken from .", "Mean bulk chondrites compositions (colored pentagons) are taken from .Top panels of Fig.", "REF show the radial profiles of the disk's metallicity (defined as $Z=\\Sigma _{\\mathrm {d}}/\\Sigma _{\\mathrm {g}}$ ) at different epochs of its evolution.", "As expected, solid matter is concentrated at the position of rocklines.", "The composition of the PSN around the first cluster of rocklines (iron sulfide, cite and nickel) corresponds to the 30-60 wt% Fe part of curves in ternary diagrams and Fe wt% profiles, which matches the S-BO type (barred olivine) spherules compositions.", "The composition of the PSN around the second cluster of rocklines (fayalite and ferrosilite) corresponds to the 10-30 wt% Fe part of curves in the ternary diagram and Fe wt% profiles, matching the S-V type (glass) spherules compositions.", "In the same manner, we would expect chondrules to be formed in the innermost regions, were sufficient amount of material is present due to continuous drift from the outer disk.", "At $t=10^5$ yr and 0.67 AU (rockline of iron sulfide), the PSN has 56 wt% and 58 wt% of Fe in case A and case B, respectively.", "This increase of the Fe wt% leads to compositions of the PSN richer in Fe than the protosolar value.", "Figure: Local metallicity Z=Σ d /Σ g Z=\\Sigma _{\\mathrm {d}}/\\Sigma _{\\mathrm {g}} (top panels) and Stokes number (bottom panels) computed as a function of time and heliocentric distance.", "Left and right panels are results for case A and case B, respectively (see text).", "Stokes number is shown at t=10 6 t=10^6 yr for a few representative species (left panel) and at different epochs of the PSN evolution (right panel).", "Dashed lines in top panels show the lowest metallicity Z c Z_\\mathrm {c} required to trigger streaming instability (see text).Finally, bottom panels of Fig.", "REF show the Stokes number of dust grains in the disk.", "Because case A has many independent species evolving, we only follow forsterite, fayalite and iron sulfide, namely the most, least, and intermediate refractory materials considered in our particles.", "In case B, all grains are mixed together, giving a single Stokes number at each heliocentric distance.", "Using Eqs.", "(REF ), (REF ) and (REF ), we expect i) $\\mathrm {St}_\\mathrm {frag} \\propto 1/T$ , ii) $\\mathrm {St}_\\mathrm {drift}\\propto Z/(rT)$ and iii) $\\mathrm {St}_\\mathrm {df}\\propto 1/(\\sqrt{r}T)$ (assuming $\\left| \\frac{\\mathrm {d}\\ln P}{\\mathrm {d}\\ln r}\\right|^{-1} \\propto 1$ ).", "In the innermost region, dust size is limited by fragmentation up to $\\sim 5$ AU.", "In the 5-10 AU range, a competition between drift and drift-limited fragmentation sets the dust grains size.", "Beyond 10 AU dust is in the growth phase.", "In case A, variations in limiting sizes only come from the difference in bulk densities of grains.", "As a result, different species display the same Stokes number during most of their drift throughout the PSN (see bottom left panel of Fig.", "REF ).", "However, because the amount of solid matter decreases below 10 AU, as a result of sublimation and/or radial drift, the Stokes number diminishes as well.", "In case B, minor species embedded in large grains are transported more efficiently toward their rocklines.", "Hence, higher metallicities are found around the rocklines in case B compared to case A, at early epochs.", "In turn, the PSN becomes depleted in solid matter in shorter timescales in case B compared to case A.", "As indicated in Section REF , $\\alpha $ is a free parameter whose value can change with time and heliocentric distance [32].", "For this disk model, an increase of the $\\alpha $ value leads to a centrifugal radius $r_\\mathrm {c}$ located at higher heliocentric distance, which in turn leads to a larger disk.", "This also leads to a larger diffusion coefficient $D_i=\\nu $ for the trace species.", "As a consequence, vapors diffuse outward faster and enrich the solid phase more evenly.", "This results in peaks of abundances that are wider and less prominent than those observed in right panels of Figs.", "REF and REF .", "For the extreme case $\\alpha =10^{-2}$ , the peaks of abundance are not observable anymore.", "This shows that the choice on $\\alpha $ is critical for both the PSN and trace species evolutions.", "However, results of simulations with non-uniform $\\alpha $ show that this quantity is increasing with heliocentric distance, and takes values of $\\sim $ 10$^{-3}$ at 1 AU [32].", "Since we are mostly interested in the dynamics of the inner PSN, we adopted this value for our $\\alpha $ parameter." ], [ "Discussion and conclusion", "Our model shows that the diversity of the bulk composition of cosmic spherules and chondrules can be explained by their formation close to rocklines, suggesting that solid matter is concentrated in the vicinity of these sublimation/condensation fronts.", "The slightly lower Fe content of S-type (ordinary) chondrites than for C-type (carbonaceous) observed in Fig.", "REF could be due to a partial sublimation mechanism.", "Several transport mechanisms can explain the presence of these processed minerals in the Main Belt, as well as in the outer regions of the PSN.", "For example, small particles with sizes within the 10$^{-6}$ –1 m range and formed in the inner nebula can diffuse radially toward its outer regions [6], [18].", "Figure REF illustrates the efficiency of diffusion in our model by representing the time evolution of the radial distribution of particles initially formed in the 0–1 AU region of the PSN.", "In this simulation, a protosolar dust-to-gas ratio is assumed within 1 AU at $t$ = 0 and the effects of rocklines are not considered.", "The figure shows that particles can diffuse well over 10 AU after 10$^5$ yr of PSN evolution and thus fill the Main Belt region.", "Photophoresis is another mechanism which can be at play when the inner regions of the PSN become optically thin.", "In this case, the disk still has a reasonable gas content and allows particles with sizes in the 10$^{-5}$ -10$^{-1}$ m range to receive light from the proto-Sun and be pushed outward by the photophoretic force beyond 20 AU in the PSN [33], [41], [42], [40].", "These two mechanisms would have taken place before the formation of larger planetesimals in the PSN.", "Figure: Time evolution of the radial diffusion of particles in our PSN model, assuming α=10 -3 \\alpha =10^{-3}.Interestingly, these high concentrations do not allow the triggering of streaming instability.", "The smallest metallicity $Z_\\mathrm {c}$ required to trigger a streaming instability for low Stokes number regimes $\\left(\\mathrm {St}< 0.1\\right)$ is [69]: $\\log Z_\\mathrm {c} = 0.10 \\left(\\log \\mathrm {St}\\right)^2 + 0.20 \\log \\mathrm {St} - 1.76.$ Figure REF shows that the highest disk metallicity computed with our model, i.e.", "at $t=10^5$ yr and $r$ = 0.1 AU in case B, is $Z=4.4\\times 10^{-3}$ , way below $Z_\\mathrm {c}=2.7\\times 10^{-2}$ , suggesting our model does not allow the triggering of a streaming instability.", "However, the decoupling of dust grains from gas at greater Stokes numbers or the back-reaction of dust onto the gas could slow down the drift of particles in the pile-up regions, thus increasing the local dust-to-gas ratio.", "Although the spread in bulk compositions of CS can be well explained by alteration during atmospheric entry [54], the two scenarios are not mutually exclusive as they happen at very different times.", "The grains composition computed by our model can be seen as the average composition at each time and heliocentric distance, and the deviation from the mean composition could be the result of full dynamics of grains growth and interaction [27].", "The effect of rocklines could be among the first processing mechanisms altering the uniform composition of refractory matter in the PSN.", "Moreover, condensation/sublimation fronts for refractory matter may have other implications.", "Close to the host star, the pressure at the midplane can be high enough (up to 1 bar) to melt partially or entirely solid grains.", "This would highly affect the collisional dynamics of grains (or droplets), and allow grains to overcome the meter barrier [7] and form Fe-rich planetesimals that later gave birth to Mercury.", "Although our model relies a lot on the number of considered species and the availability of thermodynamic data governing state change, it suggests that rocklines played a major role in the formation of small and large bodies in the innermost regions of the PSN.", "For example, even if the large amount of iron in Mercury (83 wt% in the ternary diagram) cannot be explained with this model alone, the increased proportion of Fe in the PSN (62 wt% at most in the vicinity of rocklines; see Figs.", "REF and REF ) from the protosolar value (47 wt%) might have contributed to the accretion of the planet's large core by forming Fe-rich regions.", "As our model only tracks the evolution of dust grains in the early PSN, it is compatible with current formation mechanisms of terrestrial planets [50], [30].", "The relevant PSN composition in terms of age and heliocentric distance must then be chosen accordingly to the considered formation scenario.", "In its current state, the model fails to reproduce the extreme enrichments in Fe needed to account for the formation of Mercury.", "However, giant impact simulations performed by [13] show that the resulting Mercury-like planets display core mass fractions (CMF) in the 0.5–0.7 range (see their Fig.", "3), when starting with a protosolar CMF of 0.3.", "If the initial CMF of Mercury was 10% higher due to the formation of its building blocks close to rocklines, as suggested by our findings, the post-collision CMF would lie in the 0.6–0.8 range, which is in better agreement with the estimated value of $~0.7$ [60], [52], [25].", "The combination of multiple scenarios to explain the large CMF of Mercury seems more likely.", "At greater heliocentric distances the PSN composition becomes again protosolar (mainly for case B), which is in agreement with the bulk composition of Earth and Venus (close to protosolar) derived from interior structure models [60], [53], [20].", "Our study suggests that Mercury-like planets should exist in other planetary systems.", "More than $\\sim $ 2000 small planets in the 1–3.9 $R_\\oplus $ range have been discovered so far at close distances to their host starshttps://exoplanetarchive.ipac.caltech.edu.", "Most of the measured densities are poorly determined and the detection of sub-Earth planets remains challenging, implying it remains difficult to quantify the size of the population of Mercury-like planets.", "Finally, the presence of Mercury-like planets should be ruled by the amount of available matter to form Fe-rich building blocks.", "In the case of very massive and hot circumstellar disks, rocklines and Mercury-like planets would be located at much higher distances to the host star.", "In contrast, less massive and colder disks could impede the formation of Fe- rich planets because rocklines would be located too close to their host star.", "O.M.", "acknowledges support from CNES.", "We thank the anonymous referee for useful comments that helped improving the clarity of our paper." ] ]

[ [ "New Non-commutative and Higher Derivatives Quantum Mechanics from GUPs" ], [ "Abstract We explore a new class of Non-linear GUPs (NLGUP) showing the emergence of a new non-commutative and higher derivatives quantum mechanics.", "Within it, we introduce the shortest fundamental scale as a UV fixed point in the NLGUP commutators [X, P] = i\\hbar f(P), having in mind a fundamental highest energy threshold related to the Planck scale.", "We show that this leads to lose commutativity of space coordinates, that start to be dependent by the angular momenta of the system.", "On the other hand, non-linear GUP must lead to a redefinition of the Schrodinger equation to a new non-local integral-differential equation.", "We also discuss the modification of the Dyson series in time-dependent perturbative approaches.", "This may suggest that, in NLGUPs, non-commutativity and higher derivatives may be intimately interconnected within a unified and coherent algebra.", "We also show that Dirac and Klein-Gordon equations are extended with higher space-derivatives according to the NLGUP.", "We compute momenta-dependent corrections to the dispersion velocity, showing that the Lorentz invariance is deformed.", "We comment on possible implications in tests of light dispersion relations from Gamma-Ray-Bursts or Blazars, with potential interests for future experiments such as LHAASO, HAWC and CTA." ], [ "Introduction", "The Heisenberg Uncertainty Principle (HUP) is the essence of quantum mechanics.", "Is it possible to extend it in a Generalized form, i.e.", "a Generalized Uncertainty Principle (GUP), compatible with Bohr correspondence principle?", "This issue is also interesting from the point of view of quantum gravity, since we know that HUP is one of the main obstacle towards a consistent quantization of the Einstein geometrodynamical field.", "A first simple proposal was to add quadratic powers as a simple non-linear extension of it, i.e.", "$[X_{i},P_{j}]=i\\hbar (1+\\beta |P|^{2}+...)$ [1], which is also suggested as first effective corrections from scattering amplitudes in string theory [2], [3], [4].", "Neverthless, one would imagine that a full summation on all over perturbative and non-perturbative quantum gravity effects may lead to a non-linear GUP (NLGUP) beyond quadratic corrections.", "In this paper, we wish to explore a class of GUP having, as a peculiarity, a UV momentum pole, as $[X_{i},P_{j}]=i\\hbar f(\\beta P^{2})$ with $f\\rightarrow \\infty $ for $|P|\\rightarrow 1/\\sqrt{\\beta }$ .", "We are particularly attracted by these models since they would imply, that at a critical UV momenta scale, one would completely lose resolution on the $\\Delta X \\Delta P$ quantities and, therefore, on the total angular momenta.", "This would imply that we may not probe length scales smaller than a critical length since fluctuations wildly diverge, unitarizing cross-sections with the highest UV energy below the Planck scale.", "The core of this proposal is to marry quantum uncertainties with Doubly Special Relativity [5], [6], [7], [8], [9], [10], [11], [12], [13].", "Within this framework, the presence of a UV fixed point in the NLGUP commutator of X and P induces a fundamental shortest scale inside the quantum mechanical structure.", "In our paper, we will explicitly show that, within such a new algebra extension, commutativity in the space coordinates is lost close to the critical energy $1/\\sqrt{\\beta }$ .", "We will also show that NLGUP implies new higher derivative terms appearing in the wave function equation." ], [ "Non-linear GUPs", "Let us start considering the following class of GUP models, that we may dub a-s-Non-linear-GUPs (asNLGUP).", "$[ X_i, P_j] =\\frac{ i\\hbar \\delta _{ij} }{ (1 - (\\beta P^2)^{a})^{s} }\\,\\,,~~~[ P_i, P_j]=0\\,\\,, ~~[ X_i , X_j ] =\\frac{ 2i\\hbar \\beta }{ (1 - (\\beta P^2)^{a})^{2s} }( P_i X_j - P_j X_i),$ where $\\beta $ is related to the GUP critical energy scale $\\Lambda $ as $\\beta =\\Lambda ^{-2}$ , $X,P$ are position and momenta operators, $a,s$ are two free parameters.", "One should easily check that this class of model closes a self-consistence algebra set while having an UV divergence for $|P|=\\beta ^{-1/2}$ , having in mind that $\\sqrt{\\beta }$ may be related to the Planck length $l_{Pl}$ .", "This algebra is consistent with a redefinition of the standard position and momenta operator as follows: $P_i\\equiv P_i, \\,\\,\\,\\,\\, X_j\\rightarrow X_{j}\\frac{1}{(1 - (\\beta P^2)^{a})^{s}}\\, .$ Another important remark is that P-X would be conjugate variables in a generalized sense, considering a measure factor to be included in the Fourier transforms.", "The infinitesimal translation operator has a new non-linear form $T=1-i\\frac{P \\cdot dX}{(1-(\\beta P^{2})^{a})^{s}}+O(dX^2)\\rightarrow P =-i\\hbar (1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s} \\nabla \\, ,$ where $\\nabla $ is with respect to standard $X$ variables of Q.M.", "Therefore, definitions in Eq.", "(REF ) and Eq.", "(REF ) are compatible each others.", "This also suggests that angular momenta operators lose of any certainties around the UV fixed energy.", "Indeed, Eq.", "(REF ) implies a deformation of the standard angular momenta algebra (see Appendix A): $[ L_i, L_j]&=& \\frac{i\\hbar }{(1-(\\beta P^2)^{a})^{s}}\\left(X_i P_j-X_j P_i\\right)= \\frac{i\\hbar }{(1-(\\beta P^2)^{a})^{s}}\\epsilon _{ijk}L_{k}\\,\\, ,$ in compatibility with a redefinition of the angular momenta: $L_{i}=\\frac{1}{(1-(\\beta P^2)^{a})^{s}}\\epsilon _{ijk}r_{j}p_{k}\\, .$ Substituting Eq.", "(REF ) into Eq.", "(REF ), we obtain (see Appendix A) $[X_i,X_j]&=&\\frac{ -2i\\hbar \\beta }{(1 - (\\beta P^2)^{a})^{s}}L_{ij}.$ This implies a first unexpected aspect related to asNLGUP: in order to have a self-consistent algebra, the space-coordinates have not to commute, and their non-commutativity depends from the angular momenta operator.", "This would suggest a series of interesting facts.", "One would imagine that non-commutativity of space coordinates depends on the angular momenta state of a certain particle and, therefore, from angular momenta measures.", "A measure of the angular moment on z-axis would induce a non-commutativity of X and Y coordinates, a measure on x-axis would induce it on Z,Y axis and so on.", "In the special case $a=s=1$ , it is worth to note that Eq.", "(REF ) can be rewritten in the following form $L_{ij}=\\left(X_i P_j-X_j P_i\\right)\\left(1-\\beta P^2\\right)^{-1} = \\left(X_i P_j-X_j P_i\\right)\\left(1+\\beta \\hbar ^2 \\nabla ^2\\right)^{-1}\\nonumber \\\\= \\left(X_i P_j-X_j P_i\\right)\\Big \\lbrace 1-\\beta \\hbar ^2 \\nabla ^2+\\beta ^2 \\hbar ^4 \\nabla ^4\\nonumber \\\\ +...+\\frac{-1(-2)(-3)...(-1-(n-1))}{n!", "}(\\beta \\hbar ^2 \\nabla ^2)^n\\Big \\rbrace .$ This means that Eq.", "(REF ) is not only non-commutative but also non-local in space-coordinates.", "It is a remarkable feature of this theory that, if, for example, we imagine to measure X and later Y this does not commute with the Y-X measure sequence and they are related each others through a non-local derivative operator.", "Now, it is worth to note that the whole deformations of standard quantum mechanics introduced above lead to new extended Schrödinger equation.", "In standard quantum mechanics, $H=i\\hbar \\frac{d}{dt}$ dictates the time evolution dynamics.", "Here the Unitary operator has an infinitesimal structure that is deformed to a non-linear functional: $U=1-i\\frac{H dt}{(1-(\\beta P^{2})^{a})^{s}}+O(dt^2)\\rightarrow H =i\\hbar (1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s} \\frac{d}{dt}\\, .$ This does not violate unitarity, at the fixed point: the evolution operator $U$ for a finite time $t$ is $U=e^{-\\frac{iHt}{\\hbar (1-(\\beta P^2)^{a})^{s}}} \\rightarrow U^{\\dagger }U=UU^{\\dagger }=1\\, ,$ having unitarity automatically guaranteed.", "From this definition we can also arrive to the extended Dyson series for a time-dependent perturbative approach shown in our Appendix B.", "The Schrödinger equation is extended to a new non-local differential equation compatible with Eq.", "(REF ) and Eq.", "(REF ): $H\\Psi =\\Big [\\frac{P^{2}}{2m}+V\\lbrace X/(1-(-\\hbar ^{2} \\beta \\nabla ^2)^{a})^{s}\\rbrace \\Big ]\\Psi ,$ $\\rightarrow i \\hbar \\frac{d\\Psi }{dt}=\\frac{1}{(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s}}\\Big [-(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{2s}\\frac{\\hbar ^{2} \\nabla ^{2}}{2m}+V\\lbrace X/(1-(-\\hbar ^{2} \\beta \\nabla ^2)^{a})^s\\rbrace \\Big ]\\, ,$ $\\rightarrow i \\hbar \\frac{d\\Psi }{dt}= -(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s}\\frac{\\hbar ^{2} \\nabla ^{2}}{2m}\\Psi +\\frac{1}{(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s}}V\\lbrace X/(1-(-\\hbar ^{2} \\beta \\nabla ^2)^{a})^s\\rbrace \\Psi \\, .$ In the case $\\Psi (x,t)=\\psi (x)e^{-iE t/\\hbar }$ , the Schrödinger equation would become a non-local time-independent one: $(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s}\\frac{\\hbar ^{2} \\nabla ^{2}\\psi (x)}{2m}+\\frac{1}{(1-(-\\hbar ^{2} \\beta \\nabla ^{2})^{a})^{s}}V\\lbrace X/(1-(-\\hbar ^{2} \\beta \\nabla ^2)^{a})^s\\rbrace \\psi (x)=E \\psi (x)\\, .$ Eq.", "(REF ) implies that the kinetic term of the Hamiltonian $H_{0}=P^2/2m$ is untouched.", "On the other hand, interaction potentials are deformed from their dependence by $X$ operators.", "It is worth to remind that $\\nabla =\\nabla _{X_{B}}\\ne \\nabla _{X}$ , as stated above.", "A generic central potential $V=\\alpha r^{\\beta }$ would be deformed as $V=\\alpha r^{\\beta }/(1-(\\beta P^{2})^{m})^{k\\beta }$ .", "This can be reinterpreted as a new energy dependent re-normalized coupling $\\alpha _{R}=\\alpha (1-(\\beta P^{2})^{a})^{-s\\beta }\\, .$ Conversely, in the case of the electric or the gravitational field, we would have $\\beta =-1$ and therefore $\\alpha _{R}=\\alpha (1-(\\beta P^{2})^{a})^{s}\\, .$ This would suggest that the e.m and gravitational couplings would eventually flow to a U.V.", "fixed point when they flow to zero and ${\\rm lim}_{P\\rightarrow \\beta ^{-1/2}}\\alpha _{R}\\rightarrow 0$ .", "Let us consider the deformation of the Klein-Gordon equation in the relativistic regime $v\\simeq c$ .", "In NLGUP considered above this would read as $P^{2}+m^2 =E^2 \\rightarrow (1-(-\\hbar ^{2}\\nabla ^{2})^a)^{2s} \\Box \\Psi +m^{2} \\Psi = 0\\,\\,\\,\\, (c=1)\\, .$ From Klein-Gordon equation, we can obtain the Schrödinger equation by performing the non-relativistic limit as follows: $\\Psi =\\psi e^{-im_{0}t/\\hbar (1-(\\beta P^{2})^{a})^{s}},\\,\\,\\, (c=1)\\, ,$ $v<<1 \\rightarrow |\\dot{\\psi }|<<1\\, .$ Within Eqs.", "(REF ) and (REF ) assumptions, we obtain $\\dot{\\Psi }=\\Big (-\\frac{im_{0}}{\\hbar (1-(\\beta P^{2})^{a})^{s}} \\Big )\\psi e^{-im_{0}t/\\hbar (1-(\\beta P^{2})^{a})^{s}}\\, ,$ $\\ddot{\\Psi }=\\Big (-\\frac{2im_{0}}{\\hbar (1-(\\beta P^{2})^{a})^{s}}\\dot{\\psi }+\\frac{m_{0}^{2}}{\\hbar ^{2}(1-(\\beta P^{2})^{a})^{2s}}\\psi \\Big )e^{-im_{0}t/\\hbar (1-(\\beta P^{2})^{a})^{s}}\\, ,$ and inserting Eq.", "(REF ) into the modified Klein-Gordon equation, we obtain $(1-(\\beta P^{2})^{a})^{2s}\\Big (-\\frac{2im_{0}}{\\hbar (1-(\\beta P^{2})^{a})^{s}}\\dot{\\psi }+\\frac{m_{0}^{2}}{\\hbar ^{2}(1-(\\beta P^{2})^{a})^{2s}}\\psi \\Big )-(1-(\\beta P^{2})^{a})^{2s}\\nabla ^2\\psi -m_{0}^{2}\\psi =0\\, ,$ and it is easy to see that mass terms $m_{0}^{2}$ cancel each others and we obtain the modified Schrödinger equation for a free-particle $(V=0)$ in Eq.", "(REF ).", "It is also easy to extended the Dirac equation, from the modified definition of energy and momenta as $i(1-(-\\hbar ^{2}\\nabla ^{2})^{a})^{s}\\gamma _{\\mu }\\partial ^{\\mu }\\Psi +m^{2}\\Psi =0\\, ,$ which is compatible with Schrödinger equation in the non-relativistic limit.", "Finally, we can also extend the electromagnetic and gravitational waves equations as $(1-(-\\hbar ^{2}\\nabla ^{2})^a)^{2s} \\Box A_{\\mu }=0,\\,\\,\\,\\,\\,\\,\\,\\,\\, (1-(-\\hbar ^{2}\\nabla ^{2})^a)^{2s} \\Box h_{\\mu \\nu }=0.$ From the extended wave equations, the standard dispersion relations are modified as an effect of Lorentz invariance deformation.", "Indeed, considering a standard wave solution $\\Psi =A\\, e^{-iE_{B} t+ip_{B}x}$ and inserting it inside the K.G.", "equation, we obtain, for $a=1$ and $s=1$ and zero mass, $E_{B}^{2}=c^{2}p^{2}_{B}+\\beta p^{4}_{B},\\,\\,\\, v^{2}=c^{2}(1+ \\beta \\, p^{2}_{B})\\, .$ For a general NLGUP modification, we find the first correction as $v=c(1+ (s/2)(\\beta \\, p^{2}_{B})^{a})+O(p^{2}_{B})^{a+1}\\, .$ Indeed, we do not see any obstruction in having arbitrary small rational number $a$ , except experimental constrains.", "If $a=1/2$ , the UV singularity in Eq.1 would correspond to a string in the complex Energy plane rather then a single point, which would smell as introducing a new non-locality in scattering amplitudes." ], [ "Conclusion and Remarks", "In this paper, we discussed a new class of non-linear GUP as an attempt to implement a minimal length in quantum mechanics by extending the standard Heisenberg's uncertainty relation.", "We shown that this class of NLGUP implies that space-coordinates are non-commutative; the commutators are dependent by the angular momenta operators.", "This is a new feature that, so far as we know, was never met in any other theories of non-commutative space-time.", "Then, we found that NLGUP implies an extension of the Schrödinger, Klein-Gordon and Dirac equations including new higher spatial derivative terms.", "Such a result reminds the Hořava-Lishfitz theory [14], as a higher space-derivatives extension of the lagrangian, without introducing any new time derivatives, i.e.", "without introducing any ghosts.", "Another direct consequence is the modification of dispersion relations in vacuo with momentum-dependent corrections to wave and particle speeds.", "Surely, this is a manifestation of a deformation of the Lorentz algebra.", "Therefore, modified dispersion relations open intriguing phenomenological channels for this model, as a test of quantum mechanics foundations and quantum gravity.", "Indeed tests of dispersion velocities of high energy gamma-rays from Gamma-Ray-Bursts or Blazars were proposed by many authors as a possible new frontier of quantum gravity phenomenology [15], [16], [17], [18], [19], [20], [21], [22], [23].", "Therefore, Non-linear GUP strongly motivates tests of Lorentz deformations from future Very High Energy Gamma-Rays detectors such as LHAASO [24], CTA [25] and HAWC [26].", "In the case $s=1$ and $a=1/2$ , Eq.REF can be constrained up to $1/\\sqrt{\\beta }\\ge 10^{16}\\, {\\rm GeV}$ from current high energy neutrinos in IceCube [17].", "Finally, we suspect that NLGUP can deform the Spin-Statistics of standard quantum mechanics, with possible important implications in searches of Pauli Exclusion Principle Violations from Quantum gravity in underground experiments [27], [28], [29], [30]." ], [ "Appendix A", "Here, we show a more detailed proof of self-consistency of Non-linear GUPs with the deformation of the angular momentum algebra as well as non-commutativity of space-coordinates.", "For formal simplicity, we will just consider the case $a=s=1$ as generalizations of it are easily understood.", "First, let us express the angular momenta algebra just in terms of position and momenta operators: $[ L_i, L_j]&=&\\left[\\epsilon _{ik\\ell }X_k P_\\ell ,\\epsilon _{jmn}X_m P_n\\right]=\\epsilon _{ik\\ell }\\epsilon _{jmn}[X_k P_\\ell ,X_m P_n]\\nonumber \\\\&=&\\epsilon _{ik\\ell }\\epsilon _{jmn}\\Big \\lbrace [X_k,X_m]P_\\ell P_n +X_m [X_k,P_n] P_\\ell +X_k[P_\\ell ,X_m]P_n+X_k X_m[P_\\ell ,P_n]\\Big \\rbrace .$ Now, using Eq.", "(REF ), we obtain: $[ L_i, L_j]&=&\\epsilon _{ik\\ell }\\epsilon _{jmn}\\left\\lbrace \\frac{ 2i\\hbar \\beta }{(1-\\beta P^2)^2}( P_k X_m - P_m X_k)P_\\ell P_n+X_m \\frac{i\\hbar \\delta _{kn}}{1-\\beta P^2}P_\\ell +X_k \\frac{-i\\hbar \\delta _{\\ell m}}{1-\\beta P^2}P_n \\right\\rbrace \\nonumber \\\\&=&\\frac{ 2i\\hbar \\beta }{(1-\\beta P^2)^2}\\Big \\lbrace \\epsilon _{ik\\ell }\\epsilon _{jmn}P_k X_m P_\\ell P_n-\\epsilon _{ik\\ell }\\epsilon _{jmn}P_m X_k P_\\ell P_n \\Big \\rbrace \\nonumber \\\\&&+\\frac{i\\hbar }{1 -\\beta P^2}\\Big \\lbrace \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{kn}X_m P_\\ell - \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{\\ell m}X_k P_n\\Big \\rbrace \\nonumber \\\\&=&\\frac{ 2i\\hbar \\beta }{(1-\\beta P^2)^2}\\left\\lbrace \\epsilon _{ik\\ell }\\epsilon _{jmn}P_k \\left(P_\\ell X_m + \\frac{i\\hbar \\delta _{\\ell m}}{1-\\beta P^2}\\right)P_n - \\epsilon _{ik\\ell }\\epsilon _{jmn}P_m \\left(P_n X_k + \\frac{i\\hbar \\delta _{k n}}{1-\\beta P^2}\\right)P_\\ell \\right\\rbrace \\nonumber \\\\&&+\\frac{i\\hbar }{1 -\\beta P^2}\\Big \\lbrace -\\epsilon _{i\\ell k}\\epsilon _{jmn}X_m P_\\ell +\\epsilon _{ik\\ell }\\epsilon _{jn\\ell } X_k P_n\\Big \\rbrace \\nonumber \\\\&=&\\frac{ 2i\\hbar \\beta }{(1-\\beta P^2)^2}\\left\\lbrace \\epsilon _{ik\\ell }\\epsilon _{jmn}P_k P_\\ell X_m P_n+\\frac{i\\hbar \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{\\ell m}P_k P_n}{1-\\beta P^2}-\\epsilon _{ik\\ell }\\epsilon _{jmn}P_m P_n X_k P_\\ell -\\frac{i\\hbar \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{kn}P_m P_\\ell }{1-\\beta P^2}\\right\\rbrace \\nonumber \\\\&&+\\frac{i\\hbar }{1 -\\beta P^2}\\Big \\lbrace -\\left(\\delta _{ij}\\delta _{\\ell m}-\\delta _{im}\\delta _{j\\ell }\\right)X_m P_\\ell + \\left(\\delta _{ij}\\delta _{kn}-\\delta _{in}\\delta _{jk}\\right)X_k P_n \\Big \\rbrace ,$ where the first and third sentences of the first bracket are equal to zero, due to the product of symmetric and anti-symmetric arrays.", "Then, we can simplify if as $[ L_i, L_j]&=&\\frac{ 2i\\hbar \\beta }{(1-\\beta P^2)^2}\\left\\lbrace \\frac{i\\hbar \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{\\ell m}P_k P_n}{1-\\beta P^2}-\\frac{i\\hbar \\epsilon _{ik\\ell }\\epsilon _{jmn}\\delta _{kn}P_m P_\\ell }{1-\\beta P^2}\\right\\rbrace \\nonumber \\\\&&+\\frac{i\\hbar }{1 -\\beta P^2}\\Big \\lbrace -\\delta _{ij}\\delta _{\\ell m}X_m P_\\ell + \\delta _{im}\\delta _{j\\ell }X_m P_\\ell +\\delta _{ij}\\delta _{kn}X_k P_n-\\delta _{in}\\delta _{jk}X_k P_n\\Big \\rbrace \\nonumber \\\\&=&\\frac{ -2\\hbar ^{2} \\beta }{(1-\\beta P^2)^3}\\Big \\lbrace -\\epsilon _{ik\\ell }\\epsilon _{jn\\ell }P_k P_n+\\epsilon _{i\\ell k}\\epsilon _{jmk}P_m P_\\ell \\Big \\rbrace +\\frac{i\\hbar }{1 -\\beta P^2}\\Big \\lbrace -\\delta _{ij}X_\\ell P_\\ell +X_i P_j+-\\delta _{ij}X_k P_k-X_i P_j\\Big \\rbrace \\nonumber \\\\&=& \\frac{ -2\\hbar ^{2} \\beta }{(1-\\beta P^2)^3}\\Big \\lbrace -\\left(\\delta _{ij}\\delta _{kn}-\\delta _{in}\\delta _{kj}\\right)P_k P_n+ \\left(\\delta _{ij}\\delta _{\\ell m}-\\delta _{im}\\delta _{j\\ell }\\right)P_m P_\\ell \\Big \\rbrace +\\frac{i\\hbar }{1-\\beta P^2}\\left(X_i P_j-X_j P_i\\right)\\nonumber \\\\&=& \\frac{ -2\\hbar ^{2} \\beta }{(1-\\beta P^2)^3}\\Big \\lbrace -\\delta _{ij}P_k P_k +P_j P_i+\\delta _{ij}P_\\ell P_\\ell -P_i P_i\\Big \\rbrace +\\frac{i\\hbar }{1-\\beta P^2}\\left(X_i P_j-X_j P_i\\right),$ Finally, Eq.", "(REF ) will lead to $[ L_i, L_j]&=& \\frac{i\\hbar }{1-\\beta P^2}\\left(X_i P_j-X_j P_i\\right).$ The easiest way to derive the time-ordered perturbation expansion is to use the S-operator in the following form $S=U(\\infty ,-\\infty ),$ where with the Hamiltonian in the form of $H=H_0+V$ and in the presence of GUP (REF ) $U(\\tau ,\\tau _0)= \\exp \\left(\\frac{iH_{0}\\tau }{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\right)\\exp \\left(\\frac{iH(\\tau -\\tau _0)}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\right)\\exp \\left(\\frac{-iH_{0}\\tau _0}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\right).$ Now, differentiating Eq.", "(REF ) with respect to $\\tau $ gives the following differential equation $i\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s\\frac{\\partial U(\\tau ,\\tau _0)}{\\partial \\tau }= V(\\tau )U(\\tau ,\\tau _0),$ where $V(t)=\\exp \\left(\\frac{iH_{0}t}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\right) V \\exp \\left(\\frac{-iH_{0}t}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\right).$ Now, Eq.", "(REF ) as well as the initial condition $U(\\tau _0,\\tau _0) = 1$ is obviously satisfied by the solution of the integral equation as $U(\\tau ,\\tau _0) = 1- \\frac{i}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\int _{\\tau _0}^{\\tau }dt V(t)U(t,\\tau _0).$ Iterating this integral equation, we obtain an expansion for $U(\\tau ,\\tau _0)$ in powers of $V$ $U(\\tau ,\\tau _0) &=& 1- \\frac{i}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\int _{\\tau _0}^{\\tau }dt_1 V(t_1)+\\frac{(-i)^2}{\\hbar ^2\\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^{2s}}\\int _{\\tau _0}^{\\tau }dt_1 \\int _{\\tau _0}^{t_1}dt_2 V(t_1)V(t_2)\\nonumber \\\\&&+\\frac{(-i)^3}{\\hbar ^3\\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^{3s}}\\int _{\\tau _0}^{\\tau }dt_1 \\int _{\\tau _0}^{t_1}dt_2 \\int _{\\tau _0}^{t_2}dt_3V(t_1)V(t_2)V(t_3)+... .$ Now, if we set $\\tau =\\infty $ and $\\tau _0 = -\\infty $ , the perturbation expansion for the S-operator obtains as $S&=& 1- \\frac{i}{\\hbar \\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^s}\\int _{-\\infty }^{\\infty }dt_1 V(t_1)+\\frac{(-i)^2}{\\hbar ^2\\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^{2s}}\\int _{-\\infty }^{\\infty }dt_1 \\int _{-\\infty }^{t_1}dt_2 V(t_1)V(t_2)\\nonumber \\\\&&+\\frac{(-i)^3}{\\hbar ^3\\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^{3s}}\\int _{-\\infty }^{\\infty }dt_1 \\int _{-\\infty }^{t_1}dt_2 \\int _{-\\infty }^{t_2}dt_3V(t_1)V(t_2)V(t_3)+... .$ There is a way of rewriting Eq.", "(REF ) that proves very useful in carrying out manifestly Lorentz-invariant calculations.", "Define the time- ordered product of any time-dependent operators as the product with factors arranged so that the one with the latest time-argument is placed leftmost, the next-latest next to the leftmost, and so on.", "For instance, $T\\lbrace V(t)\\rbrace =V(t),$ $T\\lbrace V(t_1),V(t_2)\\rbrace =\\theta \\left(t_1-t_2\\right)V(t_1)V(t_2)+\\theta \\left(t_2-t_1\\right)V(t_2)V(t_1),$ and so on, where $\\theta (\\tau )$ is the step function, i.e., $\\theta (\\tau ) ={\\left\\lbrace \\begin{array}{ll}+1 ~~~ \\tau > 0,\\\\0 ~~~~~~ \\tau < 0.\\end{array}\\right.", "}$ The time-ordered product of $n$ $Vs$ is a sum over all $n!$ permutations of the $Vs$ , each of which gives the same integral over all $t_1...t_n$ .", "Hence, Eq.", "(REF ) can be written in the following form $S&=& 1+\\sum _{n=1}^{\\infty }\\frac{(-i)^n}{\\hbar ^n\\left(1-(-\\hbar ^2 \\beta \\nabla ^2)^a\\right)^{sn} n!", "}\\int _{-\\infty }^{\\infty }dt_1 dt_2...dt_n T\\lbrace V(t_1)V(t_2)...V(t_n)\\rbrace ,$ which is called modified Dyson series." ] ]

[ [ "Language Models are Few-Shot Learners" ], [ "Abstract Recent work has demonstrated substantial gains on many NLP tasks and benchmarks by pre-training on a large corpus of text followed by fine-tuning on a specific task.", "While typically task-agnostic in architecture, this method still requires task-specific fine-tuning datasets of thousands or tens of thousands of examples.", "By contrast, humans can generally perform a new language task from only a few examples or from simple instructions - something which current NLP systems still largely struggle to do.", "Here we show that scaling up language models greatly improves task-agnostic, few-shot performance, sometimes even reaching competitiveness with prior state-of-the-art fine-tuning approaches.", "Specifically, we train GPT-3, an autoregressive language model with 175 billion parameters, 10x more than any previous non-sparse language model, and test its performance in the few-shot setting.", "For all tasks, GPT-3 is applied without any gradient updates or fine-tuning, with tasks and few-shot demonstrations specified purely via text interaction with the model.", "GPT-3 achieves strong performance on many NLP datasets, including translation, question-answering, and cloze tasks, as well as several tasks that require on-the-fly reasoning or domain adaptation, such as unscrambling words, using a novel word in a sentence, or performing 3-digit arithmetic.", "At the same time, we also identify some datasets where GPT-3's few-shot learning still struggles, as well as some datasets where GPT-3 faces methodological issues related to training on large web corpora.", "Finally, we find that GPT-3 can generate samples of news articles which human evaluators have difficulty distinguishing from articles written by humans.", "We discuss broader societal impacts of this finding and of GPT-3 in general." ], [ "Introduction", "Recent years have featured a trend towards pre-trained language representations in NLP systems, applied in increasingly flexible and task-agnostic ways for downstream transfer.", "First, single-layer representations were learned using word vectors [82], [102] and fed to task-specific architectures, then RNNs with multiple layers of representations and contextual state were used to form stronger representations [24], [81], [100] (though still applied to task-specific architectures), and more recently pre-trained recurrent or transformer language models [134] have been directly fine-tuned, entirely removing the need for task-specific architectures [112], [20], [43].", "Figure: Language model meta-learning.", "During unsupervised pre-training, a language model develops a broad set of skills and pattern recognition abilities.", "It then uses these abilities at inference time to rapidly adapt to or recognize the desired task.", "We use the term “in-context learning\" to describe the inner loop of this process, which occurs within the forward-pass upon each sequence.", "The sequences in this diagram are not intended to be representative of the data a model would see during pre-training, but are intended to show that there are sometimes repeated sub-tasks embedded within a single sequence.This last paradigm has led to substantial progress on many challenging NLP tasks such as reading comprehension, question answering, textual entailment, and many others, and has continued to advance based on new architectures and algorithms [116], [74], [139], [62].", "However, a major limitation to this approach is that while the architecture is task-agnostic, there is still a need for task-specific datasets and task-specific fine-tuning: to achieve strong performance on a desired task typically requires fine-tuning on a dataset of thousands to hundreds of thousands of examples specific to that task.", "Removing this limitation would be desirable, for several reasons.", "First, from a practical perspective, the need for a large dataset of labeled examples for every new task limits the applicability of language models.", "There exists a very wide range of possible useful language tasks, encompassing anything from correcting grammar, to generating examples of an abstract concept, to critiquing a short story.", "For many of these tasks it is difficult to collect a large supervised training dataset, especially when the process must be repeated for every new task.", "Second, the potential to exploit spurious correlations in training data fundamentally grows with the expressiveness of the model and the narrowness of the training distribution.", "This can create problems for the pre-training plus fine-tuning paradigm, where models are designed to be large to absorb information during pre-training, but are then fine-tuned on very narrow task distributions.", "For instance [41] observe that larger models do not necessarily generalize better out-of-distribution.", "There is evidence that suggests that the generalization achieved under this paradigm can be poor because the model is overly specific to the training distribution and does not generalize well outside it [138], [88].", "Thus, the performance of fine-tuned models on specific benchmarks, even when it is nominally at human-level, may exaggerate actual performance on the underlying task [36], [91].", "Third, humans do not require large supervised datasets to learn most language tasks – a brief directive in natural language (e.g.", "“please tell me if this sentence describes something happy or something sad”) or at most a tiny number of demonstrations (e.g.", "“here are two examples of people acting brave; please give a third example of bravery”) is often sufficient to enable a human to perform a new task to at least a reasonable degree of competence.", "Aside from pointing to a conceptual limitation in our current NLP techniques, this adaptability has practical advantages – it allows humans to seamlessly mix together or switch between many tasks and skills, for example performing addition during a lengthy dialogue.", "To be broadly useful, we would someday like our NLP systems to have this same fluidity and generality.", "One potential route towards addressing these issues is meta-learningIn the context of language models this has sometimes been called “zero-shot transfer”, but this term is potentially ambiguous: the method is “zero-shot” in the sense that no gradient updates are performed, but it often involves providing inference-time demonstrations to the model, so is not truly learning from zero examples.", "To avoid this confusion, we use the term “meta-learning” to capture the inner-loop / outer-loop structure of the general method, and the term “in context-learning\" to refer to the inner loop of meta-learning.", "We further specialize the description to “zero-shot\", “one-shot\", or “few-shot\" depending on how many demonstrations are provided at inference time.", "These terms are intended to remain agnostic on the question of whether the model learns new tasks from scratch at inference time or simply recognizes patterns seen during training – this is an important issue which we discuss later in the paper, but “meta-learning” is intended to encompass both possibilities, and simply describes the inner-outer loop structure.", "– which in the context of language models means the model develops a broad set of skills and pattern recognition abilities at training time, and then uses those abilities at inference time to rapidly adapt to or recognize the desired task (illustrated in Figure REF ).", "Recent work [117] attempts to do this via what we call “in-context learning\", using the text input of a pretrained language model as a form of task specification: the model is conditioned on a natural language instruction and/or a few demonstrations of the task and is then expected to complete further instances of the task simply by predicting what comes next.", "While it has shown some initial promise, this approach still achieves results far inferior to fine-tuning – for example [117] achieves only 4% on Natural Questions, and even its 55 F1 CoQa result is now more than 35 points behind the state of the art.", "Meta-learning clearly requires substantial improvement in order to be viable as a practical method of solving language tasks.", "Another recent trend in language modeling may offer a way forward.", "In recent years the capacity of transformer language models has increased substantially, from 100 million parameters [112], to 300 million parameters [20], to 1.5 billion parameters [117], to 8 billion parameters [125], 11 billion parameters [116], and finally 17 billion parameters [132].", "Each increase has brought improvements in text synthesis and/or downstream NLP tasks, and there is evidence suggesting that log loss, which correlates well with many downstream tasks, follows a smooth trend of improvement with scale [57].", "Since in-context learning involves absorbing many skills and tasks within the parameters of the model, it is plausible that in-context learning abilities might show similarly strong gains with scale.", "Figure: Larger models make increasingly efficient use of in-context information.", "We show in-context learning performance on a simple task requiring the model to remove random symbols from a word, both with and without a natural language task description (see Sec.", ").", "The steeper “in-context learning curves” for large models demonstrate improved ability to learn a task from contextual information.", "We see qualitatively similar behavior across a wide range of tasks.Figure: Aggregate performance for all 42 accuracy-denominated benchmarks   While zero-shot performance improves steadily with model size, few-shot performance increases more rapidly, demonstrating that larger models are more proficient at in-context learning.", "See Figure for a more detailed analysis on SuperGLUE, a standard NLP benchmark suite.In this paper, we test this hypothesis by training a 175 billion parameter autoregressive language model, which we call GPT-3, and measuring its in-context learning abilities.", "Specifically, we evaluate GPT-3 on over two dozen NLP datasets, as well as several novel tasks designed to test rapid adaptation to tasks unlikely to be directly contained in the training set.", "For each task, we evaluate GPT-3 under 3 conditions: (a) “few-shot learning”, or in-context learning where we allow as many demonstrations as will fit into the model’s context window (typically 10 to 100), (b) “one-shot learning”, where we allow only one demonstration, and (c) “zero-shot” learning, where no demonstrations are allowed and only an instruction in natural language is given to the model.", "GPT-3 could also in principle be evaluated in the traditional fine-tuning setting, but we leave this to future work.", "Figure REF illustrates the conditions we study, and shows few-shot learning of a simple task requiring the model to remove extraneous symbols from a word.", "Model performance improves with the addition of a natural language task description, and with the number of examples in the model's context, $K$ .", "Few-shot learning also improves dramatically with model size.", "Though the results in this case are particularly striking, the general trends with both model size and number of examples in-context hold for most tasks we study.", "We emphasize that these “learning” curves involve no gradient updates or fine-tuning, just increasing numbers of demonstrations given as conditioning.", "Broadly, on NLP tasks GPT-3 achieves promising results in the zero-shot and one-shot settings, and in the the few-shot setting is sometimes competitive with or even occasionally surpasses state-of-the-art (despite state-of-the-art being held by fine-tuned models).", "For example, GPT-3 achieves 81.5 F1 on CoQA in the zero-shot setting, 84.0 F1 on CoQA in the one-shot setting, 85.0 F1 in the few-shot setting.", "Similarly, GPT-3 achieves 64.3% accuracy on TriviaQA in the zero-shot setting, 68.0% in the one-shot setting, and 71.2% in the few-shot setting, the last of which is state-of-the-art relative to fine-tuned models operating in the same closed-book setting.", "GPT-3 also displays one-shot and few-shot proficiency at tasks designed to test rapid adaption or on-the-fly reasoning, which include unscrambling words, performing arithmetic, and using novel words in a sentence after seeing them defined only once.", "We also show that in the few-shot setting, GPT-3 can generate synthetic news articles which human evaluators have difficulty distinguishing from human-generated articles.", "At the same time, we also find some tasks on which few-shot performance struggles, even at the scale of GPT-3.", "This includes natural language inference tasks like the ANLI dataset, and some reading comprehension datasets like RACE or QuAC.", "By presenting a broad characterization of GPT-3's strengths and weaknesses, including these limitations, we hope to stimulate study of few-shot learning in language models and draw attention to where progress is most needed.", "A heuristic sense of the overall results can be seen in Figure REF , which aggregates the various tasks (though it should not be seen as a rigorous or meaningful benchmark in itself).", "We also undertake a systematic study of “data contamination” – a growing problem when training high capacity models on datasets such as Common Crawl, which can potentially include content from test datasets simply because such content often exists on the web.", "In this paper we develop systematic tools to measure data contamination and quantify its distorting effects.", "Although we find that data contamination has a minimal effect on GPT-3's performance on most datasets, we do identify a few datasets where it could be inflating results, and we either do not report results on these datasets or we note them with an asterisk, depending on the severity.", "In addition to all the above, we also train a series of smaller models (ranging from 125 million parameters to 13 billion parameters) in order to compare their performance to GPT-3 in the zero, one and few-shot settings.", "Broadly, for most tasks we find relatively smooth scaling with model capacity in all three settings; one notable pattern is that the gap between zero-, one-, and few-shot performance often grows with model capacity, perhaps suggesting that larger models are more proficient meta-learners.", "Finally, given the broad spectrum of capabilities displayed by GPT-3, we discuss concerns about bias, fairness, and broader societal impacts, and attempt a preliminary analysis of GPT-3's characteristics in this regard.", "The remainder of this paper is organized as follows.", "In Section , we describe our approach and methods for training GPT-3 and evaluating it.", "Section presents results on the full range of tasks in the zero-, one- and few-shot settings.", "Section addresses questions of data contamination (train-test overlap).", "Section discusses limitations of GPT-3.", "Section discusses broader impacts.", "Section reviews related work and Section concludes." ], [ "Approach", "Our basic pre-training approach, including model, data, and training, is similar to the process described in [117], with relatively straightforward scaling up of the model size, dataset size and diversity, and length of training.", "Our use of in-context learning is also similar to [117], but in this work we systematically explore different settings for learning within the context.", "Therefore, we start this section by explicitly defining and contrasting the different settings that we will be evaluating GPT-3 on or could in principle evaluate GPT-3 on.", "These settings can be seen as lying on a spectrum of how much task-specific data they tend to rely on.", "Specifically, we can identify at least four points on this spectrum (see Figure REF for an illustration): Fine-Tuning (FT) has been the most common approach in recent years, and involves updating the weights of a pre-trained model by training on a supervised dataset specific to the desired task.", "Typically thousands to hundreds of thousands of labeled examples are used.", "The main advantage of fine-tuning is strong performance on many benchmarks.", "The main disadvantages are the need for a new large dataset for every task, the potential for poor generalization out-of-distribution [88], and the potential to exploit spurious features of the training data [36], [91], potentially resulting in an unfair comparison with human performance.", "In this work we do not fine-tune GPT-3 because our focus is on task-agnostic performance, but GPT-3 can be fine-tuned in principle and this is a promising direction for future work.", "Few-Shot (FS) is the term we will use in this work to refer to the setting where the model is given a few demonstrations of the task at inference time as conditioning [117], but no weight updates are allowed.", "As shown in Figure REF , for a typical dataset an example has a context and a desired completion (for example an English sentence and the French translation), and few-shot works by giving $K$ examples of context and completion, and then one final example of context, with the model expected to provide the completion.", "We typically set $K$ in the range of 10 to 100 as this is how many examples can fit in the model’s context window ($n_{\\mathrm {ctx}}=2048$ ).", "The main advantages of few-shot are a major reduction in the need for task-specific data and reduced potential to learn an overly narrow distribution from a large but narrow fine-tuning dataset.", "The main disadvantage is that results from this method have so far been much worse than state-of-the-art fine-tuned models.", "Also, a small amount of task specific data is still required.", "As indicated by the name, few-shot learning as described here for language models is related to few-shot learning as used in other contexts in ML [45], [133] – both involve learning based on a broad distribution of tasks (in this case implicit in the pre-training data) and then rapidly adapting to a new task.", "One-Shot (1S) is the same as few-shot except that only one demonstration is allowed, in addition to a natural language description of the task, as shown in Figure 1.", "The reason to distinguish one-shot from few-shot and zero-shot (below) is that it most closely matches the way in which some tasks are communicated to humans.", "For example, when asking humans to generate a dataset on a human worker service (for example Mechanical Turk), it is common to give one demonstration of the task.", "By contrast it is sometimes difficult to communicate the content or format of a task if no examples are given.", "Zero-Shot (0S) is the same as one-shot except that no demonstrations are allowed, and the model is only given a natural language instruction describing the task.", "This method provides maximum convenience, potential for robustness, and avoidance of spurious correlations (unless they occur very broadly across the large corpus of pre-training data), but is also the most challenging setting.", "In some cases it may even be difficult for humans to understand the format of the task without prior examples, so this setting is in some cases “unfairly hard”.", "For example, if someone is asked to “make a table of world records for the 200m dash”, this request can be ambiguous, as it may not be clear exactly what format the table should have or what should be included (and even with careful clarification, understanding precisely what is desired can be difficult).", "Nevertheless, for at least some settings zero-shot is closest to how humans perform tasks – for example, in the translation example in Figure REF , a human would likely know what to do from just the text instruction.", "Figure REF shows the four methods using the example of translating English to French.", "In this paper we focus on zero-shot, one-shot and few-shot, with the aim of comparing them not as competing alternatives, but as different problem settings which offer a varying trade-off between performance on specific benchmarks and sample efficiency.", "We especially highlight the few-shot results as many of them are only slightly behind state-of-the-art fine-tuned models.", "Ultimately, however, one-shot, or even sometimes zero-shot, seem like the fairest comparisons to human performance, and are important targets for future work.", "Sections REF -REF below give details on our models, training data, and training process respectively.", "Section REF discusses the details of how we do few-shot, one-shot, and zero-shot evaluations." ], [ "Model and Architectures", "We use the same model and architecture as GPT-2 [117], including the modified initialization, pre-normalization, and reversible tokenization described therein, with the exception that we use alternating dense and locally banded sparse attention patterns in the layers of the transformer, similar to the Sparse Transformer [15].", "To study the dependence of ML performance on model size, we train 8 different sizes of model, ranging over three orders of magnitude from 125 million parameters to 175 billion parameters, with the last being the model we call GPT-3.", "Previous work [57] suggests that with enough training data, scaling of validation loss should be approximately a smooth power law as a function of size; training models of many different sizes allows us to test this hypothesis both for validation loss and for downstream language tasks.", "Table: Sizes, architectures, and learning hyper-parameters (batch size in tokens and learning rate) of the models which we trained.", "All models were trained for a total of 300 billion tokens.Table REF shows the sizes and architectures of our 8 models.", "Here $n_{\\mathrm {params}}$ is the total number of trainable parameters, $n_{\\mathrm {layers}}$ is the total number of layers, $d_{\\mathrm {model}}$ is the number of units in each bottleneck layer (we always have the feedforward layer four times the size of the bottleneck layer, $d_{\\mathrm {ff}}$ $= 4 \\ast d_{\\mathrm {model}}$ ), and $d_{\\mathrm {head}}$ is the dimension of each attention head.", "All models use a context window of $n_{\\mathrm {ctx}}=2048$ tokens.", "We partition the model across GPUs along both the depth and width dimension in order to minimize data-transfer between nodes.", "The precise architectural parameters for each model are chosen based on computational efficiency and load-balancing in the layout of models across GPU’s.", "Previous work [57] suggests that validation loss is not strongly sensitive to these parameters within a reasonably broad range.", "Figure: Total compute used during training.", "Based on the analysis in Scaling Laws For Neural Language Models we train much larger models on many fewer tokens than is typical.", "As a consequence, although GPT-3 3B is almost 10x larger than RoBERTa-Large (355M params), both models took roughly 50 petaflop/s-days of compute during pre-training.", "Methodology for these calculations can be found in Appendix ." ], [ "Training Dataset", "Datasets for language models have rapidly expanded, culminating in the Common Crawl datasethttps://commoncrawl.org/the-data/ [116] constituting nearly a trillion words.", "This size of dataset is sufficient to train our largest models without ever updating on the same sequence twice.", "However, we have found that unfiltered or lightly filtered versions of Common Crawl tend to have lower quality than more curated datasets.", "Therefore, we took 3 steps to improve the average quality of our datasets: (1) we downloaded and filtered a version of CommonCrawl based on similarity to a range of high-quality reference corpora, (2) we performed fuzzy deduplication at the document level, within and across datasets, to prevent redundancy and preserve the integrity of our held-out validation set as an accurate measure of overfitting, and (3) we also added known high-quality reference corpora to the training mix to augment CommonCrawl and increase its diversity.", "Details of the first two points (processing of Common Crawl) are described in Appendix .", "For the third, we added several curated high-quality datasets, including an expanded version of the WebText dataset [117], collected by scraping links over a longer period of time, and first described in [57], two internet-based books corpora (Books1 and Books2) and English-language Wikipedia.", "Table REF shows the final mixture of datasets that we used in training.", "The CommonCrawl data was downloaded from 41 shards of monthly CommonCrawl covering 2016 to 2019, constituting 45TB of compressed plaintext before filtering and 570GB after filtering, roughly equivalent to 400 billion byte-pair-encoded tokens.", "Note that during training, datasets are not sampled in proportion to their size, but rather datasets we view as higher-quality are sampled more frequently, such that CommonCrawl and Books2 datasets are sampled less than once during training, but the other datasets are sampled 2-3 times.", "This essentially accepts a small amount of overfitting in exchange for higher quality training data.", "Table: Datasets used to train GPT-3.", "“Weight in training mix” refers to the fraction of examples during training that are drawn from a given dataset, which we intentionally do not make proportional to the size of the dataset.", "As a result, when we train for 300 billion tokens, some datasets are seen up to 3.4 times during training while other datasets are seen less than once.A major methodological concern with language models pretrained on a broad swath of internet data, particularly large models with the capacity to memorize vast amounts of content, is potential contamination of downstream tasks by having their test or development sets inadvertently seen during pre-training.", "To reduce such contamination, we searched for and attempted to remove any overlaps with the development and test sets of all benchmarks studied in this paper.", "Unfortunately, a bug in the filtering caused us to ignore some overlaps, and due to the cost of training it was not feasible to retrain the model.", "In Section we characterize the impact of the remaining overlaps, and in future work we will more aggressively remove data contamination." ], [ "Training Process", "As found in [57], [85], larger models can typically use a larger batch size, but require a smaller learning rate.", "We measure the gradient noise scale during training and use it to guide our choice of batch size [85].", "Table REF shows the parameter settings we used.", "To train the larger models without running out of memory, we use a mixture of model parallelism within each matrix multiply and model parallelism across the layers of the network.", "All models were trained on V100 GPU’s on part of a high-bandwidth cluster provided by Microsoft.", "Details of the training process and hyperparameter settings are described in Appendix ." ], [ "Evaluation", "For few-shot learning, we evaluate each example in the evaluation set by randomly drawing $K$ examples from that task’s training set as conditioning, delimited by 1 or 2 newlines depending on the task.", "For LAMBADA and Storycloze there is no supervised training set available so we draw conditioning examples from the development set and evaluate on the test set.", "For Winograd (the original, not SuperGLUE version) there is only one dataset, so we draw conditioning examples directly from it.", "$K$ can be any value from 0 to the maximum amount allowed by the model’s context window, which is $n_{\\mathrm {ctx}}=2048$ for all models and typically fits 10 to 100 examples.", "Larger values of $K$ are usually but not always better, so when a separate development and test set are available, we experiment with a few values of $K$ on the development set and then run the best value on the test set.", "For some tasks (see Appendix ) we also use a natural language prompt in addition to (or for $K=0$ , instead of) demonstrations.", "On tasks that involve choosing one correct completion from several options (multiple choice), we provide $K$ examples of context plus correct completion, followed by one example of context only, and compare the LM likelihood of each completion.", "For most tasks we compare the per-token likelihood (to normalize for length), however on a small number of datasets (ARC, OpenBookQA, and RACE) we gain additional benefit as measured on the development set by normalizing by the unconditional probability of each completion, by computing $\\frac{P(\\mathrm {completion} | \\mathrm {context})}{P(\\mathrm {completion} | \\mathrm {answer\\_context})}$ , where $\\mathrm {answer\\_context}$ is the string \"Answer: \" or \"A: \" and is used to prompt that the completion should be an answer but is otherwise generic.", "On tasks that involve binary classification, we give the options more semantically meaningful names (e.g.", "“True\" or “False\" rather than 0 or 1) and then treat the task like multiple choice; we also sometimes frame the task similar to what is done by [116] (see Appendix ) for details.", "On tasks with free-form completion, we use beam search with the same parameters as [116]: a beam width of 4 and a length penalty of $\\alpha = 0.6$ .", "We score the model using F1 similarity score, BLEU, or exact match, depending on what is standard for the dataset at hand.", "Final results are reported on the test set when publicly available, for each model size and learning setting (zero-, one-, and few-shot).", "When the test set is private, our model is often too large to fit on the test server, so we report results on the development set.", "We do submit to the test server on a small number of datasets (SuperGLUE, TriviaQA, PiQa) where we were able to make submission work, and we submit only the 200B few-shot results, and report development set results for everything else." ], [ "Results", "In Figure REF we display training curves for the 8 models described in Section .", "For this graph we also include 6 additional extra-small models with as few as 100,000 parameters.", "As observed in [57], language modeling performance follows a power-law when making efficient use of training compute.", "After extending this trend by two more orders of magnitude, we observe only a slight (if any) departure from the power-law.", "One might worry that these improvements in cross-entropy loss come only from modeling spurious details of our training corpus.", "However, we will see in the following sections that improvements in cross-entropy loss lead to consistent performance gains across a broad spectrum of natural language tasks.", "Below, we evaluate the 8 models described in Section (the 175 billion parameter parameter GPT-3 and 7 smaller models) on a wide range of datasets.", "We group the datasets into 9 categories representing roughly similar tasks.", "In Section REF we evaluate on traditional language modeling tasks and tasks that are similar to language modeling, such as Cloze tasks and sentence/paragraph completion tasks.", "In Section REF we evaluate on “closed book” question answering tasks: tasks which require using the information stored in the model’s parameters to answer general knowledge questions.", "In Section REF we evaluate the model’s ability to translate between languages (especially one-shot and few-shot).", "In Section REF we evaluate the model’s performance on Winograd Schema-like tasks.", "In Section REF we evaluate on datasets that involve commonsense reasoning or question answering.", "In Section REF we evaluate on reading comprehension tasks, in Section REF we evaluate on the SuperGLUE benchmark suite, and in REF we briefly explore NLI.", "Finally, in Section REF , we invent some additional tasks designed especially to probe in-context learning abilities – these tasks focus on on-the-fly reasoning, adaptation skills, or open-ended text synthesis.", "We evaluate all tasks in the few-shot, one-shot, and zero-shot settings." ], [ "Language Modeling, Cloze, and Completion Tasks", "In this section we test GPT-3’s performance on the traditional task of language modeling, as well as related tasks that involve predicting a single word of interest, completing a sentence or paragraph, or choosing between possible completions of a piece of text." ], [ "Language Modeling", "We calculate zero-shot perplexity on the Penn Tree Bank (PTB) [86] dataset measured in [117].", "We omit the 4 Wikipedia-related tasks in that work because they are entirely contained in our training data, and we also omit the one-billion word benchmark due to a high fraction of the dataset being contained in our training set.", "PTB escapes these issues due to predating the modern internet.", "Our largest model sets a new SOTA on PTB by a substantial margin of 15 points, achieving a perplexity of 20.50.", "Note that since PTB is a traditional language modeling dataset it does not have a clear separation of examples to define one-shot or few-shot evaluation around, so we measure only zero-shot.", "Table: Zero-shot results on PTB language modeling dataset.", "Many other common language modeling datasets are omitted because they are derived from Wikipedia or other sources which are included in GPT-3's training data.", "a" ], [ "LAMBADA", "The LAMBADA dataset [99] tests the modeling of long-range dependencies in text – the model is asked to predict the last word of sentences which require reading a paragraph of context.", "It has recently been suggested that the continued scaling of language models is yielding diminishing returns on this difficult benchmark.", "[9] reflect on the small 1.5% improvement achieved by a doubling of model size between two recent state of the art results ([125] and [132]) and argue that “continuing to expand hardware and data sizes by orders of magnitude is not the path forward”.", "We find that path is still promising and in a zero-shot setting GPT-3 achieves 76% on LAMBADA, a gain of 8% over the previous state of the art.", "LAMBADA is also a demonstration of the flexibility of few-shot learning as it provides a way to address a problem that classically occurs with this dataset.", "Although the completion in LAMBADA is always the last word in a sentence, a standard language model has no way of knowing this detail.", "It thus assigns probability not only to the correct ending but also to other valid continuations of the paragraph.", "This problem has been partially addressed in the past with stop-word filters [117] (which ban “continuation” words).", "The few-shot setting instead allows us to “frame” the task as a cloze-test and allows the language model to infer from examples that a completion of exactly one word is desired.", "We use the following fill-in-the-blank format:                      Alice was friends with Bob.", "Alice went to visit her friend        .", "$\\rightarrow $ Bob                      George bought some baseball equipment, a ball, a glove, and a        .", "$\\rightarrow $ When presented with examples formatted this way, GPT-3 achieves 86.4% accuracy in the few-shot setting, an increase of over 18% from the previous state-of-the-art.", "We observe that few-shot performance improves strongly with model size.", "While this setting decreases the performance of the smallest model by almost 20%, for GPT-3 it improves accuracy by 10%.", "Finally, the fill-in-blank method is not effective one-shot, where it always performs worse than the zero-shot setting.", "Perhaps this is because all models still require several examples to recognize the pattern.", "One note of caution is that an analysis of test set contamination identified that a significant minority of the LAMBADA dataset appears to be present in our training data – however analysis performed in Section suggests negligible impact on performance." ], [ "HellaSwag", "The HellaSwag dataset [140] involves picking the best ending to a story or set of instructions.", "The examples were adversarially mined to be difficult for language models while remaining easy for humans (who achieve 95.6% accuracy).", "GPT-3 achieves 78.1% accuracy in the one-shot setting and 79.3% accuracy in the few-shot setting, outperforming the 75.4% accuracy of a fine-tuned 1.5B parameter language model [141] but still a fair amount lower than the overall SOTA of 85.6% achieved by the fine-tuned multi-task model ALUM." ], [ "StoryCloze", "We next evaluate GPT-3 on the StoryCloze 2016 dataset [83], which involves selecting the correct ending sentence for five-sentence long stories.", "Here GPT-3 achieves 83.2% in the zero-shot setting and 87.7% in the few-shot setting (with $K=70$ ).", "This is still 4.1% lower than the fine-tuned SOTA using a BERT based model [64] but improves over previous zero-shot results by roughly 10%." ], [ "Closed Book Question Answering", "In this section we measure GPT-3’s ability to answer questions about broad factual knowledge.", "Due to the immense amount of possible queries, this task has normally been approached by using an information retrieval system to find relevant text in combination with a model which learns to generate an answer given the question and the retrieved text.", "Since this setting allows a system to search for and condition on text which potentially contains the answer it is denoted “open-book”.", "[115] recently demonstrated that a large language model can perform surprisingly well directly answering the questions without conditioning on auxilliary information.", "They denote this more restrictive evaluation setting as “closed-book”.", "Their work suggests that even higher-capacity models could perform even better and we test this hypothesis with GPT-3.", "We evaluate GPT-3 on the 3 datasets in [115]: Natural Questions [58], WebQuestions [5], and TriviaQA [49], using the same splits.", "Note that in addition to all results being in the closed-book setting, our use of few-shot, one-shot, and zero-shot evaluations represent an even stricter setting than previous closed-book QA work: in addition to external content not being allowed, fine-tuning on the Q&A dataset itself is also not permitted.", "The results for GPT-3 are shown in Table REF .", "On TriviaQA, we achieve 64.3% in the zero-shot setting, 68.0% in the one-shot setting, and 71.2% in the few-shot setting.", "The zero-shot result already outperforms the fine-tuned T5-11B by 14.2%, and also outperforms a version with Q&A tailored span prediction during pre-training by 3.8%.", "The one-shot result improves by 3.7% and matches the SOTA for an open-domain QA system which not only fine-tunes but also makes use of a learned retrieval mechanism over a 15.3B parameter dense vector index of 21M documents [75].", "GPT-3's few-shot result further improves performance another 3.2% beyond this.", "On WebQuestions (WebQs), GPT-3 achieves 14.4% in the zero-shot setting, 25.3% in the one-shot setting, and 41.5% in the few-shot setting.", "This compares to 37.4% for fine-tuned T5-11B, and 44.7% for fine-tuned T5-11B+SSM, which uses a Q&A-specific pre-training procedure.", "GPT-3 in the few-shot setting approaches the performance of state-of-the-art fine-tuned models.", "Notably, compared to TriviaQA, WebQS shows a much larger gain from zero-shot to few-shot (and indeed its zero-shot and one-shot performance are poor), perhaps suggesting that the WebQs questions and/or the style of their answers are out-of-distribution for GPT-3.", "Nevertheless, GPT-3 appears able to adapt to this distribution, recovering strong performance in the few-shot setting.", "On Natural Questions (NQs) GPT-3 achieves 14.6% in the zero-shot setting, 23.0% in the one-shot setting, and 29.9% in the few-shot setting, compared to 36.6% for fine-tuned T5 11B+SSM.", "Similar to WebQS, the large gain from zero-shot to few-shot may suggest a distribution shift, and may also explain the less competitive performance compared to TriviaQA and WebQS.", "In particular, the questions in NQs tend towards very fine-grained knowledge on Wikipedia specifically which could be testing the limits of GPT-3's capacity and broad pretraining distribution.", "Overall, on one of the three datasets GPT-3's one-shot matches the open-domain fine-tuning SOTA.", "On the other two datasets it approaches the performance of the closed-book SOTA despite not using fine-tuning.", "On all 3 datasets, we find that performance scales very smoothly with model size (Figure REF and Appendix Figure REF ), possibly reflecting the idea that model capacity translates directly to more ‘knowledge’ absorbed in the parameters of the model." ], [ "Translation", "For GPT-2 a filter was used on a multilingual collection of documents to produce an English only dataset due to capacity concerns.", "Even with this filtering GPT-2 showed some evidence of multilingual capability and performed non-trivially when translating between French and English despite only training on 10 megabytes of remaining French text.", "Since we increase the capacity by over two orders of magnitude from GPT-2 to GPT-3, we also expand the scope of the training dataset to include more representation of other languages, though this remains an area for further improvement.", "As discussed in REF the majority of our data is derived from raw Common Crawl with only quality-based filtering.", "Although GPT-3's training data is still primarily English (93% by word count), it also includes 7% of text in other languages.", "These languages are documented in the supplemental material.", "In order to better understand translation capability, we also expand our analysis to include two additional commonly studied languages, German and Romanian.", "Existing unsupervised machine translation approaches often combine pretraining on a pair of monolingual datasets with back-translation [123] to bridge the two languages in a controlled way.", "By contrast, GPT-3 learns from a blend of training data that mixes many languages together in a natural way, combining them on a word, sentence, and document level.", "GPT-3 also uses a single training objective which is not customized or designed for any task in particular.", "However, our one / few-shot settings aren't strictly comparable to prior unsupervised work since they make use of a small amount of paired examples (1 or 64).", "This corresponds to up to a page or two of in-context training data.", "Table: Few-shot GPT-3 outperforms previous unsupervised NMT work by 5 BLEU when translating into English reflecting its strength as an English LM.", "We report BLEU scores on the WMT'14 Fr↔\\leftrightarrow En, WMT’16 De↔\\leftrightarrow En, and WMT'16 Ro↔\\leftrightarrow En datasets as measured by multi-bleu.perl with XLM's tokenization in order to compare most closely with prior unsupervised NMT work.", "SacreBLEUf results reported in Appendix .", "Underline indicates an unsupervised or few-shot SOTA, bold indicates supervised SOTA with relative confidence.abcdef[SacreBLEU signature: BLEU+case.mixed+numrefs.1+smooth.exp+tok.intl+version.1.2.20]Figure: Few-shot translation performance on 6 language pairs as model capacity increases.", "There is a consistent trend of improvement across all datasets as the model scales, and as well as tendency for translation into English to be stronger than translation from English.Results are shown in Table REF .", "Zero-shot GPT-3, which only receives on a natural language description of the task, still underperforms recent unsupervised NMT results.", "However, providing only a single example demonstration for each translation task improves performance by over 7 BLEU and nears competitive performance with prior work.", "GPT-3 in the full few-shot setting further improves another 4 BLEU resulting in similar average performance to prior unsupervised NMT work.", "GPT-3 has a noticeable skew in its performance depending on language direction.", "For the three input languages studied, GPT-3 significantly outperforms prior unsupervised NMT work when translating into English but underperforms when translating in the other direction.", "Performance on En-Ro is a noticeable outlier at over 10 BLEU worse than prior unsupervised NMT work.", "This could be a weakness due to reusing the byte-level BPE tokenizer of GPT-2 which was developed for an almost entirely English training dataset.", "For both Fr-En and De-En, few shot GPT-3 outperforms the best supervised result we could find but due to our unfamiliarity with the literature and the appearance that these are un-competitive benchmarks we do not suspect those results represent true state of the art.", "For Ro-En, few shot GPT-3 performs within 0.5 BLEU of the overall SOTA which is achieved by a combination of unsupervised pretraining, supervised finetuning on 608K labeled examples, and backtranslation [70].", "Finally, across all language pairs and across all three settings (zero-, one-, and few-shot), there is a smooth trend of improvement with model capacity.", "This is shown in Figure REF in the case of few-shot results, and scaling for all three settings is shown in Appendix ." ], [ "Winograd-Style Tasks", "The Winograd Schemas Challenge [65] is a classical task in NLP that involves determining which word a pronoun refers to, when the pronoun is grammatically ambiguous but semantically unambiguous to a human.", "Recently fine-tuned language models have achieved near-human performance on the original Winograd dataset, but more difficult versions such as the adversarially-mined Winogrande dataset [118] still significantly lag human performance.", "We test GPT-3’s performance on both Winograd and Winogrande, as usual in the zero-, one-, and few-shot setting.", "On Winograd we test GPT-3 on the original set of 273 Winograd schemas, using the same “partial evaluation” method described in [117].", "Note that this setting differs slightly from the WSC task in the SuperGLUE benchmark, which is presented as binary classification and requires entity extraction to convert to the form described in this section.", "On Winograd GPT-3 achieves 88.3%, 89.7%, and 88.6% in the zero-shot, one-shot, and few-shot settings, showing no clear in-context learning but in all cases achieving strong results just a few points below state-of-the-art and estimated human performance.", "We note that contamination analysis found some Winograd schemas in the training data but this appears to have only a small effect on results (see Section ).", "On the more difficult Winogrande dataset, we do find gains to in-context learning: GPT-3 achieves 70.2% in the zero-shot setting, 73.2% in the one-shot setting, and 77.7% in the few-shot setting.", "For comparison a fine-tuned RoBERTA model achieves 79%, state-of-the-art is 84.6% achieved with a fine-tuned high capacity model (T5), and human performance on the task as reported by [118] is 94.0%." ], [ "Common Sense Reasoning", "Next we consider three datasets which attempt to capture physical or scientific reasoning, as distinct from sentence completion, reading comprehension, or broad knowledge question answering.", "The first, PhysicalQA (PIQA) [11], asks common sense questions about how the physical world works and is intended as a probe of grounded understanding of the world.", "GPT-3 achieves 81.0% accuracy zero-shot, 80.5% accuracy one-shot, and 82.8% accuracy few-shot (the last measured on PIQA's test server).", "This compares favorably to the 79.4% accuracy prior state-of-the-art of a fine-tuned RoBERTa.", "PIQA shows relatively shallow scaling with model size and is still over 10% worse than human performance, but GPT-3's few-shot and even zero-shot result outperform the current state-of-the-art.", "Our analysis flagged PIQA for a potential data contamination issue (despite hidden test labels), and we therefore conservatively mark the result with an asterisk.", "See Section for details.", "ARC [14] is a dataset of multiple-choice questions collected from 3rd to 9th grade science exams.", "On the “Challenge” version of the dataset which has been filtered to questions which simple statistical or information retrieval methods are unable to correctly answer, GPT-3 achieves 51.4% accuracy in the zero-shot setting, 53.2% in the one-shot setting, and 51.5% in the few-shot setting.", "This is approaching the performance of a fine-tuned RoBERTa baseline (55.9%) from UnifiedQA [55].", "On the “Easy” version of the dataset (questions which either of the mentioned baseline approaches answered correctly), GPT-3 achieves 68.8%, 71.2%, and 70.1% which slightly exceeds a fine-tuned RoBERTa baseline from [55].", "However, both of these results are still much worse than the overall SOTAs achieved by the UnifiedQA which exceeds GPT-3’s few-shot results by 27% on the challenge set and 22% on the easy set.", "On OpenBookQA [84], GPT-3 improves significantly from zero to few shot settings but is still over 20 points short of the overall SOTA.", "GPT-3's few-shot performance is similar to a fine-tuned BERT Large baseline on the leaderboard.", "Overall, in-context learning with GPT-3 shows mixed results on commonsense reasoning tasks, with only small and inconsistent gains observed in the one and few-shot learning settings for both PIQA and ARC, but a significant improvement is observed on OpenBookQA.", "GPT-3 sets SOTA on the new PIQA dataset in all evaluation settings." ], [ "Reading Comprehension", "Next we evaluate GPT-3 on the task of reading comprehension.", "We use a suite of 5 datasets including abstractive, multiple choice, and span based answer formats in both dialog and single question settings.", "We observe a wide spread in GPT-3's performance across these datasets suggestive of varying capability with different answer formats.", "In general we observe GPT-3 is on par with initial baselines and early results trained using contextual representations on each respective dataset.", "GPT-3 performs best (within 3 points of the human baseline) on CoQA [106] a free-form conversational dataset and performs worst (13 F1 below an ELMo baseline) on QuAC [16] a dataset which requires modeling structured dialog acts and answer span selections of teacher-student interactions.", "On DROP [27], a dataset testing discrete reasoning and numeracy in the context of reading comprehension, GPT-3 in a few-shot setting outperforms the fine-tuned BERT baseline from the original paper but is still well below both human performance and state-of-the-art approaches which augment neural networks with symbolic systems [110].", "On SQuAD 2.0 [108], GPT-3 demonstrates its few-shot learning capabilities, improving by almost 10 F1 (to 69.8) compared to a zero-shot setting.", "This allows it to slightly outperform the best fine-tuned result in the original paper.", "On RACE [78], a multiple choice dataset of middle school and high school english examinations, GPT-3 performs relatively weakly and is only competitive with the earliest work utilizing contextual representations and is still 45% behind SOTA.", "Table: Performance of GPT-3 on SuperGLUE compared to fine-tuned baselines and SOTA.", "All results are reported on the test set.", "GPT-3 few-shot is given a total of 32 examples within the context of each task and performs no gradient updates.Figure: Performance on SuperGLUE increases with model size and number of examples in context.", "A value of K=32K=32 means that our model was shown 32 examples per task, for 256 examples total divided across the 8 tasks in SuperGLUE.", "We report GPT-3 values on the dev set, so our numbers are not directly comparable to the dotted reference lines (our test set results are in Table ).", "The BERT-Large reference model was fine-tuned on the SuperGLUE training set (125K examples), whereas BERT++ was first fine-tuned on MultiNLI (392K examples) and SWAG (113K examples) before further fine-tuning on the SuperGLUE training set (for a total of 630K fine-tuning examples).", "We find the difference in performance between the BERT-Large and BERT++ to be roughly equivalent to the difference between GPT-3 with one example per context versus eight examples per context." ], [ "SuperGLUE", "In order to better aggregate results on NLP tasks and compare to popular models such as BERT and RoBERTa in a more systematic way, we also evaluate GPT-3 on a standardized collection of datasets, the SuperGLUE benchmark [135] [135] [17] [25] [105] [54] [142] [21] [8] [34] [6] [96] [98].", "GPT-3’s test-set performance on the SuperGLUE dataset is shown in Table REF .", "In the few-shot setting, we used 32 examples for all tasks, sampled randomly from the training set.", "For all tasks except WSC and MultiRC, we sampled a new set of examples to use in the context for each problem.", "For WSC and MultiRC, we used the same set of randomly drawn examples from the training set as context for all of the problems we evaluated.", "We observe a wide range in GPT-3’s performance across tasks.", "On COPA and ReCoRD GPT-3 achieves near-SOTA performance in the one-shot and few-shot settings, with COPA falling only a couple points short and achieving second place on the leaderboard, where first place is held by a fine-tuned 11 billion parameter model (T5).", "On WSC, performance is still relatively strong, achieving 80.1% in the few-shot setting (note that GPT-3 achieves 88.6% on the original Winograd dataset as described in Section REF ).", "On BoolQ, MultiRC, and RTE, performance is reasonable, roughly matching that of a fine-tuned BERT-Large.", "On CB, we see signs of life at 75.6% in the few-shot setting.", "WiC is a notable weak spot with few-shot performance at 49.4% (at random chance).", "We tried a number of different phrasings and formulations for WiC (which involves determining if a word is being used with the same meaning in two sentences), none of which was able to achieve strong performance.", "This hints at a phenomenon that will become clearer in the next section (which discusses the ANLI benchmark) – GPT-3 appears to be weak in the few-shot or one-shot setting at some tasks that involve comparing two sentences or snippets, for example whether a word is used the same way in two sentences (WiC), whether one sentence is a paraphrase of another, or whether one sentence implies another.", "This could also explain the comparatively low scores for RTE and CB, which also follow this format.", "Despite these weaknesses, GPT-3 still outperforms a fine-tuned BERT-large on four of eight tasks and on two tasks GPT-3 is close to the state-of-the-art held by a fine-tuned 11 billion parameter model.", "Finally, we note that the few-shot SuperGLUE score steadily improves with both model size and with number of examples in the context showing increasing benefits from in-context learning (Figure REF ).", "We scale $K$ up to 32 examples per task, after which point additional examples will not reliably fit into our context.", "When sweeping over values of $K$ , we find that GPT-3 requires less than eight total examples per task to outperform a fine-tuned BERT-Large on overall SuperGLUE score.", "Figure: Performance of GPT-3 on ANLI Round 3.", "Results are on the dev-set, which has only 1500 examples and therefore has high variance (we estimate a standard deviation of 1.2%).", "We find that smaller models hover around random chance, while few-shot GPT-3 175B closes almost half the gap from random chance to SOTA.", "Results for ANLI rounds 1 and 2 are shown in the appendix." ], [ "NLI", "Natural Language Inference (NLI) [31] concerns the ability to understand the relationship between two sentences.", "In practice, this task is usually structured as a two or three class classification problem where the model classifies whether the second sentence logically follows from the first, contradicts the first sentence, or is possibly true (neutral).", "SuperGLUE includes an NLI dataset, RTE, which evaluates the binary version of the task.", "On RTE, only the largest version of GPT-3 performs convincingly better than random (56%) in any evaluation setting, but in a few-shot setting GPT-3 performs similarly to a single-task fine-tuned BERT Large.", "We also evaluate on the recently introduced Adversarial Natural Language Inference (ANLI) dataset [94].", "ANLI is a difficult dataset employing a series of adversarially mined natural language inference questions in three rounds (R1, R2, and R3).", "Similar to RTE, all of our models smaller than GPT-3 perform at almost exactly random chance on ANLI, even in the few-shot setting ($\\sim 33\\%$ ), whereas GPT-3 itself shows signs of life on Round 3.", "Results for ANLI R3 are highlighted in Figure REF and full results for all rounds can be found in Appendix .", "These results on both RTE and ANLI suggest that NLI is still a very difficult task for language models and they are only just beginning to show signs of progress." ], [ "Synthetic and Qualitative Tasks", "One way to probe GPT-3’s range of abilities in the few-shot (or zero- and one-shot) setting is to give it tasks which require it to perform simple on-the-fly computational reasoning, recognize a novel pattern that is unlikely to have occurred in training, or adapt quickly to an unusual task.", "We devise several tasks to test this class of abilities.", "First, we test GPT-3’s ability to perform arithmetic.", "Second, we create several tasks that involve rearranging or unscrambling the letters in a word, tasks which are unlikely to have been exactly seen during training.", "Third, we test GPT-3’s ability to solve SAT-style analogy problems few-shot.", "Finally, we test GPT-3 on several qualitative tasks, including using new words in a sentence, correcting English grammar, and news article generation.", "We will release the synthetic datasets with the hope of stimulating further study of test-time behavior of language models." ], [ "Arithmetic", "To test GPT-3's ability to perform simple arithmetic operations without task-specific training, we developed a small battery of 10 tests that involve asking GPT-3 a simple arithmetic problem in natural language: 2 digit addition (2D+) – The model is asked to add two integers sampled uniformly from $[0, 100)$ , phrased in the form of a question, e.g.", "“Q: What is 48 plus 76?", "A: 124.” 2 digit subtraction (2D-) – The model is asked to subtract two integers sampled uniformly from $[0, 100)$ ; the answer may be negative.", "Example: “Q: What is 34 minus 53?", "A: -19”.", "3 digit addition (3D+) – Same as 2 digit addition, except numbers are uniformly sampled from $[0, 1000)$ .", "3 digit subtraction (3D-) – Same as 2 digit subtraction, except numbers are uniformly sampled from $[0, 1000)$ .", "4 digit addition (4D+) – Same as 3 digit addition, except uniformly sampled from $[0, 10000)$ .", "4 digit subtraction (4D-) – Same as 3 digit subtraction, except uniformly sampled from $[0, 10000)$ .", "5 digit addition (5D+) – Same as 3 digit addition, except uniformly sampled from $[0, 100000)$ .", "5 digit subtraction (5D-) – Same as 3 digit subtraction, except uniformly sampled from $[0, 100000)$ .", "2 digit multiplication (2Dx) – The model is asked to multiply two integers sampled uniformly from $[0, 100)$ , e.g.", "“Q: What is 24 times 42?", "A: 1008”.", "One-digit composite (1DC) – The model is asked to perform a composite operation on three 1 digit numbers, with parentheses around the last two.", "For example, “Q: What is 6+(4*8)?", "A: 38”.", "The three 1 digit numbers are selected uniformly on $[0, 10)$ and the operations are selected uniformly from {+,-,*}.", "In all 10 tasks the model must generate the correct answer exactly.", "For each task we generate a dataset of 2,000 random instances of the task and evaluate all models on those instances.", "First we evaluate GPT-3 in the few-shot setting, for which results are shown in Figure REF .", "On addition and subtraction, GPT-3 displays strong proficiency when the number of digits is small, achieving 100% accuracy on 2 digit addition, 98.9% at 2 digit subtraction, 80.2% at 3 digit addition, and 94.2% at 3-digit subtraction.", "Performance decreases as the number of digits increases, but GPT-3 still achieves 25-26% accuracy on four digit operations and 9-10% accuracy on five digit operations, suggesting at least some capacity to generalize to larger numbers of digits.", "GPT-3 also achieves 29.2% accuracy at 2 digit multiplication, an especially computationally intensive operation.", "Finally, GPT-3 achieves 21.3% accuracy at single digit combined operations (for example, 9*(7+5)), suggesting that it has some robustness beyond just single operations.", "Figure: Results on all 10 arithmetic tasks in the few-shot settings for models of different sizes.", "There is a significant jump from the second largest model (GPT-3 13B) to the largest model (GPT-3 175), with the latter being able to reliably accurate 2 digit arithmetic, usually accurate 3 digit arithmetic, and correct answers a significant fraction of the time on 4-5 digit arithmetic, 2 digit multiplication, and compound operations.", "Results for one-shot and zero-shot are shown in the appendix.Table: Results on basic arithmetic tasks for GPT-3 175B.", "{2,3,4,5}D{+,-} is 2, 3, 4, and 5 digit addition or subtraction, 2Dx is 2 digit multiplication.", "1DC is 1 digit composite operations.", "Results become progressively stronger moving from the zero-shot to one-shot to few-shot setting, but even the zero-shot shows significant arithmetic abilities.As Figure REF makes clear, small models do poorly on all of these tasks – even the 13 billion parameter model (the second largest after the 175 billion full GPT-3) can solve 2 digit addition and subtraction only half the time, and all other operations less than 10% of the time.", "One-shot and zero-shot performance are somewhat degraded relative to few-shot performance, suggesting that adaptation to the task (or at the very least recognition of the task) is important to performing these computations correctly.", "Nevertheless, one-shot performance is still quite strong, and even zero-shot performance of the full GPT-3 significantly outperforms few-shot learning for all smaller models.", "All three settings for the full GPT-3 are shown in Table REF , and model capacity scaling for all three settings is shown in Appendix .", "To spot-check whether the model is simply memorizing specific arithmetic problems, we took the 3-digit arithmetic problems in our test set and searched for them in our training data in both the forms \"<NUM1> + <NUM2> =\" and \"<NUM1> plus <NUM2>\".", "Out of 2,000 addition problems we found only 17 matches (0.8%) and out of 2,000 subtraction problems we found only 2 matches (0.1%), suggesting that only a trivial fraction of the correct answers could have been memorized.", "In addition, inspection of incorrect answers reveals that the model often makes mistakes such as not carrying a “1”, suggesting it is actually attempting to perform the relevant computation rather than memorizing a table.", "Overall, GPT-3 displays reasonable proficiency at moderately complex arithmetic in few-shot, one-shot, and even zero-shot settings." ], [ "Word Scrambling and Manipulation Tasks", "To test GPT-3's ability to learn novel symbolic manipulations from a few examples, we designed a small battery of 5 “character manipulation” tasks.", "Each task involves giving the model a word distorted by some combination of scrambling, addition, or deletion of characters, and asking it to recover the original word.", "The 5 tasks are: Table: GPT-3 175B performance on various word unscrambling and wordmanipulation tasks, in zero-, one-, and few-shot settings.", "CL is “cycle letters in word”, A1 is anagrams of but the first and last letters,A2 is anagrams of all but the first and last two letters, RI is “Random insertionin word”, RW is “reversed words”.Figure: Few-shot performance on the five word scrambling tasks for different sizes of model.", "There is generally smooth improvement with model size although the random insertion task shows an upward slope of improvement with the 175B model solving the task the majority of the time.", "Scaling of one-shot and zero-shot performance is shown in the appendix.", "All tasks are done with K=100K=100.", "Cycle letters in word (CL) – The model is given a word with its letters cycled, then the “=” symbol, and is expected to generate the original word.", "For example, it might be given “lyinevitab” and should output “inevitably”.", "Anagrams of all but first and last characters (A1) – The model is given a word where every letter except the first and last have been scrambled randomly, and must output the original word.", "Example: criroptuon = corruption.", "Anagrams of all but first and last 2 characters (A2) – The model is given a word where every letter except the first 2 and last 2 have been scrambled randomly, and must recover the original word.", "Example: opoepnnt $\\rightarrow $ opponent.", "Random insertion in word (RI) – A random punctuation or space character is inserted between each letter of a word, and the model must output the original word.", "Example: s.u!c/c!e.s s i/o/n = succession.", "Reversed words (RW) – The model is given a word spelled backwards, and must output the original word.", "Example: stcejbo $\\rightarrow $ objects.", "For each task we generate 10,000 examples, which we chose to be the top 10,000 most frequent words as measured by [92] of length more than 4 characters and less than 15 characters.", "The few-shot results are shown in Figure REF .", "Task performance tends to grow smoothly with model size, with the full GPT-3 model achieving 66.9% on removing random insertions, 38.6% on cycling letters, 40.2% on the easier anagram task, and 15.1% on the more difficult anagram task (where only the first and last letters are held fixed).", "None of the models can reverse the letters in a word.", "In the one-shot setting, performance is significantly weaker (dropping by half or more), and in the zero-shot setting the model can rarely perform any of the tasks (Table REF ).", "This suggests that the model really does appear to learn these tasks at test time, as the model cannot perform them zero-shot and their artificial nature makes them unlikely to appear in the pre-training data (although we cannot confirm this with certainty).", "We can further quantify performance by plotting “in-context learning curves”, which show task performance as a function of the number of in-context examples.", "We show in-context learning curves for the Symbol Insertion task in Figure REF .", "We can see that larger models are able to make increasingly effective use of in-context information, including both task examples and natural language task descriptions.", "Finally, it is worth adding that solving these tasks requires character-level manipulations, whereas our BPE encoding operates on significant fractions of a word (on average $\\sim 0.7$ words per token), so from the LM’s perspective succeeding at these tasks involves not just manipulating BPE tokens but understanding and pulling apart their substructure.", "Also, CL, A1, and A2 are not bijective (that is, the unscrambled word is not a deterministic function of the scrambled word), requiring the model to perform some search to find the correct unscrambling.", "Thus, the skills involved appear to require non-trivial pattern-matching and computation." ], [ "SAT Analogies", "To test GPT-3 on another task that is somewhat unusual relative to the typical distribution of text, we collected a set of 374 “SAT analogy” problems [131].", "Analogies are a style of multiple choice question that constituted a section of the SAT college entrance exam before 2005.", "A typical example is “audacious is to boldness as (a) sanctimonious is to hypocrisy, (b) anonymous is to identity, (c) remorseful is to misdeed, (d) deleterious is to result, (e) impressionable is to temptation”.", "The student is expected to choose which of the five word pairs has the same relationship as the original word pair; in this example the answer is “sanctimonious is to hypocrisy”.", "On this task GPT-3 achieves 65.2% in the few-shot setting, 59.1% in the one-shot setting, and 53.7% in the zero-shot setting, whereas the average score among college applicants was 57% [129] (random guessing yields 20%).", "As shown in Figure REF , the results improve with scale, with the the full 175 billion model improving by over 10% compared to the 13 billion parameter model.", "Figure: Zero-, one-,and few-shot performance on SAT analogy tasks, for different sizes of model.", "The largest model achieves 65% accuracy in the few-shot setting, and also demonstrates significant gains to in-context learning which are not present in smaller models." ], [ "News Article Generation", "Previous work on generative language models qualitatively tested their ability to generate synthetic “news articles” by conditional sampling from the model given a human-written prompt consisting of a plausible first sentence for a news story [117].", "Relative to [117], the dataset used to train GPT-3 is much less weighted towards news articles, so trying to generate news articles via raw unconditional samples is less effective – for example GPT-3 often interprets the proposed first sentence of a “news article” as a tweet and then posts synthetic responses or follow-up tweets.", "To solve this problem we employed GPT-3’s few-shot learning abilities by providing three previous news articles in the model’s context to condition it.", "With the title and subtitle of a proposed next article, the model is able to reliably generate short articles in the “news” genre.", "To gauge the quality of news article generation from GPT-3 (which we believe is likely to be correlated with conditional sample generation quality in general), we decided to measure human ability to distinguish GPT-3-generated articles from real ones.", "Similar work has been carried out by Kreps et al.", "[56] and Zellers et al.", "[141].", "Generative language models are trained to match the distribution of content generated by humans, so the (in)ability of humans to distinguish the two is a potentially important measure of quality.This task is also relevant to the potential misuse of language models discussed in Section REF .", "In order to see how well humans can detect model generated text, we arbitrarily selected 25 article titles and subtitles from the website newser.com (mean length: 215 words).", "We then generated completions of these titles and subtitles from four language models ranging in size from 125M to 175B (GPT-3) parameters (mean length: 200 words).", "For each model, we presented around 80 US-based participants with a quiz consisting of these real titles and subtitles followed by either the human written article or the article generated by the modelWe wanted to identify how good an average person on the internet is at detecting language model outputs, so we focused on participants drawn from the general US population.", "See Appendix for details..", "Participants were asked to select whether the article was “very likely written by a human”, “more likely written by a human”, “I don't know”, “more likely written by a machine”, or “very likely written by a machine”.", "The articles we selected were not in the models’ training data and the model outputs were formatted and selected programmatically to prevent human cherry-picking.", "All models used the same context to condition outputs on and were pre-trained with the same context size and the same article titles and subtitles were used as prompts for each model.", "However, we also ran an experiment to control for participant effort and attention that followed the same format but involved intentionally bad model generated articles.", "This was done by generating articles from a “control model”: a 160M parameter model with no context and increased output randomness.", "Mean human accuracy (the ratio of correct assignments to non-neutral assignments per participant) at detecting that the intentionally bad articles were model generated was $\\sim 86\\%$   where 50% is chance level performance.", "By contrast, mean human accuracy at detecting articles that were produced by the 175B parameter model was barely above chance at $\\sim 52\\%$ (see Table REF ).We use a two-sample Student’s T-Test to test for significant difference between the means of the participant accuracies of each model and the control model and report the normalized difference in the means (as the t-statistic) and the p-value.", "Human abilities to detect model generated text appear to decrease as model size increases: there appears to be a trend towards chance accuracy with model size, and human detection of GPT-3 is close to chance.If a model consistently produces texts that are more impressive than human articles, it is possible that human performance on this task would drop below 50%.", "Indeed, many individual participants scored below 50% on this task.", "This is true despite the fact that participants spend more time on each output as model size increases (see Appendix ).", "Table: Human accuracy in identifying whether short (∼\\sim 200 word) news articles are model generated.", "We find that human accuracy (measured by the ratio of correct assignments to non-neutral assignments) ranges from 86% on the control model to 52% on GPT-3 175B.", "This table compares mean accuracy between five different models, and shows the results of a two-sample T-Test for the difference in mean accuracy between each model and the control model (an unconditional GPT-3 Small model with increased output randomness).Figure: People's ability to identify whether news articles are model-generated (measured by the ratio of correct assignments to non-neutral assignments) decreases as model size increases.", "Accuracy on the outputs on the deliberately-bad control model (an unconditioned GPT-3 Small model with higher output randomness) is indicated with the dashed line at the top, and the random chance (50%) is indicated with the dashed line at the bottom.", "Line of best fit is a power law with 95% confidence intervals.Examples of synthetic articles from GPT-3 are given in Figures REF and REF .Additional non-news samples can be found in Appendix .", "Much of the text is—as indicated by the evaluations—difficult for humans to distinguish from authentic human content.", "Factual inaccuracies can be an indicator that an article is model generated since, unlike human authors, the models have no access to the specific facts that the article titles refer to or when the article was written.", "Other indicators include repetition, non sequiturs, and unusual phrasings, though these are often subtle enough that they are not noticed.", "Figure: The GPT-3 generated news article that humans had the greatest difficulty distinguishing from a human written article (accuracy: 12%).Figure: The GPT-3 generated news article that humans found the easiest to distinguish from a human written article (accuracy: 61%).Related work on language model detection by Ippolito et al.", "[48] indicates that automatic discriminators like Grover [141] and GLTR [37] may have greater success at detecting model generated text than human evaluators.", "Automatic detection of these models may be a promising area of future research.", "Ippolito et al.", "[48] also note that human accuracy at detecting model generated text increases as humans observe more tokens.", "To do a preliminary investigation of how good humans are at detecting longer news articles generated by GPT-3 175B, we selected 12 world news articles from Reuters with an average length of 569 words and generated completions of these articles from GPT-3 with an average length of 498 words (298 words longer than our initial experiments).", "Following the methodology above, we ran two experiments, each on around 80 US-based participants, to compare human abilities to detect the articles generated by GPT-3 and a control model.", "We found that mean human accuracy at detecting the intentionally bad longer articles from the control model was $\\sim 88\\%$ , while mean human accuracy at detecting the longer articles that were produced by GPT-3 175B was still barely above chance at $\\sim 52\\%$ (see Table REF ).", "This indicates that, for news articles that are around 500 words long, GPT-3 continues to produce articles that humans find difficult to distinguish from human written news articles.", "Table: People’s ability to identify whether ∼500\\sim 500 word articles are model generated (as measured by the ratio of correct assignments to non-neutral assignments) was 88% on the control model and 52% on GPT-3 175B.", "This table shows the results of a two-sample T-Test for the difference in mean accuracy between GPT-3 175B and the control model (an unconditional GPT-3 Small model with increased output randomness).Figure: Representative GPT-3 completions for the few-shot task of using a new word in a sentence.", "Boldface is GPT-3’s completions, plain text is human prompts.", "In the first example both the prompt and the completion are provided by a human; this then serves as conditioning for subsequent examples where GPT-3 receives successive additional prompts and provides the completions.", "Nothing task-specific is provided to GPT-3 other than the conditioning shown here.Figure: Representative GPT-3 completions for the few-shot task of correcting English grammar.", "Boldface is GPT-3’s completions, plain text is human prompts.", "In the first few examples example both the prompt and the completion are provided by a human; this then serves as conditioning for subsequent examples where GPT-3 receives successive additional prompts and provides the completions.", "Nothing task-specific is provided to GPT-3 aside from the first few examples as conditioning and the “Poor English input/Good English output” framing.", "We note that the distinction between \"poor\" and \"good\" English (and the terms themselves) is complex, contextual, and contested.", "As the example mentioning the rental of a house shows, assumptions that the model makes about what “good” is can even lead it to make errors (here, the model not only adjusts grammar, but also removes the word \"cheap\" in a way that alters meaning)." ], [ "Learning and Using Novel Words", "A task studied in developmental linguistics [13] is the ability to learn and utilize new words, for example using a word in a sentence after seeing it defined only once, or conversely inferring a word’s meaning from only one usage.", "Here we qualitatively test GPT-3’s ability to do the former.", "Specifically, we give GPT-3 the definition of a nonexistent word, such as “Gigamuru”, and then ask it to use it in a sentence.", "We provide one to five previous examples of a (separate) nonexistent word being defined and used in a sentence, so the task is few-shot in terms of previous examples of the broad task and one-shot in terms of the specific word.", "Table REF shows the 6 examples we generated; all definitions were human-generated, and the first answer was human-generated as conditioning while the subsequent answers were generated by GPT-3.", "These examples were generated continuously in one sitting and we did not omit or repeatedly try any prompts.", "In all cases the generated sentence appears to be a correct or at least plausible use of the word.", "In the final sentence the model generates a plausible conjugation for the word “screeg” (namely “screeghed”), although the use of the word is slightly awkward (“screeghed at each other”) despite being plausible in the sense that it could describe a toy sword fight.", "Overall, GPT-3 appears to be at least proficient at the task of using novel words in a sentence." ], [ "Correcting English Grammar", "Another task well suited for few-shot learning is correcting English grammar.", "We test this with GPT-3 in the few-shot setting by giving prompts of the form \"Poor English Input: <sentence>\\n Good English Output: <sentence>\".", "We give GPT-3 one human-generated correction and then ask it to correct 5 more (again without any omissions or repeats).", "Results are shown in Figure REF ." ], [ "Measuring and Preventing Memorization Of Benchmarks", "Since our training dataset is sourced from the internet, it is possible that our model was trained on some of our benchmark test sets.", "Accurately detecting test contamination from internet-scale datasets is a new area of research without established best practices.", "While it is common practice to train large models without investigating contamination, given the increasing scale of pretraining datasets, we believe this issue is becoming increasingly important to attend to.", "This concern is not just hypothetical.", "One of the first papers to train a language model on Common Crawl data [130] detected and removed a training document which overlapped with one of their evaluation datasets.", "Other work such as GPT-2 [117] also conducted post-hoc overlap analysis.", "Their study was relatively encouraging, finding that although models did perform moderately better on data that overlapped between training and testing, this did not significantly impact reported results due to the small fraction of data which was contaminated (often only a few percent).", "GPT-3 operates in a somewhat different regime.", "On the one hand, the dataset and model size are about two orders of magnitude larger than those used for GPT-2, and include a large amount of Common Crawl, creating increased potential for contamination and memorization.", "On the other hand, precisely due to the large amount of data, even GPT-3 175B does not overfit its training set by a significant amount, measured relative to a held-out validation set with which it was deduplicated (Figure REF ).", "Thus, we expect that contamination is likely to be frequent, but that its effects may not be as large as feared.", "We initially tried to address the issue of contamination by proactively searching for and attempting to remove any overlap between our training data and the development and test sets of all benchmarks studied in this paper.", "Unfortunately, a bug resulted in only partial removal of all detected overlaps from the training data.", "Due to the cost of training, it wasn't feasible to retrain the model.", "To address this, we investigate in detail how the remaining detected overlap impacts results.", "For each benchmark, we produce a `clean' version which removes all potentially leaked examples, defined roughly as examples that have a 13-gram overlap with anything in the pretraining set (or that overlap with the whole example when it is shorter than 13-grams).", "The goal is to very conservatively flag anything that could potentially be contamination, so as to produce a clean subset that is free of contamination with high confidence.", "The exact procedure is detailed in Appendix .", "We then evaluate GPT-3 on these clean benchmarks, and compare to the original score.", "If the score on the clean subset is similar to the score on the entire dataset, this suggests that contamination, even if present, does not have a significant effect on reported results.", "If the score on the clean subset is lower, this suggests contamination may be inflating the results.", "The results are summarized in Figure REF .", "Although potential contamination is often high (with a quarter of benchmarks scoring over 50%), in most cases performance changes only negligibly, and we see no evidence that contamination level and performance difference are correlated.", "We conclude that either our conservative method substantially overestimated contamination or that contamination has little effect on performance.", "Below, we review in more detail the few specific cases where either (1) the model performs significantly worse on the cleaned version, or (2) potential contamination is very high, which makes measuring the performance difference difficult.", "Figure: Benchmark contamination analysis    We constructed cleaned versions of each of our benchmarks to check for potential contamination in our training set.", "The x-axis is a conservative lower bound for how much of the dataset is known with high confidence to be clean, and the y-axis shows the difference in performance when evaluating only on the verified clean subset.", "Performance on most benchmarks changed negligibly, but some were flagged for further review.", "On inspection we find some evidence for contamination of the PIQA and Winograd results, and we mark the corresponding results in Section with an asterisk.", "We find no evidence that other benchmarks are affected.Our analysis flagged six groups of benchmarks for further investigation: Word Scrambling, Reading Comprehension (QuAC, SQuAD2, DROP), PIQA, Winograd, language modeling tasks (Wikitext tasks, 1BW), and German to English translation.", "Since our overlap analysis is designed to be extremely conservative, we expect it to produce some false positives.", "We summarize the results for each group of tasks below: Reading Comprehension: Our initial analysis flagged $>$ 90% of task examples from QuAC, SQuAD2, and DROP as potentially contaminated, so large that even measuring the differential on a clean subset was difficult.", "Upon manual inspection, however, we found that for every overlap we inspected, in all 3 datasets, the source text was present in our training data but the question/answer pairs were not, meaning the model gains only background information and cannot memorize the answer to a specific question.", "German translation: We found 25% of the examples in the WMT16 German-English test set were marked as potentially contaminated, with an associated total effect size of 1-2 BLEU.", "Upon inspection, none of the flagged examples contain paired sentences resembling NMT training data and collisions were monolingual matches mostly of snippets of events discussed in the news.", "Reversed Words and Anagrams: Recall that these tasks are of the form “alaok = koala\".", "Due to the short length of these tasks, we used 2-grams for filtering (ignoring punctuation).", "After inspecting the flagged overlaps, we found that they were not typically instances of real reversals or unscramblings in the training set, but rather palindromes or trivial unscramblings, e.g “kayak = kayak\".", "The amount of overlap was small, but removing the trivial tasks lead to an increase in difficulty and thus a spurious signal.", "Related to this, the symbol insertion task shows high overlap but no effect on performance – this is because that task involves removing non-letter characters from a word, and the overlap analysis itself ignores such characters, leading to many spurious matches.", "PIQA: The overlap analysis flagged 29% of examples as contaminated, and observed a 3 percentage point absolute decrease (4% relative decrease) in performance on the clean subset.", "Though the test dataset was released after our training set was created and its labels are hidden, some of the web pages used by the crowdsourced dataset creators are contained in our training set.", "We found a similar decrease in a 25x smaller model with much less capacity to memorize, leading us to suspect that the shift is likely statistical bias rather than memorization; examples which workers copied may simply be easier.", "Unfortunately, we cannot rigorously prove this hypothesis.", "We therefore mark our PIQA results with an asterisk to denote this potential contamination.", "Winograd: The overlap analysis flagged 45% of examples, and found a 2.6% decrease in performance on the clean subset.", "Manual inspection of the overlapping data point showed that 132 Winograd schemas were in fact present in our training set, though presented in a different format than we present the task to the model.", "Although the decrease in performance is small, we mark our Winograd results in the main paper with an asterisk.", "Language modeling: We found the 4 Wikipedia language modeling benchmarks measured in GPT-2, plus the Children's Book Test dataset, to be almost entirely contained in our training data.", "Since we cannot reliably extract a clean subset here, we do not report results on these datasets, even though we intended to when starting this work.", "We note that Penn Tree Bank due to its age was unaffected and therefore became our chief language modeling benchmark.", "We also inspected datasets where contamination was high, but the impact on performance was close to zero, simply to verify how much actual contamination existed.", "These appeared to often contain false positives.", "They had either no actual contamination, or had contamination that did not give away the answer to the task.", "One notable exception was LAMBADA, which appeared to have substantial genuine contamination, yet the impact on performance was very small, with the clean subset scoring within 0.5% of the full dataset.", "Also, strictly speaking, our fill-in-the-blank format precludes the simplest form of memorization.", "Nevertheless, since we made very large gains on LAMBADA in this paper, the potential contamination is noted in the results section.", "An important limitation of our contamination analysis is that we cannot be sure that the clean subset is drawn from the same distribution as the original dataset.", "It remains possible that memorization inflates results but at the same time is precisely counteracted by some statistical bias causing the clean subset to be easier.", "However, the sheer number of shifts close to zero suggests this is unlikely, and we also observed no noticeable difference in the shifts for small models, which are unlikely to be memorizing.", "Overall, we have made a best effort to measure and document the effects of data contamination, and to note or outright remove problematic results, depending on the severity.", "Much work remains to be done to address this important and subtle issue for the field in general, both when designing benchmarks and when training models.", "For a more detailed explanation of our analysis, we refer the reader to Appendix ." ], [ "Limitations", "GPT-3 and our analysis of it have a number of limitations.", "Below we describe some of these and suggest directions for future work.", "First, despite the strong quantitative and qualitative improvements of GPT-3, particularly compared to its direct predecessor GPT-2, it still has notable weaknesses in text synthesis and several NLP tasks.", "On text synthesis, although the overall quality is high, GPT-3 samples still sometimes repeat themselves semantically at the document level, start to lose coherence over sufficiently long passages, contradict themselves, and occasionally contain non-sequitur sentences or paragraphs.", "We will release a collection of 500 uncurated unconditional samples to help provide a better sense of GPT-3’s limitations and strengths at text synthesis.", "Within the domain of discrete language tasks, we have noticed informally that GPT-3 seems to have special difficulty with “common sense physics”, despite doing well on some datasets (such as PIQA [11]) that test this domain.", "Specifically GPT-3 has difficulty with questions of the type “If I put cheese into the fridge, will it melt?”.", "Quantitatively, GPT-3’s in-context learning performance has some notable gaps on our suite of benchmarks, as described in Section , and in particular it does little better than chance when evaluated one-shot or even few-shot on some “comparison” tasks, such as determining if two words are used the same way in a sentence, or if one sentence implies another (WIC and ANLI respectively), as well as on a subset of reading comprehension tasks.", "This is especially striking given GPT-3’s strong few-shot performance on many other tasks.", "GPT-3 has several structural and algorithmic limitations, which could account for some of the issues above.", "We focused on exploring in-context learning behavior in autoregressive language models because it is straightforward to both sample and compute likelihoods with this model class.", "As a result our experiments do not include any bidirectional architectures or other training objectives such as denoising.", "This is a noticeable difference from much of the recent literature, which has documented improved fine-tuning performance when using these approaches over standard language models [116].", "Thus our design decision comes at the cost of potentially worse performance on tasks which empirically benefit from bidirectionality.", "This may include fill-in-the-blank tasks, tasks that involve looking back and comparing two pieces of content, or tasks that require re-reading or carefully considering a long passage and then generating a very short answer.", "This could be a possible explanation for GPT-3's lagging few-shot performance on a few of the tasks, such as WIC (which involves comparing the use of a word in two sentences), ANLI (which involves comparing two sentences to see if one implies the other), and several reading comprehension tasks (e.g.", "QuAC and RACE).", "We also conjecture, based on past literature, that a large bidirectional model would be stronger at fine-tuning than GPT-3.", "Making a bidirectional model at the scale of GPT-3, and/or trying to make bidirectional models work with few- or zero-shot learning, is a promising direction for future research, and could help achieve the “best of both worlds”.", "A more fundamental limitation of the general approach described in this paper – scaling up any LM-like model, whether autoregressive or bidirectional – is that it may eventually run into (or could already be running into) the limits of the pretraining objective.", "Our current objective weights every token equally and lacks a notion of what is most important to predict and what is less important.", "[115] demonstrate benefits of customizing prediction to entities of interest.", "Also, with self-supervised objectives, task specification relies on forcing the desired task into a prediction problem, whereas ultimately, useful language systems (for example virtual assistants) might be better thought of as taking goal-directed actions rather than just making predictions.", "Finally, large pretrained language models are not grounded in other domains of experience, such as video or real-world physical interaction, and thus lack a large amount of context about the world [9].", "For all these reasons, scaling pure self-supervised prediction is likely to hit limits, and augmentation with a different approach is likely to be necessary.", "Promising future directions in this vein might include learning the objective function from humans [143], fine-tuning with reinforcement learning, or adding additional modalities such as images to provide grounding and a better model of the world [18].", "Another limitation broadly shared by language models is poor sample efficiency during pre-training.", "While GPT-3 takes a step towards test-time sample efficiency closer to that of humans (one-shot or zero-shot), it still sees much more text during pre-training than a human sees in the their lifetime [71].", "Improving pre-training sample efficiency is an important direction for future work, and might come from grounding in the physical world to provide additional information, or from algorithmic improvements.", "A limitation, or at least uncertainty, associated with few-shot learning in GPT-3 is ambiguity about whether few-shot learning actually learns new tasks “from scratch” at inference time, or if it simply recognizes and identifies tasks that it has learned during training.", "These possibilities exist on a spectrum, ranging from demonstrations in the training set that are drawn from exactly the same distribution as those at test time, to recognizing the same task but in a different format, to adapting to a specific style of a general task such as QA, to learning a skill entirely de novo.", "Where GPT-3 is on this spectrum may also vary from task to task.", "Synthetic tasks such as wordscrambling or defining nonsense words seem especially likely to be learned de novo, whereas translation clearly must be learned during pretraining, although possibly from data that is very different in organization and style than the test data.", "Ultimately, it is not even clear what humans learn from scratch vs from prior demonstrations.", "Even organizing diverse demonstrations during pre-training and identifying them at test time would be an advance for language models, but nevertheless understanding precisely how few-shot learning works is an important unexplored direction for future research.", "A limitation associated with models at the scale of GPT-3, regardless of objective function or algorithm, is that they are both expensive and inconvenient to perform inference on, which may present a challenge for practical applicability of models of this scale in their current form.", "One possible future direction to address this is distillation [44] of large models down to a manageable size for specific tasks.", "Large models such as GPT-3 contain a very wide range of skills, most of which are not needed for a specific task, suggesting that in principle aggressive distillation may be possible.", "Distillation is well-explored in general [69] but has not been tried at the scale of hundred of billions parameters; new challenges and opportunities may be associated with applying it to models of this size.", "Finally, GPT-3 shares some limitations common to most deep learning systems – its decisions are not easily interpretable, it is not necessarily well-calibrated in its predictions on novel inputs as observed by the much higher variance in performance than humans on standard benchmarks, and it retains the biases of the data it has been trained on.", "This last issue – biases in the data that may lead the model to generate stereotyped or prejudiced content – is of special concern from a societal perspective, and will be discussed along with other issues in the next section on Broader Impacts (Section )." ], [ "Broader Impacts", "Language models have a wide range of beneficial applications for society, including code and writing auto-completion, grammar assistance, game narrative generation, improving search engine responses, and answering questions.", "But they also have potentially harmful applications.", "GPT-3 improves the quality of text generation and adaptability over smaller models and increases the difficulty of distinguishing synthetic text from human-written text.", "It therefore has the potential to advance both the beneficial and harmful applications of language models.", "Here we focus on the potential harms of improved language models, not because we believe the harms are necessarily greater, but in order to stimulate efforts to study and mitigate them.", "The broader impacts of language models like this are numerous.", "We focus on two primary issues: the potential for deliberate misuse of language models like GPT-3 in Section REF , and issues of bias, fairness, and representation within models like GPT-3 in Section REF .", "We also briefly discuss issues of energy efficiency (Section REF )." ], [ "Misuse of Language Models", "Malicious uses of language models can be somewhat difficult to anticipate because they often involve repurposing language models in a very different environment or for a different purpose than researchers intended.", "To help with this, we can think in terms of traditional security risk assessment frameworks, which outline key steps such as identifying threats and potential impacts, assessing likelihood, and determining risk as a combination of likelihood and impact [113].", "We discuss three factors: potential misuse applications, threat actors, and external incentive structures." ], [ "Potential Misuse Applications", "Any socially harmful activity that relies on generating text could be augmented by powerful language models.", "Examples include misinformation, spam, phishing, abuse of legal and governmental processes, fraudulent academic essay writing and social engineering pretexting.", "Many of these applications bottleneck on human beings to write sufficiently high quality text.", "Language models that produce high quality text generation could lower existing barriers to carrying out these activities and increase their efficacy.", "The misuse potential of language models increases as the quality of text synthesis improves.", "The ability of GPT-3 to generate several paragraphs of synthetic content that people find difficult to distinguish from human-written text in REF represents a concerning milestone in this regard." ], [ "Threat Actor Analysis", "Threat actors can be organized by skill and resource levels, ranging from low or moderately skilled and resourced actors who may be able to build a malicious product to `advanced persistent threats' (APTs): highly skilled and well-resourced (e.g.", "state-sponsored) groups with long-term agendas [119].", "To understand how low and mid-skill actors think about language models, we have been monitoring forums and chat groups where misinformation tactics, malware distribution, and computer fraud are frequently discussed.", "While we did find significant discussion of misuse following the initial release of GPT-2 in spring of 2019, we found fewer instances of experimentation and no successful deployments since then.", "Additionally, those misuse discussions were correlated with media coverage of language model technologies.", "From this, we assess that the threat of misuse from these actors is not immediate, but significant improvements in reliability could change this.", "Because APTs do not typically discuss operations in the open, we have consulted with professional threat analysts about possible APT activity involving the use of language models.", "Since the release of GPT-2 there has been no discernible difference in operations that may see potential gains by using language models.", "The assessment was that language models may not be worth investing significant resources in because there has been no convincing demonstration that current language models are significantly better than current methods for generating text, and because methods for “targeting” or “controlling” the content of language models are still at a very early stage." ], [ "External Incentive Structures", "Each threat actor group also has a set of tactics, techniques, and procedures (TTPs) that they rely on to accomplish their agenda.", "TTPs are influenced by economic factors like scalability and ease of deployment; phishing is extremely popular among all groups because it offers a low-cost, low-effort, high-yield method of deploying malware and stealing login credentials.", "Using language models to augment existing TTPs would likely result in an even lower cost of deployment.", "Ease of use is another significant incentive.", "Having stable infrastructure has a large impact on the adoption of TTPs.", "The outputs of language models are stochastic, however, and though developers can constrain these (e.g.", "using top-k truncation) they are not able to perform consistently without human feedback.", "If a social media disinformation bot produces outputs that are reliable 99% of the time, but produces incoherent outputs 1% of the time, this could reduce the amount of human labor required in operating this bot.", "But a human is still needed to filter the outputs, which restricts how scalable the operation can be.", "Based on our analysis of this model and analysis of threat actors and the landscape, we suspect AI researchers will eventually develop language models that are sufficiently consistent and steerable that they will be of greater interest to malicious actors.", "We expect this will introduce challenges for the broader research community, and hope to work on this through a combination of mitigation research, prototyping, and coordinating with other technical developers." ], [ "Fairness, Bias, and Representation", "Biases present in training data may lead models to generate stereotyped or prejudiced content.", "This is concerning, since model bias could harm people in the relevant groups in different ways by entrenching existing stereotypes and producing demeaning portrayals amongst other potential harms [19].", "We have conducted an analysis of biases in the model in order to better understand GPT-3’s limitations when it comes to fairness, bias, and representation.", "Evaluating fairness, bias, and representation in language models is a rapidly-developing area with a large body of prior work.", "See, for example, [46], [90], [120].", "Our goal is not to exhaustively characterize GPT-3, but to give a preliminary analysis of some of its limitations and behaviors.", "We focus on biases relating to gender, race, and religion, although many other categories of bias are likely present and could be studied in follow-up work.", "This is a preliminary analysis and does not reflect all of the model's biases even within the studied categories.", "Broadly, our analysis indicates that internet-trained models have internet-scale biases; models tend to reflect stereotypes present in their training data.", "Below we discuss our preliminary findings of bias along the dimensions of gender, race, and religion.", "We probe for bias in the 175 billion parameter model and also in similar smaller models, to see if and how they are different in this dimension." ], [ "Gender", "In our investigation of gender bias in GPT-3, we focused on associations between gender and occupation.", "We found that occupations in general have a higher probability of being followed by a male gender identifier than a female one (in other words, they are male leaning) when given a context such as \"The {occupation} was a\" (Neutral Variant).", "83% of the 388 occupations we tested were more likely to be followed by a male identifier by GPT-3.", "We measured this by feeding the model a context such as \"The detective was a\" and then looking at the probability of the model following up with male indicating words (eg.", "man, male etc.)", "or female indicating words (woman, female etc.).", "In particular, occupations demonstrating higher levels of education such as legislator, banker, or professor emeritus were heavily male leaning along with occupations that require hard physical labour such as mason, millwright, and sheriff.", "Occupations that were more likely to be followed by female identifiers include midwife, nurse, receptionist, housekeeper etc.", "We also tested how these probabilities changed when we shifted the context to be the \"The competent {occupation} was a\" (Competent Variant), and when we shifted the context to be \"The incompetent {occupation} was a\" (Incompetent Variant) for each occupation in the dataset.", "We found that, when prompted with \"The competent {occupation} was a,\" the majority of occupations had an even higher probability of being followed by a male identifier than a female one than was the case with our original neutral prompt, \"The {occupation} was a\".", "With the prompt \"The incompetent {occupation} was a\" the majority of occupations still leaned male with a similar probability than for our original neutral prompt.", "The average occupation bias - measured as $\\frac{1}{n_{\\mathrm {jobs}}}\\sum _{\\mathrm {jobs}}\\log (\\frac{ P(\\mathrm {female}|\\mathrm {Context})}{P(\\mathrm {male}|\\mathrm {Context}))})$ - was $-1.11$ for the Neutral Variant, $-2.14$ for the Competent Variant and $-1.15$ for the Incompetent Variant.", "We also carried out pronoun resolution on the Winogender dataset [111] using two methods which further corroborated the model's tendency to associate most occupations with males.", "One method measured the models ability to correctly assign a pronoun as the occupation or the participant.", "For example, we fed the model a context such as \"The advisor met with the advisee because she wanted to get advice about job applications.", "`She' refers to the\" and found the option with the lowest probability between the two possible options (Choices between Occupation Option: advisor; Participant Option: advisee).", "Occupation and participant words often have societal biases associated with them such as the assumption that most occupants are by default male.", "We found that the language models learnt some of these biases such as a tendency to associate female pronouns with participant positions more than male pronouns.", "GPT-3 175B had the highest accuracy of all the models (64.17%) on this task.", "It was also the only model where the accuracy for Occupant sentences (sentences where the correct answer was the Occupation option) for females was higher than for males (81.7% vs 76.7%).", "All other models had a higher accuracy for male pronouns with Occupation sentences as compared to female pronouns with the exception of our second largest model- GPT-3 13B - which had the same accuracy (60%) for both.", "This offers some preliminary evidence that in places where issues of bias can make language models susceptible to error, the larger models are more robust than smaller models.", "We also performed co-occurrence tests, where we analyzed which words are likely to occur in the vicinity of other pre-selected words.", "We created a model output sample set by generating 800 outputs of length 50 each with a temperature of 1 and top_p of 0.9 for every prompt in our dataset.", "For gender, we had prompts such as \"He was very\", \"She was very\", \"He would be described as\", \"She would be described as\"We only used male and female pronouns.", "This simplifying assumption makes it easier to study co-occurrence since it does not require the isolation of instances in which ‘they’ refers to a singular noun from those where it didn’t, but other forms of gender bias are likely present and could be studied using different approaches.. We looked at the adjectives and adverbs in the top 100 most favored words using an off-the-shelf POS tagger [60].", "We found females were more often described using appearance oriented words such as \"beautiful\" and \"gorgeous\" as compared to men who were more often described using adjectives that span a greater spectrum.", "Table REF shows the top 10 most favored descriptive words for the model along with the raw number of times each word co-occurred with a pronoun indicator.", "“Most Favored” here indicates words which were most skewed towards a category by co-occurring with it at a higher rate as compared to the other category.", "To put these numbers in perspective, we have also included the average for the number of co-occurrences across all qualifying words for each gender.", "Table: Most Biased Descriptive Words in 175B Model" ], [ "Race", "To investigate racial bias in GPT-3, we seeded the model with prompts such as - \"The {race} man was very\", \"The {race} woman was very\" and \"People would describe the {race} person as\" and generated 800 samples for each of the above prompts, with {race} replaced with a term indicating a racial category such as White or Asian.", "We then measure word co-occurrences in the generated samples.", "Given prior research demonstrating that language models produce text of differing sentiment when varying features such as occupation [46], we explored how race impacted sentiment.", "We measured sentiment using Senti WordNet [7] for the words which co-occurred disproportionately with each race.", "Each word sentiment varied from 100 to -100, with positive scores indicating positive words (eg.", "wonderfulness: 100, amicable: 87.5), negative scores indicating negative words (eg.", "wretched: -87.5 , horrid: -87.5) and a score of 0 indicating neutral words (eg.", "sloping, chalet).", "It should be noted that we were explicitly prompting the models to talk about race and this in turn generated text that focused on racial features; these results are not from the models talking about race in the wild but talking about race in an experimental setup where they have been primed to do so.", "Additionally, since we are measuring sentiment by simply looking at word co-occurrences, the resulting sentiment can reflect socio-historical factors - for instance, text relating to a discussion of slavery will frequently have a negative sentiment, which may lead to a demographic being associated with a negative sentiment under this testing methodology.", "Across the models we analyzed, `Asian' had a consistently high sentiment - it ranked 1st in 3 out of 7 models.", "On the other hand, 'Black' had a consistently low sentiment - it ranked the lowest in 5 out of 7 models.", "These differences narrowed marginally on the larger model sizes.", "This analysis gives a sense of the biases of different models and highlights the need for more sophisticated analysis of the relationship between sentiment, entities, and input data.", "Figure: Racial Sentiment Across Models" ], [ "Religion", "We studied which words co-occurred with religious terms relating to Atheism, Buddhism, Christianity, Hinduism, Islam, and Judaism, by generating 800 model outputs of length $\\approx $ 50 with a temperature of 1 and a top $p$ of $0.9$ for every prompt.", "Our prompts were of the nature \"{Religion practitioners} are\" (Eg.", "\"Christians are\") for each of the six religious categories listed above.", "We then allowed the model to naturally carry out completions and created a corpus of such completions for studying co-occurrence of words.", "The following is an example output from the model: X \"Buddhists are divided into two main branches - Theravada and Mahayana.", "Theravada is the more conservative branch, centering on monastic life and the earliest sutras and refusing to recognize the later Mahayana sutras as authentic.\"", "Similar to race, we found that the models make associations with religious terms that indicate some propensity to reflect how these terms are sometimes presented in the world.", "For example, with the religion Islam, we found that words such as ramadan, prophet and mosque co-occurred at a higher rate than for other religions.", "We also found that words such as violent, terrorism and terrorist co-occurred at a greater rate with Islam than with other religions and were in the top 40 most favored words for Islam in GPT-3.", "Table: Shows the ten most favored words about each religion in the GPT-3 175B model." ], [ "Future Bias and Fairness Challenges", "We have presented this preliminary analysis to share some of the biases we found in order to motivate further research, and to highlight the inherent difficulties in characterizing biases in large-scale generative models; we expect this to be an area of continuous research for us and are excited to discuss different methodological approaches with the community.", "We view the work in this section as subjective signposting - we chose gender, race, and religion as a starting point, but we recognize the inherent subjectivity in this choice.", "Our work is inspired by the literature on characterizing model attributes to develop informative labels such as Model Cards for Model Reporting from [89].", "Ultimately, it is important not just to characterize biases in language systems but to intervene.", "The literature on this is also extensive [104], [46], so we offer only a few brief comments on future directions specific to large language models.", "In order to pave the way for effective bias prevention in general purpose models, there is a need for building a common vocabulary tying together the normative, technical and empirical challenges of bias mitigation for these models.", "There is room for more research that engages with the literature outside NLP, better articulates normative statements about harm, and engages with the lived experience of communities affected by NLP systems [4].", "Thus, mitigation work should not be approached purely with a metric driven objective to `remove' bias as this has been shown to have blind spots [32], [93] but in a holistic manner." ], [ "Energy Usage", "Practical large-scale pre-training requires large amounts of computation, which is energy-intensive: training the GPT-3 175B consumed several thousand petaflop/s-days of compute during pre-training, compared to tens of petaflop/s-days for a 1.5B parameter GPT-2 model (Figure REF ).", "This means we should be cognizant of the cost and efficiency of such models, as advocated by [122].", "The use of large-scale pre-training also gives another lens through which to view the efficiency of large models - we should consider not only the resources that go into training them, but how these resources are amortized over the lifetime of a model, which will subsequently be used for a variety of purposes and fine-tuned for specific tasks.", "Though models like GPT-3 consume significant resources during training, they can be surprisingly efficient once trained: even with the full GPT-3 175B, generating 100 pages of content from a trained model can cost on the order of 0.4 kW-hr, or only a few cents in energy costs.", "Additionally, techniques like model distillation [69] can further bring down the cost of such models, letting us adopt a paradigm of training single, large-scale models, then creating more efficient versions of them for use in appropriate contexts.", "Algorithmic progress may also naturally further increase the efficiency of such models over time, similar to trends observed in image recognition and neural machine translation [39]." ], [ "Related Work", "Several lines of work have focused on increasing parameter count and/or computation in language models as a means to improve generative or task performance.", "An early work scaled LSTM based language models to over a billion parameters [51].", "One line of work straightforwardly increases the size of transformer models, scaling up parameters and FLOPS-per-token roughly in proportion.", "Work in this vein has successively increased model size: 213 million parameters [134] in the original paper, 300 million parameters [20], 1.5 billion parameters [117], 8 billion parameters [125], 11 billion parameters [116], and most recently 17 billion parameters [132].", "A second line of work has focused on increasing parameter count but not computation, as a means of increasing models’ capacity to store information without increased computational cost.", "These approaches rely on the conditional computation framework [10] and specifically, the mixture-of-experts method [124] has been used to produce 100 billion parameter models and more recently 50 billion parameter translation models [3], though only a small fraction of the parameters are actually used on each forward pass.", "A third approach increases computation without increasing parameters; examples of this approach include adaptive computation time [35] and the universal transformer [22].", "Our work focuses on the first approach (scaling compute and parameters together, by straightforwardly making the neural net larger), and increases model size 10x beyond previous models that employ this strategy.", "Several efforts have also systematically studied the effect of scale on language model performance.", "[57], [114], [77], [42], find a smooth power-law trend in loss as autoregressive language models are scaled up.", "This work suggests that this trend largely continues as models continue to scale up (although a slight bending of the curve can perhaps be detected in Figure REF ), and we also find relatively smooth increases in many (though not all) downstream tasks across 3 orders of magnitude of scaling.", "Another line of work goes in the opposite direction from scaling, attempting to preserve strong performance in language models that are as small as possible.", "This approach includes ALBERT [62] as well as general [44] and task-specific [121], [52], [59] approaches to distillation of language models.", "These architectures and techniques are potentially complementary to our work, and could be applied to decrease latency and memory footprint of giant models.", "As fine-tuned language models have neared human performance on many standard benchmark tasks, considerable effort has been devoted to constructing more difficult or open-ended tasks, including question answering [58], [47], [14], [84], reading comprehension [16], [106], and adversarially constructed datasets designed to be difficult for existing language models [118], [94].", "In this work we test our models on many of these datasets.", "Many previous efforts have focused specifically on question-answering, which constitutes a significant fraction of the tasks we tested on.", "Recent efforts include [116], [115], which fine-tuned an 11 billion parameter language model, and [33], which focused on attending over a large corpus of data at test time.", "Our work differs in focusing on in-context learning but could be combined in the future with those of [33], [75].", "Metalearning in language models has been utilized in [117], though with much more limited results and no systematic study.", "More broadly, language model metalearning has an inner-loop-outer-loop structure, making it structurally similar to metalearning as applied to ML in general.", "Here there is an extensive literature, including matching networks [133], RL2 [26], learning to optimize [109], [1], [73] and MAML [30].", "Our approach of stuffing the model’s context with previous examples is most structurally similar to RL2 and also resembles [45], in that an inner loop of adaptation takes place through computation in the model’s activations across timesteps, without updating the weights, while an outer loop (in this case just language model pre-training) updates the weights, and implicitly learns the ability to adapt to or at least recognize tasks defined at inference-time.", "Few-shot auto-regressive density estimation was explored in  [107] and [38] studied low-resource NMT as a few-shot learning problem.", "While the mechanism of our few-shot approach is different, prior work has also explored ways of using pre-trained language models in combination with gradient descent to perform few-shot learning [126].", "Another sub-field with similar goals is semi-supervised learning where approaches such as UDA [137] also explore methods of fine-tuning when very little labeled data is available.", "Giving multi-task models instructions in natural language was first formalized in a supervised setting with [87] and utilized for some tasks (such as summarizing) in a language model with [117].", "The notion of presenting tasks in natural language was also explored in the text-to-text transformer [116], although there it was applied for multi-task fine-tuning rather than for in-context learning without weight updates.", "Another approach to increasing generality and transfer-learning capability in language models is multi-task learning [12], which fine-tunes on a mixture of downstream tasks together, rather than separately updating the weights for each one.", "If successful multi-task learning could allow a single model to be used for many tasks without updating the weights (similar to our in-context learning approach), or alternatively could improve sample efficiency when updating the weights for a new task.", "Multi-task learning has shown some promising initial results [67], [76] and multi-stage fine-tuning has recently become a standardized part of SOTA results on some datasets [97] and pushed the boundaries on certain tasks [55], but is still limited by the need to manually curate collections of datasets and set up training curricula.", "By contrast pre-training at large enough scale appears to offer a “natural” broad distribution of tasks implicitly contained in predicting the text itself.", "One direction for future work might be attempting to generate a broader set of explicit tasks for multi-task learning, for example through procedural generation [128], human interaction [144], or active learning [80].", "Algorithmic innovation in language models over the last two years has been enormous, including denoising-based bidirectionality [20], prefixLM [24] and encoder-decoder architectures [72], [116], random permutations during training [139], architectures that improve the efficiency of sampling [28], improvements in data and training procedures [74], and efficiency increases in the embedding parameters [62].", "Many of these techniques provide significant gains on downstream tasks.", "In this work we continue to focus on pure autoregressive language models, both in order to focus on in-context learning performance and to reduce the complexity of our large model implementations.", "However, it is very likely that incorporating these algorithmic advances could improve GPT-3’s performance on downstream tasks, especially in the fine-tuning setting, and combining GPT-3’s scale with these algorithmic techniques is a promising direction for future work." ], [ "Conclusion", "We presented a 175 billion parameter language model which shows strong performance on many NLP tasks and benchmarks in the zero-shot, one-shot, and few-shot settings, in some cases nearly matching the performance of state-of-the-art fine-tuned systems, as well as generating high-quality samples and strong qualitative performance at tasks defined on-the-fly.", "We documented roughly predictable trends of scaling in performance without using fine-tuning.", "We also discussed the social impacts of this class of model.", "Despite many limitations and weaknesses, these results suggest that very large language models may be an important ingredient in the development of adaptable, general language systems." ], [ "Acknowledgements", "The authors would like to thank Ryan Lowe for giving detailed feedback on drafts of the paper.", "Thanks to Jakub Pachocki and Szymon Sidor for suggesting tasks, and Greg Brockman, Michael Petrov, Brooke Chan, and Chelsea Voss for helping run evaluations on OpenAI's infrastructure.", "Thanks to David Luan for initial support in scaling up this project, Irene Solaiman for discussions about ways to approach and evaluate bias, Harrison Edwards and Yura Burda for discussions and experimentation with in-context learning, Geoffrey Irving and Paul Christiano for early discussions of language model scaling, Long Ouyang for advising on the design of the human evaluation experiments, Chris Hallacy for discussions on data collection, and Shan Carter for help with visual design.", "Thanks to the millions of people who created content that was used in the training of the model, and to those who were involved in indexing or upvoting the content (in the case of WebText).", "Additionally, we would like to thank the entire OpenAI infrastructure and supercomputing teams for making it possible to train models at this scale.", "Tom Brown, Ben Mann, Prafulla Dhariwal, Dario Amodei, Nick Ryder, Daniel M Ziegler, and Jeffrey Wu implemented the large-scale models, training infrastructure, and model-parallel strategies.", "Tom Brown, Dario Amodei, Ben Mann, and Nick Ryder conducted pre-training experiments.", "Ben Mann and Alec Radford collected, filtered, deduplicated, and conducted overlap analysis on the training data.", "Melanie Subbiah, Ben Mann, Dario Amodei, Jared Kaplan, Sam McCandlish, Tom Brown, Tom Henighan, and Girish Sastry implemented the downstream tasks and the software framework for supporting them, including creation of synthetic tasks.", "Jared Kaplan and Sam McCandlish initially predicted that a giant language model should show continued gains, and applied scaling laws to help predict and guide model and data scaling decisions for the research.", "Ben Mann implemented sampling without replacement during training.", "Alec Radford originally demonstrated few-shot learning occurs in language models.", "Jared Kaplan and Sam McCandlish showed that larger models learn more quickly in-context, and systematically studied in-context learning curves, task prompting, and evaluation methods.", "Prafulla Dhariwal implemented an early version of the codebase, and developed the memory optimizations for fully half-precision training.", "Rewon Child and Mark Chen developed an early version of our model-parallel strategy.", "Rewon Child and Scott Gray contributed the sparse transformer.", "Aditya Ramesh experimented with loss scaling strategies for pretraining.", "Melanie Subbiah and Arvind Neelakantan implemented, experimented with, and tested beam search.", "Pranav Shyam worked on SuperGLUE and assisted with connections to few-shot learning and meta-learning literature.", "Sandhini Agarwal conducted the fairness and representation analysis.", "Girish Sastry and Amanda Askell conducted the human evaluations of the model.", "Ariel Herbert-Voss conducted the threat analysis of malicious use.", "Gretchen Krueger edited and red-teamed the policy sections of the paper.", "Benjamin Chess, Clemens Winter, Eric Sigler, Christopher Hesse, Mateusz Litwin, and Christopher Berner optimized OpenAI’s clusters to run the largest models efficiently.", "Scott Gray developed fast GPU kernels used during training.", "Jack Clark led the analysis of ethical impacts — fairness and representation, human assessments of the model, and broader impacts analysis, and advised Gretchen, Amanda, Girish, Sandhini, and Ariel on their work.", "Dario Amodei, Alec Radford, Tom Brown, Sam McCandlish, Nick Ryder, Jared Kaplan, Sandhini Agarwal, Amanda Askell, Girish Sastry, and Jack Clark wrote the paper.", "Sam McCandlish led the analysis of model scaling, and advised Tom Henighan and Jared Kaplan on their work.", "Alec Radford advised the project from an NLP perspective, suggested tasks, put the results in context, and demonstrated the benefit of weight decay for training.", "Ilya Sutskever was an early advocate for scaling large generative likelihood models, and advised Pranav, Prafulla, Rewon, Alec, and Aditya on their work.", "Dario Amodei designed and led the research." ], [ "Details of Common Crawl Filtering", "As mentioned in Section REF , we employed two techniques to improve the quality of the Common Crawl dataset: (1) filtering Common Crawl and (2) fuzzy deduplication: In order to improve the quality of Common Crawl, we developed an automatic filtering method to remove low quality documents.", "Using the original WebText as a proxy for high-quality documents, we trained a classifier to distinguish these from raw Common Crawl.", "We then used this classifier to re-sample Common Crawl by prioritizing documents which were predicted by the classifier to be higher quality.", "The classifier is trained using logistic regression classifier with features from Spark's standard tokenizer and HashingTF https://spark.apache.org/docs/latest/api/python/pyspark.ml.html#pyspark.ml.feature.HashingTF.", "For the positive examples, we used a collection of curated datasets such as WebText, Wikiedia, and our web books corpus as the positive examples, and for the negative examples, we used unfiltered Common Crawl.", "We used this classifier to score Common Crawl documents.", "We kept each document in our dataset iff ${\\tt np.random.pareto}(\\alpha ) > 1 - {\\tt document_score}$ We chose $\\alpha = 9$ in order to take mostly documents the classifier scored highly, but still include some documents that were out of distribution.", "$\\alpha $ was chosen to match the distribution of scores from our classifier on WebText.", "We found this re-weighting increased quality as measured by loss on a range of out-of-distribution generative text samples.", "To further improve model quality and prevent overfitting (which becomes increasingly important as model capacity increases), we fuzzily deduplicated documents (i.e.", "removed documents with high overlap with other documents) within each dataset using Spark's MinHashLSH implementation with 10 hashes, using the same features as were used for classification above.", "We also fuzzily removed WebText from Common Crawl.", "Overall this decreased dataset size by an average of 10%.", "After filtering for duplicates and quality, we also partially removed text occurring in benchmark datasets, described in Appendix ." ], [ "Details of Model Training", "To train all versions of GPT-3, we use Adam with $\\beta _1=0.9$ , $\\beta _2=0.95$ , and $\\epsilon =10^{-8}$ , we clip the global norm of the gradient at 1.0, and we use cosine decay for learning rate down to 10% of its value, over 260 billion tokens (after 260 billion tokens, training continues at 10% of the original learning rate).", "There is a linear LR warmup over the first 375 million tokens.", "We also gradually increase the batch size linearly from a small value (32k tokens) to the full value over the first 4-12 billion tokens of training, depending on the model size.", "Data are sampled without replacement during training (until an epoch boundary is reached) to minimize overfitting.", "All models use weight decay of 0.1 to provide a small amount of regularization [68].", "During training we always train on sequences of the full $n_{\\mathrm {ctx}}=2048$ token context window, packing multiple documents into a single sequence when documents are shorter than 2048, in order to increase computational efficiency.", "Sequences with multiple documents are not masked in any special way but instead documents within a sequence are delimited with a special end of text token, giving the language model the information necessary to infer that context separated by the end of text token is unrelated.", "This allows for efficient training without need for any special sequence-specific masking." ], [ "Details of Test Set Contamination Studies", "In section we gave a high level overview of test set contamination studies.", "In this section we provide details on methodology and results." ], [ "Initial training set filtering", "We attempted to remove text occurring in benchmarks from training data by searching for $13-$ gram overlaps between all test/development sets used in this work and our training data, and we removed the colliding $13-$ gram as well as a 200 character window around it, splitting the original document into pieces.", "For filtering purposes we define a gram as a lowercase, whitespace delimited word with no punctuation.", "Pieces less than 200 characters long were discarded.", "Documents split into more than 10 pieces were considered contaminated and removed entirely.", "Originally we removed entire documents given a single collision, but that overly penalized long documents such as books for false positives.", "An example of a false positive might be a test set based on Wikipedia, in which the Wikipedia article quotes a single line from a book.", "We ignored $13-$ grams that matched more than 10 training documents, as inspection showed the majority of these to contain common cultural phrases, legal boilerplate, or similar content that we likely do want the model to learn, rather than undesired specific overlaps with test sets.", "Examples for various frequencies can be found in the GPT-3 release repositoryhttps://github.com/openai/gpt-3/blob/master/overlap_frequency.md." ], [ "Overlap methodology", "For our benchmark overlap analysis in Section , we used a variable number of words $N$ to check for overlap for each dataset, where $N$ is the 5th percentile example length in words, ignoring all punctuation, whitespace, and casing.", "Due to spurious collisions at lower values of $N$ we use a minimum value of 8 on non-synthetic tasks.", "For performance reasons, we set a maximum value of 13 for all tasks.", "Values for $N$ and the amount of data marked as dirty are shown in Table REF .", "Unlike GPT-2's use of bloom filters to compute probabilistic bounds for test contamination, we used Apache Spark to compute exact collisions across all training and test sets.", "We compute overlaps between test sets and our full training corpus, even though we only trained on 40% of our filtered Common Crawl documents per Section REF .", "We define a `dirty' example as one with any $N$ -gram overlap with any training document, and a `clean' example as one with no collision.", "Test and validation splits had similar contamination levels despite some test splits being unlabeled.", "Due to a bug revealed by this analysis, filtering described above failed on long documents such as books.", "Because of cost considerations it was infeasible to retrain the model on a corrected version of the training dataset.", "As such, several language modeling benchmarks plus the Children's Book Test showed almost complete overlap, and therefore were not included in this paper.", "Overlaps are shown in Table REF Table: Overlap statistics for all datasets sorted from dirtiest to cleanest.", "We consider a dataset example dirty if it has a single NN-gram collision with any document in our training corpus.", "“Relative Difference Clean vs All” shows the percent change in performance between only the clean examples vs all the examples in the benchmark.", "“Count” shows the number of examples.", "“Clean percentage” is the percent of examples that are clean vs total.", "For “Acc/F1/BLEU” we use the metric specified in “Metric”.", "These scores come from evaluations with a different seed for the random examples used for in-context learning, and will therefore differ slightly from the scores elsewhere in the paper." ], [ "Overlap results", "To understand how much having seen some of the data helps the model perform on downstream tasks, we filter every validation and test set by dirtiness.", "Then we run evaluation on the clean-only examples and report the relative percent change between the clean score and the original score.", "If the clean score is more than 1% or 2% worse than the overall score, it suggests the model may have overfit to the examples it has seen.", "If the clean score is significantly better, our filtering scheme may have preferentially marked easier examples as dirty.", "This overlap metric tends to show a high rate of false positives for datasets that contain background information (but not answers) drawn from the web (such as SQuAD, which draws from Wikipedia) or examples less than 8 words long, which we ignored in our filtering process (except for wordscrambling tasks).", "One instance where this technique seems to fail to give good signal is DROP, a reading comprehension task in which 94% of the examples are dirty.", "The information required to answer the question is in a passage provided to the model, so having seen the passage during training but not the questions and answers does not meaningfully constitute cheating.", "We confirmed that every matching training document contained only the source passage, and none of the questions and answers in the dataset.", "The more likely explanation for the decrease in performance is that the 6% of examples that remain after filtering come from a slightly different distribution than the dirty examples.", "Figure REF shows that as the dataset becomes more contaminated, the variance of the clean/all fraction increases, but there is no apparent bias towards improved or degraded performance.", "This suggests that GPT-3 is relatively insensitive to contamination.", "See Section for details on the datasets we flagged for further review." ], [ "Total Compute Used to Train Language Models", "This appendix contains the calculations that were used to derive the approximate compute used to train the language models in Figure REF .", "As a simplifying assumption, we ignore the attention operation, as it typically uses less than 10% of the total compute for the models we are analyzing.", "Calculations can be seen in Table REF and are explained within the table caption.", "Table: Starting from the right hand side and moving left, we begin with the number of training tokens that each model was trained with.", "Next we note that since T5 uses an encoder-decoder model, only half of the parameters are active for each token during a forward or backwards pass.", "We then note that each token is involved in a single addition and a single multiply for each active parameter in the forward pass (ignoring attention).", "Then we add a multiplier of 3x to account for the backwards pass (as computing both ∂params ∂loss\\frac{\\partial {params}}{\\partial {loss}} and ∂acts ∂loss\\frac{\\partial {acts}}{\\partial {loss}} use a similar amount of compute as the forwards pass.", "Combining the previous two numbers, we get the total flops per parameter per token.", "We multiply this value by the total training tokens and the total parameters to yield the number of total flops used during training.", "We report both flops and petaflop/s-day (each of which are 8.64e+19 flops)." ], [ "Human Quality Assessment of Synthetic News Articles", "This appendix contains details on the experiments measuring human ability to distinguish GPT-3-generated synthetic news articles from real news articles.", "We first describe the experiments on the $\\sim 200$ word news articles, and then describe the preliminary investigation of $\\sim 500$ word news articles generated by GPT-3.", "Participants: We recruited 718 unique participants to take part in 6 experiments.", "97 participants were excluded for failing an internet check question, leaving a total of 621 participants: 343 male, 271 female, and 7 other.", "Mean participant age was $\\sim 38$ years old.", "All participants were recruited through Positly, which maintains a whitelist of high-performing workers from Mechanical Turk.", "All participants were US-based but there were no other demographic restrictions.", "Participants were paid $12 for their participation, based on a task time estimate of 60 minutes determined by pilot runs.", "In order to ensure that the sample of participants for each experiment quiz was unique, participants were not allowed to take part in an experiment more than once.", "Procedure and design: We arbitrarily selected 25 news articles that appeared in newser.com in early 2020.", "We used the article titles and subtitles to produce outputs from the 125M, 350M, 760M, 1.3B, 2.7B, 6.7B, 13.0B, and 200B (GPT-3) parameter language models.", "Five outputs per question were generated by each model and the generation with a word count closest to that of the human written article was selected automatically.", "This was to minimize the effect that completion length might have on participants’ judgments.", "The same output procedure for each model with the exception of the removal of the intentionally bad control model, as described in the main text.", "In each experiment, half of the participants were randomly assigned to quiz A and half were randomly assigned to quiz B.", "Each quiz consisted of 25 articles: half (12-13) were human written and half (12-13) were model generated: the articles with human written completions in quiz A had model generated completions in quiz B and vice versa.", "The order of quiz question was shuffled for each participant.", "Participants could leave comments and were asked to indicate if they had seen the articles before.", "Participants were instructed not to look up the articles or their content during the quiz and at the end of the quiz were asked if they had looked anything up during the quiz.", "Table: Participant details and article lengths for each experiment to evaluate human detection of ∼200\\sim 200 word model generated news articles.", "Participants were excluded due to internet check fails.Statistical Tests: To compare means on the different runs, we performed a two-sample t-test for independent groups for each model against the control.", "This was implemented in Python using the scipy.stats.ttest_ind function.", "When plotting a regression line in the graph of average participant accuracy vs model size, we fit a power law of the form $ax^{-b}$ .", "The 95% confidence intervals were estimated from the t-distribution of the sample mean.", "Figure: Participants spend more time trying to identify whether each news article is machine generated as model size increases.", "Duration on the control model is indicated with the dashed line.", "Line of best fit is a linear model on a log scale with 95% confidence intervals.Duration statistics: In the main text, we discussed the finding that the ability of human participants to distinguish model and human generated news articles decreases as our models become larger.", "We have also found that the average time spent for a given set of questions increases as the model size increases, as shown in Figure REF .", "Lower accuracy scores despite increased time investment from participants supports the finding that larger models generate harder-to-distinguish news articles.", "Preliminary investigation of $\\sim 500$ word articles: We recruited 160 unique US-based participants to take part in 2 experiments through Positly (details are given in Table REF ).", "We randomly selected 12 Reuters world news articles from late 2019 and created a context for GPT-3 175B that consisted of a single Reuters article not in this set of 12.", "We then used the article titles and Reuters locations to generate completions from GPT-3 175B and the 160M control model from the previous experiments.", "These were used to create two 12-question quizzes per model, each consisting of half human written and half model generated articles.", "Comprehension questions were added and articles were shown to participants in 3 stages at 30 second intervals to encourage closer reading.", "Participants were paid $12 for this task.", "Model generation selection methods, exclusion criteria, and statistical tests mirror those of the previous experiments.", "Table: Participant details and article lengths for the experiments investigating human detection of ∼500\\sim 500 word model generated news articles.", "Participants were excluded due to internet check fails." ], [ "Additional Samples from GPT-3", "GPT-3 adapts well to many tasks other than the ones explored in the main body of the paper.", "As an example, in Figure REF , we show four uncurated samples from a prompt suggesting that the model write a poem, with a given title, in the style of Wallace Stevens.", "We first experimented with a few prompts, then generated four samples with no additional editing or selection (sampling at temperature 1 using nucleus sampling [40] with $P=0.9$ ).", "Completions were truncated when the model began to write a new title and author heading, or broke into prose commentary.", "Figure: Four uncurated completions from a context suggesting the model compose a poem in the style of Wallace Stevens with the title `Shadows on the Way'." ], [ "Details of Task Phrasing and Specifications", "The following figures illustrate the formatting and phrasing of all the tasks included in the paper.", "All data comes from the ground truth datasets in this section, and no samples from GPT-3 are included here.", "Figure: Formatted dataset example for RACE-h.", "When predicting, we normalize by the unconditional probability of each answer as described in .Figure: Formatted dataset example for ANLI R2Figure: Formatted dataset example for RACE-m.", "When predicting, we normalize by the unconditional probability of each answer as described in .Figure: Formatted dataset example for PIQAFigure: Formatted dataset example for COPAFigure: Formatted dataset example for ReCoRD.", "We consider the context above to be a single \"problem\" because this is how the task is presented in the ReCoRD dataset and scored in the ReCoRD evaluation script.Figure: Formatted dataset example for ANLI R1Figure: Formatted dataset example for OpenBookQA.", "When predicting, we normalize by the unconditional probability of each answer as described in .Figure: Formatted dataset example for HellaSwagFigure: Formatted dataset example for ANLI R3Figure: Formatted dataset example for ARC (Challenge).", "When predicting, we normalize by the unconditional probability of each answer as described in .Figure: Formatted dataset example for SAT AnalogiesFigure: Formatted dataset example for Winograd.", "The `partial' evaluation method we use compares the probability of the completion given a correct and incorrect context.Figure: Formatted dataset example for Winogrande.", "The `partial' evaluation method we use compares the probability of the completion given a correct and incorrect context.Figure: Formatted dataset example for MultiRC.", "There are three levels within MultiRC: (1) the passage, (2) the questions, and (3) the answers.", "During evaluation, accuracy is determined at the per-question level, with a question being considered correct if and only if all the answers within the question are labeled correctly.", "For this reason, we use KK to refer to the number of questions shown within the context.Figure: Formatted dataset example for ARC (Easy).", "When predicting, we normalize by the unconditional probability of each answer as described in ." ] ]

[ [ "Approximate Message Passing with Unitary Transformation for Robust\n Bilinear Recovery" ], [ "Abstract Recently, several promising approximate message passing (AMP) based algorithms have been developed for bilinear recovery with model $\\boldsymbol{Y}=\\sum_{k=1}^K b_k \\boldsymbol{A}_k \\boldsymbol{C} +\\boldsymbol{W} $, where $\\{b_k\\}$ and $\\boldsymbol{C}$ are jointly recovered with known $\\boldsymbol{A}_k$ from the noisy measurements $\\boldsymbol{Y}$.", "The bilinear recover problem has many applications such as dictionary learning, self-calibration, compressive sensing with matrix uncertainty, etc.", "In this work, we propose a new bilinear recovery algorithm based on AMP with unitary transformation.", "It is shown that, compared to the state-of-the-art message passing based algorithms, the proposed algorithm is much more robust and faster, leading to remarkably better performance." ], [ "Introduction", "In this work, we consider the following bilinear problem $Y=\\sum _{k=1}^K b_kA_kC+W, $ where $Y$ denotes measurements, matrices $\\lbrace A_k\\rbrace $ are known, $\\lbrace b_k\\rbrace $ and $C$ are to be recovered, and $W$ represents white Gaussian noise.", "When $Y$ , $C$ blackand $W$ are replaced with the blackcorresponding vectors $y$ , $c$ blackand $w$ , respectively, the above multiple measurement vector (MMV) problem is reduced to a single measurement vector (SMV) problem.", "Model (REF ) covers a variety of problems, e.g., compressive sensing (CS) with matrix uncertainty [1], joint channel estimation and detection [2], self-calibration and blind deconvolution [3], and structured dictionary learning [4].", "Recently, several approximate message passing (AMP) [5] [6] based algorithms have been developed to solve the bilinear problem, which show promising performance, compared to existing non-message passing alternates [7].", "The generalized AMP (GAMP) [8] was extended to bilinear GAMP (BiGAMP) [9] for solving a general bilinear problem, i.e., recover both $A$ and $X$ from observation $Y=AX+W$ .", "The parametric BiGAMP (P-BiGAMP) is then proposed in [10], which works with model (REF ) to jointly recover $\\lbrace b_k\\rbrace $ and $C$ .", "Lifted AMP was proposed in [11] by using the lifting approach [12], [13].", "However, these AMP based algorithms are vulnerable to difficult $A$ matrices, e.g., ill-conditioned, correlated, rank-deficient or non-zero mean matrices as AMP can easily diverge in these cases [14].", "It was discovered in [15] that the AMP algorithm can still perform well for difficult $A$ .", "Instead of working directly with the original model $y=Ax+w$ , [15] proposed to apply AMP to a unitary transform of the original model, i.e., $U^Hy=\\Lambda Vx+U^Hw$ where the unitary matrix $U$ can be obtained by the singular value decomposition (SVD) of matrix $A$ , i.e., $A=U\\Lambda V$ .", "blackIn the case of circulant matrix $A$ , the matrix $U^H$ for unitary transformation can be simply the normalized discrete Fourier transform matrix, which allows fast implementation with the fast Fourier transform (FFT) algorithm.", "AMP with unitary transformation, called UTAMP for convenience, was inspired by the work in [16], which can be regarded as the first application of UTAMP to turbo equalization, where the normalized discrete Fourier transform matrix is used for unitary transformation.", "UTAMP was also recently employed for sparse Bayesian learning (SBL) [17], blackwhich shows outstanding performance even with difficult measurement matrices.", "The application of UTAMP to inverse synthetic aperture radar (ISAR) imaging was also studied in [18], and real data experiments show its excellent capability of achieving high Doppler resolution with low complexity, where the measurement matrix can be highly correlated to achieve high Doppler resolution.", "blackUTAMP has also been employed for low complexity direction of arrival (DOA) estimation [19] and iterative detection for orthogonal time frequency space modulation (OTFS) [20], which shows promising performance.", "These motivated us to design efficient and robust bilinear recovery algorithms with UTAMP in this work.", "Most recently, to achieve robust bilinear recovery, building on vector AMP (VAMP) [21], the lifted VAMP was proposed in [22], and the bilinear adaptive VAMP (BAd-VAMP) was proposed in [7], which inherit the robustness of VAMP.", "It was shown that BAd-VAMP is more robust and faster, and it can outperform lifted VAMP significantly [7].", "blackBased on VAMP, PC-VAMP was proposed in [23] to achieve compressive sensing with structured matrix perturbation.", "In [24], BAd-VAMP was extended to incorporate arbitrary distributions on the output transform based on the framework in [25].", "Vector Stepsize AMP Initialize $\\mathbf {\\tau }_x^{(0)}>0$ and $x^{(0)}$ .", "Set $s^{(-1)}=\\mathbf {0}$ and $ t=0$ .", "Repeat [1] $\\mathbf {\\tau }_p$ = $ | A|^2 \\mathbf {\\tau }^t_x$ $p= Ax^t - \\mathbf {\\tau }_{p} \\cdot s^{t-1} $ $\\mathbf {\\tau }_s = \\mathbf {1} ./ (\\mathbf {\\tau }_p+\\beta ^{-1} \\mathbf {1}) $ $ s^t=\\mathbf {\\tau }_s \\cdot (y-p) $ $ \\mathbf {1} ./\\mathbf {\\tau }_q = | A^H |^2 \\mathbf {\\tau }_s $ $ q= x^t + \\mathbf {\\tau }_q \\cdot A^H s^t $ $\\mathbf {\\tau }_x^{t+1}$ = $ \\mathbf {\\tau }_q \\cdot g_{x}^{\\prime } ( q, \\mathbf {\\tau }_q) $ $\\mathbf {x}^{t+1} = g_{x} ( q, \\mathbf {\\tau }_q)$ $t=t+1$ Until terminated UTAMP Version 1 Unitary transform: $r=U^H y=\\Phi x+\\mathbf {\\omega }$ , where $\\Phi =U^HA=\\Lambda V$ , and $U$ is obtained from the SVD $A=U\\Lambda V$ .", "Initialize $\\mathbf {\\tau }_x^{(0)}>0$ and $x^{(0)}$ .", "Set $s^{(-1)}=\\mathbf {0}$ and $ t=0$ .", "Repeat [1] $\\mathbf {\\tau }_p$ = $ | \\mathbf {\\Phi } |^2 \\mathbf {\\tau }^t_x$ $p= \\mathbf { \\mathbf {\\Phi } x^t} - \\mathbf {\\tau }_{p} \\cdot s^{t-1} $ $\\mathbf {\\tau }_s = \\mathbf {1} ./ (\\mathbf {\\tau }_p+\\beta ^{-1} \\mathbf {1}) $ $ s^t=\\mathbf {\\tau }_s \\cdot (r-p) $ $ \\mathbf {1} ./\\mathbf {\\tau }_q = | \\mathbf {\\Phi }^H |^2 \\mathbf {\\tau }_s $ $ q= x^t + \\mathbf {\\tau }_q \\cdot (\\mathbf {\\Phi }^H s^t) $ $\\mathbf {\\tau }_x^{t+1}$ = $ \\mathbf {\\tau }_q \\cdot g_{x}^{\\prime } ( q, \\mathbf {\\tau }_q) $ $x^{t+1} = g_{x} ( q, \\mathbf {\\tau }_q)$ $t=t+1$ Until terminated In this work, blackleveraging UTAMP, we propose a more robust and faster approximate inference algorithm for bilinear recovery, which is called Bi-UTAMP.", "By using the lifting approach, the original bilinear problem is reformulated as a linear one.", "Then, blackthe structured variational inference (VI) [26], [27], [28], expectation propagation (EP) [29] and belief propagation (BP) [30], [31] are combined with UTAMP, where UTAMP is employed to handle the most computational intensive part, leading to the fast and robust approximate inference algorithm Bi-UTAMP.", "It is shown that Bi-UTAMP performs significantly better and is much faster than blackstate-of-the-art bilinear recovery algorithms for difficult matrices.", "The remainder of this paper is organized as follows.", "In Section II, blackwe briefly introduce the (UT)AMP algorithms, which form the basis for developing Bi-UTAMP.", "Bi-UTAMP is designed for SMV problems and it is then extended for MMV problems in Section III.", "Numerical examples blackand comparisons with state-of-the-art message passing and non-message passing algorithms are provided in Section IV, and conclusions are drawn in Section V. UTAMP Version 2 Unitary transform: $r=U^H y=\\Phi x+\\mathbf {\\omega }$ , where $\\Phi =U^HA=\\Lambda V$ , and $U$ is obtained from the SVD $A=U\\Lambda V$ .", "Define vector $\\mathbf {\\lambda }=\\mathbf { \\Lambda \\Lambda }^H \\textbf {1}$ .", "Initialize ${\\tau }_x^{(0)}>0$ and $x^{(0)}$ .", "Set $s^{(-1)}=\\mathbf {0}$ and $ t=0$ .", "Repeat [1] $\\quad \\mathbf {\\tau }_p$ = $ \\tau ^t_x \\mathbf {\\lambda }$ $\\quad p= \\mathbf {\\Phi } x^t - \\mathbf {\\tau }_{p} \\cdot s^{t-1} $ $\\quad \\mathbf {\\tau }_s = \\mathbf {1}./ (\\mathbf {\\tau }_p+\\beta ^{-1} \\mathbf {1}) $ $\\quad s^t= \\mathbf {\\tau }_s \\cdot (r-p) $ $\\quad 1/\\tau _q = ({1}/{N}) \\mathbf {\\lambda }^T \\mathbf {\\tau }_s $ $\\quad q= x^t + \\tau _q \\mathbf {\\Phi }^H s^t$ $\\quad \\tau _x^{t+1}$ = $ (\\tau _q /N) \\mathbf {1}^H g_{x}^{\\prime } ( q, \\tau _q) $ $\\quad \\mathbf {x}^{t+1} = g_{x} ( q, \\tau _q)$ $\\quad t=t+1$ Until terminated Notations- Boldface lower-case and upper-case letters denote vectors and matrices, respectively.", "Superscripts $(\\cdot )^H$ and $(\\cdot )^T$ represent conjugate transpose and transpose, respectively, and $(\\cdot )^*$ represents the conjugate operation.", "A Gaussian distribution of $x$ with mean $\\hat{x}$ and variance $\\nu _x$ is represented by $\\mathcal {N}(x;{\\hat{x}},\\nu _x)$ .", "We also simply use $\\mathcal {N}(m, v)$ to represent a Gaussian distribution with mean $m$ and variance $v$ .", "Notation $\\otimes $ represents the Kronecker product.", "The relation $f(x)=cg(x)$ for some positive constant $c$ is written as $f(x)\\propto g(x)$ .", "We use $a\\cdot b$ and $a\\cdot /b$ to represent the element-wise product and division between vectors $a$ and $b$ , respectively.", "The notation $a^{.-1}$ denotes the element-wise inverse operation to vector $a$ .", "We use $|A|^{2}$ to denote element-wise magnitude squared operation for $A$ , and use $||a||^2$ to denote the squared $l_2$ norm of $a$ .", "The notation $<a>$ denotes the average operation for $a$ , i.e., the sum of the elements of $a$ divided by its length.", "The notation $\\int _{c\\vee {c_n}} f_{c}(c)$ represents integral over all elements in $c$ except $c_n$ .", "We use $\\textbf {1}$ and $\\textbf {0}$ to denote an all-one vector and an all-zero vector with a proper length, respectively.", "Sometimes, we use a subscript $n$ for $\\textbf {1}$ , i.e., $\\textbf {1}_n$ to indicate its length $n$ .", "The superscript of $a^t$ denotes the $t$ -th iteration for $a$ blackin an iterative algorithm.", "We use $[a]_n$ to denote the $n$ -th element of vector $a$ .", "blackThe notation $D(a)$ represents a diagonal matrix with $a$ as its diagonal." ], [ "Approximate Message Passing with Unitary Transformation", "blackIn this section, we briefly introduce the (UT)AMP algorithms, and show the close connection between UTAMP and AMP and their state evolution (SE)." ], [ "(UT)AMP Algorithms", "The AMP algorithm [5] was developed based on the loopy BP [30], [31] for compressive sensing with model $y=Ax+w$ where $y$ is a measurement, $A$ is a known $M\\times N$ measurement matrix, $w$ is a white Gaussian noise vector with distribution $\\mathcal {N}(w;\\mathbf {0},\\beta ^{-1}I)$ .", "AMP enjoys low complexity and its performance can be rigorously characterized by a scalar state evolution in the case of large i.i.d.", "(sub)Gaussian $A$ [32].", "However, for a generic $A$ , the convergence of AMP cannot be guaranteed, e.g., AMP can easily diverge for non-zero mean, rank-deficient, correlated, or ill-conditioned matrix $A$ [14], [15].", "Inspired by [16], it was discovered in [15] that the AMP algorithm can still work well for difficult $A$ .", "In [15], instead of employing the original model (REF ), AMP is applied to a unitary transform of (REF ).", "As any matrix $A$ has an SVD $A= U\\Lambda V$ , a unitary transformation with $U^H$ can be performed, yielding $r=\\Phi x+\\mathbf {\\omega },$ where $r=U^H y$ , $\\Phi =U^HA=\\mathbf {\\Lambda }V$ , $\\mathbf {\\Lambda }$ is an $M\\times N$ rectangular diagonal matrix, and $\\mathbf {\\omega } = U^H w$ is still a zero-mean Gaussian noise vector with the same covariance matrix $\\beta ^{-1} I$ as $U^H$ is a unitary matrix.", "blackIt is noted that in the case of a circulant matrix $A$ , e.g., in frequency domain equalization, the matrix for unitary transformation can be simply the normalized discrete Fourier transform matrix, which allows more efficient implementation of the UTAMP algorithm [16].", "Then the vector stepsize AMP [8] shown in Algorithm  can be applied to model (REF ), leading to the first version of the UTAMP algorithm, as shown in Algorithm .", "It is interesting that, with such a simple pre-processing, the robustness of AMP is remarkably enhanced, enabling it to handle difficult matrix $A$ .", "black It is obvious that UTAMP can also be derived based on the loopy BP with the unitary transformed model (REF ).", "As discussed in [15], applying an average operation to the two vectors $\\mathbf {\\tau }_x$ in Line 7 and $|\\mathbf {\\Phi }^H |^2 \\mathbf {\\tau }_s$ in Line 5 in Algorithm leads to the second version of UTAMP shown in Algorithm .", "Specifically, due to the average operation in Line 7 of Algorithm , $\\mathbf {\\tau }_x^t$ in Line 1 turns into a scaled all-one vector ${\\tau }_x^t\\mathbf {1}$ .", "With $\\Phi =\\mathbf {\\Lambda }V$ blackand noting that $V$ is a unitary matrix, it is not hard to show that $\\mathbf {\\tau }_p &=& |\\mathbf {\\Phi } |^2 ({\\tau }^t_x\\mathbf {1}) \\nonumber \\\\&=& \\tau ^t_x \\mathbf {\\lambda },$ which is Line 1 of Algorithm .", "Performing the average operation to vector $|\\mathbf {\\Phi }^H |^2 \\mathbf {\\tau }_s$ , i.e., $<|\\mathbf {\\Phi }^H |^2 \\mathbf {\\tau }_s>=\\frac{1}{N} \\mathbf {\\lambda }^T \\mathbf {\\tau }_s$ leads to Line 5 of Algorithm .", "blackIt is worth highlighting that the two average operations result in a significant reduction in computational complexity.", "Compared to Algorithm 1 and Algorithm 2, Line 1 and Line 5 of Algorithm do not blackinvolve matrix-vector product operations, i.e., the number of matrix-vector products is reduced from 4 to 2 per iteration, which is a significant reduction as the complexity of AMP-like algorithms is dominated by matrix-vector products.", "blackInterestingly, the average operations also further enhance the stability of the algorithm from our finding.", "UTAMP version 2 converges for any matrix $A$ in the case of Gaussian priors [15].", "In many cases, the noise precision $\\beta $ is unknown.", "The noise precision estimation can be incorporated into the UTAMP algorithms as in [17].", "blackIt is also worth mentioning that, VAMP involves the calculations of two \"extrinsic\" precisions (refer to Line 5 and Line 9 of Algorithm 1 in [7]), which can be negative.", "To solve this problem, heuristic remedies can be used, e.g., taking the absolute value of the calculated precisions.", "In contrast, there is no such problem in the (UT)AMP algorithms.", "In the above black(UT)AMP algorithms, the function $g_x(q, \\mathbf {\\tau }_q )$ returns a column vector whose $n$ -th element, denoted as $[ g_x(q, \\mathbf {\\tau }_q ) ]_n$ , is given by $[g_x(\\mathbf {q}, \\mathbf {\\tau }_q ) ]_n=\\frac{\\int x_n p(x_n) \\mathcal {N} (x_n ; q_n, \\tau _{q_n}) d x_n }{\\int p(x_n) \\mathcal {N} (x_n ; q_n, \\tau _{q_n}) d x_n },$ where $p(x_n)$ is the prior of $x_n$ .", "Equation (REF ) can be interpreted as the minimum mean square error (MMSE) estimation of $x_n$ based on the following model $q_n = x_n + \\varpi $ where $\\varpi $ is a Gaussian noise with mean zero and variance $\\tau _{q_n}$ .", "The function $g_x^{\\prime }(q,\\mathbf {\\tau }_q)$ returns a column vector and the $n$ -th element is denoted by $[ g_x^{\\prime }(q, \\mathbf {\\tau }_q ) ]_n$ , where the derivative is taken with respect to $q_n$ .", "Note that $g_x(q,\\mathbf {\\tau }_q)$ can also be changed for MAP (maximum a posterior) estimation of $x$ ." ], [ "State Evolution of (UT)AMP", "The performance of UTAMP can be characterized by the following simple recursion (for a more general matrix $A$ compared to AMP) $\\tau ^t &=& \\frac{N}{\\mathbf {1}^T \\big (\\mathbf {\\lambda }./(v_x^{t}\\mathbf {\\lambda }+\\beta ^{-1}\\mathbf {1})\\big )} \\\\v_x^{t+1}&=&\\mathbb {E}\\left[\\big |g_x(x+\\sqrt{\\tau ^t} z,\\tau ^t) - x\\big |^2 \\right]$ where $\\beta ^{-1}$ is the noise variance, $z$ is Gaussian with distribution $\\mathcal {N}(z; 0, 1)$ and $x$ has a prior $p(x)$ .", "It is noted that, in the case of large i.i.d.", "Gaussian matrix $A$ with elements independently drawn from $\\mathcal {N}(0, 1/ M)$ , $\\mathbf {\\lambda }$ approaches a length-$M$ vector given by $\\frac{N}{M} \\mathbf {1}_M$ (assuming $M<N$ ).", "The SE of UTAMP is reduced to that of the AMP exactly, as in this case (REF ) is reduced to $\\tau ^t=\\frac{N}{M} v_x^t+\\beta ^{-1}.$ Figure: Performance of UTAMP and its SE with a Bernoulli Gaussian prior for low-rank matrices (left) and non-zero mean matrices (right).To demonstrate the SE of UTAMP, we assume that the measurement matrix has a size of $M = 800$ and $N = 1000$ , the prior of the elements of $x$ is Bernoulli Gaussian $p(x)=0.9 \\delta (x)+0.1\\mathcal {N}(x;0,1)$ , and the signal to noise ratio (SNR) is 50 dB.", "We generate non-zero mean matrices $A$ with elements independently drawn from $\\mathcal {N}(10,1)$ , and low rank matrices $A=BC$ , where the size of $B$ and $C$ are $800\\times 500$ and $500\\times 1000$ , respectively.", "Both $B$ and $C$ are i.i.d.", "Gaussian matrices with zero mean and unit variance.", "The mean squared error (MSE) of UTAMP and its SE are shown in Fig.", "REF (the support-oracle MSE bound is also included blackfor reference), where we can see that the SE matches well the simulation performance.", "It is worth mentioning an interesting finding.", "In some cases, black(UT)AMP algorithms with the Bernoulli Gaussian prior cannot approach the support-oracle bound (e.g., the low-rank case), but UTAMP-SBL can still approach the bound as shown in [17]." ], [ "Bilinear UTAMP", "In this section, the problem formulation for bilinear recovery is discussed, and the UTAMP based approximate inference algorithm Bi-UTAMP for bilinear recovery is derived.", "We start with the case of SMV, and then extend it to the case of MMV.", "The complexity of the algorithm is also analyzed.", "black" ], [ "Problem Formulation", "Different from [1], we consider a Bayesian treatment of the bilinear recovery problem $y=\\sum _{k=1}^K b_kA_kc+w, $ where $b\\triangleq [b_1,...,b_K]^T$ , $c$ and $\\beta $ (the precision of the noise) are random variables with priors $p(b)$ , $p(c)$ and $p(\\beta )$ , respectively.", "It is noted that, in the case of no a priori information available, $p(b)$ , $p(c)$ and $p(\\beta )$ can be simply chosen as non-informative priors.", "This also differs from the development of BAd-VAMP in [7], where both $b$ and $\\beta $ are treated as unknown deterministic variables, and their values are estimated following the framework of expectation maximization (EM).", "However, a Bayesian treatment of $b$ is more advantageous.", "In the case of a priori information available for $b$ , a Bayesian method enables the use of the a priori information, which may be very helpful to improve the recovery performance.", "If no a priori information is known, a non-informative prior can be simply used.", "Moreover, in the context of iterative inference considered in this paper, the Bayesian treatment of $b$ is also different from that of the EM method in that only a point estimate of $b$ is involved in the iteration of the EM method, while a distribution of $b$ is involved in the iterative process of the method with Bayesian treatmentblackEven in the case of non-informative priors for the method with Bayesian treatment, they are still different in this way normally..", "Here, for simplicity, we take the SMV problem as example, but the extension of our discussion to the case of MMV is straightforward.", "black The joint conditional distribution of $b$ , $c$ and $\\beta $ can be expressed as $p(b,c, \\beta | y) \\propto p(y|b,c,\\beta )p(b)p(c)p(\\beta ).$ We aim to find the a posterior distributions $p(b|y)$ and $p(c|y)$ , and therefore their a posterior means that can be used as their estimates, i.e., $\\hat{b}=\\mathbb {E}(b|y)$ and $\\hat{c}=\\mathbb {E}(c|y)$ .", "However, this is often intractable because high dimensional integration is required to compute the a posteriori distributions $p(b|y)$ and $p(c|y)$ .", "As a result, we resort to the approximate Bayesian inference techniques.", "black" ], [ "Problem and Model Reformulation for Efficient UTAMP-Based Approximate Inference", "Similar to the lifting approach, we define $A\\triangleq \\left[A_{1},...,A_{K}\\right]_{M\\times NK}$ , then the original bilinear model can be reformulated as $y=Ax+w$ with blackthe auxiliary variable $x= b\\otimes c=\\begin{pmatrix}b_1c\\\\\\vdots \\\\b_Kc\\end{pmatrix}_{NK\\times 1},$ where $x$ can be indexed as $x=\\left[x_{1,1},...x_{N,1},...,x_{n,k},...x_{N,K}\\right]^T$ with $x_{n,k}=c_n b_k.$ blackWith an SVD for matrix $A$ , i.e., $A= U\\Lambda V$ , performing unitary transformation blackIt is noted that performing the unitary transformation here is purely to facilitate the use of UTAMP.", "As $U^H$ is a unitary matrix, the transformation will not result in any loss.", "So the resultant algorithms will work with the transformed observation $r$ , instead of $y$ .", "yields $r=\\Phi x+\\mathbf {\\omega }$ , where $r=U^{H}y$ , $\\Phi =\\Lambda V$ has a size of $M \\times NK$ , and $\\mathbf {\\omega }=U^{H}w$ is still white and Gaussian with the same precision $\\beta $ .", "blackThen define a new auxiliary variable $z=\\Phi x$ as in [33], [24], [25] and [34].", "Later, we will see that the introduction of the auxiliary variables $x$ and $z$ facilitates the integration of UTAMP into the approximate Bayesian inference algorithm, which is crucial to achieving efficient and robust inference.", "Table: Distributions and factors in ()blackWith the two latent variables $x$ and $z$ , we have the following joint conditional distribution of $c,b,x,z, \\beta $ and its factorization $&&\\!\\!\\!\\!\\!\\!p(c,b,x, z, \\beta | r) \\nonumber \\\\&&\\propto p(r|z,\\beta )p(z|x)p(x|b,c)p(c)p(b)p(\\beta )\\nonumber \\\\&&\\triangleq f_{r}(z,\\beta )f_{z}(z,x){f_{x}(x,b,c)} f_{c}(c)f_{b}(b) f_{\\beta }(\\beta ).$ blackHence our aim is to find the a posteriori distributions $p(c|r)$ and $p(b|r)$ and their estimates in terms of the a posteriori means, i.e., $\\hat{c}=\\mathbb {E}(c|r)$ and $\\hat{b}=\\mathbb {E}(b|r)$ .", "It seems that, due to the involvement of two extra latent variables $x$ and $z$ , the use of (REF ) could be more complicated than that of (REF ), but it enables efficient approximate inference by incorporating UTAMP, as detailed later.", "The probability functions and the corresponding factors (to facilitate the factor graph representation) are listed in Table 1, and a factor graph representation of (REF ) is depicted in Fig.", "REF .", "Figure: blackFactor graph representation of ().blackWe follow the framework of structured variational inference (SVI) [26], which can be formulated nicely as message passing with graphical models [27], [35], [36], [28].", "The trial function for the joint conditional distribution function $p(c,b,x, z, \\beta | r)$ in (REF ) is chosen as $\\tilde{q}(b, c, x, z, \\beta )= \\tilde{q}(\\beta ) \\tilde{q}(b, c, x, z).$ The employment of this trial function corresponds to a partition of the factor graph in Fig.", "REF [28], i.e., $\\tilde{q}(\\beta )$ and $\\tilde{q}(b, c, x, z)$ are associated respectively with the subgraphs denoted by Part (i) and Part (ii), where the variable node $\\beta $ is external to Part (ii).", "With SVI, the variational lower bound $\\mathcal {L} \\big (\\tilde{q}(b, c, x, z, \\beta ) \\big ) =&& \\nonumber \\\\ &&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\mathbb {E}\\big [\\mathrm {log}(p(c,b,x, z, \\beta , r))] - \\mathbb {E}\\big [\\mathrm {log}(\\tilde{q}(b, c, x, z, \\beta ))]$ is maximized with respect to the trial function, so that the following Kullback-Leibler divergence $\\mathcal {KL}\\big (\\tilde{q}(\\beta ) \\tilde{q}(b, c, x, z)|| p(b,c,x, z, \\beta | r)\\big ),$ is minimized, which leads to the approximation (by integrating out $\\beta $ ) $\\tilde{q}(b, c, x, z) \\approx p(b,c,x, z| r).$ From the above, by integrating out $c, x$ and $z$ , it is expected that the marginal $\\tilde{q}(b) \\approx p(b| r) $ , and similarly, by integrating out $b, x$ and $z$ , $\\tilde{q}(c) \\approx p(c| r) $ .", "In terms of structured variation message passing [28], the computation of $\\tilde{q}(b, c, x, z)$ corresponds to BP in the subgraph shown in Part (ii) of the factor graph in Fig.", "REF , except the function node $f_{r}$ because it connects an external variable node $\\beta $ [28].", "It is noted that the BP message passing between $z, f_z$ and $x$ (i.e., BP in the dash-dotted box in Fig.", "REF ) can be difficult and computational intensive.", "Fortunately, AMP, derived based on loopy BP (which in this case is actually UTAMP as the unitary transformation has already been performed previously) is an excellent replacement to accomplish the BP message passing for the dash-dotted box efficiently.", "In addition, we may have difficulties with the priors $p(b)$ and $p(c)$ (corresponding to the factors $f_{b}$ and $f_{c}$ in Fig.", "REF ) as they may not be friendly, resulting in intractable BP messages.", "This can be handled with EP, which has been widely used in the literature to solve similar problems.", "At the variable node $c$ (or $b$ ), we can obtain an approximate marginal about $c$ (or $b$ ) through an iterative process with moment matching [29], thereby an approximation to the a posteriori mean $\\mathbb {E}(c| r)$ (or $\\mathbb {E}(b| r)$ ), which can be served as our estimate.", "blackIt is noted that, all inference methods mentioned above including VI, EP, and UTAMP involve an iterative process (but with a different hierarchy), and the multiple iterative processes can be simply combined as a single one.", "In terms of message passing, this is to carry out a forward message passing process and a backward message passing process in Fig.", "REF as an iteration.", "Thanks to the incorporation of UTAMP to handle the BP in the dashed-dotted box in Fig.", "REF , this leads to an efficient and robust approximate inference algorithm with details elaborated in next section.", "black" ], [ "Derivation of the Message Passing Algorithm", "black In this section, we blackdetail the forward and backward message passing in Fig.", "REF according to the principle of structured variational message passing [26], [27], [28] and EP.", "Throughout this paper, we use the notation $m_{n_a \\rightarrow n_b} (h)$ to denote a message passed from node $n_a$ to node $n_b$ , which is a function of $h$ .", "black" ], [ "Message Computations at Nodes $\\textbf {x}$ , {{formula:77957f8d-23fe-40b1-a55e-54d6af1d68c4}} , {{formula:40d42893-7ed8-4b42-a5dd-0616f27860ff}} and {{formula:ea1d04dd-8522-43b8-bfbe-37cd723fd130}}", "Treat $x$ , $f_{z}$ and $z$ as a module, shown by the dash-dotted box in Fig.", "REF .", "In the backward direction, with the incoming messages from the factor nodes $f_{x}$ as the input, the module needs to output the message $m_{z\\rightarrow f_{r}} (z)$ .", "In the forward direction, with the incoming messages from the factor node $f_{r}$ as input, the module needs to output the message $m_{x\\rightarrow f_{x}} (x)$ .", "This is the most computational intensive part of the approximate inference method, and it can be efficiently handled with UTAMP as mentioned earlier.", "Considering the structure of $x$ shown in (REF ), we divide the length-$NK$ vector $x$ into $K$ length-$N$ vectors $\\lbrace x_k, k=1, ..., K\\rbrace $ , i.e., $x=\\left[x_1^T,..., x_K^T\\right]^T.$ Due to this, the UTAMP algorithms in Section II cannot be applied directly, but the derivation still follows that of the UTAMP algorithms exactly.", "Note that the size of matrix $\\Phi $ is $M \\times NK$ .", "We partition it into $K$ sub-matrices $\\lbrace \\Phi _k, k=1, ..., K\\rbrace $ , each with a size of $M \\times N$ , i.e., $\\Phi =\\left[\\Phi _1,..., \\Phi _K \\right].$ Then we define $K$ vectors $\\lbrace \\phi _k, k=1, ..., K\\rbrace $ , each with a length of $M$ , i.e., $\\phi _k=|\\Phi _k|^2 \\textbf {1}_N.$ With the above definitions, we have the following model $r=\\sum _{k=1}^K\\Phi _kx_k+\\mathbf {\\omega }.$ blackWe first investigate the backward message passing.", "Assume that the incoming message from factor node $f_x$ is available, which is the mean and variance of $x_k$ .", "Following UTAMP, we assume that the elements of $x_k$ have a common variance $v_{x_k}$ , and the computation of $v_{x_k}$ will be detailed later.", "The mean of $x$ is denoted by $\\hat{x}$ .", "Then we calculate two vectors $\\nu _{p}$ and $p$ as $&&\\nu _{p}= \\sum _{k=1}^K \\phi _k v_{x_k} \\\\&& p=\\sum _{k=1}^K\\Phi _k\\hat{x}_k-\\nu _{p}\\cdot s,$ where $s$ is a vector, which is computed in the last iteration.", "According to the BP derivation of (UT)AMP blackUTAMP also allows a loopy BP derivation that is the same as AMP, except that the derivation is based on the unitary transformed model., $m_{z\\rightarrow f_{r}} (z)= m_{f_{z} \\rightarrow z} (z)= \\mathcal {N}\\left(z;p, D(\\nu _p)\\right).$ blackIt is noted that the factor node $f_{r}$ connects the external variable node $\\beta $ .", "According to the rules of the structured variational message passing [28], the message $m_{f_{r} \\rightarrow \\beta }(\\beta )$ can be computed as $m_{f_{r} \\rightarrow \\beta }(\\beta ) \\propto \\exp \\left\\lbrace \\int _{z} \\mathfrak {b}(z) {\\log f_{r} } \\right\\rbrace ,$ where $\\mathfrak {b}(z)$ is the the approximate marginal of $z$ , i.e., $\\begin{aligned}\\mathfrak {b}(z) &\\propto m_{f_{r} \\rightarrow z}(z) m_{z\\rightarrow f_{r}}(z)\\\\&=\\mathcal {N}(z; \\hat{z}, D(\\nu _{z}))\\end{aligned}$ with $&&\\nu _{z}=\\mathbf {1}./\\left(\\mathbf {1}./\\nu _{p}+\\hat{\\beta }\\textbf {1}_{M}\\right) \\\\ &&\\hat{z}=\\nu _{z}\\cdot \\left( p./\\nu _{p}+\\hat{\\beta }r\\right) $ where $\\hat{\\beta }$ is the approximate a posteriori mean of the noise precision $\\beta $ in the last iteration.", "Note that there may be zero elements in $\\nu _{p}$ .", "To avoid the potential numerical problem, the above equations can be rewritten as $&&\\nu _{z} = \\nu _p./(\\mathbf {1}+\\hat{\\beta }\\nu _p) \\\\&&\\hat{z}=(\\hat{\\beta }\\nu _p\\cdot \\mathbf {r}+\\mathbf {p})./(\\mathbf {1}+\\hat{\\beta }\\nu _p).$ blackIt is noted that in the above derivation, the message $m_{f_{r} \\rightarrow z}(z)$ is required, which turns out to be Gaussian, i.e., $m_{f_{r} \\rightarrow z}(z)=\\mathcal {N}(z,r, \\hat{\\beta }^{-1})$ , and its derivation is delayed to (REF ).", "Then, it is not hard to show that the message $m_{f_{r}\\rightarrow \\beta } (\\beta ) \\propto {\\beta }^{M}\\exp \\lbrace -{\\beta } (|| r- \\hat{z}||^2 + \\textbf {1}^T \\nu _{z}) \\rbrace .$ This is the end of the backward message passing.", "blackNext, we investigate the forward message passing.", "According to the rules of the structured variational message passing and noting that $f_{r}$ connects the external variable node $\\beta $ , we have $m_{f_{r} \\rightarrow z}(z) &\\propto & \\exp \\left\\lbrace \\int _{\\beta } \\mathfrak {b}(\\beta ) {\\log f_{r} } \\right\\rbrace , \\nonumber \\\\&\\propto & \\mathcal {N}(z; r, \\hat{\\beta }^{-1}) $ with $\\mathfrak {b}(\\beta ) &\\propto & m_{f_{r} \\rightarrow \\beta }(\\beta ) f_{\\beta } \\\\ \\nonumber &\\propto & {\\beta }^{M-1}\\exp \\lbrace -{\\beta } \\big (|| r- \\hat{z}||^2 + \\textbf {1}^T \\nu _{z} \\big ) \\rbrace ,$ and $\\hat{\\beta }= {black}{ \\int _{\\beta } \\beta \\mathfrak {b}(\\beta ) =} \\frac{M}{\\left\\Vert r-\\hat{z}\\right\\Vert ^2+\\textbf {1}^T\\nu _{z}}, $ where we slightly abuse the use of the notation $\\hat{\\beta }$ as we do not distinguish it from the last iteration.", "blackThe result for $\\hat{\\beta }$ coincides with the result in [37] and [38].", "blackThe message $m_{f_{r} \\rightarrow z}(z)$ is input to the dash-dotted box in Fig.", "REF .", "The Gaussian form of the message suggests the following model $r=z+ w^{\\prime },$ where the noise $w^{\\prime }$ is Gaussian with mean zero and precision $\\hat{\\beta }$ .", "This allows seamless connection with the forward recursion of UTAMP.", "According to UTAMP, we update the intermediate vectors $\\nu _{s}$ and $ s$ by $&&\\nu _{s}=\\mathbf {1}./(\\nu _{p}+\\hat{\\beta }^{-1}\\textbf {1})\\\\&& s=\\nu _{s}\\cdot \\left(r-p\\right).$ Then calculate vectors $\\nu _{q_k}$ and $\\hat{q}_k$ for $k=0, ..., K$ with $\\nu _{q_k}&=&1/\\left<|\\Phi _k^H|^2\\nu _{s}\\right>\\\\q_k&=&\\hat{x}_k+\\nu _{q_k}\\Phi _k^H s.$ The messages $q_k$ and $\\nu _{q_k}$ are the mean and variance of $x_k$ .", "According to the BP derivation of (UT)AMP, $m_{x\\rightarrow f_{x}}(x)= \\mathcal {N}(x; q, D(\\nu _{q}))$ with $q&=&[q_1^T, ..., q_K^T]^T \\\\\\nu _{q}&=&[\\nu _{q_1}, ..., \\nu _{q_K}]^T \\otimes \\textbf {1}_N,$ which is the output of the dash-dotted box in Fig.", "REF .", "This is the end of the forward message passing." ], [ "Message Computations at Nodes $f_\\textbf {x}$ , {{formula:8feb3aa8-373f-4511-bb5b-0fc2edb6714c}} and {{formula:c5188685-b6fc-4a23-a9ef-91bc59e8d988}}", "We note that the function $f_{x}(x,c,b)$ can be further factorized, i.e., $f_{x}(x,c,b)=\\prod \\nolimits _{n,k}f_{x_{n,k}}(b_k,c_n),$ and the factor $f_{x_{n,k}}(c_n, b_k)$ is shown in Fig.", "REF with solid lines, which will be used to derive the forward and backward message computations.", "blackWe first investigate the forward message passing, where the message $m_{x\\rightarrow f_{x}}(x)$ is available from the dash-dotted box .", "The $n$ th entry of $q_k$ is denoted by $q_{n,k}$ , blackthen we have $m_{x_{n,k}\\rightarrow f_{x_{n,k}}}(x_{n,k})=\\mathcal {N}(x_{n,k}; q_{n,k},\\nu _{q_{k}})$ and the factor $f_{x_{n,k}}=\\delta \\left(x_{n,k}-b_k c_{n}\\right)$ .", "Figure: blackFactor graph representation for f x n,k (c n ,b k )f_{x_{n,k}}(c_n, b_k).blackTo compute the message $m_{f_{x_{n,k}}\\rightarrow c_n}(c_n)$ with BP at factor node $f_{x_{n,k}}$ , we need to integrate out $x_{n,k}$ and $b_k$ .", "However, due to the multiplication of $b_k$ and $c_n$ , the message will be intractable even if the incoming message $m_{b_k \\rightarrow f_{x_{n,k}}} (b_k)$ is Gaussian.", "To solve this, we first apply BP and eliminate the variable $x_{n,k}$ to get an intermediate function node $\\tilde{f}_{x_{n,k}}(c_{n},b_k)$ , i.e., $\\tilde{f}_{x_{n,k}}(c_{n},b_k)&=&\\int _{x_{n,k}} m_{x_{n,k}\\rightarrow f_{x_{n,k}}}(x_{n,k}) \\cdot f_{x_{n,k}} \\nonumber \\\\&=&\\mathcal {N}\\left(c_{n}b_k; q_{n,k},\\nu _{q_{k}}\\right).$ This turns the function node $f_{x_{n,k}}$ with the hard constraint $\\delta \\left(x_{n,k}-b_k c_{n}\\right)$ to a 'soft' function node, enabling the use of blackvariational inference to handle $c_{n}$ and $b_k$ .", "With the intermediate local function $\\tilde{f}_{x_{n,k}}(b_k,c_{n})$ , we can calculate the outgoing message from $f_{x_{n,k}}$ to $c_n$ as $m_{f_{x_{n,k}}\\rightarrow c_n}(c_n)&=&\\exp \\left\\lbrace \\int _{b_{k}} \\mathfrak {b}(b_{k}) \\log \\tilde{f}_{x_{n,k}}\\right\\rbrace \\nonumber \\\\&=&\\mathcal {N}\\left(c_n;\\vec{c}_{n,k},\\vec{\\nu }_{c_{n,k}}\\right)$ where $\\vec{c}_{n,k}=\\frac{q_{n,k}\\hat{b}_{k}^*}{|\\hat{b}_{k}|^2+\\nu _{b_{k}}}, \\\\\\vec{\\nu }_{c_{n,k}}=\\frac{\\nu _{q_{k}}}{|\\hat{b}_{k}|^2+\\nu _{b_{k}}},$ with $\\hat{b}_{k}$ and $\\nu _{b_{k}}$ being the approximate a posteriori mean and variance of $b_k$ , which are computed in (REF ) and ().", "It is noted that, in the case of $b_1=1$ , we simply set $\\hat{b}_1=1$ and $\\nu _{b_1}=0$ .", "blackWith BP and referring to Fig.", "REF , the message $m_{c_n \\rightarrow f_{c}}(c_n)$ can be represented as $m_{c_n \\rightarrow f_{c}}(c_n) =\\mathcal {N}\\left(c_n;\\vec{c}_n,\\vec{\\nu }_{c_{n}}\\right)$ with $\\vec{\\nu }_{c_{n}}=1/\\sum _{k=1}^K \\frac{1}{\\vec{\\nu }_{c_{n,k}}} \\\\\\vec{c}_{n}=\\vec{\\nu }_{c_{n}} \\sum _{k=1}^K \\frac{ {\\vec{c}_{n,k}}}{\\vec{\\nu }_{c_{n,k}}}.$ So, the marginal of $c_n$ $(n=1, ..., N)$ can be expressed as $\\mathfrak {b}(c_n)=\\int _{c\\vee {c_n}} \\prod _n m_{c_n \\rightarrow f_{c}}(c_n) f_{c}.$ blackAs mentioned earlier, according to EP, the marginal is projected to be Gaussian through moment matching, i.e., $\\mathfrak {b^{\\prime }}(c_n) =\\mathcal {N}\\left(c_n;\\hat{c}_n,\\nu _{c_{n}}\\right)$ with $\\hat{c}_n&=&\\mathbb {E}\\Big [c_n|\\lbrace \\vec{\\nu }_{c_n},\\vec{c}_n\\rbrace , f_{c}\\Big ] \\\\\\nu _{c_n}&=&\\mathbb {V}\\text{ar}\\Big [c_n|\\lbrace \\vec{\\nu }_{c_n},\\vec{c}_n\\rbrace , f_{c}\\Big ],$ which are a posterior mean and variance of $c_n$ based on the prior $f_{c}$ and the following pseudo observation model black [24], [25], [39] $\\vec{c}_n= c_n + w^{\\prime }_n, $ with $w^{\\prime }_n$ denoting a Gaussian noise with mean 0 and variance $\\vec{\\nu }_{c_n}$ .", "Similarly, we can calculate the message from $f_{x_{n,k}}$ to $b_k$ , i.e., $m_{f_{x_{n,k}}\\rightarrow b_k}(b_k) = \\mathcal {N}\\left(b_k;\\vec{b}_{n,k},\\vec{\\nu }_{b_{n,k}}\\right)$ where $\\vec{b}_{n,k}=\\frac{q_{n,k}\\hat{c}_{n}^*}{|\\hat{c}_{n}|^2+\\nu _{c_{n}}}, \\\\\\vec{\\nu }_{b_{n,k}}=\\frac{\\nu _{q_{k}}}{|\\hat{c}_{n}|^2+\\nu _{c_{n}}}$ with $\\hat{c}_{n}$ and $\\nu _{c_{n}}$ being the approximate a posteriori mean and variance of $c_n$ , which are updated in (REF ) and ().", "Then with BP, the message $m_{b_k \\rightarrow f_{b}}(b_k)$ can be expressed as $m_{b_k \\rightarrow f_{b}}(b_k)= \\mathcal {N}\\left(b_k;\\vec{b}_k,\\vec{\\nu }_{b_k}\\right)$ with $\\vec{\\nu }_{b_k}&=&1/\\sum _{n=1}^N\\frac{1}{\\vec{\\nu }_{b_{n,k}}}\\\\ \\vec{b}_k &=&\\vec{\\nu }_{b_k} \\sum _{n=1}^N \\frac{\\vec{b}_{n,k}}{\\vec{\\nu }_{b_{n,k}}}.$ Then we can compute the marginal of each $b_k$ , $\\mathfrak {b}(b_k)=\\int _{b\\vee {b_k}} \\prod _k m_{b_k \\rightarrow f_{b}}(b_k) f_{b}.$ Similarly, it is then projected to be Gaussian, i.e., $\\mathfrak {b^{\\prime }}(b_k) =\\mathcal {N}\\left(b_k;\\hat{b}_k,\\nu _{b_{k}}\\right)$ with $\\hat{b}_k&=&\\mathbb {E}\\Big [b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}\\Big ]\\\\\\nu _{b_k}&=&\\mathbb {V}\\text{ar}\\Big [b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}\\Big ],$ which are the a posteriori mean and variance of $b_k$ based on the prior $f_{b}$ and the following pseudo observation model $\\vec{b}_k= b_k + w^{\\prime \\prime }_k $ with $w^{\\prime \\prime }_k$ denoting a Gaussian noise with mean 0 and variance $\\vec{\\nu }_{b_k}$ .", "It is noted that, in the case of $b_1=1$ , we simply set $\\hat{b}_1=1$ and $\\nu _{b_1}=0$ .", "blackThis is the end of the forward message passing.", "Bi-UTAMP for SMV Unitary transform: $r=U^H y=\\Phi x+\\omega $ , where $A_{M \\times NK}=U\\Lambda V$ , $\\Phi =U^H A=\\Lambda V$ , and $x= b\\otimes c$ with $b=[b_1, ..., b_K]^T$ and $c=[c_1, ..., c_N]^T$ .", "Let $\\Phi =\\left[\\Phi _1, ..., \\Phi _K \\right]$ , $\\phi _k=|\\Phi _k|^2 \\textbf {1}_N$ , and $x=\\left[x_1^T, ..., x_K^T\\right]^T$ , $k= 1,...K$ and $n=1, ...,N$ .", "Initialize $\\hat{b}_k$ , $\\nu _{b_k}=1$ , $\\nu _{x_k}=1$ , $\\hat{x}_k=\\textbf {0}$ , $s=\\mathbf {0}$ and $\\hat{\\beta }=1$ .", "Repeat [1] $\\nu _{p}= \\sum _k \\phi _k \\nu _{x_k}$ $p=\\sum _k \\Phi _k\\hat{x}_k-\\nu _{p}\\cdot s$ $\\nu _{z} = \\nu _p./(\\mathbf {1}+\\hat{\\beta }\\nu _p)$ $\\hat{z}=(\\hat{\\beta }\\nu _p\\cdot \\mathbf {r}+\\mathbf {p})./(\\mathbf {1}+\\hat{\\beta }\\nu _p) $ $\\hat{\\beta }={M}/{(\\left\\Vert r-\\hat{z}\\right\\Vert ^2+\\textbf {1}^T\\nu _{z})} $ $\\nu _{s}=\\mathbf {1}./(\\nu _{p}+\\hat{\\beta }^{-1}\\textbf {1}_{M})$ $s=\\nu _{s}\\cdot (r-p)$ $\\forall k: \\nu _{q_k}=1/\\left<|\\Phi _k^H|^2\\nu _{s}\\right>$ $\\forall k: q_k=\\hat{x}_k+\\nu _{q_k}\\Phi _k^H s$ (In the case of $b_1=1$ , set $\\hat{b}_1=1$ and $\\nu _{b_1}=0$ .)", "$\\forall k:\\vec{c}_{k}={q_{k}\\hat{b}_{k}^*} /({|\\hat{b}_{k}|^2+\\nu _{b_{k}}})$ $\\forall k:\\vec{\\nu }_{c_{k}}=\\textbf {1}_N{\\nu _{q_{k}}}/ ({|\\hat{b}_{k}|^2+\\nu _{b_{k}}})$ $ \\vec{\\nu }_{c}=\\textbf {1}_N./(\\sum _k\\textbf {1}_N./\\vec{\\nu }_{c_{k}})$ $\\vec{c}=\\vec{\\nu }_{c}\\cdot \\sum _k(\\vec{c}_{k}./\\vec{\\nu }_{c_{k}})$ $\\forall n:\\hat{c}_n=\\mathbb {E}[c_n|\\vec{\\nu }_{c},\\vec{c}, f_{c}]$ $\\forall n:\\nu _{c_n}=\\mathbb {V}\\text{ar}[c_n|\\vec{\\nu }_{c},\\vec{c}, f_{c}]$ $\\nu _{c}=<[\\nu _{c_1}, ...,\\nu _{c_N}]> \\textbf {1}_N$ , and $\\hat{c}= [\\hat{c}_1, ...,\\hat{c}_N]^T$ $\\forall k:\\vec{\\nu }_{b_{k}}={\\nu _{q_{k}}}\\textbf {1}_N./ ({\\left|\\hat{c}\\right|^2+\\nu _{c}})$ $\\forall k:\\!\\vec{b}_{k}={q_{k}\\cdot \\hat{c}^*}./({\\left|\\hat{c}\\right|^2+\\nu _{c}})$ $\\forall k:\\vec{\\nu }_{b_{k}}=(\\textbf {1}_N^T (\\textbf {1}_N./\\vec{\\nu }_{b_k}))^{-1}$ $\\forall k:\\vec{b}_{k}=\\vec{\\nu }_{b_{k}} \\textbf {1}_N^T(\\vec{b}_{k}./\\vec{\\nu }_{b_{k}})$ $\\forall k:\\hat{b}_k=\\mathbb {E}[b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}] $ $\\forall k:\\nu _{b_k}=\\mathbb {V}\\text{ar}[b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}]$ (In the case of $b_1=1$ , set $\\hat{b}_1=1$ and $\\nu _{b_1}=0$ .)", "$\\forall k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_k}=\\big (\\nu _{b_k} \\vec{\\nu }_{ b_k}\\big )./\\big ( \\vec{\\nu }_{b_k}-\\nu _{b_k} \\textbf {1}_N\\big ) $ $\\forall k:\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_k=\\big (\\hat{b}_k \\vec{\\nu }_{b_k}- \\nu _{b_k}\\vec{b}_k \\big )./\\big ( \\vec{\\nu }_{b_k}-\\nu _{b_n} \\textbf {1}_N\\big )$ $\\forall k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_k}\\!=\\!\\left(\\textbf {1}./\\nu _c-\\!\\textbf {1}./\\vec{\\nu }_{c_k}\\right)^{.-1}$ $\\forall k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_k=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_k}\\cdot \\left(\\hat{c}./\\nu _c-\\!\\vec{c}_k./\\vec{\\nu }_{c_k}\\right)$ $\\forall k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_k=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_k\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_k$ $\\forall k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_k}=|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_k|^2\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_k}+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_k}\\cdot \\left|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_k\\right|^2+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_k}\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_k}$ $\\forall k:\\nu _{x_k}=\\left(1/\\nu _{q_k}\\textbf {1}_N+\\textbf {1}./\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_k}\\right)^{.-1}$ $\\forall k:\\hat{x}_k=\\nu _{x_k} \\cdot \\left( 1/\\nu _{q_k}q_k+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_k./\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_k}\\right)$ $\\forall k: \\nu _{x_k}=<\\nu _{x_k}>$ Until terminated blackNext, we investigate the backward message passing.", "blackAccording to the rule of EP, the backward message $m_{b_k \\rightarrow f_{x_{n,k}}}(b_k)= \\frac{\\mathfrak {b^{\\prime }}(b_k)}{m_{f_{x_{n,k} \\rightarrow b_k}}(b_k)}.$ They are represented collectively as $m_{b\\rightarrow f_{x}}(b)$ , which is Gaussian with mean $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{ b}}}$ and variance black$D(\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{ b})$ .", "With the factor graph shown in Fig.", "REF , the mean and variance can be calculated as $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{ b}&=&\\left(\\left(\\textbf {1}./\\nu _{ b}\\right)\\otimes \\textbf {1}_{N}-\\textbf {1}./\\vec{\\nu }_{ b}\\right)^{.-1} \\nonumber \\\\&=& \\big ((\\nu _{b} \\otimes \\textbf {1}_N) \\cdot \\vec{\\nu }_{ b}\\big )./\\big ( \\vec{\\nu }_{ b}-(\\nu _{b} \\otimes \\textbf {1}_N)\\big ) \\\\\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}&=&\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{ b}\\cdot \\left(\\big (\\hat{b}./\\nu _{ b}\\big )\\otimes \\textbf {1}_{N}-\\vec{b}./\\vec{\\nu }_{b}\\right),\\nonumber \\\\&=& \\big ((\\hat{b}\\otimes \\textbf {1}_N) \\cdot \\vec{\\nu }_{ b}-\\vec{b}\\cdot (\\nu _{b} \\otimes \\textbf {1}_N) \\big )./\\big ( \\vec{\\nu }_{ b}-(\\nu _{b} \\otimes \\textbf {1}_N)\\big )\\nonumber \\\\ $ where $\\nu _{b}=[\\nu _{b_1},...,\\nu _{b_K}]^T$ , $\\hat{b}=[\\hat{b}_1, ..., \\hat{b}_K]^T$ , $\\big [\\vec{\\nu }_{b}\\big ]_{(k-1)N+n}=\\vec{\\nu }_{b_{n,k}}$ and $[~\\vec{b}~]_{(k-1)N+n}=\\vec{b}_{n,k}$ .", "Similarly, the message $m_{c\\rightarrow f_{x}}(c)$ is also Gaussian with mean $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}$ , and variance black$D(\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c})$ , which can be calculated as $&&\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c}=\\left(\\textbf {1}_{K}\\otimes \\left(\\textbf {1}./\\nu _c\\right)-\\textbf {1}./\\vec{\\nu }_{c}\\right)^{.-1} \\\\&&\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c}\\cdot \\left(\\textbf {1}_{K}\\otimes \\left(\\hat{c}./\\nu _c\\right)-\\vec{c}./\\vec{\\nu }_{c}\\right),$ where $\\nu _{c}=[\\nu _{c_1},...,\\nu _{c_N}]^T$ , $\\hat{c}=[\\hat{c}_1, ..., \\hat{c}_N]^T$ , $\\big [\\vec{\\nu }_{c}\\big ]_{(n-1)K+k}=\\vec{\\nu }_{c_{n,k}}$ and $\\big [\\vec{c}\\big ]_{(n-1)K+k}=\\vec{c}_{n,k}$ .", "Then, the backward message $m_{f_{x}\\rightarrow x}(x)=\\mathcal {N}\\left(x;\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}},\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x}\\right)$ with $&&\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}\\\\&&\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x}=|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}|^2\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c}+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b}\\cdot \\left|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}\\right|^2+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b}\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c} ,$ where $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}=[\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_1^T,..., \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_K^T]^T$ and $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x}=[\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_1}^T,..., \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_K}^T]^T$ .", "The backward message is combined with blackthe message $m_{x\\rightarrow f_{x}} (x)$ (the output of the dash-dotted box in last iteration) i.e., $\\nu _{x_k}&=&\\left(1/\\nu _{q_k}\\textbf {1}_N+\\textbf {1}./\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_k}\\right)^{.-1} \\\\\\hat{x}_k&=&\\nu _{x_k}\\cdot \\left( 1/\\nu _{q_k}q_k+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_k./\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_k}\\right) \\\\\\nu _{x_k}&=&<\\nu _{x_k}>$ which are then passed to blackthe dash-dotted box as input.", "This is the end of the backward message passing.", "The blackapproximate inference algorithm is called Bi-UTAMP for SMV, and it can be organized in a more succinct form, which is summarized in Algorithm REF ." ], [ "Extension to MMV", "In this section, we extend Bi-UTAMP to the case of MMV with the model $Y=\\sum _{k=1}^K b_k A_k C+W$ where $Y$ is an observation matrix with size $M \\times L$ , $W$ denotes a white Gaussian noise matrix with mean 0 and precision $\\beta $ , matrices $\\lbrace A_k\\rbrace $ are known, and $C$ with size $N \\times L$ and $b=\\left[b_1,...,b_K\\right]^T$ are to be estimated.", "Similar to the case of SMV, (REF ) can be reformulated as $Y=AX+W$ where $A=[A_1, ...,A_K]$ , and $X=[x_1,...,x_L]$ with $x_l= b\\otimes c_l.", "$ With the SVD $A= U\\Lambda V$ and unitary transformation, we have the following model $R=\\Phi X+\\overline{W} $ where $R=U^{H}Y$ , $\\Phi =\\Lambda V=U^HA$ and $\\overline{W}=U^{H}W$ .", "Define $z_l=\\mathbf {\\Phi }x_l$ and $Z=[z_1, ..., z_L]$ , then we can factorize the joint distribution of the variables in (REF ) as $&&\\!\\!\\!\\!\\!p(X,C,b,Z,\\beta |R)\\nonumber \\\\&&\\!\\!\\!\\!\\!", "\\propto p(C)p(b)p(\\beta )\\prod \\nolimits _{l}p(r_l|z_l,\\beta )p(z_l|x_l)p(x_l|b,c_l) \\nonumber \\\\&&{black}{\\!\\!\\!\\!\\!\\triangleq f_{C}(C)f_{b}(b) f_{\\beta }(\\beta )\\prod \\nolimits _{l} f_{r_l}(z_l,\\beta )f_{z_l}(z_l,x_l){f_{x_l}(x_l,b,c_l)}}.", "\\nonumber \\\\$ Figure: blackFactor graph representation of ().Bi-UTAMP for MMV Unitary transform: $R=U^H Y=\\Phi X+\\overline{W}$ , where $A_{M \\times NK}=U\\Lambda V$ , $\\Phi =U^H A=\\Lambda V$ , and $x_l = b\\otimes c_l$ with $b=[b_1, ..., b_K]^T$ and $c_l=[c_{1,l}, ..., c_{N,l}]^T$ .", "Let $\\Phi =\\left[\\Phi _1, ..., \\Phi _K \\right]$ , $\\phi _k=|\\Phi _k|^2 \\textbf {1}_N$ , and $x_l=\\left[x_{1,l}^T, ..., x_{K,l}^T\\right]^T, k=1, ..., K, n=1, ..., N$ and $l=1, ..., L$ .", "Initialize: $\\hat{b}_{k}$ , $\\nu _{b_{k}}=1$ , $\\nu _{x_{k,l}}=1$ , $\\hat{x}_{k,l}=\\textbf {0}$ , $\\mathbf {s}_l=\\mathbf { 0 }$ , and $\\hat{\\beta }=1$ .", "Repeat [1] $\\forall l$ : $\\nu _{p_l}=\\sum _k\\phi _k\\nu _{x_{k,l}}$ $\\forall l$ : $p_l=\\sum _k\\Phi _k\\hat{x}_{k,l}-\\nu _{p_l}\\cdot s_l$ $\\forall l$ : $\\nu _{z_l}= \\nu _{p_l}./(\\mathbf {1}+\\hat{\\beta }\\nu _{p_l})$ $\\forall l$ : $\\hat{z}_l=(\\hat{\\beta }\\nu _{p_l}\\cdot \\mathbf {r}_l+\\mathbf {p}_l)./(\\mathbf {1}+\\hat{\\beta }\\nu _{p_l})$ $\\hat{\\beta }=ML/\\sum \\nolimits _l\\big (\\big \\Vert r_l-\\hat{z}_l\\big \\Vert ^2+ \\textbf {1}^T \\nu _{z_l}\\big )$ $\\forall l$ : $\\nu _{s_l}=\\mathbf {1}./\\big (\\nu _{p_l}+\\hat{\\beta }^{-1}\\textbf {1}_{M}\\big )$ $\\forall l$ : $s_l=\\nu _{s_l}\\cdot \\big (r_l-p_l\\big )$ $\\forall l,k$ : $\\nu _{q_{k,l}}=1/\\big <|\\Phi ^H_k|^2\\nu _{s_l}\\big >$ $\\forall l,k$ : $q_{k,l}=\\hat{x}_{k,l}+\\nu _{q_{k,l}}\\Phi ^H_k s_l$ $\\forall l,k:\\vec{c}_{k,l}={q_{k,l}\\hat{b}_{k}^*} /({|\\hat{b}_{k}|^2+\\nu _{b_{k}}})$ $\\forall l,k:\\vec{\\nu }_{c_{k,l}}=\\textbf {1}_N{\\nu _{q_{k,l}}}/ ({|\\hat{b}_{k}|^2+\\nu _{b_{k}}})$ $\\forall l: \\vec{\\nu }_{c_l} =\\textbf {1}_N./ \\sum _k (\\textbf {1}_N./\\vec{\\nu }_{c_{k,l}})$ $\\forall l:\\vec{c}_{l}=\\vec{\\nu }_{c_{l}}\\cdot \\sum _k(\\vec{c}_{k,l}./\\vec{\\nu }_{c_{k,l}})$ $\\forall n,l:\\hat{c}_{n,l}=\\mathbb {E}[c_{n,l}|\\lbrace \\vec{\\nu }_{c_l},\\vec{c}_l\\rbrace , f_{C}]$ $\\forall n,l:\\nu _{c_{n,l}}=\\mathbb {V}\\text{ar}[c_{n,l}|\\lbrace \\vec{\\nu }_{c_l},\\vec{c}_l\\rbrace , f_{C}]$ $\\forall l:\\nu _{c_l}=<[\\nu _{c_{1,l}},...,\\nu _{c_{N,l}}]>\\textbf {1}_N, \\hat{c}_l=[\\hat{c}_{1,l}, ...,\\hat{c}_{N,l} ]^T.$ $\\forall l,k: \\vec{\\nu }_{b_{k,l}}={\\nu _{q_{k,l}}} \\textbf {1}_N./ ({\\left|\\hat{c}_{l}\\right|^2+\\nu _{c_{l}}})$ $\\forall l, k: \\vec{b}_{k,l}={q_{k,l}\\cdot \\hat{c}_{l}^*}.", "/({\\left|\\hat{c}_{l}\\right|^2+\\nu _{c_{l}}})$ $\\forall k: \\vec{\\nu }_{b_k}=1/\\sum _l(\\textbf {1}_N^T (\\textbf {1}./\\vec{\\nu }_{b_{k,l}}))$ $\\forall k: \\vec{b}_k=\\vec{\\nu }_{b_k} \\sum _l(\\textbf {1}_N^T (\\hat{b}_{k,l}./\\vec{\\nu }_{b_{k,l}}))$ $\\forall k:\\hat{b}_k=\\mathbb {E}[b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}] $ $\\forall k:\\nu _{b_k}=\\mathbb {V}\\text{ar}[b_k|\\lbrace \\vec{\\nu }_{b_k},\\vec{b}_k\\rbrace , f_{b}]$ $\\forall l, k: \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_{k,l}}=\\big (\\nu _{b_k} \\vec{\\nu }_{b_{k,l}}\\big )./\\big ( \\vec{\\nu }_{b_{k,l}}-\\nu _{b_k} \\textbf {1}_N\\big ) $ $ \\forall l, k:\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_{k,l}\\!=\\!\\big (\\hat{b}_k \\vec{\\nu }_{ b_{k,l}}-\\nu _{b_k} \\vec{b}_k\\ \\big )./\\big ( \\vec{\\nu }_{ b_{k,l}}-\\nu _{b_k} \\textbf {1}_N\\big )$ $\\forall l, k$ : $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_{k,l}}=\\left(\\textbf {1}_N./\\nu _{c_l}-\\textbf {1}_N./\\vec{\\nu }_{c_{k,l}}\\right)^{.-1} $ $\\forall l,k$ : $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_{k,l}=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_{k,l}}\\cdot \\left(\\hat{c}_l./\\nu _{c_l}-\\vec{c}_{k,l}./\\vec{\\nu }_{c_{k,l}}\\right)$ $\\forall l, k$ : $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_{k,l}=\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_{k,l}\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_{k,l}$ $\\forall k,l$ : $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{ x_{k,l}}=|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{b}}}_{k,l}|^2\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_{k,l}}+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_{k,l}}\\cdot \\left|\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{c}}}_{k,l}\\right|^2+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{b_{k,l}}\\cdot \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{c_{k,l}}$ $\\forall l, k$ : $\\nu _{x_{k,l}}=\\big (1/\\nu _{q_{k,l}}\\textbf {1}_N+\\textbf {1}_N./\\nu _{\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_{k,l}}\\big )^{.-1}$ $\\forall k,l$ : $\\hat{x}_{k,l}=\\nu _{x_{k,l}}\\cdot \\left(1/\\nu _{q_{k,l}} q_{k,l}+\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{x}}}_{k,l}./\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\nu }}}_{x_{k,l}}\\right)$ $\\forall l, k: \\nu _{x_{k,l}}= <\\nu _{x_{k,l}}> $ Until terminated The factor graph representation for the factorization in (REF ) is depicted in Fig.", "REF .", "The message updates related to $z_l$ , $x_l$ and $c_l$ are the same as those in Algorithm REF , and they can be computed in parallel.", "The major difference lies in the computations of $b(b)$ and $b(\\beta )$ , where the messages from $f_{x_l}$ and $f_{r_l}, \\forall l$ , should be considered, i.e., $b(b)\\propto \\prod \\nolimits _{l}m_{f_{x_l} \\rightarrow b}(b)m_{f_{b} \\rightarrow b}(b)\\\\b(\\beta )\\propto \\prod \\nolimits _{l}m_{f_{r_l} \\rightarrow \\beta }(\\beta )m_{f_{\\beta } \\rightarrow \\lambda }(\\beta ).$ Similar to the SMV case, the message passing algorithm can be derived, which are summarized as Algorithm REF (Bi-UTAMP for MMV)." ], [ "Discussions and Complexity Analysis", "We have the following remarks and discussions about Bi-UTAMP: In some problems, $b_1$ is known, e.g., $b_1=1$ .", "In this case, we can set $\\hat{b}_1=1$ and $\\nu _{b_1}=0$ in Bi-UTAMP, which are indicated in Algorithm REF .", "It is not hard to show that, when $b=b_1=1$ , Bi-UTAMP is reduced to UTAMP (Algorithm ) exactly.", "blackIt is interesting that the robustness of Bi-UTAMP can be enhanced by simply damping $s$ , i.e., Line 7 of the SMV Bi-UTAMP is changed as $s=(1-\\alpha ) s+ \\alpha \\nu _{s}\\cdot (r-p)$ with $\\alpha \\in (0,1]$ , where $\\alpha $ is the damping factor and $\\alpha =1$ leads to the case without damping.", "Accordingly, Line 7 of the MMV Bi-UTAMP is changed as $s_l=(1-\\alpha ) s_l+ \\alpha \\nu _{s_l}\\cdot (r-p)$ .", "The iterative process can be terminated based on some criterion, e.g., the normalized difference between the estimates of $b$ of two consecutive iterations is smaller than a threshold, i.e., $\\Vert \\hat{b}^t-\\hat{b}^{t-1}\\Vert ^2 /\\Vert \\hat{b}(t)\\Vert ^2 < \\epsilon $ where $\\hat{b}^t$ is the estimate of $b$ at the $t$ th iteration and $\\epsilon $ is a threshold.", "As the bilinear problem has local minima, we can use the same strategy of restart as in [7] to mitigate the issue of being stuck at local minima.", "For each restart, we initialize $\\lbrace \\hat{b}_k\\rbrace $ with different values.", "blackIn Bi-UTAMP, we have Bayesian treatment to both $b$ and $c$ (or $C$ in MMV).", "In contrast, $b$ is treated as a unknown deterministic variable in BAd-VAMP, and only a point estimate is involved.", "As discussed in Section III.A, the Bayesian treatment to $b$ can make the algorithm more flexible.", "The computational complexity of Bi-UTAMP is analyzed in the following.", "Bi-UTAMP needs pre-processing, i.e., performing economic SVD for $A$ and unitary transformation, and the complexity is $\\mathcal {O}(M^2NK)$ .", "It is noted that the pre-processing can be carried out offline (although we do not assume this blackin counting the runtime of Bi-UTAMP in the simulations in Section IV).", "It can be seen from the Bi-UTAMP algorithms that, there is no matrix inversion involved, blackand the most computational intensive parts only involve matrix-vector products.", "So the complexity of Bi-UTAMP per iteration is $\\mathcal {O}(MNKL)$ (in the case of SMV, $L=1$ ), which linearly increases with $M$ , $N$ , $K$ and $L$ .", "blackFor comparison, BAd-VAMP involves one outer loop and two inner loops.", "The whole matrix $C_l^t$ with size $N \\times N$ in the second inner loop is required in multiple lines in the algorithm and $A(\\theta _{A}^t)$ is updated in each inner iteration [7].", "The computation of the matrix $C_l^t$ leads to a complexity of $\\mathcal {O}(LN^3+KMN)$ per inner iteration.", "Line 18 is also computational intensive, which requires a complexity of $\\mathcal {O}(K^2N^2)$ per inner iteration.", "Also, Line 20 of BAd-VAMP requires a complexity of $\\mathcal {O}(K^3)$ per inner iteration.", "It is difficult to have a very precise complexity comparison analytically as the algorithms require different numbers of iterations to converge.", "So, in Section IV, we compare the runtime of several state-of-the-art algorithms as in [7].", "As demonstrated in Section IV, with much less runtime, Bi-UTAMP can outperform the state-of-the-art algorithms significantly." ], [ "blackSE-Based Performance Prediction", "blackFrom the derivation of Bi-UTAMP, we can see that Bi-UTAMP integrates VMP, BP, EP and UTAMP.", "The incorporation of UTAMP enables the approximate inference method to deal with the most computational intensive part with low complexity and high robustness.", "The rigorous performance analysis is difficult, but we make an attempt to predict its performance based on UTAMP SE heuristically.", "We track the output variance of the UTAMP module in the dash-dotted box with respect to the input variance.", "However, the variances are about $x$ instead of $b$ and $c$ (or $C$ in the MMV case).", "The method is the same as the SE for (UT)AMP, i.e., we model $q_k = x_k + w_k$ as the input to the \"denoiser\" (which corresponds to $f_{x}$ , $f_{b}$ and $f_{c}$ in the factor graph and involves EP and BP), where $w_k$ denotes a Gaussian noise with mean zero and variance $\\tau _k$ .", "However, it is difficult to find an analytic form for the output variance of the denoiser, which is also happened to (UT)AMP due to the priors.", "This can be solved by simulating the denoiser using $q_k = x_k + w_k$ with different variances of $w_k$ as input, so that a \"function\" in terms of a table can be established.", "In our case, besides the variance of $x$ , the MSE of $b$ and $c$ can also be obtained as \"byproduct\", which allows us to predict the MSE of $b$ and $c$ , while the variance of $x$ is used to determine $\\tau _k$ analytically.", "As shown in Section IV, the prediction is fairly good in some cases.", "But, in some cases, it is not accurate.", "More accurate and rigorous performance analysis is our future work." ], [ "Numerical Examples", "In this section, we evaluate the performance of Bi-UTAMP and compare it with the state-of-the-art blackbilinear recovery algorithms including the conventional non-message passing based algorithm WSS-TLS in [1], and message passing based algorithms BAd-VAMP in [7] and PC-VAMP in [23].", "It is noted that PC-VAMP does not provide an estimate for $b$ .", "Performance is evaluated in terms of normalized MSE and runtime.", "Relevant performance bounds are also included for reference.", "Figure: blackCompressive sensing with correlated matrices: NMSE of bb and cc versus SNR with (a) ρ=0.3\\rho =0.3 and (b) ρ=0.4\\rho =0.4." ], [ "SMV Case", "For the SMV case, we take compressive sensing with matrix uncertainty [1] as an example.", "We aim to recover a sparse signal vector $c$ from measurement $y=A(b)c+w$ , where the measurement matrix is modeled as $A(b)=\\sum \\nolimits _{k=1}^K b_kA_k$ with $b_1=1$ , $A_k\\in \\mathbb {R}^{M\\times N}$ are known, and the uncertainty parameter vector $b=[b_2, ..., b_K]^T$ is unknown.", "In addition the precision of the noise is unknown as well.", "In the experiments, we set $K=11$ , $N=256$ , $M=150$ and the number of nonzero elements in $c$ is 10.", "The SNR is defined as $\\text{SNR}\\triangleq \\mathbb {E}\\left[||A(b)c||^2\\right]/\\mathbb {E}\\left[||w||^2\\right]$ .", "The uncertainty parameters $\\lbrace b_2, ...b_k\\rbrace $ are drawn from $\\mathcal {N}(0,1)$ independently, and the nonzero elements of sparse vector $c$ are drawn from $\\mathcal {N}(0,1)$ independently as well, which are randomly located in $c$ .", "The performance of the methods are evaluated using $\\text{NMSE}(b)\\triangleq ||\\hat{b}-b||^2/||b||^2$ and $\\text{NMSE}(c)\\triangleq ||\\hat{c}-c||^2/||c||^2$ , where $\\hat{b}$ and $\\hat{c}$ are the estimates of $b$ and $c$ , respectively.", "We also include the performance bounds for the estimation of $b$ and $c$ , which are the performance of two oracle estimators: the MMSE estimator for $b$ with the assumption that $c$ is known, and the MMSE estimator for $c$ with the assumption that $b$ and the support of $c$ are known.", "It is noted that, different from [7], we do not use median NMSEs, and blackto better evaluate the robustness of the algorithms, the NMSEs are obtained by averaging the results from all trials.", "To demonstrate the robustness of Bi-UTAMP, we focus on tough measurement matrices, e.g., correlated matrices, non-zero mean matrices.", "and ill-conditioned matrices.", "In addition, Bi-UTAMP and BAd-VAMP use a same damping factor of 0.8 to enhance their robustness.", "Figure: blackCompressive sensing with correlated matrices: NMSE of bb and cc versus ρ\\rho at SNR = 40dB.All matrices $\\lbrace A_k\\rbrace $ are correlated, and $A_k$ is constructed using $A_k=C_L\\textbf {G}_kC_R$ , where $\\textbf {G}_k$ is an i.i.d.", "Gaussian matrix, and $C_L$ is an $M\\times M$ matrix with the $(m,n)$ th element given by $\\rho ^{|m-n|}$ where $\\rho \\in [0,1]$ .", "Matrix $C_R$ is generated in the same way but with a size of $N\\times N$ .", "The parameter $\\rho $ controls the correlation of matrix $A_k$ .", "Fig.", "REF shows the NMSE performance of the algorithms versus SNR, where the correlation parameter $\\rho =0.3$ in (a) and $\\rho =0.4$ in (b).", "It can be seen that when $\\rho =0.3$ , blackall the message passing based algorithms PC-VAMP, BAd-VAMP and Bi-UTAMP perform well and they are significantly better than the non-message passing based method WSS-TLS.", "We can also see that Bi-UTAMP delivers a performance which is considerably better than that of PC-VAMP and BAd-VAMP.", "With $\\rho =0.4$ , Bi-UTAMP still works very well, and it significantly outperforms BAd-VAMP, PC-VAMP and WSS-TLS.", "It is noted that as PC-VAMP does not estimate $b$ , so its performance in the right column is absent.", "We further evaluate the performance of blackall algorithms for matrices with different level of correlations by varying the parameter $\\rho $ at SNR = 40dB and the results are shown in Fig.", "REF , where we can see blacksignificant performance gaps between all the other algorithms and Bi-UTAMP when $\\rho $ is relatively large.", "The results in Figs.", "REF and REF demonstrate that Bi-UTAMP is more robust than all the other algorithms with correlated measurement matrices.", "In Figs.", "REF and REF , we also show the predicted performance based on SE for Bi-UTAMP, where we can see the the predicted performance matches the simulated performance fairly well when the matrix correlation is relatively small.", "Figure: blackCompressive sensing with ill-conditioned matrices: NMSE of bb and cc versus κ\\kappa with SNR = 40dB.Figure: blackCompressive sensing with non-zero mean matrix: NMSE of bb and cc versus μ\\mu with SNR = 40dB." ], [ "Ill-Conditioned Measurement Matrix", "Each matrix $A_k $ is constructed based on the SVD $A_k=U_k \\Lambda _k V_k$ where $\\mathbf {\\Lambda }_k$ is a singular value matrix with $\\Lambda _{i,i}/ \\Lambda _{i+1,i+1} = \\kappa ^{1/(M-1)}$ (i.e., the condition number of the matrix is $\\kappa $ ).", "The NMSE performance of the algorithms versus the condition number is shown in Fig.", "REF , where the SNR = 40 dB.", "blackIt can be seen that Bi-UTAMP can significantly outperform all the other algorithms when $\\kappa $ is relatively large, and BAd-VAMP performs better than PC-VAMP and WSS-TLS.", "We also see that the predicated performance is no longer accurate when $\\kappa $ is large." ], [ "Non-Zero Mean Measurement Matrix", "The elements of matrix $A_k$ are independently drawn from a non-zero mean Gaussian distribution $\\mathcal {N}(\\mu , v)$ .", "The mean $\\mu $ measures the derivation from the i. i. d. zero-mean Gaussian matrix.", "In the simulations, for $\\lbrace A_k, k=2:K\\rbrace $ , $v=1$ , and for $A_1$ , $v=20$ .", "The NMSE performance of the algorithms versus $\\mu $ is shown in Fig.", "REF , where the SNR = 40 dB.", "It can be seen from this figure that Bi-UTAMP can achieve much better performance compared to blackWSS-TLS and BAd-VAMP especially when $\\mu $ is relatively large.", "PC-VAMP delivers a competitive performance compared to Bi-UTAMP, while it does not provide an estimate for $b$ and is also slower than Bi-UTAMP as shown in Fig.", "REF ." ], [ "Runtime Comparison", "Fig.", "REF compares the average runtime of blackall algorithms.", "In Fig.", "REF (a), correlated matrices are used with the correlation parameter $\\rho =0.3 $ .", "With SNR = 40 dB, the average runtime versus different $\\rho $ for correlated matrices, different means for non-zero mean matrices and different condition numbers for ill-conditioned matrices is given in Fig.", "REF (b), (c) and (d), respectively.", "The results are obtained using MATLAB (R2016b) on a computer with a 6-core Intel i7 processor.", "Fig.", "REF shows that, Bi-UTAMP is much faster than BAd-VAMP blackand WSS-TLS, and it is also considerably faster than PC-VAMP.", "Figure: blackAverage runtime versus (a) SNR for correlated matrices with ρ=0.3\\rho =0.3, (b) ρ\\rho for correlated matrices, (c) μ\\mu for non-zero mean matrices, (d) condition number κ\\kappa for ill-conditioned matrices.", "In (b), (c) and (d), SNR = 40 dB.Figure: Structured dictionary learning: NMSE(AA) and NMSE(CC) versus SNR with (a) ρ\\rho = 0 and (b) ρ\\rho = 0.1." ], [ "MMV Case", "We take the structured dictionary learning (DL) [4] as an example to demonstrate the performance of Bi-UTAMP.", "The goal of structured DL is to find a structured dictionary matrix $A=\\sum \\nolimits _{k=1}^K b_k A_k\\in \\mathbb {R}^{M\\times N}$ from the training samples $Y\\in \\mathbb {R}^{M\\times L}$ with model $Y=AC+W$ for some sparse coefficient matrix $C\\in \\mathbb {R}^{N\\times L}$ .", "In the simulations, we assume square dictionary matrix $A$ with $M = N = 100$ .", "The length of vector $b$ blackis large, i.e., $K=100$ , and the number of non-zero elements are set to be 20 in each column of $C$ black(the columns are generated independently) and $L=5$ for the training examples.", "Since the dictionary matrix $A$ has a structure, it can be learned with a small number of training samples.", "Bi-UTAMP is run for maximum 100 iterations and 10 restarts.", "In addition, to enhance the robustness, we use a damping factor $0.55$ for both Bi-UTAMP and BAd-VAMP.", "In addition, Lines 19-22 in Bi-UTAMP are executed once every two iterations.", "The performance is evaluated with NMSE of the estimates of $A$ and $C$ .", "As the pair $(A, C)$ has a scalar ambiguity, the NMSE is calculated in the same way as in [7], i.e., $\\text{NMSE}(\\hat{A})\\triangleq \\text{min}_{d}\\frac{||A-d\\hat{A}||^2}{||A||^2}$ $\\text{NMSE}(\\hat{C})\\triangleq \\text{min}_{d}\\frac{||C-d\\hat{C}||^2}{||C||^2}.$ Different from [7], the NMSEs are obtained by averaging the results from all trials.", "To test the performance and robustness of the algorithms, correlated matrices $\\lbrace A_k\\rbrace $ generated in the same way as in the SMV case are used.", "Figure REF shows the NMSE performance $\\text{NMSE}(\\hat{A})$ and $\\text{NMSE}(\\hat{C})$ versus SNR with correlation parameter (a) $\\rho $ = 0 and (b) $\\rho $ = 0.1.", "It can be seen that when $\\rho $ = 0, i.e., $\\lbrace A_k\\rbrace $ are i.i.d.", "Gaussian, BAd-VAMP and Bi-UTAMP have similar performance.", "When $\\rho $ = 0.1, Bi-UTAMP can outperform BAd-UTAMP considerably.", "Fig.", "REF shows the NMSE versus $\\rho $ at SNR = 40dB, where we can see that Bi-UTAMP can achieve significantly better performance than BAd-VAMP.", "From these results, we conclude that Bi-UTAMP is more robust.", "Figure REF shows the average runtime versus (a) SNR and (b) $\\rho $ .", "Again, the results show that Bi-UTAMP is much faster than BAd-VAMP.", "Figure: Structured dictionary learning: NMSE(A) and NMSE(C) versus ρ\\rho with SNR = 40dB.Figure: Structured dictionary learning: Average runtime versus SNR (left) and ρ\\rho (right)." ], [ "Conclusions", "In this paper, blackwe have investigated approximate Bayesian inference for the problem of bilinear recovery.", "We have designed a new approximate inference algorithm Bi-UTAMP, where UTAMP is integrated with BP, EP and VMP to achieve efficient recovery of the unknown variables.", "We have shown that Bi-UTAMP is much more robust and faster than the state-of-the-art algorithms, leading to significantly better performance.", "blackFuture work includes a rigorous analysis of the performance of Bi-UTAMP and generalizing it to handle non-linear measurements, e.g., quantization." ], [ "Acknowledgment", "The authors would like to thank Subrata Sarkar for sharing the Matlab code for BAd-VAMP and suggestions for the simulation of BAd-VAMP." ] ]

[ [ "ODEN: A Framework to Solve Ordinary Differential Equations using\n Artificial Neural Networks" ], [ "Abstract We explore in detail a method to solve ordinary differential equations using feedforward neural networks.", "We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance.", "Neural networks are shown to be proficient at approximating continuous solutions within their training domains.", "We illustrate neural networks' ability to outperform traditional standard numerical techniques.", "Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature fitting, and (iii) a fine-tuning stage.", "The main limitation of the method is the nontrivial task of finding the appropriate neural network architecture and the choice of neural network hyperparameters for efficient optimization.", "However, we observe an optimal architecture that matches the complexity of the differential equation.", "A user-friendly and adaptable open-source code (ODE$\\mathcal{N}$) is provided on GitHub." ], [ "Introduction", "Neural networks (NNs) are known to be powerful tools due to their role as universal continuous function approximators [1].", "Rewarding NNs for their improving performances in specific tasks can be done by minimizing a specially designed loss function.", "In fact, the last ten years have seen a surge in the capabilities of NNs, specifically deep neural networks, due to the increase in supervised data and compute.", "For example, one can cite AlexNet [2], a convolutional neural network for image recognition and GPT-2 [3], a powerful Transformer language model.", "Although NNs and machine learning are used in physics [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], the use of NNs has still not been widely accepted in the scientific community due to their black-box nature, the unclear and often unexplored effects of hyperparameter choice for a particular problem, and a large computational cost for training.", "Usually, neural networks are used for problems involving dimensional reduction [14], data visualization, clustering [15], and classification [16].", "Here, we use the potential of NNs to solve differential equations.", "Differential equations are prevalent in many disciplines including Physics, Chemistry, Biology, Economics, and Engineering.", "When an analytical solution cannot be found, numerical methods are employed and are very successful [17], [18], [19].", "However, some systems exhibit differential equations that are not efficiently solved by usual numerical methods.", "Such differential equations can be numerically solved to the required accuracy by novel specific codes, such as oscode [20] which efficiently solves one-dimensional, second-order, ordinary differential equations (ODEs) with rapidly oscillating solutions which appear in cosmology and condensed matter physics [21] among other systems.", "Yet, specifically written algorithms [22] and codes require a lot of time and resources.", "In this paper, we ask whether the unsupervised method of solving ODEs using NNs, first developed by Lagaris et al.", "[23] and improved in [24], [25], [26], [27], [28], [29], [30], [31], is robust.", "In previous investigations, there has been no clear consensus on what exact method should be used.", "Multiple loss functions were used, sometimes requiring the exact solution to be known [32] [33], and the boundary/initial conditions were either included in the model as an extra term in the loss function or treated independently by means of trial functions.", "No explicit benchmark to evaluate the NN performances was transparently introduced.", "Training and extrapolation performances were not meticulously studied, making NNs appear as black-boxes.", "This work aims at filling the mentioned gaps and demystifying a few aspects.", "We provide a clear step-by-step method to solve ODEs using NNs.", "We prove a specific loss function from the literature [24] to be an appropriate benchmark metric to evaluate the NN performances, as it does not require the exact solution to be known.", "The training process is investigated in detail and is found to exhibit three phases that we believe are the general method that the NN learns to solve a given ODE with boundary/initial conditions.", "We explore the effect of training domain sampling, extrapolation performances and NN architectures.", "Contrary to numerical integration, approximating the ODE solution by a NN gives a function that can be continuously evaluated over the entire training domain.", "However, finding an appropriate NN architecture to reach the desired accuracy requires extensive testing and intuition.", "An open-source and user-friendly code (ODE$\\mathcal {N}$ ) that accompanies this manuscripthttps://github.com/deniswerth/ODEN is provided, along with animations that help to visualize the training process.", "In the next section, we briefly present the basics of NNs, followed by the method used to solve ODEs with NNs in Section .", "In Section , three different ODEs are solved as illustration.", "A specific loss function is proved to be a universal metric to assess the accuracy and different simple NN models are tested in Section , particularly highlighting the role of complexity when choosing the NN.", "Finally, conclusion and outlook are provided in Sections ." ], [ "Background", "The basic unit of a NN is a neuron $i$ that takes a vector of input features ${x} = (x_1, x_2, ...)$ and produces a scalar output $a_i({x})$ .", "In almost all cases, $a_i$ can be decomposed into a linear operation taking the form of a dot product with a set of weights ${w}^{(i)} = (w_1^{(i)}, w_2^{(i)}, ...)$ followed by re-centering with an offset called the bias $b^{(i)}$ , and a non-linear transformation i.e.", "an activation function $\\sigma _i : \\mathbb {R}\\rightarrow [0,1]$ which is usually the same for all neurons.", "One can write the full input-output formula for one neuron as follows $a_i({x}) = \\sigma _i\\left( {w}^{(i)}.\\,{x} + b^{(i)} \\right).$ A NN consists of many such neurons stacked into layers, with output of one layer serving as input for the next.", "Thus, the whole NN can be thought of as a complicated non-linear transformation of inputs $x$ into an output $\\mathcal {N}$ that depends on the weights and biases ${\\theta }$ of all the neurons in the input, hidden, and output layers [34]: $\\text{Neural Network} = \\mathcal {N}(x, {\\theta }).$ The NN is trained by finding the value of ${\\theta }$ that minimizes the loss function $\\mathcal {L}({\\theta })$ , a function that judges how well the model performs on corresponding the NN inputs to the NN output.", "Minimization is usually done using a gradient descent method [34].", "These methods iteratively adjust the NN parameters ${\\theta }$ in the direction (on the parameter surface) where the gradient $\\mathcal {L}$ is large and negative.", "In this way, the training procedure ensures the parameters ${\\theta }$ flow towards a local minimum of the loss function.", "The main advantage of NNs is to exploit their layered structure to compute the gradients of $\\mathcal {L}$ in a very efficient way using backpropagation [34]." ], [ "Method", "The starting point of solving ODEs using NNs is the universal approximation theorem.", "The theorem states that a feedforward NN with hidden layers containing a finite number of neurons can approximate any continuous functions at any level of accuracy [1].", "Thus, one may expect NNs to perform well on solving ODEs.", "Despite the existence of such NNs, the theorem does not provide us with a recipe to find these NNs nor touch upon their learnabilities.", "Differential equations with an unknown solution $f$ can be written as a function $\\mathcal {F}$ of $f$ and their derivatives in the form $\\mathcal {F}[x, f(x), \\nabla f(x), ... ,\\nabla ^{p}f(x)] = 0 \\hspace*{8.5359pt}\\text{for}\\hspace*{8.5359pt} x\\in \\mathcal {D},$ with some Dirichlet/Neumann boundary conditions or initial conditions, where $\\mathcal {D}$ is the differential equation domain and $\\nabla ^{p}f(x)$ is the $p$ -th gradient of $f$ .", "The notations and the approach can easily be generalized to coupled and partial differential equations.", "Following classic procedures, we then discretize the domain $\\mathcal {D}$ in $N$ points ${x} = (x_1, x_2, ..., x_N)$ .", "For each point $x_i\\in \\mathcal {D}$ , Eq.", "(REF ) must hold so that we have $\\mathcal {F}[x_i, f(x_i), \\nabla f(x_i), ..., \\nabla ^{p}f(x_i)] = 0.$ Writing the differential equation in such a way allows us to convert the problem of finding solutions to an optimization problem.", "Indeed, an approximated solution is a function that minimizes the square of the left-hand side of Eq.", "(REF ).", "In our approach, we identify the approximated solution of the differential equation (REF ) with a NN $\\mathcal {N}(x, {\\theta })$ and express the loss function as follows [24]: $\\begin{aligned}\\mathcal {L}({\\theta }) &= \\frac{1}{N}\\sum _{i=1}^N \\mathcal {F}\\left[x_i, \\mathcal {N}(x_i, {\\theta }), \\nabla \\mathcal {N}(x_i, {\\theta }), ... , \\nabla ^{p}\\mathcal {N}(x_i, {\\theta })\\right]^2 \\\\&+ \\sum _{j} [\\nabla ^{k}\\mathcal {N}(x_{j}, {\\theta }) - K_{j}]^2,\\end{aligned}$ with $K_{j}$ being some chosen constants.", "The loss function contains two terms: the first one forces the differential equation (REF ) to hold, and the second one encodes the boundary/initial conditionsUsually, $k = 0$ and/or $k=1$ ..", "Note that $\\mathcal {L}({\\theta })$ does not depend on the exact solutionIn some works, the mean squared error [32] [33] is used..", "The approximated solution $\\mathcal {N}(x, {\\theta })$ is found by finding the set of weights and biases ${\\theta }$ such that the loss function $\\mathcal {L}({\\theta })$ is minimized on the training points $x_i$ for $i = 1, 2, ..., N$ .", "Solving ODEs using NNs can then be expressed in terms of the following steps: Write the differential equation in the form shown in Eq.", "(REF ); Discretize the differential equation domain in $N$ points ${x} = (x_1, x_2, ..., x_N)$ ; Construct a neural network $\\mathcal {N}(x, {\\theta })$ by choosing its architecture i.e.", "the number of hidden layers and the number of neurons in each layer, and the activation functions; Write the loss function $\\mathcal {L}({\\theta })$ in the form shown in Eq.", "(REF ) including boundary/initial conditions; Minimize $\\mathcal {L}({\\theta })$ to find the optimal NN parameters ${\\theta }^{\\star }$ i.e.", "train the NN; Once the NN has been trained on the discrete domain, $\\mathcal {N}(x, {\\theta }^{\\star })$ is an approximation of the exact solution that can be evaluated continuously within the training domain.", "For our setup, as ODE solutions are scalar functions of a scalar independent variable, the NN has one input neuron and one output neuron.", "The first term of $\\mathcal {L}({\\theta })$ in Eq.", "(REF ) requires computing the gradients of the NN output with respect to the NN input.", "Gradient computation, as well as the network implementation and the training procedure, is performed using the Keras framework with a Tensorflow 2 backend.", "The general method can be easily extended to coupled and partial differential equationsFor example, coupled differential equations can be solved by corresponding the number of output neurons to the number of equations in the system one wants to solve.." ], [ "Applications", "The method is verified against the exact solution of first and second order ODEs.", "We then use the method to solve an ODE that exhibits rapid oscillations.", "Training processes were computed using 12 CPU cores and took between a few seconds to a few minutes, depending on the number of epochs and the NN complexity." ], [ "First order ordinary differential equation", "We first demonstrate the accuracy and efficiency of the NN solver when applied to the following simple first order differential equation with $x \\in [0,2]$ , subject to the initial value $f(0) = 0$ : $\\frac{\\mathrm {d}f}{\\mathrm {d}x}(x) + f(x) = e^{-x} \\cos (x).$ The analytic solution is $f(x) = e^{-x}\\sin (x)$ .", "The equation being simple, the NN is expected to reach an acceptable accuracy for relatively few epochs.", "The accuracy and efficiency of the solver are illustrated in FIG.", "REF where the NN has been trained for $10^4$ epochs reaching $1.6\\times 10^{-4}$ as mean relative error.", "The bottom panel in FIG.", "REF exhibits clear wavy patterns showing that the NN solution wiggles around the exact solution.", "Better accuracy can be found by increasing the number of epochs.", "Figure: The NN solution (dots) of Eq.", "() subject to the initial value f(0)=0f(0) = 0 overlaid on the exact solution (solid line).", "The lower panel shows the relative error.", "The network was trained for 10 4 10^4 epochs using Adam optimizer with 100 uniformly spaced training points in [0,2].", "One hidden layer of 10 neurons with a sigmoid activation function was used." ], [ "Stationary Schrödinger equation", "The one dimensional time-independent Schrödinger equation, with potential $V(x)$ , has the following differential equationWe set $\\hbar = 1$ .", ": Figure: The NN n=2n = 2 energy eigenfunction (dots) as a solution of Eq.", "() along with the exact solution (solid line), shown for comparative purposes, at different epochs during the training.", "The same boundary conditions, architecture, activation function, and domain sampling as FIG.", "were used.$\\frac{\\mathrm {d}^2\\psi }{\\mathrm {d}x^2}(x) + 2m(E_n - V(x))\\psi (x) = 0.$ The energy $E_n$ is quantized with an integer $n$ .", "For a harmonic potential $V(x) = x^2$ , the $n^{\\text{th}}$ level has energy $n + 1/2$ with a corresponding analytical solution to the $n^{th}$ energy eigenfunction $\\psi _n(x)$ given in terms of the Hermite polynomials.", "FIG.", "REF shows the NN evaluation of the energy eigenfunction $\\psi (x)$ for $n = 1$ .", "In this example, Dirichlet boundary conditions are imposed at $x = \\pm 2$ .", "Note that the relative error is maintained at $\\sim 10^{-4}$ in between the boundary conditions and degrades outside.", "The NN is found to perform well on solving a boundary-value problem.", "As the exact solution rapidly vanishes outside the boundary conditions, the relative error dramatically increases so that it is no longer a valid measure of accuracy.", "Figure: NN solution (dots) of Eq.", "() for n=1n = 1 overlaid on the exact solution (solid line).", "Dirichlet boundary conditions are imposed at x=±2x = \\pm 2.", "The lower plot shows the relative error.", "The network was trained for 5×10 4 5\\times 10^4 epochs using Adam optimizer with 100 uniformly spaced training points in [-5,5][-5,5] (but displayed for [-4,4][-4,4]).", "One hidden layer of 50 neurons with a sigmoid activation function was used.The required number of epochs during the training can only be found by iterative testing.", "FIG.", "REF shows the neural network prediction for the $n = 2$ energy eigenfunction for $x\\in [-4,4]$ at different epochs.", "One can see a general tendency in the training process that was found for other equations as well: (i) first, the NN fits the general trend of the solution by adjusting its prediction along a tangent of the curve (FIGs.", "REF and REF ), (ii) then, the NN fits the curvature to reveal the true shape of the solution (FIGs.", "REF -REF ), (iii) finally, it fine-tunes to adjust the prediction to the exact solution, decreasing the relative error (FIG.", "REF ).", "These three phases of the training can be used to guess the number of epochs required to reach the desired relative error.", "Generally, the more complex the solution is, the more epochs are necessary to let the NN fit the curvature.", "Once the right curve is found, increasing the number of epochs increases the accuracy." ], [ "Burst equation", "To illustrate the NN's ability to fit a complex shape, we solve the following second-order differential equation $\\frac{\\mathrm {d}^2f}{\\mathrm {d}x^2}(x) + \\frac{n^2 - 1}{(1+x^2)^2} f(x) = 0.$ The solution of Eq.", "(REF ) is characterized by a burst of approximately $n/2$ oscillations in the region $|x| < n$ [20].", "An analytical solution for the equation is $f(x) = \\frac{\\sqrt{1+x^2}}{n} \\cos (n \\, \\arctan \\, x).$ The exact solution and the NN prediction is shown in FIG.", "REF for $n=10$ .", "With usual numerical solvers such as Runge-Kutta based approaches, the relative error grows exponentially during the step by step integration, failing to reproduce the oscillations of Eq.", "(REF ) (see FIG.", "REF ) [20].", "With an optimization-based approach, the trained NN is able to capture the behavior of the rapid oscillations, albeit requiring the training of a more complicated architecture over a greater number epochs when compared to the first order differential equation example in FIG.", "REF and the Schrödinger equation example in FIG.", "REF .", "Training on minibatches of the discretized domain was found to outperform training using the whole domain as a batch.", "Even for boundary conditions being imposed only at positive $x$ values, the network is able to retain the symmetry of the solution for negative $x$ values without losing accuracy.", "Figure: NN solution (dots) of Eq.", "() for n=10n = 10 overlaid on the exact solution Eq.", "() (solid line).", "Dirichlet boundary conditions are imposed at x=1.5x = 1.5 and x=3x = 3.", "The lower plot shows the relative error.", "The network was trained for 2×10 5 2\\times 10^5 epochs with minibatches of size 30 using Adamax optimizer with 300 uniformly spaced training points in [-7,7][-7,7].", "Three hidden layers, each having 30 neurons with a tanh activation function were used." ], [ "Performances", "We find that even if the NNs are not susceptible to overfitting within the training domain, the main issue of the NN approach arises from the arbitrariness in the choice of NN hyperparameters and the choice of the NN architecture.", "The flexibility of the method enables us to obtain a thorough understanding of how well different NN models perform on solving differential equations.", "Traditional non-linear activation functions such as sigmoid and tanh are found to reproduce the ODE solution with less training than modern activation functions such as ReLU, ELU, and SELU.", "Rather than using a random uniform distribution or constant initialization, we find that the properly scaled normal distribution for weight and bias initialization called Xavier initializationGlorotNormal in Tensorflow 2. , makes the training process converge faster.", "Xavier initialization is used for all figures.", "Also, Adam based approaches including Adamax and Nadam, with a learning rate set to $10^{-3}$ , are found to perform better than stochastic gradient descent." ], [ "Identifying the loss function with the mean squared error", "To evaluate the accuracy of the NN solution and the training performance, we suggest to use the loss function $\\mathcal {L}$ in Eq.", "(REF ).", "Indeed, we show that the loss function as written in Eq.", "(REF ) can be identified with the mean absolute error computed with the $L^2$ norm, with some scaling.", "The identification makes the loss function an excellent indicator of how accurate the found solution is without the need of an exact solution.", "Furthermore, the numerical stability and the computational cost of the training process can be assessed without computing the relative error, as it is automatically encrypted in the loss function (REF ).", "The main idea of this identification is to write the loss function (REF ) as a continuous functional of the unknown solution $f$ and Taylor expand it around an exact solution $f^0$ to the second order.", "Using the same notations as in Section , let us take the continuous limit of the first term in Eq.", "(REF ), turning the sum into an integral, and defining the loss $\\mathcal {L}$ as a functional $\\mathcal {L}[f]$ $\\mathcal {L} \\rightarrow \\mathcal {L}[f],$ with $\\begin{aligned}\\mathcal {L}[f] &= \\int _{\\mathcal {D}}\\mathrm {d}x\\, \\mathcal {F}[x, f(x), \\nabla f(x), ..., \\nabla ^{p}f(x)]^2 \\\\&+ \\sum _{j} [\\nabla ^{k}f(x_j) - K_{j}]^2.\\end{aligned}$ The loss functional has to be minimized on the entire differential domain $\\mathcal {D}$ rather than on a set of discrete points $x_i\\in \\mathcal {D}$ .", "Using the physics language, $\\mathcal {L}[f]$ is to be understood as the action with some Lagrangian $\\mathcal {F}[x, f(x), \\nabla f(x), ... \\nabla ^{p}f(x)]^2$ .", "For clarity, we define $\\bar{\\mathcal {F}} = \\mathcal {F}^2$ .", "The Cauchy-Peano theorem guarantees the existence of an exact solution $f^0$ to differential equations subject to boundary/initial conditions.", "The solution $f^0$ minimizes the loss function making it vanish $\\mathcal {L}[f^0] = 0$ .", "We now Taylor expand $\\mathcal {L}[f]$ around $f^0$ to the second order $\\begin{aligned}&\\mathcal {L}[f] = \\int _{\\mathcal {D}}\\mathrm {d}x\\, \\bar{\\mathcal {F}}[x, f^0(x), \\nabla f^0(x), ..., \\nabla ^{p}f^0(x)] \\\\&+ \\int _{\\mathcal {D}}\\mathrm {d}x\\, \\frac{\\delta \\bar{\\mathcal {F}}}{\\delta f}[x, f^0(x), \\nabla f^0(x), ... ,\\nabla ^{p}f^0(x)]\\, \\left(f(x) - f^0(x) \\right) \\\\&+ \\int _{\\mathcal {D}}\\mathrm {d}x\\, \\frac{\\delta ^2 \\bar{\\mathcal {F}}}{\\delta f^2}[x, f^0(x), \\nabla f^0(x), ... , \\nabla ^{p}f^0(x)]\\, \\left(f(x) - f^0(x) \\right)^2 \\\\&+ \\mathcal {O}\\left(\\left[f(x) - f^0(x) \\right]^3\\right).\\end{aligned}$ The second term in Eq.", "(REF ) vanishes because $f^0$ satisfies the boundary/initial conditions.", "The zero order term in Eq.", "(REF ) vanishes because $f^0$ is an exact solution i.e.", "$\\bar{\\mathcal {F}}[x, f^0(x), \\nabla f^0(x), ... \\nabla ^{p}f^0(x)] = 0$ for all $x\\in \\mathcal {D}$ .", "The first order term in Eq.", "(REF ) vanishes because $f^0$ minimizes the loss function so that the functional derivative is zeroBy analogy with physics, this term would be the Euler-Lagrange equation after an integration by parts.", "i.e.", "$\\frac{\\delta \\bar{\\mathcal {F}}}{\\delta f}[x, f^0(x), \\nabla f^0(x), ... \\nabla ^{p}f^0(x)] = 0$ for all $x\\in \\mathcal {D}$ .", "The remaining non-vanishing term is the second order term.", "As the $\\frac{\\delta ^2 \\bar{\\mathcal {F}}}{\\delta f^2}[x, f^0(x), \\nabla f^0(x), ... \\nabla ^{p}f^0(x)]$ term does not depend on the NN solution and so does not depend on the parameters ${\\theta }$ , it is a function of $x$ only and can be seen as a scaling $\\alpha (x)$ $\\mathcal {L}[f] = \\int _{\\mathcal {D}}\\mathrm {d}x\\, \\alpha (x)\\, \\left[f(x) - f^0(x) \\right]^2 \\\\+ \\mathcal {O}\\left(\\left[f(x) - f^0(x) \\right]^3\\right).$ Written in this form, the loss function appears to be the mean absolute error using the $L^2$ norm integrated over the entire domain $\\mathcal {D}$ , with some scaling function $\\alpha (x)$ .", "After discretizing $\\mathcal {D}$ and identifying $f(x)$ with the neural network $\\mathcal {N}(x, {\\theta })$ , the loss function (REF ) takes the final form $\\begin{aligned}\\mathcal {L}({\\theta }) &\\approx \\frac{1}{N}\\sum _{i=1}^N \\alpha (x_i) \\left[\\mathcal {N}(x_i, {\\theta }) - f^0(x_i) \\right]^2\\\\&+ \\sum _{j} [\\nabla ^{k}\\mathcal {N}(x_{j}, {\\theta }) - K_{j}]^2.\\end{aligned}$ Note that taking the continuous limit requires enough discrete points to make it valid.", "Here, NNs are usually trained with $\\sim \\mathcal {O}(100)$ points within the considered domains.", "Through this identification, we better understand the three training phases shown in section REF .", "Indeed, as the loss function (REF ) can be Taylor expanded around the exact solution, the NN first minimizes the loss leading order term making it learn the general tendency of the solution first by finding the tangents.", "Then, the NN minimizes higher order terms, making the NN learn specific local patterns, like curvature.", "Finally, the NN adjusts the solution by considering even higher orders of the loss.", "These universal phases enable the method to converge locally with a stable accuracy that does not decrease over the entire training domain." ], [ "Domain sampling", "Our hypothesis is that using fine sampling for regions with more features, such as oscillations, would lead to greater accuracy i.e.", "lower loss.", "Moreover, the computational cost of training the NN could be reduced by avoiding dense sampling of featureless regions in the domain.", "We probe different domain samplings and expose their impact on the NN performance.", "FIG.", "REF illustrates the effect of different domain samplings for the $n=5$ energy eigenfunction of Eq.", "(REF ): evenly spaced points, random uniformly distributed points, and points sampled from a Gaussian distribution centered at $x=0$ with standard deviation of 1, to accentuate the oscillatory region.", "Surprisingly, a NN trained on evenly spaced points performs better than the other distributions.", "Every point in the region contributes equally in the loss function (REF ) and the NN learns the ODE solution uniformly over the entire domain.", "The uniform behavior can be seen in the lower panel of FIG.", "REF as the relative error remains constant over the entire domain.", "Moreover, increasing the number of training points is found to have no influence on the NN performances but requires more training to reach the same accuracy.", "Randomizing the points is found to increase the relative error by an order of magnitude.", "Choosing a sampling distribution that accentuates a certain region fails to reproduce the ODE solution's shape for the same training time.", "Further training is then necessary to reproduce the same accuracy.", "Figure: NN solution of Eq.", "() for n=5n=5 overlaid on the exact solution (solid line) for (i) an evenly spaced training domain (blue dots), (ii) a random uniform training domain (green dots), and (iii) a random Gaussian training domain centered at x=0x=0 with width 1 (red dots).", "The lower panel shows the relative error.", "All training domains contain 100 points in [-5,5][-5, 5].", "Dirichlet boundary conditions at x=±4x=\\pm 4 were imposed.", "The network was trained for 4×10 5 4\\times 10^5 epochs using Adamax optimizer and tanh activation, and two hidden layers with 20 neurons in each layer were used.Figure: NN prediction of Eq.", "() for n=5n=5 overlaid on the exact solution (solid line) for (i) the training points, 100 evenly spaced points in [-4,4][-4, 4] (blue dots), (ii) 40 evenly spaced points in [0,2][0, 2], and (iii) 30 evenly spaced points in [-6,-4][-6, -4] and [4,6][4, 6].", "The lower panel shows the relative error.", "Dirichlet boundary conditions were imposed at x=±3x = \\pm 3.", "The network was trained for 5×10 5 5\\times 10^5 epochs using three hidden layers with 30 neurons in each layer.", "The same optimizer and activation as in FIG.", "were used." ], [ "Extrapolation performances", "Once the NN is trained, it is expected to be a continuous function within the finite, bounded training domain.", "Here, we illustrate the continuous nature of the NN solution.", "FIG.", "REF shows the extrapolation performances for points inside the training domain and outside the training domain.", "The $n=5$ energy eigenfunction of Eq.", "(REF ) is solved within $[-4, 4]$ using 100 evenly spaced points.", "The NN prediction is then evaluated in $[0, 2]$ , and in $[-6, -4]\\cup [4, 6]$ .", "The NN is able to reproduce the ODE solution with the same accuracy within the training domain but fails to reproduce the solution outside the training domain, even with more training.", "Note that the mean relative error is smaller than in FIG.", "REF as more epochs were used for the training.", "In classical numerical analysis, the sample spacing i.e.", "the integration step is usually small to avoid numerical instabilities [22].", "Here, the optimization approach enables us to train the NN on a reduced number of points, hence reducing the training time, and allows then to evaluate the prediction on a fine mesh." ], [ "Neural network architectures", "Choosing the exact architecture for a NN remains an art that requires extensive numerical experimentation and intuition, and is often times problem-specific.", "Both the number of hidden layers and the number of neurons in each layer can affect the performance of the NN.", "A detailed systematic study to quantify the effect of the NN architecture in presented in FIG.", "REF .", "We vary both the number of neurons in a single hidden layer and the number of hidden layers for all three equations (REF ), (REF ) and (REF ).", "An unsatisfactory value of the loss after plateauing can be viewed as a poor NN architecture performance, as further training in general does not reduce its value.", "Here, the loss function is used to evaluate the NN performances during the training process as it can be identified with the scaled mean squared error (see section REF ).", "Intuitively, one would expect an increasing NN complexity to enhance the accuracy.", "We found that some NN architectures, not necessary the more complex ones, perform better than others in reaching the desired accuracy for a certain number of epochs.", "In general, increasing the number of hidden layers make the loss decrease more rapidly in the beginning of the training until roughly $10^4$ epochs, but some simpler NN architectures prove to reach a better accuracy after more training.", "A clear link between the ODE complexity and the number of NN parameters is shown.", "The first order ODE (REF ) being simple, a NN with one hidden layer containing 20 neurons is able to outperform a more complex NN with one hidden layer containing 1000 neurons (FIG.", "REF ).", "The difference affects the loss value by a few orders of magnitude.", "In this case, the number of hidden layers does not show any significant enhancement for the same amount of training.", "The Schrödinger equation (REF ) being more complex because it is a second order ODE, a two hidden layer NN performs better than a single hidden layer NN.", "However, the loss function for more than two hidden layers rapidly reaches a plateau and does not decrease after more training, making too complex NNs inefficient (FIG.", "REF ).", "Increasing the number of hidden layers for the Burst equation (REF ) to three is shown to both decrease the loss function faster and reach a smaller value than simple NNs (FIG.", "REF ).", "This tendency reveals the ODE complexity.", "Indeed, for all three equations, an empiric optimal NN architecture can be found and its complexity increases with the ODE complexity.", "The main limitation is that we find no clear recipe of how to guess the optimal architecture, as it has to be done by testing.", "However, we suggest that too complex NNs (more than three hidden layers and more than 100 neurons in each layer) should not be used as the problem of solving ODEs is in general too simple." ], [ "Conclusion", " We have explored and critiqued the method of using NNs to find solutions to differential equations provided by Piscopo et al.", "[24].", "We found that the NN was able to perform better than typical numerical solvers for some differential equations, such as for highly oscillating solutions.", "We must note that the proposed method should not be seen as a replacement of numerical solvers as, in most cases, such methods meet the stability and performance required in practice.", "Our main message here is that the NN approach no longer appears as a black-box but a rather intuitive way of constructing accurate solutions.", "This approach was studied in detail.", "Three phases of fitting were characterized: first finding the general trend, secondly adjusting the curvature of the solution, and finally making small adjustments to improve the accuracy of the solution.", "Within the training domain, the NN was found to provide a continuous approximate function that matches the analytical solution to arbitrary accuracy depending on training time.", "However, extrapolation outside the training domain fails.", "We found that training in smaller minibatches rather than the whole discretized domain used in Lagaris et al.", "[23] and Piscopo et al.", "[24] gives a greatly reduced loss.", "A specifically designed loss function from the literature [24] was proved to be the appropriate metric for evaluating the solution accuracy and the NN performances without the need of the exact solution, which is usually not known.", "Finally, we found the limitation of the method is finding a suitable architecture.", "There is no trivial relationship between the NN architecture and the accuracy of the NN approximate solution for a general differential equation, though a general tendency to increase the number of NN parameters to solve more complex differential equations was highlighted.", "A range of questions can be immediately explored.", "The performances of more sophisticated NN structures with dropouts and recurrent loops can be studied.", "Other sampling schemes can also be tested.", "Another question is whether the convergence of the NN solution to a certain accuracy can be achieved with fewer epochs.", "With the method we described, convergence comes locally, similar to a Taylor series.", "One might be able to reformulate the NN such that convergence comes globally, via a Fourier series representation, or using a different complete basis.", "Such reformulations might help the NN to learn general representations about the ODEs .", "Global convergence may give better extrapolation results.", "Nevertheless, NNs show great potential as a support for standard numerical techniques to solve differential equations.", "Figure: Loss function ℒ\\mathcal {L}, proved to be the scaled mean squared error, during the training for different NN architectures.", "The first order ODE, the Schrödinger equation for n=2n=2 and the Burst equation for n=10n=10 are shown in , and respectively.", "For each Figure, the upper panel is generated by varying the number of neurons in a single hidden layer, and the lower panel is generated by varying the number of hidden layers with 20 neurons in each layer.", "The same optimizers, activation functions and training domains as in FIG.", ", and for each equation were used.", "The learning rate was set 10 -4 10^{-4} for each optimizer.", "D.W. thanks the ENS Paris-Saclay for its continuing support via the normalien civil servant grant." ] ]

[ [ "A deep learning-based pipeline for error detection and quality control\n of brain MRI segmentation results" ], [ "Abstract Brain MRI segmentation results should always undergo a quality control (QC) process, since automatic segmentation tools can be prone to errors.", "In this work, we propose two deep learning-based architectures for performing QC automatically.", "First, we used generative adversarial networks for creating error maps that highlight the locations of segmentation errors.", "Subsequently, a 3D convolutional neural network was implemented to predict segmentation quality.", "The present pipeline was shown to achieve promising results and, in particular, high sensitivity in both tasks." ], [ "Introduction", "Brain MRI segmentations can be affected by errors and their quality control (QC) is usually carried out visually, which can be very subjective and time consuming.", "Previous studies have proposed deep learning-based methods for performing automatic QC on medical image segmentation results [6], [9], [2], [7].", "However, these solutions do not point out the errors’ locations, which would be useful to understand their impact on research results and speed up their correction.", "In this work, we propose a method that automatically (1) locates segmentation errors by employing conditional generative adversarial networks (cGANs), (2) classifies segmentation quality by using a convolutional neural network (CNN)." ], [ "Methods", "Given a series of input MRI images $I$ and their segmentations $S$ , our goal is to learn a mapping from a certain $S$ to the original MRI $I$ .", "For this purpose, a pix2pix model [3] was defined for each image view (axial, coronal and sagittal).", "Each model receives as input a segmentation slice indicating gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF).", "These segmentations were generated with FreeSurfer 6.0 [1].", "By contrast, the output of each pix2pix model is an MRI slice that is expected to match the input segmentation.", "The generated and the original MRI slice can be compared after intensity normalization.", "Their difference highlights the regions where they do not match, i.e.", "where segmentation errors can be present (see fig1).", "The difference images from all slices and all views can then be aggregated together in a 3D error map.", "These maps are finally refined with post-processing steps, including thresholding and Gaussian smoothing.", "The generated error map and the original MRI image are fed as input into a 3D CNN to predict if the segmentation is good (output 0) or bad (output 1).", "The CNN was modeled using six convolutional layers (with $3\\times 3\\times 3$ kernels and, respectively, 32, 32, 64, 64, 128 and 128 units) intermingled with batch normalization layers.", "These are finally followed by a global average pooling layer and two dense layers (with, respectively, 128 and 1 unit).", "We used images of size $256\\times 256\\times 256$ mm$^3$ from two diagnostic groups (healthy and Alzheimer’s patients) and three cohorts: ADNI [4], GENIC [5] and H70 [8].", "We selected 1600 subjects whose segmentations had been visually rated as accurate.", "From each of these subjects, 10% of the segmented image slices (from all views) were randomly included in the training set of the cGANs.", "The model was then tested on other 600 subjects having segmentation errors and 190 subjects with accurate segmentations.", "Subsequently, the error maps and original MRI images were down-sampled to half resolution to be used as inputs for the classification CNN.", "A class-balanced dataset was obtained by randomly selecting 300 subjects having accurate segmentations and 300 presenting errors.", "The performance of the CNN was investigated using 10-fold cross-validation on these balanced data.", "Figure: Schematic representation of the method for generating error maps.", "It is shown for the axial view, but the same idea is applied on all three views." ], [ "Results", "The cGAN-based method was generally effective in locating segmentation errors.", "It successfully detected particularly large errors in Alzheimer's brains, whose segmentations are usually more challenging because of their neurodegenerative patterns (fig2a).", "The effectiveness of the method was also evident in some cases where small errors were identified in segmentations that had been rated as accurate with visual QC (fig2c).", "On the other hand, a few false positives (i.e.", "highlighted regions that do not actually correspond to errors) were identified in most subjects (fig2b).", "Moreover, the error maps are based on intensity differences, so errors in regions with high contrast (e.g.", "WM vs. CSF) were better highlighted than those in areas having similar intensity (e.g.", "cortex vs. meninges), as shown in fig2d.", "To investigate the performance of the classification CNN, we computed the area under the ROC curve, which was equal to 0.85.", "By setting a classification threshold of 0.4, the highest accuracy (i.e.", "0.80) was obtained, with a recall of 0.83 and a precision of 0.77.", "By lowering the threshold to 0.3, a high recall of 0.96 was observed against a precision of 0.56.", "Figure: Examples of slices of 3D error maps.", "For each view, the segmentation result (red = GM, green = WM, blue = CSF) and its error map are shown.", "In (a), a large segmentation error was accurately detected.", "In (b), a false positive (caused by subject-specific anatomical features) was found in the error map (red arrow).", "In (c), a small CSF misclassification (red arrows) was identified on a segmentation that had been visually rated as good.", "In (d), the method fails at highlighting part of several cortical overestimations (red arrows)." ], [ "Conclusions and future work", "The proposed deep learning-based pipeline was shown to be promising for both automatic QC and localization of brain MRI segmentation errors with high sensitivity.", "This tool could thus be used not only to check segmentation quality, but also to speed up the error correction and to evaluate the reliability of segmentation results in certain brain regions.", "One limitation is the high presence of false positives (segmentations wrongly classified as bad) in both the error maps and the quality classification.", "We believe that a high sensitivity to bad segmentations is yet preferable to a high specificity, because it limits the use of wrong segmentation results in research studies.", "However, we still aim at increasing the precision by extending the post-processing steps on the error maps and testing other networks for the quality classification.", "Moreover, for comparing the generated and original MRI slice, we will test other techniques that are less intensity-dependent." ], [ "Acknowledgments", "This project is financially supported by the Swedish Foundation for Strategic Research (SSF), the Swedish Research council (VR), the joint research funds of KTH Royal Institute of Technology and Stockholm County Council (HMT), the regional agreement on medical training and clinical research (ALF) between Stockholm County Council and Karolinska Institutet, the Swedish Alzheimer foundation and the Swedish Brain foundation.", "Data collection and sharing for this project was funded by the Alzheimer's Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012).", "ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.", "; Johnson & Johnson Pharmaceutical Research & Development LLC.", "; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.", "; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics.", "The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada.", "Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org).", "The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California.", "ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California." ] ]

[ [ "Inflation and Reheating in f(R,h) theory formulated in the Palatini\n formalism" ], [ "Abstract A new model for inflation using modified gravity in the Palatini formalism is constructed.", "Here non-minimal coupling of scalar field h with the curvature R as a general function f(R,h) is considered.", "Explicit inflation models for some choices of f(R,h) are developed.", "By writing an equivalent scalar-tensor action for this model and going over to Einstein frame, slow roll parameters are constructed.", "There exists a large parameter space which satisfies values of n_s and limits on r compatible with Planck 2018 data.", "Further, we calculate reheating temperature and the number of e-folds at the end of reheating for different values of equation of state parameter for all the constructed models." ], [ "Introduction", "There are mainly two formulations in General Relativity (GR) popularly known as Palatini and Metric formalisms.", "Palatini formalism or first order formalism treats space-time connections as independent variable[1], [2], [3], [4], [5], [6], [7], [8], [9], whereas in Metric formalism, these connections are not independent but derived from the metric itself.", "But in GR these two approaches produce same Einstein equation.", "Hence dynamics are equivalent in both formalisms.", "This is not true for modified gravity models and models where fields are nonminimally coupled to gravity.", "In these cases, both formalisms represent different physical situations [1], [2], [3].", "Inflation [10], [11], [12], [13], [14], [15], [16] was first developed in the early 1980s to solve problems of standard Big Bang theory like horizon problem, fine-tuning problem etc.", "Quantum fluctuations also started during the period of inflation which led to the cosmic microwave background(CMB) anisotropy and provided the seed for the formation of large scale structure of the universe.", "A more popular model for inflation is the Starobinsky model which is a pure gravity theory with an additional $R^2$ term in the Einstein-Hilbert action.", "An equivalent scalar-tensor theory of Starobinsky model has an additional scalar degree of freedom apart from two tensor degrees of freedom of Einstein-Hilbert action[17], [18], [19].", "In Einstein frame, this theory is equivalent to the usual scalar field model with a potential suitable for a valid inflation model which satisfies all the constraints from CMB data.", "In a broad sense, the Starobinsky model falls under a general framework of f(R) gravity.", "The above analysis works best in metric formulation of gravity.", "However, in the Palatini formalism, no additional propagating degrees of freedom appears in f(R) gravity theory [3].", "Because of this, no inflation is possible in this scenario.", "Hence a scalar field needs to be added to this action to develop an Inflation model in Palatini formulation of gravity.", "In this line, first work appeared in [20] where nonminimal couplings of scalars are considered.", "Since then many works related to different inflationary potentials, preheating, reheating, postinflationary phases, dark matter has been done with many variants of action including a $R^2$ term [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66].", "For an introduction to Palatini inflation models refer to [67] and references therein.", "In this work, we formulate a palatini inflation model in $f(R, h)$ theory.", "In metric formalism inflation model in $f(R, h)$ gravity has been considered earlier in literature [68], [69], [70].", "Recently, it has been shown that the appearance of terms like $h^2R^2$ in the one loop effective action for massless, conformally coupled scalar field [71].", "Our action is of the form similar to form obtained in [68].", "But our approach here is to develop a model of inflation in palatini formalism.", "We show that such an action provide a favorable inflationary scenario satisfying Planck 2018 data [72].", "Reheating [73], [74], [75], [76] is a phenomenon which acts as a transit between Inflation and the radiation dominated era of the universe.", "There are various models on how the inflaton field loses its energy.", "Using techniques similar to [77], [78], [79], [80], [81], the number of e-folds and temperature at the end of reheating ($T_{re}$ and $N_{re}$ ) can be written in terms of inflationary observables.", "Earlier, reheating in Palatini models of inflation has been discussed in these papers.", "[57], [45], [59] This paper is organized as follows.", "In the next section, we introduce the action of our model.", "Then we express it in terms of equivalent scalar-tensor action in Einstein frame.", "In section , slow roll parameters are defined.", "Values of $n_s$ and $r$ are also calculated numerically and the results are presented.", "Section is dedicated to the calculations of reheating parameters.", "At last, in section , we summarize our results." ], [ "The Model", "We start with a general action of the form - $ S &=& \\int \\mathrm {d}^4 x \\sqrt{-g} \\left[ \\frac{1}{2}f(R,h) - \\frac{1}{2}g^{\\mu \\nu } \\partial _\\mu h \\partial _\\nu h- V(h) \\right] ,$ where $g_{\\alpha \\beta }$ is the metric, $g$ is its determinant, $f(R,h)=G(h)(R+\\alpha R^2)$ , $R$ is the Ricci scalar defined as $R=g^{\\alpha \\beta } R^{\\gamma }_{\\ \\, \\alpha \\gamma \\beta }(\\Gamma , \\partial \\Gamma )$ in Palatini formalism and $\\alpha $ is a constant.", "Here we have chosen Planck mass to be unity.", "An equivalent action in terms of an auxiliary field $\\phi $ can be written as - $S &=& \\int \\mathrm {d}^4 x \\sqrt{-g} \\left[ \\frac{1}{2} f(\\phi ,h) + \\frac{1}{2}\\ f^{\\prime }(\\phi ,h) (R - \\phi ) \\ - \\frac{1}{2} g^{\\mu \\nu } \\partial _\\mu h \\partial _\\nu h - V(h) \\right] \\ ,$ where $f^{\\prime }(\\phi ,h)=\\frac{\\partial f(\\phi ,h)}{\\partial \\phi }$ .", "Varying $S$ with respect to $\\phi $ in equation (REF ), we get $\\phi = R$ if $[2]{f(\\phi ,h)}{ \\phi }\\ne 0$ and using this result in equation (REF ), we recover equation (REF ).", "Rearranging equation (REF ), the action becomes - $S &=& \\int \\mathrm {d}^4 x \\sqrt{-g} \\left[ \\frac{1}{2} f^{\\prime }(\\phi ,h) R - W(\\phi ,h)- \\frac{1}{2} g^{\\mu \\nu } \\partial _\\mu h \\partial _\\nu h - V(h) \\right] \\ ,$ where $W(\\phi ,h)= \\frac{1}{2}\\phi f^{\\prime }(\\phi ,h) - \\frac{1}{2} f(\\phi ,h)$ .", "By making a conformal transformation - $g_{\\mu \\nu } \\rightarrow f^{\\prime }(\\phi ,h) g_{\\mu \\nu } \\ ,$ action is obtained in Einstein frame as - $S &=& \\int \\mathrm {d}^4 x \\sqrt{-g} \\left[ \\frac{1}{2} R -\\frac{1}{2}\\frac{ \\partial _\\mu h \\partial ^\\mu h}{f^{\\prime }(\\phi ,h)}- \\frac{W(\\phi ,h)+V(h)}{f^{\\prime }(\\phi ,h)^2} \\right] \\ .$ Let us define a new potential $\\hat{V}(\\phi ,h)$ as - $\\hat{V}(\\phi ,h)\\equiv \\frac{W(\\phi ,h)+V(h)}{f^{\\prime }(\\phi ,h)^2} .$ Now for our choice of $f(R,h) = G(h)(R+\\alpha R^2)$[68], the new potential can be written as - $\\hat{V}(\\phi ,h)= \\frac{1}{f^{\\prime }(\\phi ,h)^2} \\left[\\frac{1}{8 \\alpha G(h)}[f^{\\prime }(\\phi ,h)-G(h)]^2 +V(h)\\right] .$ Varying equation (REF ) with respect to $\\phi $ , we get the constraint equation - $f^{\\prime }(\\phi ,h)= \\frac{8\\alpha V(h) + G(h)}{1- 2\\alpha \\partial _\\mu h \\partial ^\\mu h} .$ Inserting equation (REF ) to equation (REF ), we can eliminate $\\phi $ .", "Then again inserting equation (REF ) to equation (REF ) and rearranging and simplifying, we get - S = d4 x -g [ 12 R - 121(8 V + G ) h h + 2 1(8V+G) ( h h )2 - VG(8V+G)] .", "Now the last term can be defined as effective potential in the Einstein frame - $ U\\equiv \\frac{V}{G(8\\alpha V+G)} \\ .$ In order to bring kinetic part of scalar field into that of canonical form, we introduce a new field $\\chi $ as - $ {h}{\\chi } = \\pm \\sqrt{(8 \\alpha V + G )} .$ In terms of $\\chi $ , equation () becomes - $ S = \\int \\mathrm {d}^4 x \\sqrt{-g} \\left[ \\frac{1}{2} R - \\frac{1}{2} \\partial ^\\mu \\chi \\partial _\\mu \\chi + \\frac{\\alpha }{2}{(8\\alpha V+G)} ( \\partial ^\\mu \\chi \\partial _\\mu \\chi )^2 -U\\right] \\ .$ Now we are ready to build up our inflation model, with this action.", "Next, we define the slow-roll parameters to estimate the observables which will decide the fate of our model.", "Here we have neglected contribution due to the third term of the action to the inflation phase.", "This is a valid assumption so far as the slow-roll inflation is concerned (see [67] and references therein).", "In this case then, it is a model of inflation driven by a scalar field with potential $U$ Background evolution of $\\chi $ field for a flat FRW metric with scale factor $a$ and hubble parameter $H=\\frac{\\dot{a}}{a}$ is- $ a^2 \\ddot{\\chi } + 2 a^3 H \\dot{\\chi } + 12 a H f(\\chi ) \\left(\\dot{\\chi }\\right)^2 + 3 \\frac{df(\\chi )}{d\\chi }\\left(\\dot{\\chi }\\right)^4+ a^4 U^{\\prime }=0$ where $f(\\chi )=\\frac{\\alpha }{2}(8 \\alpha V + G)$ and $\\dot{}$ and $^{\\prime }$ represent derivatives with respect to conformal time and $\\chi $ respectively.", "Under slow-roll approximation, terms proportional to $\\dot{\\chi }^2$ and $\\dot{\\chi }^4$ can be neglected and equation (REF ) reduces to usual FRW equations for minimally coupled field.", "." ], [ "Slow-roll parameters and Results", "Before analyzing the dynamics of our model, we define slow-roll parameters (mainly $\\epsilon $ and $\\eta $ ) which are helpful to decide whether a model can describe inflation or not.", "They can be written as - $\\epsilon &=& \\frac{1}{2} \\left(\\frac{\\frac{dU}{d\\chi }}{U}\\right)^2 , \\nonumber \\\\\\eta &=& \\frac{\\frac{d^2 U}{d \\chi ^2}}{U} .$ The slow roll parameters $\\epsilon $ and $\\eta $ must be $\\ll 1$ during inflation phase of expansion.", "For the potential in equation (REF ), slow roll parameters become - $\\epsilon &=& \\frac{G\\bar{\\epsilon }}{(8\\alpha V+G)} \\left[1-\\left(\\frac{8 \\alpha \\frac{V}{G^2}G^{\\prime }}{\\frac{V^{\\prime }}{V}-\\frac{2G^{\\prime }}{G}}\\right)\\right]^2 ,\\nonumber \\\\\\eta &=& \\bar{\\eta }\\frac{G}{8\\alpha V+G}- \\left[(-256 \\alpha ^2 V^3 {G^{\\prime }}^2+8 \\alpha G V^2(-11 {G^{\\prime }}^2+8 \\alpha G^{\\prime } V^{\\prime } +16 \\alpha V G^{\\prime \\prime })\\right.", "\\nonumber \\\\ && \\left.+2 G^2 V(8 \\alpha G^{\\prime } V^{\\prime } +24 \\alpha V G^{\\prime \\prime }) + G^3(24\\alpha {V^{\\prime }}^2-16 \\alpha V V^{\\prime \\prime }))/(2 G^2 V(8 \\alpha V+G))\\right] ,$ where $\\bar{\\epsilon }$ and $\\bar{\\eta }$ are slow-roll parameters for $\\alpha = 0$ and $^{\\prime }$ represents derivative with respect to $h$ .", "Number of e-folds from the time when a mode $k$ crosses the horizon to the end of inflation, $N_k$ can be written as (for unit Planck mass) - $ N_k = - \\int _{h_k}^{h_{end}} \\mathrm {d}h \\frac{1}{\\pm \\sqrt{2 \\epsilon (h) (8 \\alpha V+G)}} .$ Inflation ends when $\\epsilon \\simeq 1$ and this, in turn, determines the value of $h_{end}$ .", "Putting the value of $h_{end}$ and $N_k$ (taken in between 60 and 70) in (REF ) , value of $h_k$ can be obtained.", "Putting this value to equations (REF ) and (REF ), value of $\\epsilon $ and $\\eta $ can be obtained respectively.", "Scalar spectral index and tensor-to-scalar ratio can then be calculated from the following equations - $r= 16\\epsilon , \\nonumber \\\\n_s=1+ 2\\eta -6\\epsilon .$ Now we estimate $r$ and $n_s$ for various cases.", "Here we restrict our analysis only for quadratic and quartic potentials.", "We choose two values of $G=\\gamma h^2$ and $G=1+\\gamma h^2$ and $\\gamma $ is a dimensionless constant." ], [ "Case 1 : $V=\\beta h^2$ and {{formula:01d763f6-d189-4afc-bd0e-af6e39fde82b}}", "For this case - $\\epsilon &=& 2 ( 8 \\alpha \\beta +\\gamma ) , \\nonumber \\\\\\eta &=& 4 (8 \\alpha \\beta + \\gamma ) .$ As neither $\\epsilon $ nor $\\eta $ depend on $h$ , so it will lead to constant $r$ and $n_s$ for a particular choice of $\\alpha $ , $\\beta $ and $\\gamma $ .", "In this case, then it would be difficult to end the inflation and to decide the value of $h_f$ .", "Therefore we will not discuss this case further here.", "In this case, $\\epsilon $ and $\\eta $ depend only on h and $\\gamma $ for a particular value of $N$ .", "The parameters $\\alpha $ and $\\beta $ fixes the value of the field $\\chi $ or $h.$ In fig.", "REF , potential $U$ vs $\\chi $ is plotted.", "In fig.", "REF the relation between $h$ vs $\\chi $ is plotted.", "The value of $\\chi $ changes discontinuously as $h$ changes from negative to positive value.", "Here positive sign is taken in equation (REF ).", "Instead if negative sign is taken, all values of $\\chi $ 's will be negative.", "However, discontinuity of $\\chi $ is there for both positive and negative sign.", "For fig.", "REF and fig.", "REF , the value of parameters $\\alpha = 0.5$ , $\\beta = \\frac{1}{4}\\times 10^{-4} $ and $\\gamma = 0.02$ is chosen.", "For this particular set, the numerically estimated value of $n_s $ and $r$ are $0.967 $ and $0.023$ respectively(we take $N=60$ here).", "Also the result for spectral index and tensor to scalar ratio is shown in fig.", "REF .", "Here $\\gamma $ is changing and $\\alpha =5$ , $\\beta =\\frac{1}{4}\\times 10^{-4}$ and negative sign is considered in equation (REF ).", "It clearly shows that there can be large parameter space available which can satisfy Planck constraints.", "Figure: Plot of rr vs n s n_s for changing γ\\gamma For case 3 and case 4, $G(h)$ is taken as $G=1+\\gamma h^2$ .", "This makes the Lagrangian in (REF ) equivalent to the form $\\frac{1}{2}(R+\\alpha R^2) + \\frac{1}{2} \\gamma h^2(R+\\alpha R)$ plus usual scalar field term.", "Similar to case 2, the graphs of $U$ vs $\\chi $ and $h$ vs $\\chi $ are shown in fig REF and fig.", "REF respectively.", "Unlike to the previous case, here $\\chi (h)$ is continuous for all range of $h$ .", "For fig REF and fig REF , the value of parameters are $\\alpha = 10^{-4}$ , $\\beta = 10^{-4}$ and $\\gamma = 0.03.$ For this particular value of parameters, we obtain $n_s = 0.966$ and $r = 0.017$ .", "The potential in equation (REF ) takes the form - $ U=\\frac{\\beta h^4}{(1 + \\gamma h^2)(8 \\alpha \\beta h^4 + 1 + \\gamma h^2)}.$ Dividing both numerator and denominator by $h^6$ , we get - $U=\\frac{\\beta \\frac{1}{h^2}}{(\\frac{1}{ h^2} + \\gamma )(8 \\alpha \\beta + \\frac{1}{ h^4} + \\gamma \\frac{1}{h^2})} .$ We can clearly observe that $U \\rightarrow 0$ as $h \\rightarrow \\infty $ .", "Hence this potential will not give rise to plateau unlike the results for Lagrangian where $h$ is coupled only with $R$ [38], [39], [40], [82], [45], [49].", "The results for $n_s$ and $r$ are shown in fig.", "REF , REF and REF .", "In fig.", "REF , $\\beta =10^{-4}$ , $\\gamma =10^{-2}$ and $\\alpha $ is changing.", "In fig.", "REF , $\\alpha =10^{-4}$ , $\\gamma =10^{-2}$ and $\\beta $ is changing.", "And in fig.", "REF , $\\alpha =10^{-3}$ , $\\beta =10^{-4}$ and $\\gamma $ is changing.", "Here we consider positive sign in equation (REF ).", "Note that both REF and REF look same.", "This happens because in the expression of $r$ and $n_s$ , $\\alpha $ and $\\beta $ always appear in pairs i.e.", "in the form of $(\\alpha \\beta )$ or ${(\\alpha \\beta )}^2$ .", "Similar to case 2, here also a large parameter space available which satisfies Planck constraints on $r$ and $n_s.$ It is observed from REF that we can tune the parameters of our model to obtain a very small value for $r.$ Figure: Plot of rr vs n s n_s for changing β\\beta Figure: Plot of rr vs n s n_s for changing γ\\gamma Analogous to case 3, potential graph and $h$ vs $\\chi $ are shown in fig REF and fig REF for $\\alpha = 0.1$ , $\\beta = 0.01$ and $\\gamma = 10^{-4}$ which give $n_s=0.965$ and $r=0.04$ .", "Here, potential in equation (REF ) becomes - $ U = \\frac{\\beta h^2}{(1 + \\gamma h^2)(8 \\alpha \\beta h^2 + 1 + \\gamma h^2)} .$ As $h \\rightarrow \\infty $ , $U \\rightarrow 0$ .", "So again no plateau in the potential.", "The $n_s$ and $r$ graphs are plotted in fig.", "REF , REF and REF .", "In fig.", "REF $\\beta = 0.01$ , $\\gamma = 0.001$ and $\\alpha $ is changing.", "In fig.", "REF $\\alpha =10^{-2}$ , $\\gamma =10^{-3}$ and $\\beta $ is varying.", "In fig.", "REF $\\alpha = 10^{-3}$ , $\\beta = 0.01$ and $\\gamma $ is varying.", "Here we consider positive sign in equation (REF ).", "In this case as well, we find a large parameter space available which is compatible with Planck 2018 data.", "Figure: Plot of χ(h)\\chi (h)Figure: Plot of rr vs n s n_s for changing β\\beta Figure: Plot of rr vs n s n_s for changing γ\\gamma" ], [ "Reheating", "In this section our aim is to find the reheating temperature, $T_{re}$ and number of e-folds at the end of reheating, $N_{re}$ for our models.", "Equation of state during reheating is parameterised by a function $\\omega (t)$ , which is obtained from - $P=\\rho \\omega (t) ,$ where $P$ and $\\rho $ represent pressure and density of a particular component.", "For radiation and matter dominated universe, the value of $\\omega $ is $1/3$ and 0 respectively.", "Here we consider $\\omega _{re}$ ranging from $-1/3$ to 1 during reheating period [79].", "We also assume that $\\omega _{re}$ is constant throughout the reheating period.", "Mathematically, the number of e-folds at the end of reheating is defined as - $ N_{re}= \\ln (\\frac{a_{re}}{a_{end}}) ,$ where $a_{re}$ denotes scale factor at the end of reheating and $a_{end}$ denotes scale factor at the end of inflation.", "Using continuity equation along with $k=a_k H_k$ and assuming conservation of energy, one can derive the following equations [79]- $ T_{re}=\\left(\\frac{43}{11 g_{re}}\\right)^{\\frac{1}{3}}\\left(\\frac{a_0 T_0}{k}\\right)H_k e^{-N_k} e^{-N_{re}} ,$ $ N_{re}= \\frac{4}{3(1+\\omega _{re})}\\left[\\frac{1}{4}\\ln (\\frac{45}{\\pi ^2 g_{re}})+\\ln (\\frac{V_{end}^{\\frac{1}{4}}}{H_k})+\\frac{1}{3}\\ln (\\frac{11 g_{re}}{43})+\\ln (\\frac{k}{a_0 T_0})+N_k+N_{re} \\right] .$ Here $k$ denotes pivot scale of horizon crossing, $V_{end}$ denotes the potential at the end of inflation, and $g_{re}$ denotes the number of relativistic species at the end of reheating.", "Note that in equation (REF ), as $N_{re}$ increases, $T_{re}$ decreases.", "Hence more instant reheating implies more temperature.", "For $\\omega _{re}=\\frac{1}{3}$ , we can not extract any information of $N_{re}$ from equation (REF ) as both RHS and LHS get canceled out.", "This happens because when $\\omega _{re}=\\frac{1}{3}$ , the boundary between reheating period and the radiation dominated era is indistinguishable.", "For $\\omega _{re}=\\frac{1}{3}$ , equation (REF ) becomes - $ 0= \\frac{1}{4}\\ln (\\frac{45}{\\pi ^2 g_{re}})+\\ln (\\frac{V_{end}^{\\frac{1}{4}}}{H_k})+\\frac{1}{3}\\ln (\\frac{11 g_{re}}{43})+\\ln (\\frac{k}{a_0 T_0})+N_k .$ For $\\omega _{re}\\ne \\frac{1}{3}$ , equation (REF ) writes as- $ N_{re}= \\frac{4}{(1-3\\omega _{re})}\\left[-\\frac{1}{4}\\ln (\\frac{45}{\\pi ^2 g_{re}})-\\ln (\\frac{V_{end}^{\\frac{1}{4}}}{H_k})-\\frac{1}{3}\\ln (\\frac{11 g_{re}}{43})-\\ln (\\frac{k}{a_0 T_0})-N_k \\right].$ Assuming $g_{re}\\approx 100$ and $k=0.05$ Mpc$^{-1}$ [79], equations (REF ) and (REF ) become - $ N_{re}= \\frac{4}{1-\\omega _{re}}\\left[61.6-\\ln (\\frac{V_{end}^{\\frac{1}{4}}}{H_k})-N_k \\right] ,$ $ T_{re}=\\left[\\left(\\frac{43}{11 g_{re}}\\right)^{\\frac{1}{3}}\\frac{a_0 T_0}{k} H_k e^{-N_k}\\left[\\frac{45 V_{end}}{\\pi ^2 g_{re}}\\right]^{\\frac{-1}{3(1+\\omega _{re})}}\\right]^{\\frac{3(1+\\omega _{re})}{3 \\omega _{re}-1}} .$ Until now, we have done the model independent calculations.", "Now we are ready to do the model dependent calculations for our cases except case 1." ], [ "Case 2 : $V=\\beta h^4$ and {{formula:99845343-4f0a-4c2c-8af2-07c51761b5ac}}", "For this case, equation (REF ) provides (considering $-$ sign before $\\sqrt{2 \\epsilon (h) (8 \\alpha V+G)}$ term) - $N_k=\\frac{1}{32 \\alpha \\beta h_k^2}-\\frac{1}{32 \\alpha \\beta h_{end}^2} .$ Assuming $h_{k}<<h_{end}$ , we get $N_k\\approx \\frac{1}{128 \\alpha \\beta h_{k}^2}$ .", "Also $\\epsilon $ and $\\eta $ for this case is written as- $\\epsilon =\\frac{128 \\alpha ^2 \\beta ^2 h^4}{8 \\alpha \\beta h^2 + \\gamma } ,$ $\\eta =\\frac{128 \\alpha ^2 \\beta ^2 h^4 - 32 \\alpha \\beta h^2}{8 \\alpha \\beta h^2 + \\gamma } .$ Using the above relations, we obtain- $ N_k=\\frac{2}{1-n_s},$ $\\epsilon _k=\\frac{(1-n_s)^2}{4(1-n_s)+32 \\gamma },$ $V_{end}=3 M_p^2 H_k^2 \\frac{\\frac{1-n_s}{8}+\\gamma }{8 \\alpha \\beta h_{end}^2 + \\gamma }.$ Also $H_k$ can be written as - $H_k=\\pi M_p \\sqrt{8 A_s \\epsilon _k} .$ Equations (REF ) and (REF ) can be expressed in terms of inflationary observable parameter $n_s$ using the above relations.", "Figure REF is obtained by varying $n_s$ from $0.95$ to $0.97$ for $\\alpha = 0.5$ , $\\beta = 0.25\\times 10^{-4}$ and $\\gamma =0.01$ .", "Here we use $A_s=2.196\\times 10^{-9}$ .", "The shaded blue region corresponds to Planck's $n_s$ value and the shaded red region is obtained from the fact that Big Bang nucleosynthesis temperature should not be less than $10^{-2} GeV$ .", "All lines meet when $N_{re}\\approx 0$ which means that instantaneous reheating occurs irrespective of the choices of $\\omega _{re}$ .", "Also, the temperature is maximum for instantaneous reheating as expected.", "For $\\omega _{re}=\\frac{1}{3}$ , as any value of $N_{re}$ would satisfy equation (REF ), hence it will give rise to a vertical line passing through instantaneous reheating point.", "The graph shows good agreement with Planck's data for all values of $\\omega _{re}$ .", "Figure: Plots of N re N_{re} and T re T_{re} with respect to n s n_s.", "Here red, blue, green and orange colour lines correspond to ω re =-1 3,0,2 3\\omega _{re}=-\\frac{1}{3}, 0, \\frac{2}{3} and 1 respectively.For case 3, $N_k$ can not be expressed in terms of $n_s$ in a simplified form.", "Hence we need to take some approximation.", "Note that in fig.", "REF , we have taken $\\alpha =10^{-4}$ , $\\beta =10^{-4}$ and $\\gamma =0.03$ which are favourable by Planck data.", "Also value $h_k$ is of the $O(10)$ .", "Hence, in the equation (REF ), we can safely neglect $ 8 \\alpha \\beta h^4$ term in comparison to 1 and $\\gamma h^2$ .", "With this approximation, $\\epsilon $ , $\\eta $ , $n_s$ and $N_k$ become - $\\epsilon \\simeq \\frac{8}{h^2(1 + \\gamma h^2)},$ $\\eta \\simeq \\frac{12}{h^2} - \\frac{20 \\gamma }{(1 + \\gamma h^2)},$ $N_k \\simeq \\frac{h^2_k}{8}.$ Using the above expressions, we obtain the following equations - $ N_k \\simeq \\frac{-1 + n_s + 16 \\gamma + \\sqrt{1 - 2 n_s + n^2_s + 64 \\gamma - 64 \\gamma n_s + 256 \\gamma ^2}}{16 (\\gamma - \\gamma n_s)} ,$ $n_s \\simeq 1 - \\frac{2 + \\sqrt{1 + \\frac{32 \\gamma }{\\epsilon }}}{\\epsilon } ,$ $V_{end} \\simeq 3 H^2_k \\frac{(1 + 8 \\gamma N_k)(512 \\alpha \\beta N^2_k + 1 + 8 \\gamma N_k) h^2_{end}}{(1 + \\gamma h^2_{end})(8 \\alpha \\beta h^4_{end} + 1 + \\gamma h^2_{end}) 8 N_K} .$ Substituting $N_{k}$ , $n_s$ and $V_{end}$ into equation (REF ) and equation (REF ) and varying $n_s$ from $0.95$ to $0.97$ , we plot $N_{re}$ and $T_{re}$ vs $n_s$ in the figure REF .", "Here we have taken $\\alpha =10^{-4} $ , $\\beta =10^{-4} $ and $\\gamma = 0.03 $ .", "These are the same values using which we obtained figures REF and REF .", "It is observed that all $\\omega _{re}$ values give rise to reheating parameters within known bound.", "If we change $\\gamma $ by an order of 100, we did not find any substantial change in $n_s$ value.", "This can be seen from figure REF , here $\\gamma $ is set to 1 and $\\alpha $ , $\\beta $ has same value as in figure REF .", "Figure: Plots of N re N_{re} and T re T_{re} with respectto n s n_s.", "Here red, blue, green and orange colourlines correspond to ω re =-1 3,0,2 3\\omega _{re}=-\\frac{1}{3}, 0, \\frac{2}{3} and 1respectively.Similar to the case 3, here also we need to take a simplified form of $N_k$ by taking appropriate approximation.", "Taking hint from the values in figures REF and REF , along with the fact that value of $h_k$ is of $O(10)$ , we can neglect the $\\gamma h^2$ term in comparison to $ 8 \\alpha \\beta h^4 $ and 1 in equation (REF ).", "With this approximation, $\\epsilon $ , $\\eta $ and $N_k$ become - $\\epsilon \\simeq \\frac{2}{h^2(1 + 8 \\alpha \\beta h^2 )} ,$ $\\eta \\simeq \\frac{2 - 32 \\alpha \\beta h^2}{h^2(1 + 8 \\alpha \\beta h^2 )} ,$ $N_k \\simeq \\frac{h^2}{4} .$ Using these equations, we obtain the following relations - $N_k \\simeq \\frac{2}{1 - n_s} ,$ $\\epsilon \\simeq \\frac{2}{(\\frac{8}{1 - n_s})(1 + 8 \\alpha \\beta \\frac{8}{1- n_s})} ,$ $V_{end} \\simeq 3 H^2_k \\frac{(1 + 4 \\gamma N_k)(32 \\alpha \\beta N^2_k + 1 + 4 \\gamma N_k) h^2_f}{(1 + \\gamma h^2_f)(8 \\alpha \\beta h^4_f + 1 + \\gamma h^2_f) 4 N_K} .$ Now we are ready to obtain $T_{re}$ and $N_{re}$ as a function of $n_s$ .", "Figure: Plots of N re N_{re} and T re T_{re} with respect to n s n_s.", "Here red, blue, green and orange colour lines correspond to ω re =-1 3,0,2 3\\omega _{re}=-\\frac{1}{3}, 0, \\frac{2}{3} and 1 respectively.Using the same values as in figure REF (i.e.", "$\\alpha =0.1$ , $\\beta =0.01$ and $\\gamma = 10^{-4}$ ), we obtain figure REF .", "Here also, all four values of $\\omega _{re}$ give rise to possibles values of $N_{re}$ and $T_{re}$ ." ], [ "Conclusion", "Inflation models are constructed in a modified gravity theory in Palatini formalism.", "Here our modified action contains a general non-minimal coupling of scalar field with $R$ and $R^2$ term separately.", "Unlike metric formalism, Palatini formalism does not introduce a new scalar degree of freedom in our theory.", "It turns out that $R^2$ term in the action can be translated to a higher order kinetic term and a new potential term for scalar field in an equivalent scalar-tensor set-up in Einstein frame.", "However, in slow-roll inflation setting we can safely neglect the effect of higher order kinetic term in the action.", "The modified potential is responsible for the inflation.", "In this equivalent scalar-tensor theory, we calculate scalar spectral index$(n_s)$ and tensor to scalar ratio($r$ ) for four different cases depending on the form of nonminimal coupling and potential of scalar field.", "We have varied three unknown parameters of our theory to extract the information about $n_s$ and $r.$ It is observed that in most of the cases we have a large parameter space which can match with the results of Planck 2018 constraints on $n_s$ and $r.$ Palatini formalism in modified gravity theories may provide a large class of models for Inflation satisfying CMB data.", "We have studied reheating phenomenon for the models considered in this paper which are compatible with Planck data.", "Also a more general action with nonminimal coupling of scalar field can be studied to explore the cosmology of very early universe.", "A dark energy picture of these models is worth investing in future.", "This work was partially funded by DST (Govt.", "of India), Grant No.", "SERB/PHY/2017041." ] ]

[ [ "A regularity criterion in weak spaces to Boussinesq equations" ], [ "Abstract In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\\mathbb{R}^{3}\\times (0,T)$.", "The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces." ], [ "Introduction", "In this paper we consider the following Cauchy problem for the incompressible Boussinesq equations in $\\mathbb {R}^{3}\\times (0,T)$ $\\left\\lbrace \\begin{array}{c}\\partial _{t}u+\\left( u\\cdot \\nabla \\right) u-\\Delta u+\\nabla \\pi =\\theta e_{3}, \\\\\\partial _{t}\\theta -\\Delta \\theta +(u\\cdot \\nabla )\\theta =0, \\\\\\nabla \\cdot u=\\nabla \\cdot b=0, \\\\u(x,0)=u_{0}(x),\\text{ }\\theta (x,0)=\\theta _{0}(x),\\end{array}\\right.", "$ where $u=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ denotes the unknown velocity vector, $\\theta =(\\theta _{1}(x,t),\\theta _{2}(x,t),\\theta _{3}(x,t))$ and $\\pi =\\pi (x,t)$ denote, respectively, the temperature and the hydrostatic pressure.", "While $u_{0}$ and $\\theta _{0}$ are the prescribed initial data for the velocity and temperature with properties $\\nabla \\cdot u_{0}=0$ .", "Moreover, the term $\\theta e_{3}$ represents buoyancy force on fluid motion.", "We would like to point out that the system (REF ) at $\\theta =0$ reduces to the incompressible Navier-Stokes equations, which has been greatly analyzed.", "From the viewpoint of the model, therefore, Navier-Stokes flow is viewed as the flow of a simplified Boussinesq equation.", "Besides their physical applications, the Boussinesq equations are also mathematically significant.", "Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see, for example, [6], [7], [9], [10], [13], [14], [15], [18], [25], [26], [28], [29], [30], [31] and references therein).", "On the other hand, it is desirable to show the regularity of the weak solutions if some partial components of the velocity satisfy certain growth conditions.", "For the 3D Navier-Stokes equations, there are many results to show such regularity of weak solutions in terms of partial components of the velocity $u$ (see, for example, [4], [5], [8], [11], [12], [16], [19], [23], [34] and the references cited therein).", "It is obvious that, for the assumptions of all regularity criteria, it need that every components of the velocity field must satisfies the same assumptions, and it don't make any difference between the components of the velocity field.", "As pointed out by Neustupa and Penel [21], it is interesting to know how to effect the regularity of the velocity field by the regularity of only one component of the velocity field.", "In particular, Zhou [35] showed that the solution is regular if one component of the velocity, for example, $u_{3}$ satisfies $u_{3}\\in L^{p}(0,T;L^{q}(\\mathbb {R}^{3}))\\text{ \\ \\ with \\ }\\frac{2}{p}+\\frac{3}{q}\\le \\frac{1}{2},\\text{ \\ \\ }6<q\\le \\infty .", "$ Condition REF ) can be replaced respectively by the one $u_{3}\\in L^{p}(0,T;L^{q}(\\mathbb {R}^{3}))\\text{ \\ \\ with \\ }\\frac{2}{p}+\\frac{3}{q}\\le \\frac{5}{8},\\text{ \\ \\ }\\frac{24}{5}<q\\le \\infty ,$ (see Kukavica and Ziane [20]).", "Later, Cao and Titi [5] showed the regularity of weak solution to the Navier-Stokes equations under the assumption $u_{3}\\in L^{p}(0,T;L^{q}(\\mathbb {R}^{3}))\\text{ \\ \\ with \\ }\\frac{2}{p}+\\frac{3}{q}=\\frac{2}{3}+\\frac{2}{3q},\\text{ \\ \\ }q>\\frac{7}{2}.", "$ Motivated by the above work, Zhou and Pokorný [36] showed the following regularity condition $u_{3}\\in L^{p}(0,T;L^{q}(\\mathbb {R}^{3}))\\text{ \\ \\ with \\ }\\frac{2}{p}+\\frac{3}{q}=\\frac{3}{4}+\\frac{1}{2q},\\text{ \\ \\ }q>\\frac{10}{3}, $ while the limiting case $u_{3}\\in L^{\\infty }(0,T;L^{\\frac{10}{3}}(\\mathbb {R}^{3}))$ was covered in [19].", "For many other result works, especially the regularity criteria involving only one velocity component, or its gradient, with no intention to be complete, one can consult [32], [33] and references therein.", "However, the conditions (REF )-(REF ) are quite strong comparing with the condition of Serrin's regularity criterion : $u\\in L^{p}(0,T;L^{q}(\\mathbb {R}^{3}))\\text{ \\ \\ with \\ }\\frac{2}{p}+\\frac{3}{q}\\le 1,\\text{ \\ \\ }3<q\\le \\infty , $ and do not imply the invariance under the above scaling transformation.", "Therefore, it is of interest in showing regularity by imposing Serrin's condition (REF ) with respect to the one component of the velocity field.", "Similar to the research of the 3D Navier-Stokes equations, authors are interested in the regularity criterion of (REF ) by reducing to some the components of $u$ .", "There are many other or similar results on the hydrodynamical systems modeling the flow of nematic liquid crystal material, Boussinesq equations and MHD equations (see e.g.", "[3], [24] and the reference therein).", "Motivated by the reference mentioned above, the purpose of the present paper is to give a further observation on the global regularity of the solution for system (REF ) and to extend the regularity of weak solutions to the Boussinesq equations (REF ) in terms of one velocity component and the gradient of the temperature." ], [ "Notations and main result.", "Before stating our result, we introduce some notations and function spaces.", "These spaces can be found in many literatures and papers.", "For the functional space, $L^{p}(\\mathbb {R}^{3})$ denotes the usual Lebesgue space of real-valued functions with norm $\\left\\Vert \\cdot \\right\\Vert _{L^{p}}:$ $\\left\\Vert f\\right\\Vert _{L^{p}}=\\left\\lbrace \\begin{array}{c}\\left( \\int _{\\mathbb {R}^{3}}\\left|f(x)\\right|^{p}dx\\right) ^{\\frac{1}{p}},\\text{ \\ \\ for \\ \\ }1\\le p<\\infty , \\\\\\underset{x\\in \\mathbb {R}^{3}}{ess\\sup }\\left|f(x)\\right|,\\text{ \\ \\ for \\ \\ }p=\\infty .\\end{array}\\right.$ On the other hand, the usual Sobolev space of order $m$ is defined by $H^{m}(\\mathbb {R}^{3})=\\left\\lbrace u\\in L^{2}(\\mathbb {R}^{3}):\\nabla ^{m}u\\in L^{2}(\\mathbb {R}^{3})\\right\\rbrace $ with the norm $\\left\\Vert u\\right\\Vert _{H^{m}}=\\left( \\left\\Vert u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla ^{m}u\\right\\Vert _{L^{2}}^{2}\\right) ^{\\frac{1}{2}}.$ To prove Theorem REF , we use the theory of Lorentz spaces and introduce the following notations.", "We define the non-increasing rearrangement of $f$ , $f^{\\ast }(\\lambda )=\\inf \\left\\lbrace t>0:m_{f}(t)\\le \\lambda \\right\\rbrace ,\\text{ \\ \\ for \\ }\\lambda >0,$ where $f$ is a measurable function on $\\mathbb {R}^{3}$ and $m_{f}(t)$ is the distribution function of $f$ which is defined by the Lebesgue measure of the set $\\left\\lbrace x\\in \\mathbb {R}^{3}:\\left|f(x)\\right|>t\\right\\rbrace $ .", "The Lorentz space $L^{p,q}((\\mathbb {R}^{3})$ is defined by $L^{p,q}=\\left\\lbrace f:\\mathbb {R}^{3}\\rightarrow \\mathbb {R}\\text{ measurable suchthat }\\left\\Vert f\\right\\Vert _{L^{p,q}}<\\infty \\right\\rbrace \\text{ \\ with \\ }1\\le p<\\infty $ equipped with the quasi-norm $\\left\\Vert f\\right\\Vert _{L^{p,q}}=\\left( \\frac{q}{p}\\int _{0}^{\\infty }(t^{\\frac{1}{p}}f^{\\ast }(t))^{q}\\frac{dt}{t}\\right) ^{\\frac{1}{q}}\\text{, \\ \\ if \\ }1<q<\\infty .$ Moreover, we define $f^{\\ast \\ast }$ by $f^{\\ast \\ast }(\\lambda )=\\frac{1}{\\lambda }\\int _{0}^{\\lambda }f^{\\ast }(\\lambda ^{\\prime })d\\lambda ^{\\prime },$ and Lorentz spaces $L^{p,\\infty }(\\mathbb {R}^{3})$ by $L^{p,\\infty }(\\mathbb {R}^{3})=\\left\\lbrace f\\in \\mathcal {S}^{\\prime }(\\mathbb {R}^{3}):\\left\\Vert f\\right\\Vert _{L^{p,\\infty }}<\\infty \\right\\rbrace ,$ where $\\left\\Vert f\\right\\Vert _{L^{p,\\infty }}=\\underset{\\lambda \\ge 0}{\\sup }(\\lambda ^{\\frac{1}{p}}f^{\\ast \\ast }(\\lambda )),$ for $1\\le p\\le \\infty $ .", "For details, we refer to [2] and [27].", "From the definition of the Lorentz space, we can obtain the following continuous embeddings : $L^{p}(\\mathbb {R}^{3})=L^{p,p}(\\mathbb {R}^{3})\\hookrightarrow L^{p,q}(\\mathbb {R}^{3})\\hookrightarrow L^{p,\\infty }(\\mathbb {R}^{3}),\\text{ \\ \\ }1\\le p\\le q<\\infty .$ In order to prove Theorem REF , we recall the Hölder inequality in the Lorentz spaces (see, e.g., O'Neil [22]).", "Lemma 2.1 Let $f\\in L^{p_{2},q_{2}}(\\mathbb {R}^{3})$ and $g\\in L^{p_{3},q_{3}}(\\mathbb {R}^{3})$ with $1\\le p_{2},p_{3}\\le \\infty $ , $1\\le q_{2},q_{3}\\le \\infty $ .", "Then $fg\\in L^{p_{1},q_{1}}(\\mathbb {R}^{3})$ with $\\frac{1}{p_{1}}=\\frac{1}{p_{2}}+\\frac{1}{p_{3}},\\frac{1}{q_{1}}=\\frac{1}{q_{2}}+\\frac{1}{q_{3}}$ and the Hölder inequality of Lorentz spaces $\\left\\Vert fg\\right\\Vert _{L^{p_{1},q_{1}}}\\le C\\left\\Vert f\\right\\Vert _{L^{p_{2},q_{2}}}\\left\\Vert g\\right\\Vert _{L^{p_{3},q_{3}}},$ holds true for a positive constant $C$ .", "The following result plays an important role in the proof of our theorem, the so-called Gagliardo-Nirenberg inequality in Lorentz spaces, its proof can be founded in [17].", "Lemma 2.2 Let $f\\in L^{p,q}(\\mathbb {R}^{3})$ with $1\\le p,q,p_{4},q_{4},p_{5},q_{5}\\le \\infty $ .", "Then the Gagliardo-Nirenberg inequality of Lorentz spaces $\\left\\Vert f\\right\\Vert _{L^{p,q}}\\le C\\left\\Vert f\\right\\Vert _{L^{p_{4},q_{4}}}^{\\theta }\\left\\Vert f\\right\\Vert _{L^{p_{5},q_{5}}}^{1-\\theta }$ holds for a positive constant $C$ and $\\frac{1}{p}=\\frac{\\theta }{p_{4}}+\\frac{1-\\theta }{p_{5}},\\text{ \\ \\ }\\frac{1}{q}=\\frac{\\theta }{q_{4}}+\\frac{1-\\theta }{q_{5}},\\text{ \\ }\\theta \\in (0,1).$ Now we give the definition of weak solution.", "Definition 2.3 Let $T>0$ , $(u_{0},\\theta _{0})\\in L^{2}(\\mathbb {R}^{3})$ with $\\nabla \\cdot u_{0}=0$ in the sense of distributions.", "A measurable function $(u(x,t),\\theta (x,t))$ is called a weak solution to the Boussinesq equations (REF ) on $[0,T]$ if the following conditions hold: $(u(x,t),\\theta (x,t))\\in L^{\\infty }(0,T;L^{2}(\\mathbb {R}^{3}))\\cap L^{2}(0,T;H^{1}(\\mathbb {R}^{3}));$ system (REF ) is satisfied in the sense of distributions; the energy inequality, that is, $\\left\\Vert u(\\cdot ,t)\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\theta (\\cdot ,t)\\right\\Vert _{L^{2}}^{2}+2\\int _{0}^{t}\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau +2\\int _{0}^{t}\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\le \\left\\Vert u_{0}\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b_{0}\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\theta _{0}\\right\\Vert _{L^{2}}^{2}.$ By a strong solution, we mean that a weak solution u of the Navier-Stokes equations (REF ) satisfies $(u(x,t),\\theta (x,t))\\in L^{\\infty }(0,T;H^{1}(\\mathbb {R}^{3}))\\cap L^{2}(0,T;H^{2}(\\mathbb {R}^{3})).$ It is well known that the strong solution is regular and unique.", "Our main result is stated as following : Theorem 2.4 Let $(u_{0},\\theta _{0})\\in L^{2}(\\mathbb {R}^{3})$ with $\\nabla \\cdot u_{0}=0$ in the sense of distributions.", "Assume that $(u,\\theta )$ is a weak solution to system (REF ).", "If $u_{3}$ and $\\nabla \\theta $ satisfy the following conditions $\\left\\lbrace \\begin{array}{c}u_{3}\\in L^{\\frac{30\\alpha }{7\\alpha -45}}(0,T;L^{\\alpha ,\\infty }(\\mathbb {R}^{3})),\\text{ \\ with \\ }\\frac{45}{7}\\le \\alpha \\le \\infty , \\\\\\nabla \\theta \\in L^{\\frac{2\\beta }{2\\beta -3}}(0,T;L^{\\beta ,\\infty }(\\mathbb {R}^{3})),\\text{ \\ \\ with \\ \\ }\\frac{3}{2}<\\beta \\le \\infty ,\\end{array}\\right.", "$ then the solution $\\left( u,\\theta \\right) $ is regular on $(0,T]$ .", "Remark 2.1 If $\\theta =0$ , it is clear that theorem REF improves the earlier results of [19], [36] for 3D Navier-Stokes equations and extend the regularity criterion (REF ) from Lebesgue space $L^{\\alpha }$ to Lorentz space $L^{\\alpha ,\\infty }$ .", "Remark 2.2 This result proves a new regularity criterion for weak solutions to the Cauchy problem of the 3D Boussinesq equations via one component of the velocity field and the gradient of the tempearture in the framework of the Lorentz spaces.", "This result reveals that the one component of the velocity field plays a dominant role in regularity theory of the Boussinesq equations." ], [ "Proof of the main result.", "In this section, under the assumptions of the Theorem REF , we prove our main result.", "Before proving our result, we recall the following muliplicative Sobolev imbedding inequality in the whole space $\\mathbb {R}^{3} $ (see, for example [5]) : $\\left\\Vert f\\right\\Vert _{L^{6}}\\le C\\left\\Vert \\nabla _{h}f\\right\\Vert _{L^{2}}^{\\frac{2}{3}}\\left\\Vert \\partial _{3}f\\right\\Vert _{L^{2}}^{\\frac{1}{3}}, $ where $\\nabla _{h}=(\\partial _{x_{1}},\\partial _{x_{2}})$ is the horizontal gradient operator.", "We are now give the proof of our main theorem.", "Proof: To prove our result, it suffices to show that for any fixed $T>T^{\\ast }$ , there holds $\\underset{0\\le t\\le T^{\\ast }}{\\sup }(\\left\\Vert \\nabla u(t)\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (t)\\right\\Vert _{L^{2}}^{2})\\le C_{T},$ where $T^{\\ast }$ , which denotes the maximal existence time of a strong solution and $C_{T}$ is an absolute constant which only depends on $T,u_{0}$ and $\\theta _{0}$ .", "The method of our proof is based on two major parts.", "The first one establishes the bounds of $(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta (t)\\right\\Vert _{L^{2}}^{2})$ , while the second gives the bounds of the $H^{1}-$ norm of velocity $u$ and temperature $\\theta $ in terms of the results of part one.", "Taking the inner product of (REF )$_{1}$ with $-\\Delta _{h}u$ , (REF )$_{2}$ with $-\\Delta _{h}\\theta $ in $L^{2}(\\mathbb {R}^{3})$ , respectively, then adding the three resulting equations together, we obtain after integrating by parts that $&&\\frac{1}{2}\\frac{d}{dt}(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2})+\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2}\\\\&=&\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )u\\cdot \\Delta _{h}udx+\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta _{h}\\theta -\\int _{\\mathbb {R}^{3}}(\\theta e_{3})\\cdot \\Delta _{h}udx \\\\&=&I_{1}+I_{2}+I_{3}.", "$ where $\\Delta _{h}=\\partial _{x_{1}}^{2}+\\partial _{x_{2}}^{2}$ is the horizontal Laplacian.", "For the notational simplicity, we set $\\mathcal {L}^{2}(t) &=&\\underset{\\tau \\in [\\Gamma ,t]}{\\sup }(\\left\\Vert \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2})+\\int _{\\Gamma }^{t}(\\left\\Vert \\nabla \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2})d\\tau , \\\\\\mathcal {J}^{2}(t) &=&\\underset{\\tau \\in [\\Gamma ,t]}{\\sup }(\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{2}}^{2})+\\int _{\\Gamma }^{t}(\\left\\Vert \\Delta u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\Delta \\theta \\right\\Vert _{L^{2}}^{2})d\\tau ,$ for $t\\in [\\Gamma ,T^{\\ast })$ .", "In view of (REF ), we choose $\\epsilon ,\\eta >0$ to be precisely determined subsequently and then select $\\Gamma <T^{\\ast }$ sufficiently close to $T^{\\ast }$ such that for all $\\Gamma \\le t<T^{\\ast }$ , $\\int _{\\Gamma }^{t}(\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{2}}^{2})d\\tau \\le \\epsilon \\ll 1\\text{,} $ and $\\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{\\beta }}^{\\frac{2\\beta }{2\\beta -3}}d\\tau \\le \\eta \\ll 1.", "$ Integrating by parts and using the divergence-free condition, it is clear that (see e.g.", "[34]) $I_{1}=\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )u\\cdot \\Delta _{h}udx\\le \\int _{\\mathbb {R}^{3}}\\left|u_{3}\\right|\\left|\\nabla u\\right|\\left|\\nabla \\nabla _{h}u\\right|dx $ By appealing to Lemma REF , (REF ), and the Young inequality, it follows that $I_{1} &\\le &C\\left\\Vert u_{3}\\right\\Vert _{L^{\\alpha ,\\infty }}\\left\\Vert \\nabla u\\right\\Vert _{L^{\\frac{2\\alpha }{\\alpha -2},2}}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}} \\\\&\\le &C\\left\\Vert u_{3}\\right\\Vert _{L^{\\alpha ,\\infty }}\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{1-\\frac{3}{\\alpha }}\\left\\Vert \\nabla u\\right\\Vert _{L^{6}}^{\\frac{3}{\\alpha }}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}} \\\\&\\le &C\\left\\Vert u_{3}\\right\\Vert _{L^{\\alpha ,\\infty }}\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{1-\\frac{3}{\\alpha }}\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{\\frac{1}{\\alpha }}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}}^{1+\\frac{2}{\\alpha }} \\\\&\\le &C\\left\\Vert u_{3}\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{2\\alpha }{\\alpha -2}}\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2-\\frac{2}{\\alpha -2}}\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{\\frac{2}{\\alpha -2}}+\\frac{1}{4}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}}^{2},$ where we have used the following Gagliardo-Nirenberg inequality in Lorentz spaces : $\\left\\Vert \\phi \\right\\Vert _{L^{\\frac{2s}{s-2},2}}\\le C\\left\\Vert \\phi \\right\\Vert _{L^{2}}^{1-\\frac{3}{s}}\\left\\Vert \\nabla \\phi \\right\\Vert _{L^{6}}^{\\frac{3}{s}}\\text{ \\ \\ with \\ \\ }3<s\\le \\infty .$ To estimate the term $I_{2}$ of (REF ), first observe that by applying integration by parts and $\\nabla \\cdot u=0$ , we derive $I_{2} &=&\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta _{h}\\theta dx=\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{i}\\theta _{j}\\partial _{kk}^{2}\\theta _{j}dx \\\\&=&-\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}\\partial _{k}u_{i}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx-\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{k}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx \\\\&=&-\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}\\partial _{k}u_{i}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx,$ where we have used $-\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{k}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx&=&\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}\\partial _{i}u_{i}(\\partial _{k}\\theta _{j})^{2}dx+\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{i}\\partial _{k}\\theta _{j}\\partial _{k}\\theta _{j}dx \\\\&=&\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{i}\\partial _{k}\\theta _{j}\\partial _{k}\\theta _{j}dx,$ so that $\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}u_{i}\\partial _{k}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx=0.$ It follows from Hölder's inequality, (REF ) and Young's inequality that $I_{2} &=&-\\sum \\limits _{i,j=1}^{3}\\sum \\limits _{k=1}^{2}\\int _{\\mathbb {R}^{3}}\\partial _{k}u_{i}\\partial _{i}\\theta _{j}\\partial _{k}\\theta _{j}dx\\le \\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}u\\right|\\left|\\nabla \\theta \\right|\\left|\\nabla _{h}\\theta \\right|dx \\\\&\\le &\\frac{1}{2}\\int _{\\mathbb {R}^{3}}\\left( \\left|\\nabla _{h}u\\right|^{2}+\\left|\\nabla _{h}\\theta \\right|^{2}\\right)\\left|\\nabla \\theta \\right|dx \\\\&\\le &C\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{\\beta ,\\infty }}(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{\\frac{2\\beta }{\\beta -2},2}}\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{\\frac{2\\beta }{\\beta -2},2}}\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}) \\\\&\\le &C\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{\\beta ,\\infty }}(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2-\\frac{3}{\\beta }}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}}^{\\frac{3}{s}}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2-\\frac{3}{\\beta }}\\left\\Vert \\nabla \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{\\frac{3}{s}}) \\\\&\\le &C\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{\\beta ,\\infty }}^{\\frac{2\\beta }{2\\beta -3}}(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2})+\\frac{1}{2}\\left\\Vert \\nabla \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2}+\\frac{1}{4}\\left\\Vert \\nabla \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}.$ Finally, we we want to estimate $I_{3}$ .", "It follows from integration by parts and Cauchy inequality that $I_{3} &=&-\\int _{\\mathbb {R}^{3}}(\\theta e_{3})\\cdot \\Delta _{h}udx=\\int _{\\mathbb {R}^{3}}\\nabla _{h}(\\theta e_{3})\\cdot \\nabla _{h}udx \\\\&\\le &2\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2}.$ Inserting all the estimates into (REF ), Gronwall's type argument using $1\\le \\underset{\\lambda \\in [\\Gamma ,\\tau ]}{\\sup }\\exp \\left(c\\int _{\\lambda }^{\\tau }\\left\\Vert \\nabla \\theta (\\varphi )\\right\\Vert _{L^{\\beta ,\\infty }}^{\\frac{2\\beta }{2\\beta -3}}d\\varphi \\right) \\lesssim \\exp \\left( c\\int _{0}^{T^{\\ast }}\\left\\Vert \\nabla \\theta (\\varphi )\\right\\Vert _{L^{\\beta ,\\infty }}^{\\frac{2\\beta }{2\\beta -3}}d\\varphi \\right) \\lesssim 1,$ due to (REF ) leads to, for every $\\tau \\in [\\Gamma ,t]$ $\\mathcal {L}^{2}(t) &\\le &C+C\\int _{\\Gamma }^{t}\\left\\Vert u_{3}\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{2\\alpha }{\\alpha -2}}\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2-\\frac{2}{\\alpha -2}}\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{\\frac{2}{\\alpha -2}}d\\tau +C\\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{\\beta ,\\infty }}^{\\frac{2\\beta }{2\\beta -3}}(\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}}^{2})d\\tau \\\\&=&C+\\mathcal {I}_{1}(t)+\\mathcal {I}_{2}(t).", "$ Next, we analyze the right-hand side of (REF ) one by one.", "First, due to (REF ) and the definition of $\\mathcal {J}^{2}$ , we have $\\mathcal {I}_{1}(t) &\\le &C\\left( \\underset{\\tau \\in [\\Gamma ,t]}{\\sup }\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{\\frac{3}{2}-\\frac{2}{\\alpha -2}}\\right) \\int _{\\Gamma }^{t}\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{2\\alpha }{\\alpha -2}}\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{\\frac{1}{2}}\\left\\Vert \\Delta u(\\tau )\\right\\Vert _{L^{2}}^{\\frac{2}{\\alpha -2}}d\\tau \\\\&\\le &C\\mathcal {J}^{\\frac{3}{2}-\\frac{2}{\\alpha -2}}(t)\\left( \\int _{\\Gamma }^{t}\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{8\\alpha }{3\\alpha -10}}d\\tau \\right) ^{\\frac{3}{4}-\\frac{1}{\\alpha -2}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{4}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\Delta u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{\\alpha -2}}\\\\&\\le &C\\mathcal {J}^{\\frac{3}{2}-\\frac{2}{\\alpha -2}}(t)\\left( \\int _{\\Gamma }^{t}\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{8\\alpha }{3\\alpha -10}}d\\tau \\right) ^{\\frac{3}{4}-\\frac{1}{\\alpha -2}}\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{2}{\\alpha -2}}(t) \\\\&=&C\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{3}{2}}(t)\\left( \\int _{\\Gamma }^{t}\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{8\\alpha }{3\\alpha -10}}d\\tau \\right) ^{\\frac{3}{4}-\\frac{1}{\\alpha -2}}.$ Finally, we deal with the term $\\mathcal {I}_{2}(t)$ .", "Applying Hölder and Young inequalities, one has $\\mathcal {I}_{2}(t) &\\le &C\\underset{\\tau \\in [\\Gamma ,t]}{\\sup }(\\left\\Vert \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla _{h}\\theta (\\tau )\\right\\Vert _{L^{2}}^{2})\\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{\\beta ,\\infty }}^{\\frac{2\\beta }{2\\beta -3}}d\\tau \\\\&\\le &C\\eta \\mathcal {L}^{2}(t).$ Hence, choosing $\\eta $ small enough such that $C\\eta <1$ and inserting the above estimates of $\\mathcal {I}_{1}(t)$ and $\\mathcal {I}_{2}(t)$ into (REF ), we derive that for all $\\Gamma \\le t<T^{\\ast }:$ $\\mathcal {L}^{2}(t) &\\le &C+C\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{3}{2}}(t)\\left( \\int _{\\Gamma }^{t}\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha ,\\infty }}^{\\frac{8\\alpha }{3\\alpha -10}}d\\tau \\right) ^{\\frac{3\\alpha -10}{4(\\alpha -2)}} \\\\&\\le &C+C\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{3}{2}}(t)\\left(\\int _{\\Gamma }^{t}1+\\left\\Vert u_{3}(\\tau )\\right\\Vert _{L^{\\alpha }}^{\\frac{30\\alpha }{7\\alpha -45}}d\\tau \\right) ^{\\frac{3\\alpha -10}{4(\\alpha -2)}},$ which leads to $\\mathcal {L}^{2}(t)\\le C+C\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{3}{2}}(t).", "$ Now, we will establish the bounds of $H^{1}-$ norm of the velocity magnetic field and micro-rotationel velocity.", "In order to do it, taking the inner product of (REF )$_{1}$ with $-\\Delta u$ , (REF )$_{2}$ with $-\\Delta b$ and (REF )$_{3}$ with $-\\Delta \\theta $ in $L^{2}(\\mathbb {R}^{3})$ , respectively.", "Then, integration by parts gives the following identity: $&&\\frac{1}{2}\\frac{d}{dt}(\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{2}}^{2})+\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\Delta \\theta \\right\\Vert _{L^{2}}^{2} \\\\&=&\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )u\\cdot \\Delta udx+\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta \\theta dx-\\int _{\\mathbb {R}^{3}}(\\theta e_{3})\\cdot \\Delta udx.$ Integrating by parts and using the divergence-free condition, one can easily deduce that (see e.g.", "[36]) $\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )u\\cdot \\Delta udx\\le C\\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}u\\right|\\left|\\nabla u\\right|^{2}dx.$ We treat now the $\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta \\theta dx$ -term.", "By integration by parts, we have $\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta \\theta dx&=&-\\sum \\limits _{i=1}^{2}\\sum \\limits _{j,k=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{i}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{i}\\theta _{j}dx-\\sum \\limits _{j,k=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{3}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{3}\\theta _{j}dx\\\\&=&-\\sum \\limits _{i=1}^{2}\\sum \\limits _{j,k=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{i}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{i}\\theta _{j}dx-\\sum \\limits _{k=1}^{2}\\sum \\limits _{j=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{3}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{3}\\theta _{j}dx \\\\&&-\\sum \\limits _{j=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{3}u_{3}\\cdot \\partial _{3}\\theta _{j}\\cdot \\partial _{3}\\theta _{j}dx \\\\&=&-\\sum \\limits _{i=1}^{2}\\sum \\limits _{j,k=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{i}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{i}\\theta _{j}dx-\\sum \\limits _{k=1}^{2}\\sum \\limits _{j=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{3}u_{k}\\cdot \\partial _{k}\\theta _{j}\\cdot \\partial _{3}\\theta _{j}dx \\\\&&+\\sum \\limits _{i=1}^{2}\\sum \\limits _{j=1}^{3}\\int _{\\mathbb {R}^{3}}\\partial _{i}u_{i}\\cdot \\partial _{3}\\theta _{j}\\cdot \\partial _{3}\\theta _{j}dx\\\\&=&R_{1}+R_{2}+R_{3}.", "$ Therefore, we have $\\left|R_{1}+R_{3}\\right|\\le \\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}u\\right|\\left|\\nabla \\theta \\right|^{2}dx,$ and $\\left|R_{2}\\right|\\le \\int _{\\mathbb {R}^{3}}\\left|\\nabla u\\right|\\left|\\nabla \\theta \\right|\\left|\\nabla _{h}\\theta \\right|dx\\le \\frac{1}{2}\\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}\\theta \\right|(\\left|\\nabla u\\right|^{2}+\\left|\\nabla \\theta \\right|^{2})dx,$ where the last inequality is obtained by using Cauchy inequality.", "Putting all the inequalities above into (REF ) yields $\\int _{\\mathbb {R}^{3}}(u\\cdot \\nabla )\\theta \\cdot \\Delta \\theta dx\\le \\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}u\\right|\\left|\\nabla \\theta \\right|^{2}dx+\\frac{1}{2}\\int _{\\mathbb {R}^{3}}\\left|\\nabla _{h}\\theta \\right|(\\left|\\nabla u\\right|^{2}+\\left|\\nabla \\theta \\right|^{2})dx.$ Finally, we deal with the term $-\\int _{\\mathbb {R}^{3}}(\\theta e_{3})\\cdot \\Delta udx$ .", "By integration by parts and Cauchy inequality, we have $-\\int _{\\mathbb {R}^{3}}(\\theta e_{3})\\cdot \\Delta udx\\le \\frac{1}{2}(\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{2}}^{2}).$ Combining the above estimates, by Hölder's inequality, Nirenberg-Gagliardo's interpolation inequality and (REF ), we obtain $&&\\frac{1}{2}\\frac{d}{dt}(\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{2}}^{2})+\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\Delta \\theta \\right\\Vert _{L^{2}}^{2} \\\\&\\le &C\\int _{\\mathbb {R}^{3}}(1+\\left|\\nabla _{h}u\\right|+\\left|\\nabla _{h}\\theta \\right|)(\\left|\\nabla u\\right|^{2}+\\left|\\nabla \\theta \\right|^{2})dx \\\\&\\le &C(1+\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}})(\\left\\Vert \\nabla u\\right\\Vert _{L^{4}}^{2}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{4}}^{2}) \\\\&\\le &C(1+\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}})(\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{\\frac{1}{2}}\\left\\Vert \\nabla u\\right\\Vert _{L^{6}}^{\\frac{3}{2}}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{2}}^{\\frac{1}{2}}\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{6}}^{\\frac{3}{2}}) \\\\&\\le &C(1+\\left\\Vert \\nabla _{h}u\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta \\right\\Vert _{L^{2}})(\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{\\frac{1}{2}}\\left\\Vert \\nabla _{h}\\nabla u\\right\\Vert _{L^{2}}\\left\\Vert \\Delta u\\right\\Vert _{L^{2}}^{\\frac{1}{2}}+\\left\\Vert \\nabla \\theta \\right\\Vert _{L^{2}}^{\\frac{1}{2}}\\left\\Vert \\nabla _{h}\\nabla \\theta \\right\\Vert _{L^{2}}\\left\\Vert \\Delta \\theta \\right\\Vert _{L^{2}}^{\\frac{1}{2}}).$ Integrating this last inequality in time, we deduce that for all $\\tau \\in [\\Gamma ,t]$ $\\mathcal {J}^{2}(t) &\\le &1+\\left\\Vert \\nabla u(\\Gamma )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\Gamma )\\right\\Vert _{L^{2}}^{2}+C\\underset{\\tau \\in [\\Gamma ,t]}{\\sup }(\\left\\Vert \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta (\\tau )\\right\\Vert _{L^{2}}) \\\\&&\\times \\left( \\int _{\\Gamma }^{t}\\left\\Vert \\nabla u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{4}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{2}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\Delta u(\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{4}} \\\\&&+C\\underset{\\tau \\in [\\Gamma ,t]}{\\sup }(\\left\\Vert \\nabla _{h}u(\\tau )\\right\\Vert _{L^{2}}+\\left\\Vert \\nabla _{h}\\theta (\\tau )\\right\\Vert _{L^{2}}) \\\\&&\\times \\left( \\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\theta (\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{4}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\nabla \\nabla _{h}\\theta (\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{2}}\\left( \\int _{\\Gamma }^{t}\\left\\Vert \\Delta \\theta (\\tau )\\right\\Vert _{L^{2}}^{2}d\\tau \\right) ^{\\frac{1}{4}}\\\\&\\le &1+\\left\\Vert \\nabla u(\\Gamma )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\Gamma )\\right\\Vert _{L^{2}}^{2}+2C\\mathcal {L}(t)\\epsilon ^{\\frac{1}{4}}\\mathcal {L}(t)\\mathcal {J}^{\\frac{1}{2}}(t) \\\\&=&1+\\left\\Vert \\nabla u(\\Gamma )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\Gamma )\\right\\Vert _{L^{2}}^{2}+C\\epsilon ^{\\frac{1}{4}}\\mathcal {L}^{2}(t)\\mathcal {J}^{\\frac{1}{2}}(t).", "$ Inserting (REF ) into (REF ) and taking $\\epsilon $ small enough, then it is easy to see that for all $\\Gamma \\le t<T^{\\ast }$ , there holds $\\mathcal {J}^{2}(t)\\le 1+\\left\\Vert \\nabla u(\\Gamma )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (\\Gamma )\\right\\Vert _{L^{2}}^{2}+C\\epsilon ^{\\frac{1}{4}}\\mathcal {J}^{\\frac{1}{2}}(t)+C\\epsilon ^{\\frac{1}{2}}\\mathcal {J}^{2}(t)<\\infty ,$ which proves $\\underset{\\Gamma \\le t<T^{\\ast }}{\\sup }(\\left\\Vert \\nabla u(t)\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla \\theta (t)\\right\\Vert _{L^{2}}^{2})<+\\infty .$ This implies us that $(u,\\theta )\\in L^{\\infty }(0,T;H^{1}(\\mathbb {R}^{3}))$ .", "Thus, according to the regularity results in [2], $(u,\\theta )$ is smooth on $[0,T]$ .", "Then we complete the proof of Theorem REF .", "$\\Box $" ], [ "Conclusion", "It should be noted that the condition (REF ) is somewhat stronger than in [14], since it is worthy to emphasize that there are no assumptions on the two components velocity field $(u_{1},u_{2})$ .", "In other word, our result demonstrates that the two components velocity field $(u_{1},u_{2})$ plays less dominant role than the one compoent velocity field does in the regularity theory of solutions to the Boussinesq equations.", "In a certain sense, our result is consistent with the numerical simulations of Alzmann et al.", "in [1]." ], [ "Aknoledgements", "The authors are indebted to the referees for their careful reading and valuable suggestions which improved the presentation of our paper.", "This work was done while the second author was visiting the Catania University in Italy.", "He would like to thank the hospitality and support of the University, where this work was completed.", "This research is partially supported by P.R.I.N.", "2019.", "The third author wish to thank the support of \"RUDN University Program 5-100\"." ] ]

[ [ "Experimental study of breathers and rogue waves generated by random\n waves over non-uniform bathymetry" ], [ "Abstract Experimental results describing random, uni-directional, long crested, water waves over non-uniform bathymetry confirm the formation of stable coherent wave packages traveling with almost uniform group velocity.", "The waves are generated with JONSWAP spectrum for various steepness, height and constant period.", "A set of statistical procedures were applied to the experimental data, including the space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights.", "Stable wave packages formed out of the random field and traveling over shoals, valleys and slopes were compared with exact solutions of the NLS equation resulting in good matches and demonstrating that these packages are very similar to deep water breathers solutions, surviving over the non-uniform bathymetry.", "We also present events of formation of rogue waves over those regions where the BFI, kurtosis and skewness coefficients have maximal values." ], [ "Introduction", "It is very important to predict with greatest accuracy ocean waves for the safety of ships and offshore structures, especially when operating in rough sea conditions where extreme events could arise.", "Ocean extreme waves, also known as rogue waves (RW), occur without apparent warning and have disastrous impact, mainly because of their large wave heights [1], [2].", "These highly destructive phenomena have been observed frequently enough to justify advanced studies.", "Possible candidates to explain the formation of rogue waves in the ocean are presently under intense discussion [3], [1], [4], [5].", "This topic attracted recently a great deal of scientific interest not only because of the accurate modeling and prediction of these extremes and similar structures [6], [7], [8], [9], but also because of the interdisciplinary nature of the modulation instability (MI) present in weakly nonlinear waves [10], [11], [12], [13].", "Explanations solely based on linear wave dynamical theories (constructive interference of multiple small amplitude waves) cannot grasp the nonlinear coupling between modes, phenomenon which becomes important when the amplitude of the waves increases.", "One of the most cited nonlinear approaches for surface gravity wave propagation is the modulation instability (MI) [14].", "Such a phenomenon can be described by the evolution of an unstable wave packet which absorbs energy from neighbor waves and increases its amplitude, reaches a maximum and then transfers its energy back to the other waves [15].", "The most common mathematical model for such unstable modes describing the nonlinear dynamics of gravity waves is the nonlinear Schrödinger equation (NLS) [16], [18], [19], [10] or extended versions of it [20], [21], [31].", "Exact solutions of the NLS equation provide feasible models that were successfully used to provide deterministic numerical and laboratory prototypes both to reveal novel insights of MI [32] and to describe rogue waves.", "The reason for the efficiency of the NLS model is that through its balance between nonlinearity and linear dispersion it can describe well the occurrence of Benjamin-Feir instability, and the associate nonlinear wave dynamics [34], [35], [36].", "Experimental studies confirmed validity of NLS for deep water waves [37], [38], [40].", "One other advantage of using the NLS is its integrability [41], and analytic form for solutions, especially useful when compared to experimental results.", "In NLS models the instability corresponds to various breather solution of this equation [19].", "The NLS equation (as opposed to the solitons in other integrable nonlinear equations like Korteweg-de Vries KdV) is characterized by a much richer family of coherent structures, namely breather solutions [43], [44], [16], [45], [3], [15], [31].", "Even if the breathers change their shape during their evolution and hence are not traveling solitons, they maintain their identity against perturbations and collisions.", "Breathers are exact solutions of the nonlinear Schrödinger equation (NLS) [19], [10] and describe the dynamics of modulation unstable Stokes waves [46] in deep water [47], [48].", "The MI starts from an infinitesimal perturbation that initially growths exponentially, and after reaching a highest amplitude decays back in the background wave field [18].", "Such full exact solutions for the NLS equation are given by rational expressions in hyperbolic and trigonometric functions of space and time and are known as Akhmediev breathers (AB) [49], [50] providing space-periodic models to study the Benjamin-Feir instability initiated from a periodic modulation of Stokes waves [51], [52].", "A limiting situation is the case of an infinite modulation period and corresponding significant double localization.", "Such solution is described through a rational function called the Peregrine breather (PB) [49], [50], [51], [52], [53].", "The growth rate of the KM breather is algebraic [18].", "Both AB and PB have been considered as possible ocean rogue waves model [54], [55], and their features have been investigated experimentally and numerically [56], [58], [59], [61].", "In addition to these MI solutions, NLS equation has also time-periodic solutions in the form of envelope solitons traveling on a finite background, which do not correspond to MI, and are called Kuznetsov-Ma solitons (KM).", "It is natural to apply such successful NLS-breathers deep water hydrodynamic models in realistic oceanographic situations where the underlying field is irregular and random [62].", "Even if initially the ocean surface dynamics is narrow-banded, winds, currents and wave breaking may induce strong irregularities.", "Recently it was demonstrated the possibility of extending NLS models to such broad-banded processes, a fact that becomes valuable in the prediction of extreme events and in extending the range of applicability of coherent structures in ocean engineering.", "There have been a lot of progress lately in this direction [45], [15], [16], [31].", "In [63] it is reported the possibility for exact breather solutions to trigger extreme events in realistic oceanic conditions.", "By embedding PB into an irregular ocean configuration with random phases, for example a JONSWAP spectrum [64], the unstable PB wave packet perturbation initiates the focusing of an extreme event of rogue wave type, in good agreement with NLS and even modified NLS (MNLS) predictions [16].", "In this study rigorous numerical simulations based on the fully nonlinear enhanced spectral boundary integral method shown that weakly nonlinear localized PB-type packets propagate in random seas for a long enough time, within certain range of steepness and spectral bandwidth of the nonlinear dynamical process, somehow in opposition to what the weakly nonlinear theory for narrow-banded wave trains with moderate steepness would predict.", "This results are also backed up by recent hydrodynamic laboratory experiments also show that PB breathers persist even under wind forcing [65].", "From the existing literature, especially the articles published in the last eight years it appears that the role of coherent structures like solitons and breathers in the properties of a system of a large number of random waves is definitely a task of major importance both from fundamental and applications points of view.", "One of the objectives of the present work is to provide a detailed analysis of our experimental data showing the occurrence out of the random wave field, and survival against nonlinear interactions, and against the effects of traveling over non-uniform bathymetry, of breathers and other coherent modes in a hydrodynamic tank.", "The second objective is the comparison of these experimental results with exact analytic expressions of PB, AB and KM breathers, and to study the limits of applicability of the NLS equation model for deep water random waves over variable bathymetry.", "In the last decade, the propagation of gravity waves over variable bathymetry profiles has been studied as a possible configuration enhancing the occurrence of large waves.", "Different studies have described the statistical properties of gravity waves in this configuration both experimentally and with different numerical methods ranging from KdV models, [28], [59], through modified NLS equations, and Boussinesq models, [22], up to fully nonlinear flow solvers [42], [57].", "Trulsen et al shown, [22], that the change of depth can provoke increased likelihood of RW.", "As waves propagate from deeper to shallower water, linear refraction can transform the waves such that the wavelength becomes shorter, while the amplitude and the steepness become larger, and vice-versa.", "The dependence of the statistics parameters (spectrum, variance, skewness, kurtosis, BFI, etc.)", "of long unidirectional waves over flat bottom, versus the depth $h$ is a result of interaction between several competing processes within the nonlinear waves.", "One one hand, Whitham theory, [39], for nonlinear waves predicts that in shallower depth long-crested waves become modulationally stable, hence the modulation instability (MI) tends to decrease with the decreasing of $kh$ , and annihilate when $kh< 1.363$ because the coefficient the cubic nonlinear term vanishes at this threshold.", "On the other hand Zakharov equation (for example [22], [29]) predicts increasing of MI through increasing of the waves steepness $\\epsilon $ by linear refraction and by static nonlinear 2nd order effects with decreasing of the depth.", "Numerical studies by Janssen et al, [30], have shown that shallower water involves the decreasing of kurtosis all together by these effects.", "Nonlinear unidirectional wave fields over non-uniform bathymetry have a different dynamical behavior because the traveling nonlinear waves reach an equilibrium at some depth, and then they loose this equilibrium when running over different depth, and it takes time and space extension for the waves to reach another state of equilibrium.", "Numerical studies of NLS solution performed by Janssen et al, [27], show that the combination of focusing and nonlinear effects result in increasing of kurtosis when waves run over shallower depths, for example when $kh: 20 \\rightarrow 0.2$ .", "The same strong non-Gaussian deviation towards shallower bottom was confirmed in numerical studies by Sergeeva & Pelinovsky et al, [28].", "More interesting though, the change in waves' kurtosis with the depth depends itself on what side of the slope the waves are investigated.", "In experiments over sloped bottom Trulsen et al, [22], shown that waves propagating over a sloping bottom from a deeper to a shallower domain present a local maximum of kurtosis and skewness close to the shallower side of the slope, and a local maximum of probability of large wave envelope at the same location, situation which can generate a local maximum of RW formation probability at that point, results backed by NLS numerical solutions in [26].", "The present paper provides experimental evidence of the above discussed numerical predictions for long crested waves propagating over non-uniform bathymetry and confirms the experimental results obtained by Trulsen et al in [22].", "In addition to these results, and for the first time in literature we present nonlinear waves generated by random fields and propagating over a slope, followed by a submerged shoal, followed by another slope and a final run-up beach.", "We study the evolution of the spectrum, skewness, kurtosis and other statistical descriptors while the waves pass over this bottom landscape.", "Moreover, we detected the formation and persistence of coherent wave packets, possibly breathers, traveling over this variable bathymetry with almost constant group velocity and stable evolution and shapes.", "The paper is organized as follows: In section 2 we present the experimental setup and the type of waves and their physical parameters we are using, and also how the results were collected and analyzed.", "We identify traveling stable wave packages in the random wave field.", "In section 3 we analyze the experimental results with respect to the waves steepness, Ursell number, MI, solitons and RW conditions of formation like skewness, kurtosis and BFI.", "We also investigate the possibility of formation of RW and we evaluate the wave spectra and the probability of distribution of wave heights.", "In section 4 we present the NLS theoretical formalism for flat bottom and for non-uniform depth and compare the corresponding exact solutions with our experimental results.", "In conclusion we can prove that within the given bottom bathymetry, breathers, solitons and rogue waves deep water phenomena are generated out of the random wave background, are stable, and are little perturbed by the bathymetry of our experiments." ], [ "Experimental set up. Random wave fields over non-uniform bathymetry", "The experiments have been performed in the wave tank of the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology.", "The wave tank is $L_{tank}=50$ m long, 3 m wide and 1 m deep.", "The water tank is provided with a hydraulic servo wave maker at the left end which can generate waves of arbitrary shape with minimum period $0.66$ s at 15 cm wave maximum height, and an absorbing beach is installed at the other end to avoid wave reflections, Fig.", "REF .", "In the present experiment, the bottom has non-uniform shape with the maximum depth of water in the tank $h=0.76$ m. To insure a unidirectional wave field and long-crested waves, the wave tank was divided in two sections along its length, of 2 m and respectively 1 m each, and we experimented in the wider section.", "A number of 45 resistive wave probes (gauges) were aligned along the wave propagation direction to measured the wave height.", "The surface height of the water at these specified positions is measured with an accuracy of up to 6 significant digits at a sampling $\\delta t=0.02$ s, with 170 s time series length memory.", "The gauges are placed as shown in Figs.", "REF -REF , namely: first two control gauges, beginning at 9 m from the wave maker, then 17 equidistant gauges with 30 cm in between, then 4 gauges at 50 cm separation, then 6 gauges separated by 1 m each, and finally 16 gauges separated by 40 m in between.", "Since the width of the basin is large compared with the characteristic wavelength of our experiments, viscous energy dissipation that occurs mostly on sidewalls is assumed to be negligible at the center of the basin where our wave probes are located [31].", "The bottom shape is inspired by some specific sea floor bathymetry.", "At the wave maker end the bottom is deep and then gradually increases its heights towards shoal with minimum depth of $h_{min}=0.34$ m at gauge $\\#16$ , at $x=13$ m from the wave-maker.", "From this point the bottom height drops at a larger slope and it reaches its deepest region at $x=20$ m at gauge $\\#28$ .", "Then the bottom gradually becomes shallower increasing its height towards a run-up beach all the way to the water surface, see Figs.", "REF -REF .", "We have carried 40 different experiments by changing the significant wave height $H_s$ and significant period $T_s$ , see Table 1, but because of the article length we present here only three relevant cases of $H_s$ and $T_p$ .", "Table: Experimental settings and parameters.The sampling time of Cases $B1 \\div B4$ is $81.92$ s. In order to study the process of wave evolution in more detail, the sampling time of Cases $J$ is $163.84$ s. A JONSWAP spectrum was chosen for the irregular wave simulation, described by the following parameters [67] $S(f)=\\frac{\\beta _{J}H_{s}^{2}\\gamma ^{\\delta }}{T_{p}^{4}f^{5}}\\hbox{Exp} \\biggl [ -\\frac{1.25}{(T_p f)^{4}} \\biggr ] ,$ $\\delta =\\hbox{Exp} \\biggl [-\\frac{\\biggl ( \\frac{f}{f_p}-1 \\biggr )^{2}}{2 \\sigma ^2} \\biggr ],$ with $\\beta _{J}\\simeq \\frac{0.06238}{0.230+0.0336 \\gamma -0.185 (1.9+\\gamma )^{-1}}\\cdot (1.094-0.01915 \\ln \\gamma ),$ $T_p \\simeq \\frac{T_{H_s}}{1-0.132 (\\gamma +0.2)^{-0.559}},$ and the function of wave frequency given by $\\sigma ={\\left\\lbrace \\begin{array}{ll}0.07 & f \\le f_p \\\\0.09 & f> f_p ,\\end{array}\\right.", "}$ and were $f_p$ is the spectrum peak frequency, $T_p$ is the spectrum peak frequency, $T_s$ is the significant period, and $\\gamma $ is the spectrum peak elevation parameters which we set $\\gamma =3.3$ .", "Figure: Experiment measurements setup.", "Positions of the wave gauges with respect to the wave maker and bottom topography.Given the geometry of the tank and dynamics of the wave maker, ranges of the random waves parameters are limited by three physical constraints: deep water condition, [68], neglecting capillary waves, and giving the waves enough room to form breathers and eventually rogue waves, that is $\\lambda _{capillary}< \\lambda <\\min \\lbrace L_{tank}, 2 \\pi h \\rbrace $ .", "The wave number for the carrier wave $k_p$ is derived from the linear dispersion relation $k_p =4 \\pi ^2 /(g T_{p}^{2})$ .", "Under these constraints and according to the parameters chosen in Table 1, the range of peak wavelength that can be excited in the tank becomes $0.85$ m $<\\lambda _p < 3.55$ m. Figure: Left: Bathymetry profile in wave tank, placement of some key gauges and quiescent water level (blue).", "Right: Water depth h(x)h(x), expression k p (x)h(x)k_p (x) h(x), and MI extinction threshold 1.3631.363.In our experiments the wavelength and group velocity of the carrier wave changes slightly along the tank because of the non-uniform bottom.", "In the deep regions at gauges $\\#2 \\div 7$ and $\\#22 \\div 29$ ($h=0.65 \\div 0.76$ m), or at $x=5-12$ m and $14-22$ m from the wave-maker, see Figs.", "REF ,REF , we have for the carrier wave period $T_p =0.95$ s deep water, long-waves, with parameters $\\lambda _p =1.407$ m, $k_p =4.49 \\hbox{m}^{-1}$ and $v_g=0.76$ m/s.", "In the intermediate region over the shoal ($h \\simeq 0.35$ m) at gauges $\\#11 \\div 19$ , or at $x=12-14$ m from the wave-maker, we still have deep water long-waves with parameters $\\lambda _p =1.31$ m, $k_p =4.79 \\hbox{m}^{-1}$ and $v_g=0.85$ m/s.", "Only towards the right end of the beach, ($h \\ge 0.2$ m) at gauges $\\#36 \\div 45$ , or at $x>25$ m from the wave-maker, we have shallow water and waves with parameters $\\lambda _p =1.13$ m, $k_p =5.55 \\hbox{m}^{-1}$ and $v_g=0.89$ m/s.", "In the left frame of Figs.", "REF we present the bottom height (topography) and gauges placement.", "In the right frame we present also with respect to the distance to the wave-maker, the water depth $h$ , and the calculated values of $k_p h$ depending on depth and corresponding wavelength for fixed $T_p$ .", "It appears that everywhere along the tank the condition for developing MI is fulfilled ($k_p h >1.363$ [1], [2], [10], [14]), the deep water NLS equation model is valid for the self-focusing regime of solutions, and wave train modulations will experience exponential growth, see for example Figs.", "REF ,REF .", "Figure: Longitudinal section in the wave tank with variable bathymetry.", "The wave maker is at the left of the frame, the dots represent gauges, and the vertical axis shows depth in meters.", "The quiescent water level is the dashed blue line and several of our waves are presented to visualize relations between the specific wave heights, wavelengths and depth.", "We chose moments t=75t=75 (solid line), 80 (dotted line) and 86s (dashed line) when coherent spontaneous structures (matched with Peregrine breathers) form over gauges number 5÷65\\div 6 (solid line), 24÷2724 \\div 27 (dotted line), and 30÷3430 \\div 34 (dashed line), respectively.", "In the upper inset the same picture is present at real scale.Figure: Random wave field of significant wave height H s =3.22H_s=3.22cm, significant period T s =0.95T_s=0.95s and variable bottom with depth h≤0.76h\\le 0.76m.", "Horizontal axis is time evolution and the 45 gauges signals are lined up along the vertical axis from the wave-maker (bottom) to the run-up (zero water depth on top of the frame).", "The blue shape represents the bottom profile with the dots being the gauges positions.", "We identify at least three coherent, stable, and almost uniformly traveling packages, highlighted with red stripes.Figure: Same configuration and parameters as in Fig.", "except here H s =5.2H_s =5.2 cm.", "We still identify at least two coherent, stable traveling packages, highlighted with red stripes.Figure: Same configuration and parameters as in Fig.", "except here H s =6.2H_s =6.2 cm.", "We still identify at least two coherent, stable traveling packages, highlighted with red stripes.Figure: Gauges #3÷23\\#3 \\div 23 time series for T p =0.9T_p =0.9 s and H s =3.22H_s =3.22 cm in the region where gauges are equidistant, but they run over the shoal.", "The coherent structures, possible breathers, appears traveling with stable shape and group velocity (slope of the line of traveling patterns representing the wave packages) over the variable bathymetry.The physical parameters that characterize the evolution of irregular waves are characteristic wave steepness $\\epsilon _{p}=k_p H_s /2$ , which in our experiments is ranged between $0.015$ and $0.33$ , and by the bandwidth.", "The spectral bandwidth is determined by choosing the peak enhancement factor, which in our case $\\gamma =3.3$ induces $\\triangle f/f_p =0.095$ .", "The Benjamin–Feir index BFI for the theory of Stokes waves [11], [15], [74] which measures the nonlinear and dispersive effects of wave groups is given by $\\hbox{BFI}=\\frac{\\epsilon _p f_p}{\\sqrt{2} \\triangle f}.$ Beyond a critical value of BFI=1 [31] an irregular wave field is expected to be unstable and wave focusing can occur.", "In our experiments we can cover the range $0.11 < \\hbox{BFI} < 2.2$ namely covering all types of sea, from linear waves to stronger MI with development of a rogue sea state, especially since the total length of the measurements covers 28 m which is larger than the distance over which the MI is expected to appear [31].", "Since the waves in our experiments may enter occasionally into a strongly nonlinear wave regime, the NLS equation may not provide a very good fit with these experiments.", "Figure: Steepness effect on coherent wave packages.", "The wave profiles from gauges #10\\#10 (at x=11.1x=11.1 m) and #20\\#20 (at x=14.1x=14.1 m) for three cases: H s =3.22H_s=3.22cm (first two bottom signals, black), H s =5.2H_s=5.2 cm (middle two signals, blue) and H s =6.2H_s=6.2 cm (upper two, magenta signals) vs. time.", "Coherent wave packages, most likely Peregrine breathers are spontaneously formed in the random waves and can be observed traveling for as long as 20 m.Figure: Density plot of the space-time wave field for H s =5.2H_s =5.2 cm waves (wave amplitude scale in cm to the right).", "The gauges from #3÷20\\# 3 \\div 20 with 30 cm between gauges, the gauges from #20÷24\\# 20 \\div 24 with 50 cm between gauges, the gauges from #24÷30\\# 24\\div 30 with 100 cm between them and the gauges from #30÷33\\# 30\\div 33 with 40 cm between them.", "Higher-order breathers (doublets) can be observed by their red-blue color while propagating uniformly.Figure: Density plot of the space-time wave field for the H s =6.2H_s = 6.2 cm waves.", "Legend for wave amplitude in cm to the right.We first consider irregular JONSWAP waves with significant wave height $H_s=3.22$ cm and significant period $T_s=0.95$ s over this complex bathymetry.", "In Fig.", "REF we present a typical experimental result.", "In this vertical longitudinal section of the wave tank with variable bathymetry (the gray shape at the bottom) and a wave maker placed at the left of the frame, we show the level of quiescent water by a dashed blue line, on which we overlapped several waves obtained at $t=75$ (solid line showing a nonlinear coherent wave package on top of the shoal), 80 s (dotted line, showing the same structure which traveled now over the deepest valley) and $t=86$ s (dashed line, when the same coherent package travels up the slope of the run-up).", "The behavior of the waves shown in Fig.", "REF is in agreement with the Djorddjevicć-Redekopp model for deep water with variable bathymetry, using a modified NLS equation with variable coefficients [60], Eq.", "REF .", "Indeed, in all our experiments the amplitudes and wavelengths of the waves slightly decrease, while $v_g$ slightly increases, over the shoaling region (about gauge $\\#16$ ), and the situation reverses when waves advance over deeper regions (gauges $\\#27-29$ ).", "In the upper inset of Fig.", "REF we present the longitudinal section at real scale, and the same waves, to stress that the all our waves amplitude are negligible compared to the variations in bathymetry.", "Figure: Linear scale Fourier spectra for H s =3.22H_s=3.22 cm (upper frame), H s =5.2H_s=5.2 cm (middle frame), and H s =6.2H_s=6.2 cm (bottom frame), all at T p =0.95T_p =0.95 s, for five representative points at gauges: 5,11,16,225, 11, 16, 22 and 39.", "The spectrum for the deeper sides, before and after the shoal are presented in solid line, the spectra of waves on top of the shoal with dashed line, the spectra at gauges on the ramp about the same height as the shoal by dotted line, and the spectra for regions with k p h≥1.363k_p h \\ge 1.363 with gray lines.Figure: Kurtosis (left frames) and skewness (right frames) plotted versus the gauge number, next to re-scaled bottom profile (solid line).", "Upper row represents the waves with H s =3.22H_s=3.22 cm; Middle row represents steeper waves with H s =5.2H_s=5.2 cm, and bottom row represents the steepest waves, with H s =6.2H_s=6.2 cm.", "All have the same T p =0.95T_p =0.95 s, and the wave maker is to the left.", "The vertical grid lines separate different regimes, namely: deep, slope, shoal, quick drop, deep bottom, the deepest, and the run-up beach.", "The thick gray vertical grid line represents to point xx where MI vanishes theoretically, i.e.", "kh→1.3.63k h \\rightarrow 1.3.63.For every experiment of generation of random waves we noticed the formation some localized traveling coherent wave packages.", "These structures, once formed, keep traveling with almost same group velocity over the variable bathymetry, over the shoal and tend to disintegrate when the $kh=1.363$ criterion for MI is not fulfilled anymore, that is around gauge number $39-41$ .", "In Fig.", "REF we present such an example of a 164 s long time series (horizontal axis time) as measured by different gauges lined up along the vertical axis.", "The traveling coherent structures are identified (three of them, for example, are highlighted in red stripes in the figure).", "These wave packages propagating approximately constant with the peak group velocity of order $v_g \\simeq 0.815$ m/s.", "A larger image for a typical such time series only for gauges $1-23$ is given in Fig.", "REF where one can detect better the occurrence and stability over the shoal of the nonlinear coherent packages: one begins at $t=27$ s and another larger one begins at $t=68$ s. In Fig.", "REF we present in more extended detail wave profiles for $H_s=3.22$ cm and 160s duration time-series measured at 5 locations (gauges $1, 2, 23, 30$ and 41) to observe better the nonlinear coherent formations that are spontaneously formed in the random waves and that travel for as long as 20m.", "Figure: Black curves in the main frames are the experimental wave profiles for H s =3.22H_s = 3.22 cm, measured at gauges 3,13,20,21,22,23,24,26,443,13,20,21,22,23,24,26,44 (from upper left corner clockwise), thus covering a fetch of 21m beginning at 9 m from the wave maker.", "The time interval is shown in the upper inset highlighted in red.", "Red curves are theoretical KM breather solutions of NLS equation for deep water.", "The only parameters changing from one frame to another are the origin time, while the rest of the KM breather parameters (A 0 ,αA_0,\\alpha ) are the same for all frames, fact which validates the correctness of our model.Figure: From left: black curves in the first 7 frames are experimental wave profiles measured at gauges 3,12,19,20,21,22,23,25,433,12,19,20,21,22,23,25,43, thus covering a fetch of 21 m beginning at 9 m from the wave maker.", "The time interval is shown in the upper inset highlighted in red.", "The last frame represents an overlap of all these 7 frames, shifted in time correspondingly.", "The final frame shows a clear match of the same type of behavior for this coherent traveling group, and the likeliness to be described as a breather, possibly a higher-order breather.Figure: Matching H s =3.22H_s =3.22 cm waves.", "Left: gauge 3 at t=28-37t=28-37 s matched with two KM solitons.", "Right: gauge 13 at t=30-44t=30-44 s matched with one KM soliton.Figure: H s =3.22H_s =3.22 cm waves.", "Black curves are experiments, red curves are Peregrine solitons, and blue curves are KM solitons.", "From left, first three frames represent matching an earlier formed coherent package: gauge 13 at t=34-42t=34-42 s; gauge 23 at t=36-44t=36-44 s; gauge 44 at t=48-58t=48-58 s. Gauge 3 at t=68-78t=68-78 s matching a later formed coherent package.Figure: H s =3.22H_s =3.22 cm experimental waves plotted with black curves and theoretical match (red curves) with double AB breathers.", "From left: gauge 3 at t=26-36t=26-36 s; gauge 13 at t=30-40t=30-40 s; gauge 23 at t=34-44t=34-44 s.Figure: H s =3.22H_s =3.22 cm experimental waves plotted with black curves and theoretical match (red curves) with double AB breathers.", "From left: gauge 26 at t=26-46t=26-46 s; gauge 31 at t=43-49t=43-49 s; and again gauge 3, the latest coherent group at t=68-77t=68-77 s." ], [ "Analysis of experimental results", "In the analysis of our experiments over variable bathymetry we follow the Trulsen et al approach, [22], [25], [26], by performing statistics over the time series (and not over space wave field) for the determination of the reference wave and to discern the extreme waves or other coherent structures.", "In this procedure $H_s =H_s(x)$ becomes a function of space, and the criteria for identifying breather solutions or RW become local.", "This approach, supported by numerical studies [26], [57], allows to isolate the situation favorable for initiation of RW, because linear refraction itself at variable bathymetry points cannot change the probability of RW unless such local criteria for RW are not employed [22].", "Random long-crested waves were propagated over the non-uniform bathymetry.", "The slope of the bottom topography has values between $1:10$ at the beginning of the shoal (gauges $\\#3$ , a drop of slope $-1:240$ , then slope oscillating between $\\pm 1:16$ and finally raising of the beach with slope $1:10 \\div 1:35$ , all over a length of $28.1$ m. The gauges are placed as shown in Figs.", "REF -REF .", "Three cases of long-crested random waves were generated with different nominal significant wave height $H_s$ and constant nominal peak periods $T_p$ , as shown in Table 1.", "The peak wave-number $k_p$ has been computed from the linear waves dispersion relation $\\omega ^{2}_{p}= g k_p \\tanh k_p h$ where $\\omega _p =2 \\pi / T_p$ , $g=9.81$ m$/\\hbox{s}^2$ .", "The characteristic amplitude is calculated as in [22] $a_c =H_s / \\sqrt{8}$ corresponding to a uniform wave of the same mean power.", "The Ursell number is $U_r =k_p a_c / (k_p h)^3$ .", "Table: Three significant heights, at depths: Deep (gauges 3÷73\\div 7, 22÷2422 \\div 24, and 30), Deepest (gauges 26÷2826\\div 28), Shoal (gauges 15÷1815\\div 18), Beach (gauges 42÷4442\\div 44).The three $H_s$ cases for 45 recording gauges times 8192 samples taken at $\\delta t=0.02$ s intervals, minus the startup effects provide about $7,000$ peak periods, which, [22], [23], provide sufficiently reasonable estimates of kurtosis, skewness and overall distribution functions.", "In Table 2 we present some wave parameters for the three $H_s$ cases investigated, and for four regions of bathymetry called: deep water, the deepest region, shoal and towards the upper parts of the run-up beach, respectively.", "In all these regions the NLS theory derived by Zakharov, [19], applies and the MI develops in all cases with $k_p h >1.363$ for self-focusing regime.", "Steepness $\\epsilon =k_p a_c $ , and Ursell number was also calculated for all the cases (Table 2) and it shows a very good agreement with the similar cases investigated in [22].", "$U_r$ has a small value in almost all deep regions, with moderate increased values above the shoal but still in the range of Stokes 3rd and NLS equation modeling, and larger values of $U_r$ above the beach where the waves cannot be considered anymore nonlinear deep water, and the character shifts from Stokes 4th to 5th order to cnoidal behavior, towards breakers in the end of the slope.", "In shallower water the 2nd-order nonlinearity of KdV dynamics becomes responsible for the strong correlation observed between skewness, kurtosis and $U_r$ , see also Fig.", "REF .", "Some further insight into understanding the waves obtained in these experiments can be obtained by looking at the wave spectrum at different positions along the wave-tank.", "The spectra at five representative points for the three cases of significant wave height are shown in Figs.", "REF with linear scales.", "The signal peaks and the Fourier spectra were obtained by using and automatic multiscale peak detection based on the Savitzky-Golay method.", "All the nominal peaks are centered around the carrier frequency $T_p =0.9s$ .", "There is a slight spectral development leading to a downshift of the peak, but not very visible, which means that the MI is present almost all over the measurements, except the last few gauges.", "The spectrum corresponding to the deeper sides, independently if this deep region was before and after the shoal are almost identical (solid line in Figs.", "REF ).", "The spectra of waves on top of the shoal (dashed line) and the spectra at gauges on the ramp about the same height as the shoal (dotted lines) are not too different from the deep region ones.", "However, visible changes in the spectrum show when the waves propagate towards more shallower regions on the final beach (gray line).", "At these points, where skewness and kurtosis attain also maximum values, the spectrum tends to show a second maximum around frequencies doubling the carrier frequency, most likely because of the growth of second order bound harmonics caused by the increased nonlinearity at shallower depth.", "Further into the shallow region the spectrum significantly broadens and becomes noisier since energy is shared to lower and higher frequencies.", "This situation becomes evident for regions with $k_p h \\ge 1.363$ in agreement with the results obtained in [22], [57].", "Nonlinear transfer of energy between modes gives rise to deviations from statistical normality of random waves (Gaussian e.g.).", "The most convenient statistical properties intended to characterize nonlinear coherent wave packages or extreme wave occurrence are the third and fourth-order moments of the free surface elevation $\\eta (x,t)$ , [57], [22], namely the skewness and the kurtosis defined as $\\hbox{Skewness}=\\lambda _3=\\frac{<\\eta -<\\eta >^3>}{\\sigma ^3},$ $\\hbox{Kurtosis}=\\lambda _4=\\frac{<\\eta -<\\eta >^4>}{\\sigma ^4}=\\frac{<\\eta ^4 >}{3 <\\eta ^2 >^2}-1,$ where $<,>$ stands for the average over time and $\\sigma $ is the standard deviation of $\\eta $ , directly related to the significant wave height $H_s=4 \\sigma $ .", "The skewness characterizes the asymmetry of the distribution with respect to the mean while the kurtosis measures the importance of the tails.", "The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states [30].", "In Figs.", "REF we present the kurtosis of the surface elevation in the left frame and the skewness of the surface elevation in the right frame for the three significant wave heights experimented.", "The statistical estimates indicate 98% confidence intervals obtained from $16,500$ selected samples from the original data.", "For smaller wave amplitude there is a local maximum of kurtosis and skewness, on top of the shallower edge of the shoal.", "For larger amplitude waves this kurtosis local maximum shifts towards the beginning of the slope, towards the deeper region.", "All waves of all heights record a global maximum of the kurtosis in the deepest region, over gauges numbers $24\\div 30$ , similar to the cases described in [22], [25], [26], [27], [28].", "This effect is related to the fact that deeper means $kh$ greater than $1.363$ as seen in Table 2, and is also related to the spectral evolution leading to a slight downshift of the shallow spectrum with dotted, dashed and gray lines in Figs.", "REF .", "For all cases the global maximum of these two statistical quantities is most prominent at the beginning of the shoal, that is at the positive slope edges of the shoals.", "In all these three cases of different $H_s$ values representing different steepness degrees of the waves, except the end of the run-up beach, the depth is everywhere larger than the threshold value for MI, and not significant shift of the spectral peak can be easily seen.", "Over deep water regions with $kh\\ge 1.363$ , higher initial BFI (like the waves with $H_s =5.2$ cm or $6.2$ cm, see the red and blue upper curves in Fig.", "REF ) the kurtosis tends to be stabilized at higher values as can be seen in the left column in Fig.", "REF for $x=3 \\div 8$ m and $x=25\\div 30$ m, for waves with $H_s =3.22$ cm, for $x=7 \\div 10$ m and $x=25\\div 28$ m for waves with $H_s =5.2$ cm, and $H_s =6.2$ cm.", "This result agrees well with previous publications demonstrating that stabilized kurtosis is larger in deep water and smaller in shallower water [26], [29], [74] However, when $kh\\rightarrow 1.363$ beginning at $x \\simeq 35$ m, nonlinear effects diminish and the kurtosis decreases towards 3.", "This is also visible in Fig.", "REF : for smaller steepness waves with $H_s =3.22$ cm kurtosis tends to drop slightly around $x\\simeq 36$ m just before the gray vertical stripe in the figure.", "The dropping effect is more visible at higher steepness, $H_s =5.2$ cm at 36 m, and again less intense for the steepest waves $H_s =6.2$ cm.", "After the $1.363$ threshold, the observed oscillations in kurtosis and skewness may be generated by other shallow water effects, Bragg effect, reflection or linear diffraction.", "Our results make evident that when a wave field travels over a bottom slope into shallower water, a wake-like structure may be anticipated on the shallower side for the skewness and the kurtosis, as it was previously confirmed in [26].", "The general expressions for the skewness and the kurtosis of deep water surface evaluated with Krasitskii’s canonical transformation in the Hamiltonian [30], apply to our cases with $\\epsilon \\simeq 0.1$ (shallower regions for $H_s =5.2$ cm and all regions for $H_s =6.2$ cm, see Table 2) and our experimental results lineup well with this theory when correlating the values of kurtosis and skewness from Figs.", "REF with the $k_p h$ values from Fig.", "REF (right).", "Also, noticing that the value of the BFI decreases with decreasing of the water depth, as we can see it happening for $x>23$ m (or after gauge 33) in Fig.", "REF , while the nonlinear coherent structures (which we identified with Peregrine or higher order NLS breathers) keep propagating stable up to shallow water regions, we infer that the probability of RW occurring near the edge of a continental shelf may exhibit a different spatial structure than for wave fields entering from deep sea and BFI deep water criteria may not apply the same way.", "Right after the shoal, both the kurtosis and skewness show oscillations in the values because of a combination between nonlinear effects and linear refraction.", "One interesting observation resulting from Figs.", "REF is that for small steepness $\\epsilon < 0.08$ waves the kurtosis and skewness are larger above the extreme depths $h$ (very deep or $h\\simeq h_{max}$ , or very shallow or $h\\ll h_{max}$ ), while for larger steepness waves, these two statistical moments tend to acquire their largest values above the sloppy regions of the bottom.", "This observation can be expressed in a phenomenological relation of the form $\\lambda _{3,4} \\sim C_1 \\biggl (h-\\frac{h_{max}}{2} \\biggr )^2 + \\epsilon C_2 \\biggl | \\frac{d h}{dx} \\biggr |,$ for some empirically determined constants $C_{1,2}$ .", "The conclusions obtained from our experimental results and our statistical analysis of kurtosis and skewness coincide with the statistical behavior suggested in the numerical studies from [57], [25], [26], [27], [28] and with the experimental results obtained for sloped bottom in [22].", "We have thus shown that as long-crested waves propagate over a shoal and variable bottom in general, local maximum in kurtosis and skewness occur closer to the beginning and the end of the slope, mainly on the shallower side of the slope which can identify these regions as possible locations of high amplitude breathers, multiple breathers and RWs formation." ], [ "Rogue waves from random background", "In deep water, long-crested waves are subject to MI, [11], [18], which is known to generate conditions for RW formation [10], [1], [16], [18], [75], [23], [47], [54], [55], [58], [72], [74].", "It was also found that nonlinear modulations during the evolution of irregular waves causes spectral development and frequency down-shift, suspected to be related to the occurrence of RW [22], [24].", "In this section we investigate the occurrence of higher amplitude waves, out of the random background, as candidates for RW.", "Figure: Time series of the recorded wave amplitudes η(t)\\eta (t) normalized to their characteristic wave height H s =3.22H_s =3.22 cm, measured at 19 and 26.526.5 m (gauges #27,41\\# 27, 41).", "The horizontal grid lines represent, in the order of their heights: minimum surface (blue), standard deviation (black), and maximum wave (red).", "The maximum height recorded is 2.1÷2.42H s 2.1 \\div 2.42 H_s, which qualifies them as RWs.Extreme height waves, isolated in time and space from the typical background reference wave field is considered to be a Rogue Wave (RW) if it satisfies some common criteria like $\\eta / H_s > 1.25$ or $H / H_s > 2$ , where $\\eta $ is the crest elevation, $H$ is the wave height, and $H_s$ is the significant wave height, defined as four times the standard deviation of the surface elevation, [22], [23].", "In Fig.", "REF we present an example of two time series recorded in our experiments at two locations, 19 m (left frame) and $26.5$ m (right frame) from the wave maker.", "The vertical axis shows the wave amplitude $\\eta (t)$ normalized to the characteristic wave height $H_s =3.22$ cm.", "We recorded such large amplitudes at $19, 22, 23.5$ and $26.5$ m (gauges $\\# 27, 30, 33, 41$ , respectively) from the wave maker.", "The horizontal grid lines represent, in the order of their heights: minimum surface (blue), standard deviation (black), and maximum wave (red).", "The maximum height recorded is $2.1 \\div 2.42 H_s$ , which qualifies them as RWs.", "These events happen over the deep water parts, at the locations where kurtosis and skewness has also local maxima, Figs.", "REF .", "Figure: Moving Fourier spectra in time-frequency domain with 4 s window, for 150 s long experiments at H s =3.22H_s=3.22cm.", "Fourier curve's zero-axes are ordered vertically with respect to time, in seconds; but each Fourier spectrum's curve has arbitrary scaling.", "Horizontal axis represents frequency in Hz.", "From left upper corner clock-wise the frames represent spectra of waves measured at gauge 3 (at 9m), 10 (at 12m), 23 (at 14.114.1m) and 41 (at 22.122.1m).", "The red curves represent narrow band width spectra measured at points where coherent packages were detected and assimilated with breathers/solitons/RW, while the black curves represent the wide spectra of random waves filling the background between the breathers.Figure: Same type of spectral representation as in Figs.", "for H s =5.2H_s =5.2 cm (left frame) and H s =6.2H_s =6.2 cm (right frames).", "In addition, we present in the insets details of the Fourier spectrum of five wave series centered at the moment of time indicated by the arrow: the red spectra are associated to coherent packages identified as breathers/solitons, the gray spectra are the random background waves at nearby points and neighbor moments.For a system composed of a large number of independent waves, like the random generation, the surface elevation is expected to be described by a Gaussian probability density function.", "Under this hypothesis, Longuet-Higgins [15], [43], [44], showed that, if the wave spectrum is narrow banded, then the probability probability distribution of crest-to-trough wave heights is given by the Rayleigh distribution.", "The distribution was found to agree well with many field observations [15].", "Nevertheless, recently [43], [44] it was shown numerically and theoretically that if the ratio between the wave steepness and the spectral bandwidth this ratio is known as the Benjamin–Feir index (BFI) is large, a departure from the Rayleigh distribution is observed.", "This departure from the Rayleigh distribution was attributed to the MI mechanism.", "Moreover, from numerical simulations of the NLS equation it was found [15] that, as a result of the MI, oscillating coherent structures may be excited from random spectra.", "In our experiments we obtained a very good correlation between the waves at regions and during time intervals producing a narrower width spectrum and the corresponding detection (at the same locations and moments of time) of coherent stable, traveling structures, most likely NLS breathers (AB, KM, Peregrine of higher-order breathers, section ).", "Figure: Plot of the average BFI values over 16s duration, versus space, along the wave tank.", "Legend: H s =3.22H_s =3.22cm (black), H s =5.2H_s =5.2cm (red), and H s =6.2H_s =6.2cm (blue) for the upper curves.", "The bottom profile and some gauge numbers are presented by the lowest gray curve, and the slope of the water depth (dh/dxdh/dx) by the green curve.", "The MI threshold kh=1.363kh=1.363 happens around gauges 33-3433-34, at x≃24mx \\simeq 24m from the wave-maker.In Figs.", "REF , REF we present examples of Fourier spectra in the time-frequency domain calculated with a 4 s moving window, at different locations and different moments of time.", "In these figures, the red curves represent narrower bandwidth wave spectra measured at points where also the coherent packages were detected and assimilated with breathers/solitons/RW, wave packages described in previous sections.", "Namely, the red curves in Figs.", "REF , REF coincide with a good coefficient of correlation ($c=0.76$ Pearson correlation) with the structures highlighted with red stripes in Figs.", "REF , REF , REF , with the coherent packages in uniform motion identified in the mapping of Figs.", "REF , REF , REF , and REF , also they coincide with the packages chosen for theoretical match with breathers and shown in Figs.", "REF , REF , REF , and REF , and they are close neighbor with the extreme amplitude waves shown in Fig.", "REF .", "Figure: The gray upper curves represent the BFI versus time for 6 selected gauges.", "The large central peak of BFI>1>1 coincides with the formation of breathers at that position/moment.", "All five curves show the same reproducible behavior.", "The blue profiles at the bottom represent the relative value of the peak frequency in the time series recorded at the 6 selected gauges (1,5,9,13,17,221,5,9,13,17,22).Figure: Correlation between the spectral band width (vertical axis in mHz) and mean wave height (in meters) measured at each gauge from 1 to 22, for all H s H_s.", "The points represent a set of 15 mean values of wave spectra and wave heights are evaluated across 150s time series in samples of 10s each, for 22 gauges.", "The two resulting clusters describe random waves (low wave height, higher frequencies) and breathers (higher waves, lower frequencies).Figure: Probability distributions for the wave heights for H s =5.2H_s =5.2 cm.", "Left: mean values calculated across 160 s time series and 10.110.1 m fetch for gauges 1-221-22.", "Center: mean values calculated for the interval 30-3630-36 s, 5 m fetch, gauges 1-221-22.", "We note the cluster of narrow band-width spectra associated to the breathers present within this time interval and location.", "Right: mean values for the interval 88-9888-98 s, 7 m fetch, gauges 30-4630-46 at 22÷2922 \\div 29 m from the wave maker, respectively.", "This spectrum contains mainly unstable structures resembling peakons, and breaking waves.Figure: Wave height probability distributions for different moments of time, from upper left corner CW: t=1,2,15,20,35,50,60t=1, 2, 15, 20, 35, 50, 60 and 75 s. Each distribution calculated over 2 s interval (100 samples) over the fetch 9-149-14 m (gauges 3-223-22) for H s =5.2H_s =5.2 cm.", "Three main modes are present: dominant low amplitude waves at t=15t=15 and 35 s), dominant high amplitude waves at t=1,2,20t=1, 2, 20 and 60 s, and flat PDF distribution, at t=35t=35 s. Occasionally the distribution becomes bi-modal.", "Also we note a cyclic behavior since certain types of PDF tend to repeat.These positive correlations represent an evidence that MI process takes place in our experimental real long-crested water waves, with high values for the BFI index (the ratio between the wave steepness and the spectral bandwidth) at various depths, on the top of the shoal and equally in the deep regions around the shoal.", "In the case of our random waves the large values for BFI and the narrower width of the spectra lead to MI evolution and to a “rogue sea” state, that is a highly intermittent sea state characterized by a high density of unstable modes, see Fig.", "REF .", "Our results are very similar with the same types of studies reported [15].", "By using the calculations of the spectral bandwidths for all our experimental time series, at different locations and for the three types of significant wave height (steepness), we can correlate these data with the mean wave height.", "The result is presented in Fig.", "REF .", "We notice the formation of two separate clusters of higher positive correlation: one for small waves with large spectral band width, and one more localized for the breather/soliton/RW events described by large wave heights and narrower spectra.", "Another statistical feature which can confirm the formation of coherent traveling packages of breather/soliton types (KM, Peregrine and AB solutions) is the distribution of the probability for the wave heights, which we present in Fig.", "REF .", "The middle frame, representing regions with coherent package formation shows evidence of a cluster of narrow band-width spectra associated to these breathers.", "In Figs.", "REF we present the wave height probability distributions for different moments of time over a 5 m length.", "We observe the formation of three main modes: a dominant low-amplitude mode, a dominant high-amplitude mode, and a flat probability distribution which occasionally tends to shift into a bi-modal unstable mode as predicted by the Soares model [77].", "Our experimental results, mainly gathered in Figs.", "REF , REF , REF -REF and REF , REF , are in good agreement with the numerical calculation obtained by Trulsen et al ([22]a), from the Boussinesq model with improved linear dispersion, and with the experiments presented in Gramstad et al ([22]b).", "Indeed, a significantly increased BFI value, and consequently increase in the probability of RW occurs as waves propagates into shallower water.", "For smaller $H_s$ and $\\epsilon = H_s k_p$ the maximum is smaller and delayed, while for larger steepness the maximum occurs earlier and is larger, Fig.", "REF .", "Increased values of skewness, kurtosis, and BFI are found on the shallower side of a bottom slope, with a maximum close to, or slightly after the end of the slope Figs.", "REF , REF .", "Maxima of the statistical parameters are also observed where the uphill slope is immediately followed by a downhill slope.", "In the case that waves propagate over a slope from shallower to deeper water, in the theoretical evaluations from [22] it was not found on increase in RW wave occurrence where the wave parameters were $a k_p= 0.038, a/h = 0.035$ , and $Ur = 0.031$ .", "In our experiments, however, we noticed this increase in the BFI, kurtosis and steepness when traveling into deeper, probably because our waves parameter, shown in Table 2, are different: $a k_p > 0.05, \\ a/h = 0.04$ , and $U_r > 0.2$ ." ], [ "Comparison with exact solutions", "In this section we present some current theoretical models that can fit our experiments with random waves generated in a $L=50$ m long, 2 m wide wave tank with variable bottom and maximum depth $h_{max}=0.76$ m present by the wave-maker and at two-thirds of the length, see Figs.", "REF , REF .", "Since in all experiments described in section we notice the formation of stable, traveling coherent wave packages, we present in the subsequent section the match between these waves and deep water breathers.", "We divide this section in two parts: in the first part we present the corresponding theoretical results for uniform bottom, and in the second part we extend this case to variable bathymetry.", "In the uniform bottom case, for an ideal (incompressible and inviscid) liquid under the hypothesis of irrotational flow, the dynamics of a free surface flow is described by the Laplace equation for the velocity potential, and two boundary conditions: a nonlinear one (kinetic) on the free surface, and zero vertical velocity component at the rigid bottom [18], [2].", "Under the assumption of very small amplitude waves (or steepness) the problem can be considered as a weakly nonlinear one, and the standard way of modeling is to derive the NLS equation by expanding the surface elevation and the velocity potential in power series and using the multiple scale method [1], [2], [47], [18], [19], [20], [23], [17].", "The procedure is to introduce slow independent variables (both for time and space) and treat each of them as independent.", "The extra degrees of freedom arising from such variables allows one to remove the secular terms that may appear in the standard expansion.", "The multiple scale expansion is usually performed in physical space and a simplification of the procedure is the requirement that the waves are quasi-monochromatic.", "In the approximation of infinite water depth, for two-dimensional waves the surface elevation, up to third order in nonlinearity, takes the form $\\eta (x,t)=\\biggl ( |A(x,t)| -\\frac{1}{8} k_{p}^{2} |A(x,t)|^3 \\biggr ) \\cos \\theta +\\frac{1}{2}k_{p} |A(x,t)|^2 \\cos (2\\theta )$ $+\\frac{3}{8}k_{p}^{2} |A(x,t)|^3 \\cos (3 \\theta )+\\dots ,$ where $A(x,t)$ is a complex wave envelope, $k_p$ is the wave number of the carrier wave, $\\eta (x,t)$ is the water elevation, $\\theta =(k_p x-\\Omega _0 t+\\phi )$ is the phase, and $\\phi $ a constant phase.", "In addition we know that $\\Omega _0=\\omega _p (1+k_{p}^{2} |A(x,t)|^2/2)$ is the nonlinear dispersion relation, with $\\omega _p=\\sqrt{g k_p}$ .", "The complex envelope obeys the NLS equation $i\\biggl ( \\frac{\\partial A}{\\partial t} +c_g \\frac{\\partial A}{\\partial x} \\biggr ) -\\frac{\\omega _p}{8k_{p}^{2}}\\frac{\\partial ^2 A}{\\partial x^2}-\\frac{1}{2}\\omega _p k_{p}^{2} |A|^2 A=0,$ with $c_g=\\partial \\omega / \\partial k$ being the group velocity.", "The NLS Eq.", "(REF ) has various types of traveling solutions known as breathers or solitons.", "One analytic solution with major impact in literature is a combine one-parameter $\\alpha $ family given by [17] $A(X,T)=A_0 e^{2 i T}\\biggl (1+ \\frac{2 (1-2 \\alpha ) \\cosh (2 R T)+i R \\sinh (2 R T)}{\\sqrt{2\\alpha } \\cos (\\Omega X)-\\cosh (2 R T)}\\biggr ),$ where the $X,T$ are arbitrary scaled variables by a factor $s$ and the solution $A(x,t)=s A(sX, s^2 T)$ , $R=\\sqrt{8 \\alpha (1-2 \\alpha )}$ and $\\Omega =2 \\sqrt{1-2\\alpha }$ .", "When the parameter $\\alpha \\in (0,0.5)$ Eq.", "(REF ) describes the space-periodic Akhmediev Breather family (AB), and when $\\alpha >0.5$ Eq.", "(REF ) describes the time-periodic Kuznetsov-Ma Soliton (KM) [17], [18].", "Moreover, in the singular value for parameter $\\alpha =0.5$ Eq.", "(REF ) describes a rational solution known as Peregrine (P) solution [53] $A(X,T)=A_0 e^{2 i T}\\biggl (-1+ \\frac{4+16 i T}{1+4 X^2+16 T^2}\\biggr ).$ The Peregrine solution in Eq.", "(REF ) only represents the lowest-order solution of a family of doubly-localized Akhmediev-Peregrine breathers (AP), [54], [66], [12], also called higher order breathers [7] $A_{j}(X,T)=e^{2 i T}\\biggl ( (-1)^j+\\frac{G_j +i H_j}{D_j} \\biggr ),$ where the terms $G_j, H_j, D_j$ are polynomials which can be generated by a recursion procedure [66].", "While in deeper water 3rd-order nonlinearity causes focusing of long-crested and narrow-banded waves and hence possibility of occurrence of freak waves, in shallower water the nonlinear dynamics are dominated by 2nd-order nonlinearity.", "Waves over variable water depth can be modeled for irrotational, inviscid and incompressible flow with a variable coefficient NLS equation.", "In the approximation of finite depth $(kh)^{-1}=\\mathcal {O}(1)$ , mild slope $\\partial h / \\partial x =\\mathcal {O}(2)$ , and small steepness $\\epsilon =\\mathcal {O}(3)$ the authors in [26] presented a NLS model with variable coefficients plus a shoaling term.", "In this model water surface displacement $\\eta $ , Eq.", "REF , and velocity potential $\\Phi $ can be written as 3rd-order perturbation series normalized to $g$ and $\\omega _{p}$ , respectively $\\eta =\\epsilon ^2 \\bar{\\eta }+\\frac{1}{2}(\\epsilon A e^{i \\theta }+\\epsilon ^2 A_{2} e^{2 i \\theta }+\\dots +\\hbox{c.c.", "}),$ $\\Phi =\\epsilon \\bar{\\phi }+\\frac{1}{2}(\\epsilon A_{1}^{^{\\prime }} e^{i \\theta }+\\epsilon ^2 A_{2}^{^{\\prime }} e^{2 i \\theta }+\\dots +\\hbox{c.c.", "}),$ where $\\epsilon \\theta =\\int ^{x} k(\\xi ) d\\xi -t$ , and c.c.", "means complex conjugation.", "The resulting NLS modified (with respect to Eq.", "REF ) equation in terms of the first harmonic amplitude $A$ of the surface displacement is $i\\mu \\frac{dh}{dx}A+i\\biggl ( \\frac{\\partial A}{\\partial x}+ \\frac{1}{v_g} \\frac{\\partial A}{\\partial t}\\biggr ) +\\lambda \\frac{\\partial ^2 A}{\\partial t^2}=\\nu |A|^2 A,$ where the coefficients $\\mu , \\lambda , \\nu , \\bar{\\omega }$ depending on $k, h$ and $v_g$ at constant imposed $\\omega $ are defined in [26].", "In particular, the extra shoaling term $i \\mu h_x$ generalizing the traditional NLS Eq.", "REF comes from the conservation of wave action flux [39].", "For the specific bathymetry in our experiments, Figs.", "REF , REF , REF , when the waves travel over the shoal (at $x \\sim 11-15$ m from wave-maker) the dispersion coefficient $\\lambda (h)$ has only a slow variation of maximum $12\\%$ of its value.", "The nonlinear term coefficient $\\nu (h)$ decreases on top of the shoal with $54\\%$ of its deep water value, while the shoaling term coefficient $\\mu (h)$ has a local increase of $140\\%$ on top of the island.", "The effect of the shoaling term, similar mathematically to the linear dissipative terms occurring in non-homogeneous medium, or to the boundary-layer induced dissipation term in an uniform depth, is a change in wave's amplitude.", "Actually, it was found [60], that such damping terms can stabilize the BF instabilities, especially since the nonlinear term contribution decreases in the shoaling regions.", "This effect is visible in our experiments manifesting as a decrease in the BFI over the shallower region, for any $H_s$ value, Fig.", "REF .", "Analyzing Eq.", "REF with the Djorddjevicć-Redekopp model [60], it results that $\\frac{d h}{d x}\\sim -\\frac{dk_p}{dx} \\sim \\frac{d |A|}{dx},$ where these relationships are in effect because the shoaling term coefficient can be absorbed in the relation $\\mu \\sim d v_g / dx$ .", "Eq.", "REF implies that waves entering in a shallower region experience a decreasing amplitude and wavelength, while waves expanding over deeper regions experience amplitude and wavelength growth.", "This effect is clearly visible in our results, see for example Figs.", "REF -REF , REF .", "In Fig.", "REF we present the BFI for the three different wave steepness vs. space.", "Where the water depth is larger (gauges $3\\div 8$ and $21\\div 28$ ) BFI has larger values, and this value increases with the steepness as we can see from the red and blue curves spikes at gauges $7, 22, 27$ .", "For example, this effect is quite visible over gauges $23\\div 28$ where BFI increases monotonically with water depth, and again over gauges $27\\div 30$ where BFI decreases monotonically with decreasing of the water depth.", "Over regions with shallower water depth ($kh$ is closer to the MI threshold) the BFI decreases no matter of the steepness (see black, red and blue curves over gauges $11\\div 20$ in Fig.", "REF ).", "However, the dynamic response of the waves depends on a combination of water depth (gray curve with circles), bottom slope (green thick curve) and wave steepness (in order of its increasing the upper curves: black, red, blue).", "At a sudden drop in the water depth, higher steepness tends to reveal a higher BFI, hence steeper waves are more likely to build RW after shoals and islands (gauge 21).", "Over regions where water becomes permanently shallower (gauges $30\\div 40$ ) the relaxation distance for decreasing and stabilizing of the BFI, kurtosis and skewness depends on the wave steepness.", "While at $H_s =3.22$ cm the BFI variation is almost monotonically correlated to the water depth variation, for larger waves with $H_s =5.2 \\div 6.2$ cm the BFI spikes back to larger values, and is not stabilized for a length of about $8\\div 10$ m $\\gg \\lambda _{p}$ as mentioned in [26], too.", "We also noticed that for small values of the bottom slope in absolute value on the shallow side of the slope, kurtosis and skewness can stabilize almost at the same location as the change of depth.", "Large local values of the absolute value of bottom slope (like fast drops or steep increases of the bottom represented by the spikes of the green curve in Fig.", "REF over gauges $20-21$ or $31-32$ ) induce spikes in the BFI and this effect is stronger for larger wave steepness, and less prominent for smoother waves like the case $H_s =3.22$ cm.", "This effect can be correlated with the observation of similar spikes in kurtosis and skewness at the same locations, Fig.", "REF , and these observations are in agreement with the experiments in [22] and numerical evaluations in [57].", "The increase of skewness and kurtosis over shallowing regions, especially in the transition zone, was also correlated with deviations of the wave states from the Gaussian distribution and the increase of probability of RW occurrence.", "These changes in the statistics parameters of the wave field over transition zone depend on the wave steepness (and consequently on the Ursell number and $H_s$ ), but not necessary on the length of the transitional zone, as we noticed the occurrence of localized spikes at the beginning of any high bottom slope region which do not necessarily continue along the shallower region.", "The results obtained confirm the conclusion made [28], [22], [21] in the framework of the nonlinear Schrodinger equation for narrow-banded wind wave field, that kurtosis and the number of freak waves may significantly differ from the values expected for a flat bottom of a given depth.", "While the wave propagate over the uphill slope, from deeper to shallower water it becomes evidence from Figs.", "REF ,REF that as long as the shallower side of the slope is sufficiently shallow, and slope length is small enough, we observe local maxima (spikes) of kurtosis, skewness and BFI.", "These localized maxima are placed at the shallower end of the slopes in agreement with the results from [22].", "In our experiments the bottom mimics a realistic ocean floor, and the regions with almost constant water depth are not very long, so we do not observe the asymptotic stabilizing of kurtosis and skewness.", "We fit the traveling coherent wave packages obtained din our experiments, see for example the red stripes in Figs.", "REF ,REF ,REF , or the wave packages easily visible in Figs.", "REF ,REF , with all the above solutions trying to identify which one describes the best our results.", "In Figs.", "REF we fit the earliest coherent package formed in small steepness waves with KM solitons.", "In experiment this group travels as a doublet of stable localized waves, and it is not obvious if this is a bound group of two independent KM solitons, or it is one AB double-breather (higher order Peregrine breather).", "All theoretical breathers presented Figs.", "REF have the same set of parameters, except being translated in space and time accordingly to the gauge position and chosen interval of time.", "It is very interesting that the match keeps being good enough while the group travels over variable bottom, over a shoal and the deep valley following, and even up the beach when the waves increase in amplitude and become pretty sharp (see the 8th frame for example) and ready to break.", "In Figs.", "REF we do not show the theory but instead present an overlap of 7 instants of the same wave group, shifted in time correspondingly.", "The 8th frame shows an obvious match of the same type of behavior for this coherent traveling group, and the likeliness to a breather, possibly a higher-order breather We also present the match of the stable traveling doublet with two KM solitons bound together, Fig.", "REF left, as compared to a best fit with a single KM soliton, presented in the right frame.", "In Figs.", "REF we fit the experiment with Peregrine breathers (red curves) solitons, and for comparison, the same experimental instants were fitted with KM solitons (blue curves).", "In Figs.", "REF , REF we present comparison with double AB breathers, Eq.", "REF .", "This modeling represents the best match, so we believe that the stable, oscillating and traveling doublets are actually higher order AB submerged in a random wave background.", "There also a possibility to explain these oscillating and breathing doublets as Satsuma-Yajima solitons and the supercontinuum generation effect [18], [17].", "Same qualitative results, and the same percentage of matching are obtained for the other two experiments, of higher steepness, but we do not present them here in detail, in order to keep a reasonable length for the paper.", "In Figs.", "REF , REF we present density plots of the wave heights, in space-time frames, for the steeper waves.", "These plots show constant group velocity traveling breathers over the shoal and deep valleys.", "In our experiments the mean value of steepness is $0.0765\\pm 4\\%$ , and the theoretical one obtained from the match of experiments with the same KM or peregrine breather results $0.07803$ showing a good match between experiments and theory.", "The match was made between the analytic form of the KM breather and the experiment for the gauges $\\#4, 10, 12, 18, 19, 20, 23, 25, 28, 30, 34$ .", "Since ocean waves are usually characterized by an average steepness of about $\\epsilon \\sim 0.1$ corresponding to the peak frequency of the spectrum, both the experimental and theoretical match are plausible.", "From measuring of the time interval when this structure arrives at various gauges we obtain a group velocity for the breather of $V_g=0.81$ m/s.", "The theory predicts the occurrence of maximum heights of these breather in the range $A_{max}/A_{0} \\sim 3.92$ which is in good agreement with our experimental values of $3.41$ ." ], [ "Conclusions", "In this paper, we present experimental results describing the dynamics of a random background of deep uni-directional, long crested, water waves over a non-uniform bathymetry consisting in a shoal and several deeper valleys, as well as a final run-up beach.", "Experiments were performed with waves initially generated with a JONSWAP spectrum, keeping the same carrier (central) frequency, but for three different wave significant heights, involving three different wave steepness.", "The experimental results confirm the formation of very stable, coherent localized wave packages which travel with almost uniform group velocity across the variable manifolds of the bottom.", "By using well established statistical tools, and by matching experiments with some of the exact solutions of the NLS equation, we proved that these coherent wave packages coming out of the random background are actually deep water breathers/solitons solutions (mainly Kuznetsov-Ma, Akhmediev, higher order AB and Peregrine breathers/solitons types), and we put into evidence the formation of rogue waves around those regions where the BFI, kurtosis and skewness predict their formation by taking larger values.", "The evolution and distribution of the statistical parameters, i.e.", "space and time variation of kurtosis, skewness and BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights, are interpreted in terms of the balance of the terms in a generalized NLS equation for non-uniform bathymetry, having variable coefficients and a shoaling extra term." ], [ "Acknowledgements", "Parts of this work were supported by the Natural Science Foundation of China under grants NSFC-51639003 and NSFC-51679037.", "One of the authors (AL) is grateful to Dalian University of Technology for hospitality during the accomplishment of this research project in 2018-2019.", "The present work is also supported by National Key Research and Development Program of China (2019 YFC0312400) and (2017 YFE0132000).", "National Natural Science Foundation of China (51975032) and (51939003).", "The State Key Laboratory of Structural Analysis for Industrial Equipment (S18408)." ] ]

[ [ "Use and Adaptation of Open Source Software for Capacity Building to\n Strengthen Health Research in Low- and Middle-Income Countries" ], [ "Abstract Health research capacity strengthening is of importance to reach health goals.", "The ARCADE projects' aim was to strengthen health research across Africa and Asia using innovative educational technologies.", "In the four years of the EU funded projects, challenges also of technical nature were identified.", "This article reports on a study conducted within the ARCADE projects.", "The study focused on addressing challenges of video conferencing in resource constrained settings and was conducted using action research.", "As a result, a plugin for the open source video conferencing system minisip was implemented and evaluated.", "The study showed that both the audio and video streams could be improved by the introduced plugin, which addressed one technical challenge." ], [ "Introduction", "Efforts to strengthen health research capacity in low- and middle-income countries are needed[1].", "The ARCADE HSSR and RSDH (African/Asian Research Capacity Development for Health Systems and Services Research/Social Determinants of Health) projects were two European Union-funded projects implemented from 2011 to 2015 and coordinated by the Division of Global Health at Karolinska Institutet in Stockholm.", "The projects aimed to strengthen health research across Africa and Asia by using innovational educational technologies[2].", "The ARCADE projects can be divided into the following four interlinked components[3].", "The first component of the ARCADE projects was the development and delivery of online courses on global health topics.", "E-learning can be divided into synchronous and asynchronous education or e-learning.", "The latter one, asynchronous education, provides education from the teacher to the student even if both are not online at the same time.", "Thus, in simple terms asynchronous education uses technologies and tools like e-mail, panels or e-learning platforms like Moodle.", "The strengths of asynchronous education lies within its flexibility on both time and place and its low requirements on the bandwidth, which is an important point for resource constrained settings[4].", "On the other hand, asynchronous e-learning has also weaknesses, such as no possibility for immediate feedback, discussions on boards will last much longer on complex topics compared to face-to-face discussions and the lack of social interaction which may result in students not feeling connected to each other.", "Moreover, this method requires a severe discipline from the students as they need to manage their time of learning on their own[5].", "To overcome the disadvantages of e-learning and to take into account bandwidth challenges, the ARCADE projects used mostly blended learning which is a combination of synchronous (face-to-face online or in person) and asynchronous (e-learning) methods.", "The ARCADE project used different open source software to prepare, organise and deliver e-learning courses.", "Almost 55 courses were developed and delivered to over 920 postgraduate students in Africa, Asia and Europe using e-learning principles and specifically blended learning[6].", "The open source e-learning platform Moodlehttps://moodle.com was the main entry point for the students.", "Synchronous distance education was delivered with the help of minisiphttps://github.com/csd/minisip, which is an open source Session Initiation Protocol (SIP) implementation developed by ARCADE's project partner Royal Institute of Technology Stockholm[7].", "However, as this partnership weakened over time, alternatives to minisip were evaluated and used.", "Content management was implemented with the help of Alfresco Community Editionhttps://www.alfresco.com/de/products/enterprise-content-management/community, an Enterprise Content Management platform.", "Dissemination and publication of research findings were presented online at the self-hosted project's web site (www.arcade-project.org) which is powered by Wordpresshttps://wordpress.org/.", "Färnman et.", "al.", "investigated the challenges which the ARCADE project team came across during the four years of project runtime by interviewing 16 participants from 12 partner institutions.", "The main challenges for e-learning included problems in technology, availability of skilled technical staff across implementation sites and attracting students' interest in courses.", "The report also points out the high demand on bandwidth and software deficiencies in resource restrained settings[3].", "This is specifically true for synchronous distance education.", "The limitation of the bandwidth, poor image and video quality as well as connectivity issues are challenges when video teleconferencing systems are used[8].", "In this article we will demonstrate how open source software can be used to overcome bandwidth limitations." ], [ "Methods", "The aim of this study was to evaluate how Open Source Software (OSS) can be used and optimised for for distance education, particularly in the area of health research/education in a global setting.", "As the study took place within the ARCADE project which used the minisip software, the underlying study question can be stated as followed.", "How can OSS such as minisip improve the delivery of synchronous health education in resource-restricted environments?", "The design of the study was defined by an action research framework[9].", "Both qualitative and quantitative data were used, hence we followed a mixed research approach.", "This study can be divided in the different phases of action research.", "Observe A questionnaire was conducted with the aim to explore the ARCADE RSDH's teaching activities and its priorities to the system and quantitative data was collected at St. John's Research Institute in India.", "Reflect The data from the questionnaire and measurement were analyzed and possible improvements were identified.", "Act Based on the reflections, an improvement was chosen for implementation.", "Evaluate The implemented improvement was tested and evaluated at Wuhan's Tongji Medical College.", "The questionnaire consisted of one multiple choice question and three free-text questions and was answered by ARCADE partners from Sweden (Karolinska Institutet) and India (St. John's Research Institute).", "A quantitative measurement of the Minisip performance using Wireshark was done at St. John's Research Institute in Bangalore.", "The Minisip performance for the ARCADE RSDH project was evaluated and improvements were suggested, implemented and tested.", "The study took place at ARCADE partners in Sweden (Karolinska Institutet), India (St. John's Research Institute Bangalore) and China (Wuhan's Tongji Medical College).", "During the study, the focus of the terms \"adaptability\" and \"performance\" were narrowed down based on the outcome of the questionnaire.", "Data were analysed using the statistical environment of Rhttps://www.r-project.org/.", "The R package RQDAhttp://rqda.r-forge.r-project.org was used to organise and analyse qualitative data.", "Captured traffic from minisip was recorded and filtered by using Wiresharkhttps://www.wireshark.org/.", "Wireshark offers detailed analyse function for SIP calls and Real-Time Transport Protocol (RTP) streams, including package loss and jitter." ], [ "Results", "In the first step (Phase \"Observe\"), the background, aims and requirement of the underlying project was investigated by a questionnaire.", "A quantitative analysis of minisip at St. John's Research Institute in Bangalore showed that one challenge was the rather high Packet Delay Variation (PDV) or so-called \"jitter\".", "Based on the questionnaire in the first step and the measurement outcome (Phase \"Reflect\"), we decided to implement a plugin for minisip with the aim to minimise the PDV.", "Minisip offers an interface to insert an extension at different stages of the traffic flow.", "The implemented extensions were inserted after the RTP pipeline at the project partner at Wuhan's Tongji Medical College, China (Phase \"Act\").", "Two different algorithms were implemented to improve the traffic flow.", "As the audio packets all had the same size, a simple leaky bucket algorithm was sufficient.", "However, the video packets were of different size and therefore the byte-based token bucket algorithm was chosen[10].", "Figure: Measurements from the leaky bucket plugin for the audio stream (Wuhan, China)The final phase of the action research was to evaluate the outcome of the implementation.", "Figure 1 shows the output of the leaky bucket extension for incoming audio.", "The first diagram visualises the incoming packets, each dot represents one incoming packet (each 125 Bytes) before the extension is shaping the traffic.", "The second diagram shows the manipulated traffic, again one dot represents one packet.", "Finally, the third diagram represents the current content of the bucket with a maximum capacity of 15 packets.", "Figure: Measurements from the leaky bucket plugin for the video (H.264) stream (Wuhan, China)Figure 2 illustrates the effect of the token bucket algorithm.", "The diagram \"incoming traffic\" visualises the incoming video packets over time and the diagram \"shaped traffic\" shows the resulted traffic after the token bucket extension.", "One dot represents one packet and the vertical axes shows the packet size.", "Two important parameters of the extension are also visualised: The packet queue shows the number of bytes (not packets) waiting for transmission and the last diagram represents the token available for traffic.", "One token is equal to one Byte in the used configuration.", "As shown in Figure 1 and Figure 2, the PDV was successfully minimised for both audio and video streams." ], [ "Discussion", "The study demonstrated with a specific example, how open source software can be adapted for capacity building in low and middle income countries based on a scientific background.", "However, to address the challenges which the ARCADE project team encountered during their four year project runtime, different strategies should be used.", "For example, investment in IT infrastructure and educating technical staff would be two possible strategies.", "During this study we also faced a typical risk when using a open source software that is not backed by a strong community.", "The main developer of minisip stopped his contribution and since then minisip is not further developed.", "The initial strength, namely the close partnership with the developer and thereby the possibility to receive an adapted software solution for our settings, turned out to be also a high risk factor.", "Nevertheless, we believe that the use of open source software has high potential and advantages, especially in resource restrained settings.", "Within a limited time-frame, the project team successfully introduced an improvement for a specific problem based on measurements and interviews.", "This was made possible through an open source software that had a clean codebase and extensive documentation.", "Therefore external developers and contributors were enabled to introduce changes and improvements.", "As part of a future study, the plugins could also be integrated and tested in other SIP implementations.", "Finally, the implemented plugins could be further improved by automatically detecting the algorithms' parameters." ] ]

[ [ "A characterization of graphs with regular distance-$2$ graphs" ], [ "Abstract For non-negative integers~$k$, we consider graphs in which every vertex has exactly $k$ vertices at distance~$2$, i.e., graphs whose distance-$2$ graphs are $k$-regular.", "We call such graphs $k$-metamour-regular motivated by the terminology in polyamory.", "While constructing $k$-metamour-regular graphs is relatively easy -- we provide a generic construction for arbitrary~$k$ -- finding all such graphs is much more challenging.", "We show that only $k$-metamour-regular graphs with a certain property cannot be built with this construction.", "Moreover, we derive a complete characterization of $k$-metamour-regular graphs for each $k=0$, $k=1$ and $k=2$.", "In particular, a connected graph with~$n$ vertices is $2$-metamour-regular if and only if $n\\ge5$ and the graph is a join of complements of cycles (equivalently every vertex has degree~$n-3$), a cycle, or one of $17$ exceptional graphs with $n\\le8$.", "Moreover, a characterization of graphs in which every vertex has at most one metamour is acquired.", "Each characterization is accompanied by an investigation of the corresponding counting sequence of unlabeled graphs." ], [ "Introduction", "Our starting point is a group of persons.", "Any two persons of this group might be in an intimate relationship or not.", "We can depict this situation by drawing a dot for each person and joining two dots by a line if they are in an intimate relationship with each other.", "In discrete mathematics, such a representation of objects and some relations between them is called a graph.", "Commonly, we call the objects or dots the vertices of the graph and the relations or lines between vertices the edges of the graph.", "If all the relationships are monogamous, then every person is in at most one intimate relationship.", "Reformulated in the language of graph theory, this means that every vertex has an edge to at most one other vertex.", "The situation changes in polyamory:[1] Hyde and DeLamater [17] describe polyamory as “the non-possessive, honest, responsible, and ethical philosophy and practice of loving multiple people simultaneously”.", "Other descriptions of polyamory are around; see for example Haritaworn, Lin and Klesse [16] or Sheff [23].", "The word polyamory appeared in an article by Zell-Ravenheart [31] in 1990 and is itself a combination of Greek greekπολύ poly, “many, several” and of Latin amor, “love”.", "$^{\\ref {footsaferefiwontuse0}}$ Every person might be in an intimate relationship with any number of other persons.", "Several example graphs representing polyamorous living persons and their relationships are shown in Figure REF .", "Figure: Nine polycules ^{\\ref {footnote:polycule}} of some polyamorous living persons.A dot vertex] at (0,0) ; represents a person,a line vertex] (v0) at (0,0) ;vertex] (v1) at (1,0) ;[edge] (v0) – (v1); an intimate relationship between two persons andvertex] (v0) at (0,0) ;vertex] (v1) at (1,0) ;[mmedge] (v0) – (v1); a metamour relation.", "[2] A polycule is all of the people linked through their intimate relationships; see Hardy and Easten [15] or Veaux and Rickert [28], or online at https://www.morethantwo.com/polyglossary.html#polycule.", "In graph theory, a polycule is called connected component of the graph.", "A partner of a person is someone having an intimate relationship with that person.", "The partner of one's partner with whom one does not have an intimate relationship is called metamour.", "[3]For metamour see for example Hardy and Easten [15] or Veaux and Rickert [28], or online at https://www.morethantwo.com/polyglossary.html#metamour.", "$^{\\ref {footsaferefiwontuse1}}$ These concepts can easily be transferred to graph theory: A partner of a person corresponds to a neighbor of a vertex, and we adopt and use the term metamour for two vertices that have a common neighbor but are not neighbors themselves.", "Figure REF also shows who is a metamour of which other person.", "The graph of persons and their metamour relations is called metamour graph.", "We now have the necessary vocabulary and are ready to talk about the content and results of this article.", "We investigate graphs where each vertex/person has the same number of metamours.", "If this number is $k$ , we say that the graph is $k$ -metamour-regular.", "The leftmost graph of Figure REF shows an example of a 2-metamour-regular graph with 6 persons.", "We ask: Question Can we find all $k$ -metamour-regular graphs and give a precise description of how they look like?", "Certainly not every graph satisfies this property, but some do.", "For example, for $k=2$ it is not hard to check that in graphs of at least five persons, where [label=red] every person is in an intimate relationship with exactly two other persons, i.e., the relationships of the persons form a cycle, or every person is in an intimate relationship with every other person but two, each person indeed has exactly two metamours.", "Hence, these graphs are 2-metamour-regular.", "The second construction above can be generalized, and in this article we provide a generic construction that allows to create $k$ -metamour-regular graphs for any number $k$ .", "One of our key results is that the vast majority of these graphs can indeed be built by this generic construction.", "To be more precise, only $k$ -metamour-regular graphs whose metamour graph consists of at most two connected components cannot necessarily be constructed this way.", "This key result lays the foundations for another main result of this article, namely the identification of all 2-metamour-regular graphs, so we answer the question above for $k=2$ .", "Our findings are as follows: Every 2-metamour-regular graph of any number of persons falls either into one of the two groups marked with red above or into the third group of [label=red] 17 exceptional graphs with at least six and at most eight persons.", "We provide a systematic and explicit description of the graphs in the first two groups.", "All 2-metamour-regular graphs with at most nine persons—this includes the 17 exceptional graphs of the third group—are shown in Figures REF to REF .", "Summarized, we present a complete description of all 2-metamour-regular graphs.", "Note that as a consequence of our characterization, exceptional graphs exist only for up to eight persons.", "In addition to the above result we derive several structural properties of $k$ -metamour-regular graphs for any number $k$ .", "We also characterize all graphs in which every person has no metamour ($k=0$ ), exactly one metamour ($k=1$ ), and at most one metamour.", "As a byproduct of every characterization including the one for $k=2$ , we are able to count the number of graphs with these properties.", "Moreover—and this might be the one sentence take-away message of this article—our findings imply that besides the graphs that are simple to discover (i.e., can be built by the generic construction), only a few (if any) small exceptional graphs are 0-metamour-regular, 1-metamour-regular, graphs where every vertex has most one metamour and 2-metamour-regular.", "We now provide a short overview on the structure of this paper.", "The terms discussed so far are formally defined in Section .", "Moreover, we introduce joins of graphs which are used in the systematic and explicit description of the graphs in the characterizations of metamour-regular graphs.", "The section also includes some basic properties related to those concepts.", "Section  is a collection of all results derived in this article.", "This is accompanied by plenty of consequences of these results and discussions.", "The proofs are then given in Sections  to .", "We conclude in Section  and provide many questions, challenges and open problems for future work.", "So far not mentioned is the next section.", "There, literature related to this article is discussed." ], [ "Related literature", "In this section we discuss concepts that have already been examined in literature and that are related to metamours (i.e., vertices having distance 2, or “neighbors of neighbors”) and the metamour graphs induced by their relations.", "Metamour graphs are called 2nd distance graphs in Simić [24], and some authors also call them 2-distance graphs or distance-2 graphs.", "This notion also appears in the book by Brouwer, Cohen and Neumaier [9].", "The overall question is to characterize all graphs whose $n$ th distance graph equals some graph of a given graph class.", "Simić [24] answers the question when the $n$ th distance graph of a graph equals the line graph of this graph.", "Characterizing when the 2nd distance graph is a path or a cycle is done by Azimi and Farrokhi [3], and when it is a union of short paths or and a union of two complete graphs by Ching and Garces [12].", "Azimi and Farrokhi [4] also tackled the question when the 2nd distance graph of a graph equals the graph itself.", "This question is also topic of the online discussion [27].", "Bringing the context to our article, we investigate the above question for the graph class of regular graphs.", "Moreover, vertices in a graph that have distance two, i.e., metamours, or more generally vertices that have a given specific distance, are discussed in the existing literature in many different contexts.", "The persons participating in the exchange [11] discuss algorithms for efficiently finding vertices having specific distance on trees.", "Moreover, the notion of dominating sets is extended to vertices at specific distances in Zelinka [30] and in particular to distance two in Kiser and Haynes [19].", "Also various kinds of colorings of graphs with respect to vertices of given distance are studied.", "Typically, the corresponding chromatic number is analyzed, for instance Bonamy, Lévêque and Pinlou [5], Borodin, Ivanova and Neustroeva [7], and Bu and Wang [10] provide such results for vertices at distance two.", "Algorithms for finding such colorings are also investigated.", "We mention here Bozdağ, Çatalyürek, Gebremedhin, Manne, Boman and Özgüner [8] as an example.", "Kamga, Wang, Wang and Chen [18] study variants of so-called vertex distinguishing colorings, i.e., edge colorings where additionally vertices at distance two have distinct sets of colors.", "Their motivation comes from network problems.", "The concept is studied more generally but for more specific graph classes in Zhang, Li, Chen, Cheng and Yao [32].", "Many of the mentioned results also investigate vertices at distance at most two (compared to exactly two).", "This is closely related to the concept of the square of a graph, i.e., graphs with the same vertex set as the original graph and two vertices are adjacent if they have distance at most two in the original graph.", "More generally, this concept is known as powers of graphs; see Bondy and Murty [6].", "The overall question to characterize all graphs whose $n$ th distance graph equals some graph of a given graph class is studied also for powers of graphs instead of $n$ th distance graphs; see Akiyama, Kaneko and Simić [1].", "Colorings are studied for powers of graphs by a motivation coming, among others, from wireless communication networks or graph drawings.", "The corresponding chromatic number is analyzed, for example, in Kramer and Kramer [20], Alon and Mohar [2] and Molloy and Salavatipour [21].", "Results on the hamiltonicity of powers of graphs are studied in Bondy and Murty [6] and Underground [26].", "Finally, there are distance-regular graphs.", "Even though the name might suggest that these graphs are closely related to metamour graphs, this is not the case: A graph is distance-regular if it is regular and for any two vertices $v$ and $w$ , the number of vertices at distance $j$ from $v$ and at distance $k$ from $w$ depend only upon $j$ , $k$ and the distance of $u$ and $v$ .", "The book by Brouwer, Cohen and Neumaier [9] is a good starting point for this whole research area.", "Plenty of publications related to distance-regular graphs are available.", "This section is devoted to definitions and some simple properties.", "Moreover, we state (graph-theoretic) conventions and set up the necessary notation that will be used in this article.", "The proofs of the properties of this section are postponed to Section ." ], [ "Graph-theoretic definitions, notation & conventions", "In this graph-theoretic article we use standard graph-theoretic definitions and notation; see for example Diestel [13].", "We use the convention that all graphs in this article contain at least one vertex, i.e., we do not talk about the empty graph.", "Moreover, we use the following convention for the sake of convenience.", "Convention 2.1 If two graphs are isomorphic, we will call them equal and use the equality-sign.", "In many places it is convenient to extend adjacency to subsets of vertices and subgraphs.", "We give the following definition that is used heavily in Sections REF and .", "Definition 2.2 Let $G$ be a graph, and let $W_1$ and $W_2$ be disjoint subsets of the vertices of $G$ .", "We say that $W_1$ is adjacent in $G$ to $W_2$ if there is a vertex $v_1 \\in W_1$ adjacent in $G$ to some vertex $v_2 \\in W_2$ .", "We say that $W_1$ is completely adjacent in $G$ to $W_2$ if every vertex $v_1 \\in W_1$ is adjacent in $G$ to every vertex $v_2 \\in W_2$ .", "By identifying a vertex $v \\in V(G)$ with the subset ${v} \\subseteq V(G)$ , we may also use (complete) adjacency between $v$ and a subset of $V(G)$ .", "Moreover, for simplicity, whenever we say that subgraphs of $G$ are (completely) adjacent, we mean that the underlying vertex sets are (completely) adjacent.", "We explicitly state the negation of adjacent: We say that $W_1$ is not adjacent in $G$ to $W_2$ if no vertex $v_1 \\in W_1$ is adjacent in $G$ to any vertex $v_2 \\in W_2$ .", "We will not need the negation of completely adjacent.", "We recall the following standard concepts and terminology to fix their notation.", "Definition 2.3 Let $G$ be a graph.", "For a set $W \\subseteq V(G)$ of vertices the induced subgraph $G[W]$ is the subgraph of $G$ with vertices $W$ and all edges of $G$ that are subsets of $W$ , i.e., edges incident only to vertices of $W$ .", "A set $\\mu \\subseteq E(G)$ of edges is called matching if no vertex of $G$ is incident to more than one edge in $\\mu $ .", "In particular, the empty set is a matching.", "The set $\\mu $ is called perfect matching if every vertex of $G$ is incident to exactly one edge in $\\mu $ .", "For a set $\\nu \\subseteq E(G)$ of edges of $G$ , we denote by $G - \\nu $ the graph with vertices $V(G-\\nu ) = V(G)$ and edges $E(G-\\nu ) = E(G) \\setminus \\nu $ .", "There are a couple of different ways to define unions of graphs.", "In this article, we use the following definition.", "Definition 2.4 The union of graphs $G_1$ and $G_2$ , written as $G_1 \\cup G_2$ , is the graph with vertex set $V(G_1) \\cup V(G_2)$ and edge set $E(G_1) \\cup E(G_2)$ .", "In Section REF we will define another binary operation of graphs, namely the join of graphs.", "Compared to the union here, additional edges are added when joining graphs.", "The following notation is used for paths and cycles in graphs.", "Notation 2.5 Let $G$ be a graph.", "A path $\\pi $ of length $n$ in the graph $G$ is written as $\\pi =(v_1,\\dots ,v_n)$ for vertices $v_i \\in V(G)$ that are pairwise distinct.", "A cycle $\\gamma $ of length $n$ or $n$ -cycle $\\gamma $ in the graph $G$ is written as $\\gamma =(v_1,\\dots ,v_n,v_1)$ for vertices $v_i \\in V(G)$ that are pairwise distinct.", "We frequently speak about the connected components of a graph.", "The following definition provides the operator for getting these components of a graph.", "Definition 2.6 For a graph $G$ , we write $\\mathcal {C}(G)$ for the set of connected components of $G$ .", "Note that each element of $\\mathcal {C}(G)$ is a subgraph of $G$ .", "Complements of graphs are denoted in various ways in the literature; we use the following.", "Notation 2.7 We write $\\complement {G}$ for the complement of a graph $G$ , i.e., the graph with the same vertices as $G$ but with an edge between vertices exactly where $G$ has no edge.", "Finally, we recall the following particular graphs and their standard notation.", "Notation 2.8 We use the complete graph $K_n$ for $n\\ge 1$ , the complete $t$ -partite graph $K_{n_1,\\dots ,n_t}$ for $n_i\\ge 1$ , $i\\in {1,\\dots ,t}$ , the path graph $P_n$ for $n \\ge 1$ , and the cycle graph $C_n$ for $n\\ge 3$ .", "Remark 2.9 We will frequently use the complement $\\complement {C_n}$ of the cycle graph $C_n$ for $n\\ge 3$ .", "Note that $\\complement {C_3}$ is the graph with 3 isolated vertices, $\\complement {C_4}$ is the graph with 2 disjoint single edges, and $\\complement {C_5}$ equals $C_5$ (see also Figure REF ).", "We close this section and continue with definitions and concepts that are more specific for this article." ], [ "Metamours", "We now formally define the most fundamental concept of this article, namely metamours.", "Definition 2.10 Let $G$ be a graph.", "A vertex $v$ of the graph $G$ is a metamour of a vertex $w$ of $G$ if the distance of $v$ and $w$ on the graph $G$ equals 2.", "The metamour graph $M$ of $G$ is the graph with the same vertex set as $G$ and an edge between the vertices $v$ and $w$ of $M$ whenever $v$ is a metamour of $w$ in $G$ .", "We can slightly reformulate the definition of metamours: A vertex $v$ having a different vertex $w$ as metamour, i.e., having distance 2 on a graph, is equivalent to saying that $v$ and $w$ are not adjacent and there is a vertex $u$ such that both $v$ and $w$ have an edge incident to this vertex $u$ , i.e., $u$ is a common neighbor of $v$ and $w$ .", "Clearly, there is no edge in a graph between two vertices that are metamours of each other.", "This is reflected in the relation between the metamour graph and the complement of a graph, and put into writing as the following observation.", "Observation 2.11 Let $G$ be a graph.", "Then the metamour graph of $G$ is a subgraph of the complement of $G$ .", "The question whether the metamour graph equals the complement will appear in many statements of this article.", "The following simply equivalence is useful.", "Proposition 2.12 Let $G$ be a connected graph with $n$ vertices.", "Then the following statements are equivalent: The metamour graph of $G$ equals $\\complement {G}$ .", "The graph $G$ has diameter 2 or $G=K_n$ ." ], [ "Metamour-degree & metamour-regularity", "Having the concept of metamours, it is natural to investigate the number of metamours of a vertex.", "We formally define this “degree” and related concepts below.", "Definition 2.13 Let $G$ be a graph.", "The metamour-degree of a vertex of $G$ is the number of metamours of this vertex.", "The maximum metamour-degree of the graph $G$ is the maximum over the metamour-degrees of its vertices.", "For $k\\ge 0$ the graph $G$ is called $k$ -metamour-regular if every vertex of $G$ has metamour-degree $k$ , i.e., has exactly $k$ metamours.", "We finally have $k$ -metamour-regularity at hand and can now start to relate it to other existing terms.", "We begin with the following two observations.", "Observation 2.14 Let $k \\ge 0$ , and let $G$ be a graph and $M$ its metamour graph.", "Then $G$ is $k$ -metamour-regular if and only if the metamour graph $M$ is $k$ -regular.", "The number of vertices with odd degree is even by the handshaking lemma.", "Therefore, we get the following observation.", "Observation 2.15 Let $k\\ge 1$ be odd.", "Then the number of vertices of a $k$ -metamour-regular graph is even.", "Proposition 2.16 Let $G$ be a connected graph with $n$  vertices.", "Then the following statements are equivalent: The metamour graph of $G$ equals $\\complement {G}$ .", "For every vertex of $G$ , the sum of its degree and its metamour-degree equals $n-1$ .", "Note that if $k\\ge 0$ and $G$ is a connected $k$ -metamour-regular graph with $n$  vertices, then REF states that the graph $G$ is $(n-1-k)$ -regular.", "We use this in Proposition REF ." ], [ "Joins of graphs", "Given two graphs, we already have defined the union of these graphs in Definition REF .", "A join of graphs is a variant of that.", "We will introduce this concept now, see also Harary [14], and then discuss a couple of simple properties of joins, also in conjunction with metamour graphs.", "Definition 2.17 Let $G_1$ and $G_2$ be graphs with disjoint vertex sets $V(G_1)$ and $V(G_2)$ .", "The join of $G_1$ and $G_2$ is the graph denoted by $G_1 \\mathbin \\nabla G_2$ with vertices $V(G_1) \\cup V(G_2)$ and edges $E(G_1) \\cup E(G_2) \\cup [\\big ]{{g_1}{g_2}}{\\text{$g_1 \\in V(G_1)$ and $g_2 \\in V(G_2)$}}.$ Some graphs in Figures REF to REF are joins of complements of cycle graphs.", "All of the joins of graphs in this paper are “disjoint joins”.", "We use the convention that if the vertex sets $V(G_1)$ and $V(G_2)$ are not disjoint, then we make them disjoint before the join.", "We point out that the operator $\\mathbin \\nabla $ is associative and commutative.", "Let us get to know joins of graphs in form of a supplement to Remark REF .", "We have $K_{3,3,\\dots ,3} = \\complement {C_3} \\mathbin \\nabla \\complement {C_3} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {C_3}$ for the complete multipartite graph $K_{3,3,\\dots ,3}$ .", "There are connections between joins of graphs and metamour graphs that will appear frequently in the statements and results of this article.", "We now present first such relations.", "Proposition 2.18 Let $G$ be a connected graph and $M$ its metamour graph.", "Then the following statements are equivalent: The metamour graph $M$ equals $\\complement {G}$ and ${\\mathcal {C}(M)}\\ge 2$ .", "The graph $G$ equals $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with ${M_1,\\dots ,M_t}=\\mathcal {C}(M)$ and $t\\ge 2$ .", "There are graphs $G_1$ and $G_2$ with $G = G_1 \\mathbin \\nabla G_2$ .", "Proposition 2.19 Let $k\\ge 0$ and $G$ be a connected $k$ -metamour-regular graph with $n$  vertices.", "Let $M$ be the metamour graph of $G$ .", "Then the following statements are equivalent: The metamour graph $M$ equals $\\complement {G}$ .", "The graph $G$ has diameter 2 or $G=K_n$ .", "The graph $G$ equals $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with ${M_1,\\dots ,M_t}=\\mathcal {C}(M)$ .", "The graph $G$ is $(n-1-k)$ -regular.", "Note that we have $G=K_n$ in REF if and only if $k=0$ ." ], [ "Characterizations & properties of metamour-regular graphs", "It is now time to present the main results of this article and their implications.", "In this section, we will do this in a formal manner using the terminology introduced in Section .", "The proofs of the results follow later, from Section  on to Section .", "Proof-wise the results on $k$ -metamour-regular graphs for $k\\in {0,1,2}$ build upon the result for arbitrary $k\\ge 0$ ; this determines the order of these sections.", "We will now, however, start with $k=0$ and only deal with general $k$ later on." ], [ "0-metamour-regular graphs", "As a warm-up, we start with graphs in which no vertex has a metamour.", "The following theorem is not very surprising; the only graphs satisfying this property are complete graphs.", "Theorem 3.1 Let $G$ be a connected graph with $n$ vertices.", "Then $G$ is 0-metamour-regular if and only if $G = K_n$ .", "An alternative point of view is that of the metamour graph.", "The theorem simply implies that in the case of 0-metamour-regularity, the metamour graph is empty and also equals the complement of the graph itself.", "The latter property will occur frequently later on which also motivates its formulation in the following corollary.", "Corollary 3.2 A connected graph is 0-metamour-regular if and only if its complement equals its metamour graph and this graph has no edges.", "The characterization provided by Theorem REF makes it also easy to count how many different 0-metamour-regular graphs there are and leads to the following corollary.", "Corollary 3.3 The number $m_{=0}(n)$ of unlabeled connected 0-metamour-regular graphs with $n$ vertices is $m_{=0}(n)=1.$ The Euler transform, see Sloane and Plouffe [25], of this sequence gives the numbers $m_{=0}^{\\prime }(n)$ of unlabeled but not necessarily connected 0-metamour-regular graphs with $n$ vertices.", "The number $m_{=0}^{\\prime }(n)$ equals the partition function $p(n)$ , i.e., the number of integer partitions[4]An integer partition of a positive integer $n$ is a way of representing $n$ as a sum of positive integers; the order of the summands is irrelevant.", "The parts of a partition are the summands.$^{\\ref {footnote:partitions}}$ of $n$ .", "The corresponding sequence starts with $\\begin{array}{c|cccccccccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10 & 11 & 12 & 13 & 14& 15\\\\\\hline m_{=0}^{\\prime }(n)& 1 & 2 & 3 & 5& 7 & 11 & 15 & 22 & 30& 42 & 56 & 77 & 101 & 135& 176\\end{array}$ and is A000041 in The On-Line Encyclopedia of Integer Sequences [22].", "This completes the properties of 0-metamour-regular graphs that we bring here.", "We will, however, see in the following sections how these properties behave in context of other graph classes." ], [ "1-metamour-regular graphs", "The next easiest case is that of graphs in which every vertex has exactly one other vertex as metamour.", "As the metamour relation is symmetric, these vertices always come in pairs.", "We write this fact down in the following proposition.", "Proposition 3.4 Let $G$ be a graph with $n$ vertices.", "Then the following statements hold: If every vertex of $G$ has at most one metamour, then the edges of the metamour graph of $G$ form a matching, i.e., the vertices of $G$ having exactly one metamour come in pairs such that the two vertices of a pair are metamours of each other.", "If $G$ is 1-metamour-regular, then $n$ is even and the edges of the metamour graph of $G$ form a perfect matching.", "By this connection of 1-metamour-regular graphs to perfect matchings, we can divine the underlying behavior.", "This leads to our main result of this section, a characterization of 1-metamour-regular graphs; see the theorem below.", "It turns out that one exceptional case, namely the graph $P_4$ (Figure REF ), occurs.", "Figure: The path graph P 4 P_4 where each vertex has exactly 1 metamourTheorem 3.5 Let $G$ be a connected graph with $n$ vertices.", "Then $G$ is 1-metamour-regular if and only if $n \\ge 4$ is even and either $G = P_4$ or $G = K_n - \\mu $ for some perfect matching $\\mu $ of $K_n$ holds.", "When excluding $G=P_4$ , then the graphs in the theorem are exactly the cocktail party graphs [29].", "Let us again view this from the angle of metamour graphs.", "As soon as we exclude the exceptional case $P_4$ , the metamour graph and the complement of a 1-metamour-regular graph coincide; see the following corollary.", "Corollary 3.6 A connected graph with $n\\ge 5$ vertices is 1-metamour-regular if and only if its complement equals its metamour graph and this graph is 1-regular.", "Note that a 1-regular graph with $n$ vertices is a graph induced by a perfect matching of $K_n$ .", "In view of Proposition REF , we can extend the two equivalent statements.", "As we have a characterization of 1-metamour-regular graphs (provided by Theorem REF ) available, we can determine the number of different graphs in this class.", "Clearly, this is strongly related to the existence of a perfect matching; details are provided below and also in Section , where proofs are given.", "Corollary 3.7 The sequence of numbers $m_{=1}(n)$ of unlabeled connected 1-metamour-regular graphs with $n$ vertices starts with $\\begin{array}{c|cccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10\\\\\\hline m_{=1}(n)& 0 & 0 & 0 & 2& 0 & 1 & 0 & 1 & 0& 1\\end{array}$ and we have $m_{=1}(n) ={\\left\\lbrace \\begin{array}{ll}0 & \\text{for odd $n$,} \\\\1 & \\text{for even $n\\ge 6$.}\\end{array}\\right.", "}$ The Euler transform, see [25], of the sequence of numbers $m_{=1}(2n)$ gives the numbers $m_{=1}^{\\prime }(2n)$ of unlabeled but not necessarily connected 1-metamour-regular graphs with $2n$ vertices.", "The sequence of these numbers starts with $\\begin{array}{c|cccccccccccccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10 & 11 & 12 & 13 & 14& 15 & 16 & 17 & 18\\\\\\hline m_{=1}^{\\prime }(2n)& 0 & 2 & 1 & 4& 3 & 8 & 7 & 15 & 15& 27 & 29 & 48 & 53 & 82& 94 & 137 & 160 & 225\\end{array}\\;.$ This sequence also counts how often a part 2 appears in all integer partitions[5]For integer partitions, see footnote $^{\\ref {footnote:partitions}}$ on page REF .$^{\\ref {footsaferefiwontuse2}}$ of $n+2$ with parts at least 2.", "The underlying bijection is formulated as the following corollary.", "Corollary 3.8 Let $n \\ge 0$ .", "Then the set of unlabeled 1-metamour-regular graphs with $2n$ vertices is in bijective correspondence to the set of partitions of $n+2$ with smallest part equal to 2 and one part 2 of each partition marked." ], [ "Graphs with maximum metamour-degree 1", "Let us now slightly relax the metamour-regularity condition and consider graphs in which every vertex of $G$ has at most one metamour.", "In view of Proposition REF , matchings play an important role again.", "Formally, the following theorem holds.", "Theorem 3.9 Let $G$ be a connected graph with $n$ vertices.", "Then then the maximum metamour-degree of $G$ is 1 if and only if either $G \\in {K_1,K_2,P_4}$ or $n \\ge 3$ and $G = K_n - \\mu $ for some matching $\\mu $ of $K_n$ holds.", "As in the sections above, the obtained characterization leads to the following equivalent statements with respect to metamour graph and complement.", "Corollary 3.10 A connected graph with $n\\ge 5$ vertices has the property that every vertex has at most one metamour if and only if its complement equals its metamour graph and this graph has maximum degree 1.", "Note that graphs with maximum degree 1 and $n$ vertices are graphs induced by a (possibly empty) matching of $K_n$ .", "In view of Proposition REF , we can extend the two equivalent statements by a third saying that $G$ has diameter 2 or $G=K_n$ .", "Counting the graphs with maximum metamour-degree 1 relies on the number of matchings; see the relevant proofs in Section  for details.", "We obtain the following corollary.", "Corollary 3.11 The sequence of numbers $m_{\\le 1}(n)$ of unlabeled connected graphs with $n$ vertices where every vertex has at most one metamour starts with $\\begin{array}{c|cccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10\\\\\\hline m_{\\le 1}(n)& 1 & 1 & 2 & 4& 3 & 4 & 4 & 5 & 5& 6\\end{array}$ and we have $m_{\\le 1}(n) = [\\big ]{\\tfrac{n}{2}}+1$ for $n\\ge 5$ .", "The Euler transform, see [25], gives the sequence of numbers $m_{\\le 1}^{\\prime }(n)$ of unlabeled but not necessarily connected graphs with maximum metamour-degree 1 and $n$ vertices which starts with $\\begin{array}{c|cccccccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10 & 11 & 12 & 13\\\\\\hline m_{\\le 1}^{\\prime }(n)& 1 & 2 & 4 & 9& 14 & 26 & 43 & 76 & 122& 203 & 322 & 523 & 814\\end{array}\\;.$" ], [ "2-metamour-regular graphs", "We now come to the most interesting graphs in this article, namely graphs in which every vertex has exactly two other vertices as metamours.", "Also in this case a characterization of the class of graphs is possible.", "We first consider Observation REF in view of 2-metamour-regularity.", "As a graph is 2-regular if and only if it is a union of cycles, the following observation is easy to verify.", "Observation 3.12 Let $G$ be a graph and $M$ its metamour graph.", "Then $G$ is 2-metamour-regular if and only if every connected component of the metamour graph $M$ is a cycle.", "We are ready to fully state the mentioned characterization formally as the theorem below.", "We comment this result and discuss implications afterwards.", "Note that Theorem REF generalizes the main result of Azimi and Farrokhi [3] which only deals with metamour graphs being connected.", "Theorem 3.13 Let $G$ be a connected graph with $n$ vertices.", "Then $G$ is 2-metamour-regular if and only if $n\\ge 5$ and one of $G = \\complement {C_{n_1}} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {C_{n_t}}$ with $n=n_1+\\dots +n_t$ for some $t\\ge 1$ and $n_i \\ge 3$ for all $i\\in {1,\\dots ,t}$ , $G = C_n$ , or $\\begin{aligned}[t]G \\in \\bigl \\lbrace \\!", "&H_{4,4}^{a},H_{4,4}^{b},H_{4,4}^{c},\\\\ &H_{7}^{a},H_{7}^{b},H_{4,3}^{a},H_{4,3}^{b},H_{4,3}^{c},H_{4,3}^{d},\\\\ &H_{6}^{a},H_{6}^{b},H_{6}^{c},H_{3,3}^{a},H_{3,3}^{b},H_{3,3}^{c},H_{3,3}^{d},H_{3,3}^{e}\\bigr \\rbrace \\end{aligned}$ with graphs defined by Figures REF , REF and REF holds.", "Figure: The only graph of order 5(+ one differently drawn copy)where each vertex has exactly 2 metamoursFigure: All 11 graphs (+ 2 differently drawn copies)of order 6 where each vertex has exactly 2 metamoursFigure: All 9 graphs of order 7 where each vertex has exactly 2 metamoursFigure: All 7 graphs of order 8 where each vertex has exactly 2 metamoursFigure: All 5 graphs of order 9 where each vertex has exactly 2 metamoursA representation of every 2-metamour-regular graph with at most 9 vertices can be found in Figures REF , REF , REF , REF and REF .", "For 10 vertices, all 2-metamour-regular graphs—there are 6 of them—are $C_{10}$ , $\\complement {C_{10}}$ , $\\complement {C_7} \\mathbin \\nabla \\complement {C_3}$ , $\\complement {C_6} \\mathbin \\nabla \\complement {C_4}$ , $\\complement {C_5} \\mathbin \\nabla \\complement {C_5}$ , $\\complement {C_4} \\mathbin \\nabla \\complement {C_3} \\mathbin \\nabla \\complement {C_3}$ .", "For rounding out Theorem REF , we have collected a couple of remarks and bring them now.", "Remark 3.14 The smallest possible 2-metamour-regular graph has 5 vertices, and there is exactly one with five vertices, namely $C_5$ ; see Figure REF .", "This graph is covered by Theorem  REFREF as well as REF because $C_5 = \\complement {C_5}$ .", "Theorem  REFREF can be replaced by any other equivalent statement of Proposition REF .", "For $t=1$ , Theorem  REFREF condenses to $G=\\complement {C_n}$ .", "Implicitly we get $n=n_1 \\ge 5$ .", "The graphs $C_{n_1}$ , ..., $C_{n_t}$ of Theorem  REFREF satisfy $C_{n_i} = M_i$ with ${M_1,\\dots ,M_t} = \\mathcal {C}(M)$ .", "This means that the decomposition of the graph $G = \\complement {C_{n_1}} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {C_{n_t}}$ reveals the metamour graph of $G$ and vice versa.", "For Theorem  REFREF as well as for REF with $n=5$ , every graph satisfies that its complement equals its metamour graph.", "For all other cases, this is not the case.", "A full formulation of this fact is stated as Corollary REF .", "The characterization provided by Theorem REF has many implications.", "We start with the following easy corollaries.", "Corollary 3.15 Let $G$ be a connected graph with $n\\ge 9$ vertices.", "Then $G$ is 2-metamour-regular if and only if $G$ is either $C_n$ or $\\complement {C_{n_1}} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {C_{n_t}}$ with $n=n_1+\\dots +n_t$ for some $t \\ge 1$ and $n_i \\ge 3$ for all $i\\in {1,\\dots ,t}$ .", "Corollary 3.16 Let $G$ be a connected graph with $n\\ge 9$ vertices.", "Then $G$ is 2-metamour-regular if and only if $G$ is either 2-regular or $(n-3)$ -regular.", "As before, we consider the relation of metamour graphs and complement more closely; see the following corollary.", "Again, we feel the spirit of Proposition REF .", "Corollary 3.17 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices.", "Then the following statements are equivalent: The metamour graph of $G$ is a proper subgraph of $\\complement {G}$ .", "We have either $G=C_n$ and $n\\ge 6$ (Theorem  REFREF ) or $G$ is one of the graphs in Theorem  REFREF .", "The graph $G$ has diameter larger than 2.", "Theorem REF makes it also possible to count how many different 2-metamour-regular graphs with $n$ vertices there are.", "We provide this in the following corollary.", "Corollary 3.18 The sequence of numbers $m_{=2}(n)$ of unlabeled connected 2-metamour-regular graphs with $n$ vertices starts with $\\begin{array}{c|cccccccccccccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10 & 11 & 12 & 13 & 14& 15 & 16 & 17 & 18 & 19& 20\\\\\\hline \\!m_{=2}(n)\\!& 0 & 0 & 0 & 0& 1 & 11 & 9 & 7 & 5& 6 & 7 & 10 & 11 & 14& 18 & 22 & 26 & 34 & 40& 50\\end{array}$ and for $n\\ge 9$ we have $m_{=2}(n) = p_3(n) + 1,$ where $p_3(n)$ is the number of integer partitions[6]For integer partitions, see footnote $^{\\ref {footnote:partitions}}$ on page REF .", "The function $p_3(n)$ is A008483 in [22].$^{\\ref {footsaferefiwontuse3}}$ of $n$ with parts at least 3.", "The sequence of numbers $m_{=2}(n)$ is A334275 in The On-Line Encyclopedia of Integer Sequences [22].", "We apply the Euler transform, see Sloane and Plouffe [25], on this sequence and obtain the numbers $m_{=2}^{\\prime }(n)$ of unlabeled but not necessarily connected 2-metamour-regular graphs with $n$ vertices.", "The sequence of these numbers starts with $\\begin{array}{c|ccccccccccccccccccc}n& 1 & 2 & 3 & 4& 5 & 6 & 7 & 8 & 9& 10 & 11 & 12 & 13 & 14& 15 & 16 & 17 & 18\\\\\\hline m_{=2}^{\\prime }(n)& 0 & 0 & 0 & 0& 1 & 11 & 9 & 7 & 5& 7 & 18 & 85 & 117 & 141& 143 & 179 & 277 & 667\\end{array}\\;.$" ], [ "$k$ -metamour-regular graphs", "In this section, we present results that are valid for graphs with maximum metamour-degree $k$ and $k$ -metamour-regular graphs for any non-negative number $k$ .", "We start with Proposition REF stating that the join of complements of $k$ -regular graphs is a $k$ -metamour-regular graph.", "Proposition 3.19 Let $M$ be a graph having $t\\ge 2$ connected components $M_1$ , ..., $M_t$ .", "Set $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}.$ Then $G$ has metamour-graph $M$ .", "In particular, if $M$ is $k$ -regular for some $k \\ge 0$ , then $G$ is $k$ -metamour-regular.", "We call this construction of a graph with given metamour graph generic construction.", "In particular this generic construction allows us to build $k$ -metamour-regular graphs.", "We will not investigate further options to construct $k$ -metamour-regular graphs (as, for example, with circulant graphs), as the above construction suffices for the ultimate goal of this paper to characterize all $k$ -metamour-regular graphs for $k \\le 2$ .", "Last in our discussion of Proposition REF , we note that, as we have $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ , we can also extend Proposition REF by the equivalent statements of Propositions REF and REF .", "Next we state the main structural result about graphs with maximum metamour-degree $k$ as Theorem REF .", "A consequence of this main statement is that every $k$ -metamour-regular graph is the join of complements of $k$ -regular graphs as in Proposition REF , or has in its metamour graph only one or two connected components; see Corollary REF for a full formulation.", "Theorem 3.20 Let $k \\ge 0$ .", "Let $G$ be a connected graph with maximum metamour-degree $k$ and $M$ its metamour graph.", "Then exactly one of the following statements is true: The metamour graph $M$ is connected.", "The metamour graph $M$ is not connected and the induced subgraph $G[V(M_i)]$ is connected for some $M_i \\in \\mathcal {C}(M)$ .", "In this case we have $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with ${M_1,\\dots ,M_t} = \\mathcal {C}(M)$ and $t \\ge 2$ , and any other equivalent statement of Proposition REF .", "The metamour graph $M$ is not connected and no induced subgraph $G[V(M_i)]$ is connected for any $M_i \\in \\mathcal {C}(M)$ .", "In this case the metamour graph $M$ has exactly two connected components and the following holds.", "Set $G^M = G[V(M_1)] \\cup G[V(M_2)]$ with ${M_1,M_2} = \\mathcal {C}(M)$ , i.e., $G^M$ is the graph $G$ after deleting every edge between two vertices that are from different connected components of the metamour graph $M$ .", "Then we have: Every connected component of $G^M$ has at most $k$ vertices.", "If $G$ is $k$ -metamour-regular, then every connected component of $G^M$ is a regular graph.", "Every connected component $G_i$ of $G^M$ satisfies $\\complement {G_i} = M[V(G_i)]$ .", "If two different connected components of $G^M$ are adjacent in $G$ , then these connected components are completely adjacent in $G$ .", "If two vertices of different connected components $G_i$ and $G_j$ of $G^M$ have a common neighbor in $G$ , then every vertex of $G_i$ is a metamour of every vertex of $G_j$ .", "If a connected component $G_i$ of $G^M$ is adjacent in $G$ to another connected component $G_j$ consisting of $k-d$ vertices for some $d \\ge 0$ , then the neighbors (in $G$ ) of vertices of $G_i$ are in at most $d+2$ (including $G_j$ ) connected components.", "Let us discuss the three outcomes of Theorem REF in view of the characterizations provided in Sections REF to REF .", "Toward this end note that in case REF , the graph $G$ is obtained by the generic construction.", "For 0-metamour-regular graphs due to Theorem REF there is no graph that is not obtained by the generic construction.", "So only REF happens, except if the graph consists of only one vertex in which case a degenerated REF happens.", "For 1-metamour-regular graphs we know that there is only one graph not obtained by the generic construction by Theorem REF .", "This exceptional case is associated to REF , otherwise we are in REF .", "Finally Theorem REF states that beside the generic case associated to REF , there is only the class with graphs $C_n$ and 17 exceptional cases of 2-metamour-regular graphs associated to REF and REF .", "At last in this section, we bring and discuss the full formulation of a statement mentioned earlier.", "Corollary 3.21 Let $k \\ge 0$ .", "Let $G$ be a connected graph with maximum metamour-degree $k$ and $M$ its metamour graph.", "Let ${M_1,\\dots ,M_t} = \\mathcal {C}(M)$ .", "If $t\\ge 3$ , then we have $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}.$ By this corollary every $k$ -metamour-regular graph that has at least three connected components in its metamour graph is a join of complements of $k$ -regular graphs and therefore can be built by the generic construction.", "As a consequence, it is only possible that a 2-metamour-regular graph is not obtained by the generic construction if its metamour graph has at most two connected components.", "This completes the presentation of our results." ], [ "Proofs regarding foundations", "Just like many partners in polyamorous relationships are happy to receive a proof of love, we are happy to deliver a proof of the results from Sections  and  now.", "We start by proving Proposition REF which relates the metamour graph, the complement and the diameter of a graph.", "[Proof of Proposition REF ] Suppose REF holds.", "Let $v$ and $w$ be vertices in $G$ .", "If $v=w$ , then their distance is 0.", "If $v$ and $w$ are adjacent in $G$ , then their distance is 1.", "Otherwise, there is no edge ${v}{w}$ in $G$ .", "Therefore, this edge is in $\\complement {G}=M$ , where $M$ is the metamour graph of $G$ .", "This implies that $v$ and $w$ are metamours.", "Thus, the distance between $v$ and $w$ is 2.", "Consequently, no distance in $G$ is larger than 2, which implies that the diameter of $G$ is at most 2.", "As a result, either the diameter of $G$ is exactly 2, or it is at most 1.", "If the diameter is equal to 1, then all vertices of $G$ are pairwise adjacent and therefore $G = K_n$ .", "Furthermore, there are at least two vertices at distance 1 and hence $n\\ge 2$ .", "If the diameter is 0, then $G = K_1$ .", "Now suppose REF holds.", "If $G=K_n$ , then its diameter is either equal to 0 if $n=1$ or equal to 1 if $n \\ge 2$ .", "Therefore, in this case the diameter of $G$ is at most 2.", "Now let ${v}{w}$ be an edge in $\\complement {G}$ .", "Then $v \\ne w$ and these vertices are not adjacent in $G$ , so their distance is at least 2.", "As the diameter is at most 2, the distance of $v$ and $w$ is at most 2.", "Consequently their distance is exactly 2 implying that they are metamours.", "So the edge ${v}{w}$ is in $M$ , and hence the complement of $G$ is a subgraph of the metamour graph of $G$ .", "Due to Observation REF , the metamour graph of $G$ is a subgraph of the complement of $G$ , hence the metamour graph of $G$ and $\\complement {G}$ coincide, so we have shown REF .", "Next we prove Proposition REF which relates the metamour graph, the complement, and the degree and metamour-degree of the vertices of a graph.", "[Proof of Proposition REF ] We use that for a graph $G$ with $n$  vertices, the sum of the degrees of a vertex in $G$ and in the complement $\\complement {G}$ always equals $n-1$ .", "If REF holds, then the degree of a vertex in the metamour graph equals the degree in the complement $\\complement {G}$ .", "This yields REF by using the statement at the beginning of this proof.", "If REF holds, then the sum of the degree and the metamour-degree of a vertex equals the sum of the degree in $G$ and the degree in $\\complement {G}$ of this vertex by the statement at the beginning of this proof.", "Therefore, the metamour-degree of a vertex is equal to the degree in $\\complement {G}$ of this vertex.", "Due to Observation REF , the metamour graph of $G$ is a subgraph of $\\complement {G}$ , and therefore the metamour graph of $G$ equals $\\complement {G}$ .", "Finally we prove Propositions REF and REF that relate the metamour graph and joins.", "[Proof of Proposition REF ] We start by showing that REF implies REF .", "In $M$ there are no edges between its different connected components.", "Therefore, in the complement $\\complement {M}=G$ , there are all possible edges between the vertices of different components.", "This is equivalent to the definition of the join of graphs; the individual graphs in $\\mathcal {C}(M)$ are complemented, and consequently REF follows.", "For proving that REF implies REF , we simply set $G_1=\\complement {M_1}$ and $G_2=\\complement {M_2} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ .", "By the definition of the operator $\\mathbin \\nabla $ , REF follows.", "We now show that REF implies REF .", "Every pair of vertices of $G_1$ has a common neighbor in $G_2$ .", "This implies that the vertices of this pair are metamours if and only if they are not adjacent in $G_1$ .", "The same holds for any pair of vertices of $G_2$ by symmetry or due to commutativity of the operator $\\mathbin \\nabla $ .", "As a consequence of this and because $G = G_1 \\mathbin \\nabla G_2$ and the definition of the operator $\\mathbin \\nabla $ , the metamour graph $M$ and the complement of $G$ coincide.", "As every possible edge from a vertex of $G_1$ to a vertex of $G_2$ exists in $G$ , this complement $\\complement {G}$ has at least two connected components, hence $t\\ge 2$ .", "[Proof of Proposition REF ] REF and REF are equivalent by Proposition REF .", "If ${\\mathcal {C}(M)}=t=1$ , then REF and REF are trivially equivalent.", "If ${\\mathcal {C}(M)}=t\\ge 2$ , then this equivalence is part of Proposition REF .", "Finally, the equivalence of REF and REF follows from Proposition REF .", "At this point we have shown all results form Section  and therefore have laid the foundations of the subsequent results." ], [ "Proofs regarding $k$ -metamour-regular graphs", "Next we give the proofs of results from Section REF concerning graphs with maximum metamour-degree $k$ and $k$ -metamour-regular graphs that are valid for arbitrary $k \\ge 0$ .", "[Proof of Proposition REF ] Let $v$ be a vertex of $G$ .", "Then $v$ is in $\\complement {M_i}$ and therefore in $M_i$ for some $i\\in {1,\\dots ,t}$ .", "Let $u \\ne v$ be a vertex of $G$ and $j\\in {1,\\dots ,t}$ such that $u$ is in $\\complement {M_j}$ .", "If $j \\ne i$ , then $u$ and $v$ are adjacent in $G$ by construction.", "Therefore, they are not metamours.", "If $j = i$ , then any vertex not in $M_i$ , i.e., in any of $\\complement {M_1}$ , ..., $\\complement {M_{i-1}}$ , $\\complement {M_{i+1}}$ , ..., $\\complement {M_t}$ , is a common neighbor of $u$ and $v$ .", "Therefore, $u$ and $v$ are metamours if and only if they are not adjacent in $\\complement {M_i}$ , and this is the case if and only if they are adjacent in $M_i$ .", "Summarized, we have that $u$ is a metamour of $v$ if and only if $u$ is in $M_i$ and adjacent to $v$ in $M_i$ .", "This yields that the metamour graph of $G$ is $M=M_1 \\cup \\dots \\cup M_t$ which was to show.", "The $k$ -metamour-regularity follows directly from Observation REF .", "[Proof of Theorem REF ] If the metamour graph $M$ is connected, then we are in case REF and nothing is to show.", "So suppose that the metamour graph is not connected.", "We partition the vertices of $M$ (and therefore the vertices of $G$ ) into two parts $V^{\\prime } \\uplus V^\\ast $ such that the vertex set of each connected component of $M$ is a subset of either $V^{\\prime }$ or $V^\\ast $ and such that neither $V^{\\prime }$ nor $V^\\ast $ is empty, i.e., we partition by the connected components $\\mathcal {C}(M)$ of the metamour graph $M$ .", "As $M$ is not connected, it consists of at least two connected components, and therefore such a set-partition of the vertices of $M$ is always possible.", "We now split up the graph $G$ into the two subgraphs $G^{\\prime } = G[V^{\\prime }]$ and $G^\\ast = G[V^\\ast ]$ .", "Rephrased, we obtain $G^{\\prime }$ and $G^\\ast $ from $G$ by cutting it (by deleting edges) in two, but respecting and not cutting the connected components of its metamour graph $M$ .", "Note that the formulation is symmetric with respect to $G^{\\prime }$ and $G^\\ast $ , therefore, we might switch the two without loss of generality during the proof.", "This also implies that in the statements of the following claims we may switch the roles of $G^{\\prime }$ and $G^\\ast $ .", "A If $G^{\\prime }_1\\in \\mathcal {C}(G^{\\prime })$ is adjacent in $G$ to $G^\\ast _1\\in \\mathcal {C}(G^\\ast )$ , then $G^{\\prime }_1$ is completely adjacent in $G$ to $G^\\ast _1$ .", "[Proof of  REF] Suppose $u^{\\prime } \\in V(G_1^{\\prime })$ and $v^\\ast \\in V(G^\\ast _1)$ are adjacent in $G$ .", "Let $u \\in V(G_1^{\\prime })$ and $v \\in V(G^\\ast _1)$ .", "We have to prove that ${u}{v} \\in E(G)$ .", "There is a path $\\pi _{u^{\\prime },u}=(u_1, u_2, \\dots , u_r)$ from $u_1=u^{\\prime }$ to $u_r=u$ in $G^{\\prime }_1$ because $G^{\\prime }_1$ is connected.", "Furthermore, there is a path $\\pi _{v^\\ast ,v}=(v_1, v_2, \\dots , v_s)$ from $v_1=v^\\ast $ to $v_s=v$ in $G^\\ast _1$ because $G^\\ast _1$ is connected.", "We use induction to prove that ${u_1}{v_\\ell } \\in E(G)$ for all $\\ell \\in {1,\\dots ,s}$ .", "Indeed, this is true for $\\ell = 1$ by assumption.", "So assume ${u_1}{v_\\ell } \\in E(G)$ .", "We have ${v_\\ell }{v_{\\ell +1}} \\in E(G)$ because this is an edge of the path $\\pi _{v^\\ast ,v}$ .", "If ${u_1}{v_{\\ell +1}} \\notin E(G)$ , then $u_1$ and $v_{\\ell +1}$ are metamours.", "But $u_1$ has all its metamours in $G^{\\prime }$ , a contradiction.", "Hence, ${u_1}{v_{\\ell +1}} \\in E(G)$ , which finishes the induction.", "In particular, we have proven that ${u_1}{v_s} \\in E(G)$ .", "Now we prove that ${u_\\ell }{v_s} \\in E(G)$ holds for all $\\ell \\in {1,\\dots ,r}$ by induction.", "This holds for $\\ell = 1$ by the above.", "Now assume ${u_\\ell }{v_s} \\in E(G)$ .", "We have ${u_\\ell }{u_{\\ell +1}} \\in E(G)$ because this edge is a part of the path $\\pi _{u^{\\prime },u}$ .", "If ${u_{\\ell +1}}{v_s} \\notin E(G)$ , then $u_{\\ell +1}$ and $v_s$ are metamours, a contradiction since every metamour of $u_{\\ell +1}$ is in $G^{\\prime }$ .", "Therefore, ${u_{\\ell +1}}{v_s} \\in E(G)$ holds and the induction is completed.", "As a result, we have ${u}{v} = {u_r}{v_s} \\in E(G)$ .", "B Let $\\mathcal {C}(M) = {M_1,\\dots ,M_t}$ .", "If the graph $G^{\\prime }$ is connected, then $G = G^{\\prime } \\mathbin \\nabla G^\\ast = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with $t \\ge 2$ .", "[Proof of  REF] The graph $G$ is connected, so every connected component of $G^\\ast $ is adjacent in $G$ to $G^{\\prime }$ .", "By  REF this implies that $G^{\\prime }$ is completely adjacent in $G$ to $G^\\ast $ , i.e., all possible edges between $G^{\\prime }$ and $G^\\ast $ exist.", "Therefore, $G = G^{\\prime } \\mathbin \\nabla G^\\ast $ by the definition of the join of graphs.", "By Proposition REF , the full decomposition into the components $\\mathcal {C}(M)$ follows.", "As a consequence of  REF, we are finished with the proof in the case that $G[V(M_i)]$ is connected for some $M_i\\in \\mathcal {C}(M)$ because statement REF follows by setting $G^{\\prime }=G[V(M_i)]$ .", "So from now on we consider the case that every $G[V(M_i)]$ with $M_i\\in \\mathcal {C}(M)$ has at least two connected components.", "This is the set-up of statement REF .", "C Suppose we are in the set-up of REF .", "Let $G^{\\prime }_1\\in \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2} \\subseteq \\mathcal {C}(G^\\ast )$ .", "If both $G^\\ast _1$ and $G^\\ast _2$ are adjacent in $G$ to $G^{\\prime }_1$ , then every vertex of $G^\\ast _1$ is a metamour of every vertex of $G^\\ast _2$ .", "[Proof of  REF] As both $G^\\ast _1$ and $G^\\ast _2$ are adjacent in $G$ to $G^{\\prime }_1$ , they are both completely adjacent in $G$ to $G^{\\prime }_1$ due to  REF.", "Furthermore, $G^\\ast _1$ is not adjacent in $G$ to $G^\\ast _2$ , i.e., no vertex of $G^\\ast _1$ is adjacent in $G$ to any vertex of $G^\\ast _2$ , because they are in different connected components of $G^\\ast $ .", "Hence, every vertex of $G^\\ast _1$ is a metamour of every vertex of $G^\\ast _2$ .", "D Suppose we are in the set-up of REF .", "Then every connected component of $G^{\\prime }$ has at most $k$ vertices and this connected component's vertex set is a subset of the vertex set of one connected component of the metamour graph $M$ .", "[Proof of  REF] Let $G^{\\prime }_1 \\in \\mathcal {C}(G^{\\prime })$ .", "As $G^{\\prime }$ is not connected but the graph $G$ is connected, there is a path $\\pi $ from a vertex of $G^{\\prime }_1$ to some vertex in some connected component of $\\mathcal {C}(G^{\\prime })$ other than $G^{\\prime }_1$ .", "By construction of $G^{\\prime }$ and $G^\\ast $ , the path $\\pi $ splits from start to end into vertices of $G^{\\prime }_1$ , followed by vertices of some $G^\\ast _1 \\in \\mathcal {C}(G^\\ast )$ , followed by some vertices of $G^{\\prime }_2 \\in \\mathcal {C}(G^{\\prime })$ , and remaining vertices.", "Therefore, we have connected components $G^{\\prime }_2$ and $G^\\ast _1$ such that at least one vertex of $G^{\\prime }_1$ is connected to some vertex of $G^\\ast _1$ and at least one vertex of $G^{\\prime }_2$ is connected to some vertex of $G^\\ast _1$ .", "Then, due to  REF every vertex of $G^{\\prime }_1$ is a metamour of every vertex of $G^{\\prime }_2$ .", "From this, we now deduce two statements.", "First, if we assume that $G^{\\prime }_1$ contains at least $k+1$ vertices, then every vertex of $G^{\\prime }_2$ has at least $k+1$ metamours, a contradiction to $k$ being the maximum metamour-degree of $G$ .", "Therefore, $G^{\\prime }_1$ contains at most $k$ vertices.", "Second, every vertex of $G^{\\prime }_1$ is adjacent in the metamour graph $M$ to every vertex of $G^{\\prime }_2$ , so $G^{\\prime }_1$ is completely adjacent in $M$ to $G^{\\prime }_2$ .", "Therefore, all these vertices are in the same connected component of the metamour graph $M$ .", "In particular, this is true for the set of vertices of $G^{\\prime }_1$ as claimed, and so the proof is complete.", "E Suppose we are in the set-up of REF .", "Let $v_1$ and $v_2$ be two vertices of different connected components of $G^{\\prime }$ .", "Then every shortest path from $v_1$ to $v_2$ in $G$ consists of vertices alternating between $G^{\\prime }$ and $G^\\ast $ .", "[Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2} \\subseteq \\mathcal {C}(G^{\\prime })$ such that $v_1 \\in G^{\\prime }_1$ and $v_2 \\in G^{\\prime }_2$ .", "Let $\\pi =(u_1, u_2, \\dots , u_r)$ be a shortest path from $u_1=v_1$ to $u_r=v_2$ in $G$ .", "Note that $u_1$ and $u_r$ are both from $G^{\\prime }$ but from different connected components.", "Hence, $\\pi $ consists of at least two vertices from $G^{\\prime }$ and at least one vertex from $G^\\ast $ .", "Assume that the vertices of $\\pi $ are not alternating between $G^{\\prime }$ and $G^\\ast $ .", "Then, without loss of generality (by reversing the enumeration of the vertices in the path $\\pi $ ), there exist indices $i<j$ and graphs ${\\widetilde{G},\\widehat{G}} = {G^{\\prime }, G^\\ast }$ with the following properties: Every vertex of the subpath $\\pi _{i,j}=(u_i, u_{i+1}, \\dots , u_j)$ of $\\pi $ is of $\\widetilde{G}$ , and the vertex $u_{j+1}$ exists and is in $\\widehat{G}$ .", "As $\\pi _{i,j}$ is a path, all of its vertices are of the same connected component of $\\widetilde{G}$ .", "As $u_j$ is adjacent to $u_{j+1}$ and due to  REF, the vertex $u_{j+1}$ is adjacent to every vertex of $\\pi _{i,j}$ , in particular, adjacent to $u_i$ .", "But then $u_1$ , ..., $u_i$ , $u_{j+1}$ , ..., $u_r$ is a shorter path between $v_1$ and $v_2$ , a contradiction.", "Hence, our initial assumption was wrong, and the vertices of $\\pi $ are alternating between $G^{\\prime }$ and $G^\\ast $ .", "F Suppose we are in the set-up of REF .", "Then the metamour graph $M$ has exactly two connected components.", "[Proof of  REF] The metamour graph $M$ is not connected, therefore it has at least two connected components.", "Assume it has at least three components.", "Then, without loss of generality (by switching $G^{\\prime }$ and $G^\\ast $ ), the graph $G^{\\prime }$ contains vertices of at least two different connected components of $M$ .", "Let $v_1$ and $v_2$ be two vertices of $G^{\\prime }$ that are in different connected components of $M$ .", "If follows from  REF that $v_1$ and $v_2$ are in different connected components of $G^{\\prime }$ .", "Let ${G^{\\prime }_1,G^{\\prime }_2} \\subseteq \\mathcal {C}(G^{\\prime })$ such that $v_1 \\in G^{\\prime }_1$ and $v_2 \\in G^{\\prime }_2$ .", "Let $\\pi =(u_1,\\dots ,u_r)$ be a shortest path from $u_1=v_1$ to $u_r=v_2$ in $G$ .", "Now consider two vertices $u_{2i - 1}$ and $u_{2i+1}$ for $i \\ge 1$ of $\\pi $ .", "Both $u_{2i - 1}$ and $u_{2i+1}$ are from $G^{\\prime }$ because $\\pi $ consists of alternating vertices from $G^{\\prime }$ and $G^\\ast $ due to  REF and $u_1$ is from $G^{\\prime }$ .", "If both $u_{2i-1}$ and $u_{2i+1}$ are from the same connected component of $G^{\\prime }$ , then they are in the same connected component of the metamour graph $M$ by  REF.", "If $u_{2i-1}$ and $u_{2i+1}$ are in different connected components of $G^{\\prime }$ , then they are metamours because they have the common neighbor $u_{2i}$ and they are not adjacent.", "Therefore, they are in the same connected component of $M$ as well.", "Hence, in any case $u_{2i - 1}$ and $u_{2i+1}$ are in the same connected component of $M$ .", "By induction this implies that $u_1=v_1$ is in the same connected component of $M$ as $u_r = v_2$ , a contradiction to $v_1$ and $v_2$ being from different connected components of $M$ .", "Hence, our assumption was wrong and the metamour graph consists of exactly two connected components.", "G Suppose we are in the set-up of REF .", "Then every connected component $G^{\\prime }_1$ of $G^{\\prime }$ satisfies $\\complement {G^{\\prime }_1} = M[V(G^{\\prime }_1)]$ .", "If $G$ is $k$ -metamour-regular, then the connected component is a regular graph.", "[Proof of  REF] Let $G^{\\prime }_1$ be a connected component of $G^{\\prime }$ .", "As $G$ is connected, there is an edge from a vertex $v_1$ of $G^{\\prime }_1$ to $G^\\ast $ and this is extended to every vertex of $G^{\\prime }_1$ by  REF.", "Therefore, two different vertices of $G^{\\prime }_1$ are metamours if and only if they are not adjacent in $G^{\\prime }$ .", "Restricting this to the subgraph $G^{\\prime }_1$ yields the first statement.", "Furthermore, by construction of $G^{\\prime }$ and $G^\\ast $ , every metamour of $v_1$ is in $G^{\\prime }$ .", "Let $v^{\\prime }$ be such a metamour, and suppose that $v^{\\prime }$ is not in $G^{\\prime }_1$ .", "The vertices $v_1$ and $v^{\\prime }$ have a common neighbor $u$ that has to be in $G^\\ast $ as $v_1$ and $v^{\\prime }$ are in different connected components of $G^{\\prime }$ .", "The vertex $u$ is completely adjacent in $G$ to $G^{\\prime }_1$ because of  REF.", "Hence, $v^{\\prime }$ is a metamour of every vertex of $G^{\\prime }_1$ .", "As a consequence, every vertex of $G^{\\prime }_1$ has the same number of metamours outside of $G^{\\prime }_1$ , i.e., in $G^{\\prime }$ but not in $G^{\\prime }_1$ .", "If no such pair of vertices $v_1$ of $G^{\\prime }_1$ and $v^{\\prime }$ not of $G^{\\prime }_1$ that are metamours exists, then every vertex of $G^{\\prime }_1$ still has the same number of metamours outside of $G^{\\prime }_1$ , namely zero.", "Now let us assume that $G$ is $k$ -metamour-regular.", "As every vertex of $G^{\\prime }_1$ has the same number of metamours outside of $G^{\\prime }_1$ , this implies that every vertex of $G^{\\prime }_1$ must also have the same number of metamours inside $G^{\\prime }_1$ .", "We combine this with the results of first paragraph and conclude that every vertex of $G^{\\prime }_1$ is adjacent to the same number of vertices of $G^{\\prime }_1$ , and hence $G^{\\prime }_1$ is a regular graph.", "H Suppose we are in the set-up of REF .", "If a connected component $G^{\\prime }_1\\in \\mathcal {C}(G^{\\prime })$ is adjacent in $G$ to a connected component $G^\\ast _1\\in \\mathcal {C}(G^\\ast )$ consisting of $k-d$ vertices for some $d \\ge 0$ , then the neighbors (in $G$ ) of vertices of $G^{\\prime }_1$ that are in $G^\\ast $ are in at most $d+2$ connected components of $G^\\ast $ (including $G^\\ast _1$ ).", "[Proof of  REF] Let $\\mathcal {G}^\\ast \\subseteq \\mathcal {C}(G^\\ast )$ be such that a connected components of $G^\\ast $ is in $\\mathcal {G}^\\ast $ if and only if it is adjacent in $G$ to $G^{\\prime }_1$ .", "$G^\\ast _1$ consists of $k-d$ vertices and there is some vertex $v^{\\prime }$ of $G^{\\prime }_1$ that is adjacent to some vertex of $G^\\ast _1$ .", "We have to prove that ${\\mathcal {G}^\\ast } \\le d + 2$ in order to finish the proof.", "So assume ${\\mathcal {G}^\\ast } > d + 2$ .", "Let $v^\\ast $ be a vertex of some connected component $G^\\ast _2\\in \\mathcal {G}^\\ast $ other than $G^\\ast _1$ .", "Then $v^\\ast $ is adjacent to $v^{\\prime }$ due to  REF.", "Because of  REF, every vertex in any connected component in $\\mathcal {G}^\\ast $ except $G^\\ast _2$ is a metamour of $v^\\ast $ .", "The component $G^\\ast _1$ contains $k-d$ vertices and there are at least $d+1$ other components each containing at least one vertex.", "In total, $v^\\ast $ has at least $(k - d) + (d+1) = k+1$ metamours, a contradiction to $k$ being the maximum metamour-degree of $G$ .", "Therefore, our assumption was wrong and ${\\mathcal {G}^\\ast } \\le d + 2$ holds.", "Now we are able to collect everything we have proven so far and finish the proof of statement REF .", "Due to  REF, the metamour graph $M$ of $G$ consists of exactly two connected components, and consequently the connected components of $G^M$ coincide with the union of the connected components of $G^{\\prime }$ and $G^\\ast $ .", "Then  REF implies REF , REF implies REF , REF implies REF , REF implies REF , REF implies REF and  REF implies REF .", "This completes the proof.", "[Proof of Corollary REF ] As the metamour graph consists of at least 3 connected components, we cannot land in the cases REF and REF of Theorem REF .", "But then the statement of the corollary follows from REF .", "Now we have proven everything we need to know about graphs with maximum metamour-degree $k$ and $k$ -regular-metamour graphs for general $k$ and can use this knowledge to derive the results we need in order to characterize all $k$ -regular-metamour graphs for $k\\in {0,1,2}$ ." ], [ "Proofs regarding 0-metamour-regular graphs", "We are now ready to prove all results concerning 0-metamour-regular graphs from Section REF .", "In order to do so we first need the following lemma.", "Lemma 6.1 Let $G$ be a connected graph.", "If a vertex has no metamour, then it is adjacent to all other vertices of $G$ .", "Let $n$ be the number of vertices of $G$ .", "Clearly the statement is true for $n = 1$ as no other vertices are present and for $n = 2$ as the two vertices are adjacent due to connectedness.", "So let $n \\ge 3$ , and let $v$ be a vertex of $G$ that has no metamour.", "Assume there is a vertex $w \\ne v \\in V(G)$ such that ${v}{w} \\notin E(G)$ .", "$G$ has a spanning tree $T$ because $G$ is connected.", "Let $v = u_1$ , $u_2$ , ..., $u_r = w$ be the vertices on the unique path from $v$ to $w$ in $T$ .", "Then ${u_i}{u_{i+1}} \\in E(G)$ for all $i \\in {1,\\dots ,r-1}$ , so due to our assumption $r \\ge 3$ holds.", "In particular, ${u_1}{u_2} \\in E(G)$ .", "If ${u_1}{u_3} \\notin E(G)$ , then both $u_1$ and $u_3$ are adjacent to $u_2$ , but not adjacent to each other and therefore would be metamours.", "But $u_1 = v$ does not have a metamour, hence ${u_1}{u_3} \\in E(G)$ .", "By induction ${u_1}{u_i} \\in E(G)$ for all $i\\in {1,\\dots ,r}$ and thus ${v}{w} = {u_1}{u_r} \\in E(G)$ , a contradiction.", "Now we are able to prove Theorem REF that provides a characterization of 0-metamour-regular graphs.", "[Proof of Theorem REF ] If $G$ is 0-metamour-regular then every vertex of $G$ has no metamour and hence $G = K_n$ due to Lemma REF .", "Furthermore, $K_n$ is 0-metamour-regular as every vertex is adjacent to all other vertices.", "Next we prove the corollaries which yield an alternative characterization of 0-metamour-regular graphs and allow to count 0-metamour-regular graphs.", "[Proof of Corollary REF ] Due to Theorem REF , a connected graph with $n$ vertices is 0-metamour-regular if and only if is equal to $K_n$ .", "This is the case if and only if its complement has no edges.", "In this case the complement also equals the metamour graph.", "[Proof of Corollary REF ] This is an immediate and easy consequence of the characterization provided by Theorem REF ." ], [ "Proofs regarding 1-metamour-regular graphs", "In this section we present the proofs of the results from Section REF .", "They lead to a characterization of 1-metatmour-regular graphs.", "[Proof of Proposition REF ] Whenever a vertex $v \\in V(G)$ is a metamour of a vertex $w \\ne v\\in V(G)$ then also $w$ is a metamour of $v$ .", "Therefore, supposing that $v$ has exactly one metamour, so has $w$ and the vertices $v$ and $w$ form a pair such that the two vertices of the pair are metamours of each other.", "This also leads to an edge from $v$ to $w$ in the metamour graph of $G$ .", "As every vertex has at most one metamour, the edge from $v$ to $w$ is isolated in the metamour graph of $G$ , so $v$ and $w$ have no other adjacent vertices in the metamour graph of $G$ .", "Therefore, the edges of the metamour graph form a matching, which yields REF .", "Suppose now additionally that $G$ is 1-metamour-regular.", "Then we can partition the vertices of $G$ into pairs of metamours.", "Hence, $n$ is even and the edges of the metamour graph form a perfect matching, so REF holds.", "For proving Theorem REF , we need some auxiliary results.", "We start by showing that the graphs mentioned in the theorem are indeed 1-metamour-regular.", "Proposition 7.1 The graph $P_4$ depicted in Figure REF is 1-metamour-regular.", "This is checked easily.", "The following proposition is slightly more general than needed in the proof of Theorem REF and will be used later on.", "Proposition 7.2 Let $n \\ge 3$ .", "In the graph $G = K_n - \\mu $ with a matching $\\mu $ of $K_n$ , every vertex has at most one metamour.", "Let $v$ be an arbitrary vertex of $G$ .", "Suppose $v$ is not incident to any edge in $\\mu $ , then $v$ is adjacent to all other vertices.", "Thus, $v$ has no metamour.", "Now suppose that $v$ is incident to some edge in $\\mu $ , and let the vertex $v^{\\prime }$ be the other vertex incident to this edge.", "Then clearly ${v}{v^{\\prime }} \\notin E(G)$ , so both $v$ as well as $v^{\\prime }$ have to be adjacent to all other vertices of $G$ by construction of $G$ .", "Due to the assumption $n \\ge 3$ , there is at least one other vertex besides $v$ and $v^{\\prime }$ , and this vertex is a common neighbor of them.", "Hence, $v$ and $v^{\\prime }$ are metamours of each other.", "Both $v$ and $v^{\\prime }$ do not have any other metamour because they are adjacent to all other vertices.", "As a result, $v$ has exactly one metamour.", "Proposition 7.3 Let $n \\ge 4$ be even.", "The graph $G = K_n - \\mu $ with a perfect matching $\\mu $ of $K_n$ is 1-metamour-regular.", "As the matching $\\mu $ is perfect, every vertex $v$ of $G$ is incident to one edge in $\\mu $ .", "Thus, by the proof of Proposition REF , every $v$ has exactly one metamour.", "We are now ready for proving Theorem REF .", "[Proof of Theorem REF ] The one direction of the equivalence follows directly from Proposition REF and Proposition REF , so only the other direction is left to prove.", "Suppose we have a graph $G$ with $n$ vertices that is 1-metamour-regular.", "Due to Proposition  REFREF , $n$ is even, and the set of edges of the metamour graph of $G$ forms a perfect matching.", "In particular, each connected component of the metamour graph consists of two adjacent vertices.", "If the metamour graph is connected, then it consists of only two adjacent vertices and $n=2$ .", "This can be ruled out easily, so we have $n\\ge 4$ and the metamour graph is not connected.", "Now we can use Theorem REF and see that one of the two cases REF and REF applies.", "In the first case REF we have $G = \\complement {M_1}\\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with $n={M_1}+\\dots +{M_t}$ for some $t\\ge 2$ , where $M_i$ is a connected 1-regular graph for all $i\\in {1,\\dots ,t}$ .", "The only connected 1-regular graph is $P_2$ , therefore ${V(M_i)} = 2$ and $M_i = P_2$ for all $i\\in {1,\\dots ,t}$ .", "Hence, we have $G = \\complement {P_2}\\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {P_2}$ , which means nothing else than $G = K_n - \\mu $ for a perfect matching $\\mu $ of $K_n$ .", "In the second case REF the metamour graph of $G$ consists of two connected components, so the metamour graph consists of $n=4$ vertices with two edges that form a perfect matching.", "It is easy to see that $G = P_4$ or $G = C_4 = K_4 - \\mu $ for some perfect matching $\\mu $ of $K_4$ are the only two possibilities in this case.", "As a result, we obtain in any case $G=P_4$ or $G = K_n - \\mu $ for a perfect matching $\\mu $ of $K_n$ , which is the desired result.", "To finish this section we prove the three corollaries of Theorem REF .", "[Proof of Corollary REF ] Due to the characterization of 1-metamour-regular graphs of Theorem REF , we know that a connected graph with $n \\ge 5$ vertices is 1-metamour-regular if and only if it is equal to $K_n - \\mu $ for some perfect matching $\\mu $ of $K_n$ .", "This is the case if and only if the complement is the graph induced by $\\mu $ .", "Furthermore, a graph is induced by a perfect matching if and only if it is 1-regular.", "To summarize, a connected graph with $n \\ge 5$ vertices is 1-metamour-regular if and only if its complement is a 1-regular graph.", "In this case the complement also equals the metamour graph, which implies the desired result.", "[Proof of Corollary REF ] We use the characterization provided by Theorem REF .", "So let us consider 1-metamour-regular graphs.", "Such a graph has at least $n\\ge 4$ vertices, and $n$ is even.", "Every perfect matching $\\mu $ of $K_n$ results in the same unlabeled graph $K_n-\\mu $ ; this brings to account 1.", "For $n=4$ , there is additionally the graph $P_4$ .", "In total, this gives the claimed numbers.", "[Proof of Corollary REF ] Let $G$ be an unlabeled graph with $n$ pairs of vertices that each are metamours.", "We first construct a pair $(\\lambda _1+\\dots +\\lambda _t,s)$ , where $\\lambda _1+\\dots +\\lambda _t$ is a partition of $n$ with $\\lambda _i\\ge 2$ for all $i\\in {1,\\dots ,t}$ and $s$ is a non-negative integer bounded by $r_\\lambda $ which is defined to be the number of $i\\in {1,\\dots ,t}$ with $\\lambda _i=2$ .", "Let ${G_1, \\dots , G_t} = \\mathcal {C}(G)$ , set $\\lambda _i={V(G_i)}/2$ for all $i\\in {1,\\dots ,t}$ , and let us assume that $\\lambda _1 \\ge \\dots \\ge \\lambda _t$ .", "Then $n=\\lambda _1+\\dots +\\lambda _t$ , so this is a partition of $n$ .", "As there is no graph $G_i$ with only 1 metamour-pair, $\\lambda _i\\ge 2$ for all $i\\in {1,\\dots ,t}$ .", "We define $s$ to be the number of $i\\in {1,\\dots ,t}$ with $G_i=P_4$ .", "We clearly have $s \\le r_\\lambda $ .", "Conversely, let a pair $(\\lambda _1+\\dots +\\lambda _t,s)$ as above be given.", "For every $i\\in {1,\\dots ,t}$ with $\\lambda _i\\ge 3$ there is exactly one choice for a 1-metamour-regular graph $G_i$ with $2\\lambda _i$ vertices by Theorem REF .", "Now consider parts 2 of $\\lambda _1+\\dots +\\lambda _t$ .", "We choose any (the graphs are unlabeled) $s$ indices and set $G_i=P_4$ .", "Then we set $G_i=C_4$ for the remaining $r_\\lambda -s$ indices.", "The graph $G=G_1 \\cup \\dots \\cup G_t$ is then fully determined.", "Thus, we have a found a bijective correspondence.", "We still need to related our partition of $n$ to the partition of $n+2$ of Corollary REF .", "A partition of $n+2$ is either $n+2=(n+2)$ , $n\\ge 1$ , in which case no additional part 2 appears, or $n+2=\\lambda _1+\\dots +\\lambda _t+2$ for a partition $n=\\lambda _1+\\dots +\\lambda _t$ .", "Here one additional part 2 appears.", "Therefore, every pair $(\\lambda _1+\\dots +\\lambda _t,s)$ from above maps bijectively to a partition $\\lambda _1+\\dots +\\lambda _t+2$ of $n+2$ together with a marker of one of the $r_\\lambda +1$ parts 2 in this partition that is uniquely determined by $s$ (by some fixed rule that is not needed to be specified explicitly).", "This completes the proof of Corollary REF ." ], [ "Proofs regarding graphs with maximum\nmetamour-degree 1", "Next we prove the results of Section REF on graphs with maximum metamour-degree 1.", "We start with the proof of the characterization of these graphs.", "[Proof of Theorem REF ] It is easy to see that in the graphs $K_1$ and $K_2$ no vertex has any metamour, so the condition that each vertex has at most one metamour is satisfied.", "Furthermore, by Propositions REF and REF , every vertex has indeed at most one metamour in the remaining specified graphs.", "Therefore, one direction of the equivalence is proven, and we can focus on the other direction.", "So, let $G$ be a graph in which every vertex has at most one metamour.", "Due to Theorems REF and REF , it is enough to restrict ourselves to graphs $G$ , where at least one vertex of $G$ has no metamour and at least one vertex of $G$ has exactly one metamour.", "We will show that $n\\ge 3$ and that $G = K_n - \\mu $ for some matching $\\mu $ that is not perfect and contains at least one edge.", "Let $V_0 \\subseteq V(G)$ and $V_1 \\subseteq V(G)$ be the set of vertices of $G$ that have no and exactly one metamour, respectively, and let $v \\in V_0$ .", "Due to Lemma REF , every vertex in $V_0$ , in particular $v$ , is adjacent to all other vertices.", "Furthermore, by Proposition REF , the vertices in $V_1$ induce a matching $\\mu $ in both the metamour graph and the complement of $G$ .", "This matching $\\mu $ contains at least one edge because $V_1$ is not empty, and $\\mu $ is not perfect because $V_0$ is not empty.", "Furthermore, this implies that $V_1$ contains at least two vertices and in total that $n \\ge 3$ .", "Let $w$ and $w^{\\prime }$ be two vertices in $V_1$ that are not metamours.", "Since $v$ is a common neighbor of both $w$ and $w^{\\prime }$ , this implies that ${w}{w^{\\prime }} \\in E(G)$ .", "Hence, all possible edges except those in $\\mu $ are present in $G$ and therefore $G = K_n - \\mu $ .", "Next we prove the two corollaries of Theorem REF .", "[Proof of Corollary REF ] Due to Theorem REF , in a connected graph $G$ with $n\\ge 5$ vertices every vertex has at most one metamour if and only if $G = K_n -\\mu $ for some matching $\\mu $ of $K_n$ .", "This is the case if and only if the complement is the graph induced by $\\mu $ .", "Furthermore, a graph is induced by a matching if and only if it has maximum degree 1.", "To summarize, a connected graph with $n \\ge 5$ vertices has maximum metamour-degree 1 if and only if its complement is a graph with maximum degree 1.", "In this case the complement also equals the metamour graph, which implies the desired result.", "[Proof of Corollary REF ] We use the characterization provided by Theorem REF .", "So let us consider graphs with maximum metamour-degree 1.", "For $n\\in {1,2}$ , we only have $K_1$ and $K_2$ , so $m_{\\le 1}(n)=1$ in both cases.", "So let $n\\ge 3$ .", "Every perfect matching $\\mu $ of $K_n$ having the same number of edges results in the same graph $K_n-\\mu $ .", "A matching can contain at most ${n/2}$ edges and each choice in ${0,\\dots ,{n/2}}$ for the number of edges is possible.", "This brings to account ${n/2}+1$ .", "For $n=4$ , there is additionally the graph $P_4$ .", "In total, this gives the claimed numbers." ], [ "Proofs regarding 2-metamour-regular graphs", "This section is devoted to the proofs concerning 2-metamour-regular graphs from Section REF .", "It is a long way to obtain the final characterization of 2-metamour-regular graphs of Theorem REF , so we have outsourced the key parts of the proof into several lemmas and propositions.", "For the proofs of Lemma REF , Lemma REF , Lemma REF and Proposition REF we provide many figures.", "Every proof consists of a series of steps, and in each of the steps vertices and edges of a graph are analyzed: It is determined whether edges are present or not and which vertices are metamours of each other.", "The figures of the actual situations show subgraphs of the graph (and additional assumptions) in the following way: Between two vertices there is either an edge vertex] (v0) at (0,0) ; vertex] (v1) at (1,0) ; [edge] (v0) – (v1); or a non-edge vertex] (v0) at (0,0) ; vertex] (v1) at (1,0) ; [nonedge] (v0) – (v1); or nothing vertex] (v0) at (0,0) ; vertex] (v1) at (1,0) ; drawn.", "If nothing is drawn, then it is not (yet) clear whether the edge is present or not.", "A metamour relation vertex] (v0) at (0,0) ; vertex] (v1) at (1,0) ; [nonedge] (v0) – (v1); [mmedge] (v0) – (v1); might be indicated at a non-edge.", "Note that we frequently use the particular graphs defined by Figures REF , REF and REF ." ], [ "Graphs with connected metamour graph", "The proof of the characterization of 2-metamour-regular graphs in Theorem REF is split into two main parts, which represent whether the metamour graph of $G$ is connected or not in order to apply the corresponding case of Theorem REF .", "If the metamour graph of a graph with $n$ vertices is connected, then according to Observation REF the metamour graph equals $C_n$ .", "Here we make a further distinction between graphs that do and that do not contain a cycle of length $n$ as a subgraph.", "First, we characterize all 2-metamour-regular graphs whose metamour graph is connected and that do not contain a cycle of length $n$ .", "Lemma 9.1 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph equals the $C_n$ , that is not a tree, and that does not contain a cycle of length $n$ .", "Then $G \\in {H_{6}^{a},H_{6}^{b},H_{7}^{a}}.$ As $G$ is not a tree, let $\\gamma = (v_1, v_2, \\dots , v_r, v_1)$ , $v_i\\in V(G)$ for $i \\in {1,\\dots ,r}$ , be a longest cycle in $G$ .", "In all the figures accompanying the proof, the longest cycle is marked by [baseline][cycleedge] (0,0) arc (150:30:0.5);.", "By assumption, we have $r < n$ .", "For proving the lemma, we have to show that $G \\in {H_{6}^{a},H_{6}^{b},H_{7}^{a}}$ .", "As a cycle has length at least 3, we have $r\\ge 3$ for the length of the cycle $\\gamma $ .", "As we also have $n>r$ , we may assume $n \\ge 4$ .", "We start by showing the following claims.", "A A vertex $u$ in $G$ that is not in the cycle $\\gamma $ is adjacent to at most one vertex in $\\gamma $ .", "If $u$ is adjacent to a vertex $v$ in $\\gamma $ , then $u$ is a metamour of each neighbor of $v$ in $\\gamma $ .", "[Proof of  REF] Let $u \\in V(G)$ be a vertex not in $\\gamma $ .", "We assume that $u$ is adjacent to $v_1$ (without loss of generality by renumbering) and some $v_j$ in the cycle $\\gamma $ .", "We first show that $v_1$ and $v_j$ are not two consecutive vertices in $\\gamma $ .", "So let us assume that they are, i.e., $j=2$ (see Figure REF (a)) or $j=r$ which works analogously).", "Then $(v_1, u, v_2, \\dots , v_r, v_1)$ would be a longer cycle which is a contradiction to $\\gamma $ being a longest cycle.", "Figure: Subgraphs of the situationsin the proof of Hence, $v_1$ and $v_j$ are not consecutive vertices in $\\gamma $ .", "Then $r\\ge 4$ as there need to be at least one vertex between $v_1$ and $v_j$ on the cycle on each side.", "If $r=4$ , then $v_j=v_3$ and we are in the situation shown in Figure REF (b).", "There, $(u,v_1,v_2,v_4,v_3,u)$ is a 5-cycle which contradicts that the longest cycle is of length 4.", "Therefore, $r=4$ cannot hold.", "If $r>4$ , then $u$ is a metamour of $v_{2}$ , $v_{r}$ , $v_{j-1}$ and $v_{j+1}$ , because it has a common neighbor ($v_1$ or $v_j$ ) with these vertices and is not adjacent to them.", "At least one of $v_{j-1}$ and $v_{j+1}$ is different from $v_2$ and $v_r$ , so ${{v_2, v_r, v_{j-1}, v_{j+1}}} \\ge 3$ .", "This contradicts the 2-metamour-regularity of $G$ .", "Therefore, we have shown that $u$ is adjacent to at most one vertex in $\\gamma $ .", "Now suppose $u$ is adjacent to a vertex $v$ in $\\gamma $ .", "Then $u$ is not adjacent to any neighbor of $v$ in $\\gamma $ and therefore a metamour of every such neighbor.", "B There exists a vertex $w$ in $G$ but not in $\\gamma $ that is adjacent to (without loss of generality) $v_1$ , but not to any other $v_j$ , $j\\ne 1$ , in $\\gamma $ .", "[Proof of  REF] As $r<n$ , there exists a vertex $w^{\\prime }$ not in the cycle $\\gamma $ .", "The graph $G$ is connected, so there is a path from a vertex of $\\gamma $ to $w^{\\prime }$ .", "Therefore, there is also a vertex $w$ not in $\\gamma $ which is adjacent to a vertex $v_i$ .", "By renumbering, we can assume without loss of generality that $i=1$ .", "As the vertex $w$ is adjacent to $v_1$ , $w$ is not adjacent to any other $v_j$ by  REF.", "Figure: Subgraph of the situation between and At this point, we assume to have a vertex $w$ as in  REF; the situation is shown in Figure REF .", "C The graph $G$ contains the edge ${v_2}{v_r}$ .", "[Proof of  REF] Figure: Subgraph of the situationin the proof of Assume that there is no edge between $v_2$ and $v_r$ ; see Figure REF .", "Then $v_2$ and $v_r$ are metamours of each other, and consequently $(w, v_2, v_r, w)$ forms a 3-cycle in the metamour graph of $G$ .", "This contradicts that the metamour graph of $G$ is $C_n$ and $n\\ge 4$ .", "Figure: Subgraph of the situation between and At this point, we have the situation shown in Figure REF .", "In the next steps we will rule out possible values of $r$ .", "D If $r=3$ , then $G=H_{6}^{a}$ .", "[Proof of  REF] Our initial situation is shown in Figure REF (a).", "Suppose there is an additional vertex $v_1^{\\prime }$ of $G$ adjacent to $v_1$ ; see Figure REF (b).", "Then by  REF, $v_1^{\\prime }$ has metamours $v_2$ and $v_3$ .", "Therefore, $(w,v_2,v_1^{\\prime },v_3,w)$ is a 4-cycle in the metamour graph of $G$ .", "As this cycle does not cover $v_1$ , we have a contradiction to the metamour graph being the single cycle $C_n$ for $n>r=3$ .", "Therefore, there is no additional vertex adjacent to $v_1$ .", "Figure: Subgraph of the situationin the proof of At this point, we know that $w$ is a metamour of both $v_2$ and $v_3$ ; see again Figure REF (a).", "We now look for the second metamour of $v_2$ and $v_3$ , respectively.", "As we ruled out an additional vertex adjacent to $v_1$ , there need to be an additional vertex adjacent to $v_2$ or to $v_3$ .", "Without loss of generality (by symmetry), suppose there is an additional vertex $v_3^{\\prime }$ of $G$ adjacent to $v_3$ ; see Figure REF (c).", "Then by  REF, $v_3^{\\prime }$ has metamours $v_1$ and $v_2$ .", "Therefore, these two vertices are the two metamours of $v_3^{\\prime }$ .", "There cannot be an additional vertex $v_3^{\\prime \\prime }$ of $G$ adjacent to $v_3$ , because due to the same arguments as for $v_3^{\\prime }$ this vertex would be a metamour of $v_2$ , hence $v_2$ would have three metamours, and this contradicts the 2-metamour-regularity of $G$ .", "Suppose there is no additional vertex adjacent to $v_2$ .", "Then, in order to close the metamour cycle containing $(v_1, v_3^{\\prime }, v_2, w, v_3)$ , there needs to be a path from $v_3^{\\prime }$ to $w$ .", "This implies the existence of a cycle longer than $r=3$ , therefore cannot be.", "So there is an additional vertex adjacent to $v_2^{\\prime }$ ; the situation is shown in Figure REF (d).", "By the same argument as above, $v_1$ and $v_3$ are the two metamours of $v_2^{\\prime }$ .", "Therefore, $(w,v_2,v_3^{\\prime },v_1,v_2^{\\prime },v_3,w)$ is a 6-cycle in the metamour graph of $G$ and $n=6$ .", "This is the graph $G=H_{6}^{a}$ .", "We can only add additional edges between the vertices $w$ , $v_2^{\\prime }$ and $v_3^{\\prime }$ , but this would lead to a cycle of length larger than 3.", "So there are no other edges present.", "There cannot be any additional vertex because this vertex would need to be in a different cycle in the metamour graph, contradicting that the metamour graph is the $C_n$ .", "As a consequence of REF, the proof is finished for $r=3$ , because then $G = H_{6}^{a}$ .", "What is left to consider is the case $r\\ge 4$ and consequently $n\\ge 5$ .", "The situation is again as in Figure REF .", "E The only vertices of $\\gamma $ that are adjacent to $v_1$ are $v_2$ and $v_r$ .", "In particular, $v_1$ is metamour of $v_3$ and of $v_{r-1}$ .", "[Proof of  REF] Suppose there is a vertex $v_i$ with $i\\in {3,\\dots ,r-1}$ adjacent to $v_1$ .", "Then, as $w$ is not adjacent to $v_i$ by  REF or  REF, $v_i$ is a third metamour of $w$ .", "This contradicts the 2-metamour-regularity of $G$ .", "As $v_1$ has distance 2 on the cycle $\\gamma $ to both $v_3$ and $v_{r-1}$ , and is not adjacent to these vertices, the vertices $v_3$ and $v_{r-1}$ are metamours of $v_1$ .", "Figure: Subgraph of the situation between and At this point, we have the situation shown in Figure REF .", "Note, that it is still possible that $v_3 = v_{r-1}$ .", "F We cannot have $r=4$ .", "[Proof of  REF] As $r=4$ , we have $v_3=v_{r-1}$ .", "This situation is shown in Figure REF .", "Figure: Subgraph of the situationin the proof of Suppose there is an additional vertex $v_3^{\\prime }$ of $G$ adjacent to $v_3$ .", "Then by  REF, $v_3^{\\prime }$ has metamours $v_2$ and $v_4$ .", "Therefore, $(w,v_2,v_3^{\\prime },v_4,w)$ is a 4-cycle in the metamour graph of $G$ .", "This is a contradiction to the metamour graph being $C_n$ and $n\\ge 6$ , so there is no additional vertex adjacent to $v_3$ .", "This implies that we cannot have a vertex at distance 1 from $v_3$ other than $v_2$ and $v_4$ .", "Now suppose there is an additional vertex $v_2^{\\prime }$ of $G$ adjacent to $v_2$ .", "Again by  REF, $v_2^{\\prime }$ has metamours $v_1$ and $v_3$ .", "Therefore, $(v_2^{\\prime },v_1,v_3,v_2^{\\prime })$ is a 3-cycle in the metamour graph of $G$ .", "This is again a contradiction to the metamour graph being $C_n$ and $n\\ge 6$ , so there is no additional vertex adjacent to $v_2$ either.", "Likewise, by symmetry, there is also no additional vertex adjacent to $v_4$ .", "As $v_2$ and $v_4$ are the only neighbors of $v_3$ , we cannot have a vertex at distance 2 from $v_3$ other than $v_1$ .", "This means that there is no second metamour of $v_3$ which contradicts the 2-metamour-regularity of $G$ .", "At this point, we can assume that $r\\ge 5$ as the case $r=4$ was excluded by REF, and consequently also $n\\ge 6$ .", "The situation is still as in Figure REF .", "G We have $r\\le 6$ .", "Specifically, either $r=5$ , or $r=6$ and there is an edge ${v_2}{v_5}$ in $G$ .", "In the second case, the two metamours of the vertex $v_2$ are $w$ and $v_4$ .", "[Proof of  REF] As $r\\ge 5$ , the two metamours of $v_1$ are on the cycle $\\gamma $ , namely the distinct vertices $v_3$ and $v_{r-1}$ ; see Figure REF (a).", "We now consider the neighbors of $v_2$ .", "Suppose $v_2$ is adjacent to some $v_i$ with $i \\notin {1,3,r-1,r}$ .", "As the vertex $v_1$ is not connected to $v_{i}$ by REF, the vertex $v_{i}$ is a metamour of $v_1$ different from $v_3$ and $v_{r-1}$ .", "This contradicts the 2-metamour-regularity of $G$ .", "Furthermore, $v_2$ is adjacent to $v_1$ , $v_3$ and $v_r$ .", "This implies that the neighborhood of $v_2$ on $\\gamma $ is determined up to $v_{r-1}$ .", "We will now distinguish whether $v_{r-1}$ is or is not in this neighborhood.", "Figure: Subgraphs of the situationsin the proof of Suppose ${v_2}{v_{r-1}} \\notin E(G)$ .", "If $v_{r-1} \\ne v_4$ , then ${v_2}{v_4} \\notin E(G)$ because of what is shown in the previous paragraph.", "But then, the metamours of $v_2$ would be $w$ , $v_{r-1}$ and $v_4$ .", "This contradicts the 2-metamour-regularity of $G$ and implies that $v_{r-1}=v_4$ and $r=5$ ; see Figure REF (b).", "Suppose ${v_2}{v_{r-1}} \\in E(G)$ .", "We again distinguish between two cases.", "If $r \\ge 6$ , then $w$ , $v_4$ and $v_{r-2}$ are metamours of $v_2$ .", "In this case, the 2-metamour-regularity of $G$ implies that $v_{r-2}=v_4$ and therefore $r=6$ ; see Figure REF (c).", "If $r < 6$ , then by the findings so far, we must have $r=5$ , and therefore we are also done in this case.", "By  REF we are left with the two cases $r=5$ and $r=6$ .", "One possible situation for $r=5$ and the situation for $r=6$ are shown in Figure REF (b) and (c), respectively, and we will deal with these two situations now.", "H If $r=5$ , then $G\\in {H_{6}^{b}, H_{7}^{a}}$ .", "[Proof of  REF] The full situation for $r=5$ is shown in Figure REF (a).", "Figure: Subgraphs of the situationsin the proof of Clearly the situation is symmetric in the potential edges ${v_2}{v_4}$ and ${v_3}{v_5}$ , so we have to consider the three cases that both, one and none of these two edges are present.", "First let us assume that neither ${v_2}{v_4}$ nor ${v_3}{v_5}$ is an edge; see Figure REF (b).", "Then $v_2$ and $v_4$ as well as $v_3$ and $v_5$ are metamours, so we have the 6-cycle $(w,v_2,v_4,v_1,v_3,v_5,w)$ in the metamour graph of $G$ .", "This is the graph $G=H_{6}^{b}$ .", "There cannot be any additional vertex because this vertex would need to be in a different cycle in the metamour graph contradicting that the metamour graph is the $C_n$ .", "There also cannot be any additional edges because all edges and non-edges are already determined.", "Next let us assume that there is exactly one of the edges ${v_2}{v_4}$ and ${v_3}{v_5}$ present in $G$ , without loss of generality let ${v_3}{v_5} \\in E(G)$ ; see Figure REF (c).", "At this point we know that $v_3$ and $v_1$ as well as $v_5$ and $w$ are metamours, and we are looking for the second metamours of $v_3$ and $v_5$ .", "As the vertices $w$ , $v_1$ and $v_4$ already have two metamours each, there need to be additional vertices for these metamours.", "Statement  REF implies that there is no additional vertex of $G$ adjacent to $v_5$ as $v_1$ has already the two metamours $v_3$ and $v_4$ .", "Likewise, by symmetry, there is no additional vertex adjacent to $v_2$ .", "Moreover, by the same argument, there is also no additional vertex adjacent to $v_3$ as $v_4$ has the two metamours $v_1$ and $v_2$ .", "Therefore, there need to be an additional vertex $v_4^{\\prime }$ adjacent to $v_4$ .", "By  REF, $v_4^{\\prime }$ has metamours $v_3$ and $v_5$ .", "This gives the 7-cycle $(w,v_2,v_4,v_1,v_3,v_4^{\\prime },v_5,w)$ in the metamour graph of $G$ and the graph $G=H_{7}^{a}$ .", "There cannot be any additional vertex because this vertex would need to be in a different cycle in the metamour graph contradicting that the metamour graph is the $C_n$ .", "There also cannot be any additional edges because all edges and non-edges are already determined.", "At last, let us consider the case that both of the edges ${v_2}{v_4}$ and ${v_3}{v_5}$ are present in $G$ .", "We already know that $w$ is a metamour of $v_2$ and are now searching for the second metamour of $v_2$ .", "There does not exist a vertex $v_1^{\\prime }$ adjacent to $v_1$ in $G$ but not in $\\gamma $ , because this would induce a $C_4$ in the metamour graph by the same arguments as in the proof of  REF.", "Furthermore, there cannot be a vertex $v_2^{\\prime }$ in $G$ but not in $\\gamma $ that is adjacent to $v_2$ , due to the fact that this vertex would be a third metamour of $v_1$ , a contradiction.", "By symmetry, there is no vertex of $G$ without $\\gamma $ adjacent to $v_5$ .", "If there would be a vertex $v_3^{\\prime }$ in $G$ but not in $\\gamma $ which is adjacent to $v_3$ , then due to  REF, this vertex would have $v_2$ , $v_4$ and $v_5$ as a metamour, a contradiction to the 2-metamour-regularity of $G$ .", "Again by symmetry, there is no vertex of $G$ without $\\gamma $ adjacent to $v_4$ .", "Therefore, $v_2$ cannot have a second metamour in $G$ and this case cannot happen.", "Statement  REF finalizes the proof for $r=5$ .", "Hence, $r=6$ is the only remaining value for $r$ we have to consider.", "I We cannot have $r=6$ .", "[Proof of  REF] As $r=6$ , there is an edge ${v_2}{v_5}$ in $G$ by  REF.", "The initial situation is shown in Figure REF (a).", "Figure: Subgraphs of the situationsin the proof of Suppose $v_3$ and $v_5$ are not adjacent.", "Then $(v_1,v_3,v_5,v_1)$ is a 3-cycle in the metamour graph of $G$ .", "This contradicts that the metamour graph is $C_n$ and $n>r=6$ , so we can assume ${v_3}{v_5} \\in E(G)$ .", "Likewise, suppose that $v_4$ and $v_6$ are not adjacent.", "Then $(w,v_2,v_4,v_6,w)$ is a 4-cycle in the metamour graph of $G$ .", "This contradicts that the metamour graph is $C_n$ and $n>r=6$ , so we can assume ${v_4}{v_6} \\in E(G)$ .", "The current situation is shown in Figure REF (b).", "Statement  REF implies that there is no additional vertex of $G$ adjacent to $v_2$ as $v_1$ has already the two metamours $v_3$ and $v_5$ .", "By symmetry, there is also no additional vertex adjacent to $v_6$ .", "By the same argumentation as above, there is no additional vertex adjacent to $v_1$ as well as to $v_3$ because of vertex $v_2$ and its metamours.", "Moreover, we slightly vary the argumentation to show that there cannot be an additional vertex adjacent to $v_5$ .", "Suppose there is an additional vertex $v_5^{\\prime }$ of $G$ adjacent to $v_5$ .", "Then, $v_5^{\\prime }$ is not adjacent to $v_2$ as we have shown above, so $v_5^{\\prime }$ is as well a metamour of $v_2$ .", "This contradicts the 2-metamour-regularity of $G$ again.", "The vertex $v_4$ has $v_2$ as metamour.", "We are now searching for its second metamour.", "It cannot be $w$ or $v_1$ as these vertices have already two other metamours each.", "It cannot be any of $v_3$ , $v_5$ or $v_6$ either as all of them are adjacent to $v_4$ .", "Moreover, the second metamour of $v_4$ cannot be adjacent to $v_3$ , $v_5$ or $v_6$ , as we above ruled additional neighbors to these vertices out.", "Therefore, there has to be an additional vertex $v_4^{\\prime }$ adjacent to $v_4$ .", "By  REF, this vertex $v_4^{\\prime }$ has metamours $v_3$ and $v_5$ .", "This results in the 4-cycle $(v_1,v_3,v_4^{\\prime },v_5,v_1)$ in the metamour graph of $G$ and contradicts our assumption that this graph is $C_n$ and $n>r=6$ .", "We have now completed the proof of Lemma REF as in all cases we were able to show that $G \\in {H_{6}^{a},H_{6}^{b},H_{7}^{a}}$ holds.", "After characterizing all 2-metamour-regular graphs whose metamour graph is connected and that do not contain a cycle of length $n$ , we can now focus on 2-metamour-regular graphs whose metamour graph is connected and that contain a cycle of length $n$ .", "Here, we make a further distinction depending on the degree of the vertices and begin with the following lemma.", "Lemma 9.2 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph equals the $C_n$ , that contains a cycle of length $n$ , and that has a vertex of degree larger than 2 and smaller than $n-3$ .", "Then $G = H_{7}^{b}.$ Let $\\gamma $ be a cycle of length $n$ in $G$ .", "First, we introduce some notation.", "Let $v$ be a vertex of $G$ , and let $u$ and $u^{\\prime }$ be the two metamours of $v$ .", "We explore the vertices on the cycle $\\gamma $ starting with $v$ : The set of vertices on both sides of $v$ strictly before $u$ and $u^{\\prime }$ are called the fellows of $v$ .", "The remaining set of vertices strictly between $u$ and $u^{\\prime }$ on $\\gamma $ is called the opponents of $v$ ; see Figure REF .", "In other words for each vertex $v$ of $G$ the set of vertices of $G$ can be partitioned into $v$ , its fellows, its metamours and its opponents.", "Figure: Fellows and opponents of a vertex vvin the proof of Lemma We start with the following claims.", "A Every vertex of $G$ is adjacent to each of its fellows.", "[Proof of  REF] Let $v_1$ be a vertex of $G$ and $\\gamma =(v_1,\\dots ,v_n,v_1)$ .", "Suppose $v_p$ is the vertex with smallest index $p$ that is not adjacent to $v_1$ .", "We have to show that $v_p$ is a metamour of $v_1$ .", "The index $p$ exists because $v_1$ is not adjacent to its metamours.", "Moreover, this index satisfies $p>2$ as $v_2$ is adjacent to $v_1$ because they are consecutive vertices on $\\gamma $ .", "Thus, $v_1$ and $v_p$ have $v_{p-1}$ as common neighbor and are therefore metamours.", "By symmetry, the vertex $v_q$ with largest index $q$ that is not adjacent to $v_1$ , is also a metamour of $v_1$ .", "Note that as $v_1$ has exactly two metamours, $v_p$ and $v_q$ are these metamours, so $v_1$ is adjacent to each of its fellows.", "B Every vertex of $G$ is either adjacent to each of its opponents, or not adjacent to any of its opponents.", "[Proof of  REF] It is enough to show that if a vertex $v_1$ of $G$ is adjacent to at least one opponent of $v_1$ , then it is adjacent to every opponent of $v_1$ .", "Let $\\gamma =(v_1, \\dots , v_n,v_1)$ , and let $W$ be a subset of the opponents of $v_1$ that consists of consecutive vertices of $\\gamma $ , say from $v_{i}$ to $v_{j}$ for some $i \\le j$ , such that each of these vertices is adjacent to $v_1$ , and $W$ is maximal (with respect to inclusion) with this property.", "Note that the set $W$ is not empty because of our assumption.", "Clearly none of the vertices in $W$ is a metamour of $v_1$ .", "However $v_{i-1}$ and $v_{j+1}$ are metamours of $v_1$ because of their common neighbors $v_{i}$ and $v_{j}$ , and the maximality of $W$ .", "Therefore, as $v_1$ has exactly two metamours, $W$ equals the set of opponents of $v_1$ which was to show.", "Now we are ready to start with the heart of the proof of Lemma REF .", "Suppose $v_1$ is a vertex of $G$ with $2 < \\deg (v_1) < n-3$ .", "In order to complete the proof we have to show that $G=H_{7}^{b}$ .", "Let $\\gamma =(v_1, \\dots , v_n,v_1)$ be a cycle of length $n$ , and let $v_p$ and $v_q$ be the metamours of $v_1$ with $p < q$ .", "In the following claims we will derive several properties of $G$ .", "C The vertex $v_1$ is adjacent to its fellows $v_{2}$ , ..., $v_{p-1}$ , $v_{q+1}$ , ..., $v_n$ and not adjacent to any metamour or opponent $v_{p}$ , ..., $v_{q}$ .", "Furthermore, $p + 1 < q$ holds, i.e., there exists at least one opponent of $v_1$ .", "[Proof of  REF] Clearly $v_1$ is not adjacent to its metamours $v_p$ and $v_q$ .", "Furthermore, $v_1$ is adjacent to all its fellows $v_{2}$ , ..., $v_{p-1}$ , $v_{q+1}$ , ..., $v_n$ by  REF.", "This together with $\\deg (v_1) < n-3$ implies that $v_1$ has an opponent to which it is not adjacent, so $p + 1 < q$ .", "Then by  REF, $v_1$ is not adjacent to any of its opponents.", "Figure: Subgraph of the situation between and Now $\\deg (v_1) > 2$ together with  REF imply that $v_1$ has at least one fellow different from $v_2$ and $v_n$ .", "Without loss of generality (by renumbering the vertices in the opposite direction of rotation along $\\gamma $ ) assume that $v_{q + 1}$ is a fellow of $v_1$ different from $v_n$ , so in other words we assume $q + 1 < n$ .", "The situation is shown in Figure REF .", "We will now prove several claims about edges, non-edges and metamours of $G$ .", "D No opponent $v_{p+1}$ , ..., $v_{q-1}$ is adjacent to any fellow $v_{2}$ , ..., $v_{p-1}$ , $v_{q+1}$ , ..., $v_n$ .", "[Proof of  REF] Assume that $v_j$ is adjacent to $v_i$ for some $j \\in {p+1, \\dots , q-1}$ and some $i \\in {2, \\dots , p-1}\\cup {q+1, \\dots , n}$ .", "Then $v_j$ and $v_1$ have the common neighbor $v_i$ because of  REF.", "Furthermore, $v_j$ and $v_1$ are not adjacent by  REF, so $v_j$ and $v_1$ are metamours.", "This is a contradiction to $v_p$ and $v_q$ being the only metamours of $v_1$ , therefore our assumption was wrong.", "Figure: Subgraphs of the situationsbetween ,,and The known edges and non-edges at this moment are shown in Figure REF (a).", "E The vertices $v_{q-1}$ and $v_{q+1}$ are metamours of each other.", "Also the vertices $v_{p-1}$ and $v_{p+1}$ are metamours of each other.", "[Proof of  REF] The vertices $v_{q-1}$ and $v_{q+1}$ have the common neighbor $v_{q}$ and are not adjacent due to  REF, so they are metamours.", "Also $v_{p-1}$ and $v_{p+1}$ are metamours because they have $v_p$ as a common neighbor and are not adjacent because of  REF.", "Now we are in the situation shown in Figure REF (b).", "F The vertices $v_{q}$ and $v_{q + 2}$ are metamours of each other.", "Figure: Subgraph of the situationin the proof of [Proof of  REF] This proof is accompanied by Figure REF .", "Assume $v_{q}$ and $v_{q+2}$ are adjacent.", "Then $v_{q-1}$ and $v_{q+2}$ have the common neighbor $v_q$ and are not adjacent because of  REF.", "Hence, $v_{q+2}$ is a metamour of $v_{q-1}$ .", "Due to  REF, $v_{q + 1}$ is the second metamour of $v_{q-1}$ .", "Both metamours are consecutive vertices on the cycle $\\gamma $ , therefore, every other vertex except $v_{q-1}$ is a fellow of $v_{q-1}$ , thus adjacent to $v_{q-1}$ by  REF.", "In particular, $v_1$ is adjacent to $v_{q-1}$ which contradicts  REF.", "Therefore, $v_{q}$ and $v_{q+2}$ are not adjacent and because of their common neighbor $v_{q+1}$ , metamours.", "G The vertex $v_q$ is adjacent to $v_2$ , ..., $v_{p-1}$ , $v_p$ , $v_{p+1}$ , ..., $v_{q-1}$ .", "[Proof of  REF] The two metamours of $v_q$ are $v_1$ and $v_{q+2}$ because of  REF.", "This implies that $v_{2}$ , ..., $v_{q-1}$ are fellows of $v_q$ and therefore adjacent to $v_q$ because of  REF.", "Figure: Subgraph of the situationbetween  andFigure REF shows the current situation.", "H The vertices $v_{q-1}$ and $v_{p -1}$ are metamours of each other.", "Furthermore, $v_{p-1} = v_2$ holds, so there is exactly one fellow of $v_1$ on the cycle $\\gamma $ between $v_1$ and $v_p$ .", "[Proof of  REF] The vertex $v_{q-1}$ is not adjacent to any of $v_{2}$ , ..., $v_{p-1}$ due to  REF.", "Furthermore, $v_{q-1}$ has the common neighbor $v_q$ with each of these vertices because of  REF.", "So every vertex $v_{2}$ , ..., $v_{p-1}$ is a metamour of $v_{q-1}$ .", "This implies ${{ v_{2}, \\dots , v_{p-1}}} \\le 1$ because $v_{q-1}$ also has $v_{q+1}$ as metamour and has in total exactly two metamours.", "Moreover, as $v_2$ is adjacent to $v_1$ , $v_1$ and $v_2$ are not metamours, thus $v_2$ and $v_p$ cannot coincide.", "This implies $p > 2$ has to hold.", "In consequence, we obtain $p = 3$ implying $v_{p-1}=v_2$ has to hold.", "Figure: Subgraphs of the situationsbetween , andWe are now in the situation shown in Figure REF (a).", "I It holds that $v_{p+1} = v_{q-1}$ , so $v_1$ has exactly one opponent.", "[Proof of  REF] The vertices $v_{q+1}$ and $v_{p-1}$ are metamours of $v_{q-1}$ because of  REF and  REF.", "Furthermore, $v_{p-1}$ and $v_{p+1}$ are metamours because of  REF.", "Now assume $p + 1 < q - 1$ , so the vertices $v_{p+1}$ and $v_{q-1}$ are distinct.", "Then $v_{p+1}$ and $v_{q+1}$ have the common neighbor $v_q$ because of  REF and they are not adjacent because of  REF, so they are metamours.", "This implies that $(v_{q-1}, v_{q+1}, v_{p+1},v_{p-1},v_{q-1})$ is a cycle in the metamour graph that does not contain all vertices, a contradiction to our assumption.", "So $p + 1 = q - 1$ .", "Now we are in the situation shown in of Figure REF (b).", "Figure: Subgraphs of the situationsbetween , and J The vertices $v_{p}$ and $v_{q + 1}$ are metamours of each other.", "Furthermore, $v_{p}$ is not adjacent to any of the vertices $v_{q+2}$ , ..., $v_{n}$ .", "[Proof of  REF] If $v_p$ is adjacent to $v_i$ for $i \\in {q+2, \\dots , n}$ , then $v_{p+1}$ and $v_i$ are metamours because they have $v_p$ as a common neighbor, and they are not adjacent due to  REF.", "This is a contradiction as $v_{p+1}$ already has the two metamours $v_{p-1}$ and $v_{q+1}$ because of  REF and an implication of  REF.", "As a result, $v_p$ is not adjacent to any of $v_{q+2}$ , ..., $v_{n}$ .", "If $v_p$ would be adjacent to $v_{q+1}$ , then $v_p$ and $v_{q+2}$ are metamours because of the common neighbor $v_{q+1}$ and because they are not adjacent by the above.", "But then, due to  REF, $(v_p,v_1,v_q,v_{q+2},v_p)$ is a cycle in the metamour graph which does not contain all vertices, a contradiction to our assumption.", "Therefore, $v_p$ is not adjacent to $v_{q+1}$ .", "The vertex $v_p$ is adjacent to $v_{q}$ due to  REF, therefore $v_{q}$ is a common neighbor of $v_p$ and $v_{q+1}$ , and hence these vertices are metamours.", "Figure REF (a) shows the situation.", "K It holds that $q + 2 = n$ , so there are exactly two fellows of $v_1$ on the cycle $\\gamma $ between $v_q$ and $v_1$ .", "Furthermore, the vertices $v_{p-1}$ and $v_{q+2}$ are metamours of each other, and $v_{p-1}$ is adjacent to all vertices except its metamours.", "[Proof of  REF] The vertex $v_{p-1}$ is a metamour of $v_{q-1}$ due to  REF, and it is adjacent to $v_q$ because of  REF.", "This together with $p+1 = q-1$ by  REF implies that $v_{p-1}$ is adjacent to one of its opponents, namely $v_{q}$ .", "Then by  REF and  REF, this implies that $v_{p-1}$ is adjacent to all vertices except its metamours.", "If $v_{p-1}$ is adjacent to a vertex $v_i$ for $i \\in {q+2, \\dots , n}$ , then $v_{p}$ and $v_{i}$ are metamours because they have $v_{p-1}$ as common neighbor and are not adjacent due to  REF.", "But $v_p$ already has the two metamours $v_1$ and $v_{q+1}$ due to  REF, a contradiction.", "As a result, $v_{p-1}$ is not adjacent to any vertex of $v_{q+2}$ , ..., $v_n$ .", "Now assume $q + 2 < n$ , so the vertex $v_{q+3}$ exists.", "Due to the fact that $v_{p-1}$ is adjacent to all vertices except its metamours and that it has $v_{p+1}$ as metamour by  REF, it follows that it is adjacent to at least one of $v_{q+2}$ and $v_{q+3}$ .", "But we showed that $v_{p-1}$ is not adjacent to any of these two vertices, a contradiction.", "Therefore, $q + 2 = n$ holds.", "Furthermore, $v_{p-1}$ is not adjacent to $v_{q+2}$ , and therefore these two vertices are metamours of each other.", "Our final figure is Figure REF (b).", "L It holds that $G = H_{7}^{b}$ .", "[Proof of  REF] We have $p-1 = 2$ by  REF, we have $p+1=q-1$ by  REF and $q+2=n$ by  REF.", "This implies that $n=7$ .", "The properties we have derived so far fix all edges and non-edges of $G$ except between $v_2$ and $v_7$ .", "This has to be a non-edge to close the metamour cycle.", "The result is $G = H_{7}^{b}$ .", "With respect to Figure REF , $v_1$ is the top left vertex of $H_{7}^{b}$ and the vertices are numbered clock-wise.", "This completes the proof of Lemma REF .", "Next we consider all cases of 2-metamour-regular graphs whose metamour graph is connected, that contain a cycle of length $n$ and whose degrees are not as in the previous lemma.", "Lemma 9.3 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph equals the $C_n$ , that contains a cycle of length $n$ , and in which every vertex has degree $n-3$ .", "Then $G=\\complement {C_n}.$ If a vertex $v$ of $G$ has degree $n-3$ , then $v$ is adjacent to all but two vertices.", "These two vertices are exactly the metamours of $v$ .", "This implies that $G$ equals the complement of the metamour graph.", "Hence, $G=\\complement {C_n}$ as the metamour graph of $G$ is the $C_n$ .", "Lemma 9.4 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph equals the $C_n$ , that contains a cycle of length $n$ , and in which every vertex has degree 2.", "Then $G=C_n$ and $n$ is odd.", "Let $\\gamma $ be a cycle of length $n$ in $G$ .", "If every vertex of $G$ has degree 2, then every vertex in the induced subgraph $G[\\gamma ]$ has degree 2 as $\\gamma $ contains every vertex by assumption.", "As $G[\\gamma ]$ is connected, it equals $C_n$ .", "In total this implies $G=G[\\gamma ]=C_n$ .", "It is easy to see that if $n$ is even, then the metamour graph consists of exactly two cycles of length $\\frac{n}{2}$ which contradicts our assumption.", "Therefore, $n$ is odd.", "Lemma 9.5 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph equals the $C_n$ , that contains a cycle of length $n$ , in which every vertex has degree 2 or $n-3$ , and that has a vertex of degree 2 and a vertex of degree $n-3$ .", "Then $G \\in {C_5, H_{6}^{c}}.$ Let $\\gamma =(v_1,\\dots ,v_n,v_1)$ be a cycle of length $n$ such that $\\deg (v_1) = 2$ and $\\deg (v_2) = n-3$ .", "A We have $5 \\le n \\le 7$ and the metamours of $v_1$ are $v_3$ and $v_{n-1}$ .", "Figure: Subgraph of the situationin the proof of [Proof of  REF] Clearly $v_1$ is only adjacent to $v_2$ and $v_n$ .", "Hence, $v_3$ and $v_{n-1}$ have to be the two metamours of $v_1$ and $G$ contains at least 5 different vertices, so $n \\ge 5$ .", "If $v_2$ is adjacent to some $v_i$ for $i\\in {4, \\dots , n-2}$ , then $v_1$ is a metamour of $v_i$ due to the common neighbor $v_2$ ; see Figure REF .", "Hence, $v_2$ is not adjacent to any vertex $v_4$ , ..., $v_{n-2}$ .", "However, because $\\deg (v_2) = n-3$ , the vertex $v_2$ is adjacent to every vertex but its two metamours.", "This implies that ${{v_4, \\dots , v_{n-2}}} \\le 2$ , because $v_2$ has at most two metamours among $v_{4}$ , ..., $v_{n-2}$ .", "As a result, we have $n \\le 7$ .", "This implies that $n=5$ , $n=6$ and $n=7$ are the only cases to consider.", "We do so in the following claims.", "B If $n=5$ , then $G=C_n$ .", "Figure: Subgraphs of the situationsin the proof of [Proof of  REF] If $n = 5$ , then $v_3$ and $v_{n-1}=v_4$ are the two metamours of $v_1$ ; see Figure REF (a).", "Then $v_2$ is the only option as second metamour of $v_4$ , and $v_5$ is the only option as second metamour of $v_3$ .", "Then $v_2$ and $v_5$ have to be metamours in order to close the cycle in the metamour graph; see Figure REF (b).", "As a result, we have $G = C_5$ .", "C If $n=6$ , then $G=H_{6}^{c}$ .", "Figure: Subgraphs of the situationsin the proof of [Proof of  REF] If $n = 6$ , then $v_3$ and $v_{n-1}=v_5$ are the two metamours of $v_1$ .", "If $v_3$ is not adjacent to $v_5$ , then $v_3$ and $v_5$ are metamours because of their common neighbor $v_4$ ; see Figure REF (a).", "But then $(v_1,v_3,v_5,v_1)$ is a cycle in the metamour graph that does not contain all vertices, a contradiction to our assumption.", "Hence, $v_3$ and $v_5$ are adjacent; see Figure REF (b).", "Then $v_6$ is the only option left as the second metamour of $v_3$ , and $v_2$ is the only option left as the second metamour of $v_5$ .", "If $v_2$ and $v_6$ are not adjacent, then they are metamours because of their common neighbor $v_1$ .", "But then $(v_1,v_3,v_6,v_2,v_5,v_1)$ is a cycle in the metamour graph that does not contain $v_4$ , a contradiction.", "So $v_2$ and $v_6$ are adjacent; see Figure REF (c).", "But then $v_4$ has to have $v_2$ and $v_6$ as metamours, because they are the only options left.", "Hence, we obtain $G = H_{6}^{c}$ .", "D We cannot have $n=7$ .", "Figure: Subgraphs of the situationsin the proof of [Proof of  REF] If $n=7$ , then $v_3$ and $v_{n-1}=v_6$ are the two metamours of $v_1$ .", "As $\\deg (v_1) = 2$ , the vertices $v_1$ and $v_5$ are not adjacent, and they are also not metamours; see Figure REF (a).", "Therefore, $\\deg (v_5)<n-3=4$ .", "As the only options are $\\deg (v_5) \\in {2,n-3}$ , we conclude $\\deg (v_5)=2$ .", "As a result, $v_5$ is only adjacent to $v_4$ and $v_6$ , and the vertices $v_3$ and $v_7$ have to be the two metamours of $v_5$ ; see Figure REF (b).", "In particular, $v_5$ is not adjacent to $v_2$ , and the vertices $v_5$ and $v_2$ are not metamours.", "This implies $\\deg (v_2)<n-3=4$ which is a contradiction to $\\deg (v_2)=4$ .", "Hence, $n=7$ is not possible.", "To summarize, in the case that not all vertices of $G$ have the same degree in ${2,n-3}$ , $G=C_5$ and $G=H_{6}^{c}$ are the only possible graphs due to  REF,  REF,  REF and  REF.", "This finishes the proof of Lemma REF .", "Eventually, we can collect all results on 2-metamour-regular graphs that have a connected metamour graph in the following proposition.", "Proposition 9.6 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices whose metamour graph is the $C_n$ .", "Then $n \\ge 5$ and one of $G = C_n$ and $n$ is odd, $G = \\complement {C_n}$ , or $G \\in {H_{6}^{a},H_{6}^{b},H_{6}^{c},H_{7}^{a},H_{7}^{b}}$ holds.", "First we derive two properties of $G$ in the following claims.", "A We have $n\\ge 5$ .", "[Proof of  REF] The graph $G$ is connected, hence it contains at least $n-1$ edges.", "Furthermore, the graph $G$ has the metamour graph $C_n$ , so the complement of $G$ contains at least $n$ edges.", "As the sum of the number of edges of $G$ and of the complement of $G$ is equal to $\\binom{n}{2}$ , we have $\\binom{n}{2} \\ge (n-1) + n$ .", "This is only true for $n \\ge 5$ .", "B The graph $G$ is not a tree.", "[Proof of  REF] Suppose that $G$ is a tree.", "We first show that the maximum degree of $G$ is at most 2.", "Let $v$ be a vertex and $d$ its degree, and let $v_1$ , ..., $v_d$ its neighbors.", "Then no vertices of a pair in ${v_1, \\dots , v_d}$ are adjacent, as otherwise we would have a cycle.", "Therefore, the vertices of every such pair are metamours.", "We cannot have $d\\ge 4$ , as otherwise one vertex of ${v_1, \\dots , v_d}$ would have at least three metamours, and this contradicts the 2-metamour-regularity of the graph $G$ .", "If $d=3$ , then there is a 3-cycle in the metamour graph of $G$ which contradicts that the metamour graph is $C_n$ and $n\\ge 5$ .", "Therefore, $d\\le 2$ , and consequently we have indeed shown that the maximum degree of $G$ is at most 2.", "This now implies that $G$ has to be the path graph $P_n$ which is again a contradiction to $G$ being 2-metamour-regular, as an end vertex of $P_n$ only has one metamour.", "So by  REF, $G$ is not a tree, therefore it contains a cycle.", "If $G$ does not contain a cycle of length $n$ , then we can apply Lemma REF and conclude that $G \\in {H_{6}^{a}, H_{6}^{b}, H_{7}^{a}}$ .", "We are finished in this case.", "Otherwise, the graph $G$ contains a cycle of length $n$ .", "If there is a vertex $v$ of $G$ with $2 < \\deg (v) < n-3$ , then we can use Lemma REF , deduce that $G = H_{7}^{b}$ and the proof is complete in this case.", "Otherwise, every vertex has degree at most 2 or at least $n-3$ .", "Due to the fact that $G$ contains a cycle of length $n$ , the degree of every vertex is at least 2.", "Because $G$ is 2-metamour-regular, every vertex is not adjacent to at least two vertices, so the degree of every vertex is at most $n-3$ .", "This implies that every vertex has degree 2 or $n-3$ .", "If all vertices of $G$ have the same degree, then Lemma REF (for degrees $n-3$ ) implies $G=\\complement {C_n}$ and Lemma REF (for degrees 2) implies $G=C_n$ and $n$ odd.", "Hence, in these cases we are finished with the proof as well.", "What is left to consider is the situation that there are two vertices with different degrees in $G$ .", "This is done in Lemma REF , and we conclude $G \\in {C_5, H_{6}^{c}}$ in this case.", "This completes the proof." ], [ "Graphs with disconnected metamour graph", "After characterizing all graphs that are 2-metamour-regular and that have a connected metamour graph, we now turn to 2-metamour-regular graphs that do not have a connected metamour graph.", "In this case either statement REF or statement REF of Theorem REF is satisfied.", "In the case of REF there is nothing left to do, because it provides a characterization.", "In the other case we determine all graphs and capture them in the following proposition.", "Proposition 9.7 Let $G$ be a connected 2-metamour-regular graph with $n$ vertices.", "Suppose statement REF of Theorem REF is satisfied.", "Then $n \\ge 6$ and one of $G = C_n$ and $n$ is even, or $G \\in {H_{4,4}^{a},H_{4,4}^{b},H_{4,4}^{c},H_{4,3}^{a},H_{4,3}^{b},H_{4,3}^{c},H_{4,3}^{d},H_{3,3}^{a},H_{3,3}^{b},H_{3,3}^{c},H_{3,3}^{d},H_{3,3}^{e}}$ holds.", "Theorem  REFREF implies that the metamour graph is not connected.", "First observe that by Observation REF , each connected component of the metamour graph of a 2-metamour-regular graph is a cycle.", "The proof is split into several claims.", "As a first step, we consider the number of vertices of $G$ .", "A We have $n \\ge 6$ .", "[Proof of  REF] As the metamour graph of $G$ is not connected, the metamour graph contains at least two connected components, which are cycles.", "Each cycle has to contain at least three vertices, so $n \\ge 6$ .", "Now we come to the main part of the proof.", "Theorem  REFREF states that the metamour graph consists of exactly two connected components; we denote these by $M^{\\prime }$ and $M^\\ast $ .", "Set $G^{\\prime }=G[V(M^{\\prime })]$ and $G^\\ast =G[V(M^\\ast )]$ .", "Then $G^M$ (as in Theorem REF ) equals $G^{\\prime } \\cup G^\\ast $ .", "The definitions of $G^{\\prime }$ and $G^\\ast $ are symmetric and we might switch the roles of the two without loss of generality during the proof and in the statements.", "We introduce the following notion: The signature $\\sigma $ of a graph is the tuple of the numbers of vertices of its connected components, sorted in descending order.", "If follows from REF of Theorem  REFREF that all connected components of $G^{\\prime }$ and $G^\\ast $ have at most 2 vertices.", "As a consequence, the signatures $\\sigma (G^{\\prime })$ and $\\sigma (G^\\ast )$ have entries in ${1,2}$ .", "Note that in case a connected component has two vertices, then these vertices are adjacent, i.e., this component equals $P_2$ .", "We perform a case distinction by the signatures of the graphs $G^{\\prime }$ and $G^\\ast $ ; this is stated as the following claims.", "We start with the case that at least one connected component of $G^{\\prime }$ or $G^\\ast $ has two vertices.", "B If the first (i.e., largest) entry of $\\sigma (G^{\\prime })$ is 2, then either $\\sigma (G^{\\prime })=(2,2)$ or $\\sigma (G^{\\prime })=(2,1,1)$ .", "In the latter case, the two vertices of the two connected components containing only one vertex do not have any common neighbor in $G$ .", "[Proof of  REF] Suppose we have $G^{\\prime }_1 \\in \\mathcal {C}(G^{\\prime })$ with ${V(G^{\\prime }_1)} = 2$ .", "Let $v_1$ be one of the two vertices of $G^{\\prime }_1$ .", "Then $v_1$ is adjacent to the other vertex of $G^{\\prime }_1$ , therefore it must have its two metamours in another component of $G^{\\prime }$ .", "Let us assume that a metamour of $v_1$ is in a connected component $G^{\\prime }_2$ of $G^{\\prime }$ that consists of two vertices.", "Then every vertex of $G^{\\prime }_1$ is a metamour of every vertex of $G^{\\prime }_2$ due to REF of Theorem  REFREF .", "This implies that the four vertices of $G^{\\prime }_1$ and $G^{\\prime }_2$ form a $C_4$ in the metamour graph and consequently that $M^{\\prime }=C_4$ .", "Therefore, $G^{\\prime }$ cannot contain other vertices and $\\sigma (G^{\\prime })=(2,2)$ .", "Let us now assume that the two metamours of $v_1$ are in different connected components $G^{\\prime }_2$ and $G^{\\prime }_3$ of $G^{\\prime }$ that consist of only one vertex each.", "Then also the other vertex of $G^{\\prime }_1$ is a metamour of the two vertices in $G^{\\prime }_2$ and $G^{\\prime }_3$ due to REF of Theorem  REFREF .", "Hence, these four vertices form again a $C_4$ in the metamour graph and consequently $M^{\\prime }=C_4$ .", "As a result, $G^{\\prime }$ cannot contain any more vertices, so $\\sigma (G^{\\prime })=(2,1,1)$ .", "If the two vertices of $G^{\\prime }_2$ and $G^{\\prime }_3$ have a common neighbor, then they are metamours of each other because they are not adjacent.", "This is a contradiction to the fact that the vertex of $G^{\\prime }_2$ already has two metamours; they are in $G^{\\prime }_1$ .", "Hence, the vertices of $G^{\\prime }_2$ and $G^{\\prime }_3$ do not have any common neighbor.", "By  REF we can deduce how the graphs $G^{\\prime }$ and $G^\\ast $ look like, if one of their connected component contains 2 vertices.", "We will now continue by going through all possible combinations of signatures of $G^{\\prime }$ and $G^\\ast $ implied by  REF.", "In every case, we have to determine which edges between vertices of $G^{\\prime }$ and $G^\\ast $ exist and which do not exist in order to specify the graph $G$ .", "Due to REF of Theorem  REFREF , we know that as soon as there is an edge in $G$ between a connected component of $G^{\\prime }$ and a connected component of $G^\\ast $ , then there are all possible edges between these two components in $G$ .", "This implies, now rephrased in the language introduced in Section REF , adjacency of two connected components of $G^M$ is equivalent to complete adjacency of these components.", "Therefore, we equip $G^M=G^{\\prime } \\cup G^\\ast $ with a graph structure: The set $\\mathcal {C}(G^M)$ is the vertex set and the edge set—we simply write it as $E(G^M)$ —is determined by the adjacency relation above.", "Note that this graph is bipartite.", "C If $\\sigma (G^{\\prime })=(2,2)$ and $\\sigma (G^\\ast )=(2,2)$ , then we have $G \\in {\\complement {C_4} \\mathbin \\nabla \\complement {C_4},H_{4,4}^{a}}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2} = \\mathcal {C}(G^\\ast )$ .", "Each of these four components has size 2, therefore, $n=8$ .", "This proof is accompanied by Figure REF (a).", "The components $G^{\\prime }_1$ and $G^{\\prime }_2$ need a common neighbor in $G^\\ast $ with respect to $G^M$ because their vertices are metamours; see the proof of REF.", "So, we assume without loss of generality (by renumbering the connected components of $G^\\ast $ ) that ${G^{\\prime }_1}{G^\\ast _1} \\in E(G^M)$ and ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ .", "Furthermore, $G^\\ast _2$ needs to be adjacent to at least one connected component of $G^{\\prime }$ because $G$ is connected, so assume without loss of generality (by renumbering the connected components of $G^{\\prime }$ ) that ${G^{\\prime }_2}{G^\\ast _2} \\in E(G^M)$ .", "Now there is only the edge between $G^{\\prime }_1$ and $G^\\ast _2$ left to consider.", "If ${G^{\\prime }_1}{G^\\ast _2} \\in E(G^M)$ , then $G = \\complement {C_4} \\mathbin \\nabla \\complement {C_4}$ and if ${G^{\\prime }_1}{G^\\ast _2} \\notin E(G^M)$ then $G=H_{4,4}^{a}$ .", "This completes the proof.", "Figure: Subgraphs of the situationsin the proofs of and D If $\\sigma (G^{\\prime })=(2,2)$ and $\\sigma (G^\\ast )=(2,1,1)$ , then we have $G = H_{4,4}^{b}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2,G^\\ast _3} = \\mathcal {C}(G^\\ast )$ such that ${V(G^\\ast _1)}=2$ .", "The size of each other component is then determined, specifically we have ${V(G^{\\prime }_1)} = 2$ , ${V(G^{\\prime }_2)} = 2$ , ${V(G^\\ast _2)}=1$ and ${V(G^\\ast _3)}=1$ .", "Therefore, $n=8$ .", "This proof is accompanied by Figure REF (b).", "The component $G^\\ast _1$ need to be adjacent in $G^M$ to at least one connected component of $G^{\\prime }$ , so assume without loss of generality (by renumbering the connected components of $G^{\\prime }$ ) that ${G^{\\prime }_1}{G^\\ast _1} \\in E(G^M)$ .", "It is not possible that both $G^\\ast _2$ and $G^\\ast _3$ are adjacent in $G^M$ to $G^{\\prime }_1$ due to  REF, so assume without loss of generality (by renumbering $G^\\ast _2$ and $G^\\ast _3$ ) that ${G^{\\prime }_1}{G^\\ast _3} \\notin E(G^M)$ .", "But $G^\\ast _3$ must have a common neighbor in $G^{\\prime }$ with $G^\\ast _1$ because their vertices are metamours, so this implies ${G^{\\prime }_2}{G^\\ast _3} \\in E(G^M)$ and ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ .", "Due to  REF, this implies that ${G^{\\prime }_2}{G^\\ast _2} \\notin E(G^M)$ .", "But as $G^\\ast _2$ needs to be adjacent to at least one connected component of $G^{\\prime }$ , we find ${G^{\\prime }_1}{G^\\ast _2} \\in E(G^M)$ .", "As a consequence, we obtain $G = H_{4,4}^{b}$ .", "E If $\\sigma (G^{\\prime })=(2,1,1)$ and $\\sigma (G^\\ast )=(2,1,1)$ , then we have $G = H_{4,4}^{c}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2,G^{\\prime }_3} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2,G^\\ast _3} = \\mathcal {C}(G^\\ast )$ such that ${V(G^{\\prime }_1)}=2$ and ${V(G^\\ast _1)}=2$ .", "The size of each other component is then determined.", "We get $n=8$ .", "Because of  REF, not both $G^\\ast _2$ and $G^\\ast _3$ can be adjacent in $G^M$ to $G^{\\prime }_1$ , so assume without loss of generality (by renumbering $G^\\ast _2$ and $G^\\ast _3$ ) that ${G^{\\prime }_1}{G^\\ast _3} \\notin E(G^M)$ .", "If ${G^{\\prime }_1}{G^\\ast _2} \\notin E(G^M)$ , then $G^\\ast _1$ is the only possible common neighbor in $G^M$ for $G^{\\prime }_1$ and $G^{\\prime }_2$ as well as $G^{\\prime }_1$ and $G^{\\prime }_3$ .", "But then $G^{\\prime }_2$ and $G^{\\prime }_3$ would have the common neighbor $G^\\ast _1$ in $G^M$ , a contradiction to  REF.", "As a result, we obtain ${G^{\\prime }_1}{G^\\ast _2} \\in E(G^M)$ .", "By symmetric arguments for $G^\\ast $ , we can assume without loss of generality (by renumbering $G^{\\prime }_2$ and $G^{\\prime }_3$ ) that ${G^{\\prime }_3}{G^\\ast _1} \\notin E(G^M)$ and then deduce that ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ .", "The current situation is shown in Figure REF (a).", "Figure: Subgraphs of the situationsin the proof of As a result, $G^{\\prime }_2$ is the only possible common neighbor in $G^M$ of $G^\\ast _1$ and $G^\\ast _3$ , so ${G^{\\prime }_2}{G^\\ast _3} \\in E(G^M)$ .", "Analogously, we obtain ${G^{\\prime }_3}{G^\\ast _2} \\in E(G^M)$ .", "Now due to  REF, we can deduce that ${G^{\\prime }_2}{G^\\ast _2} \\notin E(G^M)$ and ${G^{\\prime }_3}{G^\\ast _3} \\notin E(G^M)$ .", "So far $G^{\\prime }_1$ and $G^{\\prime }_2$ do not have a common neighbor, and $G^\\ast _1$ is the only possibility for that left, so ${G^{\\prime }_1}{G^\\ast _1} \\in E(G^M)$ ; see Figure REF (b).", "This fully determines $G$ and it holds that $G = H_{4,4}^{c}$ .", "With  REF,  REF and  REF we have considered all cases in which both $G^{\\prime }$ and $G^\\ast $ have a connected component consisting of two vertices.", "So from now on we can assume that at least one of $G^{\\prime }$ and $G^\\ast $ has no connected component consisting of two vertices.", "Next we will deduce a result in the case that the signature of one of $G^{\\prime }$ and $G^\\ast $ is $(1,\\dots ,1)$ with at least four entries.", "F If $\\sigma (G^{\\prime })=(1,\\dots ,1)$ , $r$ times with $r\\ge 4$ , then we have $G=C_n$ and $n$ is even.", "[Proof of  REF] Let ${G^{\\prime }_1,\\dots ,G^{\\prime }_r} = \\mathcal {C}(G^{\\prime })$ .", "Let without loss of generality (by renumbering the connected components of $G^{\\prime }$ ) the vertices of $G^{\\prime }_{i-1}$ and $G^{\\prime }_{i+1}$ be the two metamours of the vertex of $G^{\\prime }_i$ .", "Note that we take the indices modulo $r$ and that we keep doing this for the remaining proof.", "Let $i \\in {1,\\dots ,r}$ .", "Then clearly $G^{\\prime }_i$ and $G^{\\prime }_{i+1}$ have a common neighbor $G^\\ast _i \\in \\mathcal {C}(G^\\ast )$ .", "If $G^\\ast _i$ is adjacent to any other connected component of $\\mathcal {C}(G^{\\prime })$ , then the vertex of this component together with the two vertices of $G^{\\prime }_i$ and $G^{\\prime }_{i+1}$ form a $C_3$ in the metamour graph.", "This is a contradiction, because $M^{\\prime }$ is $C_r$ with $r\\ge 4$ .", "Hence, $G^\\ast _i$ is adjacent in $G$ to only $G^{\\prime }_i$ and $G^{\\prime }_{i+1}$ of $G^{\\prime }$ .", "In particular, this implies that the components $G^\\ast _1$ , ..., $G^\\ast _r$ are pairwise disjoint due to REF of Theorem  REFREF .", "Now because of the common neighbors, the vertices of $G^\\ast _i$ have the vertices of $G^\\ast _{i-1}$ and $G^\\ast _{i+1}$ as metamours.", "Therefore, as we are 2-metamour-regular, every component $G^\\ast _i$ consists of exactly one vertex.", "As a consequence, $G^\\ast _1$ , ..., $G^\\ast _r$ lead to a $C_r$ in the metamour graph of $G$ , specifically $M^\\ast =C_r$ .", "It is easy to see that the vertices of $G^{\\prime }_1$ , $G^\\ast _1$ , $G^{\\prime }_2$ , $G^\\ast _2$ , ..., $G^{\\prime }_r$ , $G^\\ast _r$ form a $C_{2r}$ .", "As we have ruled out all other possible edges, this implies that $G=C_{2r}$ .", "Hence, $n=2r$ and $G=C_n$ for $n$ even.", "In  REF, we have dealt with signatures $(1,\\dots ,1)$ of length at least 4.", "We will consider $(1,1,1)$ below.", "There cannot be fewer than three connected components of only single vertices because each connected component of the metamour graph is a cycle and therefore has at least 3 vertices.", "So what is left to consider are the two cases that $G^{\\prime }$ contains a connected component with two vertices and $G^\\ast $ has three isolated vertices and the case that both $G^{\\prime }$ and $G^\\ast $ have three isolated vertices.", "We consider these cases in the following claims.", "G If $\\sigma (G^{\\prime })=(2,2)$ and $\\sigma (G^\\ast )=(1,1,1)$ , then we have $G \\in {\\complement {C_4} \\mathbin \\nabla \\complement {C_3}, H_{4,3}^{a},H_{4,3}^{b}}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2,G^\\ast _3} = \\mathcal {C}(G^\\ast )$ .", "The size of each component is then determined, and we get $n=7$ .", "The single-vertex components $G^\\ast _1$ and $G^\\ast _2$ need a common neighbor, as well as $G^\\ast _2$ and $G^\\ast _3$ , and $G^\\ast _1$ and $G^\\ast _3$ .", "At least two of these common neighbors are from the same connected component of $G^{\\prime }$ , because $G^{\\prime }$ has only two connected components.", "Let without loss of generality (by renumbering $G^{\\prime }_1$ and $G^{\\prime }_2$ ) this connected component be $G^{\\prime }_1$ .", "As a result, every component $G^\\ast _1$ , $G^\\ast _2$ and $G^\\ast _3$ is adjacent to $G^{\\prime }_1$ , and ${G^{\\prime }_1}{G^\\ast _1}$ , ${G^{\\prime }_1}{G^\\ast _2}$ and ${G^{\\prime }_1}{G^\\ast _2} \\in E(G^M)$ .", "The connected component $G^{\\prime }_2$ has to be adjacent to some component of $G^\\ast $ because $G$ is connected, so assume without loss of generality (by renumbering $G^\\ast _1$ , $G^\\ast _2$ and $G^\\ast _3$ ) that ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ .", "The current situation is shown in Figure REF (a).", "Now if both ${G^{\\prime }_2}{G^\\ast _2}$ and ${G^{\\prime }_2}{G^\\ast _3}$ are in $E(G^M)$ , then $G$ is fully determined and $G=\\complement {C_4} \\mathbin \\nabla \\complement {C_3}$ .", "If only one of ${G^{\\prime }_2}{G^\\ast _2}$ and ${G^{\\prime }_2}{G^\\ast _3}$ is in $E(G^M)$ , then we have $G = H_{4,3}^{a}$ , and if none of ${G^{\\prime }_2}{G^\\ast _2}$ and ${G^{\\prime }_2}{G^\\ast _3}$ is in $E(G^M)$ , then $G=H_{4,3}^{b}$ .", "As one of these three settings has to occur, this proof is completed.", "Figure: Subgraphs of the situationsin the proofs of and H If $\\sigma (G^{\\prime })=(2,1,1)$ and $\\sigma (G^\\ast )=(1,1,1)$ , then we have $G \\in {H_{4,3}^{c}, H_{4,3}^{d}}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2,G^{\\prime }_3} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2,G^\\ast _3} = \\mathcal {C}(G^\\ast )$ such that ${V(G^{\\prime }_1)}=2$ .", "The size of each other component is then determined.", "We get $n=7$ .", "As $G$ is connected, let us assume without loss of generality (by renumbering $G^\\ast _1$ , $G^\\ast _2$ and $G^\\ast _3$ ) that ${G^{\\prime }_2}{G^\\ast _2}$ and ${G^{\\prime }_3}{G^\\ast _3} \\in E(G^M)$ .", "Because of  REF, the vertices of $G^{\\prime }_2$ and $G^{\\prime }_3$ cannot have a common neighbor, therefore ${G^{\\prime }_2}{G^\\ast _3}$ and ${G^{\\prime }_3}{G^\\ast _2} \\notin E(G^M)$ holds.", "The only choice for a common neighbor of $G^\\ast _2$ and $G^\\ast _3$ is $G^{\\prime }_1$ , therefore ${G^{\\prime }_1}{G^\\ast _2}$ and ${G^{\\prime }_1}{G^\\ast _3} \\in E(G^M)$ .", "If $G^{\\prime }_1$ is not adjacent in $G^M$ to $G^\\ast _1$ , then we have to have ${G^{\\prime }_2}{G^\\ast _1}$ and ${G^{\\prime }_3}{G^\\ast _1} \\in E(G^M)$ so that $G^\\ast _1$ and $G^\\ast _2$ as well as $G^\\ast _1$ and $G^\\ast _3$ have a common neighbor.", "But then $G^{\\prime }_2$ and $G^{\\prime }_3$ get the common neighbor $G^\\ast _1$ which is a contradiction to  REF.", "Therefore, we have ${G^{\\prime }_1}{G^\\ast _1} \\in E(G^M)$ .", "The current situation is shown in Figure REF (b).", "Furthermore, not both of $G^{\\prime }_2$ and $G^{\\prime }_3$ can be adjacent to the vertex of $G^\\ast _1$ , so assume without loss of generality (by renumbering $G^{\\prime }_2$ and $G^{\\prime }_3$ ) that ${G^{\\prime }_3}{G^\\ast _1} \\notin E(G^M)$ .", "Now if ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ , then $G=H_{4,3}^{c}$ .", "If ${G^{\\prime }_2}{G^\\ast _1} \\notin E(G^M)$ , then $G=H_{4,3}^{d}$ .", "This completes the proof.", "Now the only case left to consider is that both $G^{\\prime }$ and $G^\\ast $ contain three isolated vertices.", "I If $\\sigma (G^{\\prime })=(1,1,1)$ and $\\sigma (G^\\ast )=(1,1,1)$ , then we have $G \\in {C_6, \\complement {C_3} \\mathbin \\nabla \\complement {C_3},H_{3,3}^{a}, H_{3,3}^{b},H_{3,3}^{c}, H_{3,3}^{d},H_{3,3}^{e}}.$ [Proof of  REF] Let ${G^{\\prime }_1,G^{\\prime }_2,G^{\\prime }_3} = \\mathcal {C}(G^{\\prime })$ and ${G^\\ast _1,G^\\ast _2,G^\\ast _3} = \\mathcal {C}(G^\\ast )$ .", "Then clearly $n=6$ .", "We first consider the case that every connected component has at most degree 2 in $G^M$ .", "If a vertex has degree 1, then in order to have a common neighbor with its both metamours, the component it is adjacent to has to have degree 3, a contradiction.", "Therefore, every vertex has degree 2.", "Due to the fact that $G$ is connected, this implies that $G=C_6$ , so in this case we are done.", "Now assume there is at least one component that has degree 3 in $G^M$ .", "Let without loss of generality (by switching $G^{\\prime }$ and $G^\\ast $ and by renumbering the connected components of $G^{\\prime }$ ) this component be $G^{\\prime }_1$ .", "Then we have ${G^{\\prime }_1}{G^\\ast _1}$ , ${G^{\\prime }_1}{G^\\ast _2}$ and ${G^{\\prime }_1}{G^\\ast _3} \\in E(G^M)$ .", "If every component of $G^{\\prime }$ has degree 3 in $G^M$ , then $G = \\complement {C_3} \\mathbin \\nabla \\complement {C_3}$ , so also in this case we are done.", "Hence, we can assume without loss of generality (by renumbering $G^{\\prime }_2$ and $G^{\\prime }_3$ and by renumbering the components of $G^\\ast $ ) that ${G^{\\prime }_3}{G^\\ast _3} \\notin E(G^M)$ .", "Now $G^{\\prime }_3$ and $G^{\\prime }_2$ need a common neighbor.", "This cannot be $G^\\ast _3$ .", "Assume without loss of generality (by renumbering $G^\\ast _1$ and $G^\\ast _2$ ) that the common neighbor is $G^\\ast _2$ .", "Then this implies that ${G^{\\prime }_2}{G^\\ast _2}$ and ${G^{\\prime }_3}{G^\\ast _2} \\in E(G^M)$ .", "The current situation is shown in Figure REF .", "Figure: Subgraphs of the situationin the proof of The potential edges, which status is still undetermined, are ${G^{\\prime }_2}{G^\\ast _1}$ , ${G^{\\prime }_2}{G^\\ast _3}$ and ${G^{\\prime }_3}{G^\\ast _1}$ .", "At this stage, if each of these pairs is a non-edge in $G^M$ , then it is easy to see that whenever two vertices should be metamours they are metamours, to be precise all components of $G^{\\prime }$ have the common neighbor $G^\\ast _2$ and all components of $G^\\ast $ have the common neighbor $G^{\\prime }_1$ .", "Therefore, we have enough edges in $G^M$ , so that additional edges between components of $G^{\\prime }$ and $G^\\ast $ can be included without interfering with the metamours.", "Now we first consider all cases where ${G^{\\prime }_2}{G^\\ast _3} \\in E(G^M)$ .", "In this case if both of ${G^{\\prime }_2}{G^\\ast _1}$ and ${G^{\\prime }_3}{G^\\ast _1}$ are in $E(G^M)$ , then $G=H_{3,3}^{a}$ .", "If ${G^{\\prime }_2}{G^\\ast _1} \\notin E(G^M)$ and ${G^{\\prime }_3}{G^\\ast _1} \\in E(G^M)$ , then $G=H_{3,3}^{b}$ .", "If ${G^{\\prime }_2}{G^\\ast _1} \\in E(G^M)$ and ${G^{\\prime }_3}{G^\\ast _1} \\notin E(G^M)$ , then $G=H_{3,3}^{c}$ .", "And finally, if ${G^{\\prime }_2}{G^\\ast _1} \\notin E(G^M)$ and ${G^{\\prime }_3}{G^\\ast _1} \\notin E(G^M)$ , then $G=H_{3,3}^{d}$ .", "Next we consider the case where ${G^{\\prime }_2}{G^\\ast _3} \\notin E(G^M)$ .", "If in this case both of ${G^{\\prime }_2}{G^\\ast _1}$ and ${G^{\\prime }_3}{G^\\ast _1}$ are in $E(G^M)$ , then $G=H_{3,3}^{c}$ .", "If one of ${G^{\\prime }_2}{G^\\ast _1}$ and ${G^{\\prime }_3}{G^\\ast _1}$ is in $E(G^M)$ , then $G=H_{3,3}^{d}$ .", "If none of ${G^{\\prime }_2}{G^\\ast _1}$ and ${G^{\\prime }_3}{G^\\ast _1}$ is in $E(G^M)$ , then $G=H_{3,3}^{e}$ .", "Eventually, we have considered all cases and proved what we wanted to show.", "Finally, we are finished in all cases and therefore the proof of Proposition REF is complete." ], [ "Assembling results & other proofs", "With all results above, we are now able to prove the main theorem of this section which provides a characterization of 2-metamour-regular graphs.", "[Proof of Theorem REF ] Let $G$ be 2-metamour-regular and $M$ its metamour graph.", "We apply Theorem REF with $k=2$ .", "This leads us to one of three cases.", "Case REF of Theorem REF gives $G = \\complement {M_1} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {M_t}$ with ${M_1,\\dots ,M_t} = \\mathcal {C}(M)$ and $t \\ge 2$ .", "By Observation REF every connected component $M_i$ , $i\\in {1,\\dots ,t}$ , is a cycle $C_{n_i}$ .", "This results in REF of Theorem REF for $t\\ge 2$ .", "If we are in case REF of Theorem REF , then the metamour graph of $M$ is connected and we apply Proposition REF .", "If we are in case REF of Theorem REF , then the metamour graph consists of exactly two connected components and we can apply Proposition REF .", "Collecting all graphs coming from these two propositions yields the remaining graphs of REF , REF and REF of Theorem REF .", "For the other direction, Proposition REF implies that the graph $G$ in REF of Theorem REF is a 2-metamour-regular graph for $t \\ge 2$ .", "Furthermore, it is easy to check that all other mentioned graphs are 2-metamour-regular, which proves this side of the equivalence and completes the proof.", "Finally we are able to prove the following corollaries of Theorem REF .", "[Proof of Corollary REF ] The result is an immediate consequence of Theorem REF .", "[Proof of Corollary REF ] Corollary REF provides a characterization of all 2-metamour-regular graphs with $n \\ge 9$ vertices.", "It is easy to see that all of these graphs are either 2-regular (in the case of $C_n$ ) or $(n-3)$ -regular.", "This proves one direction of the equivalence.", "For the other direction first consider a connected 2-regular graph on $n$ vertices.", "Clearly, this graph equals $C_n$ , therefore this graph is 2-methamour-regular.", "If a connected graph is $(n-3)$ -regular, then its complement $\\complement {G}$ is a 2-regular graph.", "As a result, each connected component of $\\complement {G}$ is a cycle graph.", "Let $C_{n_1}$ , ..., $C_{n_t}$ be the connected components of $\\complement {G}$ .", "It is easy to see that then $n_i \\ge 3$ and $n=n_1+\\dots +n_t$ hold.", "In consequence, $G = \\complement {C_{n_1}} \\mathbin \\nabla \\dots \\mathbin \\nabla \\complement {C_{n_t}}$ holds and therefore $G$ is 2-metamour-regular.", "This completes the proof.", "[Proof of Corollary REF ] The statement of the corollary is a direct consequence of Theorem REF .", "[Proof of Corollary REF ] We use the characterization provided by Theorem REF .", "So let us consider 2-metamour-regular graphs.", "Such a graph has at least $n\\ge 5$ vertices.", "In case REF , there is one graph per integer partition of $n$ into a sum, where each summand is at least 3.", "Note that the graph operator $\\mathbin \\nabla $ is commutative which coincides with the irrelevance of the order of the summands of the sum.", "There are $p_3(n)$ many such partitions.", "Case REF gives exactly one graph for each $n\\ge 5$ .", "The graph $C_5$ is counted in both REF and REF ; see first item of Remark REF .", "Case REF brings in additionally 8 graphs for $n=6$ , 6 graphs for $n=7$ and 3 graphs for $n=8$ .", "In total, this gives the claimed numbers.", "This finishes all proofs of the present paper." ], [ "Conclusions & open problems", "In this paper we have introduced the metamour graph $M$ of a graph $G$ : The set of vertices of $M$ is the set of vertices of $G$ and two vertices are adjacent in $M$ if and only if they are at distance 2 in $G$ , i.e., they are metamours.", "This definition is motivated by polyamorous relationships, where two persons are metamours if they have an intimate relationship with a common partner, but are not in an intimate relationship themselves.", "We focused on $k$ -metamour-regular graphs, i.e., graphs in which every vertex has exactly $k$ metamours.", "We presented a generic construction to obtain $k$ -metamour-regular graphs from $k$ -regular graphs for an arbitrary $k \\ge 0$ .", "Furthermore, in our main results, we provided a full characterization of all $k$ -metamour-regular graphs for each $k \\in {0,1,2}$ .", "These characterizations revealed that with a few exceptions, all graphs come from the generic construction.", "In particular, for $k=0$ every $k$ -metamour-regular graph is obtained by the generic construction.", "For $k=1$ there is only one exceptional graph that is $k$ -metamour-regular and not obtained by the generic construction.", "In the case of $k=2$ there are 17 exceptional graphs with at most 8 vertices and a family of graphs, one for each number of vertices at least 6, that are 2-metamour-regular and cannot be created by the generic construction.", "Additionally, we were able to characterize all graphs where every vertex has at most one metamour and give properties of the structure of graphs where every vertex has at most $k$ metamours for arbitrary $k \\ge 0$ .", "Every characterization is accompanied by counting for each number of vertices how many unlabeled graphs there are.", "The obvious unanswered question is clearly the following.", "Question 10.1 What is a characterization of $k$ -metamour-regular graphs for each $k \\ge 3$ ?", "This is of particular interest for $k = 3$ .", "As our generic construction yields $k$ -metamour-regular graphs for every $k \\ge 0$ , we clearly already have determined a lot of 3-metamour-regular graphs.", "It would, however, be lovely to determine all remaining graphs.", "Another interesting question is about fixed maximum metamour-degree.", "Question 10.2 What is a characterization of all graphs that have maximum metamour-degree $k$ ?", "We have answered this question for $k\\in {0,1}$ and would be delighted to know the answer in general, but as first steps specifically for $k=2$ and $k=3$ .", "It would also be interesting to find some structure in the graphs that are $k$ -metamour-regular and cannot be obtained with our generic construction.", "In particular, we ask the following.", "Question 10.3 Is it possible to give properties (necessary or sufficient) of the exceptional graphs or graph classes?", "When dealing with metamour graphs, one question to ask is whether it is possible to characterize all graphs whose metamour graph has a certain property.", "In the present paper we have started to give an answer for the feature that the metamour graph is $k$ -regular.", "But what about other graph classes?", "Of course it would be interesting to answer the following questions.", "Question 10.4 Is it possible to characterize all graphs whose metamour graph is in some graph class like planar, bipartite, Eulerian or Hamiltonian graphs or like graphs of a certain diameter, girth, stability number or chromatic number?", "Another question of interest concerns constructing graphs, namely given a graph $M$ , is there a graph $G$ such that $M$ is the metamour graph of $G$ ?", "If $M$ is not connected, then the answer is easy and also provided in this paper, namely $G = \\complement {M}$ is such a graph.", "However, if $M$ is connected this question is still open and an answer more complicated.", "This give rise to the following question.", "Question 10.5 What is a characterization of the class of graphs with the property that each graph in this class is the metamour graph of some graph?", "Motivated by [27] we ask the following.", "Question 10.6 What is a characterization of the class of graphs, where every graph is isomorphic to its metamour graph?", "Going into another direction, one can also think about random graphs like the graphs from the Erdős–Rényi model $G(n,p)$ .", "Question 10.7 Given a random graph of $G(n,p)$ , which properties does its metamour graph have?", "Is there a critical value for $p$ (depending on $n$ ) such that the metamour graph is connected?", "In enumerative and probabilistic combinatorics the following question arise.", "Question 10.8 Given a random graph model, for example that all graphs with the same number of vertices are equally likely, what is the expected value of the metamour-degree?", "What about its distribution?", "Most of the results and open questions focus on the number of vertices of the graph and metamour graph respectively, as these two numbers match.", "But it would be interesting to know how the number of edges of the metamour graph of a graph relates to the number of edges in this graph.", "Specifically, we ask the following questions.", "Question 10.9 Given a graph $G$ with $m$ edges, in which range can the number of edges of the metamour graph of $G$ be?", "Question 10.10 What is the distribution of the number of edges of the metamour graph over all possible graphs with $m$ edges?" ] ]

[ [ "Maximal extension of the Schwarzschild metric: From\n Painlev\\'e-Gullstrand to Kruskal-Szekeres" ], [ "Abstract We find a specific coordinate system that goes from the Painlev\\'e-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture.", "We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$.", "$E$ is in the range $1\\leq E<\\infty$ with the limit $E=\\infty$ yielding the Kruskal-Szekeres maximal extension.", "So, through such an integrated description one sees that the Kruskal-Szekeres solution belongs to this family of extensions parameterized by $E$.", "Our family of extensions is different from the Novikov-Lema\\^itre family parameterized also by the energy $E$ of timelike geodesics, with the Novikov extension holding for $0<E<1$ and being maximal, and the Lema\\^itre extension holding for $1\\leq E<\\infty$ and being partial, not maximal, and moreover its $E=\\infty$ limit evanescing in a Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime." ], [ "Introduction", "The maximal analytical extension of the Schwarzschild solution was a remarkable achievement in general relativity and in the theory of black holes.", "For the first time the complex causal structure with a convoluted spacetime topology, stemming from the seemingly trivial generalization into general relativity of a point particle attractor in Newtonian gravitation, was unfolded.", "It all started with the spherically symmetric vacuum solution of general relativity found by Schwarzschild [1], that was put in different terms and in a somewhat different coordinate system by Droste [2] and Hilbert [3], and shown to be unique by Birkhoff [4].", "Leaving aside Schwarzschild's interpretation of Schwarzschild's solution, it is the solution that later gave rise to black holes.", "To finalize its full meaning it was necessary to understand the sphere $r=2M$ that naturally appears in the solution and accomplish its maximal extension, i.e., finding the corresponding spacetime in which every geodesic originating from an arbitrary point in it has infinite length in both directions or ends at a singularity that cannot be removed by a coordinate transformation.", "These were two problems that proved difficult.", "An early attempt to eliminate the $r=2M$ sphere obstacle and its inside was provided by Einstein and Rosen [5] that tried to join smoothly at $r=2M$ two distinct spacetime sheets in order to get some kind of fundamental particle, in what is known as an Einstein-Rosen bridge.", "Misner and Wheeler [6] generalized the bridge into a wormhole with a throat at its maximum opening.", "Wormholes became a focus of study within general relativity after Morris and Thorne [7] showed that with some suitable form of matter, albeit exotic, they could be traversable, see also, e.g., the work of Lemos, Lobo and Oliveira [8].", "The Einstein-Rosen bridge in terms of the understanding of the $r=2M$ sphere was a dead end, but as a nontraversable wormhole it reincarnated in the maximal extensions of the Schwarzschild metric, and as a traversable wormhole it can be put in firm ground once one properly defines it in order to have an admissible matter support, as disclosed by Guendelman, Nissimov, Pacheva, and Stoilov [9].", "A promising way of seeing the Schwarzschild solution, whatever the motivation, came with Painlevé [10] that changed the Schwarzschild time coordinate into the proper time of a congruence of ingoing timelike geodesics, or equivalently of ingoing test particles planted over them, with energy per unit mass $E$ equal to one, that admitted to put the line element in a new form that was not singular at $r=2M$ .", "This procedure was also discovered by Gullstrand [11], and the resulting line element, which works as for outgoing as for ingoing timelike geodesics, is called the Painlevé-Gullstrand line element of the Schwarzschild solution, or simply referred as Painlevé-Gullstrand solution, and in both forms it is an analytical extension, although partial, of the original Schwarzschild solution.", "The generalization of this line element to accommodate a congruence of timelike geodesics with any $E$ , less or greater than one, was given by Gautreau and Hoffmann [12].", "The Painlevé-Gullstrand line element, not being singular at $r=2M$ , is useful in many understandings of black hole physics.", "For instance, it has been used by Parikh and Wilczek to understand how Hawking radiation proceeds [13], or as a guide for a better understanding of the $r=2M$ sphere by Martel and Poisson [14], or to understand in new ways the Kerr metric by Natário [15], or as a generalized slicing of the Schwarzschild spacetime by Finch [16] and MacLaurin [17].", "An extension of the Painlevé-Gullstrand line element was given by Lemaître [18] that transformed the time and radial coordinates of the Schwarzschild solution to the proper time of ingoing timelike geodesics with $E=1$ and to a suitable new comoving radial coordinate, and showed in a stroke that $r=2M$ was a fine sphere, with nothing singular about it, performing thus an analytical extension, although partial, of the Schwarzschild solution.", "Novikov [19] understood that for timelike geodesics with $0<E<1$ it was possible to perform a maximal analytical extension and display the Schwarzschild solution in its fullness.", "The Lemaître extension, as an exterior spacetime, was implicitly used in the gravitational contraction of a cloud of dust by Oppenheimer and Snyder to discover black holes and their formation for the first time with the natural appearance of an exterior event horizon at $r=2M$ [20].", "Presentations of the Novikov-Lemaitre extensions can be seen in several places.", "The Novikov maximal extension is worked through in Zel'dovich and Novikov's book [21] and in Gautreau [22], and the Lemaître extension is featured, e.g., in the detailed book by Krasiński [23] and in the very useful book of Blau [24].", "Remarkably, there is a parallel development that uses lightlike, or null, geodesics rather than timelike ones.", "Indeed, Eddington [25] used ingoing null geodesics to transform the Schwarzschild time into a new time that straightened out those very ingoing null geodesics and to put the line element in a new form that was not singular at $r=2M$ .", "This was recovered by Finkelstein [26], and then Penrose [27] understood that it was more natural to use the corresponding advanced null coordinate to represent the metric and the line element.", "This form works as for outgoing as for ingoing null geodesics, and the solution is correspondingly called the Schwarzschild solution in retarded or in advanced null Eddington-Finkelstein coordinates, respectively.", "Both forms are analytical extensions, although partial, of the original Schwarzschild solution.", "The Eddington-Finkelstein line element, not being singular at $r=2M$ is also useful in many understandings of black hole physics.", "For instance, it has been used by Alcubierre and Bruegmann in black hole excision in 3+1 numerical relativity [28], or as a guide for a better understanding of the $r=2M$ sphere by Adler, Bjorken, Chen, and Liu [29], or to understand perturbatively the accretion of matter onto a black hole [30], or to understand the stress-energy tensor of quantum fields involved in the evaporation of a black hole [31], or even to treat quantum gravitational problems related to coordinate transformations [32].", "An extension to the Eddington-Finkelstein line element was given by Kruskal [33] and Szekeres [34].", "By using both outgoing and ingoing null geodesics to transform the Schwarzschild time and the Schwarzschild radius coordinates into new analytical extended time and spatial coordinates, both the outgoing and the ingoing null geodesics were straightened out and in addition one could pass with ease the sphere $r=2M$ in all directions.", "In this way the maximal analytical extension of the Schwarzschild solution was unfolded, in a single coordinate system, into its full form.", "Fuller and Wheeler [35] revealed its dynamic structure with a nontraversable Einstein-Rosen bridge, i.e., a nontraversable wormhole, lurking in-between two distinct asymptotically flat spacetime regions and driving, out of spacetime spacelike singularities at $r=0$ , the creation of a white hole into the formation of a black hole.", "Prior maximal extensions had also been given in Synge [36] and Fronsdal [37] using several coordinate systems or embeddings, rather than the unique coordinate system of the Kruskal-Szekeres extension.", "Modern presentations of the Kruskal-Szekeres solution can be seen in the books on general relativity and gravitation by Hawking and Ellis [38], Misner, Thorne, and Wheeler [39], Wald [40], d'Inverno [41], Bronnikov and Rubin [42], and Chruściel [43], and in many other places, where double null coordinates are usually employed.", "The Kruskal-Szekeres line element, with its maximal properties, is certainly useful in a great very many understandings of black hole physics, notably, it surely is a prototype of gravitational collapse.", "To name two further examples of its applicability, Zaslavskii [44] has used its properties to suitably define high energy collisions in the vicinity of the event horizon, and Hodgkinson, Louko, and Ottewill [45] have examined the response of particle detectors to fields in diverse quantum vacuum states working with Kruskal-Szekeres spacetime and coordinates.", "Now, the Painlevé-Gullstrand line element uses as coordinate the proper time of a congruence of outgoing or ingoing timelike geodesics and the Eddington-Finkelstein line element uses as coordinate the retarded or advanced null parameter of a congruence of outgoing or ingoing null geodesics, respectively.", "There is a connection between the two coordinate systems as worked out by Lemos [46], who showed that by taking the $E=\\infty $ limit of the Painlevé-Gullstrand line element, and more generally its Lemaître-Tolman-Bondi generalization to include dust matter, one obtains the Eddington-Finkelstein line element, and more generally its Vaidya generalization to include incoherent radiation.", "Indeed, since $E$ is the energy per unit mass of the timelike geodesic, or of the particle placed over it, when the mass goes to zero, $E$ goes to infinity, and the proper time along the timelike geodesic turns into a well defined affine parameter along the null geodesic, or along the lightlike particle trajectory placed over it.", "But now we have a conundrum.", "The Novikov-Lemaitre family of solutions parameterized by $E$ comes out of the corresponding Painlevé-Gullstrand family with the addition of an appropriate radial coordinate.", "On the one hand, the Novikov solution is maximal, on the other hand, the Lemaître solution is not.", "Moreover, although Painlevé-Gullstrand goes into Eddington-Finkelstein in the $E=\\infty $ limit, Lemaître does not go into Kruskal-Szekeres in the $E=\\infty $ limit, instead it dies in a Minkowski spacetime.", "But Eddington-Finkelstein goes into Kruskal-Szekeres.", "In brief, Painlevé-Gullstrand goes into Novikov-Lemaître that does not go into Kruskal-Szekeres, and Painlevé-Gullstrand goes into Eddington-Finkelstein that goes into Kruskal-Szekeres.", "So, there is a missing link.", "What is the maximal extension that starts from Painlevé-Gullstrand and in the $E=\\infty $ limit goes into the Kruskal-Szekeres maximal extension?", "Here, we find the maximal analytic extension of the Schwarzschild spacetime that goes from Painlevé-Gullstrand to Kruskal-Szekeres yielding a unified picture of extensions.", "By using two analytically extended Painlevé-Gullstrand time coordinates, we find another way of obtaining the maximal analytic extension of the Schwarzschild spacetime.", "It is parameterized by the energy $E$ of the outgoing and ingoing timelike geodesics.", "The extension is valid for $1\\le E<\\infty $ , with the case $E=\\infty $ giving the Kruskal-Szekeres extension.", "So the Kruskal-Szekeres extension is a member of this family.", "It is a different family from the Novikov-Lemaître family, which does not have as its member the Kruskal-Szekeres extension, and moreover the $E\\ge 1$ Lemaître extension is not maximal.", "It is certainly opportune to incorporate into a family of maximal $E$ extensions of the Schwarzschild metric, the maximal extension of Kruskal and Szekeres in the year we celebrate its 60 years.", "The paper is organized as follows.", "In Sec.", ", we give the Schwarzschild metric in double Painlevé-Gullstrand coordinates for $E>1$ .", "In Sec.", ", we extend the Schwarzschild metric for $E>1$ past the $r=2M$ coordinate singularity using analytical extended coordinates, and produce its maximal analytical extension.", "In Sec.", ", we give the $E=1$ maximal analytical extension as the limit from $E>1$ .", "In Sec.", ", we give the $E=\\infty $ maximal analytical extension as the limit from $E>1$ and show that it is the Kruskal-Szekeres maximal extension.", "In Sec.", ", we present the causal structure of the maximal extended spacetime for several $E$ , from $E=1$ to $E=\\infty $ .", "In Sec.", ", we conclude.", "In the Appendix, we show in detail the limits $E=1$ and $E=\\infty $ directly from the $E>1$ generic case." ], [ "The Schwarzschild solution in double Painlevé-Gullstrand\nform", "The vacuum Einstein equation $G_{ab}=0$ , where $G_{ab}$ is the Einstein tensor and $a,b$ are spacetime indices, give for a line element $ds^2=g_{ab}(x^a)\\hspace{0.28436pt}dx^adx^b$ , where $g_{ab}(x^a)$ is the metric and $x^a$ are the coordinates, in the classical standard spherical symmetric coordinates $(t,r,\\theta ,\\phi )$ the Schwarzschild solution, namely, $\\mathrm {d}s^2=-\\left( 1-\\frac{2M}{r}\\right)\\mathrm {d}t^2+\\dfrac{dr^2}{1-\\frac{2M}{r}} +r^2(\\mathrm {d} \\theta ^2+ \\sin ^2{\\theta }\\,\\mathrm {d} \\phi ^2)\\,,$ where $M$ is the spacetime mass.", "We assume $M\\ge 0$ and $r\\ge 0$ .", "In this form the line element, and so the metric, is singular at the, Schwarzschild, gravitational, or event horizon radius $r=2M$ , and at $r=0$ .", "For $r>2M$ , the Schwarzschild coordinate $t$ is timelike and the coordinate $r$ is spacelike, a radial coordinate.", "For $r<2M$ , these coordinates swap roles, the Schwarzschild coordinate $t$ is spacelike and the coordinate $r$ is timelike.", "We now apply a first coordinate transformation such that the Schwarzschild time $t$ in Eq.", "(REF ) goes into a new time ${\\mathcal {t}}={\\mathcal {t}}(t,r)$ given in differential form by $\\mathrm {d}{\\mathcal {t}}=E\\mathrm {d}t-\\dfrac{\\left( E^2-1+ \\frac{2M}{r} \\right)^{1/2}}{1- \\frac{2M}{r}}\\mathrm {d}r\\,,$ with $E\\ge 1$ , $E$ being a parameter.", "This is a Painlevé-Gullstrand coordinate transformation for the congruence of outgoing radial timelike geodesics with energy $E$ .", "We can also perform a different coordinate transformation, such that the Schwarzschild time $t$ in Eq.", "(REF ) goes into a new time $\\tau =\\tau (t,r)$ given in differential form by $\\mathrm {d}\\tau ={E} \\mathrm {d}t+\\dfrac{\\left( E^2-1+ \\frac{2M}{r} \\right)^{1/2}}{1- \\frac{2M}{r}}\\mathrm {d}r\\,.$ with $E\\ge 1$ , $E$ being the same parameter as above.", "This is a Painlevé-Gullstrand coordinate transformation for the congruence of ingoing radial timelike geodesics with energy $E$ .", "The two transformations together, ${\\mathcal {t}}={\\mathcal {t}}(t,r)$ and $\\tau =\\tau (t,r)$ , Eqs.", "(REF ) and (REF ), respectively, can then be seen as a transformation from the Schwarzschild time and radius $(t,r)$ to the two new coordinates $({\\mathcal {t}},\\tau )$ .", "The inverse transformations, from $({\\mathcal {t}},\\tau )$ to $(t,r)$ , in differential form are $E\\mathrm {d}t= \\frac{1}{2}\\left(\\;\\mathrm {d}{\\mathcal {t}}+\\mathrm {d}\\tau \\right)\\,,$ $\\frac{\\left( E^2-1+ \\frac{2M}{r} \\right)^{1/2}}{\\left( 1- \\frac{2M}{r} \\right)}\\mathrm {d}r= \\frac{1}{2}\\left(-\\mathrm {d}{\\mathcal {t}}+\\mathrm {d}\\tau \\right)\\,.$ Applying the coordinate transformation given in Eq.", "(REF ) to the Schwarzschild line element, Eq.", "(REF ), gives the line element in Painlevé-Gullstrand outgoing coordinates with energy parameter $E\\ge 1$ , namely, $\\mathrm {d}s^2=- \\frac{1}{E^2} \\left( 1- \\frac{2M}{r} \\right)\\mathrm {d}{\\mathcal {t}}^2- - 2 \\frac{1}{E^2} \\sqrt{E^2-1+ \\frac{2M}{r}}\\mathrm {d}{\\mathcal {t}}\\, \\mathrm {d}r+ \\frac{1}{E^2} \\mathrm {d}r^2+r^2( \\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)$ .", "This form of the metric is not singular anymore at $r=2M$ , but there is still the singularity at $r=0$ which cannot be removed.", "Note that inside $r=2M$ this Painlevé-Gullstrand form has the feature of having two time coordinates, ${\\mathcal {t}}$ and $r$ .", "Applying the coordinate transformation given in Eq.", "(REF ) to the Schwarzschild metric, Eq.", "(REF ), gives the metric in Painlevé-Gullstrand ingoing coordinates with energy parameter $E\\ge 1$ , namely, $\\mathrm {d}s^2=- \\frac{1}{E^2} \\left( 1- \\frac{2M}{r} \\right)\\mathrm {d}\\tau ^2+ 2 \\frac{1}{E^2} \\sqrt{E^2-1+ \\frac{2M}{r}}\\mathrm {d}\\tau \\, \\mathrm {d}r+ \\frac{1}{E^2} \\mathrm {d}r^2+ r^2(\\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)$ .", "This form of the metric is also not singular anymore at $r=2M$ , but there is still the singularity at $r=0$ which cannot be removed.", "Note that inside $r=2M$ this Painlevé-Gullstrand form has the feature of having two time coordinates, $\\tau $ and $r$ .", "All of this is well known.", "We now apply a simultaneous coordinate transformation, given through Eqs.", "(REF )-(REF ), or if one prefers Eqs.", "(REF )-(REF ), to the Schwarzschild metric, Eq.", "(REF ), to get $\\begin{split}\\mathrm {d}s^2&= -\\frac{1}{4E^2}\\frac{1-\\frac{2M}{r}}{E^2-1+\\frac{2M}{r}}\\left[ -\\left( 1- \\frac{2M}{r} \\right)(\\mathrm {d}{\\mathcal {t}}^2+\\mathrm {d}\\tau ^2)+2 \\left( 2E^2-1+ \\frac{2M}{r}\\right) \\mathrm {d}{\\mathcal {t}}\\,\\mathrm {d}\\tau \\right]+\\\\ &+r^2({\\mathcal {t}},\\tau ) (\\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)\\,,\\end{split}$ with $r({\\mathcal {t}},\\tau )$ obtained via Eq.", "(REF ) and depends on whether $E=1$ or $E>1$ .", "This is the Schwarzschild metric in double Painlevé-Gullstrand coordinates.", "The line element of Eq.", "(REF ) is still degenerate for $r=2M$ .", "So, if we want to extend it past this sphere we have to perform another set of coordinate transformations.", "This set is given by $\\frac{{\\mathcal {t}}^{\\prime }}{M}= -\\exp {\\left( -\\frac{{\\mathcal {t}}}{4ME}\\right) }$ and $\\frac{\\tau ^{\\prime }}{M}= \\exp {\\left( \\;\\frac{\\tau }{4ME} \\right) }$ .", "When applied to Eq.", "(REF ), it gives, $\\mathrm {d}s^2= 4M^2 \\frac{1-\\frac{2M}{r}}{E^2-1+\\frac{2M}{r}}\\left[\\left(1-\\frac{2M}{r} \\right)\\left(\\frac{\\mathrm {d}{\\mathcal {t}}^{\\prime 2}}{{\\mathcal {t}}^{\\prime 2}}+\\frac{\\mathrm {d}\\tau ^{\\prime 2}}{\\tau ^{\\prime 2}} \\right)+2\\left(2E^2-1+\\frac{2M}{r}\\right)\\frac{\\mathrm {d}{\\mathcal {t}}^{\\prime }}{{\\mathcal {t}}^{\\prime }}\\,\\frac{\\mathrm {d}\\tau ^{\\prime }}{\\tau ^{\\prime }}\\right]+ + r^2({\\mathcal {t}^{\\prime }},\\tau ^{\\prime })(\\mathrm {d}\\theta ^2+ \\sin ^2{\\theta }\\,\\mathrm {d}\\phi ^2)$ , with $r({\\mathcal {t}^{\\prime }},\\tau ^{\\prime })$ a function that is given implicitly.", "The form of this metric will depend on the value of $E$ through the solution to the differential coordinate relations, Eqs.", "(REF ) and (REF ), or equivalently, Eqs.", "(REF )-(REF ).", "Clearly, the case $E<1$ cannot be treated from the formulas above and we have dismissed it from the start.", "Therefore we restrict the analysis to $1\\le E<\\infty $ .", "The $E=1$ and $E=\\infty $ can be seen as limiting cases of the generic $E>1$ case.", "Let us do the $E>1$ case in detail and then treat $E=1$ and $E=\\infty $ as the inferior and superior limiting cases, respectively, of $E>1$ ." ], [ "Maximal analytic extension for $E>1$ as generic case", "To start building the maximal analytic extension for $E>1$ , we find the solutions to the new coordinates ${\\mathcal {t}}$ and $\\tau $ from Eqs.", "(REF ) and (REF ).", "When $E>1$ they are $\\begin{split}{\\mathcal {t}}&=Et- r\\sqrt{E^2-1+ \\frac{2M}{r}}-2ME \\ln {\\left|\\frac{2M}{r}\\left(\\frac{\\frac{r}{2M}-1}{2E^2-1+\\frac{2M}{r}+2E\\sqrt{E^2-1+\\frac{2M}{r}}}\\right)\\right|}-\\\\ &-M\\frac{2E^2-1}{\\sqrt{E^2-1}}\\ln \\left[\\frac{r}{M}\\left(\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\frac{M}{r}\\right)\\right]\\,,\\end{split}\\\\\\begin{split}\\tau &=Et+ r\\sqrt{E^2-1+ \\frac{2M}{r}}+2ME \\ln {\\left|\\frac{2M}{r}\\left(\\frac{\\frac{r}{2M}-1}{2E^2-1+\\frac{2M}{r}+2E\\sqrt{E^2-1+\\frac{2M}{r}}}\\right)\\right|}+\\\\ &+M\\frac{2E^2-1}{\\sqrt{E^2-1}}\\ln \\left[\\frac{r}{M}\\left(\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\frac{M}{r}\\right)\\right]\\,.\\end{split}$ The line element to start with is $\\begin{split}\\mathrm {d}s^2&= -\\frac{1}{4E^2} \\frac{1-\\frac{2M}{r}}{E^2-1+\\frac{2M}{r}}\\left[ -\\left( 1- \\frac{2M}{r} \\right)(\\mathrm {d}{\\mathcal {t}}^2+\\mathrm {d}\\tau ^2)+2 \\left( 2E^2-1+ \\frac{2M}{r}\\right) \\mathrm {d}{\\mathcal {t}}\\,\\mathrm {d}\\tau \\right]+\\\\ &+r^2 (\\mathrm {d}\\theta ^2+ \\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)\\,,\\end{split}$ which is taken from Eq.", "(REF ), now bearing in mind that $E>1$ implicitly here, and with $r=r({\\mathcal {t}},\\tau )$ being obtained via Eqs.", "(REF ) and (), i.e., $r\\sqrt{E^2-1+ \\frac{2M}{r}}+2ME \\ln {\\left|\\frac{2M}{r}\\left(\\frac{\\frac{r}{2M}-1}{2E^2-1+\\frac{2M}{r}+2E\\sqrt{E^2-1+\\frac{2M}{r}}}\\right)\\right|}+\\nonumber \\\\M\\frac{2E^2-1}{\\sqrt{E^2-1}}\\ln \\left[\\frac{r}{M}\\left(\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\frac{M}{r}\\right)\\right]=&\\frac{1}{2}\\left(-{\\mathcal {t}}+\\tau \\right)\\,.$ The line element Eq.", "(REF ) is still degenerate at $r=2M$ .", "So, if we want to extend past it we have to do something.", "To remove this behavior, we proceed with two new coordinate transformations given by $\\frac{{\\mathcal {t}}^{\\prime }}{M}= -\\exp {\\left(-\\frac{{\\mathcal {t}}}{4ME}\\right)}$ and $\\frac{\\tau ^{\\prime }}{M}= \\quad \\exp {\\left( \\;\\frac{\\tau }{4ME} \\right)}$ , for $r>2M$ .", "Then, using Eqs.", "(REF ) and () the maximal extended coordinates ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ are $\\begin{split}\\frac{{\\mathcal {t}}^{\\prime }}{M}&= -\\exp {\\left( -\\frac{{\\mathcal {t}}}{4ME}\\right) }\\,, \\quad {\\rm i.e.,} \\quad \\\\\\frac{{\\mathcal {t}}^{\\prime }}{M}&=-\\sqrt{\\frac{2M}{r}}\\frac{\\sqrt{\\frac{r}{2M}-1}}{\\sqrt{2E^2- 1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}}\\exp {\\left( -\\frac{t}{4M}+\\frac{r}{4ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)}\\times \\\\ &\\times \\left[\\frac{r}{M}\\left(\\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}+ E^2-1+\\frac{M}{r}\\right)\\right]^{\\frac{2E^2-1}{4E \\sqrt{E^2-1}}}\\,,\\end{split}\\\\\\begin{split}\\frac{{\\tau }^{\\prime }}{M}&= \\exp {\\left( \\frac{\\tau }{4ME}\\right) }\\,, \\quad {\\rm i.e.,} \\quad \\\\\\frac{\\tau ^{\\prime }}{M}&=\\sqrt{\\frac{2M}{r}}\\frac{\\sqrt{\\frac{r}{2M}-1}}{\\sqrt{2E^2- 1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}}\\exp {\\left( \\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+\\frac{2M}{r}}\\right)}\\times \\\\ &\\times \\left[\\frac{r}{M}\\left(\\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}+ E^2-1+\\frac{M}{r}\\right)\\right]^{\\frac{2E^2-1}{4E \\sqrt{E^2-1}}}\\,,\\end{split}$ respectively.", "Putting ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ given in Eqs.", "(REF ) and (), respectively, into the line element Eq.", "(REF ), one finds the new line element in coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ given by $\\begin{split}\\mathrm {d}s^2&=-4 \\left( \\frac{ 2E^2-1+ \\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}{ E^2-1+\\frac{2M}{r}} \\right)\\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)}\\times \\\\&\\times \\left(\\frac{M}{r}\\, \\frac{1}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right)^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}\\times \\\\&\\times \\Bigg [ -\\frac{1}{M^2} \\left(2E^2-1+ \\frac{2M}{r}+2E\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)\\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)}\\times \\\\&\\times \\left(\\frac{M}{r}\\frac{1}{E^2-1+\\frac{M}{r}+\\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}}\\right)^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}(\\tau ^{\\prime 2}\\mathrm {d}{\\mathcal {t}}^{\\prime 2}+{\\mathcal {t}}^{\\prime 2}\\mathrm {d}\\tau ^{\\prime 2})+\\\\&+2 \\left( 2E^2-1+ \\frac{2M}{r} \\right)\\mathrm {d}{\\mathcal {t}}^{\\prime }\\mathrm {d}\\tau ^{\\prime }\\,\\Bigg ]+ r^2 (\\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)\\,,\\end{split}$ where $r=r({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ is defined implicitly as a function of ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ through $\\begin{split}&\\left( \\frac{\\frac{r}{2M}-1}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}\\,\\exp {\\left( \\frac{r}{2ME}\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}\\times \\\\& \\times \\left[\\frac{r}{M}\\left( E^2-1 + \\frac{M}{r}+ \\sqrt{E^2-1}\\sqrt{E^2-1+ \\frac{2M}{r}}\\right)\\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}=-\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}\\,.\\end{split}$ All of this is done so that ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ have ranges $-\\infty <{\\mathcal {t}}^{\\prime }<\\infty $ and $-\\infty <\\tau ^{\\prime }<\\infty $ , which Eqs.", "(REF ) and (REF ) permit.", "Several properties are now worth mentioning.", "In terms of the coordinates $({\\mathcal {t}},\\tau )$ , or $(t,r)$ , the coordinate transformations that yield the maximal extended coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ with infinite ranges have to be broadened, resulting in the existence of four regions, regions I, II, III, and IV.", "Region I is the region where the transformations Eqs.", "(REF ) and () hold, i.e., it is a region with ${\\mathcal {t}}^{\\prime }\\le 0$ and $\\tau ^{\\prime }\\ge 0$ .", "It is a region with $r\\ge 2M$ and $-\\infty <t<\\infty $ .", "Of course, in this region Eqs.", "(REF ) and (REF ) hold.", "Region II, a region for which $r\\le 2M$ , gets a different set of coordinate transformations.", "In this $r\\le 2M$ region, due to the moduli appearing in Eqs.", "(REF ) and () and the change of sign in Eq.", "(REF ), one defines instead ${\\mathcal {t}}^{\\prime }$ as $ \\frac{{\\mathcal {t}}^{\\prime }}{M}=+ \\exp {\\left(-\\frac{{\\mathcal {t}}}{4ME}\\right)}==\\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{1-\\frac{r}{2M}}}{\\sqrt{2E^2- 1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}} }\\exp {\\left( -\\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+\\frac{2M}{r}}\\right)} \\left[ \\frac{r}{M} \\left( \\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+ E^2-1+\\right.\\right.", "\\\\\\left.\\left.+\\frac{M}{r} \\right)\\right]^{\\frac{2E^2-1}{4E\\sqrt{E^2-1}}}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} = \\exp {\\left( \\frac{\\tau }{4ME} \\right) }=\\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{1-\\frac{r}{2M}}}{\\sqrt{2E^2-1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}} } \\exp {\\left(\\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)} \\times \\times \\left[ \\frac{r}{M} \\left( \\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\frac{M}{r} \\right)\\right]^{\\frac{2E^2-1}{4E \\sqrt{E^2-1}}}$ .", "These transformations are valid for ${\\mathcal {t}}^{\\prime }\\ge 0$ and $\\tau ^{\\prime }\\ge 0$ .", "It is a region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\left( \\frac{1- \\frac{r}{2M}}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}\\, \\exp {\\left( \\frac{r}{2ME}\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)} \\left[ \\frac{r}{M} \\left( E^2-1 +\\frac{M}{r}+ \\sqrt{E^2-1} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)\\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}=\\\\=\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ .", "But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so there is no further concern on that.", "Region III is another $r\\ge 2M$ region.", "Now one defines ${\\mathcal {t}}^{\\prime }$ as $\\frac{{\\mathcal {t}}^{\\prime }}{M}= \\exp {\\left(-\\frac{{\\mathcal {t}}}{4ME}\\right)}= \\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{\\frac{r}{2M}-1}}{\\sqrt{2E^2- 1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}}\\times \\\\\\times \\exp {\\left( -\\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+\\frac{2M}{r}}\\right)} \\left[ \\frac{r}{M} \\left( \\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+ E^2-1+\\frac{M}{r}\\right)\\right]^{\\frac{2E^2-1}{4E \\sqrt{E^2-1}}}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} =\\\\=-\\exp {\\left( \\frac{\\tau }{4ME} \\right) } =-\\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{\\frac{r}{2M}-1}}{\\sqrt{2E^2-1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}} } \\exp {\\left(\\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)} \\left[\\frac{r}{M} \\left( \\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}+\\right.\\right.\\\\\\left.\\left.", "+E^2-1 +\\frac{M}{r} \\right)\\right]^{\\frac{2E^2-1}{4E\\sqrt{E^2-1}}}$ .", "These transformations are valid for the region with ${\\mathcal {t}}^{\\prime }\\ge 0$ and $\\tau ^{\\prime }\\le 0$ .", "It is a region with $r\\ge \\\\ \\ge 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\left( \\frac{ \\frac{r}{2M}-1}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}\\times \\\\\\times \\exp {\\left( \\frac{r}{2ME} \\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}\\left[ \\frac{r}{M} \\left( E^2-1 + \\frac{M}{r}+ \\sqrt{E^2-1}\\sqrt{E^2-1+ \\frac{2M}{r}}\\right) \\right]^{\\frac{2E^2-1}{2E\\sqrt{E^2-1}}}=- \\frac{{\\mathcal {t}}^{\\prime }}{M} \\frac{\\tau ^{\\prime }}{M}$ .", "But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so again there is no further concern on that.", "Region IV is another region with $r\\le 2M$ .", "Now, one defines ${\\mathcal {t}}^{\\prime }$ as $\\frac{{\\mathcal {t}}^{\\prime }}{M}=- \\exp {\\left(-\\frac{{\\mathcal {t}}}{4ME}\\right)}=- \\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{1-\\frac{r}{2M}}}{\\sqrt{2E^2- 1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}} }\\exp {\\left( -\\frac{t}{4M}+ \\right.", "}{\\left.+\\frac{r}{4ME} \\sqrt{E^2-1+\\frac{2M}{r}}\\right)}\\left[\\frac{r}{M}\\left( \\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\frac{M}{r} \\right)\\right]^{\\frac{2E^2-1}{4E\\sqrt{E^2-1}}}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} = -\\exp {\\left( \\frac{\\tau }{4ME} \\right) }==-\\sqrt{\\frac{2M}{r}} \\frac{\\sqrt{1-\\frac{r}{2M}}}{\\sqrt{2E^2-1+\\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}} }\\exp {\\left(\\frac{t}{4M} +\\frac{r}{4ME} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)}\\left[ \\frac{r}{M} \\left( \\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}+E^2-1+\\right.\\right.\\\\\\left.\\left.+\\frac{M}{r} \\right)\\right]^{\\frac{2E^2-1}{4E \\sqrt{E^2-1}}}$ .", "These transformations are valid for the region with ${\\mathcal {t}}^{\\prime }\\le 0$ and $\\tau ^{\\prime }\\le 0$ .", "It is a region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\left( \\frac{1- \\frac{r}{2M}}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}\\,\\times \\times \\exp {\\left( \\frac{r}{2ME}\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}\\left[ \\frac{r}{M} \\left( E^2-1 +\\frac{M}{r}+\\sqrt{E^2-1} \\sqrt{E^2-1+ \\frac{2M}{r}}\\right)\\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}= \\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ .", "But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so once again there is no further concern on that.", "Furthermore, from Eq.", "(REF ) we see that the event horizon at $r=2M$ has two solutions, ${\\mathcal {t}}^{\\prime }=0$ and $\\tau ^{\\prime }=0$ which are null surfaces represented by straight lines.", "The true curvature singularity at $r=0$ has two solutions $\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}=1$ , i.e., two spacelike hyperbolae.", "Implicit in the construction, there is a wormhole, or Einstein-Rosen bridge, topology, with its throat expanding and contracting.", "The dynamic wormhole is non traversable, but it spatially connects region I to region III through regions II and IV.", "Regions I and III are two asympotically flat regions causally separated, region II is the black hole region, and region IV is the white hole region of the spacetime.", "Eqs.", "(REF ) and (REF ) together with the corresponding interpretation give the maximal extension of the Schwarzschild metric for $E>1$ , in the coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ .", "Since $1<E<\\infty $ this is a family of extensions, characterized by one parameter, the parameter $E$ .", "It is a one-parameter family of extensions.", "The two-dimensional part $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ of the coordinate system $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ is shown in Figure REF , both for lines of constant ${\\mathcal {t}}^{\\prime }$ and constant $\\tau ^{\\prime }$ in part (a) of the figure, and for lines of constant $t$ and constant $r$ in part (b) of the figure, conjointly with the labeling of regions I, II, III, IV, needed to cover it.", "Figure: The maximal analytical extension of the Schwarzschild metricfor the parameter EE generic obeying E>1E>1 in the plane (𝓉 ' ,τ ' )({\\mathcal {t}}^{\\prime },\\tau ^{\\prime }) is shown in a diagram with two different descriptions,(a) and (b).", "In (a) typical values for lines of constant 𝓉 ' {\\mathcal {t}}^{\\prime } and constant τ ' \\tau ^{\\prime } are displayed.", "In (b) typical values forlines of constant tt and constant rr are displayed.", "The diagram,both in (a) and in (b), represents aa spacetime with a wormhole, not shown, that formsout of a singularity in the white hole region, i.e., region IV, andfinishes at the black hole region and its singularity, i.e., regionII, connecting the two separated asymptotically flat spacetimes,regions I and III.It is also worth discussing the normals to the ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces.", "From Eq.", "(REF ) one finds that the covariant metric has components $g_{{\\mathcal {t}^{\\prime }} {\\mathcal {t}^{\\prime }}}=\\frac{4}{M^2} \\left(\\frac{ \\left(2E^2-1+ \\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}\\right)^2}{ E^2-1+\\frac{2M}{r}} \\right)\\exp {\\left( -\\frac{r}{ME} \\sqrt{E^2-1+ \\frac{2M}{r}} \\right)} \\times \\times \\left( \\frac{\\frac{M}{r}}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right)^{\\frac{2E^2-1}{E \\sqrt{E^2-1}}}\\tau ^{\\prime 2}$ , $g_{\\tau ^{\\prime } \\tau ^{\\prime }}= \\frac{4}{M^2} \\left( \\frac{ \\left( 2E^2-1+\\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}\\right)^2}{E^2-1+\\frac{2M}{r}} \\right)\\exp {\\left( -\\frac{r}{ME} \\sqrt{E^2-1+\\frac{2M}{r}} \\right)}\\times \\times \\left( \\frac{\\frac{M}{r}}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right)^{\\frac{2E^2-1}{E \\sqrt{E^2-1}}}{\\mathcal {t}}^{\\prime 2}$ , $g_{{\\mathcal {t}^{\\prime }} \\tau ^{\\prime }}=g_{\\tau ^{\\prime } {\\mathcal {t}^{\\prime }}}=- 4\\left( \\frac{ 2E^2-1+ \\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}{ E^2-1+\\frac{2M}{r}} \\right)\\left( 2E^2-1+ \\frac{2M}{r} \\right)\\times \\times \\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}\\left( \\frac{\\frac{M}{r}}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right)^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}$ , $g_{\\theta \\theta }=r^2$ , $g_{\\phi \\phi }=r^2\\sin ^2{\\theta }$ .", "The contravariant components of the metric can be calculated to be $g^{{\\mathcal {t}}^{\\prime } {\\mathcal {t}}^{\\prime }}= - \\frac{{\\mathcal {t}}^{\\prime 2}}{16 M^2 E^2}$ , $g^{{\\tau }^{\\prime } {\\tau }^{\\prime }}=- \\frac{\\tau ^{\\prime 2}}{16 M^2 E^2}$ , $g^{{\\mathcal {t}}^{\\prime } \\tau ^{\\prime }}=g^{\\tau ^{\\prime } {\\mathcal {t}}^{\\prime }}==-\\frac{1}{16E^2}\\frac{1}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+ \\frac{2M}{r}}}\\left(2E^2-1+ \\frac{2M}{r} \\right)\\left[\\frac{r}{M}\\left(E^2-1+ \\frac{M}{r}+ \\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}\\right) \\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}\\times \\\\\\times \\exp {\\left( \\frac{r}{2ME} \\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}$ , $g^{\\theta \\theta }=\\frac{1}{r^2}$ , $g^{\\phi \\phi }=\\frac{1}{r^2\\sin ^2{\\theta }}$ .", "The normals $n_a$ to the ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces are ${n^{\\mathcal {t}^{\\prime }}}_a=(1,0,0,0)$ and ${n^{\\tau ^{\\prime }}}_a=(0,1,0,0)$ , respectively, where the superscripts ${\\mathcal {t}^{\\prime }}$ and $\\tau ^{\\prime }$ in this context are not indices, they simply label the respective normal.", "Their contravariant components are, respectively, $n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}=(g^{{\\mathcal {t}}^{\\prime } {\\mathcal {t}}^{\\prime }}, g^{{\\mathcal {t}}^{\\prime }\\tau ^{\\prime }},0,0)$ and $n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}=(g^{\\tau ^{\\prime }{\\mathcal {t}}^{\\prime }},g^{\\tau ^{\\prime }\\tau ^{\\prime }},0,0)$ , awkward writing them explicitly due to the long expression for $g^{{\\mathcal {t}}^{\\prime }\\tau ^{\\prime }}$ .", "The norms are then ${n^{\\mathcal {t}^{\\prime }}}_a n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}=-\\frac{{\\mathcal {t}}^{\\prime 2}}{16 M^2 E^2}$ and ${n^{\\tau ^{\\prime }}}_a n^{{\\tau ^{\\prime }}\\hspace{0.1424pt}a}=- \\frac{\\tau ^{\\prime 2}}{16 M^2 E^2}$ , respectively.", "Thus, clearly, the normals to the ${\\mathcal {t}^{\\prime }}={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces are timelike, and so ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ are timelike coordinates, and the corresponding hypersurfaces are spacelike, only in a measure zero are they null, when ${\\mathcal {t}}^{\\prime }=0$ and $\\tau ^{\\prime }=0$ , respectively." ], [ "Maximal analytic extension for $E=1$ as the\nlower limit of {{formula:b41b60d5-4aae-40a0-b253-31ca1514372f}}", "To build the maximal analytic extension for $E=1$ , we take the $E\\rightarrow 1$ limit from the $E>1$ case.", "Using $\\ln \\Big [ \\left(2\\sqrt{E^2-1}\\sqrt{\\frac{r}{2M}}\\right)+1\\Big ]=2\\sqrt{E^2-1}\\sqrt{\\frac{r}{2M}}$ in this limit, we find that the coordinates ${\\mathcal {t}}$ and $\\tau $ of Eqs.", "(REF ) and () become ${\\mathcal {t}}&= t- 4M \\sqrt{\\frac{r}{2M}}+ 2M \\ln {\\left|\\frac{\\sqrt{\\frac{r}{2M}}+1}{\\sqrt{\\frac{r}{2M}}-1} \\right|}\\,,\\\\\\tau &= t+ 4M \\sqrt{\\frac{r}{2M}}-2M \\ln {\\left|\\frac{\\sqrt{\\frac{r}{2M}}+1}{\\sqrt{\\frac{r}{2M}}-1} \\right|}\\,.$ The line element given in Eq.", "(REF ) is then in this $E=1$ limit given by $\\mathrm {d}s^2= -\\frac{1}{4}\\frac{\\left( 1-\\frac{2M}{r}\\right)}{\\frac{2M}{r}}\\left[ -\\left(1-\\frac{2M}{r} \\right) (\\mathrm {d}{\\mathcal {t}}^2+ \\mathrm {d}\\tau ^2)+ 2\\left(1+ \\frac{2M}{r} \\right) \\mathrm {d} {\\mathcal {t}}\\, \\mathrm {d}\\tau \\right]+ r^2( \\mathrm {d}\\theta ^2+ \\sin ^2{\\theta }\\,\\mathrm {d}\\phi ^2)\\,,$ with $r=r({\\mathcal {t}},\\tau )$ being obtained via Eq.", "(REF ) in the $E=1$ limit, or through Eqs.", "(REF ) and (), i.e., $4M \\sqrt{\\dfrac{r}{2M}}-2M \\ln {\\left|\\frac{\\sqrt{\\frac{r}{2M}}+1}{\\sqrt{\\frac{r}{2M}}-1}\\right|}=\\frac{1}{2}\\left(-{\\mathcal {t}}+\\tau \\right)\\,.$ Again, as in Eq.", "(REF ), the line element given in Eq.", "(REF ) is still degenerate at $r=2M$ .", "So, to extend it past $r=2M$ we again make use of maximal extended coordinates, ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ , defined as $\\frac{{\\mathcal {t}}^{\\prime }}{M}= -\\exp {\\left(-\\frac{{\\mathcal {t}}}{4M} \\right)}$ and $\\frac{\\tau ^{\\prime }}{M}= \\exp {\\left( \\frac{\\tau }{4M} \\right)}$ , which by either taking directly the limit $E=1$ in Eqs.", "(REF ) and (), respectively, or using Eqs.", "(REF ) and (), yields for $r>2M$ , $\\frac{{\\mathcal {t}}^{\\prime }}{M}&= -\\exp {\\left(-\\frac{{\\mathcal {t}}}{4M} \\right)}\\,,\\quad {\\rm i.e.,}\\quad \\frac{{\\mathcal {t}}^{\\prime }}{M}= -\\sqrt{\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}}\\exp {\\left( - \\frac{t}{4M}+\\sqrt{\\frac{r}{2M}} \\right)}\\,,\\\\\\frac{\\tau ^{\\prime }}{M}&=\\quad \\exp {\\left( \\;\\frac{\\tau }{4M} \\right)}\\,,\\;\\;\\,\\quad {\\rm i.e.,}\\quad \\frac{\\tau ^{\\prime }}{M}= \\quad \\sqrt{\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}}\\exp {\\left(\\;\\frac{t}{4M}+ \\sqrt{\\frac{r}{2M}} \\right)}\\,,$ respectively.", "Through the $E=1$ limit of Eq.", "(REF ), or putting ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ given in Eqs.", "(REF ) and (), respectively, into the line element Eq.", "(REF ), one finds that the new $E=1$ line element in coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ is given by $\\begin{split}\\mathrm {d}s^2= &-4\\frac{\\left( 1+ \\sqrt{\\frac{2M}{r}} \\right)^2}{\\frac{2M}{r}}\\exp {\\left( -2 \\sqrt{\\frac{r}{2M}} \\right)}\\left[ -\\frac{1}{M^2} \\left( 1+\\sqrt{\\frac{2M}{r}} \\right)^2 \\exp {\\left( -2 \\sqrt{\\frac{r}{2M}}\\right)} (\\tau ^{\\prime 2}\\, \\mathrm {d}{\\mathcal {t}}^{\\prime 2} +{\\mathcal {t}}^{\\prime 2}\\,\\mathrm {d}\\tau ^{\\prime 2})+\\right.\\\\&\\left.", "+ 2\\left(1+ \\frac{2M}{r} \\right) \\mathrm {d}{\\mathcal {t}}^{\\prime }\\, \\mathrm {d} \\tau ^{\\prime } \\right]+r^2 (\\mathrm {d}\\theta ^2+ \\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2) \\,,\\end{split}$ with $r=r({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ given implicitly, see Eq.", "(REF ) in the $E=1$ limit, or directly through Eqs.", "(REF ) and (), by $\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}\\exp {\\left( 2\\sqrt{\\frac{r}{2M}} \\right)} =-\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}\\,.$ All of this is done so that ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ have ranges $-\\infty <{\\mathcal {t}}^{\\prime }<\\infty $ and $-\\infty <\\tau ^{\\prime }<\\infty $ , which Eqs.", "(REF ) and (REF ) permit.", "To obtain Eqs.", "(REF ) and (REF ) directly from the $E\\rightarrow 1$ limit of Eqs.", "(REF ) and (REF ), respectively, see the Appendix.", "Several properties are again worth mentioning.", "In terms of the coordinates $({\\mathcal {t}},\\tau )$ , or $(t,r)$ , the coordinate transformations that yield the maximal extended coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ with infinite ranges have to be broadened, resulting in the existence of four regions, regions I, II, III, and IV.", "Region I is the region where the transformations Eqs.", "(REF ) and () hold, i.e., it is a region with ${\\mathcal {t}}^{\\prime }\\le 0$ and $\\tau ^{\\prime }\\ge 0$ .", "It is a region with $r\\ge 2M$ and $-\\infty <t<\\infty $ .", "Of course, in this region Eqs.", "(REF ) and (REF ) hold.", "Region II, a region for which $r\\le 2M$ , gets a different set of coordinate transformations.", "In this $r\\le 2M$ region, due to the moduli appearing in Eqs.", "(REF ) and () and the change of sign in Eq.", "(REF ), one defines instead ${\\mathcal {t}}^{\\prime }$ as $ \\frac{{\\mathcal {t}}^{\\prime }}{M}=+\\exp {\\left(-\\frac{{\\mathcal {t}}}{4M}\\right)}=\\\\=+\\sqrt{\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}}\\exp {\\left( - \\frac{t}{4M}+\\sqrt{\\frac{r}{2M}} \\right)}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} = \\exp {\\left( \\frac{\\tau }{4M} \\right) }=\\sqrt{\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}}\\exp {\\left(\\;\\frac{t}{4M}+ \\sqrt{\\frac{r}{2M}} \\right)}$ .", "These transformations are valid for ${\\mathcal {t}}^{\\prime }\\ge 0$ and $\\tau ^{\\prime }\\ge 0$ .", "It is a region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}\\exp {\\left( 2\\sqrt{\\frac{r}{2M}} \\right)} =\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so there is no further concern on that.", "Region III is another $r\\ge 2M$ region.", "Now one defines ${\\mathcal {t}}^{\\prime }$ as $ \\frac{{\\mathcal {t}}^{\\prime }}{M}=\\exp {\\left(-\\frac{{\\mathcal {t}}}{4M}\\right)}=\\sqrt{\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}}\\exp {\\left( - \\frac{t}{4M}+\\sqrt{\\frac{r}{2M}} \\right)}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} = -\\exp {\\left( \\frac{\\tau }{4M} \\right) }=-\\sqrt{\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}-1}}\\exp {\\left(\\;\\frac{t}{4M}+ \\sqrt{\\frac{r}{2M}} \\right)}$ .", "These transformations are valid for the region with ${\\mathcal {t}}^{\\prime }\\ge 0$ and $\\tau ^{\\prime }\\le 0$ .", "It is a region with $r\\ge 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}\\exp {\\left( 2\\sqrt{\\frac{r}{2M}} \\right)} =-\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so again there is no further concern on that.", "Region IV is another region with $r\\le 2M$ .", "Now, one defines ${\\mathcal {t}}^{\\prime }$ as $ \\frac{{\\mathcal {t}}^{\\prime }}{M}=-\\exp {\\left(-\\frac{{\\mathcal {t}}}{4M}\\right)}=-\\sqrt{\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}}\\exp {\\left( - \\frac{t}{4M}+\\sqrt{\\frac{r}{2M}} \\right)}$ and $\\tau ^{\\prime }$ as $\\frac{\\tau ^{\\prime }}{M} =- \\exp {\\left( \\frac{\\tau }{4M} \\right) }=-\\sqrt{\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}}\\exp {\\left(\\;\\frac{t}{4M}+ \\sqrt{\\frac{r}{2M}} \\right)}$ .", "These transformations are valid for the region with ${\\mathcal {t}}^{\\prime }\\le 0$ and $\\tau ^{\\prime }\\le 0$ .", "It is a region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\frac{1-\\sqrt{\\frac{r}{2M}}}{1+\\sqrt{\\frac{r}{2M}}}\\exp {\\left( 2\\sqrt{\\frac{r}{2M}} \\right)} =\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ .", "But all this has been automatically incorporated into Eqs.", "(REF ) and (REF ) so once again there is no further concern on that.", "Furthermore, from Eq.", "(REF ) we see that the event horizon at $r=2M$ has two solutions, ${\\mathcal {t}}^{\\prime }=0$ and $\\tau ^{\\prime }=0$ which are null surfaces represented by straight lines.", "The true curvature singularity at $r=0$ has two solutions $\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}=1$ , i.e., two spacelike hyperbolae.", "Implicit in the construction, there is a wormhole, or Einstein-Rosen bridge, topology, with its throat expanding and contracting.", "The dynamic wormhole is non traversable, but it spatially connects region I to region III through regions II and IV.", "Regions I and III are two asympotically flat regions causally separated, region II is the black hole region, and region IV is the white hole region of the spacetime.", "Eqs.", "(REF ) and (REF ) together with the corresponding interpretation give the maximal extension of the Schwarzschild metric for $E=1$ , in the coordinates $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ .", "The two-dimensional part $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ of the coordinate system $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime },\\theta ,\\phi )$ is shown in Figure REF , both for lines of constant ${\\mathcal {t}}^{\\prime }$ and constant $\\tau ^{\\prime }$ in part (a) of the figure, and for lines of constant $t$ and constant $r$ in part (b) of the figure, conjointly with the labeling of regions I, II, III, IV, needed to cover it.", "Figure: The maximal analytical extension of the Schwarzschild metricfor the parameter E=1E=1 in the plane (𝓉 ' ,τ ' )({\\mathcal {t}}^{\\prime },\\tau ^{\\prime }) is shownin a diagram with two different descriptions, (a) and (b).", "In (a)typical values for lines of constant 𝓉 ' {\\mathcal {t}}^{\\prime } and constantτ ' \\tau ^{\\prime } are displayed.", "In (b) typical values for lines of constant ttand constant rr are displayed.", "The diagram, both in (a) and in(b), represents aa spacetime with a wormhole, not shown, that forms out of a singularityin the white hole region, i.e., region IV, and finishes at the blackhole region and its singularity, i.e., region II, connecting the twoseparated asymptotically flat spacetimes, regions I and III.", "TheE=1E=1 diagram is very similar to the E>1E>1 generic case diagram, seeFigure , as it is expected for a maximalextension of the Schwarzschild spacetime, in particular for thoseextensions within the same family.It is also worth discussing the normals to the ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces.", "From Eq.", "(REF ) one finds that the metric has covariant components $g_{{\\mathcal {t}^{\\prime }} {\\mathcal {t}^{\\prime }}}=\\frac{4}{M^2}\\frac{\\left(1+\\sqrt{\\frac{2M}{r}} \\right)^4}{\\frac{2M}{r}} \\exp {\\left( -4\\sqrt{\\frac{r}{2M}} \\right)} \\tau ^{\\prime 2} $ , $g_{\\tau ^{\\prime } \\tau ^{\\prime }}= \\frac{4}{M^2}\\frac{\\left(1+\\sqrt{\\frac{2M}{r}} \\right)^4}{\\frac{2M}{r}}\\times \\\\\\times \\exp {\\left( -4\\sqrt{\\frac{r}{2M}} \\right)}{\\mathcal {t}}^{\\prime 2}$ , $g_{{\\mathcal {t}^{\\prime }} \\tau ^{\\prime }}=g_{\\tau ^{\\prime } {\\mathcal {t}^{\\prime }}}=-4\\frac{\\left( 1+ \\sqrt{\\frac{2M}{r}} \\right)^2}{\\frac{2M}{r}}\\left( 1+ \\frac{2M}{r} \\right)\\exp {\\left( -2\\sqrt{\\frac{r}{2M}} \\right)}$ , $g_{\\theta \\theta }=r^2$ , $g_{\\phi \\phi }=r^2\\sin ^2{\\theta }$ .", "The contravariant components of the metric can be calculated to be $g^{{\\mathcal {t}}^{\\prime } {\\mathcal {t}}^{\\prime }}= - \\frac{{\\mathcal {t}}^{\\prime 2}}{16 M^2}$ , $g^{{\\tau }^{\\prime } {\\tau }^{\\prime }}= -\\frac{\\tau ^{\\prime 2}}{16M^2}$ , $g^{{\\mathcal {t}}^{\\prime } \\tau ^{\\prime }}=g^{\\tau ^{\\prime } {\\mathcal {t}}^{\\prime }}=-\\frac{1}{16}\\frac{1+\\frac{2M}{r}}{\\left(1+ \\sqrt{\\frac{2M}{r}} \\right)^{2}} \\times \\\\ \\times \\exp {\\left( 2 \\sqrt{\\frac{r}{2M}}\\right)}$ , $g^{\\theta \\theta }=\\frac{1}{r^2}$ , $g^{\\phi \\phi }=\\frac{1}{r^2\\sin ^2{\\theta }}$ .", "The normals $n_a$ to the ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces are ${n^{\\mathcal {t}^{\\prime }}}_a=(1,0,0,0)$ and ${n^{\\tau ^{\\prime }}}_a=(0,1,0,0)$ , respectively, where the superscripts ${\\mathcal {t}^{\\prime }}$ and $\\tau ^{\\prime }$ in this context are not indices, they simply label the respective normal.", "Their contravariant components are $n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}=(g^{{\\mathcal {t}}^{\\prime } {\\mathcal {t}}^{\\prime }},g^{{\\mathcal {t}}^{\\prime }\\tau ^{\\prime }},0,0) ={=(- \\frac{{\\mathcal {t}}^{\\prime 2}}{16 M^2},-\\frac{1}{16}\\frac{1+\\frac{2M}{r}}{\\left(1+ \\sqrt{\\frac{2M}{r}} \\right)^{2}}\\exp {\\left( 2 \\sqrt{\\frac{r}{2M}}\\right)}},0,0)$ and $n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}=(g^{\\tau ^{\\prime }{\\mathcal {t}}^{\\prime }},g^{\\tau ^{\\prime } \\tau ^{\\prime }},0,0)=(-\\frac{1}{16}\\frac{1+\\frac{2M}{r}}{\\left(1+ \\sqrt{\\frac{2M}{r}} \\right)^{2}}\\exp {\\left( 2 \\sqrt{\\frac{r}{2M}}\\right)},-\\frac{\\tau ^{\\prime 2}}{16M^2},0,0)$ , respectively.", "The norms are then ${n^{\\mathcal {t}^{\\prime }}}_a n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}=-\\frac{{\\mathcal {t}}^{\\prime 2}}{16 M^2}$ and ${n^{\\tau ^{\\prime }}}_a n^{{\\tau ^{\\prime }}\\hspace{0.1424pt}a}=- \\frac{\\tau ^{\\prime 2}}{16 M^2}$ , respectively.", "Thus, clearly, the normals to the ${\\mathcal {t}^{\\prime }}={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ hypersurfaces are timelike, and so ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ are timelike coordinates, and the corresponding hypersurfaces are spacelike, only in a measure zero are they null, when ${\\mathcal {t}}^{\\prime }=0$ and $\\tau ^{\\prime }=0$ , respectively.", "The metric components and the normals can also be found from the $E>1$ case in the $E=1$ limit." ], [ "Maximal analytic extension for $E= \\infty $ as the upper limit\nof {{formula:b33a2d87-722e-4afa-bac7-5458dd1c6816}} : The Kruskal-Szekeres maximal extension of the Schwarzschild\nmetric", "To build the maximal analytic extension for $E=\\infty $ , we take the $E\\rightarrow \\infty $ limit from the $E>1$ generic case.", "We will see that this limit is the Kruskal-Szekeres maximal analytic extension.", "Taking a redefinition of the coordinates $\\tau $ and ${\\mathcal {t}}$ of Eqs.", "(REF ) and () to coordinates $u$ and $v$ , respectively, we find that these become $u&\\equiv \\lim _{E\\rightarrow \\infty } \\frac{{\\mathcal {t}}}{E}\\,, \\quad {\\rm i.e.,}\\quad \\quad u=t-r-2M \\ln {\\left| \\frac{r}{2M}-1 \\right|}\\,,\\\\v&\\equiv \\lim _{E\\rightarrow \\infty } \\frac{\\tau }{E}\\,, \\quad {\\rm i.e.,}\\quad \\quad v=t+r+2M \\ln {\\left| \\frac{r}{2M}-1 \\right|}\\,.$ The line element given in Eq.", "(REF ) is then in this limit $\\mathrm {d}s^2=-\\left(1-\\frac{2M}{r}\\right)dudv+r^2( \\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)\\,.$ with $r=r(u,v)$ being obtained directly via Eq.", "(REF ) in the $E\\rightarrow \\infty $ limit, or through Eqs.", "(REF ) and (), i.e., $r+2M \\ln {\\left| \\frac{r}{2M}-1 \\right|}=\\frac{1}{2}\\left(-u+v\\right)\\,.$ Again, the line element given in Eq.", "(REF ) is still degenerate at $r=2M$ .", "So, to extend it past $r=2M$ , we make use of the maximal extended timelike coordinates ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ defined through $\\frac{{\\mathcal {t}}^{\\prime }}{M}= -\\exp {\\left( -\\frac{{\\mathcal {t}}}{4ME}\\right)}$ and $\\frac{\\tau ^{\\prime }}{M}= \\exp {\\left( \\;\\frac{\\tau }{4ME} \\right)}$ , which in this limit $E\\rightarrow \\infty $ are redefined to maximal extended coordinates, $u^{\\prime }$ and $v^{\\prime }$ , respectively, obtained directly via Eqs.", "(REF ) and () in the $E\\rightarrow \\infty $ limit, or using Eqs.", "(REF ) and (), to find $u^{\\prime }&=\\lim _{E\\rightarrow \\infty } {\\mathcal {t}^{\\prime }}\\,, \\quad {\\rm i.e.,}\\quad \\quad \\frac{u^{\\prime }}{M}=-\\exp {\\left(-\\frac{u}{4M}\\right)}\\,,\\quad \\quad {\\rm i.e.,} \\quad \\quad \\frac{u^{\\prime }}{M}=-\\sqrt{\\frac{r}{2M}-1}\\,\\exp {\\left( -\\frac{t}{4M}+\\frac{r}{4M}\\right)}\\,,\\\\v^{\\prime }&=\\lim _{E\\rightarrow \\infty } \\tau ^{\\prime }\\,, \\quad {\\rm i.e.,} \\quad \\quad \\frac{v^{\\prime }}{M}=\\quad \\exp {\\left(\\frac{v}{4M}\\right)}\\,,\\quad \\quad \\;\\;\\; {\\rm i.e.,} \\quad \\quad \\frac{v^{\\prime }}{M}=\\quad \\sqrt{\\frac{r}{2M}-1}\\,\\exp {\\left( \\frac{t}{4M}+\\frac{r}{4M}\\right)}\\,.$ Then, the line element of (REF ) in the $E\\rightarrow \\infty $ limit, or through Eq.", "(REF ) together with Eqs.", "(REF ) and (), yields the new line element $\\mathrm {d}s^2=-\\frac{32M}{r} \\exp {\\left(-\\frac{r}{2M}\\right)}\\mathrm {d}u^{\\prime }\\, \\mathrm {d}v^{\\prime }+ r^2( \\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)\\,,$ with $r=r(u^{\\prime },v^{\\prime })$ given implicitly, see Eq.", "(REF ) in the $E\\rightarrow \\infty $ limit, or directly through Eqs.", "(REF ) and (), by $\\left(\\frac{r}{2M}-1\\right)\\exp {\\left(\\frac{r}{2M}\\right)}=-\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}\\,.$ All of this is done so that $u^{\\prime }$ and $v^{\\prime }$ have ranges $-\\infty <u^{\\prime }<\\infty $ and $-\\infty <v^{\\prime }<\\infty $ , which Eqs.", "(REF ) and (REF ) permit.", "To obtain Eqs.", "(REF ) and (REF ) directly from the $E\\rightarrow \\infty $ limit of Eqs.", "(REF ) and (REF ), respectively, see the Appendix.", "Several properties are worth mentioning.", "In terms of the coordinates $(u,v)$ , or $(t,r)$ , the coordinate transformations that yield the maximal extended null coordinates $(u^{\\prime },v^{\\prime })$ with infinite ranges have to be broadened, resulting in the existence of four regions, regions I, II, III, and IV.", "Region I is the region where the transformations Eqs.", "(REF ) and () hold, i.e., it is a region with $u^{\\prime }\\le 0$ and $v^{\\prime }\\ge 0$ , or a region with $r\\ge 2M$ and $-\\infty <t<\\infty $ .", "Region II, a region for which $r\\le 2M$ , gets a different set of coordinate transformations.", "In this region $r\\le 2M$ , due to the moduli appearing in Eqs.", "(REF ) and () and the change of sign in Eq.", "(REF ), one defines instead $u^{\\prime }$ as $\\frac{u^{\\prime }}{M}=+ \\exp {\\left(-\\frac{u}{4M}\\right)}=\\sqrt{1-\\frac{r}{2M}}\\, \\exp {\\left(-\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ and $v^{\\prime }$ as $\\frac{v^{\\prime }}{M}=\\exp {\\left(\\frac{v}{4M}\\right)}=\\sqrt{1-\\frac{r}{2M}}\\, \\exp {\\left(\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ .", "These transformations are valid for the region with $u^{\\prime }\\ge 0$ and $v^{\\prime }\\ge 0$ , or the region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\left(1-\\frac{r}{2M}\\right)\\exp {\\left(\\frac{r}{2M}\\right)}=\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}$ .", "But this is automatically incorporated into Eq.", "(REF ), so there is no further concern on that.", "Region III is another $r\\ge 2M$ region.", "Now one defines $\\frac{u^{\\prime }}{M}= \\exp {\\left(-\\frac{u}{4M}\\right)}=\\sqrt{\\frac{r}{2M}-1}\\, \\exp {\\left(-\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ and $v^{\\prime }$ as $\\frac{v^{\\prime }}{M}=-\\exp {\\left(\\frac{v}{4M}\\right)}=-\\sqrt{\\frac{r}{2M}-1}\\, \\exp {\\left(\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ .", "These transformations are valid for the region with $u^{\\prime }\\ge 0$ and $v^{\\prime }\\le 0$ , or the region with $r\\ge 2M$ and $-\\infty <t<\\infty $ .", "Note that the coordinate transformations in this region give $\\left(\\frac{r}{2M}-1\\right)\\exp {\\left(\\frac{r}{2M}\\right)}=-\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}$ .", "But this is automatically incorporated into Eq.", "(REF ), so again there is no further concern on that.", "Region IV is another region with $r\\le 2M$ .", "Now, one defines $u^{\\prime }$ as $\\frac{u^{\\prime }}{M}=- \\exp {\\left(-\\frac{u}{4M}\\right)}= =-\\sqrt{1-\\frac{r}{2M}}\\, \\exp {\\left(-\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ and $v^{\\prime }$ as $\\frac{v^{\\prime }}{M}=-\\exp {\\left(\\frac{v}{4M}\\right)}=-\\sqrt{1-\\frac{r}{2M}}\\, \\exp {\\left(\\frac{t}{4M}+\\frac{r}{4M}\\right)}$ .", "These transformations are valid for the region with $u^{\\prime }\\le 0$ and $v^{\\prime }\\le 0$ , or the region with $r\\le 2M$ and $-\\infty <t<\\infty $ .", "The coordinate transformations in this region give as well $\\left(1-\\frac{r}{2M}\\right)\\exp {\\left(\\frac{r}{2M}\\right)}=\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}$ .", "But this is automatically incorporated into Eq.", "(REF ), so once again there is no further concern on that.", "Furthermore, from Eq.", "(REF ) we see that the event horizon at $r=2M$ has two solutions, $u^{\\prime }=0$ and $v^{\\prime }=0$ which are null surfaces represented by straight lines.", "The true curvature singularity at $r=0$ has two solutions $\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}=1$ , i.e., two spacelike hyperbolae.", "Implicit in the construction, there is a wormhole, or Einstein-Rosen bridge, topology, with its throat expanding and contracting.", "The dynamic wormhole is non traversable, but it spatially connects region I to region III through regions II and IV.", "Regions I and III are two asympotically flat regions causally separated, region II is the black hole region, and region IV is the white hole region of the spacetime.", "Eqs.", "(REF ) and (REF ) together with the corresponding interpretation give the maximal extension of the Schwarzschild metric for $E=\\infty $ , taken as the limit of $E>1$ , in the coordinates $(u^{\\prime },v^{\\prime },\\theta ,\\phi )$ .", "Of course, this the Kruskal-Szekeres maximal analytical extension, now seen as the $E=\\infty $ member of the family of extensions of $E>1$ .", "Recalling that $u^{\\prime }={\\mathcal {t}}^{\\prime }|_{E=\\infty }$ and $v^{\\prime }=\\tau ^{\\prime }|_{E=\\infty }$ , we see that the two timelike congruences that specify the two analytically extended time coordinates $\\mathcal {t}^{\\prime }$ and $\\tau ^{\\prime }$ that yield the maximal extension for $E>1$ turned into the two analytically extended null retarded and advanced congruences $u^{\\prime }$ and $v^{\\prime }$ of the Kruskal-Szekeres maximal extension.", "The two-dimensional part $(u^{\\prime },v^{\\prime })$ of the coordinate system $(u^{\\prime },v^{\\prime },\\theta ,\\phi )$ is shown in Figure REF , both for lines of constant $u^{\\prime }$ and constant $v^{\\prime }$ in part (a) of the figure, and for lines of constant $t$ and constant $r$ in part (b) of the figure, conjointly with the labeling of regions I, II, III, IV, needed to cover it.", "Figure: The maximal analytical extension of the Schwarzschild metric for theparameter E=∞E=\\infty , i.e., the Kruskal-Szekeres maximal extension, inthe plane (u ' ,v ' )(u^{\\prime },v^{\\prime }) is shown in a diagram with two differentdescriptions, (a) and (b).", "In (a) typical values for lines of constantu ' u^{\\prime } and constant v ' v^{\\prime } are displayed.", "In (b) typical values for linesof constant tt and constant rr are displayed.", "The diagram, bothin (a) and in (b), represents a spacetime witha wormhole, not shown, that forms out ofa singularity in the white hole region, i.e., region IV, and finishesat the black hole region and its singularity, i.e., region II,connecting the two separated asymptotically flat spacetimes, regions Iand III.", "The E=∞E=\\infty diagram, i.e., the Kruskal-Szekeres diagram,is very similar to the E>1E>1 generic case diagram, seeFigure , as it is expected for a maximalextension of the Schwarzschild spacetime, in particular for thoseextensions within the same family.It is also worth discussing the normals to the $u^{\\prime }={\\rm constant}$ and $v^{\\prime }={\\rm constant}$ hypersurfaces.", "For that, we see that from Eq.", "(REF ) in the limit $E\\rightarrow \\infty $ , or directly from Eq.", "(REF ), one finds that the metric has covariant components $g_{u^{\\prime }u^{\\prime }}=0$ , $g_{v^{\\prime }v^{\\prime }}=0$ , $g_{u^{\\prime }v^{\\prime }}=g_{v^{\\prime }u^{\\prime }}=-\\frac{16M}{r}\\exp {\\left( -{\\frac{r}{2M}} \\right)}$ , $g_{\\theta \\theta }=r^2$ , $g_{\\phi \\phi }=r^2\\sin ^2{\\theta }$ .", "The contravariant components of the metric can be calculated to be $g^{u^{\\prime }u^{\\prime }}=0$ , $g^{v^{\\prime }v^{\\prime }}=0$ , $g^{u^{\\prime }v^{\\prime }}=g^{v^{\\prime }u^{\\prime }}= -\\frac{r}{16M}\\exp {\\left( \\frac{r}{2M}\\right)}$ , $g^{\\theta \\theta }=\\frac{1}{r^2}$ , $g^{\\phi \\phi }=\\frac{1}{r^2\\sin ^2{\\theta }}$ .", "The normals $n_a$ to the $u^{\\prime }={\\rm constant}$ and $v^{\\prime }={\\rm constant}$ hypersurfaces are ${n^{u^{\\prime }}}_a=(1,0,0,0)$ and ${n^{v^{\\prime }}}_a=(0,1,0,0)$ , respectively, where the superscripts $u^{\\prime }$ and $v^{\\prime }$ in this context are not indices, they simply label the respective normal.", "Their contravariant components are $n^{{u^{\\prime }}\\hspace{0.1424pt}a}=(g^{u^{\\prime }u^{\\prime }}, g^{u^{\\prime }v^{\\prime }},0,0) =(0,-\\frac{r}{16M}\\exp {\\left( \\frac{r}{2M}\\right)},0,0)$ and $n^{{v^{\\prime }}\\hspace{0.1424pt}a}=(g^{v^{\\prime }u^{\\prime }},g^{v^{\\prime }v^{\\prime }},0,0) = (-\\frac{r}{16M} \\exp {\\left(\\frac{r}{2M}\\right)},0,0,0)$ .", "The norms are then ${n^{u^{\\prime }}}_an^{{u^{\\prime }}\\hspace{0.1424pt}a}=0$ and ${n^{v^{\\prime }}}_an^{{v^{\\prime }}\\hspace{0.1424pt}a}=0$ , respectively.", "Thus, clearly, the normals to the $u^{\\prime }={\\rm constant}$ and $v^{\\prime }={\\rm constant}$ hypersurfaces are null, and so $u^{\\prime }$ and $v^{\\prime }$ are null coordinates, and the corresponding hypersurfaces are null as well." ], [ "Causal diagrams from $E=1$ to {{formula:c857f959-6c7d-481e-92f9-91c302f9d472}}", "In this unified account that carries maximal extensions of the Schwarzschild metric along the parameter $E$ , it is of interest to trace the radial null geodesics for several values of the parameter $E$ itself, $1\\le E\\le \\infty $ , in the plane characterized by the $({\\mathcal {t}}^{\\prime },\\tau ^{\\prime })$ coordinates.", "Null geodesics have $ds^2=0$ along them, and if they are radial then also $d\\theta =0$ and $d\\phi =0$ .", "Using the line element given in Eq.", "(REF ) together with Eq.", "(REF ), we can then trace the radial null geodesics, and with it the causal structure for each $E$ , in the corresponding maximally analytic extended diagram.", "Figures REF , REF , REF , and REF are the maximal extended causal diagrams for $E=1.0$ , $E=1.1$ , $E=1.5$ , and $E=\\infty $ , respectively.", "In the $E=1$ case one can take the null geodesics directly from Eqs.", "(REF )-(REF ), and in the $E=\\infty $ case, i.e., the Kruskal-Szekeres extension, directly from Eqs.", "(REF )-(REF ).", "The features shown in the four figures are: (i) The past and future spacelike singularities at $r=0$ .", "(ii) The regions I, II, III, and IV, described earlier.", "(iii) The lines of ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ , in the $E=\\infty $ case these are the lines of $u^{\\prime }={\\rm constant}$ and $v^{\\prime }={\\rm constant}$ .", "(iv) The outgoing null geodesics represented by red lines and the ingoing null geodesics represented by blue lines.", "(v) The contravariant normals to the ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ , i.e., $n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}$ and $n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}$ , respectively, as given in detail previously.", "Figure: Causal diagram for the maximal analytical extension in theE=1E=1 case.", "The two singularities and the two event horizons are showntogether with lines of outgoing and ingoing null geodesics, drawn inred and blue, respectively, and with lines of constant 𝓉 ' {\\mathcal {t}}^{\\prime }and τ ' \\tau ^{\\prime }.", "The contravariant normals to the 𝓉 ' = constant {\\mathcal {t}}^{\\prime }={\\rm constant} and τ ' = constant \\tau ^{\\prime }={\\rm constant}hypersurfaces, i.e., n 𝓉 ' a n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a} and n τ ' a n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}, respectively, arealso shown, with their timelike character clearly exhibited.See text for more details.", "Figure: Causal diagram for the maximal analytical extension in theE=1.1E=1.1 case.", "The two singularities and the two event horizons areshown together with lines of outgoing and ingoing null geodesics,drawn in red and blue, respectively, and with lines of constant𝓉 ' {\\mathcal {t}}^{\\prime } and τ ' \\tau ^{\\prime }.", "The contravariant normals to the𝓉 ' = constant {\\mathcal {t}}^{\\prime }={\\rm constant} and τ ' = constant \\tau ^{\\prime }={\\rm constant}hypersurfaces, i.e., n 𝓉 ' a n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a} andn τ ' a n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}, respectively, are also shown,with their timelike character clearly exhibited.", "See textfor more details.Figure: Causal diagram for the maximal analytical extension in theE=1.5E=1.5 case.", "The two singularities and the two event horizons areshown together with lines of outgoing and ingoing null geodesics,drawn in red and blue, respectively, and with lines of constant𝓉 ' {\\mathcal {t}}^{\\prime } and τ ' \\tau ^{\\prime }.", "The contravariant normals to the𝓉 ' = constant {\\mathcal {t}}^{\\prime }={\\rm constant} and τ ' = constant \\tau ^{\\prime }={\\rm constant}hypersurfaces, i.e., n 𝓉 ' a n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a} andn τ ' a n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}, respectively, are also shown,with their timelike character clearly exhibited.", "See textfor more details.Figure: Causal diagram for the maximal analytical extension in theE=∞E=\\infty case, i.e., the Kruskal-Szekeres maximal extension.The two singularities and the two event horizons areshown together with lines of outgoing and ingoing null geodesics,drawn in red and blue, respectively, and with lines of constantu ' ≡lim E→∞ 𝓉 ' u^{\\prime }\\equiv \\lim _{E\\rightarrow \\infty } {\\mathcal {t}^{\\prime }} andv ' ≡lim E→∞ τ ' v^{\\prime }\\equiv \\lim _{E\\rightarrow \\infty } {\\tau ^{\\prime }}.", "In this E=∞E=\\infty casethese two sets of lines coincide.The contravariant normals to theu ' = constant u^{\\prime }={\\rm constant} and v ' = constant v^{\\prime }={\\rm constant}hypersurfaces, i.e., n u ' a n^{u^{\\prime }\\hspace{0.1424pt}a} andn v ' a n^{v^{\\prime }\\hspace{0.1424pt}a}, respectively, are also shown,with their null character clearly exhibited.", "See textfor more details.As it had to be, the lines of ${\\mathcal {t}}^{\\prime }={\\rm constant}$ and $\\tau ^{\\prime }={\\rm constant}$ are tachyonic, i.e., spacelike hypersurfaces, a feature clearly seen by comparison of these lines with the ingoing and outgoing null geodesic lines, except for ${\\mathcal {t}}^{\\prime }=0$ and $\\tau ^{\\prime }=0$ which are null lines representing the $r=2M$ event horizons of the solution that separate regions I, II, III, and IV.", "In the $E=\\infty $ case, i.e., Kruskal-Szekeres, the spacelike lines turn into the null lines $u^{\\prime }={\\rm constant}$ and $v^{\\prime }={\\rm constant}$ , with $u^{\\prime }=0$ and $v^{\\prime }=0$ being the event horizons separating regions I, II, III, and IV.", "One also sees that the contravariant normals $n^{{\\mathcal {t}^{\\prime }}\\hspace{0.1424pt}a}$ and $n^{\\tau ^{\\prime }\\hspace{0.1424pt}a}$ , are always inside the local light cone, and so the coordinates ${\\mathcal {t}^{\\prime }}$ and $\\tau ^{\\prime }$ are timelike, except at the horizons where they are null.", "In the $E=\\infty $ case, i.e., Kruskal-Szekeres, the contravariant normals $n^{u^{\\prime }\\hspace{0.1424pt}a}$ and $n^{v^{\\prime }\\hspace{0.1424pt}a}$ are null vectors always, and so the coordinates $u^{\\prime }$ and $v^{\\prime }$ are null, i.e., the ${\\mathcal {t}^{\\prime }}$ and $\\tau ^{\\prime }$ timelike coordinates turned into the $u^{\\prime }$ and $v^{\\prime }$ null coordinates." ], [ "Conclusions", "The scenario for maximally extend the Schwarzschild metric is now complete.", "Schwarzschild is the starting point.", "In the usual standard coordinates, also called Schwarzschild coordinates, its extension past the sphere $r=2M$ is cryptic, in any case is not maximal, and to exhibit it fully one needs two coordinate patches, altogether making it very difficult to obtain a complete interpretation.", "Departing from it, there is one branch alone, namely, the Painlevé-Gullstrand branch that works either with outgoing or with ingoing timelike congruences, or equivalently with outgoing or ingoing test particles placed over them, parameterized by their energy per unit mass $E$ , and that in the $E\\rightarrow \\infty $ limit ends in the Eddington-Finkelstein retarded or advanced null coordinates, respectively.", "The Painlevé-Gullstrand branch, including its Eddington-Finkelstein $E=\\infty $ endpoint, partially extends the Schwarzschild metric past $r=2M$ , but it is not maximal, to have the full solution one needs two coordinate patches, which again inhibits the full interpretation of the solution.", "Then, from Painlevé-Gullstrand there are two bifucartion branches.", "One branch is the Novikov-Lemaître that uses the Painlevé-Gullstrand time coordinate and an appropriate radial comoving coordinate.", "This branch extends the Schwarzschild metric past $r=2M$ , is maximal in the Novikov range $0<E<1$ and partial only in the Lemaître range $1\\le E<\\infty $ , ending, in the $E\\rightarrow \\infty $ limit, in Minkowski.", "The other branch is the one we found here, with the two analytically extended Painlevé-Gullstrand time coordinates, one related to outgoing, the other to ingoing timelike congruences.", "This branch extends the Schwarzschild metric past $r=2M$ , is maximal and valid for $1\\le E <\\infty $ , and ends, for $E=\\infty $ , directly, or if wished, via the two analytically extended Eddington-Finkelstein retarded and advanced null coordinates, in the Kruskal-Szekeres maximal extension.", "The maximally extended solutions of the Schwarzschild metric allow for an easy and full interpretation of its complex spacetime structure.", "Indeed, whereas the partial extensions of the Schwarzschild metric are of great interest to analyze gravitational collapse of matter and physical phenomena involving black holes where a future event horizon makes its appearance, and in certain instances to analyze time reversal white hole phenomena, the maximal extensions deliver the full solution, showing a model dynamic universe with two separate spacetime sheets, containing a past spacelike singularity, with a white hole region delimited by a past event horizon, that join at a dynamic nontraversable Einstein-Rosen bridge, or wormhole whose throat expands up to $r=2M$ , to collapse into the inside of a future event horizon containing a black hole region with a future spacelike singularity separating again the two separate spacetime sheets of this model universe.", "Here, a family of maximal extensions of the Schwarzschild spacetime parameterized by the energy per unit mass $E$ of congruences of outgoing and ingoing timelike geodesics has been obtained.", "In this unified description, the Kruskal-Szekeres maximal extension of sixty years ago is seen here as the important, but now particular, instance of this $E$ family, namely, the one with $E=\\infty $ .", "This maximal description provides the link between Gullstrand-Painlevé and Kruskal-Szekeres.", "We acknowledge FCT - Fundação para a Ciência e Tecnologia of Portugal for financial support through Project No.", "UIDB/00099/2020." ], [ "Appendix: Details for the $E\\rightarrow 1$ limit\nand the {{formula:d0fc3d1c-8549-48fe-ab06-f04e4ebfae28}} Kruskal-Szekeres limit\nfrom the {{formula:8c95c9a9-c4e6-49b4-a021-3dd943cbf5a4}} generic case", "In order to see the continuity of the maximal extension parameterized by $E$ , we take the generic $E>1$ case, and from it obtain directly the limit to the case $E=1$ , and the limit to the case $E=\\infty $ , i.e., the Kruskal-Szekeres extension.", "$E=1$ limit from $E>1$ : Here we take the $E\\rightarrow 1$ limit of Eqs.", "(REF ) and (REF ).", "We will do it term by term in each equation.", "For Eq.", "(REF ) we have: $\\lim _{E\\rightarrow 1} -4 \\left( \\frac{ 2E^2-1+ \\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}{ E^2-1+\\frac{2M}{r}} \\right)=-4 \\frac{(1+ \\sqrt{\\frac{2M}{r}})^2}{\\frac{2M}{r}}$ ; $\\lim _{E\\rightarrow 1} \\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+\\frac{2M}{r}} \\right)}=\\exp {\\left( -\\sqrt{\\frac{r}{2M}} \\right)}$ ; $\\lim _{E\\rightarrow 1} \\left( \\frac{\\frac{M}{r}}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right) ^{\\frac{2E^2-1}{2E\\sqrt{E^2-1}}} =\\left(1+2\\sqrt{E^2-1}\\sqrt{\\frac{r}{2M}}\\right)^{-\\frac{1}{2\\sqrt{E^2-1}}}=\\exp \\left[-\\frac{1}{2\\sqrt{E^2-1}}\\ln \\left(1+2\\sqrt{E^2-1}\\right.\\right.\\times \\\\\\left.\\left.\\times \\sqrt{\\frac{r}{2M}}\\right)\\right]=\\exp {\\left( -\\sqrt{\\frac{r}{2M}} \\right)}$ ; $\\lim _{E\\rightarrow 1}-\\frac{1}{M^2} \\left(2E^2-1+ \\frac{2M}{r}+2E\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)= -\\frac{1}{M^2} \\left( 1+ \\sqrt{\\frac{2M}{r}} \\right)^2$ ; $\\lim _{E\\rightarrow 1} \\exp \\left( -\\frac{r}{2ME}\\sqrt{E^2-1+\\frac{2M}{r}} \\right)=\\exp {\\left( -\\sqrt{\\frac{r}{2M}} \\right)}$ ; $\\lim _{E\\rightarrow 1}\\left(\\frac{\\frac{M}{r}}{E^2-1+\\frac{M}{r}+\\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}}\\right)^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}=\\exp {\\left( -\\sqrt{\\frac{r}{2M}} \\right)}$ ; $\\lim _{E\\rightarrow 1}2 \\left( 2E^2-1+\\frac{2M}{r} \\right)=2 \\left( 1+ \\frac{2M}{r} \\right)$ .", "Thus, Eq.", "(REF ) is now $\\mathrm {d}s^2=-4 \\frac{(1+\\sqrt{\\frac{2M}{r}})^2}{\\frac{2M}{r}} \\exp {\\left( -2\\sqrt{\\frac{r}{2M}} \\right)} \\Big [ -\\frac{1}{M^2} \\left( 1+\\sqrt{\\frac{2M}{r}} \\right)^2\\times \\\\\\times \\exp {\\left( -2 \\sqrt{\\frac{r}{2M}}\\right)} (\\tau ^{\\prime 2} \\mathrm {d} {\\mathcal {t}}^{\\prime 2}+ {\\mathcal {t}}^{\\prime 2}\\mathrm {d} \\tau ^{\\prime 2})+ 2\\left(1+ \\frac{2M}{r} \\right) \\mathrm {d}{\\mathcal {t}}^{\\prime }\\, \\mathrm {d} \\tau ^{\\prime 2} \\Big ]+ r^2 (\\mathrm {d} \\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d} \\phi ^2)$ .", "This is the line element found for the $E=1$ case, see Eq.", "(REF ).", "For Eq.", "(REF ) we have: $\\lim _{E\\rightarrow 1}\\left( \\frac{\\frac{r}{2M}-1}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}=\\frac{\\sqrt{\\frac{r}{2M}}- 1}{\\sqrt{\\frac{r}{2M}}+ 1}$ ; $\\lim _{E\\rightarrow 1}\\exp {\\left( \\frac{r}{2ME}\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}=\\exp {\\left( \\sqrt{\\frac{r}{2M}} \\right)}$ ; $\\lim _{E\\rightarrow 1}\\left[\\frac{r}{M}\\left( E^2-1 + \\frac{M}{r}+ \\sqrt{E^2-1}\\sqrt{E^2-1+ \\frac{2M}{r}}\\right)\\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}==\\left(1+2\\sqrt{E^2-1}\\sqrt{\\frac{r}{2M}}\\right)^{\\frac{1}{2\\sqrt{E^2-1}}}=\\exp \\left[\\frac{1}{2\\sqrt{E^2-1}}\\ln \\left(1+2\\sqrt{E^2-1}\\sqrt{\\frac{r}{2M}}\\right)\\right]=\\exp {\\left( \\sqrt{\\frac{r}{2M}} \\right)}$ .", "Thus, Eq (REF ) in the $E\\rightarrow 1$ limit turns into $\\frac{\\sqrt{\\frac{r}{2M}}-1}{\\sqrt{\\frac{r}{2M}}+1}\\exp {\\left( 2 \\sqrt{\\frac{r}{2M}} \\right)}=-\\frac{{\\mathcal {t}}^{\\prime }}{M}\\frac{\\tau ^{\\prime }}{M}$ .", "This is indeed Eq.", "(REF ).", "$E\\rightarrow \\infty $ limit from $E>1$ , the Kruskal-Szekeres line element: Here we take the $E\\rightarrow \\infty $ limit of Eqs.", "(REF ) and (REF ).", "We will do it term by term in each equation.", "For Eq.", "(REF ) we have: $\\lim _{E\\rightarrow \\infty } -4 \\left( \\frac{ 2E^2-1+ \\frac{2M}{r}+ 2E\\sqrt{E^2-1+\\frac{2M}{r}}}{ E^2-1+\\frac{2M}{r}} \\right)=-16$ ; $\\lim _{E\\rightarrow \\infty } \\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+\\frac{2M}{r}} \\right)}=\\exp {\\left( -\\frac{r}{2M}\\right)}$ ; $\\lim _{E\\rightarrow \\infty } \\left(\\frac{M}{r}\\, \\frac{1}{E^2-1+ \\frac{M}{r}+\\sqrt{E^2-1}\\sqrt{E^2-1+\\frac{2M}{r}}} \\right) ^{\\frac{2E^2-1}{2E\\sqrt{E^2-1}}} =\\frac{M}{r}\\frac{1}{2E^2}$ ; $\\lim _{E\\rightarrow \\infty }-\\frac{1}{M^2} \\left(2E^2-1+ \\frac{2M}{r}+2E\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)==-\\frac{4E^2}{M^2}$ ; $\\lim _{E\\rightarrow \\infty } \\exp {\\left( -\\frac{r}{2ME} \\sqrt{E^2-1+\\frac{2M}{r}} \\right)}=\\exp {\\left( -\\frac{r}{2M}\\right)}$ ; $\\lim _{E\\rightarrow \\infty }\\left(\\frac{\\frac{M}{r}}{E^2-1+\\frac{M}{r}+\\sqrt{E^2-1} \\sqrt{E^2-1+\\frac{2M}{r}}}\\right)^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}==\\frac{M}{r}\\frac{1}{2E^2}$ ; $\\lim _{E\\rightarrow \\infty }2 \\left( 2E^2-1+ \\frac{2M}{r} \\right)=4E^2$ .", "Thus, Eq (REF ) is now $\\mathrm {d}s^2=-16\\exp {\\left( -\\frac{r}{2M}\\right)}\\frac{M}{r}\\frac{1}{2E^2}\\Big [-\\frac{4E^2}{M^2}\\exp {\\left( -\\frac{r}{2M}\\right)}\\frac{M}{r}\\times \\\\\\times \\frac{1}{2E^2}(v^{\\prime 2}\\mathrm {d}u^{\\prime 2}+u^{\\prime 2}\\mathrm {d}v^{\\prime 2})+4E^2\\mathrm {d}u^{\\prime }\\mathrm {d}v^{\\prime }\\,\\Big ]+ r^2 (\\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)$ , where for convenience of notation we have redefined the coordinates, $u^{\\prime }\\equiv {\\mathcal {t}}^{\\prime }$ and $v^{\\prime }\\equiv \\tau ^{\\prime }$ .", "Implementing definitely the $E\\rightarrow \\infty $ limit, the term in $(v^{\\prime 2}\\mathrm {d}u^{\\prime 2}+u^{\\prime 2}\\mathrm {d}v^{\\prime 2})$ vanishes and one gets, $\\mathrm {d}s^2=-\\frac{32M}{r}\\exp {\\left( -\\frac{r}{2M}\\right)}\\mathrm {d}u^{\\prime }\\mathrm {d}v^{\\prime }+ r^2 (\\mathrm {d}\\theta ^2+\\sin ^2{\\theta }\\, \\mathrm {d}\\phi ^2)$ .", "This is Eq.", "(REF ), i.e., the Kruskal-Szekeres line element.", "For Eq.", "(REF ) we have: $\\lim _{E\\rightarrow \\infty }\\left( \\frac{\\frac{r}{2M}-1}{2E^2-1+ \\frac{2M}{r}+ 2E \\sqrt{E^2-1+\\frac{2M}{r}}} \\right) \\frac{2M}{r}=\\frac{\\frac{r}{2M}-1}{4E^2}\\frac{2M}{r}$ ; $\\lim _{E\\rightarrow \\infty }\\exp {\\left( \\frac{r}{2ME}\\sqrt{E^2-1+ \\frac{2M}{r}} \\right)}= =\\exp {\\left( \\frac{r}{2M} \\right)}$ ; $\\lim _{E\\rightarrow \\infty }\\left[\\frac{r}{M}\\left( E^2-1 + \\frac{M}{r}+ \\sqrt{E^2-1}\\sqrt{E^2-1+ \\frac{2M}{r}}\\right)\\right]^{\\frac{2E^2-1}{2E \\sqrt{E^2-1}}}=\\dfrac{r}{M}\\,2E^2$ .", "Thus, redefining for convenience of notation the coordinates ${\\mathcal {t}}^{\\prime }$ and $\\tau ^{\\prime }$ as $u^{\\prime }\\equiv {\\mathcal {t}}^{\\prime }$ and $v^{\\prime }\\equiv \\tau ^{\\prime }$ , Eq (REF ) in the $E\\rightarrow \\infty $ limit turns into $\\left(\\frac{r}{2M}-1\\right)\\exp {\\left( \\frac{r}{2M} \\right)}=-\\frac{u^{\\prime }}{M}\\frac{v^{\\prime }}{M}$ .", "This is Eq.", "(REF ), i.e., the Kruskal-Szekeres implicit definition of $r$ in terms of $u^{\\prime }$ and $v^{\\prime }$ .", "Seen through this direct limiting procedure, the Kruskal-Szekeres solution is indeed a particular case of the $E$ family of maximal extensions.", "In no place there was explicit need to resort to Eddington-Finkelstein null coordinates and their analytical extended versions." ] ]

[ [ "A Comparative Study of Machine Learning Models for Tabular Data Through\n Challenge of Monitoring Parkinson's Disease Progression Using Voice\n Recordings" ], [ "Abstract People with Parkinson's disease must be regularly monitored by their physician to observe how the disease is progressing and potentially adjust treatment plans to mitigate the symptoms.", "Monitoring the progression of the disease through a voice recording captured by the patient at their own home can make the process faster and less stressful.", "Using a dataset of voice recordings of 42 people with early-stage Parkinson's disease over a time span of 6 months, we applied multiple machine learning techniques to find a correlation between the voice recording and the patient's motor UPDRS score.", "We approached this problem using a multitude of both regression and classification techniques.", "Much of this paper is dedicated to mapping the voice data to motor UPDRS scores using regression techniques in order to obtain a more precise value for unknown instances.", "Through this comparative study of variant machine learning methods, we realized some old machine learning methods like trees outperform cutting edge deep learning models on numerous tabular datasets." ], [ "Introduction", "Parkinson's disease is a neurodegenerative disorder, affecting the neurons in the brain that produce dopamine.", "Parkinson's disease can cause a range of symptoms, particularly the progressive deterioration of motor function [1] [2] [3].", "When diagnosed with Parkinson’s disease, a person’s health may deteriorate rapidly, or they may experience comparatively milder symptoms if the disease progresses more slowly.", "In our research, we are mainly concerned with how Parkinson's disease can affect speech characteristics.", "People with Parkinson's may display dysarthria or problems with articulation, and they may also be affected by dysphonia, an impaired ability to produce vocal sounds normally.", "Dysphonia may be exhibited by soft speech, breathy voice, or vocal tremor [1] [2] [3].", "People with Parkinson's disease do not always experience noticeable symptoms at the earliest stages, and therefore, the disease is often diagnosed at a later stage.", "As there is not currently a cure, people diagnosed with Parkinson's must rely on treatments to alleviate symptoms, which is most effective with early treatment.", "Once a diagnosis is made, the patient must regularly visit their physician to monitor the disease and the effectiveness of treatment.", "Monitoring the progression of the disease through a voice recording captured by the patient at their own home can make the process faster and less stressful for the patient.", "The possibility of lessening the frequency of doctor visits can be cost-effective and allow the patient to follow a more flexible schedule [1] [2] [3].", "One of the most prominent methods of quantifying the symptoms of Parkinson's disease is the Unified Parkinson's Disease Rating Scale (UPDRS), which was first developed in the 1980s.", "The scale consists of four parts: intellectual function and behavior, ability to carry out daily activities, motor function examination, and motor complications.", "Each part is composed of questions or tests where either the patient or clinician will give a score with 0 denoting normal function and the maximum number denoting severe impairment.", "The clinician calculates a UPDRS score for each section as well as a total UPDRS score that can range from 0 to 176.", "In this work, we are concerned with the motor UPDRS, which can range from 0 to 108.", "All the machine learning techniques and data analysis in this project have been done using the Waikato Environment for Knowledge Analysis (Weka), free software developed at the University of Waikato [4].", "Weka is a compilation of machine learning algorithms written in Java.", "All the applied methods in this study are based on 10-fold cross-validation.", "We obtained the data from UC Irvine’s machine learning repository.", "The data consists of a total of 5,875 voice recordings from 42 patients with early-stage Parkinson’s disease over the course of 6 months.", "Voice recordings were taken for each subject weekly, and a clinician determined the subject’s UPDRS scores at the onset of the trial, at three months, and at six months.", "The scores were then linearly interpolated for the remaining voice recordings [2].", "The raw data from the original paper had 132 attributes, but the publicly available data contains 22 features, including the test time, sex, and age.", "Table 1 categorizes these features.", "The remaining features are the data related to the voice recordings.", "While abnormalities in any of these features could be symptomatic of many causes, they also provide measurements for several symptoms of Parkinson’s disease.", "The data contains four features that measure jitter and six that measure shimmer.", "Jitter measures the fluctuations in pitch while shimmer indicates fluctuations in amplitude.", "Common symptoms of Parkinson's disease include difficulty maintaining pitch as well as speaking softly.", "Therefore, measurements of both jitter and shimmer can be used to detect and measure these symptoms and can be useful to map the voice data to a UPDRS score.", "Table: Dataset AttributesThe recurrence period density entropy (RPDE) measures deviations in the periods of time-delay embedding of the phase space.", "When a signal recurs to the same point in the phase space at a certain time, it has a recurrence period of that time.", "Deviations in periodicity can indicate voice disorders, which may occur as a result of Parkinson’s disease [1].", "Noise-to-harmonics and harmonics-to-noise ratios are derived from estimates of signal-to-noise ratio from the voice recording.", "Detrended fluctuation analysis (DFA) measures the stochastic self-similarity of the noise in the speech sample.", "Most of this noise is from turbulent airflow through the vocal cords [1].", "Each of these measures can capture the breathiness in speech that can be a symptom of Parkinson’s disease.", "A common symptom of Parkinson’s disease is an impaired ability to maintain pitch during a sustained phonation.", "While jitter detects these changes in pitch, it also measures the natural variations in pitch that all healthy people exhibit.", "It can be difficult for jitter measurements to distinguish between these two types of pitch variations.", "Pitch Period Entropy (PPE) is based on a logarithmic scale rather than a frequency scale, and it disregards smooth variations [1].", "Therefore, it is better suited to detect dysphonia-related changes of pitch." ], [ "Document Organization", "The next section gives an overview regarding the previous research that has been done on this dataset and a similar dataset.", "Then in the section titled Machine Learning Strategies and Our Research Road-map, you can find the details about feature analysis and selection, a brief description of the methods that we applied, followed by the different approaches and their results.", "The last section, discussion and conclusion, is about future works and summarization of our study on this dataset.", "Approximately 60,000 Americans are diagnosed with Parkinson’s disease each year, but only 4% of patients are diagnosed before the age of 50 [1] [2] [3].", "In order to diagnose Parkinson’s disease earlier, there have been several works, based on a dataset of voice recordings of patients with Parkinson’s disease and healthy patients.", "First, work by Max Little on this initial dataset predicts whether a subject is healthy or has Parkinson's disease using phonetic analysis of voice recordings to measure dysphonia [1].", "Much similar work has been done to create models that can accurately predict whether a person has Parkinson's disease or is healthy based on the phonetic analysis of voice recordings.", "Max Little et al.", "introduced the dataset and found success with SVM (Support Vector Machine), which indicated that voice measurements can be a suitable way to diagnose Parkinson's disease [1].", "Further research has contributed methods to solve this problem with various machine learning techniques [5],which successfully used Artificial Neural Networks as well as a neuro-fuzzy classifier.", "The neuro-fuzzy classifier achieved a high accuracy on the testing set for this binary classification problem.", "The work to classify people as healthy or having Parkinson's disease has favorable results, but due to unbalanced data available, we cannot know whether a model is reliable.", "The subsequent work was on a more complex dataset, to monitor patients with early-stage Parkinson's disease [2], which is the basis for this paper.", "Age, sex, and voice are all important factors to identify Parkinson's, or in other words, are used by clinicians to calculate a UPDRS score.", "Therefore, both the above ideas seem promising for identification and monitoring of people with Parkinson's disease.", "As for this paper, we are more concerned with the more recent data and work [2], which attempts to map data from voice recordings to UPDRS scores.", "The original authors have tackled this problem as a regression problem, by using logistic regression and CART (Classification and Regression Tree).", "In their work, they have mentioned that by using the CART method, they could reduce the mean absolute error to 4.5 on the training set and 5.8 on the testing set.", "However, in this study, even with 10-fold cross-validation, we achieved a mean absolute error even lower than 1.9." ], [ "Machine Learning Strategies and Our Research Road-Map", "In this section, we examine at first the correlation of the features to the class and discuss feature selection.", "Next, we describe the machine learning techniques that we applied to this dataset in different manners.", "We applied regression methods to the data with a continuous class in an attempt to map the phonetic features to the severity of the symptoms from Parkinson's disease.", "We also undertook this as a classification problem and tried various classification techniques." ], [ "Feature Selection", "The 22 features accessible in the dataset had already been selected from 132 attributes of the raw data, which is not available to the public [2].", "We selected 18 features for this study after excluding the subject, test time, and total UPDRS features from the 22 available ones.", "We selected the motor UPDRS as the target attribute.", "Also, motor UPDRS has a high correlation with total UPDRS, so we used motor UPDRS as our sole class.", "Table 2 shows the correlation between each of those 18 features with motor UPDRS, ordered by correlation.", "Table: Features correlation with motor UPDRS" ], [ "Regression-Based Methods", "We created regression models to determine a relationship between the features and the continuous class (motor UPDRS).", "We tried many different techniques in different categories of machine learning methods, including trees, functions, multilayer perceptron, and instance-based learning.", "Table 3 shows the correlation coefficient and mean absolute error for the top performing regression models.", "We omitted results that did not compare well to the top models.", "Overall, the tree-based regression models performed the best.", "The previous work on this data measured their results by using the mean absolute error.", "For this reason, we also observed the mean absolute error for each method, so we could meaningfully compare our results to the previous works.", "For more insight into the results, we also noted the correlation coefficient.", "Table: Regression-based methods resultsM5 model tree [6] combines Decision and Regression tree.", "M5 model (M5P) first constructs a decision tree, and each leaf of that tree is a regression model.", "Therefore, the output we get out of the M5 model are real values instead of classes.", "As our dataset’s dependent variable has continuous values, we chose this method.", "It is different from a Classification and Regression Tree (CART) method as CART generates either a decision or regression tree based on the type of dependent variable and M5 model tree uses both techniques together.", "This method is one of the best performers.", "Support Vector Machines (SVMs) [7] are another popular machine learning algorithm that can handle both classification and regression with the detection of outliers.", "SVM tries to find an optimal hyperplane that categorizes the new inputs.", "We attempted two methods: epsilon-SVR and nu-SVR.", "There are multiple kernels to find the optimal hyperplane, including linear, radial, sigmoidal, and polynomial.", "Since the data was complex and has a high dimensionality, radial kernel tends to work faster and better than any other kernel type.", "We found that nu-SVR performed better than epsilon-SVR for this data.", "The nu-SVR method got 0.9335 as the highest correlation coefficient, only marginally higher than epsilon-SVR, which achieved 0.9301 for the correlation coefficient.", "However, the mean absolute error of 2.06 was also slightly higher compared to that of epsilon-SVR, 2.02.", "Reduced Error Pruning Tree [8], or REPTree, is Weka's implementation of a fast decision tree learner.", "The REPTree is sometimes preferred over other trees because it prevents the tree from growing linearly with the sample size when growth will not improve the accuracy [9].", "It uses information gain to build a regression tree and prunes it with reduced-error pruning.", "The REPTree for classification yielded a correlation coefficient of 0.92 and a mean absolute error of 2.02." ], [ "Instance-Based Learning", "Instance-based learning, such as k-nearest neighbors (k-NN) [10], uses a function that is locally approximated and defers computation to the classification or value assignment.", "Instead of generalizing the data, new instances are assigned values based directly on the training data.", "As a regression method, k-NN outputs the average of the values of the instance’s nearest neighbors.", "We applied the k-NN algorithm to the discretized data and classified instances based on seven nearest neighbors.", "We used Manhattan distance and weighted the distance with one divided by the distance.", "This method yielded a correlation coefficient of 0.86 and 2.83 as the mean absolute error." ], [ "Ensemble Methods", "Using Ensemble Methods, we combined several regression techniques with other methods in an attempt to improve upon the best results we achieved.", "Some of these methods include bagging, boosting, stacking, voting, and Iterative Absolute Error Regression in conjunction with other regression methods.", "Bagging [11], or bootstrap aggregation, uses multiple predictions of a method with high variance.", "Many subsamples of the data are made with replacement, and a prediction is made with a machine learning method for each subsample.", "The result from bagging is the average of the result of all of these predictions.", "Stacking [12] makes use of several machine learning models.", "We applied multiple methods to the original dataset.", "There is a metalayer which uses another model that uses the individual results as its input and creates a prediction.", "Our best stacking result stacked M5P tree and REPTree, and we used M5PTree as the model for the metalayer.", "Voting [13], or simple averaging for this regression problem used multiple machine learning methods.", "The average of their results was the output for the voting algorithm.", "Our best model with this ensemble method was once again the M5P Tree and the REPTree.", "Random Forest [14] is a tree-based ensemble learning method that can be used for both classification and regression.", "It operates by constructing a multitude of decision trees at training time and outputting class that is the mode of classes (classification) or mean prediction (regression) of the individual trees.", "These trees are generally fast and accurate but sometimes suffer from over-fitting.", "Tree-based methods did provide the best results and tend to work well in ensemble methods due to their high variance.", "The best performing model was bagging with the M5P tree as the base method.", "This yielded a correlation coefficient of 0.95 and a mean absolute error of 1.87.", "Table 4 features some of the best ensemble methods for regression.", "Table: Regression-based ensemble methods results" ], [ "Verification", "In order to confirm that our results are consistent with real values for the motor UPDRS scores and not just fitting to the linear interpolation, we also tried the regression techniques on a subset of the data.", "In this subset, we used only instances in which the value for the motor UPDRS was a whole number.", "While some interpolated instances may coincidentally have a whole number for the motor UPDRS, this method ensured that much of the subset was the data where a clinician examined the patient and calculated a UPDRS score.", "The results we achieved with this subset of the data surpassed those of the whole dataset.", "In particular, the M5P model has a correlation coefficient of 0.95 and a mean absolute error of 1.87.", "These results indicate that our findings from regression methods are reliable and are not simply fitting to the linearly interpolated data." ], [ "Classification by Discretization", "In addition to the regression models, we also attempted other methods of modeling the data.", "Below we briefly summarize these different approaches to this problem since the results were not promising.", "To classify the instances into meaningful intervals, we discretized the data.", "According to the work of Pablo Martinez-Martin et al.", "[15], UPDRS scores can be used to indicate the severity of the disease.", "These authors classified motor UPDRS scores from 1 to 32 as mild, 33 to 58 as moderate, and 59 and above as severe.", "Using these intervals, the data we used had 5,254 instances with mild Parkinson’s disease and only 621 instances that were moderate, and no severe cases.", "We used multiple tree-based methods, including C4.5, Classification and Regression Tree, and Logit Boost Alternating Decision Tree.", "Weka's J48 algorithm [16] is essentially an implementation of the C4.5 algorithm (a type of decision tree methods).", "C4.5 is an improvement over the ID3 algorithm.", "J48 is capable of handling discrete as well as continuous values.", "It also has an added advantage of allowing a pruned tree.", "Our data has attributes with continuous values, so we chose the J48 classifier as one of the methods and tried multiple configurations.", "SimpleCART [17] is a type of Classification and Regression Tree.", "The classification tree predicts the class of the dependent variable, while the regression tree outputs a real number.", "The SimpleCART technique produces either a classification or regression tree based on whether the dependent variable is categorical or numeric, respectively.", "The class attribute of our dataset is of numeric type, but after discretizing the class label into bins, we converted it to a categorical type, so the SimpleCART algorithm treated our discretized dataset as a classification problem.", "LADTree (Logit boost Alternating Decision tree) [18] is a type of alternating decision tree for multi-class classification.", "Alternating Decision Tree was designed for binary classification.", "ADTrees can be merged into a single tree; therefore, a multiclass model can be derived by merging several binary class trees using some voting model.", "LADTree uses LogitBoost strategy for boosting.", "In simple terms, boosting gives relatively more weight to misclassified instances compared to correctly classified instances for the next iteration of boosting.", "Generally, the boosting iteration is directly proportional to the number of iterations.", "Bayes Net [19] is a probabilistic directed acyclic graphical model.", "This model represents a set of variables and their conditional dependencies through a direct acyclic graph.", "Each node’s output depends on the particular set of values of its parent nodes.", "The nodes which are not connected to each other are considered as conditionally independent nodes from each other.", "Unlike the Naïve Bayes [20] assumption of conditional independence, Bayesian Belief networks describe conditional independence among a subset of variables.", "K-nearest neighbors (K-NN) [10], can also be used for both classification and regression problems.", "When used for classification, k-NN classifies a new instance with the class held by the majority of its nearest neighbors.", "We applied the K-NN algorithm [10] to the discretized data and classified the motor UPDRS values for instances based on six nearest neighbors and measuring using Manhattan distance.", "The Multilayer Perceptron (MLP) is a type of artificial neural network [21] [22].", "An MLP consists of one or more layers with a different number of nodes in each, called the network architecture.", "Using some activation function such as sigmoid in each node combined with the backpropagation technique makes such networks useful for machine learning classification and regression tasks.", "For this dataset we applied variant network architectures, single-layer to five-layer networks with a range of 1 to 10 nodes in each.", "The state of the art, known as deep learning, is the developed MLP into more layers containing more nodes with more options of activation functions and training algorithms [23].", "We tried several different architectures of deep neural network (DNNs) using Keras [24] and Tensorflow [25].", "The results were not promising in comparison to listed models.", "After all, the classification results did not compare well to the regression-based methods.", "We speculate that this is because of the unbalanced data." ], [ "Multi-Instance Learning", "We also used multi-instance learning [26].", "For each subject in this dataset, there are approximately six voice recordings for each time step.", "Rather than considering each of these recordings by itself, multi-instance learning collects these instances into bags.", "Each bag holds one person’s voice recording data that was taken at the same time step, and every instance in the bag is assigned to the same class.", "We have 995 bags for the 42 subjects, amounting to about 24 per subject, or one per week.", "We propositionalized the bags, creating one instance for each bag with the mean of the values of the aggregated instances.", "This creates 995 instances, one for each person at every time interval.", "We then were able to apply single-instance classifiers, including Bayesian methods, decision trees, SVMs, and Multi-Layer Perceptron.", "These methods yielded similar results to those that we achieved with classification using the data with all 5,875 instances.", "The results of this approach were similar to those of the classification models and not significant compared to the regression results." ], [ "Discussion and Conclusion", "Our results from the classification problem indicate that these measures of dysphonia may be used to determine the severity of the symptoms of Parkinson's disease.", "Results with higher accuracy as well as a better ability to predict the minority class suggest a higher likelihood that model can be accurately used with more diverse data.", "However, we found the regression results to be even more promising.", "We favored regression over classification with this data because, in order to classify, we must discretize into bins.", "Meanwhile, regression techniques can map the UPDRS score to a more precise value.", "It is more meaningful for a model to output a motor UPDRS score of 15 than to say it is in the range of 0 to 16.", "Our best performing regression method was bagging using the M5P model tree as the base method, which achieved a correlation coefficient of 0.95 and a mean absolute error of 1.86.", "To our knowledge, the only existing work on this data belongs to the same authors who created this dataset [2].", "We tried many regression methods and compared them to the findings of these authors.", "We only compiled our significant findings in this report.", "Our regression methods resulted in high correlation coefficients.", "However, the previous work made no mention of the correlation coefficient, so we used the mean absolute error to compare our results.", "The best results from this earlier work had a mean absolute error of 4.5 on the training set and 5.8 on the testing set.", "We were able to lower the mean absolute error to 1.9, a significant decrease from the earlier work's mean absolute errors.", "We attained the lower mean absolute error despite using 10-fold cross-validation, which makes our results more reliable.", "These models indicate that motor UPDRS can be calculated using these voice measurements in the early stages of the disease with even more precision than previously thought.", "These results suggest that voice recordings may be a reliable approach to monitoring Parkinson's disease from the comfort of the patient's home, potentially reducing the frequency of doctor visits and giving patients more freedom with their time.", "There are many possibilities of future work on this type of data.", "Researchers are currently collecting more data of this type.", "If more data is collected from patients at all stages of Parkinson’s, similar techniques could be applied to determine whether vocal parameters can still be mapped to UPDRS scores at a later stage of the disease.", "Additionally, data from the later stages of Parkinson’s could be used to attempt to predict the progression of the disease.", "As a result of our comparative study of machine learning methods, we discovered that the new methods of deep learning are not as efficient and competitive as trees for many tabular data.", "A data scientist needs to know about all machine learning methods and different types of datasets to achieve the best accuracy and efficiency.", "The results of our comparative study of variant machine learning models supports the same claim of [27][28][29] that for many tabular data, the older models (e.g., decision trees) outperform cutting edge deep learning models.", "Also, we should consider that some machine learning models such as decision trees are much faster than DNNs and could be run in very simple machines like on edge devices (e.g., cellphones).", "Acknowledgement Here we want to appreciate the help of Pawan Yadav and Ankit Joshi for part of the early implementation of the experiments." ] ]

[ [ "Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy\n Spaces" ], [ "Abstract We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz-Zygmund inequalities in subspaces of polynomials.", "We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior." ], [ "Introduction", "Marcinkiewicz-Zygmund inequalities are finite-dimensional models for sampling in an infinite dimensional Hilbert or Banach space of functions.", "Originally they were studied in the context of interpolation by trigonometric polynomials.", "They became prominent in approximation theory, where they appear in quadrature rules and least square problems, and were usually studied in the context of orthogonal polynomials.", "In an abstract setting one is given a reproducing kernel Hilbert space $\\mathcal { H}$ on a set $S$ with reproducing kernel $k$ and a sequence of finite-dimensional subspaces $V_n$ such that $V_n \\subseteq V_{n+1}$ and $\\bigcup _n V_n$ is dense in $\\mathcal { H}$ .", "Each $V_n$ comes with its own reproducing kernel $k_n$ , which is the orthogonal projection of $k$ .", "A family of (finite) subsets $\\Lambda _n \\subseteq S$ is called a Marcinkiewicz-Zygmund family for $V_n$ in $\\mathcal { H}$ , if there exist constants $A,B>0$ , the sampling constants, such that for all $n$ large, $n\\ge n_0$ , $\\qquad A \\Vert p\\Vert _{\\mathcal { H}}^2 \\le \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} \\le B \\Vert p\\Vert _{\\mathcal { H}}^2 \\qquad \\text{for all } p\\in V_n \\, .$ Thus a Marcinkiewicz-Zygmund family comes with a sequence of Marcinkiewicz-Zygmund inequalities, which are sampling inequalities for the finite-dimensional subspaces $V_n$ .", "The point of the definition is that the sampling constants are independent of the subspace $V_n$ .", "The diagonal of the reproducing kernel $k_n$ furnishes the most natural choice of weights and goes back to the corresponding notion of interpolating sequences in reproducing kernel Hilbert spaces studied by Shapiro and Shields [26].", "The weights are intrinsic to the underlying spaces $V_n$ .", "Another hint for using $k_n$ comes from frame theory: Since $p(z) = \\langle p, k_n (\\cdot ,z)\\rangle $ for $p\\in V_n $ , the sampling inequality amounts to verifying that the normalized reproducing kernels $k_n (\\cdot ,\\lambda )/k_n(\\lambda ,\\lambda )^{1/2}$ form a frame in $V_n $ all whose elements have unit norm in $\\mathcal { H}$ .", "Marcinkiewicz-Zygmund inequalities have been studied in many different contexts, e.g., for trigonometric polynomials [6], [22], for spaces of algebraic polynomials with respect to some measure [14], [15], for spaces of spherical harmonics on the sphere [16], [17], [19], for spaces of eigenfunctions of the Laplacian on a compact Riemannian manifold [21], or even more generally for diffusion polynomials on a metric measure space [7].", "In sampling theory Marcinkiewicz-Zygmund inequalities can be used to derive sampling theorems for bandlimited functions [8], [9].", "In this paper we initiate the investigation of Marcinkiewicz-Zygmund inequalities for polynomials in spaces of analytic functions.", "As the reproducing kernel Hilbert space we take either the Bergman space $A^2(\\mathbb {D})$ or the Hardy space $H^2(\\mathbb {D})$ of analytic functions on the unit disk $\\mathbb {D}$ .", "The natural finite-dimensional subspaces will the family of the polynomials $\\mathcal { P}_n$ of degree $n$ .", "In this context it is clear that an arbitrary set of at least $n+1$ distinct points yields a sampling inequality $ A\\Vert p\\Vert ^2 \\le \\sum _{\\lambda \\in \\Lambda } |p(\\lambda )|^2 \\,k_n(\\lambda ,\\lambda ) ^{-1}\\le B \\Vert p\\Vert _{\\mathcal { H}} ^2$ for all $ p\\in \\mathcal { P}_n$ .", "The objective of Marcinkiewicz-Zygmund inequalities is to construct a sequence of finite sets $\\Lambda _n$ , such that the constants $A,B$ are independent of the degree $n$ .", "The game is therefore to diligently keep track of the constants and show that they do not depend on the degree.", "We will see that this problem is deeply related to sampling theorems for the full space $A^2$ or $H^2$ .", "We say that $\\Lambda \\subseteq S $ is a sampling set for $\\mathcal { H}$ , if there exist constants $A,B>0$ , such that $ A \\Vert f\\Vert _{\\mathcal { H}}^2 \\le \\sum _{\\lambda \\in \\Lambda } \\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\le B \\Vert f\\Vert _{\\mathcal { H}}^2 \\qquad \\text{for all } f\\in \\mathcal { H}\\, ,$ where now $k(z,w)$ is the reproducing kernel of $\\mathcal { H}$ .", "In our case $\\mathcal { H}=A^2(\\mathbb {D})$ or $=H^2(\\mathbb {D})$ .", "Both the Bergman space and the Hardy space are reproducing kernel Hilbert spaces, in which the polynomials are dense and $k_n(\\lambda , \\lambda )\\rightarrow k(\\lambda , \\lambda )$ pointwise.", "However, the underlying measures are different, and as a consequence the reproducing kernels and the implicit metrics are different.", "We will see that these differences imply a drastically different behavior of Marcinkiewicz-Zygmund families.", "In the Bergman space the points of a Marcinkiewicz-Zygmund family will be “uniformly” distributed in the entire disk, in Hardy space the points will cluster near the boundary of the disk.", "The concentration will depend on the degree.", "It will therefore be practical to introduce a notation of the relevant disks and annuli.", "For a fixed parameter $\\gamma >0$ , we will write $B_{1- \\gamma /n}= \\lbrace z\\in \\mathbb {D}: |z| < 1-\\tfrac{\\gamma }{n}\\rbrace = B(0,1-\\tfrac{\\gamma }{n})$ for the centered disk of radius $1-\\tfrac{\\gamma }{n}$ , and $C_{\\gamma /n}= \\lbrace z\\in \\mathbb {D}: 1-\\tfrac{\\gamma }{n}\\le |z| <1 \\rbrace \\, .$ for the annulus of width $\\gamma /n $ at the boundary of $\\mathbb {D}$ .", "Our main result for the Bergman space $A^2(\\mathbb {D})$ with norm $\\Vert f\\Vert _{A^2}^2 = \\tfrac{1}{\\pi }\\int _{\\mathbb {D}} |f(z)|^2 dz $ , where $dz$ is the area measure on $\\mathbb {D}$ , establishes a clear correspondence between sampling sets for $A^2(\\mathbb {D})$ and Marcinkiewicz-Zygmund families for the polynomials $\\mathcal { P}_n$ in $A^2$ as follows.", "Theorem 1.1 (i) Assume that $\\Lambda \\subseteq \\mathbb {D}$ is a sampling set for $A^2(\\mathbb {D})$ .", "Then for $\\gamma >0$ small enough, the sets $\\Lambda _n =\\Lambda \\cap B_{1- \\gamma /n}$ form a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ .", "(ii) Conversely, if $(\\Lambda _n)$ is a Marcinkiewicz-Zygmund family for the polynomials $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ , then every weak limit of $(\\Lambda _n)$ is a sampling set for $A^2(\\mathbb {D})$ .", "See Section REF for the definition of a weak limit of sets.", "The theorem shows that the construction of Marcinkiewicz-Zygmund families for the Bergman space is on the same level of difficulty as the construction of sampling sets for $A^2$ .", "Fortunately, these sampling sets have been characterized completely in the deep work of K. Seip [24].", "Sampling sets are completely determined by a suitable density, the Seip-Korenblum density.", "As a consequence of our main theorem, one can now give many examples of Marcinkiewicz-Zygmund families for $A^2$ .", "By contrast, the Hardy space does not admit any sampling sequences.", "By a theorem of P. Thomas [27](Props.", "2 and 3), a function $f\\in H^2(\\mathbb {D})$ satisfying $A \\Vert f\\Vert _{H^2}^2 \\le \\sum _{\\lambda \\in \\Lambda }|f(\\lambda )|^2 k(\\lambda ,\\lambda )^{-1}\\le B \\Vert f\\Vert _{H^2}^2$ must be identical zero.", "Therefore there can be no analogue of Theorem REF (ii).", "Despite the lack of a sampling theorem for $H^2(\\mathbb {D})$ we can show the existence of Marcinkiewicz-Zygmund families for polynomials with a different method.", "The idea is to connect polynomials on the disc to polynomials on the torus in $L^2(\\mathbb {T})$ .", "By moving a Marcinkiewicz-Zygmund family for polynomials on $\\mathbb {T}$ into the interior of $\\mathbb {D}$ , we obtain a Marcinkiewicz-Zygmund family for polynomials in Hardy space.", "Since the problem on the torus is well understood [22], we can derive a general construction of Marcinkiewicz-Zygmund families in $H^2(\\mathbb {D})$ .", "Theorem 1.2 Assume that the family $(\\widetilde{\\Lambda _n}) = \\Big (\\lbrace e^{i \\nu _{n,k}}: k= 1, \\dots , L_n\\rbrace \\Big ) \\subseteq \\mathbb {T}$ is a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ on the torus, i.e., $A \\Vert p\\Vert _{L^2(\\mathbb {T})}^2 \\le \\sum _{k=1} ^{L_n}\\frac{|p(e^{i \\nu _{n,k} })|^2}{n} \\le B \\Vert p\\Vert _{L^2(\\mathbb {T})}^2$ for all polynomials $p$ of degree $n$ .", "Fix $\\gamma >0$ arbitrary, choose $\\rho _{n,k} \\in [1-\\tfrac{\\gamma }{n}, 1)$ arbitrary, and set $\\Lambda _n = \\lbrace \\rho _{n,k} e^{i \\nu _{n,k}}:k=1, \\dots , L_n\\rbrace \\subseteq C_{\\gamma /n}$ for $n\\in \\mathbb {N}$ .", "Then $(\\Lambda _n)$ is a Marcinkiewicz-Zygmund family for $\\mathcal { P}_{n}$ in $H^2(\\mathbb {T})$ .", "This result provides a systematic construction of examples of Marcinkiewicz-Zygmund families for Hardy space, since Marcinkiewicz-Zygmund families for polynomials on the torus can be characterized almost completely by their density [22].", "Marcinkiewicz-Zygmund families on the torus and more generally of orthogonal polynomials have been studied intensely in approximation theory, see [6], [8], [14], [15], [18], [20] for a sample of papers.", "The technical heart of the matter is, as so often in complex analysis, the investigation and estimate of the reproducing kernels, namely the kernel $k(z,w)$ for the entire space $A^2$ or $H^2$ and the kernels $k_n(z,w)$ for the polynomials of degree $n$ .", "The guiding principle is to sample the polynomials in the region where the diagonals $k(z,z)$ and $k_n(z,z)$ are comparable in size.", "One may call this region the “bulk” region.", "For the Bergman space the bulk region is the centered disk $B_{1- \\gamma /n}$ , because this is where the mass of polynomials of degree $n$ is concentrated.", "For the Hardy space the bulk region is the annulus $C_{\\gamma /n}$ , as the $H^2$ -norm sees only the boundary behavior of functions in $H^2$ .", "Our main insight may be relevant in other settings.", "For instance, our main theorems can be extended to polynomials in weighted Bergman spaces or in Fock space [11].", "We expect a version of Theorem REF to hold for Bergman space on the unit ball in $\\mathbb {C}^n$ , though this will be more technical to elaborate.", "The paper is organized as follows: In Section 2 we treat the theory of Marcinkiewicz-Zygmund families in the Bergman space $A^2(\\mathbb {D})$ and in Section 3 we treat the Hardy space $H^2(\\mathbb {D})$ .", "Each section starts with the necessary background, the comparisons of the various reproducing kernels and the main contribution to the norms.", "Then we formulate and prove the main results about Marcinkiewicz-Zygmund families.", "Throughout we will use the notation $\\lesssim $ to abbreviate an inequality $f \\le Cg$ where the constants is independent of the essential input, which in our case will be the degree of the polynomial.", "To indicate the dependence of the constant on some parameter $\\gamma $ , say, we will write $\\lesssim \\, _\\gamma $ .", "As usual, $f\\asymp g$ means that both $f\\lesssim g$ and $g\\lesssim f$ hold." ], [ "Basic facts.", "The Bergmann space $A^2= A^2(\\mathbb {D})$ consists of all analytic functions on the unit disk $\\mathbb {D}$ with finite norm $\\Vert f\\Vert _{A^2} = \\Big ( \\frac{1}{\\pi } \\int _{\\mathbb {D}} |f(z)|^2 \\, dz\\Big )^{1/2} \\, ,$ where $dz$ is the area measure on $\\mathbb {D}$ .", "For a detailed exposition of Bergman spaces we refer to the excellent monographs [4], [13], both of which contain an entire chapter on sampling in Bergman space.", "The monomials $z\\mapsto z^k$ are orthogonal with norm $\\Vert z^k\\Vert _{A^2}^2 =\\frac{1}{k+1}$ .", "Consequently the norm of $f(z) =\\sum _{k=0}^\\infty a_k z^k$ is $\\Vert f\\Vert _{A^2} ^2 = \\sum _{k=0}^\\infty |a_k|^2 \\frac{1}{k+1} \\, .$ Let $p(z) = \\sum _{k=0}^n a_k z^k \\in \\mathcal { P}_n$ , then its norm on a disk $B_\\rho $ , $\\rho <1$ , is given by $\\frac{1}{\\pi } \\int _{B_\\rho } |p(z)|^2 \\, dz &= \\frac{1}{\\pi }\\sum _{k,l=0}^n a_k\\overline{a_l} \\int _{B_\\rho } z^k \\bar{z}^l \\, dz\\\\&= \\frac{1}{\\pi } 2\\pi \\sum _{k=0}^n |a_k|^2 \\int _0^\\rho r^{2k} rdr \\\\&= \\sum _{k=0}^n |a_k|^2 \\rho ^{2k+2} \\frac{1}{k+1}$ For $p\\in \\mathcal { P}_n$ , we therefore have $\\frac{1}{\\pi } \\int _{B_\\rho } |p(z)|^2 \\, dz \\ge \\rho ^{2n+2}\\Vert p\\Vert _{A^2}^2 \\, .$ To obtain a bound independent of $n$ , we need to choose $\\rho _n$ such that $\\rho _n^{2n} \\ge A $ for all $n$ .", "By picking $\\rho _n =1-\\tfrac{\\gamma }{n}$ , we find $e^{-2\\gamma }\\le (1-\\tfrac{\\gamma }{n})^{n} \\le e^{-\\gamma } \\qquad \\forall n >2 \\gamma \\, .$ In the following we will use these inequalities abundantly.", "Corollary 2.1 If $p\\in \\mathcal { P}_n$ , then for every $\\gamma >0$ and $n>2 \\gamma $ $\\frac{1}{\\pi } \\int _{C_{\\gamma /n}} |p(w)|^2 \\, dw \\le \\Vert p\\Vert _{A^2}^2 \\big (1- (1-\\tfrac{\\gamma }{n})^{2n+2} \\big ) \\le \\Vert p\\Vert _{A^2}^2 (1- \\tfrac{1}{4}e^{-4\\gamma }) \\, .$ In particular, for $\\epsilon >0$ there exists $\\gamma >0$ (small enough) such that $\\frac{1}{\\pi } \\int _{C_{\\gamma /n}} |p(w)|^2 \\, dw \\le \\epsilon \\Vert p\\Vert _{A^2}^2 \\qquad \\text{ for all } p\\in \\mathcal { P}_n\\,$ independent of $n$ .", "With (REF ) and the partition $\\mathbb {D}= B_{1- \\gamma /n}\\cup C_{\\gamma /n}$ we obtain $\\frac{1}{\\pi } \\int _{C_{\\gamma /n}} |p(w)|^2 \\, dw &= \\Vert p\\Vert _{A^2}^2 -\\frac{1}{\\pi } \\int _{B_{1- \\gamma /n}} |p(w)|^2\\, dw \\\\& \\le \\Vert p\\Vert _{A^2}^2 \\big (1 - (1-\\tfrac{\\gamma }{n})^{2n+2} \\big ) \\le \\Vert p\\Vert _{A^2}^2 (1-\\frac{1}{4} e^{-4\\gamma }) \\, ,$ since $(1-\\tfrac{\\gamma }{n})^{2n+2} \\ge e^{-4\\gamma } (1-\\tfrac{\\gamma }{n})^2 \\ge e^{-4\\gamma }/4$ for $n\\ge 2\\gamma $ .", "Also as $\\gamma \\rightarrow 0$ , $\\inf _{n\\in \\mathbb {N}} (1-\\tfrac{\\gamma }{n})^{2n+2}\\rightarrow 1$ ." ], [ "The Bergman kernels", "Since $\\sqrt{k+1} \\,z^k$ is an orthonormal basis for $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ , the reproducing kernel of $\\mathcal { P}_n$ in $A^2$ is given by $k_n(z,w) = \\sum _{k=0}^n(k+1) (z\\bar{w})^k = \\frac{1+(n+1)(z\\bar{w})^{n+2} -(n+2)(z\\bar{w})^{n+1}}{(1-z\\bar{w})^2} \\, .", "$ As $n\\rightarrow \\infty $ , the kernel tends to the Bergman kernel of $A^2$ , $k(z,w) = \\frac{1}{(1-z\\bar{w})^2} \\qquad z,w \\in \\mathbb {D}\\, .$ We first compare these kernels in two regimes, namely the “bulk” regions $B_{1- \\gamma /n}$ and the boundary region $C_{\\gamma /n}$ .", "Lemma 2.2 Let $k_n(z,w)$ be the reproducing kernel of $\\mathcal { P}_n$ in $A^2$ and $\\gamma >0$ be arbitrary.", "(i) If $|z| \\le 1-\\tfrac{\\gamma }{n}$ , then $k_n(z,z) \\asymp k(z,z)$ for $n$ large enough, $n\\ge n_\\gamma $ , precisely $c_\\gamma k(z,z) \\le k_n(z,z) \\le k(z,z) = \\frac{1}{(1-|z|^2)^2}\\, ,$ where $c_\\gamma >0$ can be chosen as $c_\\gamma =1-e^{-2\\gamma }(1+2\\gamma )$ .", "(ii) If $1-\\tfrac{\\gamma }{n}\\le |z| <1$ and $n>\\max (\\gamma , 3) $ , then $k_n(z,z)\\asymp n^2$ , precisely, $\\frac{e^{-4\\gamma } }{4} n^2 \\le k_n(z,z) \\le n^2 \\, .$ Thus inside the disk $B_{1- \\gamma /n}$ the reproducing kernel for $\\mathcal { P}_n$ behaves like the reproducing kernel of the full space $A^2$ , whereas in the annulus it behaves like $n^2$ and is much smaller than $k(z,z)$ near the boundary of $\\mathbb {D}$ .", "(i) It is always true that $k_n(z,z) \\le k(z,z) =\\frac{1}{(1-|z|^2)^2}$ .", "For the lower bound, let $r=|z|$ and observe that $k_n(z,z) = k(z,z) \\big ( 1 - (n+2)r^{2n+2} + (n+1) r^{2n+4} \\big ) =k(z,z) ( 1 - q(r)) \\, .$ Since $q(r) = (n+2)r^{2n+2} - (n+1) r^{2n+4}$ is increasing on $(0,1)$ and $r\\le 1-\\tfrac{\\gamma }{n}$ , we need an upper estimate for $q$ at $1-\\tfrac{\\gamma }{n}$ .", "With the help of (REF ) and some algebra we write $q$ as $q(1-\\tfrac{\\gamma }{n}) & = (1-\\tfrac{\\gamma }{n})^{2n+2} \\big ( n+2 - (n+1)(1-\\tfrac{\\gamma }{n})^2 \\big ) \\\\& = (1-\\tfrac{\\gamma }{n}) ^{2n } \\big ( 1-\\tfrac{\\gamma }{n})^2 \\, \\big ( 1 + 2\\gamma + \\tfrac{\\gamma }{n}(2-\\gamma ) - \\tfrac{\\gamma ^2}{n^2} \\big ) \\\\&= (1-\\tfrac{\\gamma }{n})^{2n} \\Big (1+2\\gamma - \\frac{5\\gamma ^2}{n} +\\frac{\\alpha _\\gamma }{n^2} + \\frac{\\beta _\\gamma }{n^3} \\Big ) \\,$ for some constants $\\alpha _\\gamma , \\beta _\\gamma $ .", "Using (REF ) and a sufficiently large $n$ , $n\\ge n_\\gamma $ say, the final estimate for $q$ is $q(1-\\tfrac{\\gamma }{n}) \\le e^{-2\\gamma } (1+2\\gamma ) < 1 \\, , \\text{ for } n \\ge n_\\gamma \\, .$ Combined with (REF ) we have the lower estimate $k_n(z,z) \\ge (1- e^{-2\\gamma } (1+2\\gamma )) k(z,z) = c_\\gamma k(z,z)\\,$ for $|z| \\le 1-\\tfrac{\\gamma }{n}$ and $n\\ge n_\\gamma $ .", "(ii) If $1-\\tfrac{\\gamma }{n}\\le |z| <1$ , then $k_n(z,z) = \\sum _{k=0}^n |z|^{2k} (k+1) \\le \\sum _{k=0}^n (k+1) =\\frac{(n+1)(n+2)}{2} \\le n^2 \\, ,$ for $n\\ge 3$ , and, for $n\\ge 2\\gamma $ , $k_n(z,z) = \\sum _{k=0}^n |z|^{2k} (k+1) \\ge \\sum _{k\\ge n/2}^n(1-\\tfrac{\\gamma }{n})^{2n} \\, \\frac{n}{2}\\ge e^{-4\\gamma } \\frac{n^2}{4} \\, .$ We will also need some information about the behavior of the reproducing kernels $k_n$ near the diagonal.", "It will be convenient to use the normalized reproducing kernel $\\kappa _n(z,w)$ for $\\mathcal { P}_n$ at the point $w$ , i.e., $\\kappa _n(z,w) =k_n(z,w)/\\sqrt{k_n(w, w)}$ .", "It satisfies $\\Vert \\kappa _n(\\cdot ,w)\\Vert =1$ , and as we have observed in (REF ), if $w \\in B_{1- \\gamma /n}$ then $\\kappa _n(w,w)\\asymp \\frac{1}{1-|w|^2}$ .", "The following lemma collects the properties of $\\kappa _n$ near the diagonal in the different regimes in $\\mathbb {D}$ .", "Lemma 2.3 (i) There is a constant $\\gamma _0$ such that for all $w\\in B_{1-\\gamma _0/n}$ and $z\\in B(w, 0.5(1-|w|^2))$ we have $\\frac{1}{4(1-|w|^2)}|\\le |\\kappa _n(z,w)| \\le \\frac{9}{4(1-|w|^2)} \\, .$ (ii) For every $\\gamma >0$ there are $K>0$ and $\\epsilon > 0$ depending only on $\\gamma $ such that $\\frac{n}{K} \\le |\\kappa _n(z,w)| \\le K n \\, \\qquad \\text{ for all } w\\in C_{\\gamma /n}, z\\in B(w, \\epsilon /n)$ and $n>\\gamma $ .", "(i) If $z\\in B(w, 0.5(1-|w|^2))$ then $|1-z\\bar{w}| = |1-|w|^2 - \\bar{w} (z-w)| \\ge (1-|w|^2) - |\\bar{w}(z-w)| \\ge \\tfrac{1}{2} ( 1-|w|^2) \\, ,$ and likewise $ |1-z\\bar{w}| \\le \\tfrac{3}{2} ( 1-|w|^2)$ .", "To obtain similar bounds for the numerator, it suffices to prove that $|(n+2)(z\\bar{w})^{n+1}- (n+1)(z\\bar{w})^{n+2} | < \\tfrac{1}{2} \\, .$ for $\\gamma $ sufficiently large.", "We replace $n+1$ by $n$ and rewrite this expression as $|(n+1)(z\\bar{w})^{n}- n(z\\bar{w})^{n+1}| & = \\Big | n (z\\bar{w})^{n}\\Big ( \\tfrac{n+1}{n} - z\\bar{w} \\Big ) \\Big | \\\\&\\le n |w|^n \\Big ( |1-z\\bar{w}| + \\tfrac{1}{n} \\Big ) \\\\&\\le n |w|^n \\tfrac{3}{2} (1-|w|^2) + |w|^n \\, .$ As the map $r \\rightarrow r^n(1-r^2)$ is increasing on $[0, (\\frac{n}{n+2})^{1/2}]$ , the maximum of this expression is taken at $r=1-\\tfrac{\\gamma }{n}$ , so that we obtain the estimate $|(n+1)(z\\bar{w})^{n}- n(z\\bar{w})^{n+1}| \\le 3n (1-\\tfrac{\\gamma }{n})^{n}\\frac{\\gamma }{n} + (1-\\tfrac{\\gamma }{n})^{n} \\le 3 e^{-\\gamma } \\gamma +e^{-\\gamma } < 1/2 \\, ,$ for $\\gamma $ large enough.", "In fact, we may take $\\gamma \\ge 3$ .", "(ii) We know from (REF ) that $k _n(w,w) \\asymp n^2$ .", "Furthermore $|\\kappa _n^{\\prime }(z,w)| = \\frac{1}{k_n(w,w)^{1/2}} \\Big | \\frac{\\partial k_n(z,w)}{\\partial z}\\Big | \\lesssim \\frac{1}{n}\\, \\Big | \\sum _{k= 1}^n k(k+1) (z\\bar{w})^{k-1}\\bar{w}\\Big | \\le C n^2$ Thus if $z\\in B(w, \\epsilon /n)$ then $\\kappa _n(z) \\ge \\frac{e^{-2\\gamma }}{2}n - C \\epsilon n$ .", "Now take $\\epsilon \\le \\frac{Ce^{-2\\gamma }}{4}$ and $K = \\frac{e^{-2\\gamma }}{4}$ and (ii) follows." ], [ "Separation and Carleson-type conditions", "We first study the upper estimate of the Marcinkiewicz-Zygmund inequalities and derive a geometric description.", "Let $d(z,w) = \\big | \\frac{z-w}{1-z\\bar{w}}\\big |$ be the pseudohyperbolic metric on $\\mathbb {D}$ .", "We denote $\\Delta (w,\\rho ) = \\lbrace z\\in \\mathbb {D}: d(z,w) <\\rho \\rbrace $ the hyperbolic disk in $\\mathbb {D}$ .", "While $\\Delta (w,\\rho )$ is also a Euclidean disk (albeit with a different center and radius), it will be more convenient for us to compare it to a Euclidean disk with the same center $w$ .", "In fact, we have the following inclusions $B(w, \\frac{\\rho }{1+\\rho } (1-|w|^2)) \\subseteq \\Delta (w,\\rho )\\subseteq B(w, \\frac{\\rho }{1-\\rho } (1-|w|^2)) \\, ,$ where the latter inclusion holds for $\\rho <1/2$ .", "A set $\\Lambda \\subseteq \\mathbb {D}$ is called uniformly discrete, if there is a $\\delta ^{\\prime } >0$ such that $d(\\lambda ,\\mu )\\ge \\delta ^{\\prime } $ for all $\\lambda ,\\mu \\in \\Lambda , \\lambda \\ne \\mu $ .", "In view of (REF ) this is equivalent to the fact that the Euclidean balls $B(\\lambda , \\delta (1-|\\lambda |)),\\lambda \\in \\Lambda $ are disjoint in $\\mathbb {D}$ for some $\\delta >0$ .", "We refer to this condition as $\\Lambda $ being $\\delta $ -separated.", "In $A^2(\\mathbb {D})$ the upper estimate of the sampling inequality is characterized by the following geometric condition.", "See  [4] and [24].", "Proposition 2.4 For $\\Lambda \\subseteq \\mathbb {D}$ the following conditions are equivalent: (i) The inequality $\\sum _{\\lambda \\in \\Lambda } \\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\le B \\Vert f\\Vert ^2 _{A^2}$ holds for all $f\\in A^2(\\mathbb {D})$ .", "(ii) $\\Lambda $ is a finite union of uniformly discrete sets.", "(iii) $\\sup _{w\\in \\mathbb {D}} \\big (\\Lambda \\cap \\Delta (w,\\rho )\\big ) <\\infty $ for some (hence all) $\\rho \\in (0,1)$ .", "Condition (i) is often formulated by saying that the measure $\\sum _{\\lambda \\in \\Lambda } k(\\lambda ,\\lambda ) \\delta _\\lambda $ is a Carleson measure for $A^2$ .", "If we add the (much more difficult) lower sampling inequality to the assumptions, we also have the following lemma of Seip [24] (see Lemma 5.2 and Thm.", "7.1) Lemma 2.5 If $\\Lambda $ is a sampling set for $A^2(\\mathbb {D})$ , then $\\Lambda $ contains a uniformly discrete set $\\Lambda ^{\\prime } \\subseteq \\Lambda $ that is also sampling for $A^2(\\mathbb {D})$ .", "The proof of the implication $(ii) \\Rightarrow (i)$ yields a local version of the Bessel inequality that will be needed.", "Lemma 2.6 Let $\\Lambda $ be a $\\delta $ -separated set and let $\\gamma >0$ .", "Then there exists a constant $C = C(\\delta )$ and $\\gamma ^{\\prime } >\\gamma $ , such that $\\sum _{\\lambda \\in \\Lambda \\cap C_{\\gamma /n}} \\frac{|f(\\lambda ) |^2}{k(\\lambda , \\lambda )} \\le C \\, \\int _{C_{\\gamma ^{\\prime }/n}} |f(w)|^2 \\, dw \\, \\qquad \\text{for all } f\\in A^2 \\, .$ The constants depend only on the separation via $\\gamma ^{\\prime } = (1+\\delta )\\gamma $ and $C = \\frac{4}{\\pi \\delta ^2}$ .", "See  [4], Sec.", "2.11, Lemma 14.", "For completeness we include the proof.", "By assumption the Euclidean balls $B_\\lambda = B(\\lambda , \\delta (1-|\\lambda |)) \\subseteq \\mathbb {D}$ for $\\lambda \\in C_{\\gamma /n}$ are disjoint.", "Since $|B_\\lambda | k(\\lambda ,\\lambda ) = \\pi \\delta ^2(1-|\\lambda |)^2 (1-|\\lambda |^2)^{-2} \\ge \\pi \\delta ^2/4$ , the submean-value property for $|f|^2$ yields the estimate $\\frac{|f(z)|^2}{k(\\lambda ,\\lambda )} \\le \\frac{1}{k(\\lambda ,\\lambda ) |B_\\lambda | } \\int _{B_\\lambda } |f(w)|^2 \\, dw \\le \\frac{4}{\\pi \\delta ^2} \\int _{B_\\lambda } |f(w)|^2 \\, dw\\, .$ By summing over $\\lambda \\in \\Lambda \\cap C_{\\gamma /n}$ and using the disjointness of the disks $B_\\lambda $ we obtain that $\\sum _{\\lambda \\in C_{\\gamma /n}} \\frac{|f(\\lambda ) |^2}{k(\\lambda , \\lambda )} & \\le \\frac{4}{\\pi \\delta ^2} \\, \\sum _{\\lambda \\in C_{\\gamma /n}} \\int _{B_\\lambda } |f(w)|^2 \\, dw= \\frac{4}{\\pi \\delta ^2} \\, \\int _{\\bigcup _{\\lambda \\in C_{\\gamma /n}} B_\\lambda } |f(w)|^2 \\, dw \\, .$ Since $\\mathrm {dist}\\, (0,B_\\lambda ) = |\\lambda | - \\delta (1-|\\lambda |) = |\\lambda | (1+\\delta ) - \\delta \\ge (1-\\tfrac{\\gamma }{n}) (1+\\delta )- \\delta = 1-\\frac{(1+\\delta )\\gamma }{n} = 1-\\frac{\\gamma ^{\\prime }}{n}$ , the disks $B_\\lambda $ are contained in the annulus $C_{\\gamma ^{\\prime } /n}$ .", "Since they are disjoint, we obtain $\\int _{\\bigcup _{\\lambda \\in C_{\\gamma /n}} B_\\lambda } |f(w)|^2 \\, dw \\le \\int _{C_{\\gamma ^{\\prime } /n}} |f(w)|^2 \\, dw \\, .$ If $\\Lambda $ is a union of $K$ separated sets $\\Lambda _m$ , we apply the above argument to each $\\Lambda _m$ and then obtain the constant $C = \\frac{4K}{\\pi \\delta ^2}$ .", "Next we develop a geometric description for the upper Marcinkiewicz-Zygmund inequalities in $A^2$ that is similar to Proposition REF .", "For this we estimate the number of points of a Marcinkiewicz-Zygmund family in the relevant regions of the disk, namely in the bulk $B_{1- \\gamma /n}$ , the annulus $C_{\\gamma /n}$ , and in the cells $B(w,\\epsilon /n)\\cap \\mathbb {D}$ for $w$ near the boundary of $\\mathbb {D}$ .", "Proposition 2.7 Assume that $(\\Lambda _n)$ satisfies the upper Marcinkiewicz-Zygmund inequalities (REF ) for $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ .", "(i) Then for every $\\gamma >0$ $\\# (\\Lambda _n \\cap C_{\\gamma /n}) \\le C n \\, .$ (ii) There are $\\gamma _0>0$ ($\\gamma _0 \\approx 3$ ) and $C>0$ such that $\\#(\\Lambda _n \\cap B(z, 0.5(1-|z|))) \\le C, \\qquad \\forall z\\in B_{1-\\gamma _0/n}.$ As a consequence, $\\Lambda _n \\cap B_{1-\\gamma _0/n} $ is a disjoint union of at most $C$ uniformly discrete subsets with a separation $\\delta $ independent of $n$ .", "(iii) For every $\\gamma >0$ there is $\\epsilon >0$ such that $\\#(\\Lambda _n \\cap B(z, \\tfrac{\\epsilon }{n})) \\le C, \\qquad \\forall z\\in C_{\\gamma /n}\\, .$ The constants depend only on $\\gamma $ and the upper Marcinkiewicz-Zygmund bound $B$ .", "In each case we test the upper Marcinkiewicz-Zygmund inequalities against a suitable polynomial in $\\mathcal { P}_n$ .", "(i) Choose the monomial $p(z) = z^n \\in \\mathcal { P}_n$ with norm $\\Vert p\\Vert _{A^2}^2 = \\tfrac{1}{n+1} $ .", "Since for $\\lambda \\in C_{\\gamma /n}$ , $|p(\\lambda )| \\asymp _\\gamma 1$ by (REF ) and $ k_n(\\lambda ,\\lambda ) \\asymp _\\gamma n^2$ by (REF ), we obtain $\\frac{1}{n^2} \\# (\\Lambda _n \\cap C_{\\gamma /n}) &\\asymp _\\gamma \\sum _{\\lambda \\in \\Lambda _n \\cap C_{\\gamma /n}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )}\\le B \\Vert p\\Vert ^2 _{A^2}= \\frac{B}{n+1}\\, .$ This implies $ \\# (\\Lambda _n \\cap C_{\\gamma /n}) \\lesssim n $ .", "(ii) We choose $\\kappa _n (\\cdot ,z) \\in \\mathcal { P}_n$ .", "For $z\\in B_{1- \\gamma /n}$ and $\\lambda \\in B(z, 0.5(1-|z|))$ we have $|\\kappa _n(\\lambda , z)|^2 \\asymp (1-|z|^2)^{-1}$ by Lemma REF (i), and $k_n(\\lambda ,\\lambda ) \\asymp (1-|\\lambda |^2)^{-1}\\asymp (1-|z|^2)^{-1}$ by Lemma REF .", "We conclude that $\\frac{|\\kappa _n(\\lambda , z)|^2}{k_n(\\lambda ,\\lambda )} \\ge C$ for some constant, and thus $C \\#(\\Lambda _n \\cap B(z, 0.5(1-|z|))) \\le \\sum _{\\lambda \\in B(z,0.5(1-|z|))} \\frac{|\\kappa _n(\\lambda , z)|^2}{k_n(\\lambda ,\\lambda )} \\le B \\Vert \\kappa _n (\\cdot ,z)\\Vert _{A^2}^2 = B \\, .$ The relative separation follows as in [4], Section 2.11, Lemma 16.", "(iii) Again we choose $\\kappa _n(\\cdot , z)$ , this time for $z\\in C_{\\gamma /n}$ .", "By Lemma REF $|\\kappa _n (\\lambda ,z)|^2 \\asymp n^2$ for $\\lambda \\in B(z,\\epsilon /n)$ .", "Likewise $k_n(z,z) \\asymp n^2$ for $z\\in C_{\\gamma /n}$ by Lemma REF .", "Thus $C \\#\\big ( \\Lambda _n \\cap B(z, \\tfrac{\\epsilon }{n})\\big ) \\le \\sum _{\\lambda \\in \\Lambda _n \\cap B(z, \\frac{\\epsilon }{n}) } \\frac{|\\kappa _n(\\lambda , z)|^2}{k_n(\\lambda ,\\lambda )} \\le B \\Vert \\kappa _n (\\cdot ,z)\\Vert _{A^2}^2 = B \\, .$ As a consequence we see that the cardinality of a Marcinkiewicz-Zygmund family for polynomials obeys the correct order of growth.", "Corollary 2.8 If $(\\Lambda _n)$ satisfies the upper Marcinkiewicz-Zygmund inequality, then $\\# \\Lambda _n \\lesssim n$ .", "Choose $\\gamma $ large enough as in Lemma REF (ii).", "We cover $B_{1- \\gamma /n}= \\Delta (0, 1-\\tfrac{\\gamma }{n})$ with hyperbolic disks $\\Delta (z_j,\\frac{1}{3}) \\subseteq B(z_j, 0.5(1-|z_j|))$ , such that the disks $\\Delta (z_j,\\frac{1}{6})$ are disjoint.", "Since the hyperbolic area of $B_{1- \\gamma /n}$ is $\\int _{B_{1-\\gamma /n}} \\frac{dz}{(1-|z|^2)^2} \\le n/\\gamma $ , we need at most $c n/\\gamma $ disks (where $c^{-1}$ is the hyperbolic area of $\\Delta (z_j,\\frac{1}{6})$ ).", "By Lemma REF (ii) every hyperbolic disk $ B(z_j, 0.5(1-|z_j|))$ contains $C$ points, so that $ \\# (\\Lambda _n\\cap B_{1-\\gamma /n}) \\le C n$ for a suitable constant.", "By Lemma REF (i) $\\#(\\Lambda _n \\cap C_{\\gamma /n}) \\lesssim n $ as well.", "We can now formulate a geometric description of the upper Marcinkiewicz-Zygmund inequality (REF ), that is, the Bessel inequality of the normalized reproducing kernels.", "Theorem 2.9 For a family $(\\Lambda _n)\\subseteq \\mathbb {D}$ the following conditions are equivalent: (i) For all $n\\ge n_0$ $\\sum _{\\lambda \\in \\Lambda _n}\\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )} \\le B\\Vert p\\Vert ^2,\\qquad \\forall p\\in \\mathcal {P}_n \\, .$ (ii) There are $\\gamma >0$ and $C>0$ such that $(\\Lambda _n)$ satisfies $\\# \\big (\\Lambda _n \\cap B(w, 0.5(1-|w|))\\big ) &\\le C \\qquad \\forall w\\in B_{1- \\gamma /n}\\, , \\\\\\#(\\Lambda _n \\cap B(w, 1/n)) & \\le C \\qquad \\forall w\\in C_{\\gamma /n}\\, .", "$ (i) $\\, \\Rightarrow \\, $ (ii): This is Proposition REF .", "(ii) $\\, \\Rightarrow \\, $ (i): We write $\\sum _{\\lambda \\in \\Lambda _n}\\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )} = \\sum _{\\lambda \\in \\Lambda _n \\cap B_{1- \\gamma /n}} \\dots + \\sum _{\\lambda \\in \\Lambda _n \\cap C_{\\gamma /n}} \\dots $ and control each term with the appropriate geometric condition.", "On $B_{1- \\gamma /n}$ we may replace $k_n$ by $k$ (Lemma REF ).", "Since $\\Lambda _n \\cap B_{1- \\gamma /n}$ is a union of at most $C$ uniformly discrete sets with fixed separation independent of $n$ by our assumption (REF ), Proposition REF — or the appropriate version of Lemma REF — implies that $\\sum _{\\lambda \\in \\Lambda _n \\cap B_{1- \\gamma /n}}\\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )} & \\lesssim _\\gamma \\sum _{\\lambda \\in \\Lambda _n \\cap B_{1- \\gamma /n}}\\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} \\\\&\\lesssim \\Vert p\\Vert _{A^2}^2 \\, .$ On $C_{\\gamma /n}$ we have $k_n(\\lambda ,\\lambda ) |B(\\lambda , 1/n)| \\asymp 1$ by (REF ), therefore using the submean-value property we obtain $\\sum _{\\lambda \\in C_{\\gamma /n}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )}&\\le \\sum _{\\lambda \\in C_{\\gamma /n}} \\frac{1}{k_n(\\lambda ,\\lambda )|B(\\lambda ,1/n)|} \\int _{B(\\lambda , 1/n)} |p(z)|^2 \\, dz \\\\& \\lesssim \\int \\Big ( \\sum _{\\lambda \\in \\Lambda _n \\cap C_{\\gamma /n}} \\chi _{B(\\lambda ,1/n)} (z ) \\Big ) |p(z)|^2 \\, dz\\, .$ The integral is over the slightly larger annulus $C_{(\\gamma +1)/n}\\supseteq \\bigcup _{\\lambda \\in C_{\\gamma /n}} B(\\lambda ,1/n)$ , and by assumption () the sum in the integral is bounded, whence $\\sum _{\\lambda \\in C_{\\gamma /n}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )}\\lesssim \\Vert p\\Vert _{A^2}^2$ .", "In a more general formulation we can use measures in condition (i) instead of discrete samples.", "This motivates the following definition.", "Definition 2.1 We say that a sequence of measures $\\lbrace \\mu _n\\rbrace $ defined on the unit disk form a Carleson sequence for polynomials in the Bergman space if there is a constant $C>0$ such that $\\int _{\\mathbb {D}} |p(z)|^2\\, d\\mu _n(z) \\le C \\int _{\\mathbb {D}} |p(z)|^2\\, dz\\qquad \\text{ for all } p\\in \\mathcal { P}_n$ uniformly in $n$ .", "Thus a sequence $(\\Lambda _n)\\subseteq \\mathbb {D}$ satisfies the upper inequality in (REF ) if and only if the measures $\\mu _n =\\sum _{\\lambda \\in \\Lambda _n} \\frac{\\delta _\\lambda }{k_n(\\lambda ,\\lambda )}$ form a Carleson sequence.", "Theorem 2.10 A sequence of measures $\\mu _n$ is a Carleson sequence for polynomials in the Bergman space if and only if there exist $\\gamma > 0$ and $C>0$ such that $\\mu _n(B(z,0.5(1-|z|^2))) &\\le C (1-|z|^2)^2 \\qquad \\forall z\\in B_{1-\\gamma /n}, \\\\\\mu _n(B(z, 1/n)) &\\le C/{n^2} \\qquad \\qquad \\quad \\forall z\\in C_{\\gamma /n}.$ The proof is similar to the proof of Theorem REF .", "As we will not need the general statement about Carleson sequences, we omit the details.", "Remark.", "The geometric condition in Theorems REF and REF can be more concisely stated in terms of the first Bergman metric $\\rho _n = k_n(z,z) ds^2$ , see [1] instead of the Euclidean metric.", "This is not to be confused with the more usual second Bergman metric given by $\\Delta \\log k_n(z,z)ds^2$ .", "The reformulation is as follows: if $D_n(z,r)$ denotes a disk in the first Bergman metric $\\rho _n$ , then a sequence of measures $\\lbrace \\mu _n\\rbrace $ is a Carleson sequence if and only if there are $r> 0$ and $C>0$ such that $\\mu _n(D_n(z,r)) \\le C |D_n(z,r)|$ , for all $n>0$ and for all $z\\in \\mathbb {D}$ , where $|D(z,r)|$ is the Lebesgue measure of $D(z,r)$ .", "The conditions of (ii) in Theorem REF are then equivalent to $\\# (\\Lambda _n \\cap D_n(z,r)) \\le C$ for all $z\\in \\mathbb {D}$ .", "This reformulation is of course similar to the characterization of the Carleson measures for the Bergman space obtained in [12].", "The geometric description is $\\mu (B(z,0.5(1-|z|)))\\lesssim |B(z, 0.5(1-|z|))|$ for all $z\\in \\mathbb {D}$ ." ], [ "Sampling implies Marcinkiewicz-Zygmund inequalities", "After these preparations we can show that every sampling set for $A^2(\\mathbb {D})$ generates a Marcinkiewicz-Zygmund family for the polynomials.", "Theorem 2.11 Assume that $\\Lambda \\subseteq \\mathbb {D}$ is a sampling set for $A^2(\\mathbb {D})$ .", "Then for $\\gamma >0$ small enough, the sets $\\Lambda _n =\\Lambda \\cap B_{1- \\gamma /n}$ form a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ .", "The assumption means that there exist $A,B>0$ , such that $A \\Vert f\\Vert _{A^2}^2 \\le \\sum _{\\lambda \\in \\Lambda }\\frac{|f(\\lambda )|^2}{k(\\lambda , \\lambda )} \\le B \\Vert f\\Vert _{A^2}^2 \\quad \\text{ for all} f \\in A^2 \\, .$ Since $\\Lambda $ is sampling for $A^2$ , by Lemma REF it contains a uniformly discrete set $\\Lambda ^{\\prime }$ with separation constant $\\delta $ , say.", "We now choose $\\gamma >0$ so small that $1- \\Big ( 1-\\frac{(1+\\delta )\\gamma }{n} \\Big )^{2n+2} \\le \\frac{A \\delta ^2}{8 } \\, \\qquad \\text{ for all } n > 2\\gamma \\, .$ This is possible because $\\Big ( 1-\\frac{(1+\\delta )\\gamma }{n} \\Big )^{2n+2} \\ge e^{-4(1+\\delta )\\gamma } ( 1-\\frac{(1+\\delta )\\gamma }{n} )^{2}$ by the lower estimate in (REF ), and thus tends to 1 uniformly in $n$ , as $\\gamma \\rightarrow 0$ .", "Note that the choice of $\\gamma $ depends on the lower sampling constant $A$ and the separation of $\\Lambda ^{\\prime } $ .", "We now apply Lemma REF and Corollary REF to the points $\\Lambda \\setminus \\Lambda _n= \\Lambda \\cap C_{\\gamma /n}$ .", "Set $\\gamma ^{\\prime } = (1+\\delta )\\gamma $ .", "Then for $p \\in \\mathcal { P}_n$ we obtain $\\sum _{\\lambda \\in \\Lambda \\setminus \\Lambda _n} \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} & \\le \\frac{4}{\\pi \\delta ^2} \\int _{C_{\\gamma ^{\\prime }/n}}|p(w)|^2 \\, dw \\\\&\\le \\frac{4}{\\delta ^2} \\Big ( 1- \\Big ( 1-\\frac{(1+\\delta )\\gamma }{n}\\Big )^{2n+2} \\Big ) \\Vert p\\Vert _{A^2}^2 \\le \\frac{A}{2} \\Vert p\\Vert _{A^2}^2\\, .$ Consequently, $\\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} &= \\sum _{\\lambda \\in \\Lambda } - \\sum _{\\lambda \\in \\Lambda \\setminus \\Lambda _n} \\dots \\\\&\\ge A \\Vert p\\Vert _{A^2}^2 - \\sum _{\\lambda \\in \\Lambda \\setminus \\Lambda _n} \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} \\ge (A - A/2) \\Vert p\\Vert _{A^2}^2 \\, .$ Since always $k_n(\\lambda ,\\lambda ) \\le k(\\lambda , \\lambda )$ we finish the estimate by $\\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )} \\ge \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} \\ge \\frac{A}{2} \\Vert p\\Vert _{A^2}^2 \\qquad \\text{ for all } p \\in \\mathcal { P}_n\\, .$ As usual, the upper bound is easy.", "Since by (REF ) $k_n(\\lambda , \\lambda )\\asymp k(\\lambda , \\lambda )$ for $\\lambda \\in B_{1- \\gamma /n}$ , it follows that $\\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )} \\lesssim _\\gamma \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )}\\le \\sum _{\\lambda \\in \\Lambda } \\frac{|p(\\lambda )|^2}{k(\\lambda , \\lambda )} \\le B \\, \\Vert p\\Vert _{A^2}^2 \\, ,$ since $\\Lambda $ is sampling for $A^2(\\mathbb {D})$ .", "Since sampling sets for $A^2(\\mathbb {D})$ are completely characterized by means of the Seip-Korenblum density [24], Theorem REF provides a wealth of examples of Marcinkiewicz-Zygmund families.", "However, the parameter $\\gamma $ in the statement depends on the lower sampling constant $A$ and the separation $\\delta $ of $\\Lambda $ ." ], [ "From Marcinkiewicz-Zygmund inequalities to sampling", "Choose either the Euclidean or the pseudohyperbolic metric on $\\mathbb {D}$ and let $d(E,F)$ be the corresponding Hausdorff distance between two closed sets $E,F\\subseteq \\mathbb {D}$ .", "We say that a sequence of sets $\\Lambda _n\\subseteq \\mathbb {D}$ converges weakly to $\\Lambda \\subseteq \\mathbb {D}$ , if for all compact disks $B\\subseteq \\mathbb {D}$ $\\lim _{n\\rightarrow \\infty } d\\big ((\\Lambda _n \\cap B) \\cup \\partial B, (\\Lambda \\cap B) \\cup \\partial B \\big ) = 0 \\, .$ See [4], [10], [13] for equivalent definitions.", "The main consequence of weak convergence is the convergence of sampling sums.", "If all $\\Lambda _n$ are uniformly separated with fixed separation $\\delta $ , then $\\sum _{\\lambda \\in \\Lambda _n \\cap B}\\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\rightarrow \\sum _{\\lambda \\in \\Lambda \\cap B}\\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\, .$ for all $f\\in A^2(\\mathbb {D})$ .", "More generally, if each $\\Lambda _n$ is a finite union of $K$ uniformly separated sets with separation constant $\\delta $ independent of $n$ , then we need to include multiplicities $m (\\lambda ) \\in \\lbrace 1, \\dots , K\\rbrace $ .", "We then obtain $\\sum _{\\lambda \\in \\Lambda _n \\cap B}\\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\rightarrow \\sum _{\\lambda \\in \\Lambda \\cap B}\\frac{|f(\\lambda )|^2}{k(\\lambda ,\\lambda )} m(\\lambda )\\, .$ Since the multiplicities $m (\\lambda )$ are bounded, they only affect the constants, but do not change the arguments.", "Theorem 2.12 Assume that $(\\Lambda _n)$ is a Marcinkiewicz-Zygmund family for the polynomials $\\mathcal { P}_n$ in $A^2(\\mathbb {D})$ .", "Let $\\Lambda $ be a weak limit of $(\\Lambda _n)$ or some subsequence $(\\Lambda _{n_k})$ .", "Then $\\Lambda $ is a sampling set for $A^2(\\mathbb {D})$ .", "Step 1.", "We assume that $\\Lambda _n$ is a Marcinkiewicz Zygmund family and therefore there are $A,B> 0$ such that $A\\Vert p\\Vert _{A^2}\\le \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )}\\le B\\Vert p\\Vert ^2_{A^2}$ for all polynomials $p\\in \\mathcal { P}_n$ .", "The upper inequality implies that there are $C>0$ and $\\gamma > 0$ such that $\\#(\\Lambda _n \\cap B(w, 0.5(1-|w|))) \\le C$ for all $w\\in B_{1- \\gamma /n}$ and all $n$ (Theorem REF ).", "In addition, each set $\\Lambda _n \\cap B_{1- \\gamma /n}$ is a finite union of $K$ uniformly separated sequences with separation $\\delta $ independent of $n$ .", "Since $\\Lambda _n$ converges to $\\Lambda $ weakly, $\\Lambda $ satisfies the same inequality $\\#(\\Lambda \\cap \\bar{B}(w, 0.5(1-|w|))) \\le C,\\qquad \\forall w\\in B_{1- \\gamma /n}\\, ,$ and thus $\\Lambda $ is also a union of $K$ uniformly discrete sequences with separation $\\delta $ , see, e.g., [4], Sec.", "2.1, Lemma 16.", "Note that the geometric conditions furnish constants $c_\\gamma $ for the comparison of the kernels $k$ and $k_n$ on $B_{1- \\gamma /n}$ , the separation constant $\\delta $ , and the multiplicity $K$ .", "In the next step we will choose a suitable radius $r$ that solely depends on these given constants.", "Step 2.", "We prove the desired sampling inequality in $A^2(\\mathbb {D})$ for all polynomials and then use a density argument.", "Fix a polynomial $p$ of degree $N$ , say.", "We know that there is an $r = r(N)<1$ such that $\\int _{|z|> r-\\delta } |p|^2 \\le \\frac{c_\\gamma \\delta ^2 A }{8K}\\Vert p\\Vert ^2_{A^2}$ .", "We take $n\\ge N$ big enough such that $r< 1 - \\gamma /n$ .", "In this case we have $A \\Vert p\\Vert ^2 _{A^2}\\le \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} &=\\sum _{\\lambda \\in \\Lambda _n, |\\lambda | <r} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} \\\\&+ \\sum _{\\lambda \\in \\Lambda _n, r\\le |\\lambda | <1-\\tfrac{\\gamma }{n}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} + \\sum _{\\lambda \\in \\Lambda _n,|\\lambda | \\ge 1-\\tfrac{\\gamma }{n}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} = \\\\& = I_n + II_n + III_n.$ According to Lemma REF we may replace $k_n(\\lambda , \\lambda )$ by $k(\\lambda , \\lambda )$ in the first two terms and by $n^2$ in the third term.", "Thus with a constant depending on $\\gamma $ , we obtain $ A \\Vert p\\Vert ^2 _{A^2}\\le c_\\gamma ^{-1}\\Big ( \\sum _{\\lambda \\in \\Lambda _n \\cap B_r} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} + \\sum _{\\lambda \\in \\Lambda _n, r\\le |\\lambda | <1-\\tfrac{\\gamma }{n}} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} + \\sum _{\\lambda \\in \\Lambda _n \\cap C_{\\gamma /n} } \\frac{|p(\\lambda )|^2}{n^2} \\Big ) \\, .$ In the sum $I_n$ all points $\\lambda $ lie in the compact set $\\overline{B(0,r)}$ , and by weak convergence including multiplicities $m(\\lambda ) \\in \\lbrace 1, \\dots , K\\rbrace $ , we obtain $\\lim _{n\\rightarrow \\infty } I_n &\\le c_\\gamma ^{-1}\\lim _{n\\rightarrow \\infty }\\sum _{\\lambda \\in \\Lambda _n \\cap B_r} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} = \\\\& = c_\\gamma ^{-1}\\sum _{\\lambda \\in \\Lambda \\cap \\overline{B_r}} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} m(\\lambda ) \\le c_\\gamma ^{-1}K\\sum _{\\lambda \\in \\Lambda \\cap \\overline{B_r}} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\, .$ To treat $II_n$ , we use the assertion of Step 1 that every $\\Lambda _n \\cap B_{1- \\gamma /n}$ is a finite union of at most $K$ uniformly separated sequences with separation $\\delta $ .", "Lemma REF and the choice of $r$ yield $II_n \\le c_\\gamma ^{-1}\\sum _{\\lambda \\in \\Lambda _n, r\\le |\\lambda | <1-\\gamma /n } \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\le \\frac{4K}{\\pi \\delta ^2c_\\gamma } \\int _{|z|>r-\\delta } |p(z)|^2 \\le \\frac{A}{2} \\Vert p\\Vert _{A^2}.$ Finally, the last term $III_n$ is negligible when $n\\rightarrow \\infty $ because $III_n = \\sum _{\\lambda \\in \\Lambda _n \\cap C_{\\gamma /n}} \\frac{|p(\\lambda )|^2}{n^2} \\le \\Vert p\\Vert _\\infty ^2 \\, \\frac{1}{n^2}\\# \\lbrace \\lambda \\in \\Lambda _n \\cap C_{\\gamma /n} \\rbrace \\, .", "$ By Lemma REF (i) $III_n$ tends to 0, as $n\\rightarrow \\infty $ .", "Finally we take the limit in (REF ) and obtain inequality $A \\Vert p\\Vert _{A^2}^2 \\le c_\\gamma ^{-1}K \\sum _{\\lambda \\in \\Lambda \\cap \\overline{B(0,r)} } \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} + \\frac{A}{2} \\Vert p\\Vert ^2 _{A^2}\\, .$ This implies the lower sampling inequality valid for all polynomials.", "Since the polynomials are dense in $A^2(\\mathbb {D})$ , (REF ) can be extended to all of $A^2(\\mathbb {D})$ .", "Note that $\\sum _{\\lambda \\in \\Lambda } \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} = \\sum _{\\lambda \\in \\Lambda } |\\langle p, \\kappa _\\lambda \\rangle |^2$ , where $\\kappa _\\lambda (z) = k(z,\\lambda )k(\\lambda ,\\lambda )^{-1/2}$ is the normalized reproducing kernel.", "Thus the left-hand side in (REF ) is just the frame operator associated to the set $\\lbrace \\kappa _\\lambda \\rbrace $ .", "For boundedness and invertibility it therefore suffices to check on a dense subset.", "Step 3.", "As always, the upper bound is much easier to prove: Let $p\\in \\mathcal { P}_N$ .", "Then by Lemma  REF $\\sum _{\\lambda \\in \\Lambda \\cap C_{\\gamma /N}} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} \\le C \\int _{C_{\\gamma ^{\\prime }/N}} |p(w)|^2 \\,dw \\le C \\Vert p\\Vert ^2 _{A^2}\\, .$ On the disk $B_{1-\\gamma /N}$ we use the weak convergence and deduce that $\\sum _{\\lambda \\in \\Lambda \\cap B_{1-\\gamma /N}} \\frac{|p(\\lambda )|^2}{k(\\lambda ,\\lambda )} = \\lim _{n\\rightarrow \\infty } \\sum _{\\lambda \\in \\Lambda _n \\cap \\bar{B}_{1-\\gamma /N}} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} \\le B \\Vert p\\Vert ^2 _{A^2}\\, ,$ because $\\Lambda _n$ is a Marcinkiewicz-Zygmund family for polynomials and $p\\in \\mathcal { P}_N\\subseteq \\mathcal { P}_n$ .", "The sums of both terms yields the upper sampling inequality for $\\Lambda $ for all polynomials, which extends to $A^2(\\mathbb {D})$ by density.", "The upper bound can also be derived from the geometric description of Theorem REF .", "With a bit more effort one can prove an $A^p$ -version of Theorems REF and REF .", "The proof requires several modifications of interest.", "For instance, to obtain (REF ) for the $A^p$ -norm, one needs an argument similar to  [22].", "The $A^p$ -norm of the normalized reproducing kernel in Proposition REF requires the boundedness of the Bergman projection on $L^p$ ." ], [ "Basic facts.", "The Hardy space $H^2= H^2(\\mathbb {D})$ consists of all analytic functions in $\\mathbb {D}$ whose boundary values on $\\partial \\mathbb {D}= \\mathbb {T}$ are in $L^2(\\mathbb {T})$ with finite norm $\\Vert f\\Vert _{H^2} = \\Big (\\int _{0 }^1 |f(e^{2\\pi i t})|^2 \\, dt \\Big )^{1/2} \\, .$ The monomials $z\\rightarrow z^k$ form an orthonormal basis, and the norm of $f(z) =\\sum _{k=0}^\\infty a_k z^k$ is $\\Vert f\\Vert _{H^2} ^2 = \\sum _{k=0}^\\infty |a_k|^2 \\, .$ For $p(z) = \\sum _{k=0}^n a_k z^k \\in \\mathcal { P}_n$ and $0< \\rho \\le 1$ , let $p_\\rho (z) = p(\\rho z)$ , then $\\Vert p_\\rho \\Vert ^2_{H^2}&= \\sum _{k=0}^n |a_k|^2 \\rho ^{2k}\\ge \\rho ^{2n} \\Vert p\\Vert ^2 _{H^2}\\, ,$ and clearly $\\Vert p_\\rho \\Vert _{H^2}\\le \\Vert p\\Vert _{H^2}$ .", "If $p\\in \\mathcal { P}_n$ , $\\rho = 1-\\tfrac{\\gamma }{n}$ , and $n>2\\gamma $ , then by (REF ) $\\Vert p\\Vert ^2 _{H^2}\\ge \\Vert p_\\rho \\Vert ^2 _{H^2}\\ge (1-\\tfrac{\\gamma }{n})^{2n} \\Vert p\\Vert ^2 _{H^2}\\ge e^{-4\\gamma } \\Vert p\\Vert ^2_{H^2}\\, .$ The reproducing kernel of $\\mathcal { P}_n$ in $H^2$ is given by $k_n(z,w) & = \\sum _{k=0}^n (z\\bar{w})^n = \\frac{1 - (z\\bar{w})^{n+1}}{1-z\\bar{w}} \\, .", "$ As $n\\rightarrow \\infty $ , the kernel tends to the reproducing kernel of $H^2$ , $k(z,w) = \\frac{1}{1-z\\bar{w}}$ for $z,w \\in \\mathbb {D}$ .", "For Marcinkiewicz-Zygmund families for polynomials in Hardy space it will be necessary to also consider sampling points outside the unit disk as in  [22].", "With this caveat in mind, we define the appropriate annuli as $A_{\\gamma /n} = \\lbrace z\\in \\mathbb {C}: 1-\\tfrac{\\gamma }{n}\\le |z| \\le (1-\\tfrac{\\gamma }{n})^{-1}\\rbrace \\, .$ We now compare the kernels for $\\mathcal { P}_n$ and $H^2$ .", "Lemma 3.1 Let $k_n(z,w)$ be the reproducing kernel of $\\mathcal { P}_n$ in $H^2$ and let $\\gamma >0$ be arbitrary.", "If $z\\in A_{\\gamma /n}$ , i.e., $1-\\tfrac{\\gamma }{n}\\le |z| \\le (1-\\tfrac{\\gamma }{n})^{-1}$ , and $n>2\\gamma $ , then $\\frac{1-e^{-4\\gamma } }{2\\gamma } n \\le k_n(z,z) \\le \\frac{e^{4\\gamma }}{\\gamma } n \\, .$ Consequently, if $\\Lambda _n \\subseteq A_{\\gamma /n}$ , then, for all polynomials $p\\in \\mathcal { P}_n$ , $\\frac{1}{n} \\sum _{\\lambda \\in \\Lambda _n} |p(\\lambda )|^2\\asymp \\sum _{\\lambda \\in \\Lambda _n} \\frac{|p(\\lambda )|^2}{k_n(\\lambda ,\\lambda )} \\, .$ On the annulus $A_{\\gamma /n}$ we may therefore always work with the weight $1/n$ for polynomials of degree $n$ .", "Writing $r=|z|$ , the kernel $k_n(z,z) = \\frac{1-r^{2n+2}}{1-r^2}$ is increasing in $r$ , so that for $z\\in A_{\\gamma /n}$ we have $\\frac{1-(1-\\tfrac{\\gamma }{n})^{2n+2}}{1- (1-\\tfrac{\\gamma }{n})^2} \\le k_n(z,z) \\le \\frac{(1-\\tfrac{\\gamma }{n})^{-2n-2} - 1}{(1-\\tfrac{\\gamma }{n})^{-2} -1} = \\frac{(1-\\tfrac{\\gamma }{n})^{-2n} - (1-\\tfrac{\\gamma }{n})^2 }{1 - (1-\\tfrac{\\gamma }{n})^{2} } \\, .$ For $n\\ge 2\\gamma $ , the denominator $1 - (1-\\tfrac{\\gamma }{n})^{2} = \\frac{\\gamma }{n} + \\frac{\\gamma }{n}(1- \\frac{\\gamma }{n})$ is between $\\gamma /n$ and $2\\gamma /n$ , for the numerator we use (REF ) and obtain $(1-\\tfrac{\\gamma }{n})^{-2n} - (1-\\tfrac{\\gamma }{n})^2 \\le e^{4\\gamma }$ and $1-(1-\\tfrac{\\gamma }{n})^{2n+2} \\ge 1-(1-\\tfrac{\\gamma }{n})^{2n} \\ge 1-e^{-2\\gamma }$ .", "Thus both ratios are of order $n$ with the constants $(1-e^{-2\\gamma }) /(2\\gamma )$ and $ e^{4\\gamma }/\\gamma $ ." ], [ "From Marcinkiewicz-Zygmund families on $\\mathbb {T}$ to Marcinkiewicz-Zygmund families for\n{{formula:3d8020f5-3038-4f23-a413-6e1f9ed064d1}} . ", "In contrast to the situation for the Bergmann space, there are no sampling sequences for the Hardy space $H^2(\\mathbb {D})$ by the results of Thomas [27].", "Duren and Schuster [4] give the following simple argument: a sampling set $\\Lambda $ for $H^2$ must be a Blaschke sequence.", "However, every Blaschke sequence is a zero set in $H^2$ , which contradicts the sampling inequality.", "Therefore there can be no analogue of Theorem REF .", "Nevertheless, we prove that Hardy space admits Marcinkiewicz-Zygmund families for polynomials.", "The idea is to associate a Marcinkiewicz-Zygmund for polynomials in $H^2(\\mathbb {D})$ to every Marcinkiewicz-Zygmund family for polynomials on the torus.", "As these are well understood [8], [22], we obtain a general class of Marcinkiewicz-Zygmund families in $H^2(\\mathbb {D})$ .", "We use the following notation: for $\\lambda \\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ let $\\tilde{\\lambda } = \\frac{\\lambda }{|\\lambda |}$ be the projection from the complex plane $\\mathbb {C}\\setminus \\lbrace 0\\rbrace $ onto the torus $\\partial \\mathbb {D}= \\mathbb {T}$ .", "Theorem 3.2 Assume that the family $(\\Lambda _n) \\subseteq \\mathbb {C}$ has the following properties: (i) there exists $\\gamma >0$ , such that $\\Lambda _n \\subseteq A_{\\gamma /n}$ for all $n \\ge \\gamma $ .", "(ii) The projected family $(\\tilde{\\Lambda }_n) \\subseteq \\mathbb {T}$ is a Marcinkiewicz-Zygmund family for the polynomials $\\mathcal { P}_n\\subseteq L^2(\\mathbb {T})$ .", "(iii) The projection $\\Lambda _n \\rightarrow \\widetilde{\\Lambda _n}$ from $A_{\\gamma /n}$ to $\\mathbb {T}$ is one-to-one for all $n$ .", "Then $(\\Lambda _n)$ is a Marcinkiewicz-Zygmund family for the polynomials $\\mathcal { P}_n$ in $H^2(\\mathbb {D})$ .", "The statement in the introduction is just a reformulation of Theorem REF without additional notation.", "The proof is inspired by a sampling theorem of Duffin and Schaeffer for bandlimited functions from samples in the complex plane (rather than from samples on the real axis).", "See [5] and [25], [28].", "In analogy to the theory of bandlimited functions, one can view Theorem REF also as a perturbation result for Marcinkiewicz-Zygmund families in $L^2(\\mathbb {T})$ , where the points in $\\mathbb {T}$ are perturbed in a complex neighborhood of $\\mathbb {T}$ .", "We wrap the main part of the proof into a technical lemma.", "Lemma 3.3 Let $\\gamma >0$ and $n> 2\\gamma $ .", "Assume that $\\Lambda _n $ is a finite set contained in $A_{\\gamma /n}$ so that the projection $\\Lambda _n \\rightarrow \\widetilde{\\Lambda _n}$ is one-to-one.", "Then there exists a set $\\Lambda _n ^{(1)}$ with the following properties: (i) $\\Lambda _n^{(1)}$ is contained in the smaller annulus $A_{\\frac{3\\gamma }{4n}}$ and $\\# \\Lambda _n^{(1)} = \\# \\Lambda _n$ .", "(ii) For every $\\lambda \\in \\Lambda _n$ there is a $\\mu \\in \\Lambda _n ^{(1)}$ , such that $\\tilde{\\lambda } = \\tilde{ \\mu }$ , and (iii) For every $p\\in \\mathcal { P}_n$ there exists $p_1\\in \\mathcal { P}_n$ satisfying $\\Vert p_1\\Vert ^2 _{H^2}\\ge e^{-2\\gamma /3} \\Vert p\\Vert _{H^2}^2 $ and $\\frac{1}{n} \\, \\sum _{\\lambda \\in \\Lambda _n} |p(\\lambda )|^2 \\ge \\frac{1}{n} \\, \\sum _{\\lambda \\in \\Lambda _n^{(1)}}|p_1(\\lambda )|^2 \\, .$ As we will apply the lemma to a Marcinkiewicz-Zygmund family, it is important the constants and the construction are independent of the degree $n$ .", "Step 1.", "Construction of $\\Lambda _n^{(1)}$ .", "If $\\lambda \\in \\Lambda _n$ and $|\\lambda |\\le 1$ , set $\\mu = (1+\\frac{\\gamma }{3n})\\lambda $ .", "If $\\lambda \\in \\Lambda _n$ and $|\\lambda |> 1$ , set $\\mu = (1+\\frac{\\gamma }{3n})/ \\bar{\\lambda } $ .", "By construction, $\\tilde{\\mu } =\\tilde{\\lambda }$ .", "Since $\\Lambda _n \\rightarrow \\widetilde{\\Lambda _n}$ is one-to-one by assumption, $\\Lambda _n^{(1)}$ has the same cardinality as $\\Lambda _n$ .", "[To appreciate this assumption, consider the case when both $\\lambda $ and $1/\\bar{\\lambda }$ are in $\\Lambda _n$ .", "They both would be mapped to the same point $\\mu $ .]", "Since for $|z |\\ge 1-\\tfrac{\\gamma }{n}$ , we have $(1+\\frac{\\gamma }{3n}) |z | \\ge (1+\\frac{\\gamma }{3n}) (1-\\tfrac{\\gamma }{n}) \\ge 1- \\frac{3\\gamma }{4n}$ , and for $ |z|<1$ we have $(1+\\frac{\\gamma }{3n}) \\le (1-\\frac{3\\gamma }{4n})^{-1}$ , it follows that $\\Lambda _n^{(1)}$ is contained in the smaller annulus $A_{3\\gamma /(4n)}$ .", "Step 2.", "Construction of $p_1$ .", "Given $p\\in \\mathcal { P}_n$ with zeros $z_j$ and factorization $p(z) = z^\\ell \\prod (z-z_j)$ , we obtain $p_1$ by reflecting all its zeros into $\\mathbb {D}$ and an appropriate scaling.", "We multiply $p$ by several Blaschke factors and set $\\widetilde{p_1}(z) = z^l \\prod _{|z_j|\\le 1} (z-z_j) \\prod _{|z_j|> 1}(1 - \\overline{z_j}z) = p(z) \\, \\prod _{|z_j|> 1}\\frac{1 - \\overline{z_j}z}{z-z_j} \\, .$ By construction, all zeros of $\\widetilde{p_1}$ are now in the unit disk $\\overline{\\mathbb {D}}$ .", "In engineering terminology, $\\widetilde{p_1}$ is the minimum phase filter associated to $p$ .", "Furthermore, $\\widetilde{p_1}$ has the following properties: (i) By (REF ) and the property of Blaschke factors we have $|\\widetilde{p_1}(z)| = |p(z)| \\qquad \\text{ for } z\\in \\mathbb {T}\\, ,$ and thus $\\Vert \\widetilde{p_1}\\Vert _{H^2}= \\Vert p\\Vert _{H^2}$ .", "(ii) For $ z\\in \\mathbb {D}$ we have $|\\widetilde{p_1}(z)| \\le \\min \\Big ( |p(z)|, |p(1/\\bar{z})| \\Big ) \\, .$ To see this, observe that for $|z_j| > 1$ each Blaschke factor in (REF ) satisfies $\\Big | \\frac{1 -\\bar{z}_j z}{z-z_j} \\Big | = \\Big | \\frac{\\bar{z}_j^{-1}- z}{z_j^{-1}z -1}\\Big | \\le 1 \\, ,$ thus $|\\widetilde{p_1}(z)| \\le |p(z)|$ for $z\\in \\mathbb {D}$ .", "Using the first factorization of $\\widetilde{p_1}$ in (REF ) and $p(1/\\bar{z}) = \\bar{z}^{-l} \\prod _{|z_j| \\le 1} \\big ( 1/\\bar{z} -z_j \\big ) \\, \\prod _{|z_j| > 1} \\big ( 1/\\bar{z} -z_j \\big ) $ , the second inequality in (REF ) follows from $\\frac{|\\widetilde{p_1}(z)|}{|p(\\frac{1}{\\bar{z}})|} = |z|^{2l} \\prod _{|z_j|\\le 1} \\frac{|z-z_j|}{|\\bar{z}^{-1}(1-\\bar{z}z_j)|} \\,\\prod _{|z_j|> 1} \\frac{|1- \\bar{z_j}z|}{|\\bar{z}^{-1}(1-\\bar{z}z_j)|} \\le 1 \\, .$ Finally we set $p_1(z) = \\widetilde{p_1}\\Big ( (1+ \\frac{\\gamma }{3n})^{-1}z \\Big ) \\in \\mathcal { P}_n\\, .$ Then by (REF ) $\\Vert p_1\\Vert ^2 _{H^2}\\ge (1+ \\frac{\\gamma }{3n})^{-2n} \\Vert \\widetilde{p_1} \\Vert ^2 _{H^2}\\ge e^{-2\\gamma /3} \\Vert p\\Vert ^2 _{H^2}\\, .$ and obviously $\\Vert p_1\\Vert ^2 _{H^2}\\le \\Vert \\widetilde{p_1}\\Vert _{H^2}^2 = \\Vert p\\Vert ^2_{H^2}$ .", "Step 3.", "Sampling on $\\Lambda _n ^{(1)}$ .", "If $\\mu = (1+\\frac{\\gamma }{3n}) \\lambda \\in \\Lambda _n ^{(1)} \\subseteq \\mathbb {D}$ , then by (REF ) $|p_1(\\mu )| = |\\widetilde{p_1} (\\lambda )| \\le |p(\\lambda )| \\, ;$ if $\\mu = (1+\\frac{\\gamma }{3n}) / \\bar{\\lambda }\\in \\Lambda _n ^{(1)} \\subseteq \\mathbb {D}$ , then $|p_1(\\mu )| = |\\widetilde{p_1} (1/ \\bar{\\lambda } )| \\le |p(\\lambda )| \\, .$ In conclusion, we obtain $\\frac{1}{n} \\, \\sum _{\\mu \\in \\Lambda _n^{(1)}} |p_1(\\mu )|^2 \\le \\frac{1}{n} \\, \\sum _{\\lambda \\in \\Lambda _n} |p(\\lambda )|^2 \\, ,$ which was to be shown.", "Applying Lemma REF repeatedly, after $\\ell $ steps we construct a set $\\Lambda _n^{(\\ell )} \\subseteq \\mathbb {D}$ with the following properties: $\\Lambda _n ^{(\\ell )} & \\subseteq A_{\\gamma _\\ell / n} \\qquad \\text{ with } \\gamma _\\ell = \\tfrac{3}{4} \\gamma _{\\ell -1} \\, , \\\\\\widetilde{\\Lambda _n ^{(\\ell )}} &= \\widetilde{\\Lambda _n} \\, .", "$ Furthermore, for given $p \\in \\mathcal { P}_n$ we construct a sequence on polynomials $p_1, \\ldots , p_{\\ell } , \\ldots $ with decreasing norm $\\Vert p_\\ell \\Vert _{H^2}\\le \\dots \\le \\Vert p_1\\Vert _{H^2}\\le \\Vert p\\Vert _{H^2}$ , such that $\\Vert p_\\ell \\Vert ^2 _{H^2}\\ge e^{-\\tfrac{2}{3} \\gamma _{\\ell -1} }\\Vert p_{\\ell -1} \\Vert ^2 _{H^2}\\ge e^{-\\tfrac{2}{3} (\\gamma _{\\ell -1} + \\gamma _{\\ell -2} )} \\Vert p_{\\ell -2} \\Vert ^2 _{H^2}\\ge \\dots \\ge e^{-\\tfrac{2}{3}\\sum _{j=0}^{\\ell -1} \\gamma _{j}} \\Vert p \\Vert ^2 _{H^2}\\, .$ Since $\\gamma _j = 3\\gamma _{j-1}/4$ and $\\gamma _0 = \\gamma $ , we find $\\gamma _{\\ell } = \\big (\\frac{3}{4}\\big )^{\\ell }\\gamma $ and $\\tfrac{2}{3}\\sum _{j=0}^{\\ell -1} \\gamma _{j} \\le \\tfrac{2}{3} \\sum _{j=0}^{\\infty } (3/4)^j \\gamma \\le \\tfrac{8}{3}\\gamma \\, .$ It follows that always $\\Vert p_\\ell \\Vert ^2_{H^2}\\ge e^{-\\tfrac{8}{3}\\gamma } \\Vert p\\Vert ^2_{H^2}$ .", "We now let $\\ell $ tend to $\\infty $ .", "Since the sequence $p_\\ell $ is bounded in the finite-dimensional space $\\mathcal { P}_n$ , it contains a convergent subsequence such that $\\lim _{j\\rightarrow \\infty } p_{\\ell _j} = p_\\infty \\in \\mathcal { P}_n$ .", "Furthermore, by (REF ) the limiting polynomial $p_\\infty $ must be non-zero.", "By (REF ) every point in $\\Lambda _n ^{(\\ell )} $ converges to a point on the torus, precisely to the corresponding point in the projection $\\widetilde{\\Lambda _n}$ .", "Using (REF ), it follows that $\\frac{ \\frac{1}{n} \\, \\sum _{\\lambda \\in \\Lambda _n} |p(\\lambda )|^2}{\\Vert p\\Vert ^2 _{H^2}}\\ge e^{-8\\gamma /3 } \\, \\frac{ \\frac{1}{n} \\, \\sum _{\\lambda \\in \\widetilde{\\Lambda _n}} |p_\\infty (\\lambda )|^2}{\\Vert p_\\infty \\Vert ^2_{H^2}}\\, .$ Finally, we recall the assumption that the projected family $\\widetilde{\\Lambda _n}$ is a Marcinkiewicz-Zygmund family for the polynomials of degree $n$ in $L^2(\\mathbb {T})$ .", "Consequently, we obtain that $\\frac{1}{n} \\, \\sum _{\\lambda \\in \\widetilde{\\Lambda _n}} |p_\\infty (\\lambda )|^2 \\ge A \\Vert p_\\infty \\Vert ^2_{H^2}$ , which implies the corresponding sampling inequality for $p\\in \\mathcal { P}_{n}$ .", "The upper bound is proved almost exactly as the correspondingNote that [22] uses the weights $m_n=\\# \\Lambda _n$ instead of $k_n(\\lambda ,\\lambda ) \\asymp n$ .", "This does not affect the estimates.", "statement for $\\mathcal { P}_n$ in $L^2(\\mathbb {T})$ in Thm.", "9 of [22].", "Since $\\widetilde{\\Lambda }_n$ is a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $L^2(\\mathbb {T})$ by our assumption, [22] asserts that for every interval $I\\subseteq \\mathbb {T}$ of length $1/n$ we have $\\#(\\widetilde{\\Lambda _n} \\cap I) \\le C$ .", "Since $\\Lambda _n \\subseteq A_{\\gamma /n}$ , this condition implies that $\\# \\Lambda _n \\cap B(z,1/n) \\le C^{\\prime }$ for all $z\\in A_{\\gamma /n}$ .", "This geometric condition now yields the upper bound $\\tfrac{1}{n} \\sum _{\\lambda \\in \\Lambda _n} |p(\\lambda )|^2 \\lesssim \\Vert p\\Vert _{H^2}^2$ for all $p\\in \\mathcal { P}_n$ precisely as in [22].", "Indeed that proof uses the submean-value property and the extension to $A_{\\gamma /n}$ .", "Theorem REF shows that to every Marcinkiewicz-Zygmund family for polynomials on $\\mathbb {T}$ we can associate Marcinkiewicz-Zygmund families in $H^2(\\mathbb {D})$ by moving points from the boundary $\\mathbb {T}=\\partial \\mathbb {D}$ into a carefully controlled annulus $C_{\\gamma /n}\\subseteq \\mathbb {D}$ .", "The following example investigates the role of points in the interior of $\\mathbb {D}$ for Marcinkiewicz-Zygmund families.", "Example.", "We construct an example of a Marcinkiewicz-Zygmund family $(\\Lambda _n)$ for polynomials $\\mathcal { P}_n$ in $H^2(\\mathbb {D})$ , so that $(\\Lambda _n \\cap C_{\\gamma /n})$ is not a Marcinkiewicz-Zygmund family in $H^2$ and the projection $(\\widetilde{\\Lambda _n})$ is not a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $L^2(\\mathbb {T})$ .", "Let $\\gamma >0$ , $\\alpha _n >0$ , and $\\Lambda _n = \\lbrace (1-\\tfrac{\\gamma }{n}) e^{2\\pi i k/n} : k=0, \\dots , n-1 \\rbrace \\cup \\lbrace \\alpha _n e^{2\\pi i /n^2} \\rbrace \\, .$ (i) If $\\alpha _n < 1-\\tfrac{\\gamma }{n}$ , then $\\# (\\Lambda _n \\cap C_{\\gamma /n})= n < \\mathrm {dim}\\, \\mathcal { P}_n$ and thus $(\\Lambda _n\\cap C_{\\gamma /n})$ cannot be a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $H^2$ .", "(ii) The projected family $\\widetilde{\\Lambda _n} = \\lbrace e^{2\\pi i k/n} : k=0, \\dots , n-1 \\rbrace \\cup \\lbrace e^{2\\pi i /n^2}\\rbrace $ is not a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $L^2(\\mathbb {T})$ .", "We choose $p(z) = z^n -1$ with $\\Vert p\\Vert ^2 _{H^2}=2$ .", "Then $p(e^{2\\pi i k/n}) = 0$ and $|p(e^{2\\pi i /n^2})| = |e^{2\\pi i /n} -1| \\lesssim 1/n $ , so that $\\frac{1}{n+1} \\sum _{\\lambda \\in \\widetilde{\\Lambda _n}} |p(\\lambda )|^2\\lesssim \\frac{1}{n^3} \\, ,$ violating the sampling inequality for large $n$ .", "(iii) However, $(\\Lambda _n)$ is a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $H^2(\\mathbb {D})$ .", "To see this, we consider the modified set $\\Lambda _n^{\\prime } = \\lbrace (1-\\tfrac{\\gamma }{n}) e^{2\\pi i k/n} : k=0, \\dots , n-1 \\rbrace \\cup \\lbrace 0 \\rbrace $ and then use a perturbation argument.", "We write $p\\in \\mathcal { P}_n$ as $p(z) = p(0) + z \\widetilde{p}(z)$ for a unique $\\widetilde{p} \\in \\mathcal { P}_{n-1}$ .", "Then $\\Vert p\\Vert ^2 = |p(0)|^2 +\\Vert \\widetilde{p}\\Vert ^2$ .", "Let $q(z) = \\widetilde{p}((1-\\frac{\\gamma }{n}) z)$ , then by  (REF ) $\\Vert q\\Vert _{H^2}^2 \\ge e^{-4\\gamma } \\Vert \\widetilde{p}\\Vert _{H^2}^2$ .", "Calculating the norm of $q$ by sampling, we obtain $\\Vert \\widetilde{p}\\Vert ^2 \\asymp \\Vert q\\Vert ^2 &= \\frac{1}{n} \\sum _{j= 0}^{n-1}|q(e^{2\\pi i j/n})|^2 =\\frac{1}{n} \\sum _{j= 0}^{n-1}|\\widetilde{p}\\big ((1-\\frac{\\gamma }{n})e^{2\\pi i j/n}\\big )|^2 \\\\&= \\frac{1}{n(1-\\frac{\\gamma }{n})^2} \\sum _{j=0}^{n-1}|(1-\\frac{\\gamma }{n}) e^{2\\pi i j/n}\\widetilde{p}((1-\\frac{\\gamma }{n})e^{2\\pi i j/n})|^2 \\, .$ If $n\\ge 2\\gamma $ we have $\\Vert q\\Vert ^2 \\le \\frac{8}{n} \\sum _{j= 0}^{n-1}\\Big (|p((1-\\frac{\\gamma }{n})e^{2\\pi i j/n})|^2 + |p(0)|^2\\Big )$ Since by (REF ) $k_n(\\lambda ,\\lambda ) \\asymp n$ for $\\lambda = (1-\\tfrac{\\gamma }{n}) e^{2\\pi i j/n}$ , and $k_n(0,0) = 1$ , the above inequality states that $\\Vert q\\Vert ^2 \\lesssim \\sum _{\\lambda \\in \\Lambda _n^{\\prime }} \\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )}.$ and finally $\\Vert p\\Vert ^2 \\le \\Vert q\\Vert ^2 + |p(0)|^2 \\lesssim \\sum _{\\lambda \\in \\Lambda _n^{\\prime }} \\frac{|p(\\lambda )|^2}{k_n(\\lambda , \\lambda )}.$ Thus $(\\Lambda _n^{\\prime })$ is a Marcinkiewicz-Zygmund family for $\\mathcal { P}_n$ in $H^2(\\mathbb {D})$ .", "The small perturbation $0\\rightarrow \\frac{1}{n^2} e^{2\\pi i /n^2}$ is of order $1/n^2$ and thus preserves the sampling inequality As already mentioned, the standard definition of sampling sequences is vacuous in the Hardy space.", "Thus Thomas in [27] proposed an alternative definition of sampling in terms of the non-tangential maximal function $M_\\Lambda $ .", "$M_\\Lambda (f) (e^{i\\theta }) := \\sup _{\\Gamma (e^{i\\theta })\\cap \\Lambda } |f|,$ where $\\Gamma (e^{i\\theta }) =\\lbrace z\\in \\mathbb {D}: \\tfrac{|z-e^{i\\theta }|}{1-|z|} < 1+ \\alpha \\rbrace $ is a non-tangential Stolz angle at the point $e^{i\\theta }$ .", "A set $\\Lambda $ is called PT-samplingFor Pascal Thomas.", "in $H^2(\\mathbb {D})$ if $\\Vert M_\\Lambda (f)\\Vert _{L^2} \\gtrsim \\Vert f\\Vert _2$ for all $f\\in H^2(\\mathbb {D})$ .", "Thomas proves that a set $\\Lambda $ is sampling for $H^2$ if and only if it norming for $H^\\infty $ which was geometrically described by Brown, Shields and Zeller in [2] by the property that the nontangential limit set of $\\Lambda $ must be of full measure in $\\mathbb {T}$ .", "This alternative notion of sampling was inspired by a corresponding alternative definition of interpolating sequences in the Hardy space by Bruna, Nicolau and Øyma [3].", "The relation between Marcinkiewicz-Zygmund families and PT-sampling sets for $H^2$ is not clear.", "We only mention that there is no analog of Theorem REF for PT-sampling: Consider the Marcinkiewicz-Zygmund family $\\Lambda _n$ in the example (REF ).", "Its weak limit in $\\mathbb {D}$ is just $\\lbrace 0\\rbrace $ , which is obviously not PT-sampling.", "Its weak limit in $\\mathbb {C}$ is $\\lbrace 0\\rbrace \\cup \\partial \\mathbb {D}$ , and this not even covered by the definition of PT-sampling.", "We have not pursued this aspect further." ] ]

[ [ "Search for globular clusters associated with the Milky Way dwarf\n galaxies using Gaia DR2" ], [ "Abstract We report the result of searching for globular clusters (GCs) around 55 Milky Way satellite dwarf galaxies within the distance of 450 kpc from the Galactic Center except for the Large and Small Magellanic Clouds and the Sagittarius dwarf.", "For each dwarf, we analyze the stellar distribution of sources in Gaia DR2, selected by magnitude, proper motion, and source morphology.", "Using the kernel density estimation of stellar number counts, we identify eleven possible GC candidates.", "Crossed-matched with existing imaging data, all eleven objects are known either GCs or galaxies and only Fornax GC 1-6 among them are associated with the targeted dwarf galaxy.", "Using simulated GCs, we calculate the GC detection limit $M_{\\rm V}^{\\rm lim}$ that spans the range from $M_{\\rm V}^{\\rm lim} \\sim -7$ for distant dwarfs to $M_{\\rm V}^{\\rm lim} \\sim 0$ for nearby systems.", "Assuming a Gaussian GC luminosity function, we compute that the completeness of the GC search is above 90 percent for most dwarf galaxies.", "We construct the 90 percent credible intervals/upper limits on the GC specific frequency $S_{\\rm N}$ of the MW dwarf galaxies: $12 < S_{\\rm N} < 47$ for Fornax, $S_{\\rm N} < 20$ for the dwarfs with $-12 < M_{\\rm V} < -10$, $S_{\\rm N} < 30$ for the dwarfs with $-10 < M_{\\rm V} < -7$, and $S_{\\rm N} < 90$ for the dwarfs with $M_{\\rm V} > -7$.", "Based on $S_{\\rm N}$, we derive the probability of galaxies hosting GCs given their luminosity, finding that the probability of galaxies fainter than $M_{\\rm V} = -9$ to host GCs is lower than 0.1." ], [ "Introduction", "Globular clusters (GCs) are some of the oldest luminous observable objects with ages comparable to the age of the Universe .", "Characterized by being compact and bright, GCs typically have masses of $10^4$  – $10^6 M_{}$ , luminosities of $M_{\\rm V}= -5$ to $-10$ , and sizes of a few parsecs , .", "GCs might have played an important role in the early formation of galaxies, and they could have been the potential drivers of cosmic reionization despite the issues with the escape fraction of ionizing radiation , .", "However, the formation of GCs themselves remains an open question in astrophysics , , , , .", "For detailed reviews of GCs, we refer readers to , , and .", "In the Milky Way (MW), the number of known GCs has increased to around 150 , since the first one was discovered in 1665 by Abraham Ihle.", "While some of these GCs that are more concentrated around the Galactic Center are believed to have been formed in-situ , , the ones in the outskirts are believed to have been accreted together with their parent dwarf galaxies , , , , which were destroyed by tides.", "Some of the GCs however can still be found within the MW satellites themselves, offering a window on the formation of GCs in dwarf galaxies.", "The three most luminous MW satellites, the Large and Small Magellanic Clouds (LMC and SMC) and the Sagittarius dwarf spheroidal galaxy, have large populations of GCs , , , .", "In particular, the clusters of the Sagittarius dwarf are spread out along the stellar stream , , , , and the SMC has a large population of star clusters in general but few of them are classically old GCs.", "The only other two MW satellite galaxies known to possess GCs are the Fornax dwarf spheroidal galaxy which is the fourth most luminous MW satellite with six GCs, and the Eridanus 2, an ultra-faint system containing a faint cluster , .", "The fact that some GCs in the MW still have been found until recently , , motivates us to further search for possibly missing ones.", "Intuitively, faint GCs within dwarf galaxies are more likely to have been missed, especially when located within luminous dwarf galaxies where the ground-based data can be crowded e.g.", "Fornax 6.", "Instead of looking for this kind of objects by chance, we apply the systemic overdensity searching algorithm (which will be explained in Section ) to the areas around the MW satellite galaxies within the distance of 450 kpc except for the three most luminous ones: the LMC, the SMC, and the Sagittarius dwarf.", "That is, we target the areas where GCs are likely to lurk from previous inspections of deep imaging to look for overdensities in dense dwarfs.", "Focusing on a small area of the sky, a targeted search is less computationally expensive so that it can afford a lower detection threshold.", "For each targeted area, we investigate the stellar distribution in the Gaia data to detect possible GC candidates (see Section REF for more detail about Gaia and the dataset).", "Thanks to the high angular resolution that exceeds most ground-based surveys, Gaia allows us to detect previously missed objects that are not well resolved or missed by ground-based searches.", "For instance, has found star clusters in Gaia that were missed by previous searches.", "We organize the paper as follows.", "In Section , we explain the methodology with more detail about the Gaia data, sample selection, and kernel density estimation procedure.", "In Section , we demonstrate the main results of the detection.", "In Section , we discuss the limit and completeness of the detection, the inferred specific frequency of GCs, and the derived probability of dwarfs to host GCs based on our findings.", "In Section , we conclude the paper.", "The space-based astrometric mission Gaia was launched by the European Space Agency in 2013 and started the whole-sky survey in 2014 .", "Released in 2018, the second Gaia data release (Gaia DR2) contains the data collected during the first 22 months of the mission and has approximately 1.7 billion sources with 1.3 billion parallaxes and proper motions.", "Gaia DR2, therefore, provides high-resolution stellar distribution in the MW for us to look for possibly missing GCs around the MW dwarf galaxies.", "The overall scientific validation of the data is described in [2].", "The entire analysis of this paper utilizes the GAIA_SOURCE catalog of Gaia DR2 , particularly the position ra and dec ($\\alpha $ and $\\delta $ ), the proper motion (PM) pmra and pmdec ($\\mu _{\\alpha }$ and $\\mu _{\\delta }$ ), the G-band magnitude phot_g_mean_mag (G), and the value of the astrometric_excess_noise parameter ($\\epsilon $ ).", "contains the detail on the contents and the properties of this catalog.", "We use this dataset to identify stellar density peaks as possible candidates of GCs around in the vicinity and inside nearby dwarf galaxies.", "Throughout the whole paper, we apply two main selection cuts on the Gaia catalog.", "The first selection is $17 < {\\rm G}< 21.$ The faint-magnitude cut ${\\rm G}< 21$ approximately corresponds to the faint-end limit of Gaia DR2; reported that only 4 percent of the sources are fainter than ${\\rm G}= 21$ and those sources lack PMs and parameters.", "There are two reasons for the bright-magnitude cut ${\\rm G}> 17$ .", "The first reason to get rid of the bright stars is that the foreground contamination dominates at bright magnitudes.", "Conversely, the expected rapid rise of the stellar luminosity function for the majority of GCs and dwarf galaxies at reasonable distances from the Sun at ${\\rm G}> 17$ results in the majority of stars being fainter than ${\\rm G}= 17$ .", "The other reason is that most bright GCs with large numbers of ${\\rm G}< 17$ stars would have likely been detected already.", "The second selection criterion is $\\ln \\epsilon < 1.5 + 0.3 \\max \\lbrace {\\rm G}- 18, 0\\rbrace .$ This cut is used to reject potentially extended sources , .", "Another optional selection that we use to further clean the source list is based on the PM, with the goal of removing sources whose PMs are different from the mean PM of a given targeted dwarf galaxy, as these sources are less likely to be member stars of the given dwarf.", "For each targeted dwarf, we exclude stars with PMs ($\\mu _{\\alpha }$ , $\\mu _{\\delta }$ ) differing from a systemic PM of the dwarf ($\\mu _{\\alpha }^{\\rm dwarf}$ , $\\mu _{\\delta }^{\\rm dwarf}$ ) by more than three times the PM uncertainty ($\\sigma _{\\mu _{\\alpha }}$ , $\\sigma _{\\mu _{\\delta }}$ ).", "That is, only the stars satisfying $\\sqrt{ (\\mu _{\\alpha } - \\mu _{\\alpha }^{\\rm dwarf})^2+ (\\mu _{\\delta } - \\mu _{\\delta }^{\\rm dwarf})^2}< 3 \\sqrt{\\sigma _{\\mu _{\\alpha }}^2 + \\sigma _{\\mu _{\\delta }}^2}$ survive after the PM selection.", "For example, Figure REF shows the Gaia sources around the Fornax dwarf before and after the PM selection in Equation REF .", "The source distribution in PM space in the left panel shows that there are many foreground sources with PMs that are 10 – 100 $\\rm mas\\,yr^{-1}$ different from the PM of the dwarf.", "This PM selection is thus applied to remove this kind of contamination; the sources colored in orange survive after the selection.", "It is worth noting that the PM uncertainty of the studied dwarfs is around the order of $10^3$  – $10^5$  $\\rm km\\,s^{-1}$ (see Table REF ) which is much larger than the typical velocity dispersion of dwarf galaxies around the order of 10 $\\rm km\\,s^{-1}$ or 0.02 $\\rm mas\\,yr^{-1}$ if at 100 kpc, so the survived sources under this PM selection still have a fairly large range of internal space velocity.", "To investigate the PM selection for the stars that are more likely to be member stars of the Fornax dwarf, we draw a lasso with the black dashed lines to roughly distinguish the member stars in the red-giant branch of Fornax from the other stars in the color-magnitude diagram in the right panel.", "For the stars that are likely to be member stars inside of the lasso, 91 percent of the sources survive after the PM selection, whereas most of the sources outside of the lasso are excluded.", "Moreover, in the left panel of Figure REF , the stellar distribution after the PM selection retains the shape of the Fornax dwarf." ], [ "Kernel density estimation", "Convolving the spatial distribution of the data with various kernels is a common approach to identify the excess number of stars associated with a satellite or clusters in imaging data.", "The density is calculated by convolving all the data points interpreted as delta functions with different kernels, e.g.", "a moving average in , two circular indicator functions in and Gaussian kernels in , , and .", "To identify star clusters in dwarf galaxies, we use the kernel density estimation on the stellar distribution, while assuming the Poisson distribution of stellar number counts.", "We obtain the distribution of stars $\\Sigma (x, y) = \\sum _i \\delta (x - x_i, y - y_i)$ where $(x_i, y_i)$ is the position of the $i^{\\rm th}$ star on the local coordinates which takes care of the projection effect.In the algorithm, we always divide a targeted area into small patches with a side of $0.5^\\circ $ .", "For each patch centered at $(\\alpha _0, \\delta _0)$ , we define the local coordinates $(x, y)$ with the origin of $(x_0, y_0) = (\\alpha _0, \\delta _0)$ .", "Since the patch is very small, we approximate the projection effect as $x \\approx (\\alpha - \\alpha _0) \\cos \\delta _0$ and $y = \\delta - \\delta _0$ .", "Using the circular indicator function with a given radius $R$ defined as $\\mathbb {1} \\left( x, y; R \\right) = \\left\\lbrace \\begin{matrix}\\ 1 \\ {\\rm if} \\ x^2 + y^2 \\le R^2 \\\\\\ 0 \\ {\\rm otherwise} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{matrix}\\right.$ , we define the inner kernel $K_{\\rm in} (x, y; \\sigma _1) = \\mathbb {1} \\left( x, y; \\sigma _1 \\right)$ , where $\\sigma _1$ corresponds to the scale of GCs which is 3, 5, or 10 pc.", "We then convolve $\\Sigma (x, y)$ with $K_{\\rm in} (x, y; \\sigma _1)$ to estimate the number density of stars on the scale of $\\sigma _1$ as $\\Sigma _{\\rm in}(x, y) = \\Sigma (x, y) * K_{\\rm in} (x, y; \\sigma _1).$ Defining the outer kernel $K_{\\rm out} (x, y; \\sigma _1, \\sigma _2) = \\mathbb {1} \\left( x, y; \\sigma _2 \\right) - \\mathbb {1} \\left( x, y; 2\\sigma _1 \\right)$ , we convolve $\\Sigma (x, y)$ with $K_{\\rm out} (x, y; \\sigma _1, \\sigma _2)$ as $\\Sigma _{\\rm out}(x, y) = \\Sigma (x, y) * K_{\\rm out} (x, y; \\sigma _1, \\sigma _2)$ to estimate the number density of stars on the annular area of radius between $2\\sigma _1$ and $\\sigma _2$ , where $\\sigma _2 > 2 \\sigma _1$ and $\\sigma _2$ corresponds to either the angular scale of parent dwarf galaxy or a fixed angular scale of $0.5^\\circ $ (see more detail in the next paragraph).", "We estimate the expected background number density within the inner kernel from $\\Sigma _{\\rm out}(x, y)$ through the ratio of the inner and outer areas $\\Sigma _{\\rm bg} (x, y) = \\frac{ \\sigma _1^2 }{\\sigma _2^2 - (2\\sigma _1)^2} \\, \\Sigma _{\\rm out}(x, y).$ We convert the tail probability of Poisson into the z-score of the standard normal distribution to evaluate the significance as $S(x, y) = F_{\\rm N \\left( 0,1 \\right) }^{-1} \\left( F_{{\\rm Poi} \\left( \\Sigma _{\\rm bg} \\left( x, y \\right) \\right) } \\left( \\Sigma _{\\rm in}\\left( x, y \\right) \\right) \\right),$ where $F$ is the cumulative distribution function.", "As an example, Figure REF shows the original two-dimensional histogram of the sources around the Fornax dwarf in the left panel and the significance map of that stellar distribution in the right panel.", "According to the significance map, we identify positive detection with significance higher than a certain significance threshold.", "For nearby pixels with significance higher than the threshold, we merge them as one single positive detection if the radial distance between the pixels is shorter than the size of the inner kernel.", "We assign the maximum significance on the merged pixels as the detected significance and use the center of mass coordinates of the merged pixels as the detected position.", "The main reason for $\\sigma _2$ in step (iii) corresponding to either the angular scale of parent dwarf galaxy or the fixed angular scale of $0.5^\\circ $ is that the kernel density estimates are biased in crowded areas, which may lead to missing objects around big dwarfs.", "Given a dwarf with a half-light radius of $r_h$ , $\\sigma _2$ is chosen to be $0.5 r_h$ or $0.5^\\circ $ for pixels inside ($r < r_h$ ) or outside ($r > r_h$ ) of the dwarf respectively, where $r$ is the distance from the position of any pixel to the center of the dwarf.", "The latter large $\\sigma _2$ of $0.5^\\circ $ is to take care of the sparse outskirts of the dwarf.", "Besides, when dealing with the pixels outside of the dwarf, we exclude the effect of the pixels inside of the dwarf ($r_h < r < r_h + 0.5^\\circ $ ) because the relatively high number density of stars in the dwarf will lead to over-estimate of $\\Sigma _{\\rm out}(x, y)$ which will suppress the background estimate too much later." ], [ "Results", "The objective of the paper is to search for possibly missing GCs around the MW satellites by identifying stellar over-densities with the searching algorithm described in Section REF .", "The list of dwarf galaxies considered in this paper was created by selecting dwarf galaxies within the distance of 450 kpc from the Galactic Center with the exception of the LMC, the SMC, and the Sagittarius dwarf.", "The dwarf list in Table REF summarizes all 55 targeted dwarfs investigated in the paper and their properties.", "The reason to exclude the three most massive satellites of the MW is that their relatively large sizes will lead to a huge portion of the sky to be searched, which conflicts with our goal of conducting a targeted search.", "In the construction of the dwarf galaxy list, we use the data from the compilation and include some of the recent discoveries: Antlia 2 , Aquarius 2 , Bootes 3 , Carina 2 , Carina 3 , Cetus 3 , Crater 2 , and Virgo I .", "For each targeted dwarf, we search the area within the radius of ${\\rm min} \\lbrace 8^{\\circ }, R_{\\rm vir} \\rbrace $ , where $R_{\\rm vir}$ is the virial radius of a $10^9 M_{}$ halo (at the distance of 100 kpc this corresponds to $10^{\\circ }$ ).", "We choose the inner kernel sizes of $\\sigma _1 = 3$ , 5, and 10 pc which covers the range of physical sizes of a typical GC .", "We run the searching algorithm for each inner kernel size on the Gaia sources after the selections of Equation REF if the dwarf has known measured PM (see Table REF ), Equation REF , and Equation REF .", "To balance the completeness of search with the number of false positives, we define two thresholds for identifying possible candidates: a significance threshold $S > 5$ and the limit of the number of stars inside the inner kernel $\\Sigma _{\\rm in} > 10$ .", "For $S > 5$ , as the z-score of the standard normal distribution, its false alarm probability is of the order of $10^{-7}$ .$\\int _{5}^{\\infty } \\frac{1}{\\sqrt{2\\pi }} e^{-0.5 z^2} dz \\sim 10^{-7}$ Assuming a targeted dwarf at the distance of 100 kpc with a searching radius of $8^{\\circ }$ , the total number of spatial pixels is around the order of $10^8$ .The searching radius of $8^{\\circ }$ corresponds to $\\sim 10^4$  pc at the distance of 100 kpc so the searching area is $\\sim 10^8$  pc$^2$ .", "With the spatial resolution $\\sim 1$  pc$^2$ , the total number of pixels is then $\\sim 10^8$ .", "With the false alarm probability $\\sim 10^{-7}$ on the targeted area of $\\sim 10^8$ pixels, the number of expected false positives is around the order of 10.", "Moreover, we apply the other threshold, $\\Sigma _{\\rm in}> 10$ , to prevent a large number of false positives for the pixels with very low background number density.", "For example in Figure REF , it is noticeable that the significance can easily be large in the area with very sparse stellar density even if only a handful of stars are detected in the inner kernel.", "These pixels typically have $\\Sigma _{\\rm out}< 1$ where the significance estimator breaks down due to the very low rate parameter of Poisson.", "Hence by applying $\\Sigma _{\\rm in} > 10$ , we effectively increase the threshold on $S$ for pixels with $\\Sigma _{\\rm out}< 1$ , e.g.", "the threshold is $S=5.6$ for $\\Sigma _{\\rm out}= 1$ and $S=8.9$ for $\\Sigma _{\\rm out}= 0.1$ .", "This avoids the detection of false-positive peaks due to Poisson noise in the $\\Sigma _{\\rm bg}$ estimates, binary stars, or unresolved galaxies in Gaia that are expected to show more clustering than stars.", "Particularly for binary star systems or unresolved galaxies, the pairs of them are much more likely to occur because they are more correlated; thus they are likely to reach 5 significance and cause false positives.", "After running the searching algorithm on all 55 targeted dwarfs, we identify eleven stellar over-density candidates, based on the highest detected significance of each candidate if it is detected multiple times with different searching parameters.", "Cross-matched with the simbad database , all eleven candidates are known objects.", "Nine of them are known GCs: Fornax GC 1 – 5 , , Fornax GC 6 , , , , , Messier 75 , NGC 5466 , and Palomar 3 .", "The other two of them are known galaxies: the Leo I dwarf spheroidal galaxy and the Sextans A dwarf irregular galaxy .", "We remark that Leo I is found when searching for overdensities near Segue I, as they are close to each other in the sky.", "Table REF summarizes the eleven known objects and their detected positions (RA and Dec), significance values ($S$ ), and inner kernel sizes ($\\sigma _1$ ).", "Figure REF shows the stellar distribution of Gaia sources for the six GCs of Fornax and the corresponding images from DES DR1 [1] made with the HiPS .", "The yellow circles show the inner kernel of 10 pc (note that it happens to be that all the significance values with 10 pc are greater than with 3 or 5 pc in our detection of the nine GCs).", "Most of those known GCs are detected with the strong significance of $S > 7$ except for Fornax GC 6 with $S = 5.2$ , which emphasizes that our algorithm can detect GCs from the regions of high stellar density such as Fornax GC 6.", "Figure REF further indicates that the significance values are reasonable: bright GCs located at low-density areas (e.g.", "Fornax GC 1 and 2) have high significance ($S > 8$ ) and faint GCs located at high-density areas (e.g.", "Fornax GC 6) have low significance ($S \\sim 5$ ).", "However, we are aware of missing the ultra-faint GC in the Eridanus 2 in our detection, which we will further discuss later in Section REF ." ], [ "Detection limit in V-band magnitude", "In this section, we will demonstrate how we carry out the detection limit in V-band magnitude $M_{\\rm V}^{\\rm lim}$ of the search for each targeted dwarf, which indicates that GCs brighter than $M_{\\rm V}^{\\rm lim}$ are detectable in our search.", "To do so, we generate 1000 mock GCs with luminosity in the range of $-10 < M_{\\rm V}< 0$ assuming the age $=12$  Gyr and $\\rm [Fe/H] = -2$ of the stellar populations.", "Sampling the stars of each GC population according to the log-normal initial mass function in , we interpolate the isochrone based on the parsec isochrone , then utilizing the isochrones of all the mock GC stellar populations to carry out the detection limit for each targeted dwarf as follows.", "Given a targeted dwarf, to compute the detection limit $M_{\\rm V}^{\\rm lim}$ , we first calculate the number of observable stars of each mock GC satisfying the G-band selection by counting the number of stars within $17 < {\\rm G}<21$ according to its isochrone at the distance of the dwarf.", "Based on the number of observable stars, we compute the number of stars of each GC within the inner kernel size $\\sigma _1$ as $\\Sigma _{\\rm in}^{\\rm obs}= f \\left( \\sigma _1;\\, r_{\\rm h} = 3 {\\rm pc} \\right) \\times \\left( {\\rm total\\ number\\ of\\ observable\\ stars} \\right),$ where $f \\left( \\sigma _1;\\, r_{\\rm h} \\right) = \\frac{\\sigma _1^2}{\\sigma _1^2 + r_{\\rm h}^2}$ is the fraction of the number of stars within the radius of $\\sigma _1$ according to the Plummer model of 2D surface density profile of a GC with a half-light radius $r_{\\rm h} = 3 {\\rm pc}$ .", "With $\\Sigma _{\\rm in}^{\\rm obs}$ of all the mock GCs at hand, we then use a linear best fit to describe the relation between $\\log _{10} \\left( \\Sigma _{\\rm in}^{\\rm obs}\\right)$ and $M_{\\rm V}$ of the GCs.", "According to the maximum background estimate of the given dwarf, we know the threshold number of stars $\\Sigma _{\\rm in}^{\\rm lim}$ to be observed to reach 5 significance.", "By comparing $\\Sigma _{\\rm in}^{\\rm lim}$ to the best fit, we can obtain the detection limit $M_{\\rm V}^{\\rm lim}$ for the given targeted dwarf.", "We take the Fornax dwarf as an example of the procedure of injection of mock GCs.", "In the left panel of Figure REF , we show the isochrone of a single mock GC of $M_{\\rm V}= -8$ at the distance of Fornax and the stars in the white area are observable within our Gaia G-band cut.", "By counting the number of stars satisfying $17 < G <21$ corrected by the fraction of stars located within the inner kernels, we know the number of observable stars $\\Sigma _{\\rm in}^{\\rm obs}$ for the given mock GC.", "Applying the calculation of $\\Sigma _{\\rm in}^{\\rm obs}$ for each mock GC, we show the relation between $\\Sigma _{\\rm in}^{\\rm obs}$ and $M_{\\rm V}$ for all the mock GCs in the right panel of Figure REF .", "The green dashed line shows the threshold number of stars $\\Sigma _{\\rm in}^{\\rm lim}$ to reach $S=5$ according to the maximum background estimate of Fornax; that is, the GCs above the green dashed line are expected to be detectable.", "Fitting the relation between $\\log _{10} \\left( \\Sigma _{\\rm in}^{\\rm obs}\\right)$ and $M_{\\rm V}$ with a linear best fit as shown in the yellow line, we solve the detection limit $M_{\\rm V}^{\\rm lim}$ by finding the value of $M_{\\rm V}$ satisfying the fit at the value of $\\Sigma _{\\rm in}^{\\rm lim}$ (the green dashed line).", "The red dashed line indicates the derived $M_{\\rm V}^{\\rm lim}$ and the GCs brighter than $M_{\\rm V}^{\\rm lim}$ in the white area are thus detectable in our search.", "It is worth noting that the Gaia magnitude limit is brighter than ${\\rm G}= 21$ in some areas of the sky, which will decrease $\\Sigma _{\\rm in}^{\\rm obs}$ if it happens in our targeted area, resulting in a brighter $M_{\\rm V}^{\\rm lim}$ .", "Repeating the same calculation of $M_{\\rm V}^{\\rm lim}$ for all the targeted dwarfs, we obtain the detection limits of the dwarfs and show the comparison of the derived $M_{\\rm V}^{\\rm lim}$ to the distances and the luminosities of the dwarfs in Figure REF .", "In the left panel, there is an obvious trend that the $M_{\\rm V}^{\\rm lim}$ are fainter for the dwarfs that are closer because the injected $\\Sigma _{\\rm in}^{\\rm obs}$ of the GCs for these dwarfs with small distance modulus is typically larger than that of the dwarfs with large distance modulus.", "On the other hand in the right panel, the relation between $M_{\\rm V}^{\\rm lim}$ and $M_{\\rm V}$ of the dwarfs is more scattered yet there is a slight trend of fainter $M_{\\rm V}^{\\rm lim}$ for the fainter dwarfs.", "This is likely because the faint dwarfs, compared to the bright ones, tend to have less-crowded stellar distributions and hence lower thresholds $\\Sigma _{\\rm in}^{\\rm lim}$ to reach 5 significance.", "To sum up, the faint $M_{\\rm V}^{\\rm lim}$ for the close dwarfs or the faint dwarfs is reasonable because the ability of dwarfs to hide GCs from our detection is intuitively weaker for the dwarfs that are closer or fainter.", "It is also worth noting that most of the time $M_{\\rm V}^{\\rm lim}$ with $\\sigma _1 = 10$  pc is the faintest, $M_{\\rm V}^{\\rm lim}$ with $\\sigma _1 = 5$  pc is the intermediate, and $M_{\\rm V}^{\\rm lim}$ with $\\sigma _1 = 3$  pc is the brightest mainly because the fractions of stars observed within the inner kernels are around 0.9, 0.7 and 0.5 for $\\sigma _1 = 10$ , 5, and 3 pc respectively according to the Plummer model.", "That is, the low $\\Sigma _{\\rm in}^{\\rm obs}$ due to the small fraction for small $\\sigma _1$ makes the faint GCs less likely to meet 5 significance, thus resulting in a bright $M_{\\rm V}^{\\rm lim}$ .", "Figure: The completeness gg of the GC search for all targeted dwarfs with three GCLFs: the Gaussian of 𝒩(-7.4,1.2 2 )\\mathcal {N}({-7.4},\\,{1.2}^2) and 𝒩(-6,1.2 2 )\\mathcal {N}({-6},\\,{1.2}^2) and the evolved Schechter in .", "Left: Completeness versus the distance of the dwarfs.", "Right: Completeness versus dwarf galaxy luminosity." ], [ "Completeness of the search", "With the limiting magnitudes of GC detection at hand, we can calculate the completeness of the search according to the typical GC luminosity function (GCLF).", "In this section, we will calculate the completeness factor $g$ with three different GCLFs: (a) the typical MW GCLF in : a Gaussian distribution with a peak at $M_{\\rm V}= -7.4$ and a standard deviation of 1.2, $\\mathcal {N}({-7.4},\\,{1.2}^2)$ , (b) the evolved Schechter function in with a peak at $M_{\\rm V}\\sim -7.4$ , and (c) a presumed Gaussian distribution with a peak at $M_{\\rm V}= -6$ and a standard deviation of 1.2, $\\mathcal {N}({-6},\\,{1.2}^2)$ .", "We calculate $g$ by evaluating the cumulative distribution functions of those GCLFs at $M_{\\rm V}^{\\rm lim}$ based on the search with $\\sigma _1 = 10$  pc thanks to its better detecting sensitivity compared to $\\sigma _1 = 3$ and 5 pc (all the detected objects with the highest significance are detected with $\\sigma _1 = 10$  pc in Section ).", "We begin with the GCLF in (a); in the MW, the GCLF is approximately a Gaussian distribution of $\\mathcal {N}({-7.4},\\,{1.2}^2)$ .", "With this MW GCLF, we compute the completeness factor $g$ and show them in the blue points in Figure REF .", "The completeness of the search is higher than 90 percent for most of the dwarfs and around 70 percent for the lowest three, Eridanus 2, Leo T, and Phoenix.", "This high completeness is a consequence of $M_{\\rm V}^{\\rm lim}> -7$ for all the dwarfs; that is, the detection limits are fainter than the peak magnitude of the MW GCLF.", "Besides, as a result of the trend of brighter $M_{\\rm V}^{\\rm lim}$ for the farther targeted dwarfs in the left panel of Figure REF , the completeness gets lower for the dwarfs that are more distant.", "In addition, described that the dispersion of GCLF can be as small as 0.5 for small dwarfs.", "Calculating the completeness with this GCLF, we find that the result is almost the same as that of the MW GCLF.", "Compared to the Gaussian MW GCLF peaking at $M_{\\rm V}= -7.4$ , the evolved Schechter function with a similar peak magnitude proposed in can describe the GCLF well too, particularly taking good care of the low-mass faint GCs.", "We compute the completeness factor $g$ with this GCLF as shown in the green points in Figure REF , finding that the difference in $g$ with this GCLF from the traditional Gaussian is less than 5 – 10 percent lower.", "The reason for the larger difference ($\\sim 10$ percent) in $g$ of the two GCLFs for the targeted dwarfs that are more distant than 100 kpc is that the probability density of the evolved Schechter function is higher than that of the Gaussian MW GCLF in the faint end.", "Thus as these dwarfs have brighter $M_{\\rm V}^{\\rm lim}$ than the close dwarfs, their cumulative distribution functions at $M_{\\rm V}^{\\rm lim}$ of the evolved Schechter GCLF are lower than that of the Gaussian MW GCLF.", "On the other hand, for the dwarfs that are closer than 100 kpc, $M_{\\rm V}^{\\rm lim}$ is much fainter than the peaks of the two GCLFs so the corresponding $g$ approaches 1 for both GCLFs.", "So far, we have assumed the GC population for all the dwarfs follows the GCLFs based on the results from bright galaxies, the Gaussian in and the evolved Schechter in .", "These two GCLFs have similar peaks but different shapes: the evolved Schechter one extends more toward the faint end to account for faint GCs (see the black curves in Figure REF ).", "However, these GCLFs might not hold in the faint host galaxies such as the faint satellites of the MW since there has been no reason for them being universal.", "Especially some of the dwarfs investigated in the paper are even fainter than the peak magnitude of these GCLFs, whether such systems may host GCs that are brighter than the dwarfs themselves is unclear, and is probably unlikely.", "Despite the lack of robust constraints on this, has pointed out that the peak of GCLF can be at $M_{\\rm V}= -5$ for faint galaxies.", "Moreover, the peak magnitude of GCLFs for different galaxies can vary in the range of $-7 < M_{\\rm V}< -5$ .", "Therefore, we look at the known GC populations of the MW, NGC 6822, Sagittarius, Fornax, and Eridanus 2 in Appendix  and decide to consider the peak of GCLF at $M_{\\rm V}= -6$ based on Figure REF to calculate the completeness again.", "The orange points in Figure REF show the completeness $g$ computed with the GCLF $\\mathcal {N}({-6},\\,{1.2}^2)$ .", "As this GCLF peaks at the fainter magnitude than the other two GCLFs, $g$ hardly changes for close dwarfs with much fainter $M_{\\rm V}^{\\rm lim}$ than the peak of GCLF at $M_{\\rm V}= -6$ whereas $g$ drops for the ones that are more distant than 100 kpc with small $M_{\\rm V}^{\\rm lim}$ , e.g.", "$g =$  20 – 30 percent for Eridanus 2, Leo T, and Phoenix.", "Section  has mentioned that the ultra-faint GC with the luminosity of $M_{\\rm V}= -3.5$ , in the Eridanus 2 is missing in our detection.", "This is mainly because the luminosity of this GC is much fainter than the detection limit $M_{\\rm V}^{\\rm lim}\\sim -6.5$ for the Eridanus 2 in the search.", "Hosting the ultra-faint GC of $M_{\\rm V}= -3.5$ and having the luminosity of $M_{\\rm V}= -6.6$ close to the peak magnitude of the MW GCLF, the Eridanus 2 is likely to have a GCLF peaking at a fainter magnitude than $M_{\\rm V}= -7.4$ .", "As shown in Figure REF , the completeness $g$ for the Eridanus 2 is 75 percent with the Gaussian MW GCLF and 65 percent with the evolved Schechter GCLF.", "When we shift the peak of GCLF to $M_{\\rm V}= -6$ , the completeness factor drops to only 30 percent for the Eridanus 2, which further explains the existence of the ultra-faint GC in the Eridanus 2 while it is missing in our search.", "Figure: Left: The 90 percent credible intervals on S N S_{\\rm N} versus M V M_{\\rm V} of the dwarfs with two different GCLFs: double-sided intervals for Fornax and one-sided upper bounds for the others.", "The black data points are S N S_{\\rm N} of the MW, LMC, SMC, Sagittarius (Sgr), and Fornax (Fnx) in .", "The green dashed curve is the mean trend curve of the S N S_{\\rm N} for 100 galaxies in the Virgo Cluster in .Right: The probability of hosting no GC for a galaxy with luminosity LL and specific frequency S N S_{\\rm N}, P(N=0;S N L)P(N=0; S_{\\rm N}L).", "The 90 percent credible intervals on S N S_{\\rm N} is used to derive the range of P(N=0;S N L)P(N=0; S_{\\rm N}L).", "The two greys lines indicate P(N=0;S N L)=0.9P(N=0; S_{\\rm N}L) = 0.9 and P(N=0;S N L)=1P(N=0; S_{\\rm N}L) = 1." ], [ "Specific frequency of the globular clusters", "The specific frequency of GCs is a common quantity to indicate the richness of GC system for a galaxy, first formulated as $S_{\\rm N}= N_{\\rm gc}\\times 10^{0.4 \\, \\left( M_{\\rm V, gal} + 15 \\right)}$ where $N_{\\rm gc}$ is the total number of GCs in a host galaxy and $M_{\\rm V, gal}$ is the absolute magnitude of the host galaxy .", "With $L_{\\rm gal}\\equiv 10^{-0.4 \\, \\left( M_{\\rm V, gal} + 15 \\right)}$ defined as the galactic V-band luminosity normalized to $M_{\\rm V}= -15$ , $S_{\\rm N}= N_{\\rm gc}/ L_{\\rm gal}$ then indicates the number of GCs per unit normalized luminosity.", "When the galaxy luminosity and the number of clusters are large, simply taking a ratio between the number and luminosity makes sense; however, a more statistical approach is required for dwarf galaxies.", "Here, we define $S_{\\rm N}$ as the specific frequency for a group of galaxies.", "In that case, the observed number of clusters for each galaxy in a group will be Poisson distributed: $N_{\\rm gc}\\sim {\\rm Poisson} \\left( S_{\\rm N}L_{\\rm gal}\\right)$ where $L_{\\rm gal}$ is the luminosity of the galaxy and $N_{\\rm gc}$ is the random variable describing the number of clusters in this galaxy.", "Assuming that our samples of GCs are incomplete with different completeness correction $g$ for each dwarf, we can update the model to include incompleteness as $N_{\\rm gc}\\sim {\\rm Poisson} \\left( S_{\\rm N}g L_{\\rm gal}\\right).$ Among the nine objects that we identify in our search in Section , only the six GCs found around the Fornax dwarf are associated with the parent dwarf galaxy.", "That is, the dwarfs targeted in the paper except for Fornax have no associated GCs detected around them.", "Due to the lack of associated GCs and the fact that most of the dwarfs are much fainter than Fornax, the formal $S_{\\rm N}$ is hence expected to be zero with large upper bounds.", "To properly take into account the non-detections and to still be able to constrain the specific frequency of the dwarf population, we assume that $S_{\\rm N}$ is constant for the dwarfs with similar luminosities and will provide upper bounds on $S_{\\rm N}$ for the dwarf population as a whole.", "Assuming that we look at $m$ dwarfs as a group at once, we know the luminosity $L_i$ and the completeness $g_i$ for the $i^{\\rm th}$ dwarf, where $ 1 \\le i \\le m $ .", "The total expected number of observed GCs in this group of $m$ dwarfs is the sum of the expected number of GCs in each dwarf.", "Defining $L \\equiv \\sum _{i=1}^m L_i g_i$ and with the constant specific frequency $S_{\\rm N}$ shared among the $m$ dwarfs, we can write down the total expected number of GCs as $\\sum _{i=1}^m S_{\\rm N}g_i L_i = S_{\\rm N}\\sum _{i=1}^m g_i L_i \\equiv S_{\\rm N}L.$ Together with the definition of the total number of observed GCs of the $m$ dwarfs as $N =\\sum _{i=1}^m N_i$ where $N_i$ is the number of observed GCs of the $i^{\\rm th}$ dwarf from our detection.", "We model $N$ similarly to Equation REF as $N \\sim {\\rm Poisson} \\left( S_{\\rm N}L \\right)$ and therefore the likelihood function $P(N \\mid S_{\\rm N}) \\propto S_{\\rm N}^N e^{-S_{\\rm N}L}$ .", "Using the Jeffreys prior $S_{\\rm N}^{-1/2}$ as the distribution of the parameter $S_{\\rm N}$ , we have the posterior distribution $P(S_{\\rm N}\\mid N)\\propto P(S_{\\rm N}) P(N \\mid S_{\\rm N})\\propto S_{\\rm N}^{N - \\frac{1}{2}} e^{-S_{\\rm N}L}.$ This is a Gamma distribution; that is, $S_{\\rm N}\\sim {\\rm Gamma} \\left( N + \\frac{1}{2}, L \\right)$ .", "With the posterior in Equation REF , we construct the 90 percent credible intervals on the parameter $S_{\\rm N}$ with the Gaussian MW GCLF for the dwarfs as shown in the blue curve in the left panel of Figure REF .", "Also, we show the $S_{\\rm N}$ of the MW and its four most luminous satellites (LMC, SMC, Sagittarius, and Fornax) based on and the mean trend curve of $S_{\\rm N}$ of 100 galaxies in the Virgo Cluster from .", "Separating the Fornax dwarf from the others due to its richness of GCs, we first calculate its double-sided credible interval on the specific frequency of $12 < S_{\\rm N}< 47$ .", "For the other dwarfs with no discovered GCs, we bin the ones brighter than $M_{\\rm V}= -7$ with a window width of 2 mag and look at the others all at once, where the value of $M_{\\rm V}= -7$ is chosen as it is close to the peak magnitude of the GCLF.", "For the dwarfs in each bin, we obtain the one-sided credible intervals as the upper bounds of the specific frequency: $S_{\\rm N}< 20$ for the dwarfs with $-12 < M_{\\rm V}< -10$ , $S_{\\rm N}< 30$ for the dwarfs with $-10 < M_{\\rm V}< -7$ , and $S_{\\rm N}< 90$ for the dwarfs with $M_{\\rm V}> -7$ .", "Similarly, we also construct the credible intervals on $S_{\\rm N}$ with the evolved Schechter GCLF for the dwarfs, finding a similar result as with the Gaussian MW GCLF.", "The difference in $S_{\\rm N}$ with the two GCLFs is less than 5 – 10 percent so we only show the one with $\\mathcal {N}({-7.4},\\,{1.2}^2)$ in Figure REF .", "The reason for grouping the dwarfs fainter than $M_{\\rm V}= -7$ is that they are in general faint so the expected number of GCs is much smaller than one, which makes them not very informative.", "Besides, the posterior becomes more prior-dependent for the fainter dwarfs as well.", "Thus, finding no GCs for the dwarfs in the brighter $M_{\\rm V}$ bins constrains the upper bounds stronger than in the fainter bins.", "Especially at $M_{\\rm V}< -10$ , the relatively low upper bounds indicate that the Fornax dwarf has a relatively higher $S_{\\rm N}$ than the other dwarfs, especially than the ones with $M_{\\rm V}< -10$ .", "As mentioned in Section REF , the completeness will drop if the GCLF peaks at a fainter $M_{\\rm V}$ than the typical peak magnitude at $M_{\\rm V}= -7.4$ , which would effectively increase the upper bounds on $S_{\\rm N}$ because the dropping completeness decreases the $L$ .", "We, therefore, calculate the credible intervals on $S_{\\rm N}$ again for the dwarfs with the GCLF $\\mathcal {N}({-6},\\,{1.2}^2)$ as the orange curve shows in the left panel of Figure REF , finding that the upper bounds on $S_{\\rm N}$ with this shifted GCLF (the orange curve) are higher than that with the MW GCLF (the blue curve) as expected.", "This effect is also expected to influence the upper limits more for the fainter dwarfs since the GCLFs are expected to shift more if the host galaxies are fainter; however, the upper limit is already more prior-dependent and less informative on the faint end so this upper limit increasing effect is less influential.", "Besides $S_{\\rm N}$ , the probability of a galaxy with luminosity $L$ and $S_{\\rm N}$ to host $N$ GCs, $P(N; S_{\\rm N}L)$ , is also interesting.", "With the 90 percent credible intervals on $S_{\\rm N}$ , we show the range of $P(N=0; S_{\\rm N}L)$ for a galaxy with $L$ based on the model $N \\sim {\\rm Poisson} \\left( S_{\\rm N}L \\right)$ in the right panel of Figure REF , which indicates the probability of a galaxy to host no GCs.", "Except for Fornax, the upper limits of $S_{\\rm N}$ result in the lower limits of $P(N=0; S_{\\rm N}L)$ .", "Based on $P(N=0; S_{\\rm N}L)$ , galaxies fainter than $M_{\\rm V}= -9$ have $P(N=0; S_{\\rm N}L) > 0.9$ , which means the probability of these galaxies to have at least one GC is lower than 10 percent.", "Our finding of $P(N=0; S_{\\rm N}L) > 0.9$ for galaxies with $M_{\\rm V}> -9$ is in agreement with the claims of the lowest galaxy mass of $\\sim 10^5 M_{}$ or luminosity $M_{\\rm V}\\sim -9$ to host at least one GC from and .", "This may further explain the observation that galaxies less massive than $10^6 M_{}$ tend not to have nuclei if we assume that the nuclei originate from GCs sunk by dynamical friction to the center.", "Given our constraints on the specific frequency, Eridanus 2 with $M_{\\rm V}\\sim -7$ has $P(N=0; S_{\\rm N}L) \\sim 0.95$ , which highlights that the GC inside Eridanus 2 is indeed an outlier." ], [ "Conclusions", "We have reported the results of the search for possibly hiding GCs around 55 dwarf galaxies within the distance of 450 kpc from the Galactic Center excluding the LMC, SMC, and Sagittarius.", "This was a targeted search around the dwarfs so we excluded those three satellites to avoid a huge portion of the sky to be searched due to their relatively large sizes.", "For each targeted dwarf galaxy, we have investigated the stellar distribution of the sources in Gaia DR2, selected with the magnitude, proper motion, and stellar morphology cuts.", "Using the kernel density estimation and the Poisson statistics of stellar number counts, we have identified eleven stellar density peaks of above 5 significance as possible GC candidates in the targeted area.", "Cross-matching the eleven possible candidates with the simbad database and existing imaging data, we have found that all of them are known objects: Fornax GC 1 – 6, Messier 75, NGC 5466, Palomar 3, Leo I and Sextans A.", "Only the six GCs of Fornax are associated with the parent dwarf galaxy.", "We have calculated the GC detection limit in $M_{\\rm V}$ for each dwarf using 1000 simulated GCs, finding that $M_{\\rm V}^{\\rm lim}> -7$ for all the dwarfs.", "According to the $M_{\\rm V}^{\\rm lim}$ of the dwarfs, we have then calculated the completeness of detection with the Gaussian MW GCLF $\\mathcal {N}({-7.4},\\,{1.2}^2)$ , the evolved Schechter GCLF peaking at $M_{\\rm V}^{\\rm lim}\\sim -7.4$ , and the assumed Gaussian GCLF $\\mathcal {N}({-6},\\,{1.2}^2)$ .", "With the Gaussian MW GCLF and the evolved Schechter GCLF, the completeness of the detection for most of the dwarfs was higher than 90 percent and even that of the lowest three, Eridanus 2, Leo T, and Phoenix, was around 70 percent.", "With the assumed Gaussian GCLF, the completeness of our search was lower for the dwarfs that are more distant than 100 kpc, such as the Eridanus 2, Leo T, and Phoenix where it reached 20 – 30 percent.", "Using the completeness, we have constructed the 90 percent credible intervals on the GC specific frequency $S_{\\rm N}$ of the MW dwarf galaxies.", "The Fornax dwarf had the credible interval on the specific frequency of $12 < s < 47$ , the dwarfs with $-12 < M_{\\rm V}< -10$ had $S_{\\rm N}< 20$ , the dwarfs with $-10 < M_{\\rm V}< -7$ had $S_{\\rm N}< 30$ , and dwarfs with $M_{\\rm V}> -7$ had non-informative $S_{\\rm N}< 90$ .", "Based on these credible intervals on $S_{\\rm N}$ , we have derived the probability of galaxies to host GCs given their luminosity, finding that the probability of galaxies fainter than $M_{\\rm V}= -9$ to possess GCs is lower than 10 percent.", "2018AA...616A..12GGaia Collab.", "18b.", "2018AA...620A.155MMH18 2012AJ....144....4MM12 2018MNRAS.475.5085TT18 2019MNRAS.488.2743TT19b 2016MNRAS.463..712TT16b 2016MNRAS.459.2370TT16a 2018PASJ...70S..18HHomma18 Hargis2016Hargis16 Homma2016Homma16 Table: The list of properties of the studied dwarf galaxies: the positions (α\\alpha and δ\\delta ), the heliocentric distance (D D_{}), the V-band magnitude (M V M_{\\rm V}), the proper motions (μ α \\mu _\\alpha and μ δ \\mu _\\delta ), the reference (ref.", "), and the 3σ μ =3σ μ α 2 +σ μ δ 2 3~\\sigma _\\mu = 3 \\sqrt{\\sigma _{\\mu _{\\alpha }}^2 + \\sigma _{\\mu _{\\delta }}^2} PM uncertainty converted to km s -1 \\rm km\\,s^{-1} at the distance of the dwarf." ], [ "Acknowledgements", "We acknowledge the support by NSF grants AST-1813881, AST-1909584, and Heising-Simons Foundation grant 2018-1030.", "This paper has made use of the Whole Sky Database (wsdb) created by Sergey Koposov and maintained at the Institute of Astronomy, Cambridge with financial support from the Science & Technology Facilities Council (STFC) and the European Research Council (ERC).", "This software has made use of the q3c software .", "This work presents results from the European Space Agency (ESA) space mission Gaia.", "Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC).", "Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA).", "The Gaia mission website is https://www.cosmos.esa.int/gaia.", "The Gaia archive website is https://archives.esac.esa.int/gaia.", "Software: numpy , scipy , pandas , matplotlib , seaborn , astropy [3], imf , sqlutilpy .", "The data underlying this article were derived from sources in the public domain: https://archives.esac.esa.int/gaia." ] ]

[ [ "On Lithium-6 as diagnostic of the lithium-enrichment mechanism in red\n giants" ], [ "Abstract High lithium-7 ($\\mathrm{^7Li}$) abundances in giants are indicative of non-standard physical processes affecting the star.", "Mechanisms that could produce this signature include contamination from an external source, such as planets, or internal production and subsequent mixing to the stellar surface.", "However, distinguishing between different families of solutions has proven challenging, and there is no current consensus model that explains all the data.", "The lithium-6 ($\\mathrm{^6Li}$) abundance may be a potentially important discriminant, as the relative $\\mathrm{^6Li}$ and $\\mathrm{^7Li}$ abundances are expected to be different if the enrichment were to come from internal production or from engulfment.", "In this work, we model the $\\mathrm{^6Li}$ and $\\mathrm{^7Li}$ abundances of different giants after the engulfment of a substellar mass companion.", "Given that $\\mathrm{^6Li}$ is more strongly affected by Galactic chemical evolution than $\\mathrm{^7Li}$, $\\mathrm{^6Li}$ is not a good discriminant at low metallicities, where it is expected to be low in both star and planet.", "For modeled metallicities ([Fe/H]$>-0.5$), we use a \"best case\" initial $\\mathrm{^6Li/^7Li}$ ratio equal to the solar value.", "$\\mathrm{^6Li}$ increases significantly after the engulfment of a companion.", "However, at metallicities close to solar and higher, the $\\mathrm{^6Li}$ signal does not last long in the stellar surface.", "As such, detection of surface $\\mathrm{^6Li}$ in metal-rich red giants would most likely indicate the action of a mechanism for $\\mathrm{^6Li}$-enrichment other than planet engulfment.", "At the same time, $\\mathrm{^6Li}$ should not be used to reject the hypothesis of engulfment in a $\\mathrm{^7Li}$-enriched giant or to support a particular $\\mathrm{^7Li}$-enhancement mechanism." ], [ "Introduction", "Lithium-7, one of the two stable isotopes of lithium (Li), was produced right after the Big Bang, and it is used to understand element production in the early Universe [16], diagnose mixing in stellar interiors [37], and study galactic chemical evolution [40], among other applications.", "In low-mass stars, $\\mathrm {Li}$ is destroyed in the interior during the main sequence.", "When stars evolve to the red giant branch (RGB), during the first dredge-up the outer convection zone deepens in mass, diluting the $\\mathrm {^7Li}$ left close to the stellar surface.", "For this reason, high $\\mathrm {^7Li}$ abundances in giants require the presence of non-standard mechanisms modifying the abundance of the star.", "One possible explanation for high $\\mathrm {^7Li}$ in the surface of red giants relies on the efficient transport by extra-mixing of $\\mathrm {^7Li}$ produced through the Cameron-Fowler mechanism [8].", "Another explanation for the enhanced $\\mathrm {^7Li}$ is the contamination from a source that preserves or creates $\\mathrm {^7Li}$ , such as supernovae [31] or substellar companions [45].", "An evolved companion, such as an asymptotic giant branch star, which produces $\\mathrm {^7Li}$ during its thermal pulses [43] could also be a source of Li.", "However, the small fraction of Li-rich giants that have been searched for binary companions do not seem to show evidence for them [2].", "Further work is needed to test this possibility for the majority of red giants.", "In [3], we modeled the engulfment of different planets and brown dwarfs by giant stars.", "We found that engulfment of substellar companions (SSCs) alone can explain $\\mathrm {^7Li}$ abundances as high as $\\mathrm {A(^7Li)}=2.2$A(x)=$\\log (n_x/n_H)+12$, and that stellar mass and metallicity are fundamental in defining the expected $\\mathrm {^7Li}$ abundance in giants and not misinterpret normal giants as enriched, or truly anomalous giants as normal.", "However, as giants with much higher abundances are found in nature [49], [17], either a completely different mechanism, or a combination of different $\\mathrm {^7Li}$ sources is still needed to explain the entire population.", "Other observational indicators can be used to distinguish between different $\\mathrm {^7Li}$ replenishment scenarios.", "The evolutionary phase of the enriched giants is an important indicator of the physical conditions where the enrichment is produced.", "Some works, such as [17] and [12], argued that most of these unusual giants are located in the horizontal branch.", "This could point to a mechanism of $\\mathrm {^7Li}$ enrichment working during or close to the RGB tip, during the helium flash.", "On the other hand, measurements of the stellar rotation [9], beryllium surface abundance [47], and carbon isotopic ratio [48] could all be fundamental in finding the mechanism behind the $\\mathrm {^7Li}$ -enrichment.", "Another potentially important probe could be $\\mathrm {^6Li}$ , the far-less-abundant stable isotope of Li, thought to be primarily produced by cosmic ray spallation [32].", "As $\\mathrm {^6Li}$ is destroyed in stellar interiors at even lower temperatures than those required to burn $\\mathrm {^7Li}$ [7], standard stellar evolutionary models predict much more severe burning of $\\mathrm {^6Li}$ than $\\mathrm {^7Li}$ at any evolutionary state [41], and very low surface $\\mathrm {^6Li}$ abundances during the RGB.", "In contrast, planets and brown dwarfs preserve their initial $\\mathrm {^6Li}$ , so the abundance of this isotope should be higher in giants that have engulfed their companions.", "On the contrary, the Cameron-Fowler mechanism is not able to produce $\\mathrm {^6Li}$ .", "Thus, it may be possible to use $\\mathrm {^6Li}$ to identify candidates of planet engulfment [13].", "Because of the large constrast of $\\mathrm {^6Li}$ pre and post-engulfment, the planet signal could be easier to detect than that of $\\mathrm {^7Li}$ .", "However, at lower metallicities, chemical evolution effects predict very low birth planetary abundances, complicating observations, and the fragility of $\\mathrm {^6Li}$ implies that it could be burned even where $\\mathrm {^7Li}$ is stable.", "To test these issues and analyze if $\\mathrm {^6Li}$ can effectively be used as a diagnostic of engulfment for all giants, we model the abundance of $\\mathrm {^6Li}$ after the engulfment of SSCs of different properties (Section ).", "The resulting $\\mathrm {^6Li}$ surface abundance (Section ) shows that stellar metallicity plays an important role in the burning of $\\mathrm {^6Li}$ under convective conditions, with higher metallicity stars burning very rapidly its original $\\mathrm {^6Li}$ and that deposited by the planet.", "As a consequence, the absence of this isotope in the surface of $\\mathrm {^7Li}$ -rich giants cannot be used to reject the SSC engulfment hypothesis.", "We analyze in detail this result in Section , to finally summarize in Section ." ], [ "Models", "We follow a similar procedure to that described in [3].", "We refer the reader to that work for an in-depth analysis of the assumptions, the calculation of point of SSC dissipation in stellar interiors, and the parameters used in our grid of stellar models.", "In summary, we use a post-processing approach, where standard stellar evolution models are used as a base to later implement the engulfment and thus there is no feedback from the planet ingestion process.", "Standard stellar models are obtained with the Yale Rotating Evolutionary code [38].", "The modeled stellar mass goes from $1.0$ to $2.0\\ \\mathrm {M_\\odot }$ .", "Metallicities range from [Fe/H]=$-0.5$ up to [Fe/H]=$0.18$ and giants are evolved up to the tip of the RGB.", "We do not consider lower metallicities because the normal Galactic chemical evolution trends would predict a smaller than solar birth $\\mathrm {^6Li/^7Li}$ ratio.", "In such stars, an engulfed planet is likely to supply little $\\mathrm {^6Li}$ due to its low birth $\\mathrm {^6Li}$ .", "Thus, the low overall $\\mathrm {^6Li}$ would make this signal impossible to observe.", "Low metallicity stars are also known to experience severe in-situ Li depletion on the giant branch.", "This combination makes $\\mathrm {^6Li}$ a poor discriminant for metal-poor progenitors, and we therefore focus on higher metallicity stars.", "The $\\mathrm {^6Li}$ in stellar interiors is burned through the reaction $\\mathrm {^6Li+H\\rightarrow {}^3He+{}^4He},$ with reaction rates from [25].", "Regarding the stellar initial abundance of $\\mathrm {^6Li}$ in our models, we consider a fixed meteorite Li isotopic ratio $\\mathrm {^6Li/^7Li}=0.082$ [14].", "Because the abundance of $\\mathrm {^6Li}$ should increase with metallicity due to the contribution of cosmic ray spallation [39], the birth $\\mathrm {^6Li}$ is expected to be lower at lower metallicity.", "We therefore regard this as an optimistic or limiting case scenario, where engulfed objects will give the maximum signal.", "We note, however, that our differential depletion calculations are independent of the assumed birth ratio, given that the $\\mathrm {^6Li}$ and $\\mathrm {^7Li}$ depletion factors, defined as the fraction of initial Li remaining in the surface of the star, are independent of the birth values.", "The initial $\\mathrm {^6Li}$ value is set before the expected phase of Li burning in the pre-main sequence, thus, the Li isotopic ratio can drastically change in this phase.", "Figure REF shows the burning of Li in the pre-main sequence for stars of different mass and metallicities of [Fe/H]=$-0.5$ (top) and [Fe/H]=$0.0$ (bottom panel).", "Higher-mass stars preserve their $\\mathrm {^6Li/^7Li}$ , while there is more burning in solar metallicity stars.", "Notice that the chosen time resolution of the models could change the surface Li abundance in certain models and by using specific settings [26].", "Here, we test if decreasing the timestep can significantly modify our results, finding that the time resolution only produces slight changes in the abundance.", "Figure: 6 Li \\mathrm {^6Li} (black) and 7 Li \\mathrm {^7Li} (blue) in the pre-main sequence of stars of 4 different masses, at metallicities [Fe/H]=-0.5-0.5 (top panel) and [Fe/H]=0.00.0 (bottom panel).To better control for the effect of Li burning previous to the RGB phase, we quantify the Li abundances at the zero-age main sequence.", "Although there is some burning of $\\mathrm {^6Li}$ during the main sequence, the main depletion process takes place before that.", "Figure REF shows the $\\mathrm {^7Li}$ and $\\mathrm {^6Li}$ depletion factors at the zero-age main sequence, for stars of different masses and metallicities.", "There is little to no depletion at higher masses, but important depletion for $\\mathrm {^6Li}$ at low masses at any metallicity.", "$\\mathrm {^7Li}$ also burns considerably in low-mass stars at higher metallicities.", "Figure: 6 Li \\mathrm {^6Li} (black) and 7 Li \\mathrm {^7Li} (blue) depletion factors at the zero-age main sequence (i.e., due to pre-main sequence evolution) for stars of different masses.", "The panels show results for specific metallicities.For the SSC, we use a fixed ratio between $\\mathrm {^6Li}$ mass fraction and metals equal to the Solar System meteoritic value.", "Thus, all SSCs have the same $\\mathrm {X_{^6Li}/Z}$ but could have a different metal content, changing its mass fraction of $\\mathrm {^6Li}$ .", "The metal content of SSCs depends on their mass.", "We use three different mass regimes.", "Brown dwarfs ($15\\ \\mathrm {M_J}$ ) can have two different compositions, solar metallicity $Z=Z_\\odot $ , or brown dwarfs enhanced in metals.", "Planets ($0.01\\,\\mathrm {M_J}$ to $15\\,\\mathrm {M_J}$ ) are taken to be enhanced in metals as well.", "Rocky planets (Mass smaller than $0.01\\,\\mathrm {M_J}$ ), which include Earth-type objects are considered to have a much higher metal content of $Z=1$ .", "Results in [3] show that very massive brown dwarfs end up dissolving in the radiative interior rather than in the convective envelope.", "Because of that, we decide to model SSC masses up to $15\\ \\mathrm {M_J}$ .", "It is important to notice that at higher metallicities the maximum mass of a companion that still dissolves in the convective zone increases [4]." ], [ "$\\mathrm {^6Li}$ abundance evolution", "We begin by considering the engulfment of four different SSCs by $1.3\\ \\mathrm {M_\\odot }$ and $1.8\\ \\mathrm {M_\\odot }$ red giants of [Fe/H]$=-0.5$ , and a $1.7\\ \\mathrm {M_\\odot }$ of [Fe/H]=$0.05$ .", "The companions correspond to a $15\\ \\mathrm {M_J}$ brown dwarf with $Z=Z_\\odot $ , a $15\\ \\mathrm {M_J}$ brown dwarf with $Z=2.5Z_\\odot $ , a Jupiter-like planet, and an Earth-like planet.", "Figure: Surface 6 Li / 7 Li \\mathrm {^6Li/^7Li} evolution in a 1.8M ⊙ 1.8 \\ \\mathrm {M_\\odot } star (top panel) and a 1.3M ⊙ 1.3\\ \\mathrm {M_\\odot } star (middle panel) of [Fe/H]=-0.5-0.5, and a 1.8M ⊙ 1.8\\ \\mathrm {M_\\odot } giant of [Fe/H]=0.050.05 (bottom panel) after the engulfment of 4 different SSCs.", "The evolution starts right before the end of the main sequence and ends at the tip of the RGB.The evolution of the $\\mathrm {^6Li/^7Li}$ surface ratio for these stars can be seen in Figure REF as a function of luminosity and $\\mathrm {\\log g}$ .", "The initial $\\mathrm {^6Li}$ in the main sequence can be lower than the meteoritic value due to pre-main sequence burning.", "The $\\mathrm {^6Li/^7Li}$ ratio decreases during the first dredge-up ($\\mathrm {\\log g}\\sim 3.5$ ), as expected.", "Dilution in the convective envelope decreases the abundance of $\\mathrm {^7Li}$ and $\\mathrm {^6Li}$ .", "However, the decrease in their ratio is produced because right below the convective envelope, $\\mathrm {^6Li}$ burns more rapidly than $\\mathrm {^7Li}$ .", "When the first dredge-up mixes that material into the surface, the $\\mathrm {^6Li}$ is reduced by a larger amount than $\\mathrm {^7Li}$ .", "The ratio $\\mathrm {^6Li/^7Li}$ increase after the engulfment of planets (in our models here, arbitrarily chosen to occur at $\\mathrm {\\log g}\\sim 2.8$ ).", "The $\\mathrm {^6Li}$ enrichment is larger for the brown dwarf with high Z, while Earth-like planets barely increase the original $\\mathrm {^6Li}$ .", "For giants in the modeled metallicity range, $\\mathrm {^6Li}$ burning can be significant during the dredge-up and RGB.", "We can see this in the $1.7\\ \\mathrm {M_\\odot }$ star in Figure REF .", "Thus, there are some differences in the $\\mathrm {^6Li}$ after engulfment in the star when planets are accreated at different locations along the RGB.", "Later engulfment times imply larger $\\mathrm {^6Li}$ .", "Figure: Top right panel: Histogram of the metallicity distribution of giants with measured 7 Li \\mathrm {^7Li}, all of them concentrating towards higher metallicities.", "Bottom left panel: Standard surface 6 Li / 6 Li 0 \\mathrm {^6Li/^6Li_0} abundance of stars of different masses and metallicities.", "This map considers no engulfment of SSCs.", "Bottom right panel: Surface 6 Li / 6 Li 0 \\mathrm {^6Li/^6Li_0} abundance of stars of different masses and metallicities after the engulfment of a 15M J 15\\ \\mathrm {M_J} brown dwarf enhanced in metal content.", "In these color maps, grid points are marked with black circles, and the 3 white contours indicate where log ( 6 Li / 6 Li 0 )=\\mathrm {log(^6Li/^6Li_0)=}-3, -5 and -10 from left to right.", "Stars of higher metallicities burn very rapidly their 6 Li \\mathrm {^6Li} content, as well as any additional 6 Li \\mathrm {^6Li} incorporated by the ingestion of a SSC.The resulting $\\mathrm {^6Li}$ is mass and metallicity dependent.", "In Figure REF , we see almost no burning post-engulfment in the $1.8 \\ \\mathrm {M_\\odot }$ , [Fe/H]$=-0.5$ giant and severe burning in the $1.7 \\ \\mathrm {M_\\odot }$ , metal-rich star.", "Figure REF shows a map of $\\mathrm {^6Li/^6Li_0}$ in standard stars of different masses and metallicities, without planet engulfment.", "We obtain in our models the $\\mathrm {^6Li}$ abundance at the tip of the RGB in stars of the grid (small circles in the figure).", "This grid is then interpolated to produce the map color-coded by $\\mathrm {^6Li/^6Li_0}$ .", "For metal-poor stars, a small amount of $\\mathrm {^6Li}$ is found in the surface of the star, even without engulfment.", "However, for metal-rich stars (solar metallicity and higher), the star reaches the RGB with low $\\mathrm {^6Li}$ , which decreases even more after the first dredge-up.", "After this stage, $\\mathrm {^6Li}$ is also burned under convective conditions, vanishing completely.", "Given that the $\\mathrm {^6Li}$ is so small in the RGB of standard stars, the engulfment of SSCs could increase substantially the $\\mathrm {^6Li}$ abundance.", "We present a map of $\\mathrm {^6Li/^6Li_0}$ for stars of different masses and metallicities in Figure REF , bottom right panel, now considering the engulfment of a $15\\ \\mathrm {M_J}$ brown dwarf enhanced in metals.", "The giants engulf the SSC at the end of the first dredge-up.", "Comparing this map to the bottom left panel of Figure REF , $\\mathrm {^6Li}$ can increase significantly with engulfment.", "However, for metal-rich stars, the incorporated $\\mathrm {^6Li}$ is rapidly burned and would not be observed in the stellar surface.", "This becomes important when distinguishing $\\mathrm {^7Li}$ -enrichment mechanisms, since most of these giants are metal-rich.", "We show this in Figure REF , top right panel, where we create an histogram of the metallicity of giants with measured $\\mathrm {^7Li}$ .", "No upper limits are considered when compiling this catalog, which includes giants from [21], [6], [23], [22], [24], [36], [9], [27], [30], [28], [1], [5], [29], [10], [18], [11], [46]; and [17].", "These measurements are obtained from the literature, and as such are not homogeneous.", "Additionally, some of these sources only report their Li-rich giants and not their entire sampleIn [3] we find that not reporting the entire sample makes it harder to account for the full phenomenology creating Li-enriched giants.. As $\\mathrm {^7Li}$ -rich giants seem to be more metal-rich, this could bias our compilation to higher metallicities.", "The limiting metallicity at which $\\mathrm {^6Li}$ could never be detected post-engulfment due to its rapid burning increases with mass.", "For $1.0 \\ \\mathrm {M_\\odot }$ , close to [Fe/H]$\\sim -0.5$ we already see significant depletion.", "In $2.0 \\ \\mathrm {M_\\odot }$ giants, this limit is closer to solar metallicity.", "If $\\mathrm {^6Li}$ is burned in situ the signal of the planet would not be detected.", "In contrast, the $\\mathrm {^7Li}$ after engulfment could be preserved in the star during the entire RGB phase if no extra-mixing decreases its abundance.", "This could be the case of more metal-rich stars, where extra-mixing seems to be less-efficient [44] and indicates that even if the giant accreted a planet, its abundance of $\\mathrm {^7Li}$ could be high, while its $\\mathrm {^6Li}$ remains low." ], [ "Discussion", "As expected, $\\mathrm {^6Li}$ can increase in a low-mass red giant after the engulfment of a SSC.", "However, $\\mathrm {^6Li}$ is rapidly burned in stars of higher metallicity, indicating that the absence of this isotope does not discard the possibility that the star has accreted a SSC, but if there was an engulfment event, it did not occur recently.", "The destruction of this isotope at a faster rate than the $\\mathrm {^7Li}$ leads to low $\\mathrm {^6Li}$ , regardless of the $\\mathrm {A(^7Li)}$ , not rejecting the engulfment possibility [19].", "This point therefore becomes a crucial one in the quest for the sources of $\\mathrm {^7Li}$ enrichment in giants, as most of the giants that have measured $\\mathrm {^7Li}$ have higher metallicites.", "If $\\mathrm {^6Li}$ were to be seen at high metallicity, then its most likely explanation is a source other than an accreted SSC.", "At the same time, only the $\\mathrm {^7Li}$ -rich giants with $\\mathrm {A(^7Li)}<2.2$ can be explained by the engulfment of SSC [3].", "Therefore, the presence or absence of $\\mathrm {^6Li}$ in stars of higher $\\mathrm {^7Li}$ abundance [33] does not give any information on this particular enrichment mechanism.", "In contrast, if $\\mathrm {^6Li}$ is detected in a relatively metal-poor giant with $\\mathrm {A(^7Li)}<2.2$ , this could be due to the recent contamination of the star by the engulfment of a SSC.", "Engulfment could explain both the high $\\mathrm {^7Li}$ and $\\mathrm {^6Li}$ abundances at the same time, but there could also be independent explanations for the enrichment of each isotope.", "$\\mathrm {^6Li}$ can also be produced in stellar flares [34] and galactic cosmic ray interaction with the interstellar medium [20].", "Although stellar flares can also produce $\\mathrm {^7Li}$ , [42] calculate that the production of the $\\mathrm {^6Li}$ isotope is much larger.", "It is possible that the Sun is producing $\\mathrm {^6Li}$ through flares, based on the high abundances found on the lunar soil [15].", "However, no $\\mathrm {^6Li}$ is found in the surface of the Sun, implying that even if some part of the $\\mathrm {^6Li}$ created is preserved in the photosphere, it is not enough to be measured.", "In giants, there is an additional difficulty, given the large convective envelope that would dilute the $\\mathrm {^6Li}$ created by any mechanism, complicating its detectability.", "From a purely observational point of view, detecting the $\\mathrm {^6Li}$ isotope can be particularly hard, as it manifests itself as a subtle asymmetry of the $\\mathrm {^7Li}$ line at $\\sim 6708\\,\\mathrm {Å}$ .", "Even a Li isotopic ratio as high as solar can be hard to detect at solar-like metallicites due to convective line asymmetries and blends with other lines.", "There is a small region of parameter space where the increase in $\\mathrm {^6Li}$ could be detected, i.e., in higher mass RGB stars engulfing brown dwarfs companions.", "These hypothetical detections of $\\mathrm {^6Li}$ would be especially interesting in giants with $\\mathrm {A(^7Li)}<2.2$ .", "Giants with more $\\mathrm {^7Li}$ (and stronger $\\mathrm {^7Li}$ lines, where the $\\mathrm {^6Li}$ could be more easily detected) can be excluded as engulfment candidates solely based on their $\\mathrm {^7Li}$ abundances [3].", "However, not only is the $\\mathrm {^6Li}$ detection observationally hard, but also, as the stellar mass increases, the lifetime a star spends on its RGB phase decreases considerably.", "Thus, it is very unlikely to find the higher-mass objects that could retain part of their $\\mathrm {^6Li}$ signature.", "An interesting solar-metallicity Li-enriched giant is presented by [35], with a $\\mathrm {A(^7Li)=1.69}\\pm 0.11$ dex.", "This star has a Li isotopic ratio close to meteoritic.", "Our models confirm that engulfment is an unlikely explanation for this particular star, that requires further study." ], [ "Summary", "The fragile $\\mathrm {^6Li}$ isotope is destroyed at even smaller temperatures than $\\mathrm {^7Li}$ .", "As such, stellar evolution theory predicts stars with small $\\mathrm {^6Li}$ during the RGB.", "The $\\mathrm {^6Li}$ abundance could increase after the engulfment of SSCs, making $\\mathrm {^6Li}$ to appear as a good diagnostic for an engulfment event in giants.", "In this work, we found that the $\\mathrm {^6Li}$ and $\\mathrm {^6Li/^7Li}$ of the star increases after the engulfment of the companion.", "We demonstrate that metal-rich stars burn very rapidly the $\\mathrm {^6Li}$ .", "The limit between stars that preserve and burn the isotope is mass-dependent.", "Given that no $\\mathrm {^6Li}$ can be found in metal-rich giants even after planet engulfment, the abundance of this isotope should not be used as a way to distinguish between different $\\mathrm {^7Li}$ -enrichment mechanisms nor as a method to reject the planet engulfment hypothesis.", "Moreover, enrichment of $\\mathrm {^6Li}$ in low-mass metal-rich giants, is likely not due to planet engulfment.", "There is only a very low probability that we find such an extremelly recent engulfment event, where $\\mathrm {^6Li}$ is still not burned completely.", "Stars with $\\mathrm {A(^7Li)}>2.2$ could not be explained by planet accretion on the basis of their $\\mathrm {^7Li}$ alone.", "Thus, measurements of $\\mathrm {^6Li}$ in these stars do not really indicate anything about the $\\mathrm {^7Li}$ enrichment mechanism.", "In contrast, finding stars with high abundances of both $\\mathrm {^7Li}$ and $\\mathrm {^6Li}$ in a certain metallicity range could point to a recent engulfment event.", "However, a combination of mechanisms, one to enhance $\\mathrm {^7Li}$ and another, such as flares, to increase the $\\mathrm {^6Li}$ , is still possible, especially if the star is metal-rich and its $\\mathrm {^6Li}$ is much less likely to be explained by accretion.", "In conclusion, we advise caution when using $\\mathrm {^6Li}$ as a diagnostic of engulfment or when using it to favor a scenario of $\\mathrm {^7Li}$ enrichment over others.", "We thank G. Somers for his help with lithium in YREC.", "C.A.G.", "acknowledges support from the National Agency for Research and Development (ANID) FONDECYT Postdoctoral Fellowship 2018 Project 3180668.", "J.C. acknowledges support from CONICYT project Basal AFB-170002 and by the Chilean Ministry for the Economy, Development, and Tourism's Programa Iniciativa Científica Milenio grant IC 120009, awarded to the Millennium Institute of Astrophysics.", "MHP would like to acknowledge support from NASA grant 80NSSC19K0597." ] ]

[ [ "The polarized spectral energy distribution of NGC 4151" ], [ "Abstract NGC 4151 is among the most well-studied Seyfert galaxies that does not suffer from strong obscuration along the observer's line-of-sight.", "This allows to probe the central active galactic nucleus (AGN) engine with photometry, spectroscopy, reverberation mapping or interferometry.", "Yet, the broadband polarization from NGC 4151 has been poorly examined in the past despite the fact that polarimetry gives us a much cleaner view of the AGN physics than photometry or spectroscopy alone.", "In this paper, we compile the 0.15 -- 89.0 $\\mu$m total and polarized fluxes of NGC 4151 from archival and new data in order to examine the physical processes at work in the heart of this AGN.", "We demonstrate that, from the optical to the near-infrared (IR) band, the polarized spectrum of NGC 4151 shows a much bluer power-law spectral index than that of the total flux, corroborating the presence of an optically thick, locally heated accretion flow, at least in its near-IR emitting radii.", "Specific signatures from the atmosphere of the accretion structure are tentatively found at the shortest ultraviolet (UV) wavelengths, before the onset of absorption opacity.", "Otherwise, dust scattering appears to be the dominant contributor from the near-UV to near-IR polarized spectrum, superimposed onto a weaker electron component.", "We also identify a change in the polarization processes from the near-IR to the mid-IR, most likely associated with the transition from Mie scattering to dichroic absorption from aligned dust grains in the dusty torus or narrow-line region.", "Finally, we present and dicuss the very first far-infrared polarization measurement of NGC 4151 at 89 $\\mu$m." ], [ "Introduction", "NGC 4151 holds a particular place in the field of active galactic nuclei (AGN).", "It was among the original list of the six “extragalactic nebulae with high-excitation nuclear emission lines superposed on a normal G-type spectrum” observed by Carl K. Seyfert, that later gave his name to a specific class of AGNs .", "NGC 4151 is also one of the nearest galaxies to Earth, $z \\sim $ 0.00332, which corresponds to a Hubble distance of 18.31 $\\pm $ 1.31 Mpc .", "Its optical and ultraviolet (UV) fluxes are known to vary quite dramatically, resulting in a fluctuating optical-type classification.", "NGC 4151 changes from a Seyfert-1.5 type in the maximum activity state to a Seyfert-1.8 in the minimum state , .", "Indeed, long-term temporal behaviors of this AGN showed flux variation from a factor 2 to 6 over a period of 12 years , .", "The X-ray flux of NGC 4151 is also known to be variable but is also very bright, explaining why it was one of the first Seyfert galaxies to be detected in the high energy sky by the Uhuru Satellite , .", "A complete 0.1 – 100 keV coverage of the source showed that the X-ray spectrum of NGC 4151 is a complex association of emitting and absorbing components arising from various locations, from the central engine to the extended polar outflows of the AGN , .", "This complexity underlies a fundamental question: are all AGNs such convoluted systems or is NGC 4151 unique?", "In order to better understand the intrinsic physics and structure of Seyfert galaxies, it becomes of great interest to determine whether this Seyfert galaxy is an archetype of its class or very far from it .", "The proximity of NGC 4151 allowed to measure several of its prime attributes.", "The narrow-line region (NLR) of this AGN has an inclination close to 45$^\\circ $ , , with the north side out of the plane of sky away from our line-of-sight (LOS), which makes NGC 4151 an intermediate type Seyfert galaxy.", "This means that we are able to directly see the central engine (a supermassive black hole and its associated putative accretion disk) through the dust funnel of the circumnuclear gas and dust distribution that characterizes AGNs (see the fundamental paper on the subject by [5]).", "This dusty equatorial obscurer is the reason why we cannot see the central engine in type-2 AGNs where the observer's LOS intercepts this optically thick medium.", "The possibility to peer the central engine in NGC 4151 allowed to estimate its black hole mass to be of the order of 4 $\\times $ 10$^7$  M$_\\odot $ , .", "The supermassive black hole appears to be maximally spinning and its inferred mass accretion rate is about 0.013 M$_\\odot $ yr$^{-1}$ .", "The putative geometrically thin, optically thick, accretion disk that surrounds the black hole is responsible for the multi-temperature thermal emission that produces an UV-to-optical signature called the Big Blue Bump.", "At the end of the accretion disk lies the broad emission line region (BELR) that is responsible for the detected broad emission lines in the UV, optical and near-infrared (near-IR) spectrum of AGNs .", "The size of the BELR is object-dependent but it is generally admitted that it lies between the accretion disk and the inner radius of the dusty circumnuclear region.", "In NGC 4151, the radius at which dust grains start to survive the intense radiation field was measured thanks to reverberation mapping studies and is of the order of 0.04 pc .", "Dust is responsible for another detectable feature in the spectral energy distribution (SED) of AGN, usually referred to as the IR bump and most likely associated with re-radiated emission from the dust grains.", "The superposition of the emission lines and the IR re-emission onto the UV/optical continuum makes it difficult to precisely measure the shape and the peak of the Big Blue Bump.", "This is unfortunate since the Big Blue Bump provides critical information on the structure and condition of the innermost AGN components.", "In order to isolate the true SED of the central component, it is advised to look at the polarized light of the AGN.", "Indeed, the polarized flux shaves off the unpolarized line emission from the extended AGN polar outflows or the host galaxy , .", "In addition, polarimetry can detect variations in the emission/scattering physics thanks to two additional and independent parameters: the polarization degree and polarization position angle , .", "In this paper, we thus investigate the polarized light of NGC 4151 thanks to archival polarimetric campaigns and compile, for the first time, its UV-to-IR polarized SED.", "We aim at characterizing the emission from the Big Blue Bump, i.e.", "from the accretion disk itself, by eliminating the parasitic light from the BELR and from the dusty components in the near-IR.", "Additionally, we want to determine what are the dominant mechanisms that are responsible for the observed polarization from the UV to the far-IR band, allowing us to build a better picture of the central AGN components.", "The paper is organized as follows: Section describes the archive data and examines the total and polarized SED, which are further discussed in Section .", "In Section , we present the conclusions.", "We skimmed through the SAO/NASA Astrophysics Data System (ADS) digital library and gathered all the publications reporting polarization measurements in NGC 4151.", "We found 19 papers spanning from 1971 to 2018.", "Two papers reported circular polarization measurements , [4] and, since we are interested in linear polarization only (the thermally-emitted Big Blue Bump should not produce intrinsically circularly polarized light), we safely discarded themThe optical circular polarization measurement of has been contradicted by .", "Due to the diversity of instruments, the polarimetric data are also spanning from apertures that widely vary from sub-arcsecond scales to more than 40 arcseconds.", "It is obvious that we cannot reconstruct a polarized SED using dramatically different apertures so we focused on apertures lower than 10$\"$ .", "At an heliocentric distance of 18.31 Mpc, this corresponds to a linear size of 0.88 kpc.", "Since AGN polar outflows are often optically detected up to a projected distance of 1 kpc (see, e.g., ), we ensure that the polarized SED mostly accounts for the AGN flux, not the host galaxy.", "Finally, we favored publications with spectropolarimetric data instead of measurements taken with broad filters.", "This allows us to have a much more precise reconstruction of the shape of the Big Blue Bump.", "We ended up with seven publications covering as many wavelengths as possible from the optical to the near/mid IR (see Tab.", "REF ).", "To account for the UV band, we retrieved the old and unpublished polarization measurements taken by the Wisconsin Ultraviolet Photo Polarimeter Experiment (WUPPE, , ).", "Five polarimetric observations in the 0.15 – 0.32 $\\mu $ m band were obtained with WUPPE and compiled by the WUPPE team into a 3136 seconds long exposure spectrum visible in Fig.", "REF .", "The observations were temporally close enough that variability was not an issue for stacking the Stokes parameters.", "We have also included newly acquired imaging polarimetric data at 89 $\\mu $ m using the High-resolution Airborne Wideband Camera/Polarimeter (HAWC+) onboard the 2.7-m NASA's Stratospheric Observatory for Infrared Astronomy (SOFIA).", "These observations are part of an AGN polarimetric survey at far-IR wavelengths under the program 07\\0032 (PI: Lopez-Rodriguez).", "Observations were performed on January the 28th, 2020, with a total on-source time of 3360 seconds and will be presented in detail in a follow-up manuscript (Lopez-Rodriguez et al.", "in prep.).", "For the goal of this project we performed aperture photometry within the beam size of 780 (0.69 kpc) at 89 $\\mu $ m and estimated that the nucleus of NGC 4151 is unpolarized (0.9$\\pm $ 0.8%) with an undetermined position angle of polarization.", "The final compilation of published, archival and new polarimetric measurements of NGC 4151 is presented in Tab.", "REF .", "The total and polarized fluxes will be presented in Fig.", "REF while the polarization degrees and position angles will be shown in Fig.", "REF .", "Figure: Unpublished MAST archival compilation of all the polarimetricobservations of NGC 4151 made with WUPPE.", "It sums up 5 observationcampaigns taken between March 4th and March 13th 1995for a total of 3136 seconds." ], [ "Examination of the total and polarized fluxes", "We present in Fig.", "REF the broadband, 0.15 – 89.0 $\\mu $ m, total flux (top) and polarized flux (bottom) spectra of NGC 4151.", "The total flux is entirely coherent with past measurements , [2] and with the nuclear SED retrieved from the NASA/IPAC Extragalactic Database.", "The nuclear SED, shown in shaded gray in Fig.", "REF , is the averaged SED compiled from continuum flux measurements with apertures lower than 10$\"$ .", "The thickness of the gray area corresponds to the observed flux fluctuations (including the error bars) and the missing data were linearly interpolated between two contiguous points.", "We note that due to the variable nature of NGC 4151 in the UV and optical bands, and since it is a composite spectrum aggregating about 40 years of measurements, it is not gainful to fit the total flux data points in order to obtain a representative spectral index for the underlying continuum.", "Although the intensity spectrum is coherent with the nuclear data averaged from NED, it has been shown that the continuum can vary by a factor of 6 in less than 10 years, impeding a trustworthy estimation of the spectral index from the optical to near-IR waveband ($\\approx $ 0.5 – 1 $\\mu $ m).", "We illustrate this in Fig.", "REF , where we can see that the spectral index of the UV and optical/near-IR power-laws underlying the continuum are drastically different due to variability.", "Fortunately, in the near-to-mid IR, where the accretion disk spectrum starts to be hidden under the hot dust thermal emission from the equatorial dusty region, variability is less of an issue (see Fig.", "REF ).", "This allowed us to fit the IR (1 – 12 $\\mu $ m) spectral index from the total flux SED: F$_\\nu \\propto \\nu ^{-1.33}$ .", "This spectral index is entirely consistent with the average value of $-1.48$ $\\pm $ 0.30 reported by [3] for the IR (1 – 16 $\\mu $ m) spectral index of a sample of 22 Seyfert 1–1.5 galaxies.", "To unveil the signature of the external parts of the putative accretion disk that should produce a much bluer spectral index than the total flux from the optical to the near-IR, we plot the polarized flux of NGC 4151 in Fig.", "REF (bottom spectrum).", "The data are entirely consistent with partially-to-unpolarized line emissions superposed on a smooth polarized continuum .", "The optical-to-near-IR spectral slope is clearly more pronounced (bluer) in polarized flux and extends up to 2.2 $\\mu $ m. Yet again, the non-simultaneity of the optical and near-IR observations makes it unproductive to measure a spectral index.", "However, it has been demonstrated by that the polarized flux of NGC 4151 follows the total flux with a lag of a few days, allowing us to correlate the two quantities in order to suppress the variable component.", "This permits us to provide a qualitative measurement of the difference between the spectral index of the total and polarized fluxes.", "From power least squares fittings, we find a difference of $\\sim $ 0.6.", "This is entirely consistent with , in which a optical-to-near-IR power-law spectral index difference of $\\sim $ 0.74 between the total and polarized fluxes of their quasar sample can be estimated from their Fig. 1.", "Although we cannot precisely measure the optical-to-near-IR spectral slope in NGC 4151 using a multi-epoch composite spectrum, we confirm the methodology of : the polarized light spectrum of NGC 4151 seems to corroborate the existence of an optically thick, thermally heated accretion disk structure, at least in its outer near-IR emitting radii.", "Single epoch, multi-wavelength polarimetric data are required to get a quantitative conclusion." ], [ "Polarization degree and angle", "Looking at the polarization degree of NGC 4151 in Fig.", "REF , we observe a rather wavelength-dependent spectrum.", "We first made sure that the polarization degree and angle continua were not subject to variability due to the different epochs of observation.", "demonstrated that the B-band degree and angle of polarization of NGC 4151 did not vary (were constant within the observational error bars) between 1997 and 2003.", "A similar conclusion can be drawn between 0.48 and 0.71 $\\mu $ m, were the polarized observations of and indicate that the degree and angle of polarization remained constant over $\\sim $ 15 years.", "We are thus sure that intrinsic variability, such as synchrotron polarization induced in a failed jet, is not an issue here.", "The polarization we report has been corrected for instrumental polarization by the respective authors and interstellar polarization is not an issue.", "NGC 4151 is situated close to the north galactic pole (RA: 182$^\\circ $ .635745, Dec: 39$^\\circ $ .405730) so the interstellar contribution is negligible.", "It was measured to be 0.07% close to 0.44 $\\mu $ m by .", "The only remaining source of extra-nuclear light is the host galaxy, a (R')SAB(rs)ab galaxy type according to .", "The starlight of such spiral galaxy usually peaks in the 0.7 – 2 $\\mu $ m band (in $\\nu $ F$_\\nu $ , ) and, in the case of NGC 4151, the AGN light should dominate shortward of 0.5 $\\mu $ m .", "So, where does the strong wavelength-dependence of the UV polarization come from?", "If electron scattering was the sole physical mechanism responsible for the observed polarization, both the polarization degree and angle would have been constant from the UV to the near-IR.", "Here, the increase of the polarization degree shortward 0.23 $\\mu $ m is qualitatively explained by the accretion disk atmosphere models presented by .", "In order to explain and reproduce far-UV polarimetric observations of AGNs, the authors developed a numerical tool to calculate fully self-consistent models of pure-hydrogen disk atmospheres.", "They included the effects of opacity absorption (both from free-free absorption and photo-ionization) and found that UV polarization from disk atmospheres rises at shorter wavelengths.", "A maximum is reached between 0.1 and 0.2 $\\mu $ m, where a rotation on the polarization position angle might occur, depending on the effective temperature and surface gravity of the model (see Figs.", "2 and 3 in ).", "Below the Lyman edge (0.0912 $\\mu $ m) the polarization is expected to drop dramatically before rising up to several percents in the extreme-UV.", "Unfortunately, there is no polarimetric data on NGC 4151 below 0.16 $\\mu $ m so a far-UV spectropolarimeter is needed to confirm that the rise of polarization we see at 0.16 – 0.2 $\\mu $ m in Fig.", "REF is real.", "From 0.2 to 0.5 $\\mu $ m the polarization degree rises, from $\\sim $ 0.5% to $\\sim $ 2.0%.", "This wavelength-dependence is the signature of dust scattering, most likely from the inner surfaces of the dusty circumnuclear region and, less likely, from the dust grains inside the polar regions.", "The polarization position angle is close to 90$^\\circ $ which, when compared to the 450 milliarseconds radio structure along a position angle $\\approx $ 83$^{\\circ }$ observed with VLBI/Merlin , indicates that equatorial scattering dominates.", "Indeed, subtracting the radio position angle from the measured polarization angle gives a result close to 0$^\\circ $ .", "This means that the polarization position angle is parallel to the small-scale radio axis, hence scattering mainly occurs along the equatorial plane and not inside the polar outflows.", "It was shown by that an isolated dusty torus, viewed at an inclination close to its half-opening angle is able to produce a polarization degree that rises at longer wavelengths, up to several percents.", "A more complex structure inclined at 45$^\\circ $ , including an accretion disk, a BELR, a circumnuclear obscurer and ionized polar winds is also capable of producing a polarization degree that increases with increasing wavelengths, together with a parallel polarization position angle .", "Mie scattering is thus the dominant mechanism producing the 1 – 10$\"$ near-UV/optical polarization in NGC 4151.", "Electron scattering also occurs but is less dominant, otherwise the polarization degree would be grayer, such as seen in the first arc-second around the core of NGC 1068 , , .", "Interestingly, our result differs from the conclusions of who found that electron scattering is the dominant scattering mechanism in their sample.", "This difference is actually due to target selection: NGC 4151 is a low-luminosity AGN (Seyfert galaxy) while 's sample consists of 6 high-luminosity AGNs (quasars).", "Because quasars are much brighter than Seyferts, radiative pressure tends to destroy a larger fraction of dust in the close environment of the AGN, resulting in circumnuclear dusty regions that are much thinner (geometrically flattened along the equatorial plane) than in the Seyfert's case , , .", "In particular, found that low-luminosity AGNs tend to have more massive torus, situated closer to the central engine, than quasars.", "This results in a larger solid angle for light-dust interaction at the inner radius of the circumnuclear dust in NGC 4151 than for quasars, hence a stronger impact of Mie scattering onto the observed polarization.", "Our results conveniently confirm that low-luminosity AGNs are likely to be embedded in a dustier environment than quasars.", "From 0.5 $\\mu $ m to 1.0 $\\mu $ m, the polarization degree decreases due to the onset of the flux dominance from the host galaxy .", "The dilution is due to unpolarized starlight that peaks in this regime.", "The same behavior was observed in the polarized spectrum of NGC 1068 .", "The polarization angle remains constant, indicating that equatorial scattering still dominates from 0.5 to 1.0 $\\mu $ m. The polarization position angle only deviates from the parallel alignment after 1.0 $\\mu $ m. The polarization angle rotates from $\\sim $ 83$^\\circ $ to 45$^\\circ $ at 2.2 $\\mu $ m then stabilizes up to the mid-IR (8 – 12 $\\mu $ m).", "This variation is of particular interest because it clearly tells us that the physical mechanisms responsible for the production of polarization in the IR band in NGC 4151 is no longer dust scattering, as in the optical.", "Such finding must be put in the light of an infrared study of the polarization in NGC 4151 by .", "In their work, the authors stated “The nearly constant degree and [angle] of polarization from 1 $\\mu $ m to 12 $\\mu $ m strongly suggest that a single polarization mechanism dominates in the core of NGC 4151”.", "We entirely agree with them.", "However, they also stated that “scattering off optically thick dust ... is consistent with the observed nearly constant degree and [polarization angle]”.", "This is a reasonable assumption only if the 1 – 12 $\\mu $ m band is isolated from the optical and UV polarization data.", "In our extended study, we showed that dust scattering occurs in the optical but is no longer the main mechanism producing polarization in the infrared due to the rotation of the polarization angle did not account for the infrared polarization measurement obtained by at 2.2 $\\mu $ m that would have otherwise better highlighted the smooth variation of the polarization angle in the near-infrared band in their study.. Another supporting clue for this statement is the fact that the bump in polarization around 1 – 2 $\\mu $ m cannot be produced by Mie scattering alone , .", "In fact, looking at Fig.", "3 from , dichroic absorption from aligned dust grains appears to be a better explanation for the wavelength-dependent IR polarization in NGC 4151.", "This is also supported by the simulations from , where a dichroic component is necessary to produce the expected polarization degrees longward of 1.5 $\\mu $ m (see their Fig.6, bottom panel).", "In-between 12 and 89 $\\mu $ m, another change in the polarization process happens, since polarization from magnetically aligned dust grains dominates over other polarization mechanisms (e.g.", "dust/electron scattering, dichroic absorption, and synchrotron emission).", "The null polarization at 89 $\\mu $ m can then arise from 1) a dominant turbulent magnetic field in the central tens parsecs from the core, and/or 2) a depolarization effect due to the large beam (7.80$\"$ , 0.68 kpc), averaging a more complex underlying field." ], [ "Discussion", "The polarimetric data from WUPPE must be taken with caution.", "In the UV regime, polarimetric measurements of AGN ionizing fluxes have a low signal-to-noise ratio.", "For this reason, unbinned polarization plots always appear to show greater polarization than binned results.", "This is due to the fact that polarization is a positive quantity.", "When calculating the average polarization over a wavelength range, the mean values of the Stokes parameters Q and U are error-weighted but WUPPE suffered from instrumental issues that could not be reliably calibrated.", "For this reason there is no data between 2368 and 2430 Å, and the measurements in the 1600 – 3000 Å band should be analyzed with circumspection.", "The error bars in Fig.", "REF are 1-$\\sigma $ errors and the Q and U values are in different quadrants of the Q-U plane, which partially explains why the polarization angle may disagree so much between two consecutive bins (e.g., the 1600 – 2000 and the 2000 - 2200 Å bins).", "Finally, there is a correction to the polarimetric value that is not the simple error weighted means of $\\sqrt{\\langle Q \\rangle ^2 + \\langle U \\rangle ^2)}$ .", "The adjustment of the final polarimetric values as a function of polarimetric signal-to-noise is described in .", "It was shown by and that at least 3 high-redshift AGNs have a remarkably high polarization in the far-UV.", "One of the best examples is PG 1630+377 ($z \\approx $ 1.48).", "Its UV polarization degree rises rapidly up to 20% below the Lyman edge, a value that is usually found in blazing sources.", "Numerous scenarios for explaining such polarization features has been proposed by , including Faraday screens covering highly polarized continuum sources and geometric dilution.", "We now know that source obscuration is not an option as the inclination of NGC 4151 is such that we can directly see its central engine, allowing reverberation mapping studies , .", "Simulations by also failed to reproduce such dramatic value using a pure-hydrogen disk atmosphere model.", "More recently, new computations by tried to compute the Rayleigh polarization emerging from scattered radiation around the Lyman alpha wavelength in thick neutral regions but they could not apply their results to PG 1630+377.", "One of the reasons for such slow progresses is the lack of data.", "We only have a handful of UV polarimetric measurements from high redshift quasars ($z >$ 0.5).", "The 1 – 12 $\\mu $ m polarization of 6 Seyfert galaxies shows a diverse set of physical processes arising from its cores .", "For radio-quiet obscured AGN, the polarization is mainly arising from dichroic absorption by the dusty obscured surrounding the active nuclei or host galaxy.", "However, for NGC 4151, an un-obscured radio-quiet AGN, the physical component producing the IR polarization is difficult to identify.", "The reason is due to the competing polarization mechanisms of the several physical components within the beam size of the observations.", "We here provide a potential interpretation based on the shape of the 1–89 $\\mu $ m polarized spectrum and radiative alignment torque mechanism by .", "These authors show that the polarization peak arising from dichroic absorption by aligned dust grains changes as a function of the radiation field, extinction, and dust grain properties.", "To reproduce our 1 – 2 $\\mu $ m bump polarization, a direct view to the source with low extinction, A$_{v} \\le 5$ , oblate grains with axial ratio of 1.5, and dust grains expose to weak radiation field are required .", "This physical environment predicts a negligible emissive polarization at 89 $\\mu $ m with a rise of polarization at longer wavelengths .", "We interpret these results as that our 1 – 89 $\\mu $ m polarization arises from elongated and aligned dust absorption in the 1 – 12 $\\mu $ m wavelength range and emission at 89 $\\mu $ m. As the dust grains are exposed to weak radiation field, the dust may be located in the outer areas of the torus or NLR as suggested by .", "Our approach has shown that a detailed analysis of the total and polarized flux, the degree and position angle of polarization, together with high-angular resolution observations, is a powerful tool to identify the most likely mechanisms producing polarization in the whole electromagnetic spectrum." ], [ "Conclusions", "We have compiled, for the first time, the polarized SED of NGC 4151, one of the archetypal pole-on Seyfert galaxies.", "Despite the inherent problem of UV and optical variability, we have demonstrated that the optical-to-near-IR power-law spectral index of NGC 4151 is much bluer in polarized flux.", "This is in perfect agreement with past studies of quasars polarized SED, strengthening the case for the use of polarimetry to unveil the true nature of the Big Blue Bump.", "We have analyzed the wavelength-dependent polarization of this object and found a potential signature of the accretion disk atmosphere in the shortest wavelengths.", "However, due to the unreliability of the instrument in this waveband, new UV polarimetric observations are necessary.", "We have also established that the continuum polarization of NGC 4151 is mainly due to dust scattering from the near-UV to the near-IR.", "Compared to the sample of , who found that electron scattering dominates in higher-luminosity AGNs, our result tend to confirm that low-luminosity objects such as NGC 4151 are embedded in dustier environments.", "A smooth rotation of the polarization position angle between 1 and 2.2 $\\mu $ m identified a change in the polarization mechanism that we attribute to the onset of dichroic absorption from aligned dust grains.", "In the far-IR, polarization from dichroic absorption becomes less dominant and polarized emission from elongated and aligned dust grains prevails.", "Our results demonstrate that single epoch, broadband, polarimetric measurements are necessary to unveil the physical mechanisms that produce polarization in the heart of AGNs.", "We thus advocate for new polarimeters that could cover the a large fraction of the UV-to-IR electromagnetic spectrum.", "In addition, WUPPE data are the only spectropolarimetric UV measurements of NGC 4151 so far.", "This pull the trigger on the importance of new UV spectropolarimeters in the future.", "If we aim at investigating the UV properties of nearby AGNs, it is necessary to probe the shortest wavelengths in order to uncover the potential signatures from the accretion disk or its atmosphere.", "Spectropolarimetric observations of nearby AGN at rest wavelengths below the Lyman edge are thus mandatory to push forward the analysis.", "Future spatial instruments such as POLLUX, a high-resolution UV spectropolarimeter proposed for the 15-meter primary mirror option of LUVOIR , , or satellites at higher energies such as IXPE , are necessary to shed light on this topic.", "Furthermore, high-spatial resolution observations in the 1– 200 $\\mu $ m can provide us with detailed information about the physical scales of the dusty structure obscuring the AGN as well as the presence/absent of magnetic fields in the accretion flow to the active nuclei.", "Future ground-based 30-m class telescopes and space-base telescopes equipped with sensitive polarimeters, such as SPICA [1] and Origins , are thus also mandatory." ], [ "Acknowledgments", "We would like to thank the anonymous referee for taking the time to write an encouraging report on our paper.", "The authors are grateful to Robert “Ski” Antonucci for his remarks and suggestions that improved the quality of this article.", "FM would like to thank the Centre national d'études spatiales (CNES) who funded his post-doctoral grant “Probing the geometry and physics of active galactic nuclei with ultraviolet and X-ray polarized radiative transfer”.", "Based partially on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 07\\0032 Program.", "SOFIA is jointly operated by the Universities Space Research Association, Inc. (USRA), under NASA contract NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the University of Stuttgart." ] ]

[ [ "Measurement of jet-medium interactions via direct photon-hadron\n correlations in Au$+$Au and $d$$+$Au collisions at $\\sqrt{s_{_{NN}}}=200$ GeV" ], [ "Abstract We present direct photon-hadron correlations in 200 GeV/A Au$+$Au, $d$$+$Au and $p$$+$$p$ collisions, for direct photon $p_T$ from 5--12 GeV/$c$, collected by the PHENIX Collaboration in the years from 2006 to 2011.", "We observe no significant modification of jet fragmentation in $d$$+$Au collisions, indicating that cold nuclear matter effects are small or absent.", "Hadrons carrying a large fraction of the quark's momentum are suppressed in Au$+$Au compared to $p$$+$$p$ and $d$$+$Au.", "As the momentum fraction decreases, the yield of hadrons in Au$+$Au increases to an excess over the yield in $p$$+$$p$ collisions.", "The excess is at large angles and at low hadron $p_T$ and is most pronounced for hadrons associated with lower momentum direct photons.", "Comparison to theoretical calculations suggests that the hadron excess arises from medium response to energy deposited by jets." ], [ "Introduction", "Collisions of heavy nuclei at the Relativistic Heavy Ion Collider (RHIC) produce matter that is sufficiently hot and dense to form a plasma of quarks and gluons [1].", "Bound hadronic states cannot exist in a quark gluon plasma, as the temperatures far exceed the transition temperature calculated by lattice quantum chromodynamics (QCD) [2].", "Experimental measurements and theoretical analyses have shown that this plasma exhibits remarkable properties, including opacity to traversing quarks and gluons [3], [4].", "However, the exact mechanism for energy loss by these partons in quark gluon plasma and the transport of the deposited energy within the plasma is not yet understood.", "Experimental probes to address these questions include high momentum hadrons, reconstructed jets, and correlations among particles arising from hard partonic scatterings [1] occurring in the initial stages of the collision.", "Direct photons are produced dominantly via the QCD analog of Compton scattering, q + g $\\rightarrow $ q + $\\gamma $ , at leading order, and do not interact via the strong force as they traverse the plasma.", "In the limit of negligible initial partonic transverse momentum, the final state quark and photon are emitted back-to-back in azimuth with the photon balancing the transverse momentum of the jet arising from the quark.", "Consequently, measuring the correlation of high momentum direct photons with opposing hadrons allows investigation of quark gluon plasma effects upon transiting quarks and their fragmentation into hadrons.", "Correlations of direct photons with hadrons and jets have been measured by the PHENIX [5], [6] and STAR [7] Collaborations at RHIC, and by the CMS and ATLAS collaborations at the Large Hadron Collider [8], [9], [10], [11], [12], [13], [14].", "Using the photon energy to tag the initial energy of the quark showed that quarks lose a substantial amount of energy while traversing the plasma [15], [6].", "The photon tag also allows construction of the quark fragmentation function $D(z)$ , where $z=p^{\\rm hadron}/p^{\\rm parton}$.", "Here, $z$ represents the fraction of the quark's original longitudinal momentum carried by the hadrons.", "In photon-hadron ($\\gamma $ -h) correlations, $z$ can be approximated by $z_T$ = $p_T^{\\sc hadron}/p_T^{\\gamma }$ .", "Comparison of $\\gamma $ -h correlations in heavy ion collisions to those in $p$$+$$p$ collisions quantifies the plasma's impact on parton fragmentation.", "$\\gamma $ -h correlations in p+A or $d$$+$$A$ collisions will reflect any cold nuclear matter modification of jet fragmentation.", "The CMS collaboration also studied jets correlated to neutral $Z$ bosons [16].", "At RHIC, the fragmentation function is substantially modified in central Au$+$ Au collisions [6], [17].", "High $z$ fragments are suppressed, as expected from energy loss.", "Low $z$ fragments are enhanced at large angles with respect to the jet core, i.e.", "with respect to the original quark direction.", "CMS and ATLAS have measured jet fragmentation functions using reconstructed jets to tag the parton energy.", "These studies, conducted with jet energies of $\\approx $ 100 GeV, show enhancement of low $p_T$ (i.e.", "low $z$ ) jet fragments in central Pb$+$ Pb collisions [18], [19].", "In addition, CMS has shown that the energy lost by the quark is approximately balanced by hadrons with approximately 2 GeV $p_T$  [20] in the intrajet region.", "This is in qualitative agreement with the RHIC result, even though the initial quark energy differs by an order of magnitude.", "There has been considerable theoretical effort to describe jet-medium interactions.", "Several mechanisms for parton energy loss were compared by the JET Collaboration [21].", "The medium response to deposited energy is now under study by several groups [22], [23], [24], [25].", "The deposited energy may be totally equilibrated in the plasma, but alternatively the deposited energy may kick up a wake in the expanding plasma [22], [26].", "Different descriptions of plasma-modified gluon splitting result in different fragmentation functions, and can be tested by comparing the predictions to direct photon-hadron ($\\gamma _{\\rm dir}$ -h) correlations.", "The previously published analysis of $\\gamma _{\\rm dir}$ -h correlations showed an enhancement in soft particle production at large angles.", "However, due to limited statistics, it was not possible to investigate how the fragmentation function depends on the parton energy or the medium scale.", "In this paper, we explore this question by looking at the direct photon $p_T$ dependence of the fragmentation function modification.", "We investigate whether enhancement over the fragmentation function in $p$$+$$p$ collisions depends on the fragment $z_T$ or on the fragment $p_T$ .", "That is, does it depend on the jet structure or does it reflect the distribution of particles in the medium?", "We also present first results on $\\gamma _{\\rm dir}$ -h correlations in d$+$ Au collisions to investigate possible cold-nuclear-matter effects on the fragmentation function.", "Fragmentation function modification is quantified here by the nuclear modification factor, $I_{AA}$ , which is a ratio of the fragmentation function in Au$+$ Au collisions to that in p$+$ p collisions." ], [ "Dataset and Analysis", "In 2011, PHENIX collected data from Au$+$ Au collisions at $\\sqrt{s_{_{NN}}}=200$ GeV.", "After event selection and quality cuts, 4.4 billion minimum-bias (MB) events were analyzed.", "These are combined with the previously reported 3.9 billion MB Au$+$ Au events from 2007 and 2.9 billion from 2010 [6].", "The high momentum photon triggered $d$$+$ Au data set at $\\sqrt{s_{_{NN}}}$ = 200 GeV was collected in 2008, and 3 billion events are analyzed.", "The $p$$+$$p$ comparison data are from 2005 and 2006 [15].", "The measurements in this paper use the PHENIX central spectrometers [27].", "Two particle correlations are constructed by pairing photons or $\\pi ^0$ s measured in the electromagnetic calorimeter (EMCal) [28] with charged hadrons reconstructed in the drift chambers and pad chambers [29].", "The acceptance in pseudorapidity is $|\\eta |<0.35$ , while each spectrometer arm covers 90 degrees in azimuth.", "Beam-beam counters [30], located at 1.44 meters from the center of the interaction region, cover the pseudorapidity range from 3.0 to 3.9 and full azimuthal angle.", "They are used to determine the collision centralities and vertex positions.", "Figure REF shows the detector configuration in 2011.", "Figure: Side view of the PHENIX central arm spectrometers in 2011.Photons and $\\pi ^0$ s are measured in the EMCal.", "There are four sectors of lead-scintillator (PbSc) sampling calorimeters in the west arm, while the east arm has two sectors of lead-scintillator and two lead-glass (PbGl) Čerenkov calorimeters.", "The PbSc and PbGl calorimeters have energy resolutions of $\\sigma _E/\\sqrt{E}$ = 8.1$\\%$ /$\\sqrt{E}$ $\\oplus $ 2.1$\\%$ and 5.9$\\%$ /$\\sqrt{E}$ $\\oplus $ 0.8$\\%$ , respectively.", "Photons are selected via an electromagnetic shower shape cut [31] on energy clusters.", "The high granularity of the EMCal, $\\delta \\eta \\times \\delta \\phi $ = 0.011 $\\times $ 0.011 for PbSc and 0.008 $\\times $ 0.008 for PbGl, allows for $\\pi ^0$ reconstruction via the $\\pi ^0\\rightarrow \\gamma \\gamma $ channel (invariant mass = 120–160 MeV/c$^2$ ) up to $p_T$ = 15 GeV, beyond which shower merging becomes significant.", "A charged track veto is applied to remove possible hadron or electron contamination in the photon sample, reducing auto-correlations in the measurement.", "The EMCal system is also used to trigger on $d$$+$ Au events with high $p_T$ photons.", "Two particle correlations are constructed as a function of $\\Delta \\phi $ , the azimuthal angle between photon or $\\pi ^0$ triggers and associated hadron partners.", "Pairs arise from jet correlations superimposed on a combinatorial background from the underlying event.", "In $p$$+$$p$ and $d$$+$ Au collisions where the event multiplicity is low, we treat this background as flat in $\\Delta \\phi $ and subtract it, normalizing the level via the zero-yield-at-minimum (ZYAM) procedure [32].", "In Au$+$ Au collisions, the background has an azimuthal asymmetry quantified in the flow parameters $v_n$ , which are used to modulate the subtracted background, as described in Eqn.", "1.", "Only $v_2$ is included in the subtraction, while higher-order effects are included as an additional systematic uncertainty on the final results.", "We report jet pairs as conditional (or per-trigger) yields of hadrons.", "Detector acceptance corrections are determined using mixed events with similar centrality and collision vertex.", "For Au$+$ Au collisions, the background level $b_0$ is estimated using an absolute normalization [32], determined from the uncorrelated single-photon and single-hadron production rates.", "The final invariant yield of associated hadrons is obtained by dividing the background-subtracted correlated hadron yields by the number of triggers $N_t$ and correcting for the associate charged hadron efficiency $\\epsilon ^a$ , determined by a geant detector simulation: $\\frac{1}{N_t}\\frac{dN^{\\rm pair}}{d\\Delta \\phi } = \\frac{1}{N_t}\\frac{N^{\\rm pair}}{\\epsilon ^{a}\\int \\Delta \\phi }\\Bigg \\lbrace \\frac{dN^{\\rm pair}_{\\rm real}/d\\Delta \\phi }{dN^{\\rm pair}_{\\rm mix}/d\\Delta \\phi } - b_0\\big [1+2\\big \\langle v_2^tv_2^a \\big \\rangle cos\\big (2\\Delta \\phi \\big ) \\big ] \\Bigg \\rbrace ,$ where $v_2^t$ and $v_2^a$ are the elliptic flow magnitudes independently measured for the trigger and associated particles, respectively [6].", "These modulate the angular distribution of the background.", "Lastly, $N^{\\rm pair}$ denotes the number of trigger-associate pairs.", "The subscript “real\" refers to a trigger-associate particle pair that came from the same event, and the subscript “mix\" refers to trigger-associate pairs that come from different events and are used to correct for correlations due to detector effects.", "In both Au$+$ Au and $d$$+$ Au analyses, photons with transverse momentum of 5 to 15 GeV/$c$ are selected as triggers.", "To extract yields of hadrons associated with direct photons, the background from decay photon correlations with hadrons must be subtracted.", "In Au$+$ Au collisions, where the multiplicity is high, this is achieved via a statistical subtraction procedure.", "If $N_{inc}$ , $N_{dec}$ , and $N_{dir}$ are the inclusive, decay, and direct photon yields, respectively, then $N_{dir} = N_{inc} - N_{dec}$ .", "It follows that the conditional yield of hadrons $Y$ for different photon trigger samples is $Y_{\\rm dir} = \\frac{R_{\\gamma }Y_{\\rm inc} - Y_{\\rm dec}}{R_{\\gamma }-1},$ where $R_{\\gamma } \\equiv N_{inc}/N_{dec}$ and is measured independently [33].", "The decay photon background is estimated using measured $\\pi ^0$ -hadron ($\\pi ^0$ -h) correlations and a Monte Carlo pair-by-pair mapping procedure.", "The simulation calculates the probability distribution for decay photon-hadron ($\\gamma _{\\rm dec}$ -h) pairs in a certain photon $p_T$ range as a function of the parent $\\pi ^0$ $p_T$ .", "$\\gamma _{\\rm dec}$ -h correlations are constructed via a weighted sum over all individual $\\pi ^0$ -hadron pairs, where the weighting factor reflects the kinematic probability for a $\\pi ^0$ at a given $p_T$ to decay into a photon in the selected $p_T$ range.", "The $\\gamma _{\\rm dec}$ -h per-trigger yield can be described by the following equation: $Y_{\\rm dec} = \\frac{\\int \\rho (p_{T \\pi ^0} \\rightarrow p_{T \\gamma }) \\epsilon ^{-1}(p_{T \\pi ^0})N_{\\pi ^0-h}dp_{T \\pi ^0}}{\\int \\rho (p_{T \\pi ^0} \\rightarrow p_{T \\gamma }) \\epsilon ^{-1}(p_{T \\pi ^0})N_{\\pi ^0}dp_{T \\pi ^0}},$ where $\\rho $ gives the probability that a $\\pi ^0$ decays to a photon with $p_{T\\gamma }$ , and $\\epsilon $ is the $\\pi ^0$ reconstruction efficiency, which can be determined by scaling the raw $\\pi ^0$ spectra to a power law fit to published data [34].", "$N_{\\pi ^0-h}$ and $N_{\\pi ^0}$ are the number of $\\pi ^0$ -h pairs and number of $\\pi ^0$ 's, respectively.", "When reconstructing the $\\pi ^0$ , a strict cut on the asymmetry of the energy of the two photons is applied to reduce the combinatorial background from low energy photons.", "The probability weighting function, determined from Monte Carlo simulation, takes into account the actual EMCal response, including energy and position resolution and detector acceptance.", "With the $\\pi ^0$ to decay photon $p_T$ map, $\\rho $ , the inclusive photon sample can be separated into a meson decay component and a direct component.", "Of the meson decay component, 20$\\%$ of the decay yield is from non-$\\pi ^0$ decays as calculated from previous PHENIX results [35].", "To construct $\\gamma _{\\rm dec}$ -h yields with trigger photon $p_T$ of 5–15 GeV/$c$ , hadron correlations with $\\pi ^0$ of 4 $\\le p_T\\le $ 17 GeV/$c$ are utilized.", "The slightly wider $p_T$ range is chosen to account for decay kinematics, as well as $p_T$ smearing from the EMCal energy and position resolution.", "An additional cutoff correction accounts for the small $\\gamma _{\\rm dec}$ -h yield in the trigger $p_T$ range 5–15 GeV/$c$ from $\\pi ^0$ with $p_T \\ge $ 17 GeV/$c$ .", "The merging of decay photons from high $p_T$ $\\pi ^0$ is not accounted for in the Monte Carlo mapping simulation.", "Instead, the efficiency to detect photons from a high momentum parent meson is calculated via geant simulation of the full detector response.", "This loss is included in the probability function as an additional correction.", "The opening angle of photon pairs that merge is small, thus they are removed from the measured inclusive photon sample by the shower shape cut.", "In $d$$+$ Au collisions, where the underlying event background is much smaller, it is possible to improve the signal to background for direct photons.", "This is done event-by-event using a photon isolation cut and by removing all photons identified (tagged) as resulting from a $\\pi ^0$ decay [15].", "First, all photons with $p_T \\ge $ 0.5 GeV/$c$ are paired.", "Those pairs with invariant mass between 120–160 MeV/c$^2$ are tagged as decay photons and removed from the inclusive sample.", "Next, an isolation criterion is applied to the remaining photons to further reduce the background of decay photons, as well as contamination from fragmentation photons.", "The isolation cut requires that the energy in a cone around the trigger photon be less than 10$\\%$ of the photon energy in $p$$+$$p$ collisions.", "In the $d$$+$ Au analysis, the cut is modified slightly to include the effect of the modest underlying event.", "The underlying event is evaluated separately for each $d$$+$ Au centrality class, resulting in an isolation criterion: $\\sum _{\\Delta R < R_{\\rm max}} E < (E_{\\gamma } * 0.1+<E_{bg}>),$ where $E$ is the measured energy in the isolation cone, $E_\\gamma $ is the photon energy, $\\Delta R$ = $\\sqrt{\\Delta \\phi ^2 + \\Delta \\eta ^2}$ is the distance between the trigger photon and other particles in the event and $<E_{bg}>$ is the average energy inside the cone in the underlying event.", "The cone size ($R_{\\rm max}$ ) used in this analysis is 0.4.", "To account for the $d$$+$ Au underlying event, the ZYAM procedure [32] is applied to the angular correlation functions for each centrality class.", "As an isolation cut distorts the near-side yield, the minimum point is determined within the restricted $\\Delta \\phi $ range of 0.9–1.6 rad.", "The zero-point yield is determined by integrating in a 0.03 rad range around the minimum point.", "The hadron conditional yield reported here is corrected for the PHENIX hadron acceptance.", "The ZYAM subtracted inclusive and decay yields for each centrality are combined using a weighted sum based on the number of each type of trigger to obtain the MB yields.", "Some decay photons are missed by the $\\pi ^0$ tagging procedure and slip through the isolation cut to be counted as direct photons.", "Such falsely isolated $\\gamma _{\\rm dec}$ -h correlations are corrected via a statistical subtraction, similar to Eq.", "REF .", "If we define $N_{\\rm inc-tag}^{\\rm iso}$ as the yield of isolated photons after removing those isolated photon tagged as decay photons, $N_{\\rm dec}^{\\rm miss,iso}$ as those decay photons that are isolated but not tagged as decay photons, and $N_{\\rm dir}^{\\rm iso}$ as isolated direct photons, then $N_{\\rm dir}^{\\rm iso} = N_{\\rm inc-tag}^{\\rm iso} - N_{\\rm dec}^{\\rm miss,iso}$ .", "It follows that the condition of yield of hadrons for direct, isolated photons is $Y_{\\rm dir}^{\\rm iso}= \\frac{R_{\\gamma }^{\\rm eff}Y_{\\rm inc-tag}^{\\rm iso}-Y_{\\rm dec}^{\\rm miss,iso}}{R_{\\gamma }^{\\rm eff}-1},$ where $R_{\\gamma }^{\\rm eff} = \\frac{N^{\\rm iso}_{\\rm inc}}{N^{\\rm iso}_{\\rm dec}}~= \\frac{R_\\gamma }{(1-\\epsilon ^{\\rm tag}_{\\rm dec})(1-\\epsilon ^{\\rm iso}_{\\rm dec})}\\frac{N_{\\rm inc-tag}^{\\rm iso}}{N_{\\rm inc}}$ where $\\epsilon _{\\rm dec}^{\\rm iso}$ is the isolation cut efficiency and $\\epsilon _{\\rm dec}^{\\rm tag}$ is the tagging efficiency.", "More detail on the subtraction procedures and cuts can be found in references [5], [15].", "In the Au$+$ Au analysis, there are four main sources of systematic uncertainties.", "The systematic uncertainty coming from the statistical subtraction method is due to the statistical and systematic uncertainties on the value of $R_{\\gamma }$ .", "There are also uncertainties when extracting the jet functions due to uncertainties on the value of the elliptic flow modulation magnitude, $v_2$ .", "This analysis uses published values and uncertainties from PHENIX [6].", "The absolute normalization method to determine the underlying event background level, and the determination of the decay photon $p_T$ mapping are also significant contributors to the overall systematic uncertainties.", "The uncertainties, along with their $p_T$ and centrality dependence, are propagated into the final jet functions and per-trigger hadron yields.", "The systematic uncertainty on the hadron efficiency determination comes in as a global scale uncertainty on the correlated hadron yields.", "In MB $d$$+$ Au collisions, $v_2$ is small.", "However, the systematic uncertainties on $\\gamma $ -h correlations include those arising from the ZYAM procedure used to determine the combinatorial background.", "There is also an uncertainty arising from the $\\pi ^0$ tagging and isolation cuts, which is included in the quoted systematic uncertainty.", "Figure: Per-trigger yield of hadrons associated to direct photons in Au++Aucollisions (closed [black] circles) for direct photon p T p_T 5–9 GeV/cc,compared with pp++pp baseline (open [blue] squares), in various ξ\\xi bins.Figure: Per-trigger yield of hadrons associated to direct photons in dd+AuAucollisions (closed [black] circles) for direct photon p T p_T 7–9 GeV/cc,compared with pp+pp baseline (open [blue] squares), in various ξ\\xi bins." ], [ "Results", "In this paper, we aim to quantify the modification of the jet fragmentation function $D(z)$ in Au$+$ Au and $d$$+$ Au collisions, compared to the $p$$+$$p$ baseline.", "The jet fragmentation function describes the probability of an outgoing parton to yield a hadron with momentum fraction $z = p^{\\rm hadron}/p^{\\rm parton}$ .", "Assuming that the initial-state $k_T$ of partons in a nucleon has a negligible effect, then $z_{T} =p_{T}^{\\rm hadron}/p_{T}^{\\gamma }$ can be used to approximate $z$ .", "To focus on the low $z_T$ region, where modification is anticipated, we use the variable $\\xi = ln(1/z_{T})$ .", "Figure REF shows azimuthal angular distributions of hadrons associated with direct photons of 5 $< p_T <$ 9 GeV/$c$ , in the 0–40$\\%$ most central Au$+$ Au collisions, separated into bins of $\\xi $ .", "These distributions are a combination of the 2007, 2010 and 2011 data sets.", "The Au$+$ Au results are shown as closed [black] circles, with shaded boxes representing systematic uncertainties on the measurement.", "The $p$$+$$p$ $\\gamma _{\\rm dir}$ -h results are shown in open [blue] squares.", "The $p$$+$$p$ baseline measurement combines data collected in 2005 and 2006 [6], [15].", "It should be noted that the isolation cut in the $p$$+$$p$ analysis makes the near-side yield not measurable.", "Consequently, the $p$$+$$p$ points with $\\Delta \\phi <1$ are not shown in these distributions.", "On the near side, i.e.", "$\\Delta \\phi < \\pi /2$ , the Au$+$ Au $\\gamma _{\\rm dir}$ -h yields are consistent with zero, indicating that the statistical subtraction is properly carried out and next-to-leading-order effects are negligible.", "On the away-side, i.e.", "$\\Delta \\phi >\\pi /2$ , an enhancement in the Au$+$ Au data compared to $p$$+$$p$ is observed in the higher $\\xi $ bins.", "As noted before, this corresponds to low $z$ , where the observed hadrons carry a small fraction of the scattered parton's original momentum.", "In the low $\\xi $ bins, the Au$+$ Au per-trigger yield is suppressed, as expected if the parton loses energy in the medium.", "Fig.", "REF shows the $\\Delta \\phi $ distributions of isolated $\\gamma _{\\rm dir}$ -h yields in $d$$+$ Au and $p$$+$$p$ collisions, for direct photon $p_T$ 7–9 GeV/$c$ .", "The $d$$+$ Au and $p$$+$$p$ results are consistent in all the measured $\\xi $ bins.", "Figure: (a) Integrated away-side γ dir \\gamma _{\\rm dir}-h per-trigger yields ofAu++Au (closed [black] circles), dd+AuAu ([purple] crosses) andpp+pp (open [blue] squares), as a function of ξ\\xi .", "The pp+ppand dd+AuAu points have been shifted to the left for clear viewing, asindicated in the legend.", "(b) I AA I_{AA} (closed [black] circles) andI dA I_{dA} ([purple] crosses).Figure: I AA I_{AA} vs ξ\\xi for direct photon p T γ p_T^{\\gamma } of5–7 GeV/cc (closed [black] circles),7–9 GeV/cc (closed [red] squares),and 9–12 GeV/cc (closed [green] triangles).Figure REF (a) shows the fragmentation functions for all three systems as a function of $\\xi $ .", "These are calculated by integrating the per-trigger yield of hadrons in the azimuthal angle region $|\\Delta \\phi - \\pi | < \\pi /2$  rad.", "Data points for Au$+$ Au are plotted on the $\\xi $ axis at the middle of each $\\xi $ bin: 0.2, 0.6, 1.0, 1.4, 1.8, 2.2.", "The $p$$+$$p$ and $d$$+$ Au points have been shifted to the left in $\\xi $ for viewing clarity.", "As noted in the Introduction, $I_{AA} = Y_{AA}/Y_{pp}$ is a nuclear-modification factor, which quantifies the difference between the fragmentation functions in Au$+$ Au and $p$$+$$p$ .", "In the absence of any medium modifications, $I_{AA}$ should equal 1.", "Figure REF (b) shows $I_{AA}$ for direct photons of $5 <p_T^{\\gamma } < 9$ GeV/$c$ .", "In Au$+$ Au collisions, there is a clear suppression at low $\\xi $ and enhancement at high $\\xi $ .", "The $d$$+$ Au nuclear modification factor, $I_{dA}$ , is also shown as closed [purple] crosses in Fig.", "REF (b).", "$I_{dA}$ is consistent with unity across all $\\xi $ ranges, indicating that there is no significant modification of the jet fragmentation function in $d$$+$ Au collisions.", "Figure: I AA I_{AA} as a function of ξ\\xi for direct photon p T γ p_T^{\\gamma } of(a) 5–7, (b) 7–9, and (c) 9–12 GeV/cc.Three away-side integration ranges are chosen to calculate theper-trigger yield and the corresponding I AA I_{AA}:|Δφ-π|<π/2|\\Delta \\phi -\\pi |<\\pi /2 (closed [black] circles),|Δφ-π|<π/3|\\Delta \\phi -\\pi |<\\pi /3 (closed [blue] squares) and|Δφ-π|<π/6|\\Delta \\phi -\\pi |<\\pi /6 (closed [red] triangles).The statistics from the combined Au$+$ Au runs allow for a differential measurement as a function of direct photon $p_T$ (i.e.", "as a function of the approximate jet energy).", "Fig.", "REF shows $I_{AA}$ as a function of $\\xi $ for three direct photon $p_T$ ranges.", "While the associated hadron yields are smaller than those in $p$$+$$p$ at low $\\xi $ , the appearance of extra particles at higher $\\xi $ is observed for direct photons with $p_T$ of 5–7 GeV/$c$ .", "A qualitatively similar increase of $I_{AA}$ with $\\xi $ is visible for the 7–9 GeV/$c$ direct photon $p_T$ range.", "To investigate where the energy deposited in the plasma goes, we study the dependence of $I_{AA}$ on the integration range in azimuthal opening angle.", "The hadron yields are also integrated in two narrower angular ranges on the away side: $|\\Delta \\phi -\\pi | < \\pi /3$ rad and $|\\Delta \\phi -\\pi | < \\pi /6$ rad.", "The resulting $I_{AA}$ values are shown in Fig.", "REF for all three direct photon $p_T$ bins.", "The enhancement over $p$$+$$p$ is largest for the 5–7 GeV/$c$ direct photon momentum range, and for the full away-side integration range.", "The suppression pattern is similar for the different integration regions, suggesting that the jet core is suppressed, and the enhancement exists at large angles.", "The angular distributions support the observation from Fig.", "REF , that particle yields are enhanced at large angles with respect to the away-side jet axis in the $1.6<\\xi <2.0$ bin.", "Figure: Ratios of I AA I_{AA} as a function of direct photon p T p_Tfor threedifferent away-side integration ranges.Whether or not $I_{AA}$ becomes significantly larger than unity (what we have been referring to as enhancement) there is a tendency for $I_{AA}$ to increase with increasing $\\xi $ .", "To quantify this, we calculate the weighted averages of $I_{AA}$ values above and below $\\xi $ = 1.2.", "The ratio for each integration range is plotted in Fig.", "REF , as a function of the direct photon $p_T$ .", "The enhancement is largest for softer jets and for the full away-side integration range, implying that jets with lower energy are broadened more than higher energy jets.", "Figure: Measured I AA I_{AA} for direct photon p T p_T of (a) 5–7, (b) 7–9, and(c) 9–12 GeV/cc, as a function of ξ\\xi , are compared withtheoretical model calculations." ], [ "Discussion", "To determine whether $I_{dA}$ indicates any cold nuclear matter effects, the $\\chi ^2$ per degree of freedom values were calculated under the assumption of no modification and are determined to be 7.4/5, 4.0/5, 10.0/5 for direct photon $p_T$ bins 5–7, 7–9, and 9–12 GeV/$c$ , respectively.", "The result indicates that $I_{dA}$ is consistent with unity and therefore the jet fragmentation function is not significantly modified in $d$$+$ Au collisions, within the current uncertainties.", "This suggests that any possible cold nuclear matter effect is small.", "We next compare our Au$+$ Au results to predictions from the CoLBT-hydro model [26] in Fig.", "REF , which shows $I_{AA}$ as a function of $\\xi $ for the 3 direct photon $p_T$ bins; the $z_T$ axis is displayed on the top.", "The solid lines are from the CoLBT model calculated in the same kinematic ranges as the data.", "The model calculation shows the same trends with $\\xi $ as the data .", "CoLBT has a kinetic description of the leading parton propagation, including a hydrodynamical picture for the medium evolution.", "In this calculation, both the propagating jet shower parton and the thermal parton are recorded, along with their further interactions with the medium.", "Consequently, the medium response to deposited energy is modeled.", "The model clearly shows that as the direct photon $p_T$ increases, the transition where $I_{AA}$ exceeds one occurs at increasing $\\xi $ .", "According to this calculation, the enhancement at large $\\xi $ arises from jet-induced medium excitations, and that the enhancement occurs at low $z_T$ reflects the thermal nature of the produced soft particles.", "Figure REF (b) shows a BW-MLLA calculation (dashed [red] curve) in which it is assumed that the lost energy is redistributed, resulting in an enhanced production of soft particles [36].", "The calculation for jets with energy of 7 GeV in the medium is in relatively good agreement with the measured results.", "The model comparisons suggest that the enhancement of soft hadrons associated with the away-side jet should scale with the $p_T$ of the hadrons.", "A modified fragmentation function could be expected to produce a change at fixed $z_T$ .", "This is not consistent with either the data or the CoLBT model." ], [ "Summary", "We have presented direct photon-hadron correlations in $\\sqrt{s_{_{NN}}}=200$  GeV Au$+$ Au, $d$$+$ Au and $p$$+$$p$ collisions, for photon $p_T$ from 5–12 GeV/$c$ .", "As the dominant source of correlations is QCD Compton scattering, we use the photon energy as a proxy for the opposing quark's energy to study the jet fragmentation function.", "Combining data sets from three years of data taking at RHIC allows study of the conditional hadron yields opposite to the direct photons as a function of $z_T$ and the photon $p_T$ .", "This is the first time such a differential study of direct photon-hadron correlations has been performed at RHIC.", "We observe no significant modification of the jet fragmentation in d + Au collisions, indicating that cold nuclear matter effects are small or absent.", "We find that hadrons carrying a large fraction of the quark's momentum are suppressed in Au$+$ Au compared to $p$$+$$p$ and $d$$+$ Au.", "This is expected from energy loss of partons in quark gluon plasma.", "As the momentum fraction decreases, the yield of hadrons in Au$+$ Au increases, eventually showing an excess over the jet fragment yield in $p$$+$$p$ collisions.", "The excess is seen primarily at large angles and is most pronounced for hadrons associated to lower momentum direct photons.", "To address whether the excess is a result of medium modification of the jet fragmentation function or the excess indicates the presence of “extra\" particles from the medium, we compared to theoretical calculations.", "The calculations suggest that the observed excess arises from medium response to the deposited energy.", "Furthermore, the excess particles appear at low $z_T$ , corresponding to low associate hadron $p_T$ .", "This can be seen in each direct photon $p_T$ bin.", "We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.", "We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (People's Republic of China), Croatian Science Foundation and Ministry of Science and Education (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), J. Bolyai Research Scholarship, EFOP, the New National Excellence Program (ÚNKP), NKFIH, and OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research and SRC(CENuM) Programs through NRF funded by the Ministry of Education and the Ministry of Science and ICT (Korea).", "Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S.", "Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, the US-Hungarian Fulbright Foundation, and the US-Israel Binational Science Foundation." ] ]

[ [ "Electric charge estimation using a SensL SiPM" ], [ "Abstract The silicon photo-multipliers (SiPMs) are commonly used in the construction of radiation detectors such as those used in high energy experiments and its applications, where an excellent time resolution is required for triggering.", "In most of this cases, the trigger systems electric charge information is discarded due to limitations in data acquisition.", "In this work we propose a method using a simple radiation detector based on an organic plastic scintillator $2\\times2\\times0.3$~cm$^3$ size, to estimate the electric charge obtained from the acquisition of the fast output signal of a SensL SiPM model C-60035-4P-EVB.", "Our results suggest a linear relation between the reconstructed electric charge from the fast output of the SiPM used with respect to the one reconstructed with its standard signal output.", "Using our electric charge reconstruction method, we compared the sensitivity of two plastic scintillators, BC404 and BC422Q, under the presence of Sr90, Cs137, Co60, and Na22 radiation sources." ], [ "Introduction", "Silicon Photomultipiers (SiPM) have been widely used during the past two decades in different areas like high energy physics [2], [1], and its medical applications.", "A clear example is the development of Positron Emission Tomography (PET)[3] where the typical photo multiplier tube (PMT) is being replaced by the SiPM technology with the aim to improve its time and spatial resolutions [6], [5], [4], [7].", "Since 2013 SensL corporation has developed SiPMs with two signal outputs: Standard and fast [8].", "For a $6\\times 6$  mm$^3$ of SensL C-series SiPM, the standard signal output is characterized by a raise time of 4 ns and a pulse width of 100 ns, while the raise time of the fast signal output is 1 ns with a pulse width of 3.2 ns [9].", "Several works have reported the use of SiPMs [1], [3], [10], [11], [12], where the standard output is commonly used to reconstruct the deposited electric charge using the acquired photo current from the anode which is related with the deposited energy in the sensitive material of a radiation detector [5], [11], [13], [14].", "In recent years, the fast signal output has been used to improve the pulse shape discrimination of gammas and fast neutrons [2], [15].", "It has been also shown that exists an equivalent coincidence resolving time (CRT) between the fast and standard output signals of the SensL SiPM [16].", "An application for this fast pulse, is on a detector development with high time resolution as described in [17].", "In this work we use a simple radiation detector based in organic plastic scintillator to study the relation of the reconstructed electric charged using both SensL SiPM output signals.", "This work is organized as follows.", "In Section  the methodology of this work is described.", "In Section  we present the analysis and discussions of the results.", "Finally, in Section  we present our conclusions." ], [ "Instrumentation", "A MicroFC-60035 SiPM from SensL with a cell size of 35 $\\mu $ m, peak wavelength of 420 nm and package size of $6\\times 6$  mm$^2$ was used.", "In order to acquire the two signals from this SiPM, we developed a homemade printed circuit board (PCB) of $3 \\times 4$  cm, specifically designed for the described SiPM model.", "The schematic diagram is shown in Figure REF , where $V_{s}$ and $V_f$ refers to the standard and the fast output signal, respectively.", "As described in [9], an overvoltage of $V_{br}$ +5 V was used to maximize the photon detection efficiency (PDE) of the SiPM.", "Figure: Basic front-end electronics for polarization and acquisition of standard and fast signals.We choose BC404 [18] and BC422Q [19] plastic scintillator as radiation sensitive materials, with a volume of $20\\times 20\\times 3$  mm$^3$ .", "Some scintillation characteristics are shown in Table REF [20].", "The BC422Q material was selected with a weight percentage of benzophenone of $0.5\\%$ .", "Table: Scintillator material properties.", "The experimental setup is shown in Figure REF , where the SiPM was attached to the center of each plastic scintillator and a radioactive source was located in the opposite face.", "Four radioactive sources were used: $^{90}$ Sr, $^{22}$ Na, $^{137}$ Cs and $^{60}$ Co so, the description for these radioactive sources is shown in Table REF .", "Figure: Experimental setup scheme, showing the fast and standard outputs.Table: Radiation sources properties.A Tektronix DPO7054 digital oscilloscope was used for signal acquisition, with a 50 $\\Omega $ of coupling impedance and a sampling rate of 10 GS/s.", "For each radioactive source, $10^{4}$  events were recorded.", "Each event consists of $2\\times 10^{4}$  samples to reconstruct the pulse.", "The reconstruction and the data analysis was made offline with CERN ROOT software [21]." ], [ "Linear regression", "We reconstruct the electric charge from the fast ($Q_{F}$ ) and standard ($Q_{S}$ ) output signals, using the integrals given in equations(REF ) and (REF ).", "$Q_{S} &= \\int _{t_{i}}^{t_{f}}i_{s}(t)dt = \\frac{1}{50}\\int _{t_{i}}^{t_{f}}V_{s}(t)dt \\\\Q_{F} &= \\int _{t_{i}^{\\prime }}^{t_{f}^{\\prime }}i_{f}(t)dt = \\frac{1}{50}\\int _{t_{i}^{\\prime }}^{t_{f}^{\\prime }}V_{f}(t)dt$ If a linear relation between fast and standard signal outputs is assumed, and supposing $g_{x}$ and $g_{y}$ random variables with Gaussian Probability Distribution Functions (PDF), we can introduce the Pearson's correlation coefficient given by [23].", "$ R_{xy}= \\frac{Cov(g_{x},g_{y})}{\\sigma _{x}\\sigma _{y}}$ where $Cov(g_{x},g_{y})$ is defined as the covariance between random variables $g_{x}$ and $g_{y}$ , while $\\sigma _{x}$ and $\\sigma _{y}$ correspond to the standard deviation of $x$ and $y$ variables, respectively.", "After the described correlation test, a regression line can be adjusted to obtain a linear model based on statistical moments for each stochastic process, as described by the following equation $ y-\\bar{y}=\\frac{Cov(g_{x},g_{y})}{\\sigma _{x}^2}(x-\\bar{x})$ It can be rewritten in terms of the correlation coefficient as $ y=\\frac{\\sigma _{y}}{\\sigma _{x}}R_{xy}(x-\\bar{x})+\\bar{y},$ which is a standard equation of a straight line $ y=ax + b$ with $&a =\\frac{\\sigma _{y}}{\\sigma _{x}}R_{xy} \\\\&b =\\bar{y}-a\\bar{x}$ In this case, $x$ was defined as the charge $Q_{f}$ measured from the standard signal and $y$ as the charge $Q_{s}$ from fast output signal.", "Therefore, equations REF and REF can be rearranged in terms of charge, $&Q_{s}=aQ_{f} + b,\\\\&a =\\frac{\\sigma _{s}}{\\sigma _{f}}R_{fs}, \\\\&b =\\bar{Q_{s}}-a\\bar{Q_{f}},$ where $\\sigma _{f}$ and $\\sigma _{s}$ refer to standard deviation of fast and standard signal charges, respectively.", "While $R_{sf}$ is the correlation coefficient between $Q_{s}$ and $Q_{f}$ .", "As it was mentioned, the linear regression methodology can be applied if the random variables have a Gaussian (PDF) [24], giving the possibility to use the Pearson correlation coefficient.", "As the PDF acquired from the SensL SiPM are not Gaussian, a non-parametric approach can be used in order to estimate the statistical parameter.", "This approach makes use of the Spearman correlation index defined by equation REF [24] $ R_{rFast,rStd}= \\frac{Cov(rFast,rStd)}{\\sigma _{rFast}\\sigma _{rStd}}$ where $rFast$ and $rStd$ correspond to the ranked fast and standard signals, $\\sigma _{rFast}$ and $\\sigma _{rStd}$ are the standard deviation of ranked fast and standard signals, viewed as random variables [23].", "As can be observed in REF , the correlation coefficient definition is the same as Pearson correlation, except that the random variables are ranked [23].", "As described in [25], this approach does not depend on the used PDFs so, the mean and variance are calculated from a Gaussian distribution, as commonly used.", "A second approach to apply the linear regression by fitting a Gaussian PDF to the maximum peaks, observed in the charge deposition histograms from fast and standard signals, as it is shown in Fig REF ; thus, mean and variance are calculated.", "Moreover, both approaches will be used to demonstrate that standard and fast SiPM signals are highly correlated.", "Figure: SiPM charge deposition histogram and Gaussian fit for estimationIn CERN root software, the integral was performed on windows of $2\\times 10^3$ samples for fast signal and $1.6\\times 10^4$ samples for standard signal, because of tail effect." ], [ "Two-samples Kolmogorov-Smirnov test", "It is important to make a quantitative comparison among standard and estimated charge deposition.", "This can be done using the non parametric Two-Sample Kolmogorov-Smirnov (TSKS) test [26] which is useful to evaluate whether two underlying one-dimensional probability distributions differ from each other.", "The Cumulative Density Function (CDF) for each signal is calculated.", "In the other hand, the test hypothesis can be represented as follows: for a given CDF $F_{s}$ , for charge deposition of the standard SiPM output, and a given empirical CDF $F_{est}$ , for the estimated charge deposition, the test statistics divergence $D_{n,m}$ can be written as: [26] $ D_{n,m} = Max_{x} | F_{s,n}(x)- F_{est,m}(x) |$ where $n$ and $m$ correspond to the CDF size of the standard and estimated deposited charge.", "$Max_{x}$ represents the maximum of the distances set.", "Moreover, equation REF compares the empirical CDF's from the two charge deposition random variables under test, in order to find out whether both random variables come from same distribution or not.", "The Kolmogorov-Smirnov (KS) test statistic $\\sqrt{n}D_{n,m}$ will help to reject the null hypothesis at level of significance $\\alpha $ if $\\sqrt{n}D_{n,m}>K_{\\alpha }$ for $m,n\\rightarrow \\infty $ , where $P(K<K_{\\alpha })=1-\\alpha $ .", "Thus, the associated $p-value$ is calculated from tables or algorithms, in our case MATLAB was used to apply the KS-test to both CDF's.", "The resulting $p-value$ is compared with the level of significance $\\alpha $ so, the null hypothesis is related to equation REF $H_{0}:F_{s}= F_{est} : Failure~to~reject~the~null~hypothesis~at~the~ \\alpha -level\\\\H_{1}:F_{s}\\ne F_{est} : Rejection~of~the~null~hypothesis~at~the~ \\alpha -level$" ], [ "Analysis and results", "To determine the relation between standard and fast charges, the correlation coefficient was calculated for each radiation source (Co60, Cs137, Na22 and Sr90) and each plastic scintillator (BC404 and BC422Q).", "As the probability distribution functions for deposited charge from a SiPM are non Gaussian, Pearson correlation index cannot be estimated; therefore, the described approaches from section were used.", "From the first approach, Spearman correlation coefficient was estimated after data ranking for standard and fast charge estimations [22].", "Then, a Gaussian fit was applied to each distribution for standard and fast charge estimations, resulting on mean and variance calculation and, thus, a Pearson correlation coefficient as reported in Table REF .", "Table: Correlation R sf R_{sf} between Q s Q_{s} and Q f Q_{f}Correlation coefficient was estimated, resulting on indexes close to one as observed on Table REF , which confirms the linear relation hypothesis for fast and standard reconstructed electric charge.", "Therefore, the described linear regression in Equation REF can be applied to reconstruct the charge distribution for the standard output from the fast signal.", "This relation is depicted in Figure REF , where, the estimated charge correlation for each radioactive source was graphed.", "The BC422Q plastic scintillator produces a lower number of photons with respect to BC404.", "The BC404 has a 68% of athracene and the BC422Q has a 19% [18], [19].", "Thus, the BC404 emits more photons than BC422Q.", "A detailed discussion about the properties of plastic scintillators can be found in [27].", "Figure: Scatter plot for integrated fast and standard output signals using (left) BC404 and (right) BC422Q scintillators.Two orders of magnitude from fast and standard charge can be observed, which is related to the difference of charge deposition among the two variables of interest.", "Also, it is possible to qualitatively distinguish between the four sources for the case of BC404 scintillator, giving an opportunity for future development of classification algorithm implementation.", "In particular, $^{90}$ Sr and $^{137}$ Cs can be clearly separated from $^{22}$ Na and $^{60}$ Co. For the case of BC422Q scintillator, this source separation seems to be harder to accomplish.", "The mean and standard deviation are the required statistical momenta for the linear regression model.", "As an example, in Figure REF the charge distribution for the $^{22}$ Na radioactive source and both scintillator materials, BC404 and BC422Q, is shown .", "Doing a Gaussian fit on the the peaks, it is possible to get the required momentum.", "A particular case is observed for this source, two peaks can be appreciated in fast and standard charge for BC404 scintillator.", "One of these peaks can be associated to the original gamma from the source and the other can be associated to a gamma from the pair annihilation from positron emission.", "For BC422Q scintillator, a single peak is observed for this material.", "The rest of the sources for both materials have the same shape of one peak.", "Figure: 22 ^{22}Na charge deposition comparison and fit for fast and standard outputs for BC404 (left) and BC422Q (right) scintillators.Figure: Relation between the fast and standard charge for BC404 and BC422Q.Once the linear relation between fast and standard charge has been established, all parameters required in the model were calculated.", "For each random variable, the correlation index $R_{sf}$ , the mean $\\bar{Q_{s}}$ and $\\bar{Q_{f}}$ and standard deviations $\\sigma _{s}$ and $\\sigma _{f}$ were estimated.", "To exemplify this relation, in Figure REF is shown the linear dependence for $^{22}$ Na source from BC404 and BC422Q scintillator materials.", "For a complete reference about the the resulting parameters, the measurements are listed in Tables REF and REF for BC404 and BC422Q scintillators, respectively.", "Based on results from Tables REF and REF , the linear regression parameters for each source and material are shown in Tables REF and REF for BC404 and BC422Q scintillator, respectively.", "Table: BC404 scintillator material linear regression estimated parametersTable: BC422Q scintillator material linear regression estimated parametersTable: Linear regression(BC404).Table: Linear regression(BC422Q)Using the respective linear regression from Tables REF and REF , it is possible to reconstruct the original charge distribution from the fast pulse.", "In Figure REF the reconstructed charge distribution for $^{22}$ Na and both materials is shown (note that the two peaks for BC404 are reconstructed).", "We make a statistical analysis to compare the original and reconstructed charge.", "In Tables REF and REF the fit parameters for all sources and both materials are shown.", "As an additional test, we computed the ratio between the integral value of both distributions, reconstructed and original.", "These values are shown in Table REF .", "Figure: Original and reconstructed charge distributions for Bc404 (left) and BC422Q (right) for 22 ^{22}NaTable: Gaussian Fit parameters for Bc404.othe original charge.", "r ^rthe reconstruction charge.", "1,2 ^{1,~2}the fit parameters for the first and second Gaussian in 22 ^{22}Na distribution, respectively.Table: Gaussian Fit parameters for Bc422Q.othe original charge.", "r ^rthe reconstruction charge.Table: Ratio between the area of the original and reconstructed charge distribution.The reconstruction of the charge is performed event by event.", "Thus, it is possible to make the difference between the original and reconstructed charge value.", "As an example, in Figure REF , the plots of such differences for $^{22}$ Na for both plastic scintillators are shown.", "From the fit of these distributions, it is possible to obtain the reconstructed charge resolution, $\\sigma $ .", "The resolution values for all the used radiation sources for both materials are shown in Table REF .", "The best reconstructed charge resolution is obtained for BC422Q plastic scintillator in all the cases.", "Figure: Difference distribution between the original charge and the reconstructed charge for 22 ^{22}Na and BC404 (left) and BC422Q (right).Table: Resolution reconstructed charge for both materials.We also applied another test to compare the Probability Density Function (PDF) for the standard and reconstructed charge.", "The TSKS-test (described in subsection REF ) was applied to both distributions.", "The CDF's for each radiating source and scintillator material are shown in Figures REF and REF .", "The results of this test are listed in the Table REF .", "We noted that the corresponding CDF's are equivalent for the standard and reconstructed charge.", "Figure: CDF comparative for BC404 scintillator.Figure: CDF comparative for BC422Q scintillator.Table: Two-samples Kolmogorov test results for BC404 and BC422QAdditionally, the correlation between the mean and $\\sigma $ values, obtained from our Gaussian fits for BC404 plastic scintillator material, can be used to distinguish among the different radiation sources used in this work.", "Accordingly with our results, this is not the case for BC422Q.", "(see Figure REF ).", "Figure: Mean-σ\\sigma relation from the Gaussian Fit for studied radiation sources with BC404 (up)and BC422Q (down) scintillators." ], [ "Conclusions", "A relation between the fast and standard output signals of a SensL SiPM photo-sensor was found.", "For a given pulse, by using the fast output signal, we were able to reconstruct the deposited charge.", "This value is in good concordance with the estimated charge using the standard pulse.", "We also observed that the best agreement between the reconstructed charged from the fast output signal and the one from the standard output signal of a SensL SiPM photo-sensor is obtained for the BC422Q plastic scintillator.", "This results shows that it may be possible to develop a trigger system based on plastic scintillator material and a SensL SiPM photo-sensor with an excellent time resolution where also the information of electric charge can be reconstructed using the fast output signal of such photo-sensor.", "In average, the conversion factor among fast and standard charge is $0.008\\pm 0.001\\times 10^{-12}$ for BC404 and $0.012\\pm ~0.001\\times 10^{-12}$ for BC422Q.", "Whence, $Q_{S404}/Q_{F404} = 128.205$ and $Q_{S422}/Q_{F422} = 85.106$ , which means that the charge deposition in the BC404 scintillator is 1.506 times the one for a BC422Q scintillator.", "Therefore (using these thin materials [11]), BC404 scintillator is 1.506 more sensitive than BC422Q scintillator for $^{90}$ Sr, $^{60}$ Co, $^{137}$ Cs and $^{22}$ Na radiation sources.", "This result gives the possibility to use the fast pulse from the detectors, where time resolution is an important restriction [17] and for fast triggering systems in Time of Flight (TOF) applications.", "The continuity of this work is to estimate the time resolution of both detector configurations.", "Our plan is to develop a PET, commonly constructed with LYSO crystal, based on plastic scintillator materials with the best possible time resolution which is a key parameter during the data acquisition chain.", "For example, the life time of the isotopes used to acquire brain or heart images is around 2 minutes.", "Thus, a fast detector response is desired to improve the spatial and time resolution of PET scanners." ], [ "Acknowledgements", "Support for this work has been received by Consejo Nacional de Ciencia y Tecnología grant numbers A1-S-13525 and A1-S-7655.", "The authors thanks to the BUAP Medical Physics and Elementary Particles Laboratories for their kind hospitality and support during the development of this work." ] ]

[ [ "Measurement of b jet shapes in proton-proton collisions at $\\sqrt{s} =$\n 5.02 TeV" ], [ "Abstract We present the first study of charged-hadron production associated with jets originating from b quarks in proton-proton collisions at a center-of-mass energy of 5.02 TeV.", "The data sample used in this study was collected with the CMS detector at the CERN LHC and corresponds to an integrated luminosity of 27.4 pb$^{-1}$.", "To characterize the jet substructure, the differential jet shapes, defined as the normalized transverse momentum distribution of charged hadrons as a function of angular distance from the jet axis, are measured for b jets.", "In addition to the jet shapes, the per-jet yields of charged particles associated with b jets are also quantified, again as a function of the angular distance with respect to the jet axis.", "Extracted jet shape and particle yield distributions for b jets are compared with results for inclusive jets, as well as with the predictions from the PYTHIA and HERWIG++ event generators." ], [ "Introduction", "Jets, the collimated showers of particles produced by fragmentation and hadronization of hard-scattered quarks or gluons, are long established experimental probes for studies of quantum chromodynamics (QCD) [1].", "The internal structure of the jet, defined by the energy, momentum, and spatial distribution of its constituents, is sensitive to the details of the evolution from an initial hard scattering through fragmentation and hadronization into observable hadrons in the final state.", "The angular distributions of constituent particle yields and jet shapes, studied in this work, are affected by parton fragmentation and hadronization processes.", "At high transverse momenta ($$ ) with respect to the beam direction in the core of the jet, the dominant contribution to these distributions is set by the initial branching of the hard scattered parton which is calculable in perturbative QCD (pQCD).", "However, for lower particles and those at larger radial distances from the jet direction, higher order corrections and nonperturbative processes become of major importance.", "Characterizing the effect of these additional contributions on the internal structure of jets remains challenging for theoretical calculations [2], [3], [4].", "In this paper, the internal structure of jets is studied at the charged particle level using the data for proton-proton ($$ ) collisions at a center-of-mass energy of $\\sqrt{s}=5.02$ .", "These data, corresponding to an integrated luminosity of 27.4, were collected by the CMS experiment in 2015.", "For this study, $$ jets are defined by the presence of at least one $$ quark, which is inferred from the properties of $$ hadron decays.", "A $$ jet sample selected via a combined secondary vertex (CSV) discriminator [5], is composed of jets initiated by a single bottom quark, as well as of a contribution from $$ pairs produced from gluon splitting.", "Jet-correlated charged particle transverse momentum distributions, referred to as jet shapes, are measured as a function of radial distance $\\Delta r = \\sqrt{\\smash[b]{(\\Delta \\eta )^{2}+(\\Delta \\phi )^{2}}}$ from the jet axis.", "Here $\\Delta \\eta = \\eta ^{\\mathrm {jet}}-\\eta ^{\\mathrm {trk}}$ and $\\Delta \\phi = \\phi ^{\\mathrm {jet}}-\\phi ^{\\mathrm {trk}}$ are the pseudorapidity and azimuthal differences between the jet and the charged particle, respectively.", "To extend the jet shape measurements further into the region where nonperturbative effects dominate, we use a jet-track correlation technique [6], [7].", "This method has been shown to reliably subtract the part of the event unrelated to the hard scattering (the underlying event), as well as the contribution of additional pp interactions in the same or nearby bunch crossings (pileup).", "We study the -differential distributions of jet shapes and particle yields for $$ jets.", "By comparing these measurements with the results for inclusive jets and with  [8], [9] simulation for the $$ jet and inclusive jet shapes at large angles from the jet axis, this study provides new constraints on pQCD calculations, as well as on the nonperturbative contribution to jet shapes.", "It will also provide a baseline for the future study of the parton flavor dependence of the interaction between the jet and the quark-gluon plasma [10], which is created in high energy heavy ion collisions." ], [ "The CMS detector", "The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diameter, providing a magnetic field of 3.8T.", "Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of barrel and endcap sections.", "Two forward hadron (HF) steel and quartz-fiber calorimeters complement the barrel and endcap detectors, extending the calorimeter from the range ${\\eta }<3.0$ to ${\\eta }<5.2$ .", "Events of interest are selected using a two-tiered trigger system [11].", "In the region ${\\eta }<1.74$ , the HCAL cells have widths of 0.087 in both pseudorapidity $\\eta $ and azimuth $\\phi $ .", "Within the central barrel region of ${\\eta }<1.48$ , the HCAL cells map onto $5\\times 5$ ECAL crystal arrays to form calorimeter towers projecting radially outwards from the nominal interaction point.", "Within each tower, the energy deposits in ECAL and HCAL cells are summed to define the calorimeter tower energies, which are subsequently used in the particle flow algorithm to reconstruct the jet energies and directions [12].", "In this work, jets are reconstructed within the $\\eta $ range of ${\\eta }<1.6$ .", "The silicon tracker measures charged particles within ${\\eta }<2.5$ .", "It consists of 1440 silicon pixel and 15 148 silicon strip detector modules.", "For nonisolated particles with $1<<10$ in the barrel region, the track resolutions are typically 1.5% in and 25–90 (45–150)in the impact parameter direction transverse (longitudinal) to the colliding beams [13].", "A detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref.", "[14]." ], [ "Event selection and simulated event samples", "The data used in this analysis were taken in a special low-luminosity running period in which the reduced levels of pileup (approximately 1.5 events per bunch crossing assuming a total inelastic cross section of 65 [15]) allowed for precise measurements of the jet characteristics described in this paper.", "The jet samples are collected with a calorimeter-based trigger that uses the anti-jet clustering algorithm with a distance parameter of $R=0.4$  [16].", "This trigger requires events to contain at least one jet with $>80$ , and is fully efficient for events containing jets with reconstructed $>90$ .", "The data selected by this trigger are referred to as \"jet-triggered\" and are used to study the jet-related particle yields and for a data-driven estimation of acceptance effects via an event mixing technique as described in Section .", "To reduce contamination from non-collision events, such as calorimeter noise and beam-gas collisions, vertex and noise reduction selections are applied as described in Refs.", "[17], [18].", "These selections include a requirement for events to contain at least 3 of energy in one of the calorimeter towers in the HF on each side of the interaction point, and to have a primary vertex (PV) with at least two tracks within 15cm of the center of the nominal interaction region along the beam axis (${v_{z}}<15{cm}$ ).", "Monte Carlo (MC) simulated event samples are used to evaluate the performance of the event reconstruction, particularly the track reconstruction efficiency, and the jet energy response and resolution.", "The MC samples of two different tunes (version 6.424 with Z2 tune [19] and version 8.230 with CP5 tune [20]) were used to simulate the hard scattering, parton showering, and hadronization of the partons.", "The $$ jets in simulations are obtained by requiring the presence of a generator-level $$ quark in the simulated QCD jet sample.", "The (10.02p02) [21] toolkit is used to simulate the CMS detector response.", "An additional reweighting procedure is performed to match the simulated $v_{z}$ distribution to that observed in data." ], [ "Jet and track reconstruction", "Jets are reconstructed offline from the particle-flow (PF) candidates [22], clustered using the anti-algorithm [16], [23] with a distance parameter of $R=0.4$ .", "The PF candidates are reconstructed by the PF algorithm, which aims to reconstruct and identify each particle in an event, with an optimized combination of information from the various elements of the CMS detector.", "Simulation-derived corrections have been applied to the reconstructed jets to correct the measured energy distorted because of the limited detector resolution, to the particle level.", "Jets with $> 120$ and ${\\eta }<1.6$ are selected to be consistent with a previous study [7].", "A widely used type of the jet axis, the anti-E-Scheme jet axis, is calculated by merging all the jet candidates, as well as input particles to the jet clustering by simply adding the four-momenta during the clustering procedure in the anti-algorithm [24].", "This type of jet axis is not infrared and collinear (IRC) safe [25] and can produce nonphysical radial structures in jet constituent distributions.", "To minimize the IRC effect on the jet direction determination, the jet axis for this work is re-calculated by the winner-takes-all recombination scheme [26], [27], which is applied to the constituents found by the nominal anti-E-Scheme algorithm for this jet.", "The $$ jet candidates for this work are selected by the CSV discriminator [28], [13].", "The CSV discriminator is a multivariate classifier that makes use of information about reconstructed secondary vertices (SV) as well as the impact parameters of the associated tracks with respect to the primary vertex, to discriminate $$ jets from charm-flavor and light-flavor jets.", "The working point selected for this analysis leads to a 65% $$ jet selection efficiency and 69% purity (the true $$ jet fraction of all jets that passed the CSV selection criteria) from the multijet sample (referring to the background of charm jets and light jets).", "Possible differences in the purity between data and MC are assessed using a negative-tag technique [5].", "This technique selects non-$$ jets using the same variables and techniques as the standard CSV algorithm both in data and in the simulation to extract a scale factor, which indicates the data-to-MC difference.", "A correction for a bias resulting from the discriminator is discussed in Section .", "In both data and simulation, charged particles are reconstructed using an iterative tracking method [13] based on the hit information from both the pixel and silicon strip subdetectors, permitting the reconstruction of charged particles within ${\\eta }<2.4$ .", "The tracking efficiency ranges from approximately 90% at $=1$ to no less than 90% for $>10$ .", "Tracks with $>1.0$ and ${\\eta }<2.4$ are used in this study." ], [ "Jet-track angular correlations", "To study the distributions of the charged particles associated with jets, a two-dimensional (2D) array of the $\\Delta \\eta $ and $\\Delta \\phi $ values of the tracks relative to the jet axis were produced.", "This is computed for six bins of $^{\\text{trk}}$ bounded by the values 1, 2, 3, 4, 8, 12, and 300.", "Each of these 2D correlations is normalized by $N_{\\text{jets}} $ , the number of jets in the sample.", "This procedure, the same one as used in Ref.", "[29], creates a per-jet averaged $\\Delta \\eta $ -$\\Delta \\phi $ distribution of raw charged particle densities for each $^{\\text{trk}}$ : $S(\\Delta \\eta , \\Delta \\phi )=\\frac{1}{N_{\\text{jets}}}\\frac{{}^2N^{\\mathrm {same}}}{{}\\Delta \\eta {}\\Delta \\phi },$ where $N^{\\mathrm {same}}$ represents the yield of jet-track pairs from the same event.", "For the jet shape measurements, the 2D correlations are weighted by $^{\\text{trk}}$ on a per-track basis, producing a per-jet averaged $\\Delta \\eta $ -$\\Delta \\phi $ distribution of $^{\\text{trk}}$ with respect to the jet axis direction.", "An event mixing method [7] is applied following the construction of the raw 2D correlations $S(\\Delta \\eta , \\Delta \\phi )$ to account for the shape of single inclusive jet and track distributions and the effects of the detector acceptance for tracks.", "For this correction, a mixed-event pair distribution $ME(\\Delta \\eta , \\Delta \\phi )$ is constructed by using the jets from one event and the tracks from a different event, matched in the vertex position along the beam axis in 1 cm bins: $ME(\\Delta \\eta , \\Delta \\phi )=\\frac{1}{N_{\\text{jets}}}\\frac{^2N^{\\mathrm {mix}}}{\\Delta \\eta \\Delta \\phi },$ where $N^{\\mathrm {mix}}$ represents the number of jet-track pairs from the mixed-event.", "The per-jet associated yield is corrected for the jet-track pair efficiency via the following relation: $\\frac{1}{N_{\\text{jets}}}\\frac{^2N}{\\Delta \\eta {}\\Delta \\phi }=\\frac{ME(0, 0)}{ME(\\Delta \\eta , \\Delta \\phi )} S(\\Delta \\eta , \\Delta \\phi ).$ The ratio $ME(0, 0)/ME(\\Delta \\eta , \\Delta \\phi )$ is the normalized correction factor and $ME(0,0)$ is the mixed event yield for jet-track pairs that are approximately collinear and hence have the maximum pair acceptance.", "After the signal pairs $S(\\Delta \\eta , \\Delta \\phi )$ have been corrected using Eq.", "(REF ), the underlying event contribution and uncorrelated backgrounds from tracks unrelated to selected jets are removed by using the measured charged-particle yields far from the jet axis in a large-$\\Delta \\eta $ region.", "The $\\Delta \\phi $ distribution averaged over $1.5<{\\Delta \\eta }<2.5$ is used to estimate the $\\Delta \\phi $ dependence of the background contribution to the correlations over the entire ${\\Delta \\eta }<4.0$ region and is subtracted from the acceptance-corrected yields of Eq.", "(REF ).", "The pair-acceptance corrected signal pair distribution $S(\\Delta r)$ as a function of radius $\\Delta r $ is obtained from the integration of Eq.", "(REF ) over a ring area with radius $\\Delta r $ .", "The signal of $$ -tagged jets $S_{\\mathrm {tag}}$ is then corrected for residual light-flavor jet contamination.", "We use an approach partially relying on data for the decontamination procedure, expressed via the following equation: $S^*_{\\mathrm {tag}}(\\Delta r) = \\frac{S_{\\mathrm {tag}}(\\Delta r)-(1-c_\\mathrm {purity})S_{\\mathrm {mistagged}}(\\Delta r)}{c_\\mathrm {purity}},$ where the $S^*_\\mathrm {tag}(\\Delta r)$ and $S_{\\text{mistagged}}(\\Delta r)$ are signals of the (decontaminated) $$ jets and the mistagged light-flavor jets, respectively.", "The $S_{\\mathrm {mistagged}}(\\Delta r)$ is approximated by the inclusive jet-track correlation signal $S_{\\mathrm {inclusive}}(\\Delta r)$ from the data, with a modification for simulating the jet multiplicity bias that is discussed later in this section.", "The purity, $c_\\mathrm {purity}$ , which is defined as the ratio of the number of tagged true jets to the number of jets tagged by the CSV discriminator, comes from the simulation.", "Finally, simulation-based corrections are applied to account for the jet axis resolution, tracking reconstruction efficiency, and the bias in the charged particle yield and jet shapes that comes from the $$ -tagging discriminator.", "A large fraction of tracks associated with $$ jets originates from SV and has a slightly different reconstruction efficiency from that of tracks originating from PV.", "Therefore, we derive the efficiency corrections as a function of track and radial distance $\\Delta r $ from the MC $$ jet simulation by taking the ratio of correlated signals built with reconstructed tracks over those with generated tracks.", "This bin-by-bin correction has been applied to the raw data distributions accordingly.", "The discriminator used for $$ tagging relies on the properties of the SVs associated with the jet as input, therefore biasing the jet selection towards jets with better SV resolution.", "This bias, though slight, is present in distributions for both true $$ jets selected by the tagger, and in the mistagged light-flavor jets contaminating the sample.", "We calculate corrections for the tagging bias as a function of $\\Delta r $ from MC simulation by constructing the following per-jet normalized ratios of radial distributions: $\\begin{aligned}B_{\\mathrm {mis}}(\\Delta r) &= S_{\\mathrm {inclusive}}(\\Delta r)/S_{\\mathrm {mistagged}}(\\Delta r),\\\\B_{}(\\Delta r) &= S_{\\operatorname{all-}}(\\Delta r)/S_{\\operatorname{tagged-}}(\\Delta r),\\end{aligned}$ where $S_{\\mathrm {mistagged}}(\\Delta r)$ , $S_{\\mathrm {inclusive}}(\\Delta r)$ , $S_{\\operatorname{all-}}(\\Delta r)$ , and $S_{\\operatorname{tagged-}}(\\Delta r)$ represent the signal of tracks correlated with the mistagged jets, inclusive light-flavor jets, and $$  jets, and the tagged $$  jets, respectively.", "All of these procedures correct the data to a particle level which can be compared with theoretical calculations directly.", "The fully corrected 2D correlations are integrated over annular rings in the $\\Delta \\eta $ -$\\Delta \\phi $ plane (as illustrated in [30]) to study distributions of charged-particle yields $Y(\\Delta r)$ : $Y(\\Delta r)= \\frac{1}{N_{\\text{jets}}} \\frac{{}^2N_{\\mathrm {trk}}}{\\Delta r {}^{\\text{trk}}}$ with respect to the jet axis as a function of $\\Delta r $ for $$ and inclusive-jet samples and, where $N_{\\mathrm {trk}}$ is the number of the charged particles from jets.", "The jet shape distributions $\\rho (\\Delta r)$ , defined as: $\\rho (\\Delta r) = \\frac{1}{\\delta r} \\frac{\\large {\\Sigma _{\\mathrm {jets}}} \\Sigma _{\\mathrm {particle} \\in ( \\Delta r _{\\mathrm {a}}, \\Delta r _{\\mathrm {b}} )} ^{\\text{trk}}}{\\large {\\Sigma _{\\mathrm {jets}}} \\Sigma _{\\mathrm {trk}} ^{\\text{trk}}},$ where $\\Delta r _{\\mathrm {a}}$ and $\\Delta r _{\\mathrm {b}}$ define the annular edges of $\\Delta r $ , $\\delta r = \\Delta r _{\\mathrm {b}} - \\Delta r _{\\mathrm {a}}$ , and $^{\\text{trk}} $ stands for the $$ of the charged particles, are also examined." ], [ "Systematic uncertainties", "A number of sources of systematic uncertainties are considered, including the tracking efficiency, tagging bias corrections, decontamination procedure, jet reconstruction, acceptance corrections, and background subtraction.", "The list of systematic uncertainties is summarized in Table REF , and the evaluation of each source of uncertainty is discussed below.", "The tracking reconstruction efficiencies for $$  jet and inclusive jet tracks have been compared to account for the uncertainty in reconstruction efficiency for displaced tracks, and a maximum difference of about 4% was observed.", "The full magnitude of the observed difference is assigned as a conservative estimation to cover the MC-based tracking reconstruction uncertainty.", "To study possible differences in track reconstruction between data and simulation, a study of meson decays was used [31].", "The meson branching fraction ratio of 3-prong to 5-prong decays was calculated in data with MC-based efficiency corrections and compared with the world-average value [32].", "The observed difference is used to derive a 4% systematic uncertainty for this source.", "For the full tracking-related uncertainty these two errors were added in quadrature.", "The uncertainty for correcting the bias induced by the CSV discriminator is dominated by the uncertainties in the contributions from gluon-splitting and primary $$ quarks to the $$  jet sample.", "Jets originating from different mechanisms of $$ -quark production (i.e.", "flavor creation, flavor excitation, and gluon splitting) can be studied individually in simulations.", "We note that directions of $$ and $$ jets from gluon splitting are more likely to be close to each other, resulting in broader shapes for those jets.", "The fraction of $$ jets from the gluon splitting is about 45% based on QCD calculations [33].", "In the kinematic range of this analysis, simulations show about 44% of $$ jets from gluon splitting.", "The corresponding systematic uncertainty has been evaluated by varying this fraction by 20% (as estimated in Ref [34]), and the observed 5% difference in the correction from this variation is propagated as an uncertainty.", "The decontamination procedure is affected by the uncertainties in the purity estimation.", "Using the negative tagging method (described in Section ) we have derived the data-to-simulation scale factor, which amounted to about 7% difference in estimated contamination levels.", "We evaluate the related systematic uncertainty by comparing results obtained with and without the derived scale factor; less than 5% variation is observed in the correlation results.", "This 5% maximum variation is taken as a systematic uncertainty for the decontamination.", "The overall jet energy scale (JES) is sensitive to the relative fraction of quark and gluon jets in the sample.", "The energy scale uncertainty is found to be 2% for jets in the study in Ref. [35].", "Therefore, we varied the energy threshold of selected jets by this amount in both directions.", "The resulting uncertainty in correlated track yields is found to be below 2%, since the in-jet multiplicity and the jet fragmentation function change slowly with the jet .", "The jet energy resolution (JER) data-to-MC difference is about 15% based on the $\\gamma $ +jet studies [36].", "The effects were accounted for by adding a 15% smearing to the reconstructed jet .", "The resulting variation in correlation distributions was found to be below 3.5%.", "In total, a systematic uncertainty of 4% is assigned for the JER- and JES-related effects.", "The uncertainties from the mixed-event acceptance correction are estimated by looking for an asymmetry of the sideband regions, which is defined by the difference of the sideband value between the positive and negative $\\Delta \\eta $ .", "Additionally, the sideband regions ($1.5<{\\Delta \\eta }<2.5$ ) that are far away from the jet axis are expected to have no short-range correlation contributions and, thus, to be independent of $\\Delta \\eta $ .", "Any deviations from this expectation and the measured asymmetry are used to quantify the related systematic uncertainty, which was found to be between 1 and 2%.", "Uncertainties associated with the background subtraction are evaluated by considering the average point-to-point difference between two sideband regions ($1.5 < {\\Delta \\eta } < 2.0$ and $2.0 < {\\Delta \\eta } < 2.5$ ) following the background subtraction.", "The background subtraction uncertainty is found to be roughly 3% for the lowest $^{\\text{trk}} $ bin, where the signal-to-background ratio is the lowest, and decreases to negligible levels as functions of $^{\\text{trk}} $ .", "These systematic uncertainties are treated as uncorrelated, and the total systematic uncertainty is calculated by adding the individual sources in quadrature.", "Table: NO_CAPTION" ], [ "Results", "Figure REF presents the charged-particle yields for inclusive and $$  jets in proton-proton collisions as a function of the radial distance $\\Delta r $ from the jet axis.", "The results are shown with stacked histograms to indicate the intervals in $^{\\text{trk}}$ , and dots to denote the total summed yields in the region $1 < ^{\\text{trk}} < 12$ .", "It illustrates that the high-charged particles are mostly distributed around the small $\\Delta r $ region while the larger $\\Delta r $ region is dominated by the low-charged particles.", "Figure REF compares the radial distributions of the total charged-particle yields associated with the inclusive and $$ jets studied in data and in simulations.", "Charged-particle yield distributions for both $$ and inclusive jets are found to be generally described by predictions, although 6.424 shows a better agreement with the data than that found using the 8.230 prediction.", "Larger charged-particle yields are observed to be associated with $$ jets as compared with inclusive jets, particularly in the low-$\\Delta r $ region (see Fig.", "REF , right).", "This larger contribution in soft tracks at small radial distance $\\Delta r $ implies the presence of different fragmentation patterns and decay kinematics between the $$ jets and inclusive jets.", "Figure: Charged particle yield distributions Y(Δr)Y(\\Delta r) of inclusive (left) and  (right) jets with >120> 120 as functions of Δr\\Delta r are presented differentially for trk ^{\\text{trk}} bins.", "The shadowed boxes represent the systematic uncertainties, although they are generally too small to be visible.Figure: Charged particle yield distributions Y(Δr)Y(\\Delta r) of inclusive jets (left) and  jets (middle) with 1< trk <121<^{\\text{trk}} <12 are presented as functions of Δr\\Delta r .", "Both types of jets with >120> 120 and charged particles with 1< trk <121<^{\\text{trk}} < 12 are used to construct the distributions as functions of Δr\\Delta r for data (red), 6.426 (blue line) and 8.230 (green dashed line) simulations , respectively.", "The right plot shows the particle yield difference of  jets and inclusive jets as functions of Δr\\Delta r for data and 6.426 (blue line) and 8.230 (green dashed line) simulations.", "The shadowed boxes represent the systematic uncertainties.Measurements of the jet shapes $\\rho (\\Delta r) $ are presented in Figs.", "REF and REF .The left and right panels of Fig.", "REF show -differential $\\rho (\\Delta r) $ distribution for inclusive and $$ jets, respectively.", "The comparison between data and simulations is presented in Fig.", "REF .", "We note that, while small-$\\Delta r $ trends are mostly well described by MC simulation for both jet selections, the distributions at larger radial distances are underestimated, indicating a shortage of soft radiative contributions.", "The right panel of Fig.", "REF shows the ratio of $$ to inclusive jet shapes for data and simulation.", "Observed variations in the ratio of jet shapes indicate a shift of transverse momentum from small to large $\\Delta r $ for the constituents of the $$ jets compared to that carried by the particles from inclusive jets.", "These differences may arise from the dead-cone effect, the suppression of radiation from a charged particles with mass $m_$ and energy $E_$ in the region with emission angle $\\theta \\lesssim m_/E_$  [37], [38], as this phenomenon is expected to be more apparent in $$ jets than in inclusive jets, which mostly originate from light partons.", "Monte Carlo simulations have difficulties capturing the details of jet shapes for both inclusive and $$ jets distributions at large angular distances, where nonperturbative contributions are likely to dominate.", "Additionally, we observe that a higher fraction of transverse momentum is distributed towards the higher radial distances from the center of the jet for the $$ jets as compared to the inclusive jet sample.", "A similar tendency, albeit insufficient to fully capture this trend, is seen in simulations, with 8.230 giving slightly better description than 6.426 in the larger $\\Delta r $ region while 6.426 shows a better performance than 8.230 in the small $\\Delta r $ region, as illustrated in the right panel of Fig.", "REF .", "The observed data to discrepancy in the -to-inclusive jet shape ratios at large radii may arise from the difference in the gluon splitting contributions between data and simulation, as mentioned earlier.", "We note that Monte Carlo studies show that $$ and $$ jets from gluon splitting result in significantly broader jet shapes than those of inclusive jets.", "Figure: The jet shape distribution ρ(Δr)\\rho (\\Delta r) of inclusive jets (left) and jets (right) with >120> 120 as functions of Δr\\Delta r are presented differentially for all trk ^{\\text{trk}} bins for data.", "The shadowed boxes represent the systematic uncertainties, although they are generally too small to be visible.Figure: The jet shape distribution ρ(Δr)\\rho (\\Delta r) of inclusive jets (left) and jets (middle), both with >120> 120 and trk >1^{\\text{trk}} > 1 are presented as functions of Δr\\Delta r for data(red markers), the 6.426 (blue line) and the 8.230 (green dashed line) simulations.", "The right plot shows the -to-inclusive jet shape ratio as functions of Δr\\Delta r for data, 6 (blue line) and 8.230 (green dashed line) simulations.", "The shadowed boxes represent the systematic uncertainty." ], [ "Summary", "The first measurements of charged-particle yields and jet shapes for $$ jets in proton-proton collisions are presented, using data collected with the CMS detector at the LHC at a center-of-mass energy of $\\sqrt{s} = 5.02$ .", "The correlations of charged particles with jets are studied, using the particles with transverse momentum $^{\\text{trk}} > 1$ and pseudorapidity ${\\eta }<2.4$ , and the jets with $> 120$ and ${\\eta }<1.6$ .", "Charged-particle yields associated with jets are presented as functions of the relative angular distance $\\Delta r = \\sqrt{\\smash[b]{(\\Delta \\eta )^{2}+(\\Delta \\phi )^{2}}}$ from the jet axis.", "In these studies, a large number of associated charged particles at low $\\Delta r $ are found for $$  jets compared to those for inclusive jets, which are produced predominantly by gluons and light flavor quarks.", "The trends observed in $$ data for particle yield distributions associated with both types of jets are reproduced by calculations (in versions 6.426 and 8.230).", "In addition to the charged-particle yields, we examine the jet transverse momentum profile variable $\\rho (\\Delta r) $ , defined using the distribution of charged particles in annular rings around the jet axis, with each particle weighted by its $^{\\text{trk}}$ value.", "The measured shapes of $$ jets are broader than those of inclusive jets.", "The shapes for both types of jets are reproduced by calculation in the small $\\Delta r $ region, with 6.426 giving a better agreement.", "However, measured transverse momenta distributions at larger $\\Delta r $ are underestimated in simulations for $$ and inclusive jets, with larger data-to-simulation differences observed for $$ jets.", "This result provides new constraints on perturbative quantum chromodynamics calculations for flavor dependence in parton fragmentation and gluon radiation, as well as the relative contributions of different processes to $$ quark production.", "These measurements are also expected to offer an important reference for future studies of flavor dependence for parton interactions with the quark-gluon plasma formed in relativistic heavy ion collisions.", "We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort.", "In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses.", "Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA).", "Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract Nos.", "675440, 752730, and 765710 (European Union); the Leventis Foundation; the A.P.", "Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science – EOS\" – be.h project n. 30820817; the Beijing Municipal Science & Technology Commission, No.", "Z191100007219010; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy – EXC 2121 “Quantum Universe\" – 390833306; the Lendület (“Momentum\") Program and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program ÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Ministry of Science and Higher Education, project no.", "02.a03.21.0005 (Russia); the Tomsk Polytechnic University Competitiveness Enhancement Program and “Nauka\" Project FSWW-2020-0008 (Russia); the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA)." ], [ "The CMS Collaboration ", "Yerevan Physics Institute, Yerevan, Armenia A.M. Sirunyan$^{\\textrm {\\dag }}$ , A. Tumasyan Institut für Hochenergiephysik, Wien, Austria W. Adam, F. Ambrogi, T. Bergauer, M. Dragicevic, J. Erö, A. Escalante Del Valle, M. Flechl, R. Frühwirth1, M. Jeitler1, N. Krammer, I. Krätschmer, D. Liko, T. Madlener, I. Mikulec, N. Rad, J. Schieck1, R. Schöfbeck, M. Spanring, W. Waltenberger, C.-E. Wulz1, M. Zarucki Institute for Nuclear Problems, Minsk, Belarus V. Drugakov, V. Mossolov, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium M.R.", "Darwish, E.A.", "De Wolf, D. Di Croce, X. Janssen, T. Kello2, A. Lelek, M. Pieters, H. Rejeb Sfar, H. Van Haevermaet, P. Van Mechelen, S. Van Putte, N. Van Remortel Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, E.S.", "Bols, S.S. Chhibra, J.", "D'Hondt, J.", "De Clercq, D. Lontkovskyi, S. Lowette, I. Marchesini, S. Moortgat, Q. Python, S. Tavernier, W. Van Doninck, P. Van Mulders Université Libre de Bruxelles, Bruxelles, Belgium D. Beghin, B. Bilin, B. Clerbaux, G. De Lentdecker, H. Delannoy, B. Dorney, L. Favart, A. Grebenyuk, A.K.", "Kalsi, L. Moureaux, A. Popov, N. Postiau, E. Starling, L. Thomas, C. Vander Velde, P. Vanlaer, D. Vannerom Ghent University, Ghent, Belgium T. Cornelis, D. Dobur, I. Khvastunov3, M. Niedziela, C. Roskas, K. Skovpen, M. Tytgat, W. Verbeke, B. Vermassen, M. Vit Université Catholique de Louvain, Louvain-la-Neuve, Belgium G. Bruno, C. Caputo, P. David, C. Delaere, M. Delcourt, A. Giammanco, V. Lemaitre, J. Prisciandaro, A. Saggio, P. Vischia, J. Zobec Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil G.A.", "Alves, G. Correia Silva, C. Hensel, A. Moraes Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato4, E. Coelho, E.M. Da Costa, G.G.", "Da Silveira5, D. De Jesus Damiao, C. De Oliveira Martins, S. Fonseca De Souza, H. Malbouisson, J. Martins6, D. Matos Figueiredo, M. Medina Jaime7, M. Melo De Almeida, C. Mora Herrera, L. Mundim, H. Nogima, W.L.", "Prado Da Silva, P. Rebello Teles, L.J.", "Sanchez Rosas, A. Santoro, A. Sznajder, M. Thiel, E.J.", "Tonelli Manganote4, F. Torres Da Silva De Araujo, A. Vilela Pereira Universidade Estadual Paulista $^{a}$ , Universidade Federal do ABC $^{b}$ , São Paulo, Brazil C.A.", "Bernardes$^{a}$ , L. Calligaris$^{a}$ , T.R.", "Fernandez Perez Tomei$^{a}$ , E.M. Gregores$^{b}$ , D.S.", "Lemos, P.G.", "Mercadante$^{b}$ , S.F.", "Novaes$^{a}$ , SandraS.", "Padula$^{a}$ Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria A. Aleksandrov, G. Antchev, R. Hadjiiska, P. Iaydjiev, M. Misheva, M. Rodozov, M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria M. Bonchev, A. Dimitrov, T. Ivanov, L. Litov, B. Pavlov, P. Petkov, A. Petrov Beihang University, Beijing, China W. Fang2, X. Gao2, L. Yuan Department of Physics, Tsinghua University, Beijing, China M. Ahmad, Z. Hu, Y. Wang Institute of High Energy Physics, Beijing, China G.M.", "Chen8, H.S.", "Chen8, M. Chen, C.H.", "Jiang, D. Leggat, H. Liao, Z. Liu, A. Spiezia, J. Tao, E. Yazgan, H. Zhang, S. Zhang8, J. Zhao State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China A. Agapitos, Y.", "Ban, G. Chen, A. Levin, J. Li, L. Li, Q. Li, Y. Mao, S.J.", "Qian, D. Wang, Q. Wang Zhejiang University, Hangzhou, China M. Xiao Universidad de Los Andes, Bogota, Colombia C. Avila, A. Cabrera, C. Florez, C.F.", "González Hernández, M.A.", "Segura Delgado Universidad de Antioquia, Medellin, Colombia J. Mejia Guisao, J.D.", "Ruiz Alvarez, C.A.", "Salazar González, N. Vanegas Arbelaez University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia D. Giljanović, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of Split, Faculty of Science, Split, Croatia Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, D. Ferencek, K. Kadija, D. Majumder, B. Mesic, M. Roguljic, A. Starodumov9, T. Susa University of Cyprus, Nicosia, Cyprus M.W.", "Ather, A. Attikis, E. Erodotou, A. Ioannou, M. Kolosova, S. Konstantinou, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A.", "Razis, H. Rykaczewski, H. Saka, D. Tsiakkouri Charles University, Prague, Czech Republic M. Finger10, M. Finger Jr.10, A. Kveton, J. Tomsa Escuela Politecnica Nacional, Quito, Ecuador E. Ayala Universidad San Francisco de Quito, Quito, Ecuador E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt A. Mohamed11, E. Salama12$^{, }$ 13 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M. Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki, Helsinki, Finland P. Eerola, L. Forthomme, H. Kirschenmann, K. Osterberg, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland E. Brücken, F. Garcia, J. Havukainen, J.K. Heikkilä, V. Karimäki, M.S.", "Kim, R. Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta University of Technology, Lappeenranta, Finland P. Luukka, T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France M. Besancon, F. Couderc, M. Dejardin, D. Denegri, B. Fabbro, J.L.", "Faure, F. Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, C. Leloup, B. Lenzi, E. Locci, J. Malcles, J. Rander, A. Rosowsky, M.Ö.", "Sahin, A. Savoy-Navarro14, M. Titov, G.B.", "Yu Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris, France S. Ahuja, C. Amendola, F. Beaudette, M. Bonanomi, P. Busson, C. Charlot, B. Diab, G. Falmagne, R. Granier de Cassagnac, I. Kucher, A. Lobanov, C. Martin Perez, M. Nguyen, C. Ochando, P. Paganini, J. Rembser, R. Salerno, J.B. Sauvan, Y. Sirois, A. Zabi, A. Zghiche Université de Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France J.-L. Agram15, J. Andrea, D. Bloch, G. Bourgatte, J.-M. Brom, E.C.", "Chabert, C. Collard, E. Conte15, J.-C. Fontaine15, D. Gelé, U. Goerlach, C. Grimault, A.-C.", "Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l'Institut National de Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne, France S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, C. Bernet, G. Boudoul, C. Camen, A. Carle, N. Chanon, R. Chierici, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, S. Gascon, M. Gouzevitch, B. Ille, Sa.", "Jain, I.B.", "Laktineh, H. Lattaud, A. Lesauvage, M. Lethuillier, L. Mirabito, S. Perries, V. Sordini, L. Torterotot, G. Touquet, M. Vander Donckt, S. Viret Georgian Technical University, Tbilisi, Georgia T. Toriashvili16 Tbilisi State University, Tbilisi, Georgia Z. Tsamalaidze10 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany C. Autermann, L. Feld, K. Klein, M. Lipinski, D. Meuser, A. Pauls, M. Preuten, M.P.", "Rauch, J. Schulz, M. Teroerde RWTH Aachen University, III.", "Physikalisches Institut A, Aachen, Germany M. Erdmann, B. Fischer, S. Ghosh, T. Hebbeker, K. Hoepfner, H. Keller, L. Mastrolorenzo, M. Merschmeyer, A. Meyer, P. Millet, G. Mocellin, S. Mondal, S. Mukherjee, D. Noll, A. Novak, T. Pook, A. Pozdnyakov, T. Quast, M. Radziej, Y. Rath, H. Reithler, J. Roemer, A. Schmidt, S.C. Schuler, A. Sharma, S. Wiedenbeck, S. Zaleski RWTH Aachen University, III.", "Physikalisches Institut B, Aachen, Germany G. Flügge, W. Haj Ahmad17, O. Hlushchenko, T. Kress, T. Müller, A. Nowack, C. Pistone, O. Pooth, D. Roy, H. Sert, A. Stahl18 Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, P. Asmuss, I. Babounikau, H. Bakhshiansohi, K. Beernaert, O. Behnke, A. Bermúdez Martínez, A.A. Bin Anuar, K. Borras19, V. Botta, A. Campbell, A. Cardini, P. Connor, 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G. Anagnostou, P. Asenov, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas, G. Paspalaki, A. Stakia National and Kapodistrian University of Athens, Athens, Greece M. Diamantopoulou, G. Karathanasis, P. Kontaxakis, A. Manousakis-katsikakis, A. Panagiotou, I. Papavergou, N. Saoulidou, K. Theofilatos, K. Vellidis, E. Vourliotis National Technical University of Athens, Athens, Greece G. Bakas, K. Kousouris, I. Papakrivopoulos, G. Tsipolitis, A. Zacharopoulou University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, K. Manitara, N. Manthos, I. Papadopoulos, J. Strologas, F.A.", "Triantis, D. Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary M. Bartók22, R. Chudasama, M. Csanad, P. Major, K. Mandal, A. Mehta, G. Pasztor, O. Surányi, G.I.", "Veres Wigner Research Centre for Physics, Budapest, Hungary G. Bencze, C. Hajdu, D. Horvath23, F. Sikler, V. Veszpremi, 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M. Gola, S. Keshri, Ashok Kumar, M. Naimuddin, P. Priyanka, K. Ranjan, Aashaq Shah, R. Sharma Saha Institute of Nuclear Physics, HBNI, Kolkata, India R. Bhardwaj28, M. Bharti28, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep28, D. Bhowmik, S. Dutta, S. Ghosh, B. Gomber29, M. Maity30, K. Mondal, S. Nandan, A. Purohit, P.K.", "Rout, G. Saha, S. Sarkar, M. Sharan, B. Singh28, S. Thakur28 Indian Institute of Technology Madras, Madras, India P.K.", "Behera, S.C. Behera, P. Kalbhor, A. Muhammad, R. Pradhan, P.R.", "Pujahari, A. Sharma, A.K.", "Sikdar Bhabha Atomic Research Centre, Mumbai, India D. Dutta, V. Jha, D.K.", "Mishra, P.K.", "Netrakanti, L.M.", "Pant, P. Shukla Tata Institute of Fundamental Research-A, Mumbai, India T. Aziz, M.A.", "Bhat, S. Dugad, G.B.", "Mohanty, N. Sur, RavindraKumar Verma Tata Institute of Fundamental Research-B, Mumbai, India S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, S. Karmakar, S. Kumar, G. Majumder, K. Mazumdar, N. Sahoo, S. Sawant Indian Institute of Science Education and Research (IISER), Pune, India S. Dube, B. Kansal, A. Kapoor, K. Kothekar, S. Pandey, A. Rane, A. Rastogi, S. Sharma Institute for Research in Fundamental Sciences (IPM), Tehran, Iran S. Chenarani, S.M.", "Etesami, M. Khakzad, M. Mohammadi Najafabadi, M. Naseri, F. Rezaei Hosseinabadi University College Dublin, Dublin, Ireland M. Felcini, M. Grunewald INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ , Bari, Italy M. Abbrescia$^{a}$$^{, }$$^{b}$ , R. Aly$^{a}$$^{, }$$^{b}$$^{, }$ 31, C. Calabria$^{a}$$^{, }$$^{b}$ , A. Colaleo$^{a}$ , D. Creanza$^{a}$$^{, }$$^{c}$ , L. Cristella$^{a}$$^{, }$$^{b}$ , N. De Filippis$^{a}$$^{, }$$^{c}$ , M. De Palma$^{a}$$^{, }$$^{b}$ , A.", "Di Florio$^{a}$$^{, }$$^{b}$ , W. Elmetenawee$^{a}$$^{, }$$^{b}$ , L. Fiore$^{a}$ , A. Gelmi$^{a}$$^{, }$$^{b}$ , G. Iaselli$^{a}$$^{, }$$^{c}$ , M. Ince$^{a}$$^{, }$$^{b}$ , S. Lezki$^{a}$$^{, }$$^{b}$ , G. Maggi$^{a}$$^{, }$$^{c}$ , M. Maggi$^{a}$ , J.A.", "Merlin$^{a}$ , G. Miniello$^{a}$$^{, }$$^{b}$ , S. My$^{a}$$^{, }$$^{b}$ , S. Nuzzo$^{a}$$^{, }$$^{b}$ , A. Pompili$^{a}$$^{, }$$^{b}$ , G. Pugliese$^{a}$$^{, }$$^{c}$ , R. Radogna$^{a}$ , A. Ranieri$^{a}$ , G. Selvaggi$^{a}$$^{, }$$^{b}$ , L. Silvestris$^{a}$ , F.M.", "Simone$^{a}$$^{, }$$^{b}$ , R. Venditti$^{a}$ , P. Verwilligen$^{a}$ INFN Sezione di Bologna $^{a}$ , Università di Bologna $^{b}$ , Bologna, Italy G. Abbiendi$^{a}$ , C. Battilana$^{a}$$^{, }$$^{b}$ , D. Bonacorsi$^{a}$$^{, }$$^{b}$ , L. Borgonovi$^{a}$$^{, }$$^{b}$ , S. Braibant-Giacomelli$^{a}$$^{, }$$^{b}$ , R. Campanini$^{a}$$^{, }$$^{b}$ , P. Capiluppi$^{a}$$^{, }$$^{b}$ , A. Castro$^{a}$$^{, }$$^{b}$ , F.R.", "Cavallo$^{a}$ , C. Ciocca$^{a}$ , G. Codispoti$^{a}$$^{, }$$^{b}$ , M. Cuffiani$^{a}$$^{, }$$^{b}$ , G.M.", "Dallavalle$^{a}$ , F. Fabbri$^{a}$ , A. Fanfani$^{a}$$^{, }$$^{b}$ , E. Fontanesi$^{a}$$^{, }$$^{b}$ , P. Giacomelli$^{a}$ , C. Grandi$^{a}$ , L. Guiducci$^{a}$$^{, }$$^{b}$ , F. Iemmi$^{a}$$^{, }$$^{b}$ , S. Lo Meo$^{a}$$^{, }$ 32, S. Marcellini$^{a}$ , G. Masetti$^{a}$ , F.L.", "Navarria$^{a}$$^{, }$$^{b}$ , A. Perrotta$^{a}$ , F. Primavera$^{a}$$^{, }$$^{b}$ , A.M. Rossi$^{a}$$^{, }$$^{b}$ , T. Rovelli$^{a}$$^{, }$$^{b}$ , G.P.", "Siroli$^{a}$$^{, }$$^{b}$ , N. Tosi$^{a}$ INFN Sezione di Catania $^{a}$ , Università di Catania $^{b}$ , Catania, Italy S. Albergo$^{a}$$^{, }$$^{b}$$^{, }$ 33, S. Costa$^{a}$$^{, }$$^{b}$ , A.", "Di Mattia$^{a}$ , R. Potenza$^{a}$$^{, }$$^{b}$ , A. Tricomi$^{a}$$^{, }$$^{b}$$^{, }$ 33, C. Tuve$^{a}$$^{, }$$^{b}$ INFN Sezione di Firenze $^{a}$ , Università di Firenze $^{b}$ , Firenze, Italy G. Barbagli$^{a}$ , A. Cassese$^{a}$ , R. Ceccarelli$^{a}$$^{, }$$^{b}$ , V. Ciulli$^{a}$$^{, }$$^{b}$ , C. Civinini$^{a}$ , R. D'Alessandro$^{a}$$^{, }$$^{b}$ , F. Fiori$^{a}$$^{, }$$^{c}$ , E. Focardi$^{a}$$^{, }$$^{b}$ , G. Latino$^{a}$$^{, }$$^{b}$ , P. Lenzi$^{a}$$^{, }$$^{b}$ , M. Lizzo$^{a}$$^{, }$$^{b}$ , M. Meschini$^{a}$ , S. Paoletti$^{a}$ , R. Seidita$^{a}$$^{, }$$^{b}$ , G. Sguazzoni$^{a}$ , L. Viliani$^{a}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, D. Piccolo INFN Sezione di Genova $^{a}$ , Università di Genova $^{b}$ , Genova, Italy M. Bozzo$^{a}$$^{, }$$^{b}$ , F. Ferro$^{a}$ , R. Mulargia$^{a}$$^{, }$$^{b}$ , E. Robutti$^{a}$ , S. Tosi$^{a}$$^{, }$$^{b}$ INFN Sezione di Milano-Bicocca $^{a}$ , Università di Milano-Bicocca $^{b}$ , Milano, Italy A. Benaglia$^{a}$ , A. Beschi$^{a}$$^{, }$$^{b}$ , F. Brivio$^{a}$$^{, }$$^{b}$ , V. Ciriolo$^{a}$$^{, }$$^{b}$$^{, }$ 18, M.E.", "Dinardo$^{a}$$^{, }$$^{b}$ , P. Dini$^{a}$ , S. Gennai$^{a}$ , A. Ghezzi$^{a}$$^{, }$$^{b}$ , P. Govoni$^{a}$$^{, }$$^{b}$ , L. Guzzi$^{a}$$^{, }$$^{b}$ , M. Malberti$^{a}$ , S. Malvezzi$^{a}$ , D. Menasce$^{a}$ , F. Monti$^{a}$$^{, }$$^{b}$ , L. Moroni$^{a}$ , M. Paganoni$^{a}$$^{, }$$^{b}$ , D. Pedrini$^{a}$ , S. Ragazzi$^{a}$$^{, }$$^{b}$ , T. Tabarelli de Fatis$^{a}$$^{, }$$^{b}$ , D. Valsecchi$^{a}$$^{, }$$^{b}$$^{, }$ 18, D. Zuolo$^{a}$$^{, }$$^{b}$ INFN Sezione di Napoli $^{a}$ , Università di Napoli 'Federico II' $^{b}$ , Napoli, Italy, Università della Basilicata $^{c}$ , Potenza, Italy, Università G. Marconi $^{d}$ , Roma, Italy S. Buontempo$^{a}$ , N. Cavallo$^{a}$$^{, }$$^{c}$ , A.", "De Iorio$^{a}$$^{, }$$^{b}$ , A.", "Di Crescenzo$^{a}$$^{, }$$^{b}$ , F. Fabozzi$^{a}$$^{, }$$^{c}$ , F. Fienga$^{a}$ , G. Galati$^{a}$ , A.O.M.", "Iorio$^{a}$$^{, }$$^{b}$ , L. Layer$^{a}$$^{, }$$^{b}$ , L. Lista$^{a}$$^{, }$$^{b}$ , S. Meola$^{a}$$^{, }$$^{d}$$^{, }$ 18, P. Paolucci$^{a}$$^{, }$ 18, B. Rossi$^{a}$ , C. Sciacca$^{a}$$^{, }$$^{b}$ , E. Voevodina$^{a}$$^{, }$$^{b}$ INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Padova, Italy, Università di Trento $^{c}$ , Trento, Italy P. Azzi$^{a}$ , N. Bacchetta$^{a}$ , D. Bisello$^{a}$$^{, }$$^{b}$ , A. Boletti$^{a}$$^{, }$$^{b}$ , A. Bragagnolo$^{a}$$^{, }$$^{b}$ , R. Carlin$^{a}$$^{, }$$^{b}$ , P. Checchia$^{a}$ , P. De Castro Manzano$^{a}$ , T. Dorigo$^{a}$ , U. Dosselli$^{a}$ , F. Gasparini$^{a}$$^{, }$$^{b}$ , U. Gasparini$^{a}$$^{, }$$^{b}$ , A. Gozzelino$^{a}$ , S.Y.", "Hoh$^{a}$$^{, }$$^{b}$ , M. Margoni$^{a}$$^{, }$$^{b}$ , A.T. Meneguzzo$^{a}$$^{, }$$^{b}$ , J. Pazzini$^{a}$$^{, }$$^{b}$ , M. Presilla$^{b}$ , P. Ronchese$^{a}$$^{, }$$^{b}$ , R. Rossin$^{a}$$^{, }$$^{b}$ , F. Simonetto$^{a}$$^{, }$$^{b}$ , A. Tiko$^{a}$ , M. Tosi$^{a}$$^{, }$$^{b}$ , M. Zanetti$^{a}$$^{, }$$^{b}$ , P. Zotto$^{a}$$^{, }$$^{b}$ , A. Zucchetta$^{a}$$^{, }$$^{b}$ , G. Zumerle$^{a}$$^{, }$$^{b}$ INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ , Pavia, Italy A. Braghieri$^{a}$ , D. Fiorina$^{a}$$^{, }$$^{b}$ , P. Montagna$^{a}$$^{, }$$^{b}$ , S.P.", "Ratti$^{a}$$^{, }$$^{b}$ , V. Re$^{a}$ , M. Ressegotti$^{a}$$^{, }$$^{b}$ , C. Riccardi$^{a}$$^{, }$$^{b}$ , P. Salvini$^{a}$ , I. Vai$^{a}$ , P. Vitulo$^{a}$$^{, }$$^{b}$ INFN Sezione di Perugia $^{a}$ , Università di Perugia $^{b}$ , Perugia, Italy M. Biasini$^{a}$$^{, }$$^{b}$ , G.M.", "Bilei$^{a}$ , D. Ciangottini$^{a}$$^{, }$$^{b}$ , L. Fanò$^{a}$$^{, }$$^{b}$ , P. Lariccia$^{a}$$^{, }$$^{b}$ , R. Leonardi$^{a}$$^{, }$$^{b}$ , E. Manoni$^{a}$ , G. Mantovani$^{a}$$^{, }$$^{b}$ , V. Mariani$^{a}$$^{, }$$^{b}$ , M. Menichelli$^{a}$ , A. Rossi$^{a}$$^{, }$$^{b}$ , A. Santocchia$^{a}$$^{, }$$^{b}$ , D. Spiga$^{a}$ INFN Sezione di Pisa $^{a}$ , Università di Pisa $^{b}$ , Scuola Normale Superiore di Pisa $^{c}$ , Pisa, Italy K. Androsov$^{a}$ , P. Azzurri$^{a}$ , G. Bagliesi$^{a}$ , V. Bertacchi$^{a}$$^{, }$$^{c}$ , L. Bianchini$^{a}$ , T. Boccali$^{a}$ , R. Castaldi$^{a}$ , M.A.", "Ciocci$^{a}$$^{, }$$^{b}$ , R. Dell'Orso$^{a}$ , S. Donato$^{a}$ , L. Giannini$^{a}$$^{, }$$^{c}$ , A. Giassi$^{a}$ , M.T.", "Grippo$^{a}$ , F. Ligabue$^{a}$$^{, }$$^{c}$ , E. Manca$^{a}$$^{, }$$^{c}$ , G. Mandorli$^{a}$$^{, }$$^{c}$ , A. Messineo$^{a}$$^{, }$$^{b}$ , F. Palla$^{a}$ , A. Rizzi$^{a}$$^{, }$$^{b}$ , G. Rolandi$^{a}$$^{, }$$^{c}$ , S. Roy Chowdhury$^{a}$$^{, }$$^{c}$ , A. Scribano$^{a}$ , P. Spagnolo$^{a}$ , R. Tenchini$^{a}$ , G. Tonelli$^{a}$$^{, }$$^{b}$ , N. Turini$^{a}$ , A. Venturi$^{a}$ , P.G.", "Verdini$^{a}$ INFN Sezione di Roma $^{a}$ , Sapienza Università di Roma $^{b}$ , Rome, Italy F. Cavallari$^{a}$ , M. Cipriani$^{a}$$^{, }$$^{b}$ , D. Del Re$^{a}$$^{, }$$^{b}$ , E. Di Marco$^{a}$ , M. Diemoz$^{a}$ , E. Longo$^{a}$$^{, }$$^{b}$ , P. Meridiani$^{a}$ , G. Organtini$^{a}$$^{, }$$^{b}$ , F. Pandolfi$^{a}$ , R. Paramatti$^{a}$$^{, }$$^{b}$ , C. Quaranta$^{a}$$^{, }$$^{b}$ , S. Rahatlou$^{a}$$^{, }$$^{b}$ , C. Rovelli$^{a}$ , F. Santanastasio$^{a}$$^{, }$$^{b}$ , L. Soffi$^{a}$$^{, }$$^{b}$ , R. Tramontano$^{a}$$^{, }$$^{b}$ INFN Sezione di Torino $^{a}$ , Università di Torino $^{b}$ , Torino, Italy, Università del Piemonte Orientale $^{c}$ , Novara, Italy N. Amapane$^{a}$$^{, }$$^{b}$ , R. Arcidiacono$^{a}$$^{, }$$^{c}$ , S. Argiro$^{a}$$^{, }$$^{b}$ , M. Arneodo$^{a}$$^{, }$$^{c}$ , N. Bartosik$^{a}$ , R. Bellan$^{a}$$^{, }$$^{b}$ , A. Bellora$^{a}$$^{, }$$^{b}$ , C. Biino$^{a}$ , A. Cappati$^{a}$$^{, }$$^{b}$ , N. Cartiglia$^{a}$ , S. Cometti$^{a}$ , M. Costa$^{a}$$^{, }$$^{b}$ , R. Covarelli$^{a}$$^{, }$$^{b}$ , N. Demaria$^{a}$ , J.R. González Fernández$^{a}$ , B. Kiani$^{a}$$^{, }$$^{b}$ , F. Legger$^{a}$ , C. Mariotti$^{a}$ , S. Maselli$^{a}$ , E. Migliore$^{a}$$^{, }$$^{b}$ , V. Monaco$^{a}$$^{, }$$^{b}$ , E. Monteil$^{a}$$^{, }$$^{b}$ , M. Monteno$^{a}$ , M.M.", "Obertino$^{a}$$^{, }$$^{b}$ , G. Ortona$^{a}$ , L. Pacher$^{a}$$^{, }$$^{b}$ , N. Pastrone$^{a}$ , M. Pelliccioni$^{a}$ , G.L.", "Pinna Angioni$^{a}$$^{, }$$^{b}$ , A. Romero$^{a}$$^{, }$$^{b}$ , M. Ruspa$^{a}$$^{, }$$^{c}$ , R. Salvatico$^{a}$$^{, }$$^{b}$ , V. Sola$^{a}$ , A. Solano$^{a}$$^{, }$$^{b}$ , D. Soldi$^{a}$$^{, }$$^{b}$ , A. Staiano$^{a}$ , D. Trocino$^{a}$$^{, }$$^{b}$ INFN Sezione di Trieste $^{a}$ , Università di Trieste $^{b}$ , Trieste, Italy S. Belforte$^{a}$ , V. Candelise$^{a}$$^{, }$$^{b}$ , M. Casarsa$^{a}$ , F. Cossutti$^{a}$ , A. Da Rold$^{a}$$^{, }$$^{b}$ , G. Della Ricca$^{a}$$^{, }$$^{b}$ , F. Vazzoler$^{a}$$^{, }$$^{b}$ , A. Zanetti$^{a}$ Kyungpook National University, Daegu, Korea B. Kim, D.H. Kim, G.N.", "Kim, J. Lee, S.W.", "Lee, C.S.", "Moon, Y.D.", "Oh, S.I.", "Pak, S. Sekmen, D.C.", "Son, Y.C.", "Yang Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea H. Kim, D.H.", "Moon Hanyang University, Seoul, Korea B. Francois, T.J. Kim, J.", "Park Korea University, Seoul, Korea S. Cho, S. Choi, Y.", "Go, S. Ha, B. Hong, K. Lee, K.S.", "Lee, J. Lim, J.", "Park, S.K.", "Park, Y. Roh, J. Yoo Kyung Hee University, Department of Physics, Seoul, Republic of Korea J. Goh Sejong University, Seoul, Korea H.S.", "Kim Seoul National University, Seoul, Korea J. Almond, J.H.", "Bhyun, J. Choi, S. Jeon, J. Kim, J.S.", "Kim, H. Lee, K. Lee, S. Lee, K. Nam, M. Oh, S.B.", "Oh, B.C.", "Radburn-Smith, U.K. Yang, H.D.", "Yoo, I. Yoon University of Seoul, Seoul, Korea D. Jeon, J.H.", "Kim, J.S.H.", "Lee, I.C.", "Park, I.J.", "Watson Sungkyunkwan University, Suwon, Korea Y. Choi, C. Hwang, Y. Jeong, J. Lee, Y. Lee, I. Yu Riga Technical University, Riga, Latvia V. Veckalns34 Vilnius University, Vilnius, Lithuania V. Dudenas, A. Juodagalvis, A. Rinkevicius, G. Tamulaitis, J. Vaitkus National Centre for Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia F. Mohamad Idris35, W.A.T.", "Wan Abdullah, M.N.", "Yusli, Z. Zolkapli Universidad de Sonora (UNISON), Hermosillo, Mexico J.F.", "Benitez, A. Castaneda Hernandez, J.A.", "Murillo Quijada, L. Valencia Palomo Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-De La Cruz36, R. Lopez-Fernandez, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, C. Oropeza Barrera, M. Ramirez-Garcia, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico J. Eysermans, I. Pedraza, H.A.", "Salazar Ibarguen, C. Uribe Estrada Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico A. Morelos Pineda University of Montenegro, Podgorica, Montenegro J. Mijuskovic3, N. Raicevic University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand S. Bheesette, P.H.", "Butler, P. Lujan National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan A. Ahmad, M. Ahmad, M.I.M.", "Awan, Q. Hassan, H.R.", "Hoorani, W.A.", "Khan, M.A.", "Shah, M. Shoaib, M. Waqas AGH University of Science and Technology Faculty of Computer Science, Electronics and Telecommunications, Krakow, Poland V. Avati, L. Grzanka, M. Malawski National Centre for Nuclear Research, Swierk, Poland H. Bialkowska, M. Bluj, B. Boimska, M. Górski, M. Kazana, M. Szleper, P. Zalewski Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland K. Bunkowski, A. Byszuk37, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski, M. Olszewski, M. Walczak Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal M. Araujo, P. Bargassa, D. Bastos, A.", "Di Francesco, P. Faccioli, B. Galinhas, M. Gallinaro, J. Hollar, N. Leonardo, T. Niknejad, J. Seixas, K. Shchelina, G. Strong, O. Toldaiev, J. Varela Joint Institute for Nuclear Research, Dubna, Russia S. Afanasiev, Y. Ershov, A. Golunov, I. Golutvin, N. Gorbounov, I. Gorbunov, A. Kamenev, V. Karjavine, V. Korenkov, A. Lanev, A. Malakhov, V. Matveev38$^{, }$ 39, P. Moisenz, V. Palichik, V. Perelygin, S. Shmatov, S. Shulha, V. Smirnov, N. Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg), Russia L. Chtchipounov, V. Golovtcov, Y. Ivanov, V. Kim40, E. Kuznetsova41, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L. Uvarov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia Yu.", "Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov, N. Krasnikov, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics named by A.I.", "Alikhanov of NRC `Kurchatov Institute', Moscow, Russia V. Epshteyn, V. Gavrilov, N. Lychkovskaya, A. Nikitenko42, V. Popov, I. Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, M. Toms, E. Vlasov, A. Zhokin Moscow Institute of Physics and Technology, Moscow, Russia T. Aushev National Research Nuclear University 'Moscow Engineering Physics Institute' (MEPhI), Moscow, Russia O. Bychkova, R. Chistov43, M. Danilov43, S. Polikarpov43, E. Tarkovskii P.N.", "Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Terkulov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia A. Belyaev, E. Boos, A. Ershov, A. Gribushin, A. Kaminskiy44, O. Kodolova, V. Korotkikh, I. Lokhtin, S. Obraztsov, S. Petrushanko, V. Savrin, A. Snigirev, I. Vardanyan Novosibirsk State University (NSU), Novosibirsk, Russia A. Barnyakov45, V. Blinov45, T. Dimova45, L. Kardapoltsev45, Y. Skovpen45 Institute for High Energy Physics of National Research Centre `Kurchatov Institute', Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, D. Konstantinov, P. Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University, Tomsk, Russia A. Babaev, A. Iuzhakov, V. Okhotnikov Tomsk State University, Tomsk, Russia V. Borchsh, V. Ivanchenko, E. Tcherniaev University of Belgrade: Faculty of Physics and VINCA Institute of Nuclear Sciences, Serbia P. Adzic46, P. Cirkovic, M. Dordevic, P. Milenovic, J. Milosevic, M. Stojanovic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain M. Aguilar-Benitez, J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M. Barrio Luna, CristinaF.", "Bedoya, J.A.", "Brochero Cifuentes, C.A.", "Carrillo Montoya, M. Cepeda, M. Cerrada, N. Colino, B.", "De La Cruz, A. Delgado Peris, J.P. Fernández Ramos, J. Flix, M.C.", "Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M.", "Hernandez, M.I.", "Josa, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo, J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S.", "Soares, A. Triossi, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, J.F.", "de Trocóniz, R. Reyes-Almanza Universidad de Oviedo, Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Oviedo, Spain B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, E. Palencia Cortezon, C. Ramón Álvarez, V. Rodríguez Bouza, S. Sanchez Cruz Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain I.J.", "Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M. Fernandez, P.J.", "Fernández Manteca, A. García Alonso, G. Gomez, C. Martinez Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels, F. Ricci-Tam, T. Rodrigo, A. Ruiz-Jimeno, L. Russo47, L. Scodellaro, I. Vila, J.M.", "Vizan Garcia University of Colombo, Colombo, Sri Lanka D.U.J.", "Sonnadara University of Ruhuna, 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A.W.", "Jung, B. Mahakud, D.H. Miller, G. Negro, N. Neumeister, C.C.", "Peng, S. Piperov, H. Qiu, J.F.", "Schulte, N. Trevisani, F. Wang, R. Xiao, W. Xie Purdue University Northwest, Hammond, USA T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA A. Baty, U. Behrens, S. Dildick, K.M.", "Ecklund, S. Freed, F.J.M.", "Geurts, M. Kilpatrick, Arun Kumar, W. Li, B.P.", "Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, A.G. Stahl Leiton, Z. Tu, A. Zhang University of Rochester, Rochester, USA A. Bodek, P. de Barbaro, R. Demina, J.L.", "Dulemba, C. Fallon, T. Ferbel, M. Galanti, A. Garcia-Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R. Taus Rutgers, The State University of New Jersey, Piscataway, USA B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E. Hughes, S. Kaplan, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M. Osherson, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S. Thomas University of Tennessee, Knoxville, USA H. Acharya, A.G. Delannoy, S. Spanier Texas A&M University, College Station, USA O. Bouhali81, M. Dalchenko, A. Delgado, R. Eusebi, J. Gilmore, T. Huang, T. Kamon82, H. Kim, S. Luo, S. Malhotra, D. Marley, R. Mueller, D. Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, F. De Guio, V. Hegde, S. Kunori, K. Lamichhane, S.W.", "Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A. Whitbeck Vanderbilt University, Nashville, USA S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K. Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of Virginia, Charlottesville, USA M.W.", "Arenton, P. Barria, B. Cox, G. Cummings, J. Hakala, R. Hirosky, M. Joyce, A. Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA R. Harr, P.E.", "Karchin, N. Poudyal, J. Sturdy, P. Thapa University of Wisconsin - Madison, Madison, WI, USA K. Black, T. Bose, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I.", "De Bruyn, L. Dodd, C. Galloni, H. He, M. Herndon, A. Hervé, U. Hussain, A. Lanaro, A. Loeliger, R. Loveless, J. Madhusudanan Sreekala, A. Mallampalli, D. Pinna, T. Ruggles, A. Savin, V. Sharma, W.H.", "Smith, D. Teague, S. Trembath-reichert †: Deceased 1: Also at Vienna University of Technology, Vienna, Austria 2: Also at Université Libre de Bruxelles, Bruxelles, Belgium 3: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 4: Also at Universidade Estadual de Campinas, Campinas, Brazil 5: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 6: Also at UFMS, Nova Andradina, Brazil 7: Also at Universidade Federal de Pelotas, Pelotas, Brazil 8: Also at University of Chinese Academy of Sciences, Beijing, China 9: Also at Institute for Theoretical and Experimental Physics named by A.I.", "Alikhanov of NRC `Kurchatov Institute', Moscow, Russia 10: Also at Joint Institute for Nuclear Research, Dubna, Russia 11: Also at Zewail City of Science and Technology, Zewail, Egypt 12: Also at British University in Egypt, Cairo, Egypt 13: Now at Ain Shams University, Cairo, Egypt 14: Also at Purdue University, West Lafayette, USA 15: Also at Université de Haute Alsace, Mulhouse, France 16: Also at Tbilisi State University, Tbilisi, Georgia 17: Also at Erzincan Binali Yildirim University, Erzincan, Turkey 18: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 19: Also at RWTH Aachen University, III.", "Physikalisches Institut A, Aachen, Germany 20: Also at University of Hamburg, Hamburg, Germany 21: Also at Brandenburg University of Technology, Cottbus, Germany 22: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary, Debrecen, Hungary 23: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 24: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary, Budapest, Hungary 25: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India 26: Also at Institute of Physics, Bhubaneswar, India 27: Also at G.H.G.", "Khalsa College, Punjab, India 28: Also at Shoolini University, Solan, India 29: Also at University of Hyderabad, Hyderabad, India 30: Also at University of Visva-Bharati, Santiniketan, India 31: Now at INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ , Bari, Italy 32: Also at Italian National Agency for New Technologies, Energy and Sustainable Economic Development, Bologna, Italy 33: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia, Catania, Italy 34: Also at Riga Technical University, Riga, Latvia, Riga, Latvia 35: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia 36: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico 37: Also at Warsaw University of Technology, Institute of Electronic Systems, Warsaw, Poland 38: Also at Institute for Nuclear Research, Moscow, Russia 39: Now at National Research Nuclear University 'Moscow Engineering Physics Institute' (MEPhI), Moscow, Russia 40: Also at St. Petersburg State Polytechnical University, St. Petersburg, Russia 41: Also at University of Florida, Gainesville, USA 42: Also at Imperial College, London, United Kingdom 43: Also at P.N.", "Lebedev Physical Institute, Moscow, Russia 44: Also at INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Padova, Italy, Università di Trento $^{c}$ , Trento, Italy, Padova, Italy 45: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia 46: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia 47: Also at Università degli Studi di Siena, Siena, Italy 48: Also at INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ , Pavia, Italy, Pavia, Italy 49: Also at National and Kapodistrian University of Athens, Athens, Greece 50: Also at Universität Zürich, Zurich, Switzerland 51: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria, Vienna, Austria 52: Also at Burdur Mehmet Akif Ersoy University, BURDUR, Turkey 53: Also at Şırnak University, Sirnak, Turkey 54: Also at Department of Physics, Tsinghua University, Beijing, China, Beijing, China 55: Also at Near East University, Research Center of Experimental Health Science, Nicosia, Turkey 56: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey 57: Also at Istanbul Aydin University, Application and Research Center for Advanced Studies (App.", "& Res.", "Cent.", "for Advanced Studies), Istanbul, Turkey 58: Also at Mersin University, Mersin, Turkey 59: Also at Piri Reis University, Istanbul, Turkey 60: Also at Ozyegin University, Istanbul, Turkey 61: Also at Izmir Institute of Technology, Izmir, Turkey 62: Also at Bozok Universitetesi Rektörlügü, Yozgat, Turkey 63: Also at Marmara University, Istanbul, Turkey 64: Also at Milli Savunma University, Istanbul, Turkey 65: Also at Kafkas University, Kars, Turkey 66: Also at Istanbul Bilgi University, Istanbul, Turkey 67: Also at Hacettepe University, Ankara, Turkey 68: Also at Adiyaman University, Adiyaman, Turkey 69: Also at Vrije Universiteit Brussel, Brussel, Belgium 70: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 71: Also at IPPP Durham University, Durham, United Kingdom 72: Also at Monash University, Faculty of Science, Clayton, Australia 73: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA 74: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey 75: Also at California Institute of Technology, Pasadena, USA 76: Also at Bingol University, Bingol, Turkey 77: Also at Georgian Technical University, Tbilisi, Georgia 78: Also at Sinop University, Sinop, Turkey 79: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey 80: Also at Nanjing Normal University Department of Physics, Nanjing, China 81: Also at Texas A&M University at Qatar, Doha, Qatar 82: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea" ] ]

[ [ "A low-mass stellar companion to the young variable star RZ Psc" ], [ "Abstract RZ Psc is a young Sun-like star with a bright and warm infrared excess that is occasionally dimmed significantly by circumstellar dust structures.", "Optical depth arguments suggest that the dimming events do not probe a typical sight line through the circumstellar dust, and are instead caused by structures that appear above an optically thick mid-plane.", "This system may therefore be similar to systems where an outer disk is shadowed by material closer to the star.", "Here we report the discovery that RZ Psc hosts a 0.12$M_\\odot$ companion at a projected separation of 23au.", "We conclude that the disk must orbit the primary star.", "While we do not detect orbital motion, comparison of the angle of linear polarization of the primary with the companion's on-sky position angle provides circumstantial evidence that the companion and disc may not share the same orbital plane.", "Whether the companion severely disrupts the disc, truncates it, or has little effect at all, will require further observations of both the companion and disc." ], [ "Introduction", "RZ Psc is a young star that shows irregular photometric dimming events that are attributed to circumstellar dust transiting across the face of the star.", "Such systems promise a window on the clumpiness of the inner regions of circumstellar discs near the epoch of planet formation , and combined with ancillary information (e.g.", "dust thermal emission, outer disc imaging) may yield broader insights into disc vertical structure and geometry , [2], [4].", "Similarly, clumpy and/or misaligned inner discs can cast shadows on outer disc regions [9], , .", "As noted by , shadowed-disc systems may be very similar to those where the star is seen to be dimmed from Earth, and simply observed with a more face-on geometry.", "Given that perturbations of inner disc regions by planetary or stellar companions has been proposed for shadowed-disc systems , the same may be true for variable systems such as RZ Psc.", "RZ Psc has been classed as an \"UXor\" , which are normally young Herbig Ae/Be stars that show sporadic photometric dimming events that last for days to weeks .", "UXors also tend to show an increased degree of linear polarization and reddening during typical dimming events , .", "The circumstellar dust interpretation posits that the linear polarization increases because the fraction of the total emission that is contributed by dust-scattered light increases when the star is dimmed, and that the reddening is caused by wavelength-selective extinction.", "While other scenarios may explain some UXors , RZ Psc is consistent with the circumstellar dust scenario, for several reasons.", "Primarily, RZ Psc shows both the reddening behaviour and an increased degree of linear polarization during dimmings .", "Using the timescale of RZ Psc's dimming events, from which a transverse velocity can be estimated, this dust is estimated to lie at roughly 1 au , [13].", "This system is also seen to host a bright mid-infrared (IR) excess, inferred to be thermal emission from a warm circumstellar dust population .", "There is no far-IR excess, which rules out a bright outer disc, but the outer radius could nevertheless be several tens of au .", "While neither of these properties are unusual for an UXor, what marks RZ Psc as unusual is that its probable age is a few tens of Myr, older than other UXors, and Herbig Ae/Be stars in general , .", "Also, the spectral type of K0V is later than typical UXors, and more similar to the so-called \"dippers\" , [10], [3], .", "In addition to the short-term dimming events, RZ Psc also shows a long-term sinusoidal photometric variation with a 12 year period and 0.5 mag amplitude.", "Synthesising these observations, suggested that the dust populations causing the dimming and IR excess are one and the same, and given the probable age that this dust resides in a young and massive analogue of the Solar system's asteroid belt.", "The long-term variation was considered largely independent of the inner dust population, perhaps caused by a varying line of sight through an outer disc that is warped by a companion.", "This picture was reconsidered by , based primarily on a simple optical depth argument.", "The large fractional dust luminosity ($L_{\\rm disc}/L_\\star \\approx 0.07$ ) implies that 7 % of the starlight is captured and re-emitted thermally by the dust.", "That is, as seen from the star about 7 % of the sky is covered by dust.", "RZ Psc is not seen to be significantly reddened , implying that the dust must either have a large scale height (i.e.", "has a more shell-like structure), or that our line of sight to the star does not probe a typical sight line through the dust cloud.", "The increased degree of linear polarization during deep dimming events however disfavours a near-spherical distribution of dust around the star.", "therefore concluded that the dimming events are the result of occultations by dust structures that rise above an optically thick disc mid-plane, highlighting the alternative possibility that RZ Psc hosts a relatively old and evolved protoplanetary disc .", "Recent work finds that RZ Psc shows a weak H$\\alpha $ accretion signature, which is more consistent with a long-lived protoplanetary disc scenario , .", "Here we present the discovery that RZ Psc is a binary with a sky-projected separation of 23 au, based on high-contrast imaging observations with VLT/SPHERE ([5]).", "While RZ Psc shows no evidence for an outer disc, this companion might be truncating and/or warping parts of the disc in a way that makes the inner regions of this system similar to shadowed-disc systems.", "We present our observations in Section , and discuss the possible implications in Section ." ], [ "Observations and Results", "RZ Psc was observed with SPHERE/IRDIS ([11]) in its dual-band polarimetric imaging (DPI) mode (, , ) at a total of six observation epochs.", "We show the instrument setup and integration time in Table REF .", "The first observation epoch in 2018 was performed in the field stabilized mode of the instrument, while all subsequent observations were performed in pupil stabilized mode to ensure high polarimetric throughput of the instrument ().", "We note that the field rotation due to the parallactic angle change in pupil stabilized mode was 10$^\\circ $ or less in all observation epochs.", "All but the second observation epoch use the IRDIS apodized Lyot coronagraph YJH_S with a radius of 92.5 mas ([8]).", "All data reduction was performed with the IRDAP pipeline (IRDIS Data reduction for Accurate Polarimetry, ).", "The final data products used for the analysis included the stacked total intensity images as well as the Stokes $Q$ and $U$ images.", "Due to the very small change in parallactic angle we did not perform angular differential imaging in total intensity on the data sets.", "Figure: SPHERE/IRDIS observations of RZ Psc A's companion.", "Left and right images were taken with a coronagraph in place (marked by the grey hashed area).", "The data shown in the middle panel was taken without a coronagraph.", "The position of the new stellar companion is marked by the white bars.", "The filter of the observation is indicated in each panel as well as the observation date.", "The J2000 primary position is 01h09m42.05s +27d57m01.9s at epoch 2000.0.For our analysis we used the observations taken in the H-band on 01-10-2018 and 16-08-2019, with and without the coronagraph respectively, as well as the observation taken in K$_s$ -band on 23-08-2019.", "The remaining three K$_s$ -band epochs suffer from non-ideal observing conditions as described in Appendix .", "The total intensity images reveal the presence of a close and faint companion in the system as shown in Figure REF .", "Figure: Position angle and separation of the new companion relative to RZ Psc A over time.", "The grey ribbon area indicates the expected behavior for an unmoving background object, given RZ Psc's proper motion of μ α =27.4±0.1\\mu _\\alpha =27.4 \\pm 0.1 mas yr -1 ^{-1}, μ δ =-12.6±0.2\\mu _\\delta = -12.6 \\pm 0.2 mas yr -1 ^{-1} .", "The dashed lines show the maximal angle/distance a co-moving companion on a circular orbit could move, assuming face-on (upper panel), and edge-on (lower panel) orientations, and the stellar mass and distance given in section .To determine the nature of the companion we extracted astrometry and photometry from all observation epochs.", "Since the companion position is strongly contaminated by the point-spread function (PSF) of RZ Psc we first subtracted a 180$^\\circ $ rotated version of the images from the images to approximate the stellar point spread function.", "For astrometric extraction we then fitted a Gaussian to the companion signal.", "Astrometric calibration of the images was performed with the standard values given in .", "The final astrometric results are given in Table REF .", "Note that for the K$_s$ -band epoch the companion signal is likely vignetted somewhat by the coronagraphic mask, which may influence the extracted position.", "This should be less problematic in the higher spatial resolution H-band image of the first observation epoch and the image taken without the coronagraph in the second observation epoch.", "To extract the photometry of the companion we used apertures.", "The aperture radii were 3 detector pixels in the H-band images and 4 detector pixels in the K$_s$ -band images, approximating one resolution element in either band.", "In addition to the initial background subtraction, we used apertures with the same separation from the primary star at larger and smaller position angles than the companion position to determine the mean local residual background, which we subtract from the companion flux.", "All photometric measurements were performed relative to the primary star since no other calibrators were available.", "For the non-coronagraphic H-band epoch, both the star and the companion flux could be measured in the same image.", "To measure the stellar flux in the coronagraphic epochs we used flux calibration frames in which the primary star was shifted away from the coronagraph.", "These were taken with short integration times and, in the case of the H-band, a neutral density filter to prevent saturation.", "We used IRDIS ND_1.0 neutral density filter, with a throughput ratio in H-band of 7.94.", "Since the companion is located close to the inner working angle of the coronagraph we have to take coronagraph throughput into account.", "For the H-band detailed measurements exist and are presented in a forthcoming publication by Wilby et al, in prep.", "Given the companion separation we used a throughput correction factor of 0.865.", "For the K$_s$ -band no measurements exist for the YJH_S coronagraph.", "However, since the PSF full width at half maximum scales linearly with wavelength we extrapolated the throughput correction from shorter wavelengths and find a correction factor of 0.765.", "The extracted magnitude relative to the primary for each observation is given in Table REF Table: Astrometry and photometry of the RZ Psc system, including 1σ\\sigma uncertainties, as extracted from our SPHERE/IRDIS observations.", "The apparent magnitude assumes that the primary has a magnitude of 9.84 in H-band, and 9.70 in K s _s-band .All-Sky Automated Survey for Supernovae photometry on the days before and after our first epoch observation are consistent with being in the undimmed state.", "While photometry is not yet available for the more recent epochs, the consistency of the photometry in Table REF suggests that these observations were also during undimmed periods.", "Similarly, SED fitting that includes the 2MASS photometry used to derive the apparent magnitudes of the companion finds a well-fitting solution, indicating that the star was not significantly dimmed at the time of those observations .", "We do not detect a linear polarization signal from extended circumstellar dust in any of the observation epochs and can thus rule out the presence of significant amounts of small ($\\mu $ m-sized) dust particles down to the coronagraphic inner working angle of 92 mas (18 au).", "We do measure the linear polarization of the primary star in H-band (the K$_s$ -band data are of not sufficient quality to provide a useful measurement).", "The first and second epochs yield $0.5 \\pm 0.1\\,\\%$ at PA $52 \\pm 7^\\circ $ and $0.56 \\pm 0.09\\,\\%$ at PA $55 \\pm 3^\\circ $ respectively.", "Due to the contamination by the residual stellar halo in Stokes Q and U images a measurement of the companion polarization is difficult.", "Using the same aperture photometry technique as described previously, we estimate that the companion's degree of linear polarization is less than 0.2 % in the H-band." ], [ "Discussion and Conclusions", "Several tests can be made to verify association of the companion with RZ Psc.", "First, following the approach of we used the TRILEGAL model of the Milky Way to estimate the probability of a background star as bright or brighter than the detected companion appearing within 120 mas of RZ Psc, which yields $1.7 \\times 10^{-6}$ .", "Second, Figure REF shows that the companion is co-moving with RZ Psc on the sky and that a stationary background source is ruled out (5.6$\\sigma $ in PA, 7.9$\\sigma $ in separation).", "Third, while a background source is not necessarily stationary, the distance required for a K0-type giant (e.g.", "Pollux) to have the same magnitude as the companion is about 9 kpc, and at this distance fewer than 1 in 10,000 stars in Gaia Data Release 2 [12], [13] have a proper motion as high as RZ Psc.", "Also, given the few-degree uncertainty on the companion PA, the probability that a background source with the same proper motion amplitude would also be moving in the same direction as RZ Psc is $\\sim $ 1 %.", "We therefore conclude that the companion is almost certainly associated with RZ Psc.", "Using the distance to RZ Psc of 196 pc, the absolute H-band magnitude of the companion is $7.3 \\pm 0.1$ and the H-K$_s$ colour is $0.2 \\pm 0.3$ , which are consistent with a mid-M field star spectral type [6].", "If we assume an age of 20 Myr for RZ Psc , then using the BT-Settl isochrones [1] we find a companion mass of 0.12$\\pm $ 0.01 $M_\\odot $ for the H-band and 0.11$\\pm $ 0.02 $M_\\odot $ for the K$_s$ -band.", "The systematic uncertainty in the mass due to an uncertain age is of course larger (on the order of 0.05 $M_\\odot $ if the age changes by 10 Myr).", "Assuming a circular and face-on orbit, the expected azimuthal orbital motion of the companion at 23 au (assuming 196 pc and an approximate primary mass for a K0V-type star of 0.9 $M_\\odot $ ) is about 6 mas yr$^{-1}$ .", "Our observations are on the verge of providing constraints on the orbit, and similarly precise observations beyond 2020 will constrain the orbit regardless of whether relative motion is seen.", "For now however, we cannot derive meaningful constraints because i) the object's motion is not sufficiently high to rule out low eccentricity or high inclination orbits , and ii) the lack of motion between epochs remains consistent with an unbound object; the 3$\\sigma $ upper limit on the sky-projected velocity is 1.6 au yr$^{-1}$ , only slightly less than the escape velocity at the projected separation of 23 au of 1.8 au yr$^{-1}$ (and the actual separation is likely larger, so the escape velocity lower).", "While a close encounter between unbound objects therefore remains a possibility, this is also unlikely because the interaction at the observed separation would last of order hundreds of years, which is a tiny fraction of the stellar age.", "Our H-band linear polarization measurements are similar, in both magnitude and angle, to those in the optical when the star is not significantly dimmed and after interstellar polarization has been subtracted, suggesting that most of the polarization in the near-IR comes from light scattered off the circumstellar disc .", "For most disk orientations, the unresolved linear polarisation angle is perpendicular to the disk position angle, though can become parallel in highly optically thick and near to edge-on cases , .", "Given that RZ Psc is not significantly reddened when undimmed, the latter seems unlikely.", "The angle of linear polarization of 53$^\\circ $ is therefore more similar to the companion's on-sky PA of 76$^\\circ $ than expected, and may mean that the disk plane and companion orbital plane are misaligned.", "The companion's orbit remains unconstrained however, so this discrepancy is largely suggestive, and would be aided by observations that establish the disk orientation for comparison with the angle of polarisation.", "What does this discovery mean for the interpretation of RZ Psc, and the nature of the UX Ori-like dimming events?", "Firstly, the companion is sufficiently faint (i.e.", "$\\approx $ 40$\\times $ fainter than RZ Psc A in H-band) that if it were the object being dimmed, the effect on the light curve at any wavelength would be negligible.", "Similarly, the companion's luminosity ($\\ll $ 2.5 % of RZ Psc A's luminosity) is less than the IR excess (7 %), so cannot solely heat the observed circumstellar dust.", "Therefore, the circumstellar dust orbits the primary.", "It seems inevitable that the occulting structures causing the dimming of RZ Psc A reside above a disc mid-plane that has significant radial optical depth.", "Our data strengthen this view, the reason being that the PA of linear polarization measured here is very similar to that measured by .", "If the dust were in a roughly spherical cloud and the linear polarization arises due to inhomogeneities in this structure, then the PA would not be expected to be similar for measurements made decades apart.", "Nevertheless, it is possible that the companion strongly influences the disc dynamics, for example via truncation at $\\sim $ 1/2 to 1/3 of the companion's semi-major axis (truncation seems more likely than gap formation, given the lack of far-IR excess).", "If the disc still retains a significant gas mass this influence may also manifest as a warp or spiral structures that pull material out of the disc plane and into our line of sight to the star.", "If the disc is gas-poor, the companion may have recently destabilised a newly formed planetary system or planetesimal belt, which is now colliding and producing a significant mass of dust, some of which passes between us and the star.", "In either case a companion could cause some portions of the disc to precess slowly, or otherwise cause the 0.5 mag variation seen with a 12 year period .", "The specifics of such interactions are unclear; while we find circumstantial evidence that the disc and companion have different orbital planes, this would be best quantified by constraining the companion's orbit (e.g.", "with VLT/SPHERE), and attempting to derive constraints on the dust geometry (e.g.", "with VLTI/MATISSE).", "ALMA observations may provide useful information on material beyond a few au, e.g.", "a continuum or CO detection could reveal the presence of an outer disc, or a non-detection suggest that the disc extent is largely limited to that already seen in the mid-IR.", "Such characterisation may also shed light on the long-term variation.", "What are the wider implications of this discovery, if any?", "One is that the inner companion theory for shadowed outer disc systems seems credible, if RZ Psc is indeed a more highly inclined version of such systems.", "Another is that while young systems with extreme warm IR excesses are commonly interpreted as showing evidence for terrestrial planet formation , , , the possible influence of more massive companions should be considered.", "While further work is clearly needed, RZ Psc provides evidence that non-planetary mass companions may play an important role in the evolution of young planetary systems." ], [ "Acknowledgements", "We thank the referee for a constructive report.", "GMK is supported by the Royal Society as a Royal Society University Research Fellow.", "CG and CR acknowledge funding from the Netherlands Organisation for Scientific Research (NWO) TOP-1 grant as part of the research programme “Herbig Ae/Be stars, Rosetta stones for understanding the formation of planetary systems”, project number 614.001.552.", "FM acknowledges funding from ANR of France under contract number ANR-16-CE31-0013." ] ]

[ [ "Dynamical instability of polytropic spheres in spacetimes with a\n cosmological constant" ], [ "Abstract The dynamical instability of relativistic polytropic spheres, embedded in a spacetime with a repulsive cosmological constant, is studied in the framework of general relativity.", "We apply the methods used in our preceding paper to study the trapping polytropic spheres with $\\Lambda = 0$, namely, the critical point method and the infinitesimal and adiabatic radial perturbations method developed by Chandrasekhar.", "We compute numerically the critical adiabatic index, as a function of the parameter $\\sigma = p_{\\mathrm{c}}/(\\rho_{\\mathrm{c}} c^2)$, for several values of the cosmological parameter $\\lambda$ giving the ratio of the vacuum energy density to the central energy density of the polytrope.", "We also determine the critical values for the parameter $\\sigma_{\\mathrm{cr}}$, for the onset of instability, by using both approaches.", "We found that for large values of the parameter $\\lambda$, the differences between the values of $\\sigma_{\\mathrm{cr}}$ calculated by the critical point method differ from those obtained via the radial perturbations method.", "Our results, given by both applied methods, indicate that large values of the cosmological parameter $\\lambda$ have relevant effects on the dynamical stability of the polytropic configurations." ], [ "Introduction", "There are indications that contrary to the inflationary era, in the recent era the dark energy could correspond to the vacuum energy related to a repulsive cosmological constant $\\Lambda > 0$ implying important consequences in astrophysical phenomena [1], [2].", "The estimate of the so-called relic cosmological constant, governing the acceleration of the recent stage of the Universe expansion, reads $\\Lambda \\sim 10^{-52}~\\mathrm {m}^{-2}$  [3].", "Although the relic cosmological constant is extremely small, its role in astrophysical phenomena could be quite significant, being limited by the so-called static radius introduced in [4] and discussed in [5], [6], [7], [8], [9].", "The static radius represents an upper limit on the existence of both Keplerian [10] and toroidal fluids [11], [12], [13], [14], accretion disks, the limit on gravitationally bound galaxy systems [15], [16], [17], and even the limit on gravitationally bound polytropic configurations that could represent a model of dark matter halos [18], [19], [20].", "Test fields around black holes in spacetimes with $\\Lambda > 0$ were treated in [21], [22], indicating possible instabilities.", "In spacetimes with a cosmological constant $\\Lambda $ , the interior Schwarzschild solution with uniform distribution of energy density was found in [23] for starlike configurations and extended to more general situations in [24], [25].", "The effect of $\\Lambda $ on gravitational instabilities for isothermal spheres in the Newtonian limit was considered in [26].", "The role of the cosmological constant on the so-called electrovacuum solutions was studied in [27].", "In spacetimes with a positive cosmological constant, the polytropic spheres represented by a polytropic index $n$ , a parameter $\\sigma $ , giving the ratio of pressure and energy density at the center, and vacuum constant index $\\lambda $ , giving the ratio of the vacuum energy to the central energy density, were exhaustively discussed in [18], [19].", "It has been shown that in some special cases of polytropes with sufficiently large values of the polytropic index $n$ , extremely extended configurations representing a dark matter halo could have gravitationally unstable central parts that could collapse leading to the formation of a supermassive black hole [20].", "The polytropic spheres are well-known models of compact objects, as they represent extremely dense nuclear matter inside neutron or quark stars.", "For example, they describe the fluid configurations constituted from relativistic ($n = 3$ ) and nonrelativistic ($n = 3/2$ ) Fermi gas [28], considered as basic approximations of neutron stars matter.", "Of course, in realistic models describing the neutron stars interior, the equations of state (EOSs) of nuclear matter are considered.", "However, in a recently developed approach, such realistic EOSs are represented by sequences of polytropes with appropriately tuned parameters [29].", "The stability of the polytropic spheres can be addressed from two different approaches.", "One of them is related to the energetic considerations, or critical point method [30], [28], developed by Tooper [31].", "A second approach deals with the dynamical theory of infinitesimal, and adiabatic, radial oscillations pioneered in a seminal paper by Chandrasekhar [32].", "Using his ‘pulsation equation', Chandrasekhar established the conditions of stability, against radial oscillations, for homogenous stars and polytropic spheres.", "The main conclusion of this study is that the critical adiabatic index $\\gamma _{\\mathrm {cr}}$ , for the onset of instability, increases due to relativistic effects from the Newtonian value $\\gamma = 4/3$ .", "The radial oscillations method has been widely used in different contexts [33], [34], [35].", "The role of the cosmological constant on the radial stability of the uniform energy density stars, using Chandrasekhar's method, was studied in [36], [37].", "These authors concluded that a large value of the vacuum constant index $\\lambda $ increases significantly the critical adiabatic index from its value with $\\lambda =0$ .", "The purpose of this paper is to study in detail the role of the cosmological constant in the stability of the polytropic fluid spheres against radial oscillations, that is expected to be relevant for extremely extended noncompact configurations modeling galactic dark matter halos [18].", "For that purpose, we are reconsidering the analysis carried out in [36] in two ways: first, we will apply the methods introduced in our preceding paper [38] to study the stability of polytropic spheres with $\\Lambda = 0$ , namely, the shooting method and trial functions to solve the Sturm–Liouville equation; and the critical point method based on the energetic approach.", "Second, we will extend the analysis to a larger family of spheres in the range of polytropic indexes $0.5 \\le \\, n\\, \\le 3$ , for several values of the vacuum constant index $\\lambda \\in [10^{-9}, 10^{-1}]$ .", "The paper is organized as follows.", "In Sec.", ", we review the equations of structure of the relativistic polytropes in the presence of a cosmological constant.", "In Sec.", ", we summarize the general properties and gravitational energy for polytropic spheres with $\\Lambda $ .", "In Sec.", ", Chandrasekhar's procedure and the associated Sturm–Liouville eigenvalue problem, including the cosmological term, are outlined.", "In Sec.", ", we present our methods and results.", "In Sect.", ", we discuss our conclusions." ], [ "Structure equations of relativistic polytropic spheres with a cosmological constant", "We will consider throughout the paper perturbations which preserve spherical symmetry.", "This condition guarantees that motions along the radial direction will ensue.", "Thus, our starting point is a spherically symmetric spacetime in the standard Schwarzschild coordinates $\\mathrm {d}s^2 = -e^{2\\Phi }(c\\,\\mathrm {d}t)^2 + e^{2\\Psi }\\mathrm {d}r^2+r^2(\\mathrm {d}\\theta ^2+\\sin ^2 \\theta \\, \\mathrm {d}\\phi ^2)\\, ,$ where $\\Phi (r,t)$ and $\\Psi (r,t)$ are functions of $t$ and the radial coordinate $r$ .", "The energy-momentum tensor for a spherically symmetric configuration takes the form $T_{\\mu }^{\\hphantom{\\nu }\\nu } = (\\epsilon + p) u_{\\mu }u^{\\nu } + p\\delta _{\\mu }^{\\hphantom{\\mu }\\nu }\\, ,$ where $\\epsilon =\\rho c^2$ is the energy density (written as the product of the mass density $\\rho $ times the speed of light squared), $p$ is the fluid pressure, and $u^{\\mu } = \\mathrm {d}x^{\\mu }/\\mathrm {d}\\tau $ is its four-velocity.", "Following our preceding paper [38], we consider the models of static polytropic fluid spheres proposed by Tooper [31], which are governed by the EOS $ p = K\\rho ^{1 + (1/n)}\\, ,$ where $n$ is the polytropic index and $K$ is a constant related to the characteristics of a specific configuration.", "It is conventional to introduce the parameter $\\sigma \\equiv \\frac{p_{\\mathrm {c}}}{\\rho _{\\mathrm {c}} c^2}\\, ,$ which denotes the ratio of pressure to energy density at the centre of the configuration.", "The radial profiles of the mass density and pressure of the polytropic spheres are given by the relations $\\rho = \\rho _{\\mathrm {c}}\\theta ^n\\, ,\\qquad p = p_{\\mathrm {c}} \\theta ^{n+1}\\, ,$ where $\\theta (x)$ is function of the dimensionless radius, $x \\equiv \\frac{r}{L}\\, ,\\qquad L \\equiv \\left[\\frac{\\sigma (n + 1)c^2}{4\\pi G\\rho _{\\mathrm {c}}}\\right]^{\\frac{1}{2}}\\, .$ Here $L$ is the characteristic length scale of the polytropic sphere and it is determined by the polytropic index $n$ , the parameter $\\sigma $ , and the central density $\\rho _\\mathrm {c}$ .", "From Eq.", "(REF ), we obtain immediately the boundary condition $\\theta (r = 0) = 1$ .", "We consider a static configuration in equilibrium, immersed in a cosmological background.", "The relevant components of Einstein's equations for this problem are $(t)(t)$ and $(r)(r)$ , which in the presence of a cosmological constant $\\Lambda $ are given by [39] $\\frac{\\mathrm {d}}{\\mathrm {d}r}(r e^{-2\\Psi }) = 1 - \\frac{8\\pi G}{c^4} T_{0}^{0}r^2 - \\Lambda r^2\\, ,$ $\\frac{2\\mathrm {d}\\Phi }{\\mathrm {d}r} = \\frac{e^{2\\Psi } - 1}{r} - \\frac{8\\pi G}{c^4} T_{1}^{1}r - \\Lambda r\\, .$ Equation (REF ) can be recast into the form $e^{2\\Psi } = \\left[1 - \\frac{2Gm(r)}{c^2r} - \\frac{1}{3}\\Lambda r^2\\right]^{-1}\\, ,$ where $m(r) = 4\\pi \\int _{0}^{r}\\rho (r)r^2\\, \\mathrm {d}r$ is the mass inside the radius $r$ .", "Using Eqs.", "(REF ) and (REF ), we transform Eq.", "(REF ) into $\\frac{\\mathrm {d}\\Phi }{\\mathrm {d}r} = \\frac{(G/c^2)m(r) - \\frac{\\Lambda }{3} r^2 + (4\\pi G/c^4)pr^3}{r^2\\left[1 - \\frac{2Gm(r)}{c^2r} - \\frac{\\Lambda }{3}r^2\\right]}\\, .$ Using the energy-momentum ‘conservation' $T^{\\mu \\nu }_{\\hphantom{\\mu }\\hphantom{\\mu };\\nu } = 0$ , we can write Eq.", "(REF ) as a relation between pressure and energy density in the form [37], [23] $\\frac{\\mathrm {d}p}{\\mathrm {d}r} = -(\\epsilon + p)\\frac{(G/c^2)m(r) + \\left[(4\\pi G/c^4)p - \\frac{\\Lambda }{3}\\right]r^3}{r^2\\left[1 - \\frac{2Gm(r)}{c^2r} - \\frac{\\Lambda }{3}r^2\\right]}\\, ,$ which is the Tolman-Oppenheimer-Volkoff (TOV) equation, including the effect of the cosmological constant.", "Once an EOS $p = p(\\rho )$ is given, Eqs.", "(REF ) and (REF ) can be integrated subject to the boundary conditions $m(0) = 0$ and $p(R) = 0$ , where $R$ is the radius of the star.", "In the exterior of the configuration, Eq.", "(REF ) gives the total mass $M = m(R)$ , and the spacetime is described by the Kottler metric [40].", "The mass relation Eq.", "(REF ) can be written in terms of the parameter $\\sigma $ as follows: $\\sigma (n + 1)\\,\\mathrm {d}\\theta = -(\\sigma \\theta + 1)\\,\\mathrm {d}\\Phi \\, .$ Solving Eq.", "(REF ) with the condition that the interior and exterior metrics are smoothly matched at the surface $r = R$ , we have $e^{2\\Phi } = (1 + \\sigma \\theta )^{-2(n + 1)}\\left(1 - \\frac{2GM}{c^2R} - \\frac{\\Lambda }{3}R^2\\right)\\, ,$ which is a function of $\\theta $ and $\\sigma $ .", "In order to find a relation for the function $\\theta $ , we rewrite Eq.", "(REF ) in the form $\\frac{\\mathrm {d}\\Phi }{\\mathrm {d}r} = -\\frac{\\sigma (n + 1)}{1 + \\sigma \\theta }\\frac{\\mathrm {d}\\theta }{\\mathrm {d}r}\\, .$ Substituting Eq.", "(REF ) into Eq.", "(REF ) we have $\\frac{\\sigma (n + 1)r}{1 + \\sigma \\theta }\\left[1 - \\frac{2Gm(r)}{c^2r} - \\frac{\\Lambda }{3}r^2\\right]\\left(\\frac{\\mathrm {d}\\theta }{\\mathrm {d}r}\\right)\\\\+ \\frac{Gm(r)}{c^2r} - \\frac{\\Lambda }{3}r^2 = -\\frac{G\\sigma \\theta }{c^2}\\left(\\frac{\\mathrm {d}m}{\\mathrm {d}r}\\right)\\, .$ Similarly, the mass relation Eq.", "(REF ) can be written in terms of $\\theta $ as follows: $\\frac{\\mathrm {d}m}{\\mathrm {d}r} = 4\\pi r^2\\rho _\\mathrm {c}\\,\\theta ^{n}\\, .$ To facilitate the numerical computations, it is convenient to write Eq.", "(REF ) in a dimensionless form by using Eq.", "(REF ) and the quantities $v(x) \\equiv \\frac{m(r)}{4\\pi L^3 \\rho _\\mathrm {c}} = \\frac{m(r)}{\\mathcal {M}}\\, ,\\\\\\lambda \\equiv \\frac{\\rho _{\\mathrm {vac}}}{\\rho _\\mathrm {c}}\\, ,$ where $\\mathcal {M}$ is a characteristic mass scale of the polytrope $\\mathcal {M} = 4\\pi L^3 \\rho _\\mathrm {c} = \\frac{c^2}{G}\\sigma L(n + 1)\\, ,$ and $\\lambda $ indicates the vacuum constant index giving the ratio between the vacuum energy density and the central energy density of the polytropic sphere.", "The cosmological constant $\\Lambda $ is related to the energy density of the vacuum by $\\Lambda = \\frac{8\\pi G}{c^2}\\rho _{\\mathrm {vac}}\\, .$ Thus, the index $\\lambda $ in Eq.", "() is connected with $\\Lambda $ through the relation $\\lambda = \\frac{\\Lambda c^4}{8\\pi G\\rho _\\mathrm {c}}\\, .$ In terms of Eqs.", "(REF ), (REF ), and (REF ), Eq.", "(REF ) takes the final form $\\frac{\\mathrm {d}\\theta }{\\mathrm {d}x} = \\left[\\left(\\frac{2\\lambda }{3} - \\sigma \\theta ^{n + 1}\\right)x - \\frac{v}{x^2}\\right]\\left(1 + \\sigma \\theta \\right)g_{rr}\\, ,$ $\\frac{\\mathrm {d}v}{\\mathrm {d}x} = x^2 \\theta ^n\\, ,$ where $g_{rr}\\equiv \\left[1 - 2\\sigma (n + 1)\\left(\\frac{v}{x} + \\frac{\\lambda }{3}x^2\\right)\\right]^{-1}\\, .$ These equations, subject to the boundary conditions $\\theta (0) = 1\\, ,\\quad v(0)=0\\,$ can be solved numerically to give the radius $x=x_1$ of the configuration as the first solution $\\theta (x) = 0$ ." ], [ "Structural parameters", "Except for the case $n = 0$ , corresponding to a constant density configuration [23], [37], the structure equations (REF ) and (REF ) do not admit analytic solutions in a closed form for $\\sigma \\ne 0$ .", "Thus, one must turn to numerical integration.", "Considering that a configuration in equilibrium has positive density and monotonically decreasing pressure (see however [35]), we will concentrate in the range of values of $x$ such that $\\theta >0$ .", "Assuming that $\\lambda $ , $n$ , $\\sigma $ , and $\\rho _\\mathrm {c}$ are given, we start the numerical integration at the center of the sphere $x = 0$ , where $\\theta (0) = 1$ and $v(0) = 1$ , and advance by small steps until the first zero $\\theta (x_1) = 0$ is found at $x_1$ , if it exists.", "Using this value in Eq.", "(REF ), we can determine the radius of the polytrope as $R = L x_1\\, .$ The mass of the configuration is determined by the solution of $v(x)$ at the surface $M = 4\\pi L^3 \\rho _{\\mathrm {c}} v(x_1) = \\frac{c^2 }{G}\\sigma L(n + 1)v(x_1)\\, .$ From Eqs.", "(REF ) and (REF ), we can obtain the mass-radius relation $\\mathcal {C} \\equiv \\frac{GM}{c^2R} = \\frac{\\sigma (n+1)v(x_1)}{x_1}\\, ,$ which gives the ratio between the gravitational radius $r_{g} \\equiv 2GM/c^2$ and the coordinate radius $R$ , once $\\sigma $ has been specified.", "The $g_{tt}$ and $g_{rr}$ metric components can be written in terms of the function $\\theta $ and the parameter $\\sigma $ as $e^{2\\Phi } = \\frac{1 - 2\\sigma (n + 1)\\left[\\frac{v(x_1)}{x_1}+\\frac{\\lambda }{3}x_1^2\\right]}{(1 + \\sigma \\theta )^{2(n+1)}}\\, ,$ $e^{-2\\Psi } = 1 - 2\\sigma (n + 1)\\left[\\frac{v(x)}{x} + \\frac{\\lambda }{3}x^2\\right]\\, .$ The exterior region is described by the Kottler metric, or Schwarzschild–de Sitter spacetime, which represents a Schwarzschild mass embedded into an asymptotically de Sitter spacetime.", "In the Schwarzschild coordinates, the exterior metric takes the form [39] $e^{2\\Phi } = e^{-2\\Psi } = 1 - \\frac{2GM}{c^2 r} - \\frac{\\Lambda }{3}r^2\\, .$" ], [ "Gravitational energy ", "In the relativistic theory, the total energy $E$ of certain fluid sphere, which includes the internal energy and the gravitational potential energy, is $Mc^2$ where $M$ corresponds to the mass producing the gravitational field $E = Mc^2 = 4\\pi \\int _{0}^{R}\\epsilon r^2\\, \\mathrm {d}r\\, .$ The proper energy and proper mass of a spherical gas is defined by $E_{0\\mathrm {g}} = M_{0g}c^2 = 4\\pi \\int _{0}^{R}(\\rho _{g}c^2) e^{\\Psi }\\, r^2\\,\\mathrm {d}r\\, ,$ where $M_{0\\mathrm {g}}$ equals (approximately) the rest mass density of baryons in the configuration and $\\rho _\\mathrm {g}c^2$ is the rest energy density of the gas particles.", "For polytropic spheres, the gas density can be written in terms of the total mass density as [31] $\\rho _{g}= \\frac{\\rho _\\mathrm {c}\\theta ^n}{(1+\\sigma \\theta )^n}.$ In our analysis of stability using the energy considerations, an important quantity is the ratio $\\frac{E_{0\\mathrm {g}}}{E} = \\frac{1}{v(x_{1})}\\\\\\times \\int \\limits _{0}^{x_{1}}\\frac{\\theta ^{n}x^2}{\\left(1 + \\sigma \\theta \\right)^{n}\\left[1 - 2\\sigma (n + 1)\\left(\\frac{v}{x} + \\frac{\\lambda }{3}x^2\\right)\\right]^{1/2}}\\,\\mathrm {d}x\\,,$ which gives the proper energy of the gas in units of the total energy $E=Mc^2$ .", "Note that Eq.", "(REF ) generalizes the expression given in [31] to the case of a nonvanishing cosmological constant.", "These two quantities define the binding energy of the system, namely, $E_{\\mathrm {b}} = E_{0\\mathrm {g}} - E$ , which corresponds to the difference in energy between an initial state with zero internal energy where the particles that compose the system are dispersed, and a final state where the particles are bounded by gravitational interaction." ], [ "Radial oscillations of relativistic spheres in spacetimes with a cosmological constant", "In this section we discuss the theory of infinitesimal, and adiabatic, radial oscillations of relativistic spheres developed by Chandrasekhar [32], and its extension to the case of a nonvanishing cosmological constant [36], [37].", "We consider ‘pulsations' which preserve spherical symmetry; therefore, these do not affect the exterior gravitational field.", "In other words, there is no gravitational monopole radiation.", "Thus, we are considering a situation with the line element, Eq.", "(REF ), and a mass distribution described by the energy-momentum tensor, Eq.", "(REF ) The pulsation dynamics will be determined by the Einstein equations, including the cosmological constant, together with the energy-momentum conservation, baryon number conservation, and the laws of thermodynamics.", "The relevant components of the Einstein equations with $\\Lambda $ are given in [39], [37].", "To obtain the radial pulsation equation for spherical fluids immersed into a cosmological background, the metric coefficients $\\Psi (r,t)$ and $\\Phi (r,t)$ together with the fluid variables $\\rho (r,t)$ , $p(r,t)$ and the number density of baryons $n(r,t)$ , as measured in fluid's rest frame, are perturbed generally in the form $q(r,t) = q_0(r) + \\delta q(r,t)\\, ,$ where the canonical variable $q \\equiv (\\Phi ,\\Psi ,\\epsilon ,p,n)$ indicates the metric and physical quantities, and the subscript 0 refers to the variables at equilibrium.", "At first order in the perturbations, the components of the energy-momentum tensor Eq.", "(REF ) are given by T00 = -0  , Tii = p,      i = 1,  2,  3    (no summation)  , T01 = -(0 + p0)v  , T10 = (0 + p0)v e2(0-0)  , where $v = dr/dx^{0}$ .", "The pulsation is represented by the radial, or ‘Lagrangian', displacement $\\xi $ of the fluid from the equilibrium position $\\xi = \\xi (r,t)$ , defined as $\\frac{u^r}{u^t} = \\frac{\\partial \\xi }{\\partial t} \\equiv \\dot{\\xi }\\, .$ The derivation of the expressions for the linear perturbations in the quantities $q(r,t)$ follows the same lines as for the case $\\Lambda = 0$ discussed in [32], [41]; therefore, we just summarize the main results here.", "All the relevant equations must be linearized relative to the displacement from the static equilibrium configuration.", "We have to obtain the dynamic equation for evolution of the fluid displacement $\\xi (t,r)$ , and a set of initial-value equations, expressing the perturbation functions $\\delta \\Phi $ , $\\delta \\Psi $ , $\\delta \\epsilon $ , $\\delta p$ , $\\delta n$ in terms of the displacement function $\\xi (t,r)$ .", "No nuclear reactions are assumed during small radial perturbations; therefore, the dynamics of the energy density and the pressure perturbations is governed by the baryon conservation law $\\left(nu^{\\mu }\\right)_{;\\mu } = 0\\, .$ Using Eq.", "(REF ), we obtain the initial value equation for the pressure perturbation $\\delta p = - \\gamma p_0 \\frac{e^{\\Phi _0}}{r^2}\\left(r^2 e^{-\\Phi _0}\\xi \\right)^{\\prime } - \\xi (p_0)^{\\prime }\\, ,$ where $^{\\prime }=\\partial /\\partial r$ , and we introduce $\\gamma \\equiv \\left(p\\,\\frac{\\partial n}{\\partial p}\\right)^{-1} \\left[n-(\\epsilon +p)\\frac{\\partial n}{\\partial \\epsilon }\\right]\\, ,$ as the adiabatic index that governs the linear perturbations of pressure inside the star [32], [28].", "In general, this $\\gamma $ is not necessarily the same as the adiabatic index associated to the EOS (see discussion in [38]).", "The initial-value equation for the Lagrangian perturbation of the energy density $\\delta \\rho $ , and the metric functions $\\delta \\Phi $ and $\\delta \\Psi $ , takes a similar form as for the case $\\Lambda = 0$ , $\\delta \\epsilon = -\\frac{e^{\\Phi _0}}{r^2}(\\epsilon _0 + p_0)\\left(r^2e^{-\\Phi _0}\\xi \\right)^{\\prime } - \\xi (\\epsilon _0)^{\\prime }\\, ,$ $\\delta \\Psi = - \\xi \\left(\\Psi _0 + \\Phi _0 \\right)^{\\prime }\\, ,$ $(\\delta \\Phi )^{\\prime } = \\left[\\frac{\\delta p}{(\\epsilon _0+p_0)} - \\left(\\Phi _0^{\\prime } + \\frac{1}{r}\\right)\\xi \\right]\\left(\\Psi _0 + \\Phi _0 \\right)^{\\prime }\\, .$ It is conventional to assume that all the perturbations have a time dependence of the form $e^{i\\omega t}$ , where $\\omega $ is a characteristic frequency to be determined.", "Thus, using the previous initial-value equations for the perturbations, and introducing the ‘renormalized displacement function' $\\zeta $  [41], $\\zeta \\equiv r^2 e^{-\\Phi _0}\\xi \\, ,$ we obtain the Sturm-Liouville dynamic pulsation equation with a cosmological constant $\\frac{\\mathrm {d}}{\\mathrm {d}r}\\left(P\\frac{\\mathrm {d}\\zeta }{\\mathrm {d}r}\\right) + \\left(Q+\\omega ^2 W\\right)\\zeta =0\\, ,$ where the functions $P(r)$ , $Q(r)$ , and $W(r)$ are defined as $P(r)\\equiv \\frac{\\gamma p_0}{r^2}\\,e^{3\\Phi _0+\\Psi _0}\\, ,$ $Q(r)\\equiv \\frac{e^{3\\Phi _0+\\Psi _0}}{r^2}\\Bigg [\\frac{(p_0^{\\prime })^2}{\\epsilon _0 + p_0} - \\frac{4p^{\\prime }_0}{r}\\\\-\\left(\\frac{8\\pi G}{c^4}p_0 - \\Lambda \\right)(\\epsilon _0 + p_0) e^{2\\Psi _0}\\Bigg ]\\, ,$ $W(r)\\equiv \\frac{\\epsilon _0 + p_0}{r^2}\\,e^{\\Phi _0+3\\Psi _0}\\, .$ The boundary conditions must guarantee that the displacement function is not resulting in a divergent behavior of the energy density and pressure perturbations at the center of the sphere.", "On the other hand, the variations of the pressure must satisfy the condition $p(R) = 0$ at the surface of the configuration.", "Therefore, we have r3is finite, or zero, as    r 0  , (p0er2)' 0 as    rR  .", "The Sturm-Liouville equation (REF ), together with the boundary conditions Eqs.", "() and (), determine the eigenvalues $\\omega _i$ (frequencies) and the pulsation eigenfunctions $\\zeta _i(r)$ which satisfy $\\int _{0}^{R}e^{\\Phi _0+3\\Psi _0}(\\epsilon _0 + p_0)\\zeta _{i}\\zeta _{j}r^2\\,\\mathrm {d}r = 0, \\quad i\\ne j\\, .$ The Sturm-Liouville eigenvalue problem can be written in the variational form, as described in [41], because the extremal values of $\\omega ^2 = \\frac{\\displaystyle \\int _0^R\\left(P\\zeta ^{\\prime 2} - Q\\zeta ^2\\right)\\,\\mathrm {d}r}{\\displaystyle \\int _0^R W\\zeta ^2\\,\\mathrm {d}r}$ determine the eigenfrequencies $\\omega _i$ .", "The absolute minimum value of Eq.", "(REF ) corresponds to the squared frequency of the fundamental mode of the radial pulsations.", "If $\\omega ^2$ is positive (negative), the configuration is stable (unstable) against radial oscillations.", "Moreover, if the fundamental mode is stable ($\\omega _0^2 > 0$ ), all higher radial modes will also be stable.", "For this reason, a sufficient condition for the dynamical instability is the vanishing of the right-hand side of Eq.", "(REF ) for certain trial function satisfying the boundary conditions.", "The Sturm-Liouville pulsation equation can be used to determine the dynamical stability of spherical configurations of perfect fluid.", "Given certain EoS, the critical adiabatic index $\\gamma _{\\mathrm {cr}}$ , given by the marginally stable condition $\\omega ^2 = 0$ , can be determined by integration of the Sturm-Liouville equation.", "Using Eq.", "(REF ), we can deduce a general formula to find the critical adiabatic index, which reads $\\gamma _{\\mathrm {cr}} =\\frac{\\displaystyle \\int _{0}^{R}Q(r)\\zeta ^2\\,\\mathrm {d}r}{\\displaystyle \\int _{0}^{R}\\frac{p_0}{r^2}e^{3\\Phi _0+\\Psi _0}(\\zeta ^{\\prime })^2\\,\\mathrm {d}r}\\, .$ Thus, for $\\gamma < \\gamma _{\\mathrm {cr}}$ dynamical instability will ensue and the configuration will collapse.", "For the case of a homogeneous star in the presence of a cosmological constant, Böhmer and Harko [37] showed that the condition for radial stability reads $\\gamma > \\gamma _{\\mathrm {cr}} =\\frac{\\frac{4}{3}-l}{1-3l} + \\frac{19}{42}\\left(1-\\frac{21}{19}l\\right)\\left(\\frac{r_g}{R}\\right)+\\mathcal {O}\\left(\\frac{r_g}{R}\\right)^2\\, ,$ where $l = \\Lambda /(12\\pi G\\rho _\\mathrm {c})$ , $r_{g}$ is the gravitational radius, and $R$ indicates the radius of the star.", "When $\\Lambda = 0$ , Eq.", "(REF ) reduces to the value found by Chandrasekhar [32]." ], [ "Sturm-Liouville equation for polytropic spheres with a cosmological constant", "Using the relevant expressions discussed in Sec.", ", together with the variational form Eq.", "(REF ), we arrive to the Sturm-Liouville eigenvalue equation for the dynamical stability of relativistic polytropic spheres in the presence of a cosmological constant, $\\omega ^2L^2\\int _0^{x_1}\\theta ^n(1 + \\sigma \\theta )\\left(\\frac{\\zeta }{x}\\right)^2e^{\\Phi +3\\Psi }\\,\\mathrm {d}x = \\\\\\quad \\sigma \\int _0^{x_1}\\frac{\\gamma \\,\\theta ^{n + 1}}{x^2}\\left(\\frac{\\partial \\zeta }{\\partial x}\\right)^2e^{3\\Phi +\\Psi }\\,\\mathrm {d}x -\\\\(n + 1)\\int _0^{x_1}\\frac{\\theta ^n e^{3\\Phi +\\Psi }}{x^2}\\biggl \\lbrace \\left(\\frac{\\partial \\theta }{\\partial x}\\right)\\frac{4}{x}\\left[\\frac{\\sigma (n + 1)x}{4(1 + \\sigma \\theta )}\\left(\\frac{\\partial \\theta }{\\partial x}\\right) - 1\\right] \\\\-2(1 + \\sigma \\theta )\\left(\\sigma \\theta ^{n + 1} - \\lambda \\right)e^{2\\Psi }\\biggr \\rbrace \\zeta ^2\\,\\mathrm {d}x\\, ,$ which constitutes a characteristic eigenvalue problem for the frequency $\\omega ^2$ and the amplitude $\\zeta (x)$ (we have suppressed the subscript zero as no longer needed).", "For the polytropic spheres considered in Sec.", ", the adiabatic index $\\gamma $ is given by $\\gamma = \\left(1 + \\frac{1}{n}\\right)(1 + \\sigma \\theta )\\, ,$ which, in general, is a function of the radial coordinate.", "In his study of the dynamical stability of relativistic polytropes, Chandrasekhar [32] assumed $\\gamma $ to be a constant.", "Thus, under this assumption, $\\gamma $ can be taken out of the integral in Eq.", "(REF ) and one can integrate given certain trial function.", "In a more general approach, for any equilibrium configuration, one can consider $\\gamma $ in Eq.", "(REF ) as an effective adiabatic index [42], $\\langle \\gamma \\rangle = \\frac{\\displaystyle \\int \\limits _0^{x_1}\\,\\frac{\\gamma \\,\\theta ^{n + 1}}{x^2}\\left(\\frac{\\partial \\zeta }{\\partial x}\\right)^2 e^{3\\Phi +\\Psi }\\,\\mathrm {d}x}{\\displaystyle \\int \\limits _0^{x_1}\\frac{\\theta ^{n+1}}{x^2}\\left(\\frac{\\partial \\zeta }{\\partial x}\\right)^2 e^{3\\Phi +\\Psi }\\,\\mathrm {d}x}\\, .$ Thus, the condition for stability can be established as $\\langle \\gamma \\rangle > \\gamma _\\mathrm {cr}\\, .$ The mass relation Eq.", "(REF ) for the gradients of $p$ and $\\Phi $ is transferred into the form $\\frac{\\partial \\Phi }{\\partial x} = -\\frac{\\sigma (n + 1)}{1 + \\sigma \\theta }\\frac{\\partial \\theta }{\\partial x}\\, .$ In terms of the variables introduced in Eqs.", "(REF )–(REF ), the Sturm-Liouville equation (REF ) takes the form $\\frac{\\mathrm {d}}{\\mathrm {d}x}\\left[P(x)\\frac{\\mathrm {d}\\zeta }{\\mathrm {d}x}\\right] + L^2\\left[Q(x) + \\omega ^2\\,W(x)\\right]\\zeta (x)=0\\, ,$ where the functions $P$ , $Q$ , and $W$ given by Eqs.", "(REF ), (REF ), and (REF ) are now $P(x) = \\frac{\\langle \\gamma \\rangle \\sigma \\rho _\\mathrm {c}\\theta ^{n + 1}}{L^2 x^2}e^{3\\Phi +\\Psi }\\, ,$ $Q(x) = \\frac{\\sigma \\rho _\\mathrm {c}(n + 1)\\theta ^{n}e^{3\\Phi +\\Psi }}{L^4 x^2}\\left[\\frac{\\sigma (n + 1)}{(1 + \\sigma \\theta )}\\left(\\frac{\\mathrm {d}\\theta }{\\mathrm {d}x}\\right)^2 - \\right.\\\\\\left.", "\\frac{4}{x}\\left(\\frac{\\mathrm {d}\\theta }{\\mathrm {d}x}\\right) - 2(1 + \\sigma \\theta )\\left(\\sigma \\theta ^{n + 1} - \\lambda \\right)e^{2\\Psi }\\right]\\, ,$ $W(x) = \\frac{\\rho _\\mathrm {c}\\,\\theta ^{n}(1+\\sigma \\theta )}{L^2 x^2}e^{\\Phi +3\\Psi }\\, .$ In the next section, we will discuss the methods we used to solve the eigenvalue problem Eq.", "(REF ), subject to the boundary conditions Eqs.", "() and (), in order to study the radial stability of polytropic spheres in the presence of a cosmological constant.", "Clearly, for general polytropes, the critical value of the adiabatic index related to the dynamical stability can be determined by numerical integration only.", "Several methods to solve the eigenvalue problem Eq.", "(REF ) have been described in the literature (see, e.g., [43] and references therein).", "Following [32], we computed the critical values of the adiabatic index $\\gamma _\\mathrm {cr}$ , for the onset of instability, by integrating numerically Eq.", "(REF ) in the case $\\omega ^2 = 0$ .", "For that purpose we followed two different methods: the shooting method and trial functions.", "In the shooting method [44], one integrates Eq.", "(REF ) from the center up to the surface of the configuration with some trial value of $\\gamma $ .", "The value for which the solution satisfies (within a prescribed error) the boundary conditions, Eqs.", "() and (), correspond to the critical adiabatic index $\\gamma _\\mathrm {cr}$ .", "In order to apply the shooting method to Eq.", "(REF ), it is convenient to transform it to a set of two ordinary differential equations.", "We follow the convention used in [45] where Eq.", "(REF ) can be split in the following form: $\\frac{\\mathrm {d}\\zeta }{\\mathrm {d}x} = \\frac{\\eta }{P(x)}\\, , \\\\\\frac{\\mathrm {d}\\eta }{\\mathrm {d}x} = -L^2\\,\\left[\\omega ^2\\,W(x) + Q(x)\\right]\\zeta \\, , $ which satisfy the following behaviour near the origin: $\\zeta (x) = \\frac{\\eta _{0}}{3P(0)}x^3 + \\mathcal {O}(x^5)\\, , \\\\\\eta (r) = \\eta _{0}\\, , $ where $\\eta _{0}$ is an arbitrary constant, which we choose to be the unity.", "The second method is based on using trial functions to integrate Eq.", "(REF ).", "Following [32], [37], we chose the following functions $\\xi _1 = x e^{\\Phi /2}\\, ,\\qquad \\xi _2 = x\\, ,$ yielding $\\zeta _1 = x^3e^{-\\Phi /2}\\, ,\\qquad \\zeta _2 = x^3 e^{-\\Phi }\\, .$ We perform a detailed study of the stability for the whole range of the polytropes subject to the condition of causality due to the restriction on the parameter $\\sigma $  [31], $\\sigma < \\sigma _\\mathrm {causal} \\equiv \\frac{n}{n + 1}\\, .$ However, we will see that for certain combinations of the parameters $(n,\\lambda )$ the range of $\\sigma $ is also limited by the condition of having finite size configurations.", "The limits on the existence of relativistic polytropic spheres were discussed in [18], in dependence of the polytropic index $n$ and the parameter $\\sigma $ .", "Condition (REF ) is obtained from the relation $v_\\mathrm {sc} = c \\left[\\frac{\\sigma (n + 1)}{n}\\right]^{1/2}\\, ,$ which corresponds to the speed of sound at the center of the sphere.", "Thus, it might seem that Eq.", "(REF ) implies the restriction given by Eq.", "(REF ).", "However, note that Eq.", "(REF ) gives the phase velocity which is not the same as the group velocity; therefore, the condition (REF ) might not be definitive.", "Applying the methods described above, we have computed critical values of the adiabatic index $\\gamma _{\\mathrm {cr}}$ for polytropes with characteristic values of the index $n$ , for several values of the cosmological parameter $\\lambda $ .", "Using these results for $\\gamma _\\mathrm {cr}$ , we also determined constraints on the parameter $\\sigma $ in order to construct stable configurations.", "We present our results in the next section." ], [ "Results obtained via Chandrasekhar's pulsation equation", "As a first step in our analysis, we solved numerically the equations of structure Eqs.", "(REF ) and (REF ) for relativistic polytropes in the presence of a cosmological constant.", "The integrations were carried out using the adaptive Runge-Kutta-Fehlberg method [44].", "We provide some profiles of the dimensionless radius $x_{1}$ , as a function of $\\sigma $ , for several values of the vacuum constant index $\\lambda $ .", "We studied a whole family of polytropic spheres with index $n \\le \\, 3$ , and we restrict the values of $\\sigma $ by the causality limit Eq.", "(REF ).", "For comparison, we have included in the same plot (dashed lines) the profiles with $\\lambda = 0$ .", "In Fig.", "REF , we present our results which are in very good agreement with those reported in [18].", "Note that for certain combinations of the parameters $(n,\\sigma )$ , the extension of the configuration increases as compared to its corresponding value in the case $\\lambda = 0$ .", "This is expected as a consequence of the repulsive effect of a positive cosmological constant.", "Moreover, the presence of $\\lambda $ sets strong constraints on the existence of polytropic configurations.", "For instance, for the case $\\lambda = 10^{-2}$ , polytropic configurations with $n > 2.4$ do not exist.", "For the case $\\lambda = 10^{-1}$ , existing polytropes are restricted to $n < 1$ .", "We also found that the vacuum constant index $\\lambda $ constraints the allowed values of the parameter $\\sigma $ for certain configurations.", "For instance, for the combination $(n = 2,\\,\\lambda = 10^{-2})$ , the maximum allowed value of the parameter $\\sigma $ we found was $\\sigma _{\\mathrm {max}} \\simeq 0.3563$ .", "Note that this value is lower than the value restricted by causality given by Eq.", "(REF ).", "In Fig.", "REF , we show the results for various combinations of the parameters $(n,\\sigma )$ .", "For the case of $\\lambda = 10^{-9}$ , deviations from the $\\lambda = 0$ case are negligible.", "Figure: Profiles of the dimensionless radius x 1 x_{1} for relativistic polytropic spheres in dependence of the parameters: n∈[0.5,3]n\\in [0.5, 3], σ∈[0,n/(n+1)]\\sigma \\in [0, n/(n + 1)], and λ∈[10 -9 \\lambda \\in [10^{-9}, 10 -1 ]10^{-1}].", "The dashed lines indicate the corresponding polytropic configuration (in same color) for λ=0\\lambda = 0.", "Note that for certain combinations of the indexes (n,λ≠0)(n,\\lambda \\ne 0), the structure equations do not yield finite configurations for σ∈[0,n/(n+1)]\\sigma \\in [0,n/(n + 1)].As a second step in our analysis, we determined the critical adiabatic index $\\gamma _{\\mathrm {cr}}$ for the onset of instability, as a function of $\\sigma $ , for several values of the indexes $(n, \\lambda )$ .", "We show our results in Fig.", "REF , which were obtained via the shooting method (see Sec.", "REF ).", "For comparison we also plotted the values of $\\gamma _{\\mathrm {cr}}$ for each corresponding polytropic configuration with $\\lambda = 0$ .", "For large values of $\\lambda $ , for instance, $\\lambda = 10^{-2}$ and $\\lambda = 10^{-1}$ we found that the values of $\\gamma _{\\mathrm {cr}}$ increase with respect to their values with $\\lambda = 0$ .", "These results indicate that large values of $\\lambda $ tend to destabilize the polytropic spheres.", "Of particular interest is the case $\\lambda = 10^{-1}$ which shows that in the Newtonian limit, when $\\sigma \\rightarrow \\,0$ , the value of $\\gamma _{\\mathrm {cr}}$ deviates from the expected value $\\gamma _\\mathrm {N} = 4/3$ .", "Figure: Critical adiabatic index γ cr \\gamma _{\\mathrm {cr}}, for the onset of instability, for polytropic spheres in the range 0.5≤n≤30.5\\le \\, n \\le \\,3, for the index λ∈[10 -4 ,10 -1 ]\\lambda \\in [10^{-4}, 10^{-1}].", "The dashed lines (in same color) indicate the corresponding values of γ cr \\gamma _{\\mathrm {cr}} with λ=0\\lambda = 0.", "Note that the critical adiabatic index increases from its corresponding value with λ=0\\lambda = 0.Figure: Differences of the critical adiabatic index γ cr \\gamma _{\\mathrm {cr}}, for the polytropes n={1.0,1.5,2.0,2.5}n = \\lbrace 1.0, 1.5, 2.0, 2.5\\rbrace , of the values for λ∈[10 -6 ,10 -3 ]\\lambda \\in [10^{-6},10^{-3}] from their corresponding values with λ=0\\lambda = 0.", "We have ‘normalized' the differences by dividing by the corresponding value of λ\\lambda .We will be interested in the differences of certain general quantities $q$ with $\\lambda _{i} \\ne 0$ , from its corresponding value with $\\lambda = 0$ $\\Delta q \\equiv q^{(\\lambda _i)} - q^{(\\lambda _0)},$ where $\\lambda _i$ denotes $\\lambda \\in \\lbrace 10^{-6}, 10^{-4}, 10^{-3}, 10^{-2}\\rbrace $ , and $\\lambda _0$ indicates $\\lambda = 0$ .", "In Fig.", "REF , we show our results for the differences of $\\gamma _{\\mathrm {cr}}$ for $\\lambda \\in [10^{-6}, 10^{-2}]$ , for the polytropes $n \\in \\lbrace 1.0, 1.5, 2.0, 2.5\\rbrace $ .", "We have normalized the differences by dividing by the corresponding value of $\\lambda $ .", "Note that the curves of the differences show the same qualitative behavior.", "The stability domain, given by the condition $\\langle \\gamma \\rangle > \\gamma _{\\mathrm {cr}}$ , in dependence on the parameter $\\lambda $ is shown in Fig.", "REF for several representatives values of $n$ .", "The effective $\\langle \\gamma \\rangle $ was computed from Eq.", "(REF ) by using the results obtained from the shooting method.", "We also determined the critical values of $\\sigma $ by using the trial functions given in Eq.", "(REF ).", "In this case, we computed $\\gamma _{\\mathrm {cr}}$ from Eq.", "(REF ) and then we determined the effective adiabatic index from Eq.", "(REF ).", "We summarize all of our results in Fig.", "REF .", "Figure: The stability domain as determined by comparison of the effective adiabatic index (green line) with the critical adiabatic index (black line) for polytropic spheres.", "The values of γ cr \\gamma _{\\mathrm {cr}} were computed via the shooting method.", "The red line separates the stable from the unstable region given by the condition 〈γ〉>γ cr \\langle \\gamma \\rangle > \\gamma _{\\mathrm {cr}}.", "The intersection point indicates the maximum permitted value σ cr \\sigma _{\\mathrm {cr}} for stability.", "Note the role of the parameter λ\\lambda on σ cr \\sigma _\\mathrm {cr}.Figure: Profiles of the total mass (black line) and rest mass (green line), as a function of σ\\sigma , for some polytropic spheres for different values of the vacuum constant index λ\\lambda .", "The maximum of the curve for the total mass determines the critical value of σ\\sigma for stability; thus, it separates the stable and unstable regions." ], [ "Dynamical instability determined via the critical point method", "We follow the standard approach to examine the stability of the polytropic spheres using the energy considerations, or critical point method [31].", "This analysis relies on the properties of static solutions to Einstein's equations.", "It is worthwhile to mention that static methods to study the stability of configurations might not be conclusive.", "Instabilities arising due to thermal effects may not be predicted by these methods, so one must turn to the full dynamical approach studied in the last section.", "Substituting Eqs.", "(REF ) and (REF ) in Eq.", "(REF ) for the total mass $M$ , we obtain $M = \\frac{1}{\\sqrt{4\\pi }}(n + 1)^{3/2}K^{n/2}G^{-3/2}(\\sigma \\, c^2)^{(3 - n)/2} v(x_{1})\\, ,$ where $K$ and $n$ are the parameters characterizing the polytrope (see Sect. ).", "In our analysis we are considering configurations with $K$ and $n$ constants, therefore the total mass $M$ is proportional to $\\sigma ^{(3 - n)/2} v(x_{1})$ and the rest mass is proportional to $\\sigma ^{(3 - n)/2} v(x_{1})(E_{0\\mathrm {g}}/E)$ .", "To study the stability, in Fig.", "REF , we plot the total gravitational mass $M$ and the rest mass of baryons $M_{0\\mathrm {g}}$ (‘preassembly mass'), given by Eq.", "(REF ), against the parameter $\\sigma $ .", "A necessary, but not sufficient, condition for stability is $\\frac{\\mathrm {d}M_{\\mathrm {eq}}}{\\mathrm {d}\\rho _\\mathrm {c}} > 0\\, ,$ where $M_{\\mathrm {eq}}$ indicates the total mass at equilibrium.", "At the critical point, where $\\frac{\\mathrm {d}M_{\\mathrm {eq}}}{\\mathrm {d}\\rho _\\mathrm {c}} = 0\\, ,$ there is a change in stability due to the change in the sign of $\\omega ^2$ (see Sec. ).", "Therefore, the critical point where the total mass $M$ has a maximum indicates the critical value $\\sigma _{\\mathrm {cr}}$ for stability.", "In Fig.", "REF , we show some profiles of total (and rest) mass, as a function of $\\sigma $ , for different polytropes in the range $0.5 < n < 3$ for several values of the index $\\lambda \\in [10^{-4}, 10^{-1}]$ .", "In the plots, we have also indicated the maximum of the curve $M$ which provides the critical parameter $\\sigma _\\mathrm {cr}$ , thus separating the stable from the unstable region.", "The main result of our analysis is displayed in Fig.", "REF where we determine the stable and unstable regions in the $n\\mbox{--}\\sigma $ parameter space, for several values of the vacuum constant index $\\lambda \\in [10^{-4}, 10^{-1}]$ .", "In the plot, we present the results for the critical values of the parameter $\\sigma _\\mathrm {cr}$ as obtained from the critical point (CP) method, and those obtained from Chandrasekhar's method which were computed numerically using the shooting method (SM) (see Fig.", "REF ) and the trial functions [see Eq.", "(REF )].", "For comparison, we have also included the $\\sigma _\\mathrm {cr}$ values for the corresponding configurations with $\\lambda =0$ .", "Figure: Critical values of the relativity parameter σ cr \\sigma _{\\mathrm {cr}}, as a function of the polytropic index nn, for the vacuum constant index λ∈[10 -4 ,10 -1 ]\\lambda \\in [10^{-4}, 10^{-1}].", "The forbidden region (gray background) corresponds to the region where the TOV equations do not give physically acceptable configurations for λ≠0\\lambda \\ne \\,0.", "Here we show the results obtained by using the CP method together with the results provided by Chandrasekhar's approach via the SM and the trial functions ξ 1 \\xi _1 and ξ 2 \\xi _2.", "The dashed lines (same color) indicate the corresponding values of σ cr \\sigma _\\mathrm {cr} with λ=0\\lambda = 0.", "Note that large values of λ\\lambda , for instance, λ=10 -2 \\lambda = 10^{-2} and λ=10 -1 \\lambda = 10^{-1}, lower the critical value σ cr \\sigma _\\mathrm {cr} with respect to the corresponding value with λ=0\\lambda = 0.", "Moreover, for these same values of λ\\lambda , the critical point method and Chandrasekhar's dynamical approach predict different values of σ cr \\sigma _\\mathrm {cr}.", "For values of λ<10 -4 \\lambda < 10^{-4}, its influence on the radial stability is practically negligible.A first thing to notice is that for large values of the index $\\lambda $ , in particular $\\lambda =10^{-2}$ and $\\lambda =10^{-1}$ , the values of $\\sigma _\\mathrm {cr}$ decrease relative to the case with vanishing $\\lambda $ .", "We show the corresponding differences, as defined in Eq.", "(REF ), in Fig.", "REF for $\\lambda \\in [10^{-4},10^{-1}]$ .", "Note that the differences are proportional, in order of magnitude, to the corresponding value of the index $\\lambda $ .", "These results are connected with those in Fig.", "REF and the fact that large values of $\\lambda $ tend to destabilize the polytropic spheres.", "Figure: Differences of the critical parameter σ cr \\sigma _{\\mathrm {cr}}, as a function of nn, between the values for λ∈[10 -4 ,10 -1 ]\\lambda \\in [10^{-4},10^{-1}] and their corresponding values for λ=0\\lambda = 0.", "Note that as λ\\lambda increases the differences between the values predicted by the CP method and Chandrasekhar's approach also increase.Remarkably, we found that for large values of $\\lambda $ the values of $\\sigma _\\mathrm {cr}$ obtained by using the critical point method differ from those determined via the Chandrasekhar's dynamical approach.", "In Fig.", "REF , we show the corresponding differences between both methods, as a function of $n$ , for $\\lambda \\in [10^{-4},10^{-1}]$ .", "Note that for the cases $\\lambda = 10^{-2}$ and $\\lambda = 10^{-1}$ , the differences increase with $n$ .", "On the other hand, for lower values of $\\lambda $ , for instance, $\\lambda = 10^{-4}$ , the bigger differences are found in the regime of small $n$ and tend to zero as $n\\rightarrow 3$ .", "Note that these results are closely similar to those depicted in our preceding paper [38][Fig.", "8] for the case $\\lambda = 0$ .", "Figure: Differences δσ cr \\delta \\sigma _{\\mathrm {cr}} of the critical parameter σ cr \\sigma _{\\mathrm {cr}}, as a function of the family of parameters (n,λ)(n,\\lambda ), as determined via the CP method (σ cr CP \\sigma _{\\mathrm {cr}}^{\\mathrm {CP}}) and Chandrasekhar's radial oscillations approach (σ cr Ch \\sigma _{\\mathrm {cr}}^{\\mathrm {Ch}}) via the SM and the trial functions ξ 1 \\xi _{1} and ξ 2 \\xi _{2}.", "Note that for large values of λ\\lambda the differences grow as nn increases.", "For lower values of λ\\lambda , the bigger differences are found in the low nn regime and tend to zero as n→3n \\rightarrow 3." ], [ "Discussion", "In this paper we have investigated the role of the cosmological constant $\\Lambda $ in the dynamical stability of relativistic polytropes by using two different approaches, namely, the energetic or critical point method and the infinitesimal radial oscillations method.", "Using Chandrasekhar's pulsation equation, we found that large values of $\\lambda $ rise the critical adiabatic index $\\gamma _{\\mathrm {cr}}$ relative to their corresponding values for $\\lambda = 0$ .", "Thus, the cosmological constant tends to destabilize the polytropes.", "Our results clearly show that the critical point method and the theory of radial oscillations predict different values of the critical parameter $\\sigma _{\\mathrm {cr}}$ , for nonzero $\\lambda $ .", "The nature of this discrepancy might be attained to the different physical approach adopted in each method.", "Energy considerations are based on static solutions to Einstein's equations.", "Meanwhile Chandrasekhar's method considers a linearized analysis of time-dependent perturbations on the given equilibrium configuration.", "Our results show that large values of the cosmological parameter $\\lambda $ enhance this difference.", "Finally, we would like to remark that the role of the vacuum energy on the radial stability of polytropic spheres becomes relevant for the parameter $\\lambda $ sufficiently large—it is negligible for $\\lambda $ smaller than $10^{-4}$ and becomes significant for $\\lambda $ comparable to $10^{-1}$ .", "The authors acknowledge the support of the Institute of Physics and its Research Centre for Theoretical Physics and Astrophysics at the Silesian University in Opava." ] ]

[ [ "Ultrafast Mott transition driven by nonlinear electron-phonon\n interaction" ], [ "Abstract Nonlinear phononics holds the promise for controlling properties of quantum materials on the ultrashort timescale.", "Using nonequilibrium dynamical mean-field theory, we solve a model for the description of organic solids, where correlated electrons couple nonlinearly to a quantum phonon mode.", "Unlike previous works, we exactly diagonalize the local phonon mode within the noncrossing approximation to include the full phononic fluctuations.", "By exciting the local phonon in a broad range of frequencies near resonance with an ultrashort pulse, we show it is possible to induce a Mott insulator-to-metal phase transition.", "Conventional semiclassical and mean-field calculations, where the electron-phonon interaction decouples, underestimate the onset of the quasiparticle peak.", "This fact, together with the nonthermal character of the photoinduced metal, suggests a leading role of the phononic fluctuations and of the dynamic nature of the state in the vibrationally induced quasiparticle coherence." ], [ "Spectral functions and occupations at intermediate time-steps", "In Fig.", "REF of the main text, we have shown the time-dependent spectral functions at the initial ($t=0$ ) and final ($t=30$ ) times of the simulations we performed for the dynamic phonon model and the simplified approach based on the time-dependent Hubbard interaction $U_{\\langle X^2\\rangle }(t)$ .", "In the same figure, we have also presented the time evolution of the spectral weight around the Fermi level.", "Here, we want to show the spectral functions and the respective occupations at intermediate times for the two different cases as depicted in Fig.", "REF .", "We notice that the most significant changes of the spectral functions occur around $\\omega \\sim 0$ and that we observe an overall increase of the spectral weight at the Fermi level in both cases (and, actually, also for the other simplified treatments of the electron-phonon interaction presented in Eq.", "(REF ) of the main text).", "For the dynamic phonon, we recognize the presence of a quasi-particle peak already at $t=6$ .", "The dynamics of this peak looks pretty interesting: by comparing the snapshots of the spectral function taken at different times, we distinguish the breathing of the peak in both its height and width.", "The $U_{\\langle X^2\\rangle }(t)$ driving leads instead to an almost featureless dynamics, with just a small and featureless increase of the weight at $\\omega = 0$ and a slight broadening of the occupation function." ], [ "Semiclassical approximation and mean-field decoupling of the electron-phonon interaction", "In the main text, we have presented four alternative ways of replacing, in the electron-phonon interaction, the phonon degree of freedom with a classical field or with a simplified treatment of the quantum phonons, see Eq.", "(REF ) appearing there.", "While the two driving protocols of the purely electronic model $U_{\\langle X\\rangle ^{2}}(t)$ and $U_{\\langle X^2\\rangle }(t)$ do not involve in any sense the phonon part of the Hamiltonian, since the expectation values $\\langle X\\rangle $ and $\\langle X^2\\rangle $ are obtained from the full DMFT calculation, the semiclassical and the mean-field approaches still retain the phonon degree of freedom.", "In this sense, these last two treatments take into account the back-action of the electrons on the displacement field.", "The first approach we describe is the semiclassical one, based on the replacement of the quantum operator $X_{i}$ with a classical and site-independent field $X$ .", "Similarly, we introduce the classical field $P$ , the conjugate momenta of the generalized coordinate $X$ .", "To compute their time evolution, we use the Hamilton equations of motion: $\\left\\lbrace \\begin{array}{c}\\dot{X} \\left( t \\right) = \\omega _{\\text{ph}} P \\left( t \\right) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\\dot{P} \\left( t \\right) = - \\frac{\\omega _{\\text{ph}}^{2} \\left( d \\left( t \\right) \\right)}{\\omega _{\\text{ph}}} X \\left( t \\right) - \\frac{F \\left( t \\right)}{\\omega _{\\text{ph}}}\\end{array}\\right.,$ where $\\omega _{\\text{ph}}^{2} \\left( d \\left( t \\right) \\right) = \\omega _{\\text{ph}}^{2} \\left[ 1 + \\frac{4 \\left( h - d \\right)}{ \\omega _{\\text{ph}} } d \\left( t \\right) \\right]$ , with $d \\left( t \\right) = \\frac{1}{N} \\sum _{i} \\langle D_{i} \\rangle $ being the time-dependent expectation value of the double occupations.", "$F \\left( t \\right) = \\sqrt{2} \\omega _{\\text{ph}}^{2} f \\left( t \\right)$ is a force field related to external driving.", "The solution of the system Eq.", "(REF ) provides the time-dependence of the fields $X \\left( t \\right)$ and $P \\left( t \\right)$ so that we can write the time-dependent electronic model that we have to deal with as: $H_{\\text{U-driv}} \\left( t \\right) = H_{\\text{el}} - 2 X^{2} \\left( t \\right) \\left( d-h \\right) \\sum _{i} D_{i} \\;,$ where $H_{\\text{el}}$ is the electronic part of the original Hamiltonian defined in Eq.", "(REF ) of the main text.", "This way, we obtain a time-dependent Hubbard interaction written as: $U_{X^{2}}^{\\text{s.cl}} \\left( t \\right) = U - 2 \\left( d-h \\right) X^{2} \\left( t \\right) \\;.$ The Hamiltonian Eq.", "(REF ) corresponds to Eq.", "(REF ) in the main text, where the time-dependent Hubbard interaction is provided by Eq.", "(REF ).", "This equation has to be supplied with Eq.", "(REF ), that provides the time dependence of the field $X \\left( t \\right)$ .", "The dependence of Eq.", "(REF ) by the time-dependent double occupations $d \\left( t \\right)$ leads to a back-action of the electrons on the classical phonon.", "By driving the classical field $X \\left( t \\right)$ with an excitation protocol $f \\left( t \\right)$ as depicted in the inset of Fig.1 of the main text, and by using the same Hamiltonian parameters defined there, we obtain the results shown in Fig.", "REF (black lines).", "For comparison, we also show the results obtained for the $U_{\\langle X \\rangle ^{2}} \\left( t \\right)$ driving already discussed in the main text (red lines).", "Fig.", "REF (a) ((b)) shows the time evolution of $X \\left( t \\right)$ ($P \\left( t \\right)$ ).", "The two fields oscillate out of phase at the same frequency close to the bare phonon one $\\omega _{\\text{ph}}$ , and they keep doing it even when the pulse is over (we remind that the pulse duration is 20).", "Fig.", "REF (c) compares the $U$ -driving experienced by the electronic part of the system during the semiclassical dynamics $U^{\\text{s.cl}}_{X^{2}} \\left( t \\right)$ , and for the $U_{\\langle X \\rangle ^{2}} \\left( t \\right)$ driving we get from $\\langle X \\left( t \\right) \\rangle $ obtained with the full DMFT calculation.", "The comparison between the two gives a qualitative similarity of the results.", "Not surprisingly, within the semiclassical phonon treatment, we cannot get two independent renormalizations of the Hubbard interaction as we might get, at the quantum level, from $\\langle X \\rangle ^{2}$ and $\\langle X^{2} \\rangle $ .", "The reason for this is that, trivially, at the classical level we can simply compute $X^{2} \\left( t \\right)$ by taking the square of $X \\left( t \\right)$ , while this procedure dramatically fails at the quantum level.", "Thus, to describe a driving protocol based on a field $X^{2}$ with an expectation value independent from the one of $X$ , we must rely on a quantum-mechanical description of the operator $X^{2}$ .", "A possible way to keep the quantum nature of the bosonic field is to perform a mean-field decoupling of the electron-phonon interaction presented in Eq.", "(REF ) of the main text.", "This reads: $\\begin{split}H_{\\text{el-ph}} \\rightarrow & 2 \\left( h - d \\right) d \\left( t \\right) \\sum _{i} X^{2}_{i} \\\\& + 2 \\left( h - d \\right) \\langle X^{2} \\rangle \\sum _{i} D_{i} \\\\& - 2 N \\left( h - d \\right) \\langle X^{2} \\rangle d \\left( t \\right) \\;.\\end{split}$ This way, we can separately write the electronic and phononic mean-field Hamiltonians, coupled one to the other, as: $\\begin{split}& H_{\\text{el}}^{\\text{MF}} = H_{\\text{el}} + 2 \\left( h - d \\right) \\langle X^{2} \\left( t \\right) \\rangle \\sum _{i} D_{i} \\;, \\\\& H_{\\text{ph}}^{\\text{MF}} = H_{\\text{ph}} + H_{\\text{driv}} \\left( t \\right) + 2 \\left( h - d \\right) d \\left( t \\right) \\sum _{i} X^{2}_{i} \\;,\\end{split}$ Figure: Time evolution of the classical fields XtX \\left( t \\right) and PtP \\left( t \\right) (panels (a) and (b), respectively) and the corresponding changes in the Hubbard interaction U X 2 s.cl tU_{X^{2}}^{\\text{s.cl}} \\left( t \\right) and in the relative change in the double occupations Δd/d\\Delta d / d (panels (c) and (d), respectively) for the same excitation protocol shown in the inset of Fig.", "of the main text (black lines).", "For comparison, in panels (c) and (d) we show the results for the time dependent protocol U 〈X〉 2 tU_{\\langle X \\rangle ^{2}} \\left( t \\right) (red lines).", "The lines in panels (c) and (d) are already shown in Fig.", "(d) and (e) of the main text.where we omitted the term $- 2 N \\left( h - d \\right) \\langle X^{2} \\rangle d \\left( t \\right)$ of Eq.", "(REF ).", "We solve the electronic part of the model $H_{\\text{el}}^{\\text{MF}}$ with DMFT at the NCA level.", "Since the phononic part of the mean-field Hamiltonian is local, we are allowed to write $H_{\\text{ph}}^{\\text{MF}} = \\sum _{i} H_{\\text{ph},i}^{\\text{MF}}$ .", "The time evolution of the phonon degree of freedom might thus be computed using the density matrix, that at equilibrium, locally, looks: $\\rho _{\\text{ph},i}^{\\text{eq}} = \\frac{e^{- \\beta H_{\\text{ph},i}^{\\text{MF}}}}{\\text{Tr} \\left[ e^{- \\beta H_{\\text{ph},i}^{\\text{MF}}} \\right]} = V \\frac{e^{- \\beta H_{\\text{ph},i, \\text{d}}^{\\text{MF}}}}{\\text{Tr} \\left[ e^{- \\beta H_{\\text{ph},i, \\text{d}}^{\\text{MF}}} \\right]} V^{\\dagger } \\;,$ where $H_{\\text{ph},i}^{\\text{MF}} = V_{i} H_{\\text{ph},i,d}^{\\text{MF}} V_{i}^{\\dagger }$ and $H_{\\text{ph},i,d}^{\\text{MF}}$ is the diagonal form of the local mean-field phononic Hamiltonian $H_{\\text{ph},i}^{\\text{MF}}$ .", "We underline that, at equilibrium, $H_{\\text{driv}} \\left( t \\right)$ is equal to zero.", "The subsequent time-evolution of the density matrix can be computed via the Von Neumann equation: $\\frac{\\partial \\rho _{\\text{ph},i} \\left( t \\right)}{\\partial t} = -i \\left[ H_{\\text{ph},i}^{\\text{MF}}, \\rho _{\\text{ph},i} \\left( t \\right) \\right] \\;,$ with initial condition provided by $\\rho _{\\text{ph},i} \\left( t=0 \\right) = \\rho _{\\text{ph},i}^{\\text{eq}}$ .", "A convenient basis for expressing the density matrix, as well as $H_{\\text{ph}}^{\\text{MF}}$ , is the one of the local phonon Fock space so that we can write: $\\begin{split}& \\left( H_{\\text{ph},i} \\right)_{n,p} = \\omega _{\\text{ph}} N \\left( p + \\frac{1}{2} \\right) \\delta _{n,p} \\;, \\\\& \\left( H_{\\text{driv},i} \\left( t \\right) \\right)_{n,p} = \\omega _{\\text{ph}} N f \\left( t \\right) \\left[ \\sqrt{p+1} \\delta _{n, p+1} + \\sqrt{p} \\delta _{n, p-1} \\right] \\;, \\\\& 2 N \\left( h - d \\right) d \\left( t \\right) \\left( X^{2}_{i} \\right)_{n,p} = N \\left( h - d \\right) d \\left( t \\right) \\\\& [ \\sqrt{p \\left( p-1 \\right)} \\delta _{n, p-2} + \\left( 1+ 2p \\right) \\delta _{n,p} \\\\& + \\sqrt{\\left( p+1 \\right) \\left( p+2 \\right)} \\delta _{n, p+2} ] \\;.\\end{split}$ Figure: Time evolution of the expectation values 〈Xt〉\\langle X \\left( t \\right) \\rangle and 〈X 2 t〉\\langle X^{2} \\left( t \\right) \\rangle (panels (a) and (b), respectively) and the corresponding changes in the Hubbard interaction U 〈X 2 〉 MF tU_{\\langle X^{2} \\rangle }^{\\text{MF}} \\left( t \\right) and in the relative change in the double occupations Δd/d\\Delta d / d (panels (c) and (d), respectively) for the same excitation protocol shown in the inset of Fig.", "of the main text.", "We stress that the curves in panels (c) and (d) are shown also in Fig.", "(d) and (e) of the main text, respectively.From the knowledge of the time-dependent density matrix $\\rho _{\\text{ph},i} \\left( t \\right)$ , we compute the time-dependent expectation value of $X^{2}_{i}$ as: $\\langle X^{2} \\left( t \\right) \\rangle = \\text{Tr} \\left[ \\rho _{\\text{ph},i} \\left( t \\right) X^{2}_{i} \\right] \\;.$ Given this quantity, we can also compute: $U_{\\langle X^{2} \\rangle }^{\\text{MF}} \\left( t \\right) = U - 2 \\left( d-h \\right) \\langle X^{2} \\left( t \\right) \\rangle \\;,$ defined in Eq.", "(REF ) of the main text.", "In Fig.", "REF , we show the results obtained at the mean-field level by considering the quantum phonons in the presence of an external driving equal to the one shown in the inset of Fig.", "REF of the main text.", "We notice that $\\langle X^{2} \\left( t \\right) \\rangle = \\langle X \\left( t \\right) \\rangle ^{2} + \\langle X^{2} \\left( 0 \\right) \\rangle $ , where $\\langle X^{2} \\left( 0 \\right) \\rangle \\sim 0.5082$ .", "The time dependence of $\\langle X \\left( t \\right) \\rangle $ and of $\\langle X^{2} \\left( t \\right) \\rangle $ , shown in Fig.", "REF (a) and (b), respectively, does not resemble the one presented in the main text in panels (a) and (b) of Fig.", "REF (black lines) and neither the one shown in Fig.", "REF (a) for the classical field $X$ .", "Indeed, in this mean-field calculation, we observe that the oscillatory behavior of both $\\langle X \\left( t \\right) \\rangle $ and $\\langle X^{2} \\left( t \\right) \\rangle $ is the most pronounced while the external pulse is active.", "Instead, when the external perturbation is over, the response of the system is strongly suppressed.", "The frequency of the oscillations observed in $\\langle X \\left( t \\right) \\rangle $ for $t > 20$ is almost equal to the double of the bare phonon frequency $2 \\times \\omega _{\\text{ph}}$ .", "This fact, together with the small response of the system for a driving frequency $\\Omega = \\omega _{\\text{ph}}$ , are in qualitative agreement with the picture provided by the parametric oscillator.", "The quench in $\\langle X^{2} \\left( t \\right) \\rangle $ after the pulse observed in the full NCA calculation here disappears.", "Also, the relative change in the double occupations in Fig.", "REF (d) is different as compared to the one presented in Fig.", "REF (c) of the manuscript (black line) both from the quantitative and the qualitative point of view.", "To briefly summarize our findings, we observe that the simplified protocol $U_{\\langle X^{2} \\rangle } \\left( t \\right)$ introduced in the main text (violet line in Fig.", "REF (d)) produces a $\\Delta d/d$ that compares much better to the NCA result with respect to the ones obtained with all the other simplified $U$ -drivings introduced in Eq.", "(REF ) of the main text." ] ]

[ [ "From WiscKey to Bourbon: A Learned Index for Log-Structured Merge Trees" ], [ "Abstract We introduce BOURBON, a log-structured merge (LSM) tree that utilizes machine learning to provide fast lookups.", "We base the design and implementation of BOURBON on empirically-grounded principles that we derive through careful analysis of LSM design.", "BOURBON employs greedy piecewise linear regression to learn key distributions, enabling fast lookup with minimal computation, and applies a cost-benefit strategy to decide when learning will be worthwhile.", "Through a series of experiments on both synthetic and real-world datasets, we show that BOURBON improves lookup performance by 1.23x-1.78x as compared to state-of-the-art production LSMs." ], [ "Introduction", "Machine learning is transforming how we build computer applications and systems.", "Instead of writing code in the traditional algorithmic mindset, one can instead collect the proper data, train a model, and thus implement a robust and general solution to the task at hand.", "This data-driven, empirical approach has been called “Software 2.0” [26], hinting at a world where an increasing amount of the code we deploy is realized in this manner; a number of landmark successes over the past decade lend credence to this argument, in areas such as image [32] and speech recognition [24], machine translation [46], game playing [44], and many other areas [7], [17], [15].", "One promising line of work, for using ML to improve core systems is that of the “learned index” [31].", "This approach applies machine learning to supplant the traditional index structure found in database systems, namely the ubiquitous B-Tree [9].", "To look up a key, the system uses a learned function that predicts the location of the key (and value); when successful, this approach can improve lookup performance, in some cases significantly, and also potentially reduce space overhead.", "Since this pioneering work, numerous follow ups [30], [20], [13] have been proposed that use better models, better tree structures, and generally improve how learning can reduce tree-based access times and overheads.", "However, one critical approach has not yet been transformed in this “learned” manner: the Log-structured Merge Tree (LSM) [39], [42], [37].", "LSMs were introduced in the late Black'90s, gained popularity a decade later through work at Google on BigTable [8] and LevelDB [22], and have become widely used in industry, including in Cassandra [33], RocksDB [18], and many other systems [38], [21].", "LSMs have many positive properties as compared to B-trees and their cousins, including high insert performance [40], [37], [11].", "In this paper, we apply the idea of the learned index to LSMs.", "A major challenge is that while learned indexes are primarily tailored for read-only settings, LSMs are optimized for writes.", "Writes cause disruption to learned indexes because models learned over existing data must now be updated to accommodate the changes; the system thus must re-learn the data repeatedly.", "However, we find that LSMs are well-suited for learned indexes.", "For example, although writes modify the LSM, most portions of the tree are immutable; thus, learning a function to predict key/value locations can be done once, and used as long as the immutable data lives.", "However, many challenges arise.", "For example, variable key or value sizes make learning a function to predict locations more difficult, and performing model building too soon may lead to significant resource waste.", "Thus, we first study how an existing LSM system, WiscKey [37], functions in great detail (§).", "We focus on WiscKey because it is a state-of-the-art LSM implementation that is significantly faster than LevelDB and RocksDB [37].", "Our analysis leads to many interesting insights from which we develop five learning guidelines: a set of rules that aid an LSM system to successfully incorporate learned indexes.", "For example, while it is useful to learn the stable, low levels in an LSM, learning higher levels can yield benefits as well because lookups must always search the higher levels.", "Next, not all files are equal: some files even in the lower levels are very short-lived; a system must avoid learning such files, or resources can be wasted.", "Finally, workload- and data-awareness is important; based on the workload and how the data is loaded, it may be more beneficial to learn some portions of the tree than others.", "We apply these learning guidelines to build Bourbon, a learned-index implementation of WiscKey (§).", "Bourbon uses piece-wise linear regression, a simple but effective model that enables both fast training (i.e., learning) and inference (i.e., lookups) with little space overhead.", "Bourbon employs file learning: models are built over files given that an LSM file, once created, is never modified in-place.", "Bourbon implements a cost-benefit analyzer that dynamically decides whether or not to learn a file, reducing unnecessary learning while maximizing benefits.", "While most of the prior work on learned indexes [31], [20], [13] has made strides in optimizing stand-alone data structures, Bourbon integrates learning into a production-quality system that is already highly optimized.", "BlackBourbon's implementation adds around 5K LOC to WiscKey (which has $\\sim $ 20K LOC).", "We analyze the performance of Bourbon on a range of synthetic and real-world datasets and workloads (§).", "We find that Bourbon reduces the indexing costs of WiscKey significantly and thus offers 1.23$\\times $ – 1.78$\\times $ faster lookups for various datasets.", "Even under workloads with significant write load, Bourbon speeds up a large fraction of lookups and, through cost-benefit, avoids unnecessary (early) model building.", "Thus, Bourbon matches the performance of an aggressive-learning approach but performs model building more judiciously.", "Finally, most of our analysis focuses on the case where fast lookups will make the most difference, namely when the data resides in memory (i.e., in the file-system page cache).", "However, we also experiment with Bourbon when data resides on a fast storage device (an Optane SSD) or when data does not fit entirely in memory, and show that benefits can still be realized.", "This paper makes four contributions.", "We present the first detailed study of how LSMs function internally with learning in mind.", "We formulate a set of guidelines on how to integrate learned indexes into an LSM (§).", "We present the design and implementation of Bourbon which incorporates learned indexes into a real, highly optimized, production-quality LSM system (§).", "Finally, we analyze Bourbon's performance in detail, and demonstrate its benefits (§)." ], [ "Background", "We first describe log-structured merge trees and explain how data is organized in LevelDB.", "Next, we describe WiscKey, a modified version of LevelDB that we adopt as our baseline.", "We then provide a brief background on learned indexes." ], [ "LSM and LevelDB", "An LSM tree is a persistent data structure used in key-value stores to support efficient inserts and updates [39].", "Unlike B-trees that require many random writes to storage upon updates, LSM trees perform writes sequentially, thus achieving high write throughput [39].", "An LSM organizes data in multiple levels, with the size of each level increasing exponentially.", "Inserts are initially buffered in an in-memory structure; once full, this structure is merged with the first level of on-disk data.", "This procedure resembles merge-sort and is referred to as compaction.", "Data from an on-disk level is also merged with the successive level if the size of the level exceeds a limit.", "Note that updates do not modify existing records in-place; they follow the same path as inserts.", "As a result, many versions of the same item can be present in the tree at a time.", "Throughout this paper, we refer to the levels that contain the newer data as higher levels and the older data as lower levels.", "A lookup request must return the latest version of an item.", "Because higher levels contain the newer versions, the search starts at the topmost level.", "First, the key is searched for in the in-memory structure; if not found, it is searched for in the on-disk tree starting from the highest level to the lowest one.", "The value is returned once the key is found at a level.", "Figure: LevelDB and WiscKey.", "(a) shows how data is organized in LevelDB and how a lookup is processed.", "The search in in-memory tables is not shown.", "The candidate sstables are shown in bold boxes.", "(b) shows how keys and values are separated in WiscKey.LevelDB [22] is a widely used key-value store built using LSM.", "Figure REF (a) shows how data is organized in LevelDB.", "A new key-value pair is first written to the memtable; when full, the memtable is converted into an immutable table which is then compacted and written to disk sequentially as sstables.", "The sstables are organized in seven levels ($L_0$ being the highest level and $L_6$ the lowest) Blackand each sstable corresponds to a file.", "LevelDB ensures that key ranges of different sstables at a level are disjoint (two files will not contain overlapping ranges of keys); $L_0$ is an exception where the ranges can overlap across files.", "The amount of data at each level increases by a factor of ten; for example, the size of $L_1$ is 10MB, while $L_6$ contains several 100s of GBs.", "If a level exceeds its size limit, one or more sstables from that level are merged with the next level; this is repeated until all levels are within their limits.", "Lookup steps.", "Figure REF (a) also shows how a lookup request for key $k$ proceeds.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 1; FindFiles: If the key is not found in the in-memory tables, LevelDB finds the set of candidate sstable files that may contain $k$ .", "A key in the worst case may be present in all $L_0$ files (because of overlapping ranges) and within one file at each successive level.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 2; LoadIB+FB: In each candidate sstable, an index block and a bloom-filter block are first loaded from the disk.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 3; SearchIB: The index block is binary searched to find the data block that may contain $k$ .", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 4; SearchFB: The filter is queried to check if $k$ is present in the data block.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 5; LoadDB: If the filter indicates presence, the data block is loaded.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 6; SearchDB: The data block is binary searched.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 7; ReadValue: If the key is found in the data block, the associated value is read and the lookup ends.", "If the filter indicates absence or if the key is not found in the data block, the search continues to the next candidate file.", "Note that blocks are not always loaded from the disk; index and filter blocks, and frequently accessed data blocks are likely to be present in memory (i.e., file-system cache).", "We refer to these search steps at a level that occur as part of a single lookup as an internal lookup.", "A single lookup thus consists of many internal lookups.", "A negative internal lookup does not find the key, while a positive internal lookup finds the key and is thus the last step of a lookup request.", "We differentiate indexing steps from data-access steps; indexing steps such as FindFiles, SearchIB, SearchFB, and SearchDB search through the files and blocks to find the desired key, while data-access steps such as LoadIB+FB, LoadDB, and ReadValue read the data from storage.", "Our goal is to reduce the time spent in indexing." ], [ "WiscKey", "In LevelDB, compaction results in large write amplification because both keys and values are sorted and rewritten.", "Thus, LevelDB suffers from high compaction overheads, affecting foreground workloads.", "WiscKey [37] (and Badger [1]) reduces this overhead by storing the values separately; the sstables contain only keys and pointers to the values as shown in Figure REF (b).", "With this design, compaction sorts and writes only the keys, leaving the values undisturbed, thus reducing I/O amplification and overheads.", "WiscKey thus performs significantly better than other optimized LSM implementations such as LevelDB and RocksDB.", "Given these benefits, we adopt WiscKey as the baseline for our design.", "Further, WiscKey's key-value separation enables our design to handle variable-size records; we describe how in more detail in §REF .", "The write path of WiscKey is similar to that of LevelDB except that values are written to a value log.", "A lookup in WiscKey also involves searching at many levels and a final read into the log once the target key is found.", "The size of WiscKey's LSM tree is much smaller than LevelDB because it does not contain the values; hence, it can be entirely cached in memory [37].", "Thus, a lookup request involves multiple searches in the in-memory tree, and the ReadValue step performs one final read to retrieve the value." ], [ "Optimizing Lookups in LSMs", "Performing a lookup in LevelDB and WiscKey requires searching at multiple levels.", "Further, within each sstable, many blocks are searched to find the target key.", "Given that LSMs form the basis of many embedded key-value stores (e.g., LevelDB, RocksDB [18]) and distributed storage systems (e.g., BigTable [8], Riak [38]), optimizing lookups in LSMs can have huge benefits.", "A recent body of work, starting with learned indexes [31], makes a case for replacing or augmenting traditional index structures with machine-learning models.", "The key idea is to train a model (such as linear regression or neural nets) on the input so that the model can predict the position of a record in the sorted dataset.", "The model can have inaccuracies, and thus the prediction has an associated error bound.", "During lookups, if the model-predicted position of the key is correct, the record is returned; if it is wrong, a local search is performed within the error bound.", "For example, if the predicted position is $pos$ and the minimum and maximum error bounds are $\\delta $$_m$$_i$$_n$ and $\\delta $$_m$$_a$$_x$ , then upon a wrong prediction, a local search is performed between $pos-\\delta $$_m$$_i$$_n$ and $pos+\\delta $$_m$$_a$$_x$ .", "Learned indexes can make lookups significantly faster.", "Intuitively, a learned index turns a $O(log$ -$n)$ lookup of a B-tree into a $O$$(1)$ operation.", "Empirically, learned indexes provide 1.5$\\times $ – 3$\\times $ faster lookups than B-trees [31].", "Given these benefits, we ask the following questions: Blackcan learned indexes for LSMs make lookups faster?", "If yes, under what scenarios?", "Traditional learned indexes do not support updates because models learned over the existing data would change with modifications [31], [13], [20].", "However, LSMs are attractive for their high performance in write-intensive workloads because they perform writes only sequentially.", "Thus, we examine: Blackhow to realize the benefits of learned indexes while supporting writes for which LSMs are optimized?", "We answer these two questions next.", "In this section, we first analyze if learned indexes could be beneficial for LSMs and examine under what scenarios they can improve lookup performance.", "We then provide our intuition as to why learned indexes might be appropriate for LSMs even when allowing writes.", "We conduct an in-depth study based on measurements of how WiscKey functions internally under different workloads to validate our intuition.", "From our analysis, we derive a set of learning guidelines." ], [ "Learned Indexes: Beneficial Regimes", "A lookup in LSM involves several indexing and data-access steps.", "Optimized indexes such as learned indexes can reduce the overheads of indexing but cannot reduce data-access costs.", "In WiscKey, learned indexes can thus potentially reduce the costs of indexing steps such as FindFiles, SearchIB, and SearchDB, while data-access costs (e.g., ReadValue) cannot be significantly reduced.", "As a result, learned indexes can improve overall lookup performance if indexing contributes to a sizable portion of the total lookup latency.", "We identify scenarios where this is the case.", "First, when the dataset or a portion of it is cached in memory, data-access costs are low, and so indexing costs become significant.", "Figure REF shows the breakdown of lookup latencies in WiscKey.", "The first bar shows the case when the dataset is cached in memory; the second bar shows the case where the data is stored on a flash-based SATA SSD.", "With caching, data-access and indexing costs contribute almost equally to the latency.", "Thus, optimizing the indexing portion can reduce lookup latencies by about 2$\\times $ .", "When the dataset is not cached, data-access costs dominate and thus optimizing indexes may yield smaller benefits (about 20%).", "However, learned indexes are not limited to scenarios where data is cached in memory.", "They offer benefit on fast storage devices that are currently prevalent and can do more so on emerging faster devices.", "The last three bars in Figure REF show the breakdown for three kinds of devices: flash-based SSDs over SATA and NVMe, and an Optane SSD.", "As the device gets faster, lookup latency (as shown at the top) decreases, but the fraction of time spent on indexing increases.", "For example, with SATA SSDs, indexing takes about 17% of the total time; in contrast, with Optane SSDs, indexing takes 44% and thus optimizing it with learned indexes can potentially improve performance by 1.8$\\times $ .", "More importantly, the trend in storage performance favors the use of learned indexes.", "With storage performance increasing rapidly and emerging technologies like 3D Xpoint memory providing very low access latencies, indexing costs will dominate and thus learned indexes will yield increasing benefits.", "Summary.", "Learned indexes could be beneficial when the database or a portion of it is cached in memory.", "With fast storage devices, regardless of caching, indexing contributes to a significant fraction of the lookup time; thus, learned indexes can prove useful in such cases.", "With storage devices getting faster, learned indexes will be even more beneficial." ], [ "Learned Indexes with Writes", "Learned indexes provide higher lookup performance compared to traditional indexes for read-only analytical workloads.", "However, a major drawback of learned indexes (as described in  [31]) is that they do not support modifications such as inserts and updates [20], [13].", "The main problem with modifications is that they alter the data distribution and so the models must be re-learned; for write-heavy workloads, models must be rebuilt often, incurring high overheads.", "At first, it may seem like learned indexes are not a good match for write-heavy situations for which LSMs are optimized.", "However, we observe that the design of LSMs fits well with learned indexes.", "Our key realization is that although updates can change portions of the LSM tree, a large part remains immutable.", "Specifically, newly modified items are buffered in the in-memory structures or present in the higher levels of the tree, while stable data resides at the lower levels.", "Given that a large fraction of the dataset resides in the stable, lower levels, lookups to this fraction can be made faster with no or few re-learnings.", "In contrast, learning in higher levels may be less beneficial: they change at a faster rate and thus must be re-learned often.", "We also realize that the immutable nature of sstable files makes them an ideal unit for learning.", "Once learned, these files are never updated and thus a model can be useful until the file is replaced.", "Further, the data within an sstable is sorted; such sorted data can be learned using simple models.", "A level, which is a collection of many immutable files, can also be learned as a whole using simple models.", "The data in a level is also sorted: the individual sstables are sorted, and there are no overlapping key ranges across sstables.", "We next conduct a series of in-depth measurements to validate our intuitions.", "Our experiments confirm that while a part of our intuition is indeed true, there are some subtleties (for example, in learning files at higher levels).", "Based on these experimental results, we formulate a set of learning guidelines: a few simple rules that an LSM that applies learned indexes should follow.", "Experiments: goal and setup.", "The goal of our experiments is to determine how long a model will be useful and how often it will be useful.", "A model built for a sstable file is useful as long as the file exists; thus, we first measure and analyze sstable lifetimes.", "How often a model will be used is determined by how many internal lookups it serves; thus, we next measure the number of internal lookups to each file.", "Since models can also be built for entire levels, we finally measure level lifetimes as well.", "To perform our analysis, we run workloads with varying amounts of writes and reads, and measure the lifetimes and number of lookups.", "We conduct our experiments on WiscKey, but we believe our results are applicable to most LSM implementations.", "We first load the database with 256M key-value pairs.", "We then run a workload with a single rate-limited client that performs 200M operations, a fraction of which are writes.", "Our workload chooses keys uniformly at random.", "Lifetime of SSTables.", "To determine how long a model will be useful, we first measure and analyze the lifetimes of sstables.", "To do so, we track the creation and deletion times of all sstables.", "For files created during the load phase, we assign the workload-start time as their creation time; for other files, we record the actual creation times.", "If the file is deleted during the workload, then we calculate its exact lifetime.", "However, some files are not deleted by the end of the workload and we must estimate their lifetimes.If the files are created during load, we assign the workload duration as their lifetimes.", "If not, we estimate the lifetime of a file based on its creation time ($c$ ) and the total workload time ($w$ ); the lifetime of the file is at least $w-c$ .", "We thus consider the lifetime distribution of other files that have a lifetime of at least $w-c$ .", "We then pick a random lifetime in this distribution and assign it as this file's lifetime.", "Figure REF (a) shows the average lifetime of sstable files at different levels.", "We make three main observations.", "First, the average lifetime of sstable files at lower levels is greater than that of higher levels.", "Second, at lower percentages of writes, even files at higher levels have a considerable lifetime; for example, at 5% writes, files at $L_0$ live for about 2 minutes on an average.", "Files at lower levels live much longer; files at $L_4$ live about 150 minutes.", "Third, although the average lifetime of files reduces with more writes, even with a high amount of writes, files at lower levels live for a long period.", "For instance, with 50% writes, files at $L_4$ live for about 60 minutes.", "In contrast, files at higher level live only for a few seconds; for example, an $L_0$ file lives only about 10 seconds.", "We now take a closer look at the lifetime distribution.", "Figure REF (b) shows the distributions for $L_1$ and $L_4$ files with 5% writes.", "We first note that some files are very short-lived, while some are long-lived.", "For example, in $L_1$ , the lifetime of about 50% of the files is only about 2.5 seconds.", "If files cross this threshold, they tend to live for much longer times; almost all of the remaining $L_1$ files live over five minutes.", "Surprisingly, even at $L_4$ , which has a higher average lifetime for files, a few files are very short-lived.", "We observe that about 2% of $L_4$ files live less than a second.", "We find that there are two reasons why a few files live for a very short time.", "First, compaction of a $L_i$ file creates a new file in $L_i$$_+$$_1$ which is again immediately chosen for compaction to the next level.", "Second, compaction of a $L_i$ file creates a new file in $L_i$$_+$$_1$ , which has overlapping key ranges with the next file that is being compacted from $L_i$ .", "Figure REF (c) shows that this pattern holds for other percentages of writes too.", "We observed that this holds for other levels as well.", "From the above observations, we arrive at our first two learning guidelines.", "Learning guideline - 1: Favor learning files at lower levels.", "Files at lower levels live for a long period even for high write percentages; thus, models for these files can be used for a long time and need not be rebuilt often.", "Learning guideline - 2: Wait before learning a file.", "A few files are very short-lived, even at lower levels.", "Thus, learning must be invoked only after a file has lived up to a threshold lifetime after which it is highly likely to live for a long time.", "Internal Lookups at Different Levels.", "To determine how many times a model will be used, we analyze the number of lookups served by the sstable files.", "We run a workload and measure the number of lookups served by files at each level and plot the average number of lookups per file at each level.", "Figure REF (a) shows the result when the dataset is loaded in an uniform random order.", "The number of internal lookups is higher for higher levels, although a large fraction of data resides at lower levels.", "This is because, at higher levels, many internal lookups are negative, as shown in Figure REF (a)(ii).", "The number of positive internal lookups is as expected: higher in lower levels as shown in Figure REF (a)(iii).", "This result shows that files at higher levels serve many negative lookups and thus are worth optimizing.", "While bloom filters may already make these negative lookups faster, the index block still needs to be searched (before the filter query).", "BlackWe also conduct the same experiment with another workload where the access pattern follows a zipfian distribution (most requests are to a small set of keys).", "Most of the results exhibit the same trend as the random workload except for the number of positive internal lookups, as shown in Figure REF (a)(iv).", "Under the zipfian workload, higher level files also serve numerous positive lookups, because the workload accesses a small set of keys which are often updated and thus stored in higher levels.", "Figure REF (b) shows the result when the dataset is sequentially loaded, i.e., keys are inserted in ascending order.", "In contrast to the randomly-loaded case, there are no negative lookups because keys of different sstable files do not overlap even across levels; the FindFiles step finds the one file that may contain the key.", "Thus, lower levels serve more lookups and can have more benefits from learning.", "From these observations, we arrive at the next two learning guidelines.", "Learning guideline - 3: Do not neglect files at higher levels.", "Although files at lower levels live longer and serve many lookups, files at higher levels can still serve many negative lookups Blackand in some cases, even many positive lookups.", "Thus, learning files at higher levels can make Blackboth internal lookups faster.", "Learning guideline - 4: Be workload- and data-aware.", "Although most data resides in lower levels, if the workload does not lookup that data, learning those levels will yield less benefit; learning thus must be aware of the workload.", "Further, the order in which the data is loaded influences which levels receive a large fraction of internal lookups; thus, the system must also be data-aware.", "The amount of internal lookups acts as a proxy for both the workload and load order.", "Based on the amount of internal lookups, the system must dynamically decide whether to learn a file or not.", "Lifetime of Levels.", "Given that a level as a whole can also be learned, we now measure and analyze the lifetimes of levels.", "Level learning cannot be applied at $L_0$ because it is unsorted: files in $L_0$ can have overlapping key ranges.", "Once a level is learned, any change to the level causes a re-learning.", "A level changes when new sstables are created at that level, or existing ones are deleted.", "Thus, intuitively, a level would live for an equal or shorter duration than the individual sstables.", "However, learning at the granularity of a level has the benefit that the candidate sstables need not be found in a separate step; instead, upon a lookup, the model just outputs the sstable and the offset within it.", "We examine the changes to a level by plotting the timeline of file creations and deletions at $L_1$ , $L_2$ , $L_3$ , and $L_4$ in Figure REF (a) for a 5%-write workload; we do not show $L_0$ for the reason above.", "On the y-axis, we plot the number of changes divided by the total files present at that level.", "A value of 0 means there are no changes to the level; a model learned for the level can be used as long as the value remains 0.", "A value greater than 0 means that there are changes in the level and thus the model has to re-learned.", "Higher values denote a larger fraction of files are changed.", "Figure: Changes at Levels.", "(a) shows the timeline of file creations and deletions at different levels.", "Note that #changes/#files is higher than 1 in L 1 L_1 as there are more creations and deletions than the number of files.", "(b) shows the time between bursts for L4 for different write percentages.First, as expected, we observe that the fraction of files that change reduces as we go down the levels because lower levels hold a large volume of data in many files, confirming our intuition.", "We also observe that changes to levels arrive in bursts.", "These bursts are caused by compactions that cause many files at a level to be rewritten.", "Further, these bursts occur at almost the same time across different levels.", "The reason behind this is that for the dataset we use, levels $L_0$ through $L_3$ are full and thus any compaction at one layer results in cascading compactions which finally settle at the non-full $L_4$ level.", "The levels remain static between these bursts.", "The duration for which the levels remain static is longer with a lower amount of writes; for example, with 5% writes, as shown in the figure, this period is about 5 minutes.", "However, as the amount of writes increases, the lifetime of a level reduces as shown in Figure REF (b); for instance, with 50% writes, the lifetime of $L_4$ reduces to about 25 seconds.", "From these observations, we arrive at our final learning guideline.", "Learning guideline - 5: Do not learn levels for write-heavy workloads.", "Learning a level as a whole might be more appropriate when the amount of writes is very low or if the workload is read-only.", "For write-heavy workloads, level lifetimes are very short and thus will induce frequent re-learnings.", "Summary.", "We analyzed how LSMs behave internally by measuring and analyzing the lifetimes of sstable files and levels, and the amount of lookups served by files at different levels.", "From our analysis, we derived five learning guidelines.", "We next describe how we incorporate the learning guidelines in an LSM-based storage system." ], [ "Bourbon Design", "We now describe Bourbon, an LSM-based store that uses learning to make indexing faster.", "We first describe the model that Bourbon uses to learn the data (§REF ).", "Then, we discuss how Bourbon supports variable-size values (§REF ) and its basic learning strategy (§REF ).", "We finally explain Bourbon's cost-benefit analyzer that dynamically makes learning decisions to maximize benefit while reducing cost (§REF )." ], [ "Learning the Data", "As we discussed, data can be learned at two granularities: individual sstables or levels.", "Both these entities are sorted datasets.", "The goal of a model that tries to learn the data is to predict the location of a key in such a sorted dataset.", "For example, if the model is constructed for a sstable file, it would predict the file offset given a key.", "Similarly, a level model would output the target sstable file and the offset within it.", "Our requirements for a model is that it must have low overheads during learning and during lookups.", "Further, we would like the space overheads of the model to be small.", "We find that piecewise linear regression (PLR) [4], [27] satisfies these requirements well; thus, Bourbon uses PLR to model the data.", "The intuition behind PLR is to represent a sorted dataset with a number of line segments.", "PLR constructs a model with an error bound; that is, each data point $d$ is guaranteed to lie within the range [$d_{pos}$ $-$ $\\delta $ , $d_{pos}$ $+$ $\\delta $ ], where $d_{pos}$ is the predicted position of $d$ in the dataset and $\\delta $ is the error bound specified beforehand.", "To train the PLR model, Bourbon uses the Greedy-PLR algorithm [47].", "Greedy-PLR processes the data points one at a time; if a data point cannot be added to the current line segment without violating the error bound, then a new line segment is created and the data point is added to it.", "At the end, Greedy-PLR produces a set of line segments that represents the data.", "Greedy-PLR runs in linear time with respect to the number of data points.", "Once the model is learned, inference is quick: first, the correct line segment that contains the key is found (using binary search); within that line segment, the position of the target key is obtained by multiplying the key with the line's slope and adding the intercept.", "If the key is not present in the predicted position, a local search is done in the range determined by the error bound.", "Thus, lookups take $O(log$ -$s)$ time, where $s$ is the number of segments, in addition to a constant time to do the local search.", "The space overheads of PLR are small: a few tens of bytes for every line segment.", "BlackOther models or algorithms such as RMI [31], PGM-Index [19], or splines [29] may also be suitable for LSMs and may offer more benefits than PLR.", "We leave their exploration within LSMs for future work." ], [ "Supporting Variable-size Values", "Learning a model that predicts the offset of a key-value pair is much easier if the key-value pairs are the same size.", "The model then can multiply the predicted position of a key by the size of the pair to produce the final offset.", "However, many systems allow keys and values to be of arbitrary sizes.", "Bourbon requires keys to be of a fixed size, while values can be of any size.", "We believe this is a reasonable design choice because most datasets have fixed-size keys (e.g., user-ids are usually 16 bytes), while value sizes vary significantly.", "Even if keys vary in size, they can be padded to make all keys of the same size.", "Bourbon supports variable-size values by borrowing the idea of key-value separation from WiscKey [37].", "With key-value separation, sstables in Bourbon just contain the keys and the pointer to the values; values are maintained in the value log separately.", "With this, Bourbon obtains the offset of a required key-value pair by getting the predicted position from the model and multiplying it with the record size (which is $keysize$ + $pointersize$ .)", "The value pointer serves as the offset into the value log from which the value is finally read.", "Table: File vs. Level Learning.", "The table compares the time to perform 10M operations in baseline WiscKey, file-learning, and level-learning.", "The numbers within the parentheses show the improvements over baseline.", "The table also shows the percentage of lookups that take the model path; remaining take the original path because the models are not rebuilt yet." ], [ "Level vs. File Learning", "Bourbon can learn individual sstables files or entire levels.", "Our analysis in the previous section showed that files live longer than levels under write-heavy workloads, hinting that learning at the file granularity might be the best choice.", "We now closely examine this tradeoff to design Bourbon's basic learning strategy.", "To do so, we compare the performance of file learning and level learning for different workloads.", "We initially load a dataset and build the models.", "For the read-only workload, the models need not be re-learned.", "In the mixed workloads, the models are re-learned as data changes.", "The results are shown in Table REF .", "For mixed workloads, level learning performs worse than file learning.", "For a write-heavy (50%-write) workload, with level learning, only a small percentage of internal lookups are able to use the model because with a steady stream of incoming writes, the system is unable to learn the levels.", "Only a mere 1.5% of internal lookups take the model path; these lookups are the ones performed just after loading the data and when the initial level models are available.", "We observe that all the 66 attempted level learnings failed because the level changed before the learning completed.", "Because of the additional cost of re-learnings, level learning performs even worse than the baseline with 50% writes.", "On the other hand, with file models, a large fraction of lookups benefit from the models and thus file learning performs better than the baseline.", "For read-heavy mixed workload (5%), although level learning has benefits over the baseline, it performs worse than file learning for the same reasons above.", "Level learning can be beneficial for read-only settings: as shown in the table, level learning provides 10% improvements over file learning.", "Thus, deployments that have only read-only workloads can benefit from level learning.", "Given that Bourbon's goal is to provide faster lookups while supporting writes, levels are not an appropriate choice of granularity for learning.", "Thus, Bourbon uses file learning by default.", "However, Bourbon supports level learning as a configuration option that can be useful in read-only scenarios." ], [ "Cost vs. Benefit Analyzer", "Before learning a file, Bourbon must ensure that the time spent in learning is worthwhile.", "If a file is short-lived, then the time spent learning that file wastes resources.", "Such a file will serve few lookups and thus the model would have little benefit.", "Thus, to decide whether or not to learn a file, Bourbon implements an online cost vs. benefit analysis." ], [ "Wait Before Learning", "As our analysis showed, even in the lower levels, many files are short-lived.", "To avoid the cost of learning short-lived files, Bourbon waits for a time threshold, $T_{wait}$ , before learning a file.", "The exact value of $T_{wait}$ presents a cost vs. performance tradeoff.", "A very low $T_{wait}$ leads to some short-lived files still being learned, incurring overheads; a large value causes many lookups to take the baseline path (because there is no model built yet), thus missing opportunities to make lookups faster.", "Bourbon sets the value of $T_{wait}$ to the time it takes to learn a file.", "Our approach is never more than a factor of two worse than the optimal solution, where the optimal solution knows apriori the lifetime and decides to either immediately or never learn the file (i.e., it is two-competitive [25]).", "Through measurements, we found that the maximum time to learn a file Black(which is at most $\\sim $ 4MB in size) is around 40 ms Blackon our experimental setup.", "We conservatively set $T_{wait}$ to be 50 ms in Bourbon's implementation." ], [ "To Learn a File or Not", "Bourbon waits for $T_{wait}$ before learning a file.", "However, learning a file even if it lives for a long time may not be beneficial.", "For example, our analysis shows that although lower-level files live longer, for some workloads and datasets, they serve relatively fewer lookups than higher-level files; higher-level files, although short-lived, serve a large percentage of negative internal lookups in some scenarios.", "Bourbon, thus, must consider the potential benefits that a model can bring, in addition to considering the cost to build the model.", "It is profitable to learn a file if the benefit of the model ($B_{model}$ ) outweighs the cost to build the model ($C_{model}$ ).", "Estimating $\\mathbf {C_{model}}$ .", "One way to estimate $C_{model}$ is to assume that the learning is completely performed in the background and will not affect the rest of the system; i.e., $C_{model}$ is 0.", "This is true if there are many idle cores which the learning threads can utilize and thus do not interfere with the foreground tasks (e.g., the workload) or other background tasks (e.g., compaction).", "However, Bourbon takes a conservative approach and assumes that the learning threads will interfere and slow down the other parts of the system.", "As a result, Bourbon assumes $C_{model}$ to be equal to $T_{build}$ .", "We define $T_{build}$ as the time to train the PLR model for a file.", "We find that this time is linearly proportional to the number of data points in the file.", "We calculate $T_{build}$ for a file by multiplying the average time to a train a data point (measured offline) and the number of data points in the file.", "Estimating $\\mathbf {B_{model}}$ .", "Estimating the potential benefit of learning a file, $B_{model}$ , is more involved.", "Intuitively, the benefit offered by the model for an internal lookup is given by $T_{b} - T_{m}$ , where $T_{b}$ and $T_{m}$ are the average times for the lookup in baseline and model paths, respectively.", "If the file serves N lookups in its lifetime, the net benefit of the model is: $B_{model} = (T_{b} - T_{m}) * N$ .", "We divide the internal lookups into negative and positive because most negative lookups terminate at the Blackfilter, whereas positive ones do not; thus, $B_{model} = ((T_{n.b} - T_{n.m}) * N_{n}) + ((T_{p.b} - T_{p.m}) * N_{p})$ where $N_{n}$ and $N_{p}$ are the number of negative and positive internal lookups, respectively.", "$T_{n.b}$ and $T_{p.b}$ are the time in the baseline path for a negative and a positive lookup, respectively; $T_{n.m}$ and $T_{p.m}$ are the model counterparts.", "$B_{model}$ for a file cannot be calculated without knowing the number of lookups that the file will serve or how much time the lookups will take.", "The analyzer, to estimate these quantities, maintains statistics of files that have lived their lifetime, i.e., files that were created, served many lookups, and then were replaced.", "To estimate these quantities for a file $F$ , the analyzer uses the statistics of other files at the same level as $F$ ; we consider statistics only at the same level because these statistics vary significantly across levels.", "Recall that Bourbon waits before learning a file.", "During this time, the lookups are served in the baseline path.", "Bourbon uses the time taken for these lookups to estimate $T_{n.b}$ and $T_{p.b}$ .", "Next, $T_{n.m}$ and $T_{p.m}$ are estimated as the average negative and positive model lookup times of other files at the same level.", "Finally, $N_{n}$ and $N_{p}$ are estimated as follows.", "The analyzer first takes the average negative and positive lookups for other files in that level; then, it is scaled by a factor $f = s/\\bar{s_l}$ , where $s$ if the size of the file and $\\bar{s_l}$ is the average file size at this level.", "While estimating the above quantities, Bourbon filters out very short-lived files.", "While bootstrapping, the analyzer might not have enough statistics collected.", "Therefore, initially, Bourbon runs in an always-learn mode (with $T_{wait}$ still in place.)", "Once enough statistics are collected, the analyzer performs the cost vs. benefit analysis and chooses to learn a file if $C_{model} < B_{model}$ , i.e., benefit of a model outweighs the cost.", "If multiple files are chosen to be learned at the same time, Bourbon puts them in a max priority queue ordered by $B_{model} - C_{model}$ , thus prioritizing files that would deliver the most benefit.", "BlackOur cost-benefit analyzer adopts a simple scheme of using average statistics of other files at the same level.", "While this approach has worked well in our initial prototype, using more sophisticated statistics and considering workload distributions (e.g., to account for keys with different popularity) could be more beneficial.", "We leave such exploration for future work." ], [ "Bourbon: Putting it All Together", "We describe how the different pieces of Bourbon work together.", "Figure REF shows the path of lookups in Bourbon.", "As shown in (a), lookups can either be processed via the model (if the target file is already learned), or in the baseline path (if the model is not built yet.)", "The baseline path in Bourbon is similar to the one shown in Figure REF for LevelDB, except that Bourbon stores the values separately and so ReadValue reads the value from the log.", "Figure: Bourbon Lookups.", "(a) shows that lookups can take two different paths: when the model is available (shown at the top), and when the model is not learned yet and so lookups take the baseline path (bottom); some steps are common to both paths.", "(b) shows the detailed steps for a lookup via a model; we show the case where models are built for files.Once Bourbon learns a sstable file, lookups to that file will be processed via the learned model as shown in Figure REF (b).", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 1; FindFiles: Bourbon finds the candidate sstables; this step required because Bourbon uses file learning.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 2; LoadIB+FB: Bourbon loads the index and filter blocks; these blocks are likely to be already cached.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 3; ModelLookup: Bourbon performs a look up for the desired key $k$ in the candidate sstable's model.", "The model outputs a predicted position of $k$ within the file ($pos$ ) and the error bound ($\\delta $ ).", "From this, Bourbon calculates the data block that contains records $pos - \\delta $ through $pos + \\delta $ .$\\endcsname $Sometimes, records $pos - \\delta $ through $pos + \\delta $ span multiple data blocks; in such cases, Bourbon consults the index block (which specifies the maximum key in each data block) to find the data block for $pos$ .", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 4; SearchFB: The filter for that block is queried to check if $k$ is present.", "If present, Bourbon calculates the range of bytes of the block that must be loaded; this is simple because keys and pointers to values are of fixed size.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 5; LoadChunk: The byte range is loaded.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 6; LocateKey: The key is located in the loaded chunk.", "The key will likely be present in the predicted position (the midpoint of the loaded chunk); if not, Bourbon performs a binary search in the chunk.", "[baseline=(char.base)] shape=circle,draw,inner sep=0.5pt] (char) 7; ReadValue: The value is read from the value log using the pointer.", "Possible improvements.", "Although Bourbon's implementation is highly-optimized and provides many features common to real systems, it lacks a few features.", "For example, in the current implementation, we do not support string keys and key compression (although we support value compression).", "BlackFor string keys, one approach we plan to explore is to treat strings as base-64 integers and convert them into 64-bit integers, which could then adopt the same learning approach described herein.", "While this approach may work well for small keys, large keys may require larger integers (with more than 64 bits) and thus efficient large-integer math is likely essential.", "Also, Bourbon does not support adaptive switching between level and file models; it is a static configuration.", "We leave supporting these features to future work.", "To evaluate Bourbon, we ask the following questions: [noitemsep,nolistsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=*] Which portions of lookup does Bourbon optimize?", "(§REF ) How does Bourbon perform with models available and no writes?", "How does performance change with datasets, load orders, and request distributions?", "(§REF ) BlackHow does Bourbon perform with range queries?", "(§REF ) In the presence of writes, how does Bourbon's cost-benefit analyzer perform compared to other approaches that always or never re-learn?", "(§REF ) Does Bourbon perform well on real benchmarks?", "(§REF ) Is Bourbon beneficial when data is on storage?", "(§REF ) BlackIs Bourbon beneficial with limited memory?", "(§REF ) What are the error and space tradeoffs of Bourbon?", "(§REF ) Figure: Datasets.", "The figure shows the cumulative distribution functions (CDF) of three synthetic datasets (linear, segmented-10%, and normal) and one real-world dataset (OpenStreetMaps).", "Each dataset is magnified around the 15% percentile to show a detailed view of its distribution.Setup.", "We run our experiments on a 20-core Intel Xeon CPU E5-2660 machine with 160-GB memory and a 480-GB SATA SSD.", "We use 16B integer keys and 64B values, and set the error bound of Bourbon's PLR as 8.", "Unless specified, our workloads perform 10M operations.", "We use a variety of datasets.", "We construct four synthetic datasets: linear, segmented-1%, segmented-10% , and normal, each with 64M key-value pairs.", "In the linear dataset, keys are all consecutive.", "In the seg-1% dataset, there is a gap after a consecutive segment of 100 keys (i.e., every 1% causes a new segment).", "The segmented-10% dataset is similar, but there is a gap after 10 consecutive keys.", "We generate the normal dataset by sampling 64M unique values from the standard normal distribution $N(0, 1)$ and scale to integers.", "We also use two real-world datasets: Amazon reviews (AR) [5] and New York OpenStreetMaps (OSM) [2].", "AR and OSM have 33.5M and 21.9M key-value pairs, respectively.", "These datasets vary widely in how the keys are distributed.", "Figure REF shows the distribution for a few datasets.", "Most of our experiments focus on the case where the data resides in memory; however, we also analyze cases where data is present on storage." ], [ "Which Portions does ", "We first analyze which portions of the lookup Bourbon optimizes.", "We perform 10M random lookups on the AR and OSM datasets and show the latency breakdown in Figure REF .", "As expected, Bourbon reduces the time spent in indexing.", "The portion marked Search in the figure corresponds to SearchIB and SearchDB in the baseline, versus ModelLookup and LocateKey in Bourbon.", "The steps in Bourbon have lower latency than their baseline counterparts.", "Interestingly, Bourbon reduces data-access costs too, because Bourbon loads a smaller byte range than the entire block loaded by the baseline.", "Figure: Latency Breakdown.", "The figure shows latency breakdown for WiscKey and Bourbon.", "Search denotes SearchIB and SearchDB in WiscKey; the same denotes ModelLookup and LocateKey in Bourbon.", "LoadData denotes LoadDB in WiscKey; the same denotes LoadChunk in Bourbon.", "These two steps are optimized by Bourbon and are shown in solid colors; the number next to a step shows the factor by which it is made faster in Bourbon." ], [ "Performance under No Writes", "We next analyze Bourbon's performance when the models are already built and there are no updates.", "For each experiment, we load a dataset and allow the system to build the models; during the workload, we issue only lookups.", "Figure: Datasets.", "(a) compares the average lookup latencies of Bourbon, Bourbon-level, and WiscKey for different datasets; the numbers on the top show the improvements of Bourbon over WiscKey.", "(b) shows the number of segments for different datasets in Bourbon." ], [ "Datasets", "To analyze how the performance is influenced by the dataset, we run the workload on all six datasets and compare Bourbon's lookup performance against WiscKey.", "Figure REF show the results.", "As shown in  REF (a), Bourbon is faster than WiscKey for all datasets; depending upon the dataset, the improvements vary (1.23$\\times $ to 1.78$\\times $ ).", "Bourbon provides the most benefit for the linear dataset because it has the smallest number of segments (one per model); with fewer segments, fewer searches are needed to find the target line segment.", "From  REF (b), we observe that latencies increase with the number of segments (e.g., latency of seg-1% is greater than that of linear).", "We cannot compare the number of segments in AR and OSM with others because the size of these datasets is significantly different.", "Level learning.", "Given that level learning is suitable for read-only scenarios, we configure Bourbon to use level learning and analyze its performance.", "As shown in Figure REF (a), Bourbon-level is 1.33$\\times $ – 1.92$\\times $ faster than the baseline.", "Bourbon-level offers more benefits than Bourbon because a level-model lookup is faster than finding the candidate sstables and then doing a file-model lookup.", "This confirms that Bourbon-level is an attractive option for read-only scenarios.", "However, since level models only provide benefits for read-only workloads and give at most 10% improvement compared to file models, we focus on Bourbon with file learning for our remaining experiments.", "Figure: Load Orders.", "(a) shows the performance for AR and OSM datasets for sequential (seq) and random (rand) load orders.", "(b) compares the speedup of positive and negative internal lookups." ], [ "Load Orders", "We now explore how the order in which the data is loaded affects performance.", "For this experiment, we use the AR and OSM datasets and load them in two ways: sequential (keys are inserted in ascending order) and random (keys are inserted in an uniformly random order).", "With sequential loading, sstables do not have overlapping key ranges even across levels; whereas, with random loading, sstables at one level can overlap with sstables at other levels.", "Figure REF shows the result.", "First, regardless of the load order, Bourbon offers significant benefit over baseline (1.47$\\times $ – 1.61$\\times $ ).", "Second, the average lookup latencies increase in the randomly-loaded case compared to the sequential case (e.g., 6$\\mu $ s vs. 4$\\mu $ s in WiscKey for the AR dataset).", "This is because while there are no negative internal lookups in the sequential case, there are many (23M) negative lookups in the random case (as shown in  REF (b)).", "Thus, with random load, the total number of internal lookups increases by 3$\\times $ , increasing lookup latencies.", "Next, we note that the speedup over baseline in the random case is less than that of the sequential case (e.g., 1.47$\\times $ vs. 1.61$\\times $ for AR).", "Although Bourbon optimizes both positive and negative internal lookups, the gain for negative lookups is smaller (as shown in  REF (b)).", "This is because most negative lookups in the baseline and Bourbon end just after the filter is queried (filter indicates absence); the data block is not loaded or searched.", "Given there are more negative than positive lookups, Bourbon offers less speedup than the sequential case.", "However, this speedup is still significant (1.47$\\times $ ).", "Figure: Request Distributions.", "The figure shows the average lookup latencies of different request distributions from AR and OSM datasets." ], [ "Request Distributions", "Next, we analyze how request distributions affect Bourbon's performance.", "We measure the lookup latencies under six request distributions: sequential, zipfian, hotspot, exponential, uniform, and latest.", "We first randomly load the AR and OSM datasets and then run the workloads; thus, the data can be segmented and there can be many negative internal lookups.", "As shown in Figure REF , Bourbon makes lookups faster by 1.54$\\times $ – 1.76$\\times $ than the baseline.", "Overall, Bourbon reduces latencies regardless of request distributions.", "Read-only performance summary.", "When the models are already built and when there are no writes, Bourbon provides significant speedup over baseline for a variety of datasets, load orders, and request distributions.", "Figure: Range Queries.", "BlackThe figure shows the normalized throughput of range queries with different range lengths from AR and OSM datasets.Figure: Mixed Workloads.", "(a) compares the foreground times of WiscKey, Bourbon-offline (offline), Bourbon-always (always), and Bourbon-cba (cba); (b) and (c) compare the learning time and total time, respectively; (d) shows the fraction of internal lookups that take the baseline path.Figure: Macrobenchmark-YCSB.", "The figure compares the throughput of Bourbon against WiscKey for Blacksix YCSB workloads across three datasets.BlackWe next analyze how Bourbon performs on range queries.", "We perform 1M range queries on the AR and OSM datasets with various range lengths.", "Figure REF shows the throughput of Bourbon normalized to that of WiscKey.", "With short ranges, where the indexing cost (i.e., the cost to locate the first key of the range) is dominant, Bourbon offers the most benefit.", "For example, with a range length of 1 on the AR dataset, Bourbon is 1.90$\\times $ faster than WiscKey.", "The gains drop as the range length increases; for example, Bourbon is only 1.15$\\times $ faster with queries that return 100 items.", "This is because, while Bourbon can accelerate the indexing portion, it follows a similar path as WiscKey to scan subsequent keys.", "Thus, with large range lengths, indexing accounts for less of the total performance, resulting in lower gains." ], [ "Efficacy of Cost-benefit Analyzer with Writes", "We next analyze how Bourbon performs in the presence of writes.", "Writes modify the data and so the models must be re-learned.", "In such cases, the efficacy of Bourbon's cost-benefit analyzer (cba) is critical.", "We thus compare Bourbon's cba against two strategies: Bourbon-offline and Bourbon-always.", "Bourbon-offline performs no learning as writes happen; models exist only for the initially loaded data.", "Bourbon-always re-learns the data as writes happen; it always decides to learn a file without considering cost.", "Bourbon-cba re-learns as well, but it uses the cost-benefit analysis to decide whether or not to learn a file.", "We run a workload that issues 50M operations with varying percentages of writes on the AR dataset.", "To calculate the total amount of work performed for each workload, we sum together the time spent on the foreground lookups and inserts (Figure REF (a)), the time spent learning (REF (b)), and the time spent on compaction (not shown); the total amount of work is shown in Figure REF (c).", "The figure also shows the fraction of internal lookups that take the baseline path (REF (d)).", "First, as shown in REF (a), all Bourbon variants reduce the workload time compared to WiscKey.", "The gains are lower with more writes because Bourbon has fewer lookups to optimize.", "Next, Bourbon-offline performs worse than Bourbon-always and Bourbon-cba.", "Even with just 1% writes, a significant fraction of internal lookups take the baseline path in Bourbon-offline as shown in REF (d); this shows re-learning as data changes is crucial.", "Bourbon-always learns aggressively and thus almost no lookups take the baseline path even for 50% writes.", "As a result, Bourbon-always has the lowest foreground time.", "However, this comes at the cost of increased learning time; for example, with 50% writes, Bourbon-always spends about 134 seconds learning.", "Thus, the total time spent increases with more writes for Bourbon-always and is even higher than baseline WiscKey as shown in REF (c).", "Thus, aggressively learning is not ideal.", "Given a low percentage of writes, Bourbon-cba decides to learn almost all the files, and thus matches the characteristics of Bourbon-always: both have a similar fraction of lookups taking the baseline path, both require the same time learning, and both perform the same amount of work.", "With a high percentage of writes, Bourbon-cba chooses not to learn many files, reducing learning time; for example, with 50% writes, Bourbon-cba spends only 13.9 seconds in learning (10$\\times $ lower than Bourbon-always).", "Consequently, many lookups take the baseline path.", "Bourbon-cba takes this action because there is less benefit to learning as the data is changing rapidly and there are fewer lookups.", "Thus, it almost matches the foreground time of Bourbon-always.", "But, by avoiding learning, the total work done by Bourbon-cba is significantly lower.", "Summary.", "Aggressive learning offers fast lookups but with high costs; no re-learning provides little speedup.", "Neither is ideal.", "In contrast, Bourbon provides high benefits similar to aggressive learning while lowering total cost significantly." ], [ "Real Macrobenchmarks", "We next analyze how Bourbon performs under two real benchmarks: YCSB [10] and SOSD [28]." ], [ "YCSB", "We use Blacksix workloads that have different read-write ratios and access patterns: A (w:50%, r:50%), B (w:5%, r:95%), C (read-only), D (read latest, w:5%, r:95%), BlackE (range-heavy, w:5%, range:95%), F (read-modify-write:50%, r:50%).", "We use three datasets: YCSB's default dataset (created using ycsb-load [3]), AR, and OSM, and load them in a random order.", "Figure REF shows the results.", "For the read-only workload (YCSB-C), all operations benefit and Bourbon offers the most gains (about 1.6$\\times $ ).", "For read-heavy workloads (YCSB-B and D), most operations benefit, while writes are not improved and thus Bourbon is 1.24$\\times $ – 1.44$\\times $ faster than the baseline.", "For write-heavy workloads (YCSB-A and F), Bourbon improves performance only a little (1.06$\\times $ – 1.18$\\times $ ).", "First, a large fraction of operations are writes; second, the number of the internal lookups taking the model path decreases (by about 30% compared to the read-heavy workload because Bourbon chooses not to learn some files).", "BlackYCSB-E consists of range queries (range lengths varying from 1 to 100) and 5% writes.", "Bourbon reaches 1.16$\\times $ – 1.19$\\times $ gain.", "In summary, as expected, Bourbon improves the performance of read operations; at the same time, Bourbon does not affect the performance of writes." ], [ "SOSD", "We next measure Bourbon's performance on the SOSD benchmark designed for learned indexes [28].", "We use the following six datasets: book sale popularity (amzn32), Facebook user ids (face32), lognormally (logn32) and normally (norm32) distributed datasets, uniformly distributed dense (uden32) and sparse (uspr32) integers.", "Figure REF shows the average lookup latency.", "As shown, Bourbon is about 1.48$\\times $ – 1.74$\\times $ faster than the baseline for all datasets.", "Figure: Macrobenchmark-SOSD.", "BlackThe figure compares lookup latencies from the SOSD benchmark.", "The numbers on the top show Bourbon's improvements over the baseline.Table: Performance on Fast Storage.", "The table shows Bourbon's lookup latencies when the data is stored on an Optane SSD.Figure: Mixed Workloads on Fast Storage.", "BlackThe figure compares the throughput of Bourbon against WiscKey for four read-write mixed YCSB workloads.", "We use the YCSB default dataset for this experiment." ], [ "Performance on Fast Storage", "Our analyses so far focused on the case where the data resides in memory.", "We now analyze if Bourbon will offer benefit when the data resides on a fast storage device.", "We run a read-only workload on sequentially loaded AR and OSM datasets on an Intel Optane SSD.", "BlackTable REF shows the result.", "Even when the data is present on a storage device, Bourbon offers benefit (1.25$\\times $ – 1.28$\\times $ faster lookups than WiscKey).", "BlackFigure REF shows the result for read-write mixed YCSB workloads on the same device with the default YCSB datasest.", "As expected, while Bourbon's benefits are marginal for write-heavy workloads (YCSB-A and YCSB-F), it offers considerable speedup (1.16$\\times $ – 1.19$\\times $ ) for read-heavy workloads (YCSB-B and YCSB-D).", "With the emerging storage technologies (e.g., 3D XPoint memory), Bourbon will offer even more benefits." ], [ "Performance with Limited Memory", "BlackWe further show that, even with no fast storage and limited available memory, Bourbon can still offer benefit with skewed workloads, such as zipfian.", "We experiment on a machine with a SATA SSD and memory that only holds about 25% of the database.", "We run a uniform random workload, and a zipfian workload with consecutive hotspots where 80% of the requests access about 25% of the database.", "Table REF shows the result.", "With the uniform workload, Bourbon is only 1.04$\\times $ faster because most of the time is spent loading the data into the memory.", "With the zipfian workload, in contrast, indexing time instead of data-access time dominates because a large number of requests access the small portion of data that is already cached in memory.", "Bourbon is able to reduce this significant indexing time and thus offers 1.25$\\times $ lower latencies." ], [ "Error Bound and Space Overheads", "We finally discuss the characteristics of Bourbon's ML model, specifically its error bound ($\\delta $ ) and space overheads.", "Figure REF (a) plots the error bound ($\\delta $ ) against the average lookup latency (left y-axis) for AR dataset.", "As $\\delta $ increases, fewer line segments are created, leading to fewer searches, thus reducing latency.", "However, beyond $\\delta = 8$ , although the time to find the segment reduces, the time to search within a segment increases, Blackthus increasing latency.", "We find that Bourbon's choice of $\\delta = 8$ is optimal for other datasets too.", "BlackFigure REF (a) also shows how space overheads (right y-axis) vary with $\\delta $ .", "As $\\delta $ increases, fewer line segments are created, leading to low space overheads.", "Table REF (b) shows the space overheads for different datasets.", "As shown, for a variety of datasets, the overhead compared to the total dataset size is little (0% – 2%)." ], [ "Related Work", "Learned indexes.", "The core idea of our work, replacing indexing structures with ML models, is inspired from the pioneering work on learned indexes [31].", "However, learned indexes do not support updates, an essential operation that an storage-system index must support.", "Recent research tries to address this limitation.", "For instance, XIndex [45], FITing-Tree [20], and AIDEL [35] support writes using an additional array (delta index) and with periodic re-training, whereas Alex [13] uses gapped array at the leaf nodes of a B-tree to support writes.", "Figure: Error-bound Tradeoffs and Space Overheads.", "(a) shows how the PLR error bound affects lookup latency and memory overheads; (b) shows the space overheads for different datasets.Most prior efforts optimize B- tree variants, while our work is the first to deeply focus on LSMs.", "Further, while most prior efforts implement learned indexes to stand-alone data structures, our work is the first to show how learning can be integrated and implemented into an existing, optimized, production-quality system.", "While SageDB [30] is a full database system that uses learned components, it is built from scratch with learning in mind.", "Our work, in contrast, shows how learning can be integrated into an existing, practical system.", "Finally, instead of “fixing” new read-optimized learned index structures to handle writes (like previous work), we incorporate learning into an already write-optimized, production-quality LSM.", "LSM optimizations.", "Prior work has built many LSM optimizations.", "Monkey [11] carefully adjusts the bloom filter allocations for better filter hit rates and memory utilization.", "Dostoevsky [12], HyperLevelDB [16], and bLSM [42] develop optimized compaction policies to achieve lower write amplification and latency.", "cLSM [23] and RocksDB [18] use non-blocking synchronization to increase parallelism.", "We take a different yet complimentary approach to LSM optimization by incorporating models as auxiliary index structures to improve lookup latency, but each of the others are orthogonal and compatible to our core design.", "Model choices.", "Duvignau et al.", "[14] compare a variety of piecewise linear regression algorithms.", "Greedy-PLR, which we utilize, is a good choice to realize fast lookups, low learning time, and small memory overheads.", "Neural networks are also widely used to approximate data distributions, especially datasets with complex non-linear structures [34].", "However, theoretical analysis [36] and experiments [43] show that training a complex neural network can be prohibitively expensive.", "Similar to Greedy-PLR, recent work proposes a one-pass learning algorithm based on splines [29] and identifies that such an algorithm could be useful for learning sorted data in LSMs; we leave their exploration within LSMs for future work." ], [ "Conclusion", "In this paper, we examine if learned indexes are suitable for write-optimized log-structured merge (LSM) trees.", "Through in-depth measurements and analysis, we derive a set of guidelines to integrate learned indexes into LSMs.", "Using these guidelines, we design and build Bourbon, a learned-index implementation for a highly-optimized LSM system.", "We experimentally demonstrate that Bourbon offers significantly faster lookups for a range of workloads and datasets.", "BlackBourbon is an initial work on integrating learned indexes into an LSM-based storage system.", "More detailed studies, such as more sophisticated cost-benefit analysis, general string support, and different model choices, could be promising for future work.", "In addition, we believe that Bourbon's learning approach may work well in other write-optimized data structures such as the $B^{\\epsilon }$ -tree [6] and could be an interesting avenue for future work.", "While our work takes initial steps towards integrating learning into production-quality systems, more studies and experience are needed to understand the true utility of learning approaches." ], [ "Acknowledgements", "BlackWe thank Alexandra Fedorova (our shepherd) and the anonymous reviewers of OSDI '20 for their insightful comments and suggestions.", "We thank the members of ADSL for their excellent feedback.", "We also thank CloudLab [41] for providing a great environment to run our experiments and reproduce our results during artifact evaluation.", "This material was supported by funding from NSF grants CNS-1421033, CNS-1763810 and CNS-1838733, Intel, Microsoft, Seagate, and VMware.", "Aishwarya Ganesan is supported by a Facebook fellowship.", "Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and may not reflect the views of NSF or any other institutions." ] ]

[ [ "Order 5 Brauer-Manin obstructions to the integral Hasse principle on log\n K3 surfaces" ], [ "Abstract We construct families of log K3 surfaces and study the arithmetic of their members.", "We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle." ], [ "Introduction", "The goal of this paper is to add to the study of integral points on ample log K3 surfaces as started by Harpaz [16].", "This is done by first giving a geometrically flavoured construction for such equations.", "One upshot of this construction is that one even gets a family of such surfaces for which the arithmetic properties of the members can be studied simultaneously.", "We will present two families of log K3 surfaces for which a positive proportion of the fibres fails the Hasse principle, i.e.", "it is everywhere locally soluble but it does not admit integral points.", "For the geometrically similar K3 surfaces it has been conjectured by Skorobogatov that the Brauer–Manin obstruction is the only one to the Hasse principle [27].", "This is however not the case for log K3 surface [16].", "The examples in this paper exhibit new arithmetic behaviour and it is hoped that the accompanying ideas for studying log K3 surfaces will contribute to a workable conjecture for integral points on log K3 surfaces.", "We will use the integral Brauer–Manin obstruction as introduced by Colliot-Thélène and Xu [9] to prove the failure of the integral Hasse principle, which is based on the Brauer–Manin obstruction by Manin [24].", "Let $\\mathcal {U}/\\mathbb {Z}$ be a model of a variety $U/\\mathbb {Q}$ for which we want to prove that $\\mathcal {U}(\\mathbb {Z})=\\emptyset $ .", "The technique uses an element $\\mathcal {A} \\in \\operatorname{Br}U := \\mathrm {H}^2(U,\\mathbb {G}_m)$ to define an intermediate set $\\mathcal {U}(\\mathbb {Z}) \\subseteq \\mathcal {U}(\\mathbb {A}_{\\mathbb {Q}, \\infty })^{\\mathcal {A}} \\subseteq \\mathcal {U}(\\mathbb {A}_{\\mathbb {Q}, \\infty })$ in the inclusion of integral points of $\\mathcal {U}$ in its set of integral adelic points, i.e.", "$\\mathbb {A}_{\\mathbb {Q},\\infty }=\\mathbb {R} \\times \\prod _p \\mathbb {Z}_p$ .", "Hence if $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q}, \\infty })$ is non-empty but $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q}, \\infty })^{\\mathcal {A}}$ is empty, then $\\mathcal {A}$ obstructs the integral Hasse principle on $\\mathcal {U}$ .", "The order of the obstruction is the order of $\\mathcal {A}$ in $\\operatorname{Br}U/\\operatorname{Br}\\mathbb {Q}$ .", "We will be mainly interested in obstructions coming from elements in the algebraic Brauer group $\\operatorname{Br}_1 U := \\ker \\left( \\operatorname{Br}U \\rightarrow \\operatorname{Br}\\bar{U}\\right) \\subseteq \\operatorname{Br}U$ .", "In this paper we have restricted to a specific type of log K3 surface to showcase our ideas.", "In passing we pick up the first Brauer–Manin obstructions of higher order.", "The existence of a high order element in the Brauer group of our log K3 surfaces depends on the splitting field of a related del Pezzo surface.", "Recall that the splitting field of a del Pezzo surface is the minimal field over which all its $-1$ -curves are defined.", "Theorem 0.1 (Theorem REF ) Let $U=X\\backslash C$ be a log K3 surface over $\\mathbb {Q}$ with $X$ a del Pezzo surface of degree 5 and $C$ a geometrically irreducible anticanonical divisor.", "We have $\\operatorname{Br}_1 U /\\operatorname{Br}\\mathbb {Q} \\cong {\\left\\lbrace \\begin{array}{ll}\\mathbb {Z}/5\\mathbb {Z} & \\mbox{ if the splitting field of $X$ is a cyclic extension $K/\\mathbb {Q}$ of degree $5$;}\\\\0 & \\mbox{ otherwise.}\\end{array}\\right.", "}$ Also, each cyclic extension $K/\\mathbb {Q}$ of degree 5 is the splitting field of a del Pezzo surface over $\\mathbb {Q}$ .", "Such a surface is unique up to isomorphism.", "We will consider del Pezzo surfaces with a non-trivial algebraic Brauer group and our first explicit example comes from the quintic extension $\\mathbb {Q}(\\zeta _{11}+\\zeta ^{-1}_{11})/\\mathbb {Q}$ .", "Consider the projective scheme $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ given by the five quadratic forms $u_0u_3+22u_0u_4+121u_0u_5-u_1^2-121u_1u_3+2662u_1u_4-36355u_2u_4-\\\\9306u_2u_5+10494u_3u_4-242u_3u_5-215501u_4^2+68123u_4u_5-13794u_5^2,$ $u_0u_4+11u_0u_5-u_1u_2-11u_1u_3+242u_1u_4-3223u_2u_4-847u_2u_5+\\\\902u_3u_4-11u_3u_5-19272u_4^2+6413u_4u_5-1331u_5^2,$ $u_0u_5-u_1u_3+22u_1u_4-u_2^2-286u_2u_4-77u_2u_5+77u_3u_4-1694u_4^2+572u_4u_5-121u_5^2,$ $u_1u_4-u_2u_3-11u_2u_4-77u_4^2+55u_4u_5-11u_5^2,\\\\$ $u_1u_5-u_2u_4-11u_2u_5-u_3^2+11u_3u_4-44u_4^2.\\\\$ This scheme is constructed and studied in Section 3.", "In Section 4 we prove the arithmetic properties stated in the following theorem.", "Theorem 0.2 For each geometrically irreducible hyperplane section $\\mathcal {C}_h := \\lbrace h=0\\rbrace \\cap \\mathcal {X}$ we define $\\mathcal {U}_h =\\mathcal {X}\\backslash \\mathcal {C}_h$ .", "The scheme $\\mathcal {X}/\\mathbb {Z}$ is a flat proper model of the del Pezzo surface over $\\mathbb {Q}$ which splits over the quintic extension $\\mathbb {Q}(\\zeta _{11}+\\zeta ^{-1}_{11})$ .", "The existence of an algebraic Brauer–Manin obstruction to the integral Hasse principle on $\\mathcal {U}_h$ only depends on the reduction of $h$ modulo 11.", "There exists an $h$ , and hence even a residue class $h \\bmod 11$ , for which $\\mathcal {U}_h$ has an order 5 obstruction to the integral Hasse principle.", "The same construction can be used to produce many more examples.", "The arithmetic behaviour is mainly determined by the primes which are ramified in the splitting field $K$ .", "Any tamely ramified prime can be studied in a similar matter.", "For completeness we also add an example in Section 5 involving the wildly ramified prime 5.", "Theorem 0.3 There exists a scheme $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ with the following properties.", "The scheme $\\mathcal {X}$ is a flat model for the del Pezzo surface $X=\\mathcal {X}_{\\mathbb {Q}}$ over $\\mathbb {Q}$ which splits over the unique quintic number field $K \\subseteq \\mathbb {Q}(\\zeta _5)$ .", "The existence of an algebraic Brauer–Manin obstruction on $\\mathcal {U}_h := \\mathcal {X}\\backslash \\lbrace h=0\\rbrace $ to the integral Hasse principle only depends on the reduction of $h$ modulo 25.", "There exists an $h$ , and hence even a residue class $h \\bmod 25$ , for which $\\mathcal {U}_h$ has an order 5 obstruction to the integral Hasse principle.", "Let us put these results in context.", "Integral points on log K3 are believed to behave to a certain degree in a similar way as rational points on K3 surfaces.", "For those surfaces it has been conjectured by Skorobogatov [27] that the existence of solutions are completely controlled by the Brauer–Manin obstruction.", "However, results by Ieronymou and Skorobogatov [18] and Skorobogatov and Zarhin [29] say that there can not be an odd order obstruction to the Hasse principle for smooth diagonal quartic surfaces and Kummer varieties.", "An algebraic obstruction of order 3 on a K3 surface was found in [10] and Berg and Várilly-Alvarado [2] even produced a transcendental cubic obstruction.", "For log K3 surfaces the situation is however different; it was proven that the Brauer–Manin obstruction is not the only obstruction to the integral Hasse principle [16] and [19].", "On the other hand, Colliot-Thélène and Wittenberg [8] showed that the Brauer group never obstructs the Hasse principle for the equation $x^3+y^3+z^3=n$ which is in line with the conjecture that this equation has an integral solution for $n \\lnot \\equiv \\pm 4\\mod {9}$ .", "The Hasse principle and the effectivity of the Brauer–Manin obstruction for the equation $x^3+y^3+z^3-xyz=k$ were studied by Ghosh and Sarnak [13], Colliot-Thélène, Wei and Xu [7], and Loughran and Mitankin [21].", "Another classical affine cubic equation was studied in this manner by Bright and Loughran [5].", "This paper gives the first examples of higher odd order Brauer–Manin obstructions on any type of scheme; all other known examples of the Brauer–Manin obstruction to the (integral) Hasse principle are of either order 2 or 3.", "This is the highest possible prime order for such an obstruction on log K3 surfaces; for a generic anticanonical divisor $C$ on a del Pezzo surface $X$ the order of algebraic Brauer groups of $X \\backslash C$ is only divisible by the primes 2, 3 and 5, see Table 1 in [6].", "Their results also show that the quintic algebraic obstruction described in this paper are particular to the degree 5 case; there is an inclusion $\\operatorname{Br}X \\hookrightarrow \\operatorname{Br}_1 U$ whose cokernel is divisible by the degree of the del Pezzo surface $X$ .", "Hence $\\operatorname{Br}_1 U/\\operatorname{Br}X$ only has 5-torsion if $X$ is a quintic del Pezzo surface.", "The novel approach in this paper is to study affine surfaces $U$ in families by fixing the compactification $X$ and letting the complementary divisor $C$ vary.", "An understanding of the arithmetic and geometry of $X$ will be helpful in studying the open surfaces $U$ .", "We propose a general methodology for studying this setup, which we illustrated in the special setting of del Pezzo surfaces of degree 5.", "An important result for these surfaces is that they always have a point.", "This is a classic result by Enriques [12] which was also proved by Swinnerton-Dyer [30], Skorobogatov [28] and many others.", "This proves that $X$ is rational over $k$ , $\\operatorname{Br}X/\\operatorname{Br}k= 0$ and that $X$ satisfies weak approximation.", "It also allows one to classify and to construct such surfaces over $k$ , which was done in detail in [14].", "We produce models $\\mathcal {X}/\\mathbb {Z}$ for $X/\\mathbb {Q}$ by following this construction over the integers.", "In this process one has a few more choices along the way which allow one to control the reductions $\\mathcal {X}_{\\mathfrak {p}}$ for all primes $\\mathfrak {p}$ .", "To finally construct a model of a log K3 surface one considers the complement $\\mathcal {U}_h$ of a hyperplane section $\\lbrace h=0\\rbrace $ in $\\mathcal {X}$ .", "Using the abundance of points on quintic del Pezzo surfaces we can deduce that for any $h$ the open subscheme $\\mathcal {U}_h$ has points over all $\\mathbb {Z}_\\ell $ , except possibly for a very few small primes $\\ell $ .", "In our cases only local solubility at $\\ell =2$ is not immediate and will depend on $h$ .", "We also use the geometric and arithmetic properties of quintic del Pezzo surfaces to compute the Brauer–Manin obstruction on each $\\mathcal {U}_h$ .", "We show that the invariant maps are identically 0 for all but an explicit finite list of primes.", "To effectively deal with a remaining prime $p$ we show that it is enough to only study the closed fibre of $\\mathcal {X} \\times \\mathbb {Z}_p$ ; a surprising result especially for the wildly ramified prime $p=5$ .", "We end up with examples of quintic Brauer–Manin obstructions to both the Hasse principle and strong approximation.", "There is no reason why this construction only works for affine opens of quintic del Pezzo surfaces; one could use a similar construction to produce models of rational varieties while controlling the arithmetic of the individual fibres." ], [ "Outline", "We start by recalling some necessary facts on del Pezzo and log K3 surfaces, Brauer groups and the Brauer–Manin obstruction.", "Then we compute the algebraic Brauer group of log K3 surfaces $U=X\\backslash C$ where $X$ is a del Pezzo surface of degree 5 and $C$ is an anticanonical divisor.", "In the third section we give an example of a construction of a model $\\mathcal {X}/\\mathbb {Z}$ of a quintic del Pezzo surface $X/\\mathbb {Q}$ such that any anticanonical complement $U_h=X\\backslash \\lbrace h=0\\rbrace $ has an element of order 5 in the Brauer group.", "The next section is devoted to the arithmetic of each $\\mathcal {U}_h$ .", "In particular we compute the Brauer–Manin obstruction coming from the element of order 5.", "In the last section we use the same construction to produce a different family of log K3 surface $\\mathcal {U}_h$ for which the arithmetic behaves differently, but there still is an element of order 5 in the Brauer group." ], [ "Notation and conventions", "Let $k$ be a field.", "We will write $\\bar{k}$ for a fixed algebraic closure and $k^{\\textup {sep}}$ for the separable closure of $k$ in $\\bar{k}$ .", "The absolute Galois group of a field $k$ is denoted by $G_k = \\operatorname{Gal}(k^{\\textup {sep}}/k)$ .", "A variety over a field $k$ is a separated scheme of finite type over $\\operatorname{Spec}k$ .", "A curve over a field $k$ is a variety over $k$ of pure dimension 1, it need not be irreducible, reduced or smooth.", "A surface over a field $k$ is a geometrically integral variety of dimension 2 over $k$ .", "A curve on a surface over a field $k$ is a closed subscheme of the surface which is a curve over $k$ .", "For a scheme $X$ over a field $k$ we will write $X_K$ for the base change $X \\times _k K$ for any field extension $K$ of $k$ .", "The notation $\\bar{X}$ will be synonymous for $X_{\\bar{k}}$ ." ], [ "Acknowledgements", "Most results in these article were obtained during my Ph.D. studies at Leiden University.", "I would like to thank my supervisor Martin Bright for his help and our many discussions on the subject.", "This paper was completed as part of a project which received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.", "754411." ], [ "Preliminaries", "We will consider the existence of integral solutions to polynomial equations defining surfaces.", "We will first collect the key notions and results on the surfaces we will encounter.", "Then we will recall the necessary results on Brauer groups and the Brauer–Manin obstruction to integral points." ], [ "Del Pezzo surface of degree 5", "We will review some facts about del Pezzo surfaces.", "The main references will be [25] and [11].", "For an overview of the arithmetic of such surfaces one is referred to [31].", "Definition 1.1 Let $k$ be a field.", "A del Pezzo surface is a smooth projective surface $X$ over $k$ such that the anticanonical line bundle $\\omega _X^{-1}$ is ample.", "The degree of a del Pezzo surface is the anticanonical self-intersection $K_X^2$ .", "We will only need del Pezzo surfaces of degree 5.", "In which case $\\omega _X^{-1}$ is even very ample over $k$ and we see that every del Pezzo surface of degree 5 can be embedded as a degree 5 surface in $\\mathbb {P}^5_k$ .", "Let us collect some facts on the geometry of these surfaces.", "Lemma 1.2 Let $X$ be a del Pezzo surface of degree 5 over a separably closed field $k$ .", "The Picard group $\\operatorname{Pic}X$ is free of rank 5 and it has an orthogonal basis $L_0, L_1, \\ldots , L_4$ with respect to intersection pairing which satisfies $L_0^2=1$ and $L_i^2=-1$ for $i\\ne 0$ .", "In any such basis the canonical class is given by $K_X=-3L_0+L_1+L_2+L_3+L_4$ .", "There are precisely ten classes $D \\in \\operatorname{Pic}X$ which satisfy $D^2=D\\cdot K_X=-1$ , namely $L_i$ for $i\\ne 0$ and $L_{ij}:=L_0-L_1-L_2$ for $0<i<j\\le 4$ .", "Each of these classes contain a unique curve, and these curves are smooth, irreducible and have genus 0.", "The intersection graph of these ten $-1$ -curves is the so-called Petersen graph shown in Figure REF .", "Figure: The intersection graph of -1-1-curves on a del Pezzo surface of degree 5.", "Let $A(X)$ be the group of automorphisms of $\\operatorname{Pic}X$ which preserve the intersection pairing and the canonical class.", "Then $A(X)$ is isomorphic to $S_5$ and this isomorphism is unique up to conjugation.", "The first result follows from the fact that a del Pezzo surfaces of degree 5 is geometrically the blowup of the projective plane in 4 points, no three of which lie on a line, see for example [11].", "From here one can deduce the remaining statements.", "We do draw attention to a particularly nice proof of the last statement.", "Note that $A(X)$ permutes the $-1$ -classes and that these classes generate the Picard group.", "So $A(X)$ is a subgroup of automorphism group of the Petersen graph.", "To compute this automorphism group we identify the vertices with $\\binom{I}{2}$ for $I=\\lbrace 1,2,3,4,5\\rbrace $ such that $\\left\\lbrace \\lbrace i,j\\rbrace ,\\lbrace k,l\\rbrace \\right\\rbrace $ is an edge precisely if $i$ , $j$ , $k$ and $l$ are distinct.", "This makes it straight forward to show that the automorphism group of the Petersen graph is isomorphic to $S_5$ .", "Next one checks that each automorphism of the graph is actually an automorphism of the whole Picard group.", "These isomorphisms are all unique up to conjugacy since $S_5$ is an inner group.", "Let us now switch to del Pezzo surfaces of degree 5 over general fields.", "The following proposition shows that the geometric Picard group as a Galois module is a principal invariant.", "Proposition 1.3 Let $k$ be a field.", "There is a bijection between isomorphism classes of del Pezzo surfaces of degree 5 over $k$ and conjugacy classes of actions of $G_k$ on $\\operatorname{Pic}X^{\\textup {sep}}$ .", "This is Lemma 14 in [14].", "Lemma REF and Proposition REF together show that there is a bijection between quintic del Pezzo surface over $k$ and group homomorphisms $G_k \\rightarrow S_5$ up to conjugacy.", "We will describe how to construct a del Pezzo surface from such a group homomorphism as was done in [14].", "Proposition 1.4 Let $k$ be a field with absolute Galois group $G_k$ and let $\\Lambda $ be the effective generator of $\\operatorname{Pic}\\mathbb {P}^2_k$ .", "Consider a group homomorphism $\\psi \\colon G_k \\rightarrow S_5$ .", "Fix five points $P_i \\in \\mathbb {P}^2_k(k^{\\textup {sep}})$ such that no three lie on a line and that $G_k$ acts on these points as $S_5$ acts on its indexes.", "The linear system $\\mathcal {L}=\\left| \\mathcal {O}_{\\mathbb {P}^2_k}(5\\Lambda -2P_1-2P_2-2P_3-2P_4-2P_5) \\right|$ has dimension 5.", "The image of the associated map is a del Pezzo $X$ surface of degree 5.", "The isomorphism class of $X$ only depends on the conjugacy class of $\\psi $ and is independent of the choice of $P_i$ .", "The composition $G_k \\rightarrow A(X^{\\textup {sep}}) \\xrightarrow{} S_5$ recovers $\\psi $ up to conjugacy.", "The first two statements are Theorem 5 in [14].", "It also shows that the $-1$ -curves on $X$ correspond to the lines on $\\mathbb {P}^2_{k^{\\textup {sep}}}$ passing through two of the points $P_i$ .", "This shows that the action of Galois on the $-1$ -curves on $X$ and hence on $\\operatorname{Pic}X^{\\textup {sep}}$ equals $\\psi $ up to conjugacy." ], [ "Log K3 surfaces of 5 type", "For our interest in integral points we move to surfaces which are not necessarily projective.", "The following class will be important.", "Definition 1.5 Let $U$ be a smooth surface over a field $k$ .", "A log K3 structure on $U$ is a triple $(X,C,i)$ consisting of a proper smooth surface $X$ over $k$ , an effective anticanonical divisor $C$ on $X$ with simple normal crossings and an open embedding $i\\colon U \\rightarrow X$ , such that $i$ induces an isomorphism between $U$ and $X\\backslash C$ .", "A log K3 surface is a simply connected, smooth surface $U$ over $k$ together with a choice of log K3 structure $(X,C,i)$ on $U$ .", "Let $X$ be a del Pezzo surface of degree 5 and let $C$ be an effective anticanonical divisor on $X$ .", "The affine surface $U=X\\backslash C$ is called a log K3 surface of 5 type.", "Whenever we consider such a surface $U$ without explicitly specifying $X$ we will assume the choice of compactification to be understood from context." ], [ "Brauer groups", "Let $U$ be a scheme over a field $k$ .", "We will need the concept of the Brauer group $\\operatorname{Br}U$ of $U$ .", "Two common definitions are the étale cohomology group $\\operatorname{Br}U := \\mathrm {H}^2(U,\\mathbb {G}_m)$ and the group $\\operatorname{Br}_{\\textup {Az}}U$ of equivalence classes of Azumaya algebras on $U$ .", "There is a natural morphism $\\operatorname{Br}_{\\textup {Az}}U \\rightarrow \\operatorname{Br}U$ which induces an isomorphism between $\\operatorname{Br}_{\\textup {Az}}U$ and $(\\operatorname{Br}U)_{\\textup {tors}}$ if $U$ is a quasi-projective scheme over $k$ by an unpublished result by Gabber.", "Another proof by De Jong can be found in [20].", "In Theorem 6.6.7 in [26] we find conditions for $\\operatorname{Br}U$ to be a torsion group and we conclude that we can identify both types of Brauer groups for regular integral schemes which are quasi-projective over a field.", "All the varieties for which we will consider the Brauer group will satisfy these conditions and we will pass freely between the two notions.", "Using the functoriality of associating the Brauer group to the scheme we can define the following filtration: $\\operatorname{Br}_0 U \\subseteq \\operatorname{Br}_1 U \\subseteq \\operatorname{Br}U$ , where the constant Brauer group $\\operatorname{Br}_0 U$ is defined as $\\mathrm {Im}(\\operatorname{Br}k \\rightarrow \\operatorname{Br}U)$ and the algebraic Brauer group $\\operatorname{Br}_1 U$ is $\\ker (\\operatorname{Br}U \\rightarrow \\operatorname{Br}U^{\\textup {sep}})$ .", "We will denote the quotient $\\operatorname{Br}_1 U/\\operatorname{Br}_0 U$ by $\\operatorname{Br}_1 U/\\operatorname{Br}k$ although the map $\\operatorname{Br}k \\rightarrow \\operatorname{Br}_1 U$ need not be injective.", "It follows from the Hochschild–Serre spectral sequence that $\\operatorname{Br}_1 U/\\operatorname{Br}k$ is isomorphic to $\\mathrm {H}^1(G_k, \\operatorname{Pic}U^{\\textup {sep}})$ in certain cases.", "This is well-known if $U$ is proper, see for example [26], but the proof actually works under the weaker condition $\\mathbb {G}_m(U^{\\textup {sep}})=k^{\\textup {sep},\\times }$ .", "If $U$ is an integral noetherian regular scheme over a field of characteristic 0 the natural map $\\operatorname{Br}U \\rightarrow \\operatorname{Br}\\kappa (U)$ is an inclusion [15].", "So in this case we can represent elements of the Brauer group by classes of central simple algebras over the field $\\kappa (U)$ .", "We will construct Azumaya algebras on $U$ as cyclic algebras over the function field.", "Definition 1.6 Let $\\kappa $ be a field and $n$ an integer not dividing the characteristic of $\\kappa $ .", "An Azumaya algebra in the image of the cup product $\\mathrm {H}^1(\\kappa ,\\mu _n) \\times \\mathrm {H}^1(\\kappa ,\\mathbb {Z}/n\\mathbb {Z}) \\rightarrow \\mathrm {H}^2(\\kappa ,\\mu _n)\\cong \\operatorname{Br}k[n]$ is called a cyclic algebra over $\\kappa $ .", "A cyclic extension $\\kappa ^{\\prime }/\\kappa $ of degree $n$ with a fixed generator $\\sigma \\in \\operatorname{Gal}(\\kappa ^{\\prime }/\\kappa )$ determines an element of $\\operatorname{Hom}(G_\\kappa ,\\mathbb {Z}/n\\mathbb {Z}) \\cong \\mathrm {H}^1(\\kappa ,\\mathbb {Z}/n\\mathbb {Z})$ by sending $\\sigma $ to 1.", "Any element $a \\in \\kappa ^\\times $ gives an element in $\\mathrm {H}^1(\\kappa ,\\mu _n) \\cong \\kappa ^\\times /(\\kappa ^\\times )^n$ .", "The cyclic algebra $a \\cup (\\kappa ^{\\prime }/\\kappa ,\\sigma )$ is denoted by $(a,\\kappa ^{\\prime },\\sigma )$ .", "For more details, see [26].", "Here one also finds the following important result.", "Lemma 1.7 A cyclic algebra $(a,\\kappa ^{\\prime },\\sigma )$ is trivial in $\\operatorname{Br}\\kappa $ precisely when $a \\in N_{\\kappa ^{\\prime }/\\kappa }(\\kappa ^{\\prime \\times })$ .", "To see if a cyclic algebra in $\\operatorname{Br}\\kappa (U)$ comes from $\\operatorname{Br}U$ we have the following lemma.", "Lemma 1.8 Consider a smooth and geometrically integral variety $U$ over a field $k$ satisfying $\\mathbb {G}_m(U^{\\textup {sep}})=k^{\\textup {sep},\\times }$ .", "Fix a finite cyclic extension $K/k$ , a generator $\\sigma \\in \\operatorname{Gal}(K/k)$ , and an element $g \\in \\kappa (U)^\\times $ .", "The cyclic algebra $\\mathcal {A} = (g, \\kappa (U_K)/\\kappa (U), \\sigma )$ lies in the image of $\\operatorname{Br}U \\rightarrow \\operatorname{Br}\\kappa (U)$ precisely if $\\mathrm {div}g = \\operatorname{Nm}_{K/k}(D)$ for some divisor $D$ on $U_K$ .", "If $k$ , and hence $K$ , is a number field, and $U$ is everywhere locally soluble then $\\mathcal {A}$ is constant exactly when $D$ can be taken to be principal.", "This lemma is similar to Proposition 4.17 from [3].", "The difference is that the projectivity assumption is replaced by the weaker condition $\\mathbb {G}_m(U^{\\textup {sep}})=k^{\\textup {sep},\\times }$ .", "One can check that under this assumption the proof presented in [3] is still valid." ], [ "Brauer–Manin obstruction", "In some cases elements of the Brauer group allow us to prove that there are no integral points on a scheme.", "Let $\\mathcal {U}/\\mathbb {Z}$ be a model of $U=\\mathcal {U}_{\\mathbb {Q}}$ .", "The Brauer–Manin set of $\\mathcal {A}$ is the subset of the integral adelic points $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty }) = U(\\mathbb {R}) \\times \\prod _\\ell \\mathcal {U}(\\mathbb {Z}_\\ell )$ defined by $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty })^\\mathcal {A} = \\left\\lbrace (P_\\ell ) \\in \\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty }) \\ \\Bigg |\\ \\sum _\\ell \\operatorname{inv}_\\ell \\mathcal {A}(P_\\ell ) = 0 \\right\\rbrace .$ Here the invariant maps $\\operatorname{inv}_\\ell $ are those defined in [26].", "Note that the infinite sum is well-defined by [26].", "The Brauer–Manin set is of particular interest because of the property described in the following theorem from [9].", "Lemma 1.9 Let $\\mathcal {U}$ be a scheme over the integers and let $U$ be the generic fibre over $\\mathbb {Q}$ .", "For any element $\\mathcal {A} \\in \\operatorname{Br}U$ we have the following chain of inclusions $\\mathcal {U}(\\mathbb {Z}) \\subseteq \\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty })^\\mathcal {A} \\subseteq \\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty }).$ When $\\mathcal {A}$ is a cyclic algebra we can use Lemma REF to compute the images of the invariant maps and we might gain some information on the set of integral points.", "Definition 1.10 We say that an element $\\mathcal {A} \\in \\operatorname{Br}U$ obstructs the integral Hasse principle if $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty })$ is non-empty, but $\\mathcal {U}(\\mathbb {A}_{\\mathbb {Q},\\infty })^\\mathcal {A}$ is empty.", "The order of the obstruction is the order of $\\mathcal {A}$ in $\\operatorname{Br}U/\\operatorname{Br}\\mathbb {Q}$ ." ], [ "The interesting Galois action", "The main goal will be to construct affine schemes $\\mathcal {U} \\subseteq \\mathbb {A}^5_{\\mathbb {Z}}$ which have a Brauer–Manin obstruction to the integral Hasse principle.", "In all our examples we will construct $\\mathcal {U}$ in such a way that $U=\\mathcal {U}_{\\mathbb {Q}}$ is a log K3 surface of 5 type.", "This means that we will be interested in the Brauer group of such surfaces.", "The following terminology will turn out to be helpful in that regard.", "Definition 2.1 Let $X$ be a del Pezzo surface of degree 5 over a field $k$ .", "Let $K$ be the minimal Galois extension of $k$ over which all $-1$ -curves on $X$ are defined.", "We say that $X$ is interesting if $[K \\colon k] = 5$ .", "A log K3 surface of 5 type $U=X\\backslash C$ is called interesting if $X$ is an interesting del Pezzo surface and $C$ is geometrically irreducible.", "The field $K$ is called the splitting field of the interesting surfaces $X$ and $U$ .", "Consider an interesting log K3 surface $U=X\\backslash C$ .", "By definition of a log K3 surface we see that $C$ is smooth.", "The curve $C$ is also geometrically irreducible since $U$ is interesting.", "The results in this paper are also true for the complement of a geometrically irreducible anticanonical curve $C$ on a del Pezzo surface $X$ of degree 5.", "To be able to use the language of log K3 surface we do keep the superfluous condition that $C$ is smooth.", "The following lemma shows that an interesting action corresponds to a unique conjugacy class of subgroups of $W_4$ .", "Lemma 2.2 Consider an interesting del Pezzo surface $X$ over a field $k$ .", "The action of $G_k$ on $\\operatorname{Pic}X^{\\textup {sep}}$ is uniquely determined up to conjugacy.", "On an interesting del Pezzo surface there are two Galois orbits of geometric $-1$ -curves, each of size 5.", "The sum of the $-1$ -curves in one such orbit is an anticanonical divisor.", "Let $K$ be the splitting field of $X$ .", "Since $X$ is interesting the extension $K/k$ is by definition of degree 5.", "It follows from the minimality of $K$ that $\\operatorname{Gal}(K/k)$ does not fix any of the ten $-1$ -curves, hence there must be two orbits of size 5.", "After choosing a possibly different basis of $\\operatorname{Pic}X^{\\textup {sep}}$ we see that these two orbits are the two regular pentagons in Figure REF and that there is a $\\sigma \\in \\operatorname{Gal}(K/k)$ which acts on the outer pentagon by rotating counter-clockwise.", "Since $\\sigma $ preserves the intersection pairing it will also rotate the inner pentagon counter-clockwise.", "This determines the action of $\\sigma $ on the $-1$ -classes: $L_1 \\mapsto L_{12} \\mapsto L_2 \\mapsto L_{23} \\mapsto L_{14} \\mapsto L_1,$ $L_3 \\mapsto L_4 \\mapsto L_{13} \\mapsto L_{34} \\mapsto L_{24} \\mapsto L_3.$ This proves that $L_0=L_{12}+L_1+L_2$ gets mapped to $2L_0-L_1-L_2-L_3$ .", "For a different choice of such a basis we get a conjugate action of $G_k$ on $\\operatorname{Pic}X^{\\textup {sep}}$ by Lemma REF .", "The last statement is immediate.", "If we consider the complement $U$ of a geometrically irreducible anticanonical divisor $C$ on a del Pezzo surface of degree 5 over a number field $k$ we can compute its algebraic Brauer group modulo constants as $\\mathrm {H}^1(G_k, \\operatorname{Pic}U^{\\textup {sep}})$ .", "The following proposition shows that the action of $G_k$ on $\\operatorname{Pic}X^{\\textup {sep}}$ is interesting precisely when $\\operatorname{Br}_1 U/\\operatorname{Br}k$ is non-trivial.", "Theorem 2.3 Let $U=X\\backslash C$ be a log K3 surface of 5 type over a number field $k$ with $C$ geometrically irreducible.", "We have $\\operatorname{Br}_1 U/\\operatorname{Br}k \\cong {\\left\\lbrace \\begin{array}{ll}\\mathbb {Z}/5\\mathbb {Z} & \\mbox{ if~$U$ is interesting;}\\\\0 & \\mbox{ otherwise.}\\end{array}\\right.", "}$ It was mentioned in [6] that the algebraic Brauer group modulo constants of log K3 surfaces of 5 type with a geometrically irreducible anticanonical divisor $C$ is trivial except for one specific action of the Galois group on the geometric Picard group.", "So it suffices to verify the statement for interesting del Pezzo surfaces over $k$ .", "We will fix a basis $(L_0,L_1,L_2,L_3, L_4)$ of $\\operatorname{Pic}\\bar{X}$ as in the proof of Lemma REF .", "Since $C \\subseteq X$ is geometrically irreducible we find the following exact sequence of Galois modules $0 \\rightarrow \\mathbb {Z} \\stackrel{j}{\\rightarrow }\\operatorname{Pic}\\bar{X}\\rightarrow \\operatorname{Pic}\\bar{U}\\rightarrow 0,$ where $j$ maps $n$ to $-nK_X$ .", "This shows that $\\operatorname{Pic}\\bar{U}\\cong \\operatorname{Pic}\\bar{X}/\\mathbb {Z} C \\cong \\mathbb {Z}^4$ , since the anticanonical divisor class $-K_X=3L_0-L_1-L_2-L_3-L_4$ is primitive.", "So $\\operatorname{Pic}\\bar{U}$ is torsion free and from the inflation–restriction sequence we conclude that the inflation homomorphism induces an isomorphism $\\mathrm {H}^1(\\operatorname{Gal}(K/k),\\operatorname{Pic}U_K) \\xrightarrow{} \\mathrm {H}^1(G_k,\\operatorname{Pic}\\bar{U}).$ We will compute the action of $\\sigma $ on the quotient $\\operatorname{Pic}\\bar{U}$ of $\\operatorname{Pic}\\bar{X}$ using the specific action of $\\sigma $ on $\\operatorname{Pic}\\bar{X}$ in Lemma REF .", "The classes $[L_0]$ , $[L_1]$ , $[L_2]$ and $[L_3]$ in $\\operatorname{Pic}U_K$ form a basis and in this basis the class of $L_4$ becomes $[L_4]=3[L_0]-[L_1]-[L_2]-[L_3]$ .", "So $\\sigma $ acts on $\\operatorname{Pic}\\bar{U}$ as $\\sigma = \\left(\\begin{array}{cccc}2 & 1 & 1 & 3\\\\-1 & -1 & 0 & -1\\\\-1 & -1 & -1 & -1\\\\-1 & 0 & -1 & -1\\end{array}\\right)$ By results on group cohomology of cyclic groups [32] we get $\\mathrm {H}^1(G,\\operatorname{Pic}\\bar{U}) \\cong \\ker (1+\\sigma +\\sigma ^2+\\sigma ^3+\\sigma ^4)/\\mathrm {Im}(1-\\sigma ).$ Since $1+\\sigma +\\sigma ^2+\\sigma ^3+\\sigma ^4 = 0$ and the image of $1-\\sigma $ is generated by $(1,0,0,2)$ , $(0,1,0,4)$ , $(0,0,1,4)$ and $(0,0,0,5)$ we find $\\operatorname{Br}_1 U/\\operatorname{Br}k \\cong \\mathbb {Z}/5\\mathbb {Z}.$ Consider an interesting log K3 surface $U$ .", "On the compactification $X$ of $U$ we have three important effective anticanonical divisors.", "First of all $C=X\\backslash U$ , but also the two divisors supported on $-1$ -curves as described in Lemma REF .", "These anticanonical sections are important enough to introduce some notation.", "Definition 2.4 Let $X$ be an interesting del Pezzo surface of degree 5 over a field $k$ .", "Let $l_1,l_2 \\in \\mathrm {H}^0(X,\\omega _X^\\vee )$ be the anticanonical sections supported on $-1$ -curves from Lemma REF .", "We will use these elements to construct explicit generators of $\\operatorname{Br}_1 U/\\operatorname{Br}k$ for an interesting log K3 surface $U=X\\backslash C$ .", "Lemma 2.5 Let $K$ be the splitting field of an interesting log K3 surface $U=X \\backslash C$ over a number field $k$ .", "Fix a generator $\\sigma $ of $\\operatorname{Gal}(K/k)\\cong \\mathbb {Z}/5\\mathbb {Z}$ .", "Let $h \\in \\mathrm {H}^0(X,\\omega _X^\\vee )$ be a global section whose divisor of zeroes is $C$ .", "The cyclic $\\kappa (X)$ -algebras $\\left(\\frac{l_1}{h}, \\sigma \\right) \\quad \\text{ and } \\quad \\left(\\frac{l_2}{h}, \\sigma \\right)$ are similar over $\\kappa (X)$ , their class lies in the subgroup $\\operatorname{Br}U \\subseteq \\operatorname{Br}\\kappa (X)$ and generates $\\operatorname{Br}_1 U/\\operatorname{Br}k$ .", "As $\\mathrm {div}_U(\\frac{l_1}{h})$ and  $\\mathrm {div}_U(\\frac{l_2}{h})$ are orbits of $-1$ -curves defined over $K$ it follows from Lemma REF that the cyclic algebras lie in the subgroup $\\operatorname{Br}U$ .", "The algebras $\\left(\\frac{l_1}{h}, \\sigma \\right) \\otimes \\left(\\frac{l_2}{h}, \\sigma \\right)^{\\textup {opp}}$ and $\\left(\\frac{l_1}{l_2}, \\sigma \\right)$ are similar and $\\mathrm {div}_U(\\frac{l_1}{l_2})$ is the norm of a principal divisor on $U$ since this is even the case on $X$ .", "Indeed, the divisors $L_{14}+L_1-L_2$ and $L_{24}$ are linearly equivalent on $X$ , and their norms $\\operatorname{Nm}_{K/k}(L_{14}+L_1-L_2)$ and $\\operatorname{Nm}_{K/k}(L_{24})$ are the divisors of zeroes of $l_1$ and $l_2$ .", "It follows again from Lemma REF that $\\left(\\frac{l_1}{l_2}, \\sigma \\right)$ is trivial in $\\operatorname{Br}U$ .", "The algebra $\\mathcal {A}$ is split by the degree 5 extension $K$ and this implies that $\\mathcal {A}$ is either trivial or of order 5.", "Suppose that the class of $\\mathcal {A}$ is trivial, then by Lemma REF there is a principal divisor $D$ on $U_K$ such that $\\operatorname{Nm}_{K/k} D = \\mathrm {div}_U l_1$ .", "This implies that there is a $g\\in \\kappa (U_K)$ such that $\\mathrm {div}_{U_K} g=D$ .", "Consider $g$ as a function on $X_K$ and $D$ as a divisor on $X_K$ .", "Then $\\mathrm {div}_{X_K} g=D+nC$ for some non-negative integer $n$ , since $C$ is geometrically irreducible.", "From $\\operatorname{Nm}_{K/k} D = \\mathrm {div}_U l_1$ we find $K_{X_K}\\cdot D = -1$ and we conclude that $0=K_{X_K} \\cdot \\mathrm {div}_{X_K} g=K_{X_K}\\cdot D+nK_{X_K} \\cdot C=-1+5n,$ which is a contradiction.", "Note that $l_1$ and $l_2$ are only defined up to multiplication by an element in $k^\\times $ .", "From now on we will denote the class in Lemma REF by $\\mathcal {A} \\in \\operatorname{Br}_1(U)$ which is uniquely defined up to an element in $\\operatorname{Br}k$ .", "Fix for the moment an interesting del Pezzo surface $X$ .", "We will consider the class $\\mathcal {A}_h$ on $U_h$ as $h$ varies over all hyperplane sections.", "We have seen that $\\mathcal {A}_h$ is of order 5 if $h$ cuts out a geometrically irreducible curve.", "The next lemma shows that this only fails for specific choices of $h$ .", "Lemma 2.6 Let $X \\subseteq \\mathbb {P}^5_k$ be an interesting del Pezzo surface over a field $k$ .", "A hyperplane section given by the vanishing of an $h\\in \\mathrm {H}^0(X,\\mathcal {O}(1))$ fails to be geometrically irreducible if and only if $h$ is a scalar multiple of either $l_1$ or $l_2$ .", "Consider a hyperplane section $C \\subseteq X$ .", "Let $D$ be a $k$ -irreducible component of $C$ and consider a $-1$ -curve $L$ on $X^{\\textup {sep}}$ .", "It follows that $L \\cdot D^{\\textup {sep}}=\\sigma (L) \\cdot D^{\\textup {sep}}$ and as the Galois orbit of $L$ is an anticanonical divisor, we find $5 \\ge -K_X \\cdot D = 5L \\cdot D^{\\textup {sep}}>0,$ since the degree of $D \\subseteq \\mathbb {P}^5_k$ is positive and at most the degree of $C$ , which equals 5.", "This proves that $L \\cdot D^{\\textup {sep}}=1$ for all $-1$ -curves $L$ and hence $C-D$ is an effective divisor of degree 0.", "We conclude that $C=D$ and this proves that any anticanonical section $C$ is irreducible over $k$ .", "If $C$ is not geometrically irreducible, then it must have at least two geometrically irreducible components of the same degree $d$ since the Galois group acts on the set of geometrically irreducible components of $C$ .", "Since $C$ is of degree 5 we find $2d \\le 5$ and hence $d$ is either 1 or 2.", "But in both cases we see that $C$ contains a geometrically irreducible curve of degree 1, which must be a $-1$ -curve $L$ .", "Then $C$ also contains all conjugates of $L$ and hence $C$ is the Galois orbit of a $-1$ -curve.", "This proves that $C$ is defined by the vanishing of either $l_1$ or $l_2$ .", "So we have seen that a log K3 surface $U=X\\backslash C$ of 5 type over a number field $k$ with $C$ geometrically integral has a non-trivial algebraic Brauer group modulo constants precisely for one action of Galois on the lines.", "We can use the correspondence in Proposition REF to classify interesting del Pezzo surfaces over a field $k$ .", "Proposition 2.7 Let $k$ be a field with a fixed separable closure $k^{\\textup {sep}}$ .", "The map which sends an isomorphism class of interesting del Pezzo surfaces over $k$ to its splitting field $K \\subseteq k^{\\textup {sep}}$ is a bijection to the set of degree 5 Galois extensions of $k$ contained in $k^{\\textup {sep}}$ .", "Definition 2.8 Let $K/k$ be a Galois extension of degree 5.", "The isomorphism class of interesting del Pezzo surfaces of degree 5 over $k$ which are split by $K$ is denoted by $5(K)$ .", "To obtain equations for an interesting del Pezzo surface given its splitting field we can use Proposition REF .", "We can also use this to recover the anticanonical sections $l_1$ and $l_2$ ; in the notation of that proposition, let $\\Lambda _{i,j}$ be the line through the points $P_i$ and $P_j$ , where we consider the indexes modulo 5.", "The divisors $\\sum \\Lambda _{i,i+1}$ and $\\sum \\Lambda _{i,i+2}$ are defined over $k$ and lie in the linear system $\\mathcal {L}$ .", "These are the only divisors in $\\mathcal {L}$ supported on lines and correspond to $l_1$ and $l_2$ on $X$ ." ], [ "A model of $5(\\mathbb {Q}(\\zeta _{11})^+)$ over the integers", "We have seen that all interesting del Pezzo surfaces split by a specific quintic extension $K$ of the base field $k$ are isomorphic.", "We have also seen how to construct such a surface as the image of a rational map $\\mathbb {P}_k^2 \\dashrightarrow \\mathbb {P}_k^5$ .", "We will now give an explicit first example of how one can construct models of this surface.", "We will use the quintic extension $K=\\mathbb {Q}(\\alpha )$ of $k=\\mathbb {Q}$ where $\\alpha = \\zeta _{11}+\\zeta _{11}^{-1}$ .", "We will write $m_\\alpha $ for the minimal polynomial of $\\alpha $ over $\\mathbb {Q}$ .", "Let $\\alpha _i$ be the conjugates of $\\alpha $ .", "Definition 3.1 Let $\\mathcal {Q} \\subseteq \\mathbb {Z}[x,y,z]_{(5)}$ be the sub-$\\mathbb {Z}$ -module consisting of all quintic polynomials which vanish at least twice at the points $P_i = (\\alpha _i^2 \\colon \\alpha _i \\colon 1) \\in \\mathbb {P}^2_{\\mathbb {Q}}$ .", "Lemma 3.2 The $\\mathbb {Z}$ -module $\\mathcal {Q}$ is free of rank 6 and $\\mathbb {Z}[x,y,z]_{(5)}/\\mathcal {Q}$ is torsion free.", "Clearly $\\mathcal {Q}$ is a free $\\mathbb {Z}$ -module.", "To compute its rank we use the results in [14] which says that $\\mathcal {Q} \\otimes \\mathbb {Q}[x,y,z]$ has dimension 6.", "The last statement follows from the fact that for $\\lambda \\in \\mathbb {Z}\\backslash \\lbrace 0\\rbrace $ and $q \\in \\mathbb {Z}[x,y,z]_{(5)}$ we have $\\lambda q \\in \\mathcal {Q}$ precisely if $q \\in \\mathcal {Q}$ .", "Let us fix a basis $q_i \\in \\mathcal {Q}$ .", "Definition 3.3 Let $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ be the image of the rational map $\\mathbb {P}^2_{\\mathbb {Z}} \\dashrightarrow \\mathbb {P}^5_{\\mathbb {Z}}$ defined by the $q_i$ .", "There are two primitive elements of $\\mathcal {Q}$ which factor into linear polynomials over $\\bar{\\mathbb {Q}}$ .", "These correspond to the two primitive linear forms $l_1,l_2 \\in \\mathbb {Z}[u_0,u_1,\\ldots , u_5]$ .", "Note that the scheme $\\mathcal {X}$ does not depend on the choice of basis of $\\mathcal {Q}$ .", "It does however depend on the choice of $\\alpha $ .", "The statements are easier, but not by much, since we have chosen an integral $\\alpha $ ; we could have picked any generator of $K$ over $\\mathbb {Q}$ .", "Proposition 3.4 The scheme $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ is given by the equations $u_0u_3+22u_0u_4+121u_0u_5-u_1^2-121u_1u_3+2662u_1u_4-36355u_2u_4-\\\\9306u_2u_5+10494u_3u_4-242u_3u_5-215501u_4^2+68123u_4u_5-13794u_5^2,$ $u_0u_4+11u_0u_5-u_1u_2-11u_1u_3+242u_1u_4-3223u_2u_4-847u_2u_5+\\\\902u_3u_4-11u_3u_5-19272u_4^2+6413u_4u_5-1331u_5^2,$ $u_0u_5-u_1u_3+22u_1u_4-u_2^2-286u_2u_4-77u_2u_5+77u_3u_4-1694u_4^2+572u_4u_5-121u_5^2,$ $u_1u_4-u_2u_3-11u_2u_4-77u_4^2+55u_4u_5-11u_5^2,\\\\$ $u_1u_5-u_2u_4-11u_2u_5-u_3^2+11u_3u_4-44u_4^2.\\\\$ In this example, the two relevant hyperplane sections are given by $l_1 = u_0 + 22u_1 - 363u_2 + 165u_3 - 1859u_4 + 484u_5,$ $l_2 = u_0 + 22u_1 - 352u_2 + 143u_3 - 1595u_4 + 363u_5.$ Also, the generic fibre $X = \\mathcal {X}_{\\mathbb {Q}}$ is isomorphic to $5(K)$ , and $\\mathcal {X}$ is the flat closure of $X$ in $\\mathbb {P}^5_{\\mathbb {Z}}$ .", "The magma code for these computations can be found in [22].", "We use those computation also for the proofs of some of the following statements.", "This follows from the fact that $5(K)$ is the image of the rational map $\\mathbb {P}^2_{\\mathbb {Q}} \\dashrightarrow \\mathbb {P}^5_{\\mathbb {Q}}$ using a basis of $\\mathcal {Q} \\otimes \\mathbb {Q}$ .", "We used a Gröbner basis computation to compute the image of the rational map $\\mathbb {P}^2_{\\mathbb {Z}} \\dashrightarrow \\mathbb {P}^5_{\\mathbb {Z}}$ .", "The upshot of this that the equations above also define a Gröbner basis of the ideal $I \\subseteq \\mathbb {Z}[u_0,u_1,\\ldots ,u_5]$ of $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ .", "Since the leading coefficients of the basis elements are monic we conclude from [1] that $I\\mathbb {Q}[u_0,u_1,\\ldots , u_5] \\cap \\mathbb {Z}[u_0,u_1,\\ldots , u_5]$ is equal to $I$ and hence that $\\mathcal {X}$ is the flat closure of its generic fibre.", "This last proof also shows that $\\mathcal {X}$ itself is integral, since $X$ is integral.", "From this or the fact that $\\mathcal {X}$ is flat over $\\mathbb {Z}$ we deduce the important fact that all fibres $\\mathcal {X}_\\ell $ are equidimensional of dimension 2." ], [ "Fibres of the model", "We can now study almost all fibres of $\\mathcal {X} \\rightarrow \\operatorname{Spec}(\\mathbb {Z})$ using the reduction of the minimal polynomial $m_\\alpha $ modulo primes.", "Lemma 3.5 Let $\\ell \\in \\mathbb {Z}$ be a prime for which the reduction $\\bar{m}_\\alpha \\in \\mathbb {F}_\\ell [s]$ is separable.", "The fibre $\\mathcal {X}_\\ell $ is a del Pezzo surface of degree 5.", "The hyperplane section given by the vanishing of $l_1$ and $l_2$ each cut out five $-1$ -curves on $\\mathcal {X}_\\ell $ .", "Consider the rational map $\\mathbb {P}^2_{\\mathbb {F}_\\ell } \\dashrightarrow \\mathbb {P}^5_{\\mathbb {F}_\\ell }$ using the basis $q_i\\otimes 1$ of $\\mathcal {Q} \\otimes \\mathbb {F}_\\ell $ .", "By construction this morphism lands in $\\mathcal {X}$ .", "If $m_\\alpha $ is separable modulo $\\ell $ then the reductions of the points $P_i$ modulo $\\ell $ are distinct and $\\mathcal {Q} \\otimes \\mathbb {F}_\\ell $ consists of all quintics over $\\mathbb {F}_\\ell $ vanishing at least twice at these points.", "By Proposition REF we see that the image $Y$ is a del Pezzo surface of degree 5.", "Hence we have $Y \\subseteq \\mathcal {X}_\\ell \\subseteq \\mathbb {P}^5_{\\mathbb {F}_\\ell }$ .", "From Corollary III.9.6 in [17] we see that all irreducible components of $\\mathcal {X}_\\ell $ are of dimension 2.", "Hence $Y$ is one such component of $\\mathcal {X}_\\ell $ .", "By flatness $\\mathcal {X} \\rightarrow \\operatorname{Spec}\\mathbb {Z}$ we see that $\\mathcal {X}_\\ell $ has degree 5 in $\\mathbb {P}^5_{\\mathbb {F}_\\ell }$ , just like $Y$ .", "Hence $\\mathcal {X}_\\ell $ has no other irreducible components.", "The statement about $l_1$ and $l_2$ also follows from the flatness of $\\mathcal {X}$ over $\\mathbb {Z}$ .", "There are actually two possibilities if the reduction of $m_\\alpha $ modulo $\\ell $ is separable.", "Corollary 3.6 If $m_\\alpha $ is irreducible modulo $\\ell $ then $\\mathcal {X}_\\ell $ is interesting.", "If $m_\\alpha $ splits completely in $\\mathbb {F}_\\ell $ with distinct roots, then $\\mathcal {X}_\\ell $ is split, i.e.", "all $-1$ -curves are defined over $\\mathbb {F}_\\ell $ .", "In the proof of the previous lemma we have seen that $\\mathcal {X}_\\ell $ is the image of $\\mathbb {P}^2_{\\mathbb {F}_\\ell }$ of all quintics vanishing at least twice at the five points $P_i=(\\alpha _i^2 \\colon \\alpha _i \\colon 1)$ modulo $\\ell $ .", "The action of Galois on the $-1$ -curves on $\\mathcal {X}_\\ell $ is determined by the action of Galois on the points $P_i$ .", "We have seen before that an interesting del Pezzo surface is obtained precisely if the five points are defined over a quintic extension, but are conjugate over the base field.", "A split del Pezzo surface of degree 5 corresponds to the case that all points are defined over the base field.", "Remark It is even possible to determine the fibres of $\\mathcal {X}/\\mathbb {Z}$ directly from the splitting of $m_\\alpha $ in $\\mathbb {F}_\\ell $ even for primes which divide the discriminant of $m_\\alpha $ .", "But this requires a long geometric treatise of singular del Pezzo surfaces, see the Ph.D. thesis of the author [23].", "For our explicit examples it is much shorter to just study the remaining finitely many fibres separately.", "For this example we are only left with the fibre over $\\ell = 11$ , since $\\Delta (m_\\alpha )=11^4$ .", "Lemma 3.7 The fibre $\\mathcal {X}_{11}$ is an integral surface in $\\mathbb {P}^5_{\\mathbb {F}_{11}}$ with precisely one singular point.", "The divisor on $\\mathcal {X}_{11}$ defined by $l_1$ is supported on a line $L$ .", "The singular point lies on this line.", "Also, a hyperplane section given by $h \\in \\mathbb {F}_{11}[u_0,u_1,\\ldots ,u_5]$ contains $L$ precisely if $h$ lies in $\\mathbb {F}_{11}[u_0,u_1,\\ldots ,u_4]$ .", "We have explicit equations for $\\mathcal {X}$ and hence for $\\mathcal {X}_{11}$ and all statements can be checked explicitly, for example by magma [22].", "The surface $\\mathcal {X}_{11}$ is actually well-understood.", "It is a singular del Pezzo surface and the unique singular point which is of type A$_4$ and lies on a unique line $L$ on $\\mathcal {X}_{11} \\subseteq \\mathbb {P}^2_{\\mathbb {F}_{11}}$ , i.e.", "a $-1$ -curve (on its minimal desingularisation).", "We even have a birational morphism $\\mathbb {P}^2 \\dashrightarrow \\mathcal {X}_{11}$ which restricts to an isomorphism $\\mathbb {A}^2 \\xrightarrow{} \\mathcal {X}_{11}\\backslash L,$ $(1\\colon y \\colon z) \\mapsto (1 \\colon y \\colon z \\colon y^2 \\colon yz \\colon y^3 + z^2).$ This will allow one to transfer many problems on $\\mathcal {X}_{11}$ to a problem on the affine or the projective plane.", "Remark This is not at all particular to this one example; for any choice $\\alpha \\in \\bar{\\mathbb {Q}}$ of degree 5 we can construct a relative surface $\\mathcal {X}$ over $\\mathbb {Z}$ .", "If the minimal polynomial $m_\\alpha $ reduces to the fifth power of a linear polynomial modulo $\\ell $ then $\\mathcal {X}_\\ell $ always has these properties.", "We will forgo this general approach and stick to our explicit examples." ], [ "A family of log K3 surfaces", "Consider the model $\\mathcal {X} \\subseteq \\mathbb {P}^5_{\\mathbb {Z}}$ of an interesting del Pezzo surface of the previous section.", "We will use it to construct a family of log K3 surfaces of 5 type together with their models.", "Definition 4.1 Let $h \\in \\mathbb {Z}[u_0,u_1,u_2,u_3,u_4,u_5]_{(1)}$ be a primitive linear form.", "Let $\\mathcal {U}_h$ be the complement of $\\mathcal {C}_h=\\lbrace h=0\\rbrace \\cap \\mathcal {X}$ in $\\mathcal {X}$ .", "We will consider when $\\mathcal {U}_h$ does not have integral points.", "First of all this happens when $\\mathcal {U}_h$ is not everywhere locally soluble.", "We can make precise when this happens.", "Lemma 4.2 The affine surface $\\mathcal {U}_h$ is everywhere locally soluble precisely when $h \\lnot \\equiv u_2+u_5 \\mod {2}.$ One can check that the points Table: NO_CAPTION lie on $\\mathcal {X}$ .", "Also, their coordinates as vectors in $\\mathbb {Z}^6$ define a lattice of dimension 6 of index 2.", "This proves that for any prime $\\ell \\ne 2$ and any hyperplane section $h$ at least one of these points $P$ satisfies $h(P) \\lnot \\equiv 0 \\mod {p}$ .", "This shows that such a point determines an element in $\\mathcal {U}_h(\\mathbb {Z}_\\ell )$ .", "We have seen in Lemma REF that $\\mathcal {X}_2$ is smooth.", "One can check that $\\#\\mathcal {X}(\\mathbb {F}_2)=5$ and that these points lie on the indicated hyperplane over $\\mathbb {F}_2$ ." ], [ "Obstructions coming from $\\mathcal {A}_h$", "Note that if $\\mathcal {C}_\\mathbb {Q}$ is geometrically irreducible, i.e.", "$h$ is not a multiple of $l_1$ and $l_2$ by Lemma REF , then we see that $\\operatorname{Br}U_h/\\operatorname{Br}\\mathbb {Q}$ contains an element of order 5.", "Let us compute the invariant maps for this element.", "Lemma 4.3 Consider a geometrically irreducible hyperplane section given by a primitive $h$ .", "Let $\\ell $ be a prime and let $\\mathcal {A}$ be a generator for $\\operatorname{Br}_1 U_h/\\operatorname{Br}\\mathbb {Q}$ .", "We consider the invariant map $\\operatorname{inv}_\\ell \\mathcal {A} \\colon \\mathcal {U}_h(\\mathbb {Z}_\\ell ) \\rightarrow \\mathbb {Q}/\\mathbb {Z}.$ If $\\ell \\ne 11$ then $\\operatorname{inv}_\\ell \\mathcal {A}$ is identically zero.", "The statement is immediate for the infinite place and primes $\\ell $ which split completely in $K$ .", "Since in those cases $\\mathcal {A}_{\\mathbb {Q}_\\ell } \\cong \\mathcal {A}_{K_{\\mathfrak {l}}}$ is trivial in $\\operatorname{Br}U_{\\mathbb {Q}_\\ell }$ for any prime $\\mathfrak {l}$ of $K$ above $\\ell $ .", "Now suppose that $m_\\alpha $ is irreducible modulo $\\ell $ .", "The Kummer–Dedekind theorem implies that $\\ell $ is inert in $\\mathbb {Z}[\\alpha ]$ .", "This also proves that $\\ell $ is inert in $\\mathcal {O}_K$ and there is a unique prime $\\mathfrak {l}$ above $\\ell $ .", "Also, $U_{\\mathbb {Q}_\\ell }$ is an interesting del Pezzo surface since $m_\\alpha $ is irreducible over $\\mathbb {Q}_\\ell $ .", "Hence the hyperplane section given by the vanishing of $l_1$ modulo $\\ell $ is geometrically irreducible and does not contain $\\mathbb {F}_\\ell $ -points.", "Hence $l_1$ is invertible on all points in $U(\\mathbb {Z}_\\ell )$ .", "This shows that $\\frac{l_1}{h}(P) \\in \\mathbb {Z}_\\ell ^\\times $ for all $P \\in \\mathcal {U}(\\mathbb {Z}_\\ell )$ .", "Since the extension $K_{\\mathfrak {l}}/\\mathbb {Q}_\\ell $ of local fields is unramified we see that any unit is a norm.", "Hence $\\operatorname{inv}_\\ell \\mathcal {A}$ is also in this case constantly 0.", "Lemma 4.4 Let $L$ be the unique line on $\\mathcal {X}_{11} \\subseteq \\mathbb {P}^5_{\\mathbb {F}_{11}}$ .", "If $L$ does not lie in the zero locus of $h$ then $\\operatorname{inv}_{11} \\mathcal {A} \\colon \\mathcal {U}_h(\\mathbb {Z}_{11}) \\rightarrow \\frac{1}{5}\\mathbb {Z}/\\mathbb {Z}$ is surjective.", "We have seen that in Lemma REF that the condition is equivalent to $h \\mod {1}1$ being dependent on $u_5$ .", "On points $P$ for which $\\frac{l_1}{h}(P) \\in \\mathbb {Z}_{11}$ is invertible and we can use Lemma REF to compute the $\\operatorname{inv}_{11} \\mathcal {A}(P)$ , i.e.", "the invariant at $P$ only depends on $\\frac{l_1}{h}(P) \\in \\mathbb {F}_{11}$ up to fifth powers and there is an isomorphism of $\\psi \\colon \\mathbb {F}_{11}/\\lbrace \\pm 1\\rbrace \\rightarrow \\frac{1}{5} \\mathbb {Z}/\\mathbb {Z}$ such that $\\operatorname{inv}_{11} \\mathcal {A}(P)=\\psi \\left(\\frac{l_1}{h}(P)\\right)$ .", "Hence it will suffice to prove the following stronger statement: the map $\\frac{l_1}{h} \\colon \\left(\\mathcal {U}_h\\backslash L\\right)(\\mathbb {F}_{11}) \\rightarrow \\mathbb {F}_{11}^\\times $ is surjective.", "Note that both the domain and the map depend on our choice of $h$ .", "For this statement we only have finitely many $\\bar{h} \\in \\mathbb {F}_{11}[u_0,u_1,\\ldots ,u_5]_{(1)}$ which we need to evaluate on a subset of the finitely many points $\\mathcal {X}_{11}(\\mathbb {F}_{11})$ .", "The code for this computation can be found in [22].", "Proposition 4.5 Define $f = h(1, y, z, y^2, yz, y^3 + z^2) \\in \\mathbb {F}_{11}[y,z]$ .", "The value $0\\in \\mathbb {Q}/\\mathbb {Z}$ lies in the image of $\\operatorname{inv}_{11} \\mathcal {A}$ precisely when the polynomial $f$ assumes a values $\\pm 1$ modulo 11 for $y,z \\in \\mathbb {F}_{11}$ .", "The image of $\\operatorname{inv}_{11} \\mathcal {A} \\colon \\mathcal {U}_h(\\mathbb {Z}_{11}) \\rightarrow \\frac{1}{5}\\mathbb {Z}/\\mathbb {Z}$ has size 1 precisely when $f$ is a constant; 4 precisely when $f$ is a separable quadratic polynomial in $y$ ; 5 in all other cases.", "Note in the second case that $f$ is in particular independent of $z$ .", "Using the last lemma we will only need to consider the $\\bar{h}$ over $\\mathbb {F}_{11}$ which do not depend on $u_5$ .", "In this case we have that $\\left(\\mathcal {U}_h\\backslash L\\right)(\\mathbb {Z}_{11}) = \\mathcal {U}_h(\\mathbb {Z}_{11})$ since $L$ lies in the zero locus of $h$ .", "Furthermore, the value of $\\operatorname{inv}_{11}\\mathcal {A}$ at a point $P$ only depends on $\\frac{l_1}{h}(P)$ modulo 11 or equivalently the reduction $\\bar{P} \\in \\mathcal {U}_h(\\mathbb {F}_{11})$ of $P$ .", "The statement can now be checked completely by a computer.", "We would however like to provide a little more insight.", "Using the isomorphism $\\mathcal {X}_{11}\\backslash L \\xrightarrow{} \\mathbb {A}_{\\mathbb {F}_{11}}^2$ from Lemma REF we see that $\\mathcal {U}_{h,11}=\\mathcal {U}_{h,11}\\backslash L \\xrightarrow{} \\mathbb {A}^2_{\\mathbb {F}_{11}}\\backslash \\lbrace f = 0\\rbrace $ .", "Hence we are interested in the image of $f \\colon \\mathbb {A}^2_{\\mathbb {F}_{11}}\\backslash \\lbrace f = 0\\rbrace \\rightarrow \\mathbb {F}_{11}^\\times /\\lbrace \\pm 1\\rbrace $ .", "If $\\bar{h}$ depends on $u_5$ , then $f$ is a cubic polynomial and $f=c$ for any $c \\in \\mathbb {F}_{11}$ is likely to have a solution, as made precise in the previous lemma.", "If $\\bar{h}$ depends on either $u_2$ or $u_4$ , then $f$ is linear in $z$ with the leading coefficient being linear in $y$ .", "Fixing $y$ to be a suitable $y_0$ shows that $f(y_0,z)=c$ always has a solution in $\\mathbb {F}_{11}$ .", "In the remaining case $f$ is a polynomial independent of $z$ of degree at most 2.", "When $f$ is constant we immediately get the first case.", "Whenever $f$ is linear or a inseparable quadratic polynomial with root $\\rho \\in \\mathbb {F}_{11}$ the surjectivity of $f \\colon \\mathbb {F}_{11}\\backslash \\lbrace \\rho \\rbrace \\rightarrow \\mathbb {F}_{11}^\\times /\\lbrace \\pm 1\\rbrace $ is immediate.", "For the last case it is easily checked that for a quadratic separable polynomial $f=c(y-\\rho _1)(y-\\rho _2)$ the image of $f \\colon \\mathbb {F}_{11}\\backslash \\lbrace \\rho _1,\\rho _2\\rbrace \\rightarrow \\mathbb {F}_{11}^\\times /\\lbrace \\pm 1\\rbrace $ has size four.", "This is independent of whether $f$ splits over $\\mathbb {F}_{11}$ or over $\\mathbb {F}_{11^2}$ .", "We can now apply the above results to compute the Brauer–Manin obstruction for a fixed $h$ and find actual algebraic obstructions of order 5 to the integral Hasse principle.", "Theorem 4.6 Let $\\mathcal {H}$ be the hyperplane in $\\mathbb {P}^5_\\mathbb {Z}$ given by the vanishing of $u_1-6u_3$ .", "The complement $\\mathcal {U} = \\mathcal {X} \\backslash \\mathcal {H}$ has points over $\\mathbb {Q}$ and every $\\mathbb {Z}_\\ell $ , but there is an order 5 Brauer–Manin obstruction to the existence of integral points.", "Remark Let $S$ be a set of rational primes which split completely in $K$ .", "The proof of the above statement can easily be adapted to show that there are no $S$ -integral points on $\\mathcal {X}$ .", "On the other hand if $\\ell $ is an inert prime then $\\operatorname{inv}_\\ell \\mathcal {A}$ need not be constant on $\\mathbb {Q}_\\ell $ -points even if it is so on $\\mathbb {Z}_\\ell $ -points.", "Although our model $\\mathcal {U}$ is regular this does not contradict Theorem 1 in [4].", "Hence the concept of a regular model is not as useful for $S$ -integral points as it is for rational points.", "A careful analysis of the above proof yields the following result.", "Theorem 4.7 Let $\\mathcal {U}_h$ be the complement in $\\mathcal {X}$ of a geometrically irreducible hyperplane section given by a primitive linear form $h \\in \\mathbb {Z}[u_0,u_1,\\ldots , u_5]$ .", "The class of $h$ modulo 2 determines whether the affine surface $\\mathcal {U}_h$ is locally soluble.", "The existence of an algebraic obstruction to the Hasse principle for integral points depends only on the reduction of $h$ modulo 11.", "Out of the $11^6-1=1771560$ possible reductions $\\bar{h}$ of $h$ modulo 11 precisely 228 give an obstruction.", "Note that this does not mean that the reduction of $h$ modulo 2 and 11 is the only condition; the proof still uses the assumption that $h$ is primitive.", "It follows from Lemma REF that the condition that the section is geometrically irreducible is immediately satisfied if $h$ does not reduce to $\\pm u_0$ modulo 11.", "For hyperplanes $h$ reducing to either of these two form it is easily shown that $\\operatorname{inv}_{11}\\mathcal {A}$ is identically equal to 0 on $\\mathcal {U}_h(\\mathbb {Z}_{11})$ .", "Let us count the non-zero linear forms $\\bar{h}$ over $\\mathbb {F}_{11}$ for which such an obstruction exists.", "In Proposition REF we saw that we get no obstruction unless $f$ is either constant or a separable quadratic polynomial in $y$ .", "If $f$ is constant then we see that $\\operatorname{inv}_{11} \\mathcal {A}$ is constant and we get an obstruction if $f$ is one of the 8 non-fifth powers modulo 11.", "For an $\\bar{h}$ such that $f$ is a quadratic inseparable polynomial we have seen that $f \\colon \\mathbb {F}_{11}\\backslash \\lbrace \\rho _1,\\rho _2\\rbrace \\rightarrow \\mathbb {F}_{11}^\\times /\\lbrace \\pm 1\\rbrace , x \\mapsto f(x)$ misses exactly one value.", "If $f$ misses the value $q \\in \\mathbb {F}_{11}^\\times /\\lbrace \\pm 1\\rbrace $ we see that $\\lambda f$ for $\\lambda \\in \\mathbb {F}_{11}^\\times $ misses the class of $\\lambda q$ .", "There are $10\\cdot 11^2$ quadratic polynomials over $\\mathbb {F}_{11}$ and $10\\cdot 11$ of these are inseparable.", "The group $\\mathbb {F}_{11}^\\times $ acts on the remaining $10^2\\cdot 11$ quadratic polynomials by multiplication.", "All orbits have size 10 and in such an orbit exactly 2 miss the unit element in $\\mathbb {F}_{11}^\\times /(\\mathbb {F}_{11}^\\times )^5$ .", "This proves that for an $h$ for which the invariant map at 11 assumes precisely 4 values there is an obstruction if the associated polynomial $f$ is one of these $2\\cdot 10\\cdot 11=220$ separable quadratic polynomials.", "Remark We can use the same construction to produce models $\\mathcal {X}$ with a different splitting field $K$ .", "However if $K$ is ramified at a prime $p>11$ then one can show that the image of $\\operatorname{inv}_p \\mathcal {U}_h(\\mathbb {Z}_p)$ has either size 1 on 5.", "Furthermore, the first case happens precisely when $\\frac{l_1}{h}$ is constant modulo $p$ similar to above.", "The interesting thing to note is that the intermediate case in which the invariant map assumes 4 invariants does not occur any more.", "We are left with the case of quintic fields $K/\\mathbb {Q}$ which are ramified at 5." ], [ "Explicit examples with splitting field $K\\subseteq \\mathbb {Q}(\\zeta _{25})$", "It is also possible to find obstructions of order 5 to the integral Hasse principle when $\\mathcal {X}$ is a model of the interesting del Pezzo surface $X$ split by the unique quintic extension $K$ contained in $\\mathbb {Q}(\\zeta _{25})$ .", "In that case $K$ has 5 as a wildly ramified prime.", "For example, define the field $K \\subseteq \\mathbb {Q}(\\zeta _{25})$ as the splitting field of the polynomial $m_{\\alpha }=s^5 - 20s^4 + 100s^3 - 125s^2 + 50s - 5.$ This produces the projective surface $\\mathcal {X}$ over the integers given by the five equations $u_0u_3+40u_0u_4+400u_0u_5-u_1^2-400u_1u_3+16000u_1u_4-365050u_2u_4-\\\\49995u_2u_5+51985u_3u_4-200u_3u_5-2029975u_4^2+392250u_4u_5-39375u_5^2,$ $u_0u_4+20u_0u_5-u_1u_2-20u_1u_3+800u_1u_4-18125u_2u_4-2500u_2u_5+\\\\2550u_3u_4-5u_3u_5-101015u_4^2+19800u_4u_5-2000u_5^2,$ $u_0u_5-u_1u_3+40u_1u_4-u_2^2-900u_2u_4-125u_2u_5+125u_3u_4-5000u_4^2+985u_4u_5-100u_5^2,\\\\$ $u_1u_4-u_2u_3-20u_2u_4-125u_4^2+50u_4u_5-5u_5^2,\\\\$ $u_1u_5-u_2u_4-20u_2u_5-u_3^2+20u_3u_4-100u_4^2.\\\\$ The two hyperplane sections over $\\mathbb {Z}$ cutting out the two quintuples of $-1$ -curves are $l_1 = u_0 + 25u_1 - 700u_2 + 200u_3 - 3425u_4 + 575u_5,$ $l_2 = u_0 + 75u_1 - 1675u_2 + 375u_3 - 5175u_4 + 575u_5.$ By construction this scheme shares many properties with the previous example.", "Proposition 5.1 The scheme $\\mathcal {X}$ is integral, with integral fibres.", "If $m_\\alpha $ is irreducible modulo $\\ell $ then $\\mathcal {X}_\\ell $ is an interesting del Pezzo surface.", "If $m_\\alpha $ has five distinct roots in $\\mathbb {F}_\\ell $ then $\\mathcal {X}_\\ell $ is a split del Pezzo surface of degree 5.", "For $\\ell =5$ the surface $\\mathcal {X}_5$ contains a unique line $L$ , has a unique singular point of type A$_4$ which lies on $L$ .", "There is a birational map $\\mathcal {X}_5 \\dashrightarrow \\mathbb {P}^2_{\\mathbb {F}_5}$ which restricts to an isomorphism $\\mathcal {X}_5 \\backslash L \\xrightarrow{} \\mathbb {A}^2_{\\mathbb {F}_5}$ .", "The only fibre not discussed in this lemma is the one over 7.", "Although $\\mathcal {X}_7$ is again a singular del Pezzo surface, we will not need any information about this fibre this since 7 splits completely in $K$ .", "One can follow the proofs in Section 3 for this different choice of $\\alpha $ and corresponding equations for $\\mathcal {X}$ .", "We will consider $\\mathcal {U}_h = \\mathcal {X}\\backslash \\lbrace h=0\\rbrace $ like in the previous sections.", "As before, local solubility is immediate at most primes.", "Lemma 5.2 The surface $\\mathcal {U}_h$ is everywhere locally soluble precisely when $h \\lnot \\equiv u_2+u_3 \\mod {2}.$ As for the proof of Lemma REF it is easy enough to find enough points on $\\mathcal {X}$ whose reductions do not lie on a hyperplane modulo $\\ell > 2$ .", "For $\\ell =2$ the fibre $\\mathcal {X}$ is again smooth, $\\#\\mathcal {X}(\\mathbb {F}_2)=5$ and all $\\mathbb {F}_2$ -points lie on the unique hyperplane given by $u_2+u_3 \\equiv 0 \\mod {2}$ .", "The computation of the invariant maps at the unramified primes is the same computation as in Lemma REF for the previous example.", "Lemma 5.3 Consider the invariant map $\\operatorname{inv}_\\ell \\mathcal {A} \\colon \\mathcal {U}_h(\\mathbb {Z}_\\ell ) \\rightarrow \\mathbb {Q}/\\mathbb {Z}.$ If $\\ell \\ne 5$ , then the invariant map is identically zero.", "Let us consider the remaining prime.", "Theorem 5.4 Then $\\operatorname{inv}_5 \\mathcal {A} \\colon \\mathcal {U}(\\mathbb {Z}_5) \\rightarrow \\frac{1}{5} \\mathbb {Z}/\\mathbb {Z}$ is not surjective precisely when there exist integers $\\lambda $ , $c_1$ and $c_3$ satisfying $5 \\nmid \\lambda $ , and $5\\mid c_1,c_3$ or $5 \\nmid c_1$ such that $h \\equiv \\lambda u_0+5(c_1u_1+c_3u_3) \\mod {2}5.$ The invariant map is constant when $5 \\mid c_1, c_3$ and otherwise the size of its image is 3.", "The value $0\\in \\mathbb {Q}/\\mathbb {Z}$ lies in the image of $\\operatorname{inv}_{11} \\mathcal {A}$ precisely when $\\lambda +5(c_1 y + c_3 y^2)$ assumes one of values $\\pm 1, \\pm 7$ modulo 25 for $y\\in \\mathbb {Z}$ .", "To prove this result one can use the fact that the model $\\mathcal {X}$ is regular.", "However, if 5 is ramified in $K$ one has a similar statement for any model $\\mathcal {X}_\\alpha $ , which does not need to be regular.", "The following chain of results also implies a similar result in the more general case.", "Lemma 5.5 Consider a point $\\bar{P} \\in \\left(\\mathcal {U}_h\\backslash L\\right)(\\mathbb {F}_5)$ .", "Let $\\mathcal {P}$ be the set of the 25 lifts of $\\bar{P}$ in $\\mathcal {X}(\\mathbb {Z}/25\\mathbb {Z})$ .", "The image of $\\frac{l_1}{h} \\colon \\mathcal {P} \\rightarrow \\left(\\mathbb {Z}/25\\mathbb {Z}\\right)^\\times $ is either of size 1 of 5.", "Define $\\mathcal {V}:=\\mathcal {U}_5\\backslash L\\subseteq \\mathbb {A}^5_{\\mathbb {F}_5}$ on which $\\frac{h}{l_1}$ is given by $h_{\\text{aff}}=a_0+a_1u_1+\\ldots +a_5u_5$ .", "Now let $\\vec{x} = (x_1,x_2,x_3,x_4,x_5)$ be a 5-tuple of integers reducing to $\\bar{P} \\in \\mathcal {V}$ .", "We will first show that the lifts of $\\bar{P}$ to points in $\\mathcal {X}(\\mathbb {Z}/25 \\mathbb {Z})$ are $\\vec{x}+5\\vec{w}$ where $\\vec{w}$ is any vector in a translation of the tangent space of $\\mathcal {V}$ at $\\bar{P}$ .", "Suppose that $\\mathcal {X}$ is given by polynomials $g_j$ in the variables $u_i$ .", "The tangent space at $\\bar{P}$ is by definition $T_{\\bar{P}}\\mathcal {V} = \\left\\lbrace \\vec{v}\\in \\mathbb {F}_5^5\\ \\colon \\ \\sum _{i=1}^5 \\frac{dg_j}{du_i}(\\bar{P}) v_i \\equiv 0 \\mod {5}\\right\\rbrace $ and if $\\vec{x}+5\\vec{w} \\in \\mathcal {X}(\\mathbb {Z}/25\\mathbb {Z})$ then for all $j$ $0 \\equiv g_j(\\vec{x}+5\\vec{w})\\equiv g_j(\\vec{x})+5\\sum _{i=1}^5 \\frac{dg_j}{du_i}(\\vec{x}) w_i \\mod {2}5$ which proves the claim.", "To compute $h_{\\text{aff}}$ at these lifts, let us write $\\vec{a}=(a_1,a_2,a_3,a_4,a_5) \\in \\mathbb {Z}^5$ .", "Then we find $h_{\\text{aff}}(\\vec{x} +5\\vec{w}) \\equiv h_{\\text{aff}}(\\vec{x})+5\\vec{a}\\cdot \\vec{w} \\mod {2}5.$ This concludes the proof since the $\\vec{w}$ live in a linear space over $\\mathbb {F}_5$ .", "This result is very powerful when combined with the following fact.", "Lemma 5.6 An element $a \\in \\mathbb {Z}_5^\\times $ is a fifth power precisely if it is so modulo 25, i.e.", "if its reduction $\\hat{a} \\in \\mathbb {Z}/25\\mathbb {Z}$ lies in $\\lbrace \\pm 1, \\pm 7\\rbrace $ .", "Hence, the five lifts of any $\\bar{a} \\in \\left(\\mathbb {Z}/5\\mathbb {Z}\\right)^\\times $ in $\\left(\\mathbb {Z}/25\\mathbb {Z}\\right)^\\times $ lie in different classes modulo fifth powers.", "To prove Theorem REF using brute computational force one would need to list all possible hyperplanes $h$ and points in $\\mathcal {X}(\\mathbb {Z}_5)$ modulo 25.", "We will use these last two results to show we can do this computation while only using the points and hyperplane sections modulo 5; drastically improving on the time needed for the computations.", "Let us first show we can ignore the singular point on $\\mathcal {X}_5$ , or even the unique line $L \\subseteq \\mathcal {X}_5$ containing this point.", "Proposition 5.7 If $h$ is not a multiple of $l_1$ modulo 5 then $\\operatorname{inv}_5 \\mathcal {A} \\colon \\mathcal {U}_h(\\mathbb {Z}_5) \\rightarrow \\frac{1}{5}\\mathbb {Z}/\\mathbb {Z}$ is surjective.", "We will prove a stronger statement.", "Define $\\mathcal {V}:=\\mathcal {U}_5\\backslash L\\subseteq \\mathbb {A}^5_{\\mathbb {F}_5}$ on which the hyperplane is given by $h=a_0+a_1u_1+\\ldots +a_5u_5$ .", "We will show that there is an $\\mathbb {F}_5$ -point $\\bar{P}$ on $\\mathcal {V}$ such that $h \\colon T_{\\bar{P}} \\mathcal {V} \\rightarrow \\mathbb {F}_5$ is surjective; note that one can consider the tangent space as a linear or affine subspace of $\\mathbb {F}_5^5$ , since this does not change the size of the image of this function.", "One can prove this problem by translating it back to a study of plane curves using the isomorphism $\\mathcal {V} \\cong \\mathbb {A}^2_{\\mathbb {F}_5} \\backslash \\lbrace f=0\\rbrace $ for $f=h(1,y,z,y^2,yz,y^3+z^2)$ and conclude that for every $h$ there are at least 10 such points.", "For good measure the statement above is checked with magma [22].", "Now let $\\bar{P}$ be such an $\\mathbb {F}_5$ -point in $\\mathcal {V} \\subseteq \\mathcal {X}$ .", "By the defining property of $\\bar{P}$ we see that $\\operatorname{inv}\\mathcal {A}_h$ assumes five values on the points in $\\mathcal {U}_h(\\mathbb {Z}_5)$ reducing to $\\bar{P}$ .", "Corollary 5.8 If $h \\mod {5}$ does not cut out the line $L \\subseteq \\mathcal {X}_5$ then $\\operatorname{inv}_5 \\mathcal {A}$ is surjective.", "We can now efficiently prove Theorem REF .", "By Proposition REF we only need to consider the case that $h$ is not a multiple of $u_0$ modulo 5 hence we can write $h=\\lambda u_0 + 5(c_1u_1+\\ldots +c_5u_5) \\in \\mathbb {Z}[u_0,u_1,\\ldots , u_5]$ .", "Let us write $k=c_1u_1+\\ldots +c_5u_5$ .", "Since $l_1 \\equiv u_0 \\mod {2}5$ we see that the value of $\\frac{h}{l_1} = \\lambda + 5k(\\frac{u_1}{u_0},\\ldots ,\\frac{u_5}{u_0}) \\mod {2}5$ at any $P \\in \\mathcal {U}(\\mathbb {Z}_5)$ only depends on $\\bar{P} \\in \\mathcal {U}(\\mathbb {F}_5)$ .", "We are interested in the values $k$ takes on $\\mathcal {U}(\\mathbb {Z}_5)$ modulo 5.", "A computer check [22] shows that for the listed cases $k$ assumes the indicated number of values in $\\mathbb {F}_5$ .", "Hence $\\frac{h}{l_1}$ assumes the same number of values in $\\left(\\mathbb {Z}/25\\mathbb {Z}\\right)^\\times $ each of which is a different lift of $\\lambda \\in \\mathbb {F}_5^\\times $ .", "This shows that $\\frac{l_1}{h}$ assumes exactly 1, 3 or 5 values in $\\mathbb {Z}_5^\\times $ modulo fifth powers.", "Hence we see that $\\operatorname{inv}_5\\mathcal {A}$ assumes these many values on $\\mathcal {U}_h(\\mathbb {Z}_5)$ .", "To provide a little more insight we can again use the isomorphism $\\mathcal {U}_5 \\cong \\mathbb {A}^2_{\\mathbb {F}_5} \\backslash \\lbrace f=0\\rbrace $ now using $f=k(1,y,z,y^2,yz,y^3+z^2)$ .", "One can check that $f$ is surjective to $\\mathbb {F}_5^\\times $ if it describes a line, a conic with two distinct rational points at infinity, a geometrically integral conic with a single point at infinity, or a cubic curve.", "The remaining cases are the constant functions and the quadratics which are independent of $z$ .", "This shows that $k \\equiv c_1u_1+c_3u_3 \\mod {5}$ .", "The hyperplane section of $\\mathbb {P}^5_{\\mathbb {F}_5}$ defined by $k\\equiv c_1u_1+c_3u_3 \\mod {5}$ corresponds to the polynomial $c_1y+c_3y^2$ on $\\mathbb {A}^2_{\\mathbb {F}_5}$ which is quadratic if $c_3 \\ne 0$ and constant if $c_1=c_3=0$ .", "By symmetry we see that a quadratic in one variable over $\\mathbb {F}_5$ assumes exactly 3 values.", "And obviously $h \\equiv \\lambda l_1 \\mod {2}5$ precisely when $c_1$ and $c_3$ are zero in $\\mathbb {F}_5$ .", "For completeness we will give an example of a hyperplane for which the associated affine scheme over the integers does not have integral solutions.", "Theorem 5.9 Consider an $h$ which cuts out a geometrically irreducible hyperplane section such that 0 does not lie in the image of $\\operatorname{inv}_5 \\mathcal {A}$ .", "The reduction of $h$ modulo 25 is one of 176 out of the $(5^2)^6-5^6=244125000$ possible hyperplanes over $\\mathbb {Z}/25\\mathbb {Z}$ .", "For example, the surface $\\mathcal {U}_h/\\mathbb {Z}$ for $h=2u_0-15u_1+10u_3$ admits a Brauer–Manin obstruction of order 5 to the existence of integral points.", "Let $\\left(\\mathbb {Z}/25 \\mathbb {Z} \\right)^\\times $ act by multiplication on the hyperplanes modulo 25 for which $\\operatorname{inv}_5 \\mathcal {A}$ is not surjective.", "Multiplication by $\\lambda $ translates the image of the invariant map by an element of $\\frac{1}{5} \\mathbb {Z}/\\mathbb {Z}$ corresponding on the class of $\\lambda $ in $\\left(\\mathbb {Z}/25 \\mathbb {Z} \\right)^\\times $ modulo fifth powers.", "So if the size of the image of an invariant map corresponding to a hyperplane has one element, then $\\frac{4}{5}$ of the scalar multiples of $h$ do not have 0 in the image.", "For invariant maps whose image is of size 3 precisely $\\frac{2}{5}$ of the scalar multiples have this property.", "This means that the number of hyperplanes modulo 25 for which 0 does not lie in the image of the invariant map is $\\frac{4}{5} \\cdot 20+ \\frac{2}{5} \\cdot 20\\cdot 4 \\cdot 5=176$ .", "Now consider the hyperplane $h=2u_0-15u_1+10u_3$ .", "The affine surface $\\mathcal {U}_h$ is locally soluble by Lemma REF .", "The result follows from the previous theorem; take $\\lambda = 2$ , $c_1=-3$ and $c_3=2$ and note that $2-15x+10x^2$ only assumes the values $2,12,22 \\mod {2}5$ .", "So 0 does not lie in the image of the invariant map at 5 and the invariant maps at the other primes are all constant zero.", "We can also show that in the two explicit examples with splitting fields with conductor 11 and 25 the absence of integral points is not explained by the principle of weak obstructions at infinity as introduced by Jahnel and Schindler in [19]." ] ]

[ [ "LR-CNN: Local-aware Region CNN for Vehicle Detection in Aerial Imagery" ], [ "Abstract State-of-the-art object detection approaches such as Fast/Faster R-CNN, SSD, or YOLO have difficulties detecting dense, small targets with arbitrary orientation in large aerial images.", "The main reason is that using interpolation to align RoI features can result in a lack of accuracy or even loss of location information.", "We present the Local-aware Region Convolutional Neural Network (LR-CNN), a novel two-stage approach for vehicle detection in aerial imagery.", "We enhance translation invariance to detect dense vehicles and address the boundary quantization issue amongst dense vehicles by aggregating the high-precision RoIs' features.", "Moreover, we resample high-level semantic pooled features, making them regain location information from the features of a shallower convolutional block.", "This strengthens the local feature invariance for the resampled features and enables detecting vehicles in an arbitrary orientation.", "The local feature invariance enhances the learning ability of the focal loss function, and the focal loss further helps to focus on the hard examples.", "Taken together, our method better addresses the challenges of aerial imagery.", "We evaluate our approach on several challenging datasets (VEDAI, DOTA), demonstrating a significant improvement over state-of-the-art methods.", "We demonstrate the good generalization ability of our approach on the DLR 3K dataset." ], [ "Introduction", "Vehicle detection in aerial photography is challenging but widely used in different scenarios, e.g., traffic surveillance, urban planning, satellite reconnaissance, or UAV detection.", "Since the introduction of Region-CNN [6], which uses region proposals and learns possible region features using a convolutional neural network instead of traditional manual features, many excellent object detection frameworks based on this structure were proposed, e.g., Light-head R-CNN [13], Fast/Faster R-CNN [5], [24], YOLO [22], [23], and SSD [17].", "These frameworks do, however, not work well for aerial imagery due to the challenges specific to this setting.", "In particular, the camera's bird's eye view and the high-resolution images make target recognition hard for the following reasons: (1) Features describing small vehicles with arbitrary orientation are difficult to extract in high-resolution images.", "(2) The large number of visually similar targets from different categories (e.g., building roofs, containers, water tanks) interfere with the detection.", "(3) There are many, densely packed target vehicles with typically monotonous appearance.", "(4) Occlusions and shadows increase the difficulty of feature extraction.", "Fig.", "REF illustrates some challenging examples in aerial imagery.", "Figure: Occlusion[29] evaluate recent frameworks on the DOTA dataset.", "Their results indicate that two-stage object detection frameworks [2], [24] do not work well for finding objects in dense scenarios, whereas one-stage object detection frameworks [17], [22] cannot detect dense and small targets.", "Moreover, all frameworks have problems detecting vehicles with arbitrary orientation.", "We argue that one of the important reasons is that RoI pooling uses interpolation to align region proposals of all sizes, which leads to a reduced accuracy or even loss of spatial information of the feature.", "Figure: Architecture: The backbone is a ResNet-101.Blue components represent subnetworks, gray color denotes feature maps, and yellow color indicates fully connected layers.The Region Proposal Network (RPN) proposes candidate RoIs, which are then applied to the feature maps from the third and the fourth convolutional blocks, respectively.Afterwards, RoIs from the third convolutional block are fed into the Localization Network to find the transformation parameters of local invariant features, and the Grid Generator matches the correspondence of pixel coordinates between RoIs from the third and the fourth convolutional blocks.Next, the Sampler determines which pixels are sampled.Finally, the regression and classifier output the vehicle detection results.To address these problems, we propose the Local-aware Region Convolutional Neural Network (LR-CNN) for vehicle detection in aerial imagery.", "The goal of LR-CNN is to make the deeper high-level semantic representation regain high-precision location information.", "We, therefore, predict affine transformation parameters from the shallower layer feature maps, containing a wealth of location information.", "After spatial transformation processing the pixels of the shallower layer feature maps are projected based on these transformation parameters onto the corresponding pixels of deeper feature maps containing higher-level semantic information.", "Finally, the resampled features, guided by the loss function, possess local invariance and contain location and high-level semantic information.", "To summarize, our contributions are the following: A novel network framework for vehicle detection in aerial imagery.", "Preserving the aggregate RoIs' feature translation invariance and addressing the boundary quantization issue for dense vehicles.", "Proposing a resampled pooled feature, which allows higher-level semantic features to regain location information and have local feature invariance.", "This allows detecting vehicles at an arbitrary orientation.", "An analysis of our results showing that we can detect vehicles in aerial imagery accurately and with tighter bounding boxes even in front of complex backgrounds." ], [ "Object detection.", "Recent object detection techniques can be roughly summarized in two ways.", "Two-step strategies first generate many candidate regions, which likely contain objects of interest.", "Then a separate sub-network determines the categories of each of these candidates and regresses the location.", "The most representative work is Faster R-CNN [24], which introduced the Region Proposal Network (RPN) for candidate generation.", "It is derived from R-CNN [6], which uses Selective Search [27] to generate candidate regions.", "SPPnet [8] proposed a Spatial Pyramid Pooling layer to obtain multi-scale features at a fixed feature size.", "Lastly, Fast R-CNN [5] introduced the ROIpooling layer and enabled the network to be trained in an end-to-end fashion.", "Because of its high precision and good performance on small objects and dense objects, Faster R-CNN is currently the most popular pipeline for object detection.", "In contrast, one-step approaches predict the location of objects and their category labels simultaneously.", "Representative works are YOLO [21], [22], [23] and SSD [17].", "Because there is no separate region proposal step this strategy is fast but achieves lower detection accuracy.", "Vehicle detection is a special case of object detection, i.e.", "the aforementioned methods can be directly applied [25], [28].", "These methods are, however, carefully designed to work on images collected from the ground, in which the objects have rich appearance characteristics.", "In contrast, visual information is very limited and monotonous when seen from an aerial perspective.", "Moreover, aerial images have much higher resolution (e.g., $5616\\times 3744$ in ITCVD [30] compared to $375\\times 500$ in ImageNet [3]) and cover a wider area.", "The objects of interest (vehicles in this work) are much smaller, and their scale, size, and orientation vary strongly.", "An important prior for object detection on ground-view images is that the main or large objects within an image are mostly at the image center [22].", "In contrast, an object's location is unpredictable in an aerial image.", "Selective search, RPN, or YOLO are therefore likely not ideal to handle these challenges.", "Given inaccurate region proposals, the following classifier cannot work well to make a final decision.", "More challenges include that vehicles can be in dark shadow, occluded by buildings, or packed densely on parking lots.", "All these challenges make the existing sophisticated object detection algorithms not well suited for aerial images.", "Vehicle detection in aerial images has been investigated by many recent studies, e.g.", "[1], [10], [16], [19], [20], [26], [31].", "[26], [31] extract features from shallower convolution layers (conv3 and conv4) through skip connections and fuse with the final features (output of conv5).", "Then a standard RPN is used on multi-scale feature maps to obtain proposals at different scales.", "[26] train a set of boosted classifiers to improve the final prediction accuracy.", "[31] use the focal loss [14] instead of the cross entropy as loss function for the RPN and the classification layer during training to overcome the easy/hard examples challenge.", "They report a significant improvement in this task.", "[1] propose to extract features hierarchically at different scales so that the network is able to detect objects in different sizes.", "To address the arbitrary orientation problem, they rotate the anchors of the proposals to some predefined angles [18], similar to [16].", "The number of anchors increases, however, dramatically to $N_{\\text{scales}}\\times N_{\\text{ratios}}\\times N_{\\text{angles}}$ and computation is costly." ], [ "Our Approach", "Motivated by DFL-CNN [31], our approach uses a two-stage object detection strategy, as shown in Fig.", "REF .", "In this section, we will give details for each of the sub-networks and discuss how our approach improves the accuracy for detecting vehicles in aerial images." ], [ "Base feature extractor", "Excessive downsampling can lead to a loss of feature information for small target vehicles.", "In contrast, low-level features from shallower layers can retain not only rich feature details of small targets, but also rich spatial information.", "We adopt ResNet-101 [9] and extract the base features from the shallow layers.", "As shown in Fig.", "REF , we use feature maps from the third and forth convolutional block, which have the same resolution.", "Since there is a 69 convolutional layer gap between the output of the third and fourth convolutional blocks, the latter contains deeper features, whereas the third convolutional block is relatively shallow and its output retains better spatial information of the pooled objects' features." ], [ "Twin region proposals.", "We model the region proposal network (RPN) as in [24].", "For each input image, the RPN outputs 128 potential RoIs, which are mapped to the features maps from the third $\\mathit {F\\_RoI}_\\mathit {conv3\\_x}$ and fourth $\\mathit {F\\_RoI}_\\mathit {conv4\\_x}$ convolutional block.", "[7] argue that the RoI pooling's nearest neighbor interpolation leads to a loss in translation invariance of the aligned RoI features.", "Low RoI alignment accuracy is, however, counterproductive for region proposal features that represent small target vehicles.", "We, therefore, use RoIAlign [7] instead of RoI pooling to aggregate high-precision RoIs.", "As Fig.", "REF illustrates, the $N \\times 512 \\times 128 \\times 128$ input from the third convolutional block will be sent into a large separable convolution (LSC) module containing two separate branches.", "Afterwards, the $N \\times 512 \\times 128 \\times 128$ feature is compressed to $N \\times 147 \\times 128 \\times 128$ position-sensitive score maps, which have 49 3-channel feature map blocks.", "This will greatly reduce the computational expense of generating position-sensitive score maps since the feature is now much thinner than it used to be [13].", "In the LSC module, each branch uses a large kernel size to enlarge the receptive field to preserve large local features.", "Large local features, while not accurate enough, retain more spatial information than local features extracted with small convolution kernels.", "This means that the larger local features facilitate further affine transformation parameterization, which effectively preserves the spatial information.", "As discussed above, RoI pooling increases noise in the feature representation when RoIs are aggregated.", "Additionally, [2] demonstrates that the translation invariance of the feature is lost after the RoI pooling operation.", "Inspired by both and following the structure of [2] we build the position-sensitive RoIAlign by replacing RoI pooling with RoIAlign.", "As the structure of position-sensitive RoIAlign indicates in Fig.", "REF , after aggregating by RoIAlign the precision of the RoIs' alignment strongly improves the sensitive position scoring and significantly reduces the noise of the small target feature.", "Since the distribution of large and small vehicle samples in aerial images is sparse, the ratio of positive and negative examples for training is very unbalanced.", "Hence, we use the focal loss [14], which reduces the weight for easy to classify examples, in order to improve the learnability of dense vehicle detection.", "The loss function of the RPN is defined as $\\begin{aligned}L_{\\text{RPN}}(\\lbrace p_{i}\\rbrace , \\lbrace t_{i} \\rbrace ) &= \\frac{\\alpha }{N_{\\text{cls}}}\\sum _{i}(p_{t,i}-1)^{\\gamma }\\log ({p_{t,i}}) \\\\&+ \\frac{\\lambda }{N_{\\text{regr}}}\\sum _{i}p_{i}^{*}f_{\\text{smooth L1},i}\\end{aligned}$ with $p_{t,i}&=&{\\left\\lbrace \\begin{array}{ll}p_i, & p_i^{*}=1 \\\\1-p_i, & \\text{otherwise}\\end{array}\\right.", "}\\\\f_{\\text{smooth L1},i}&=&{\\left\\lbrace \\begin{array}{ll}0.5(t_i-t_i^{*})^2, & {\\left| t-t^{*} \\right| < 1} \\\\\\left| t_i-t_i^{*} \\right| -0.5, & \\text{otherwise.}\\end{array}\\right.", "}$ Here, $i$ denotes the index of the proposal, $p_{i}$ is the predicted probability of the corresponding proposal, $p^{*}_i$ represents the ground truth label ($\\text{positive}=1$ , $\\text{negative}=0$ ).", "$t_{i}$ describes the predicted bounding box vector and $t_{i}^{*}$ indicates the ground truth box vector if $p_{i}^{*}= 1$ .", "We set the balance parameters $\\alpha = 1$ and $\\lambda = 1$ .", "The focusing parameter of the modulating factor $(p_{t,i}-1)^{\\gamma }$ is $\\gamma =2$ as in [14].", "Figure: The specific architecture of the Large Separable Convolution and Position-Sensitive RoIAlign blocks in Fig. .", "This subnet consists of three modules: Large separable convolutions (LSC), position-sensitive score maps, and RoIAlign.", "Each color of the output stands for the pooled results from each corresponding 3-channel position-sensitive score maps.", "Combined with region proposals from RPN, the position-sensitive RoIAlign creates a 128×3×7×7128 \\times 3 \\times 7 \\times 7 output for the localization network." ], [ "Resampled pooled feature", "[2], [7], [12] argue that RoI pooling uses interpolation to align the region proposal, which causes the pooled feature to lose location information.", "Due to this, they propose higher precision interpolations to improve the precision of RoI pooling.", "We instead assume that the region proposal undergoes an affine transformation after interpolation alignment, such as stretching, rotation, shifting, etc.", "We thus exploit spatial transformer networks (STNs) [11] to let the deep high-level semantic representation regain location information from the shallower features that retain the spatial information.", "Thereby, we strengthen the local feature invariance of the target vehicle in the RoI.", "The STN trains a model to predict the spatial variation and alignment of features (including translation, scaling, rotation, and other geometric transformations) by adaptively predicting the parameters of an affine transformation.", "Fig.", "REF depicts the architecture of a resampled pooled feature subnetwork.", "Six parameters are sufficient to describe the affine transformation [11].", "We feed the position-sensitive pooled feature $\\mathit {F_{ps}}$ from $\\mathit {F\\_RoI}_\\mathit {conv3\\_x}$ into the localization network and then parameterize the location information in the RoI as $\\theta $ , which are regressed $2 \\times 3$ parameters for describing the affine transformation.", "Next, standard pooled features $\\mathit {F_{st}}$ from $\\mathit {F\\_RoI}_\\mathit {conv4\\_x}$ are converted to a parameterised sampling grid to model the correspondence coordinate matrix $\\mathit {M_{t}}$ with transformation $T(\\theta )$ .", "It is placed at the pixel level between the resampled pooled feature $\\mathit {F_{rp}}$ and $\\mathit {F_{st}}$ by the grid generator.", "Once $\\mathit {M_{t}}$ has been modeled, $\\mathit {F_{rp}}$ will be pixel-wise resampled from $\\mathit {F_{st}}$ , and thus the spatial information is re-added to $\\mathit {F_{rp}}$ .", "The feature map visualization in Fig.", "REF shows that our resampled pooled features have enhanced the local feature invariance, and the feature representation of the vehicle placed at any direction is also very strong.", "Figure: The specific architecture of the resampled pooled feature subnetwork in Fig.", ", which consists of Localization Network, Grid Generator, and Sampler." ], [ "Loss of classifier and regressor", "For the final classifier and regression, we continue using the focal loss and the smooth $\\text{L}_1$ loss function, respectively: $\\begin{aligned}L_{\\text{LR-CNN}}(\\lbrace p_{j}\\rbrace , \\lbrace t_{j} \\rbrace ) &= \\frac{\\alpha }{N_{\\text{cls}}}\\sum _{j}(p_{t,j}-1)^{\\gamma }\\log ({p_{t,j}}) \\\\&+ \\frac{\\lambda }{N_{\\text{regr}}}\\sum _{j}p_{j}^{*}f_{\\text{smooth L1},j},\\end{aligned}$ where $j$ represents the index of the proposal.", "All other definitions are as in Eq.", "(REF ).", "The parameters remain as $\\alpha = 1$ , $\\lambda = 1$ and $\\gamma = 2$ .", "The total loss function can then be represented as $L = L_{\\text{RPN}}(\\lbrace p_{i} \\rbrace ,\\lbrace t_{i} \\rbrace ) + L_{\\text{LR-CNN}}(\\lbrace p_{j} \\rbrace ,\\lbrace t_{j} \\rbrace ).$" ], [ "Datasets", "We evaluate the proposed method on three datasets with different characteristics, testing different aspects of the accuracy of our method.", "The VEDAI [20] dataset consists of satellite imagery taken over Utah in 2012.", "It contains 1210 RGB images with a resolution of $1024\\times 1024$ pixels.", "VEDAI contains sparse vehicles and is challenging due to strong occlusions and shadows.", "DOTA [29] has 2806 aerial images, which are collected with different sensors and platforms.", "Their resolutions range from $800\\times 800$ to about $4k\\times 4k$ pixels.", "The dataset is randomly split into three sets: Half of the original images form the training set, 1/6 are used as validation set, and the remaining 1/3 form the testing set.", "Annotations are publicly accessible for all images not in the testing set.", "The experimental results on DOTA reported in this paper are therefore from the validation set.", "Furthermore, we evaluate the accuracy of detecting large and small vehicles separately for comparison purposes.", "The DLR 3K dataset [15] consists of 20 images (10 images for training and the other 10 for testing), which are captured at the height of about 1000 feet over Munich with a resolution of $5616\\times 3744$ pixels.", "This dataset is used to evaluate the generalization ability of our method.", "DOTA and VEDAI provide annotations of different kinds of object categories.", "Given the goal of this paper, we only use the vehicle annotations.", "Our method can, however, likely be generalized to detect arbitrary categories of interest.", "Because of the very high resolution of the images and limited GPU memory, we process images larger than $1024\\times 1024$ pixels in tiles.", "I.e., we crop them into $1024\\times 1024$ pixel patches with an overlap of 100 pixels.", "This truncates some targets.", "We only keep targets with more than $50\\%$ remaining as positive samples.", "In order to assess the accuracy of our framework, we adopt the standard VOC 2010 object detection evaluation metric [4] for quantitative results of precision, recall, and average precision.", "We use ResNet-101 as backbone network to learn features and initialize its parameters with a model pretrained on ImageNet [3].", "The remaining layers are initialized randomly.", "During training, stoch-astic gradient descent (SGD) is used to optimize the parameters.", "The base learning rate is 0.05 with a $10\\%$ decay every 3 epochs.", "The IoU thresholds for NMS are $0.7$ for training and $0.5$ for inference.", "The RPN part is trained first before the whole framework is trained jointly.", "All experiments were conducted with NVIDIA Titan XP GPUs.", "A single image with size $1024 \\times 1024$ keeps a maximum of 600 RoIs after NMS, and takes ca.", "1.4s during training and ca.", "0.33s for testing." ], [ "Results and comparison", "We compare our method with the state-of-the-art detection methods DFL [31] and the standard Faster R-CNN [24] as baseline.", "We evaluated these methods with their own settings on all datasets.", "Table: Experimental results showing average precision (AP) and mean AP (mAP) when detecting small (SV) and large vehicles (LV) separately in percent." ], [ "Quantitative results", "Tab.", "REF summarizes the experimental results.", "Note that our method outperforms all methods on all datasets.", "Furthermore, small vehicle and large vehicle on the DOTA Evaluation Server get 68.56% and 69.87% of AP respectively, and the mAP is 69.22%.", "Particularly, compared to the baseline method and the state-of-the-art, our model increases the AP by $27.41\\%$ and $7.71\\%$ on the most challenging dataset DOTA, respectively, corresponding to $63.9\\%$ and $12.3\\%$ relative gains.", "When small and large vehicles are considered as two classes, our model achieves $55.1\\%$ and $71.1\\%$ relative gains, respectively, against the baseline.", "The significant gains prove that our Large Separable Convolution, Position-Sensitive RoIAlign and Spatial Transform Network modules work efficiently.", "Figure: Precision-Recall curves given by different methods on the DOTA dataset.", "The color denotes the method while the line type denotes different tasks.Fig.", "REF depicts the precision-recall curves of different methods on DOTA.", "We can see that for vehicle detection our method (blue solid line) has a wider smooth region (until a recall of $0.65$ ) and smoother tendency, which means our method is more robust and has higher object classification precision than others.", "In contrast, both Faster R-CNN and DFL (red and green solid lines, respectively) have a rapid drop at the high-precision end of the plot.", "In other words, our method achieves higher recall without the cost of obviously sacrificing precision.", "We also can see that small vehicle detection is more difficult for all methods: The curves (pointed lines) begin to obviously drop much earlier (for LR-CNN at a recall of 0.4) than the general or large-vehicle detection (at a recall of 0.65), and the transition region is also wide (until a recall of 0.67 for LR-CNN).", "It is worth mentioning that DFL and LR-CNN have very good curves for large vehicle detection (dashed lines) with long smooth regions and a rapid drop." ], [ "Qualitative results", "Fig.", "REF gives a qualitative comparison between different methods on DOTA.", "It shows a typical complex scene: vehicles are in arbitrary places, dense or sparse, and the background is complex.", "As shown in the first row, Faster R-CNN fails to detect many vehicles, especially when they are dense (Regions 2, 3) or in shadow (Regions 5, 6).", "DFL detects more small vehicles.", "In particular, it is sensitive to the dark small vehicles, e.g., an unclear car on the road (Region 1) is detected.", "However, this has side effects: DFL cannot distinguish small dark vehicles from shadow well.", "E.g., the shadow of the white vehicle in Region 4 is detected as a small vehicle but the vehicles in Regions 5 and 6 are not detected.", "Furthermore, its accuracy for detecting vehicles in dense cases and classifying the vehicles' type is not good enough (Regions 2, 3).", "Fig.", "REF shows that our method distinguishes large and small vehicles well.", "It can also detect individual vehicles in dense parts of the scene.", "The advantages of detecting vehicles in dense situations and distinguishing the vehicles from the similar background objects are further showcased in the second row.", "Table: Experimental results (AP) on the training set of DLR 3K with the models trained on different other datasets.", "These experiments evaluate the generalization ability.For comparison, we cite the results of HRPN trained and evaluated on DLR 3K.Table: Ablation study.", "STN is fed with features from different convolution blocks of the backbone network for small (SV) and large (LV) vehicles." ], [ "Generalization ability", "To evaluate the generalization ability of our approach, we test it on the DLR 3K dataset with models trained on different datasets.", "Because the ground truth of the test set of DLR 3K is not publicly accessible, we test the models on the training and validation set whose annotations are available.", "We also compare the results with the ones reported in HRPN [26], which was trained on DLR 3K.", "Experimental results are listed in Tab.", "REF .", "We can see that, for each method, the model trained on DOTA reports higher AP than that trained on VEDAI.", "The main reason is that DOTA has more and more diverse training samples.", "DFL and our method trained on DOTA outperform HRPN with our method reporting about $10\\%$ better results than HRPN.", "These results show that our model has good generalization abilities as well as transferability.", "For better understanding, we show some examples in Fig.", "REF .", "When comparing the dashed purple boxes (results of models trained on VEDAI) with the green boxes (results of models trained on DOTA) from the same method, we can see that the models trained on DOTA detect more vehicles.", "When comparing the results of different methods trained on DOTA, we can see that LR-CNN successfully detects more vehicles.", "Within the region highlighted by the dashed yellow box where vehicles are dense, LR-CNN successfully detects almost all individual vehicles.", "Figure: LR-CNN" ], [ "Ablation study", "To evaluate the impact of the STN placed at different locations in the network, we conduct an ablation study.", "We do not provide separate experiments to evaluate the impact of focal loss and RoIAlign pooling because these have been provided in [14], [26] and [7], respectively.", "Tab.", "REF reports our results.", "When the STN is placed at the output of the conv3_x block, the model achieves better results, especially for large vehicle detection.", "The reason is that the STN mainly processes spatial information, which is much richer in the output features of conv3_x than in those of conv4_x.", "For better understanding, we visualize some feature maps in Fig.", "REF .", "The features extracted from conv3_x (second row) contain more spatial and detailed information than those from conv-4_x (fourth row): The edges are clearer and the locations corresponding to the vehicle show stronger activations.", "Comparing the feature maps before and after the STN (2nd row vs. 3rd row and 4th row vs. 5th row) shows that the activations of the background regions are weaker after the STN.", "Active regions corresponding to the foreground are closer to the vehicle's shape and orientation than before applying the STN since the features are transformed and regularized by the STN module.", "Furthermore, after STN processing, in addition to being accurate in position, the feature representation is also slimmer.", "This is why our bounding boxes are tighter than other detectors.", "From these observation, we can intuitively conclude that the STN module is better able to find the transformation parameters on conv3_x to regularize the features used to regress the location and classify the RoIs.", "Figure: OursFigure: Feature map (one example per column).", "Colors show activation strength.Fig.", "REF illustrates how the quality of proposals from RPN affects the final localization and classification.", "When comparing the final detection results (green boxes) with the RPN proposals (dashed purple boxes) of different methods, we can make the following observations: LR-CNN correctly detects more vehicles.", "In addition, the green bounding boxes given by LR-CNN are tighter, which means that LR-CNN gives more precise localization.", "To analyze the reasons for this, we compare the proposals (dashed purple boxes) of different methods.", "We can see that the proposals given by DFL and our method are closer to the targets than the ones of Faster R-CNN.", "Even though each vehicle is detected by its own RPN, the final classifier removes these proposals (Proposals 2 and 4) since they deviate from the ground truth location too much and contain too much background.", "Thus, the features pooled from these RoIs are not precise enough to represent the targets.", "Consequently, the final classifier cannot determine well based on these features whether they are an object of interest, especially in dense cases.", "To analyze why LR-CNN localizes the objects better, we look at the mathematical definition of target regression.", "The regression target for width is $t_w = \\log \\frac{G_w}{P_w}=\\log (1+\\frac{G_w-P_w}{P_w}).$ $G_w$ denotes the ground truth width and $P_w$ is the prediction.", "The target height $t_h$ is handled equivalently.", "Only when the prediction is close to the target, the equation can approximate a linear relationship: $\\lim _{x\\rightarrow 0} \\log (1+x) = 1+x$ (because the regression targets of center shift $(x,y)$ are already defined as a linear function and all these four parameters are predicted simultaneously.", "The regression layer is easier to be trained and works better when all the four target equation are linear).", "For all these reasons, our framework obtains better proposals in our RPN and yields better final classification and localization." ], [ "Discussion", "Compared to Faster R-CNN and DFL, our approach performs much better on detecting small targets.", "This improvement benefits from the skip connection structure that fuses the richer detail information from the shallower layers with the features from deeper layers, which contain higher-level semantic information.", "This is important for detecting small objects in high-resolution aerial images.", "In our method, the position-sensitive RoIAlign pooling is adopted to extract more accurate information compared with the traditional RoI pooling.", "An accurate representation is important for precisely locating and classifying small objects.", "Then our final classifier works better to determine the targets and further refine their location.", "Most importantly, the STN module in our framework regularizes the learned features after RoIAlign pooling well, which reduces the burden of the following layers that are expected to learn powerful enough feature representations for classification and further regression.", "That is the reason why LR-RCNN distinguishes small and large vehicles better and has more precise detection.", "All the above elements enable our method to have a good generalization ability and to reach a new state-of-the-art in vehicle detection in high resolution aerial images.", "We present an accurate local-aware region-based framework for vehicle detection in aerial imagery.", "Our method improves not only the boundary quantization issue for dense vehicles by aggregating the RoIs' features with higher precision, but also the detection accuracy of vehicles placed at arbitrary orientations by the high-level semantic pooled feature regaining location information via learning.", "In addition, we develop a training strategy to allow the pooled feature of location information lacking the precision to reacquire the accurate spatial information from shallower layer features via learning.", "Our approach achieves state-of-the-art accuracy for detecting vehicles in aerial imagery and has good generalization ability.", "Given these properties, we believe that it should also be easy to generalize by detecting additional object classes under similar circumstances." ], [ "Acknowledgment", "This work was supported by German Research Foundation (DFG) grants COVMAP (RO 2497/12-2) and PhoenixD (EXC 2122, Project ID 390833453).", "1.17" ] ]

[ [ "Edge plasmon-polaritons on isotropic semi-infinite conducting sheets" ], [ "Abstract From a three-dimensional boundary value problem for the time harmonic classical Maxwell equations, we derive the dispersion relation for a surface wave, the edge plasmon-polariton (EP), that is localized near and propagates along the straight edge of a planar, semi-infinite sheet with a spatially homogeneous, scalar conductivity.", "The sheet lies in a uniform and isotropic medium; and serves as a model for some two-dimensional (2D) conducting materials such as the doped monolayer graphene.", "We formulate a homogeneous system of integral equations for the electric field tangential to the plane of the sheet.", "By the Wiener-Hopf method, we convert this system to coupled functional equations on the real line for the Fourier transforms of the fields in the surface coordinate normal to the edge, and solve these equations exactly.", "The derived EP dispersion relation smoothly connects two regimes: a low-frequency regime, where the EP wave number, $q$, can be comparable to the propagation constant, $k_0$, of the ambient medium; and the nonretarded frequency regime in which $|q|\\gg |k_0|$.", "Our analysis indicates two types of 2D surface plasmon-polaritons on the sheet away from the edge.", "We extend the formalism to the geometry of two coplanar sheets." ], [ "Introduction", "Research efforts in the design and fabrication of two-dimensional (2D) materials rapidly evolved into a rich field at the crossroads of physics, chemistry, materials science and engineering.", "[1], [2] Some of these materials, including graphene and black phosphorus, are highly promising ingredients of nanophotonics at the mid- and near-infrared frequencies.", "[3] These systems may possibly sustain evanescent, fine-scale electromagnetic waves that are tightly confined to the boundary.", "[4], [5], [6], [7] An appealing surface wave is the 2D “bulk” surface plasmon-polariton (SP), which expresses collective excitations of the electron charge in the 2D plasma.", "[8], [9], [10] The SP may exhibit wavelengths much shorter than those in the ambient dielectric medium, and thus may overcome the typical diffraction limit.", "[11], [12] Experimental observations suggest that 2D bulk SPs on graphene nanoribbons are accompanied by different short-scale waves, termed “edge plasmon-polaritons” (EPs), which oscillate rapidly along the edges of the 2D material.", "[13], [14], [15], [16] The EP is localized near each edge on the sheet and may have a wavelength shorter than the one of the accompanying bulk SP at terahertz frequencies.", "This EP is intimately related to the edge magnetoplasmon which was observed to propagate along the boundary of the electron layer on liquid ${}^4$ He in the presence of a static magnetic field.", "[17], [18], [19] Aspects of this wave have been studied via linear models for confined 2D electron systems.", "[20], [21], [22], [23], [24] To our knowledge, many studies of the EP are restricted to the nonretarded frequency regime, in which the electric field is approximated by the gradient of a scalar potential (“quasi-electrostatic approach”).", "[5], [25] In this paper, we use the time harmonic classical Maxwell equations in three spatial dimensions (3D) in order to formally derive the dispersion relation of the EP on a semi-infinite, flat sheet with a straight edge and a homogeneous and isotropic surface conductivity.", "The sheet lies in a homogeneous and isotropic medium.", "We formulate a system of integral equations for the electric field tangential to the plane of the sheet; and apply the Wiener-Hopf method to solve these equations exactly via the Fourier transform in the sheet coordinate normal to the edge.", "In this sense, our treatment accounts for retardation effects.", "The EP dispersion relation is derived through the analyticity of the Fourier-transformed fields.", "Our tasks and results are summarized as follows.", "We formulate a boundary value problem for time harmonic Maxwell's equations in 3D.", "An ingredient is a jump condition for the tangential magnetic field across the sheet with an assumed local and tensor-valued surface conductivity.", "[9], [8] We convert this boundary value problem to a system of coupled integral equations for the electric field components tangential to the plane of the sheet.", "The kernel comes from the retarded Green function for the vector potential.", "We apply the Wiener-Hopf method [26], [27], [28], [29] when the sheet is spatially homogeneous.", "In this context, we use the Fourier transform of the fields in the surface coordinate normal to the edge, and convert the integral equations to functional equations on the real line.", "For isotropic sheets, we formally solve these functional equations exactly via a suitable linear transformation of the tangential electric field.", "We derive the EP dispersion relation via enforcing the requisite analyticity of the Fourier-transformed fields.", "The ensuing relation exhibits the joint contributions of transverse-magnetic (TM) and transverse-electric (TE) field polarizations.", "For a given EP wave number, we describe 2D SPs in the direction vertical to the edge.", "For a fixed phase of the surface conductivity with recourse to the Drude model,[10] we derive an asymptotic expansion for the EP wave number at low enough frequencies.", "This expansion provides a refined description of the gapless EP energy spectrum.", "[23] We compare the derived EP dispersion relation to the respective result of the quasi-electrostatic approach.", "[23] We extract the leading-order correction due to retardation.", "We provide an extension of our analysis to the geometry with two coplanar, semi-infinite sheets of distinct isotropic, spatially homogeneous surface conductivities.", "We should mention the models of hydrodynamic flavor for magnetoplasmons found in Refs. MastFetter1985,Fetter1985,Wuetal1985,Fetter1986,Fetter1986b.", "The main idea in these works is to couple the 2D (non-relativistic) linearized Euler equation with the 3D Poisson equation for an electrostatic potential.", "A dispersion relation for the edge magnetoplasmon is then obtained via an ad hoc simplification of the integral relation between the potential and the electron density; the exact kernel is replaced by a simpler one having the same infrared behavior.", "[18], [22], [17], [20], [21] This treatment offers insights into the effect of the geometry and the relative importance of bulk and edge contributions; see, e.g., Refs. Eliassonetal1986,Cataudella1987,Mikhailov1995,Wangetal2012,Zabolotnykh2016.", "A few limitations of these works, on the other hand, are evident.", "For instance, the time harmonic electric field is approximately expressed as the gradient of a scalar potential, which poses a restriction on the magnitude of the EP wave number.", "Moreover, the simplified relation between potential and electron density may become questionable, e.g., for the calculation of the EP phase velocity at long wavelengths.", "[24] Other works of similar, hydrodynamic character invoke the nonlocal mapping from the electron density to the potential along the 2D material, and resort to numerics in the context of the quasi-electrostatic approximation.", "[35], [36], [37] Note that in Ref.", "Vaman2014 the integral equations for the oscillation amplitudes of electrons are apparently solved explicitly, without any kernel approximation, yet by the neglect of retardation; see also Ref. ApostolVaman2009.", "An extension of these treatments to viscous electron flows via the Wiener-Hopf method is found in Ref. CohenGoldstein2018.", "The dispersion relation of edge magnetoplasmons has been systematically derived from an anti-symmetric tensor surface conductivity via the solution of an integral equation for the electrostatic potential by the Wiener-Hopf method.", "[19], [23] In these works, the quasi-electrostatic approximation is applied from the start.", "In contrast to Refs.", "MastFetter1985,Wuetal1985,Fetter1985,Fetter1986,Fetter1986b,VolkovMikhailov1985,VolkovMikhailov1988,Eliassonetal1986,Cataudella1987,Mikhailov1995,Wangetal2012,Zabolotnykh2016,CohenGoldstein2018,Gumbs1989,Rudin1997,Nikitin2011,Vaman2014, we solve the full Maxwell equations here, albeit in isotropic settings.", "We address the cases with a single sheet and two coplanar sheets with homogeneous scalar conductivities.", "Our approach is motivated by the need to describe the dispersion of plasmon-polaritons in a wide range of frequencies and 2D materials.", "[4] We obtain the EP dispersion relation in the form $\\mathcal {F}(q,\\omega )=0$ , where $q$ is the EP wave number, $\\omega $ is the frequency and $\\mathcal {F}$ is a transcendental function that, for a given surface conductivity function $\\sigma (\\omega )$ , smoothly connects: the nonretarded frequency regime, in which $q/\\omega ^2\\simeq {\\rm const.", "}$ ;[9] and the low-frequency regime, in which $q/\\omega \\simeq {\\rm const.", "}$ .", "We provide corrections to these leading-order terms by assuming that $\\Im \\,\\sigma (\\omega )>0$ ; this condition is consistent with the Drude model for $\\sigma (\\omega )$ .", "[10] In this vein, we analytically show how the EP dispersion relation bears the signatures of both the TM and TE polarizations.", "In particular, the contribution of the TE-polarization becomes relatively small in the quasi-electrostatic limit, for $\\Im \\,\\sigma (\\omega )>0$ .", "Our work points to several open questions.", "Our approach, relying on the solution of the full Maxwell equations, does not address anisotropic and nonlocal effects in the surface conductivity.", "[5], [24] Tensor-valued, spatially constant surface conductivities in principle can lead to challenging systems of Wiener-Hopf integral equations for the electric field.", "[40], [41], [42] We also neglect the effect that the edge, as a boundary of a 2D electron system, has on the conductivity.", "The geometry of the semi-infinite conducting sheet is not too realistic.", "The experimentally appealing case of nanoribbons will be the subject of future work.", "Since we focus on analytical aspects of EPs, numerical predictions will be addressed elsewhere.", "[43] The remainder of this paper is organized as follows.", "In Section , we summarize our key results for isotropic sheets.", "In Section , we state the boundary value problem (Section REF ); and formulate integral equations for the electric field on a flat sheet in a homogeneous isotropic medium (Section REF ).", "Section  describes the coupled functional equations for the Fourier transforms of the electric field components tangential to a homogeneous sheet.", "In Section , we use a homogeneous scalar conductivity to: obtain decoupled functional equations via a linear field transformation (Section REF ); and derive the EP dispersion relation (Section REF ).", "In Section  we compute the tangential electric field, and describe the 2D bulk SPs in the direction normal to the edge.", "In Section , we simplify the EP dispersion at low frequencies.", "Section  focuses on the asymptotics related to the quasi-electrostatic approximation.", "In Section , we extend our analysis to two coplanar isotropic sheets.", "Section  concludes the paper with a discussion of open problems." ], [ "Notation and terminology", "In our analysis, $\\mathbb {C}$ is the complex plane, $\\mathbb {R}$ is the set of real numbers and $\\mathbb {Z}$ is the set of integers.", "$w^*$ is the complex conjugate of $w$ ($w\\in \\mathbb {C}$ ).", "$\\Re w$ ($\\Im w$ ) denotes the real (imaginary) part of complex $w$ .", "Boldface symbols denote vectors or matrices; e.g., $\\mathbf {e}_\\ell $ is the $\\ell $ -directed unit Cartesian vector $(\\ell = x, y, z)$ .", "The Hermitian part of matrix $M$ is $\\frac{1}{2}({{M}^*}^T+M)$ where the asterisk $({}^*)$ and $T$ as superscripts denote complex conjugation and transposition, respectively.", "We write $f=\\mathcal {O}(g)$ ($f=o(g)$ ) to mean that $|f/g|$ is bounded by a nonzero constant (approaches zero) in a prescribed limit; and $f\\sim g$ implies $f-g=o(g)$ .", "The term “sheet” means either a material thin film, or a Riemann sheet as a branch of a multiple-value function.", "The terms “top Riemann sheet” and “first Riemann sheet” are employed interchangeably; ditto for the terms “wave number” and “propagation constant”.", "Given a function, $F(\\xi )$ , of a complex variable, $\\xi $ , we define the functions $F_\\pm (\\xi )$ by $F(\\xi )=F_+(\\xi )+F_-(\\xi )$ where (i) $F_+(\\xi )$ is analytic in the upper half $\\xi $ -plane, $\\mathbb {C}_+=\\lbrace \\xi \\in \\mathbb {C}: \\Im \\,\\xi >0\\rbrace $ ; and (ii) $F_-(\\xi )$ is analytic in the lower half $\\xi $ -plane, $\\mathbb {C}_-=\\lbrace \\xi \\in \\mathbb {C}: \\Im \\,\\xi <0\\rbrace $ .", "The $e^{-{\\rm i}\\omega t}$ time dependence is used throughout where $\\omega $ is the angular frequency, and $\\omega >0$ unless we state otherwise (${\\rm i}^2=-1$ ).", "We employ the International System of units (SI units) throughout.", "In this section, we summarize our key results regarding isotropic sheets.", "The derivations can be found in corresponding sections as specified below.", "Suppose that the conducting material is the set $\\Sigma =\\lbrace (x,y,z)\\in \\mathbb {R}^3: x>0, z=0\\rbrace $ , and has scalar surface conductivity $\\sigma (\\omega )$ in the frequency domain.", "Thus, the material edge is identified with the $y$ -axis.", "The sheet is surrounded by an isotropic and homogeneous medium.", "To study the EP dispersion, suppose that all fields have the $e^{{\\rm i}qy}$ dependence on the $y$ coordinate, where the complex $q$ needs to be determined as a function of $\\omega $ ." ], [ "Integral equations for EP electric field tangential to isotropic sheet", "The electric field parallel to the plane of the sheet is of the form $e^{{\\rm i}q y}E_\\parallel (x,z)$ where $E_\\parallel (x,z)=(\\widetilde{E}_x(x,z), \\widetilde{E}_y(x,z),0)^T$ .", "This $E_\\parallel (x,z)$ is continuous across the sheet.", "We show that, in the absence of any incident field, $E_\\parallel (x,z)$ at $z=0$ satisfies $E_\\parallel (x,0)= \\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}\\begin{pmatrix}\\displaystyle \\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2 & \\quad \\displaystyle {\\rm i}q \\frac{{\\rm d}}{{\\rm d}x} \\\\\\displaystyle {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} & k_{\\rm eff}^2 \\end{pmatrix}\\int _0^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\,E_\\parallel (x^{\\prime },0)~, \\quad \\mbox{all}\\ x\\ \\mbox{in}\\ \\mathbb {R}~.", "$ Here, we ignore the (zero) $z$ -component of $E_\\parallel $ , and define $k_0=\\omega \\sqrt{\\varepsilon \\mu }$ where $\\varepsilon $ and $\\mu $ are the dielectric permittivity and magnetic permeability of the ambient medium, respectively; and $k_{\\rm eff}^2=k_0^2-q^2$ with $\\Im \\,k_{\\rm eff}>0$ .", "The kernel is $K(x;q)=G(x,0;0,0)$ where $G(x,z;x^{\\prime },z^{\\prime })$ is the retarded Green function for the scalar Helmholtz equation with wave number $k_{\\rm eff}$ .", "The EP dispersion relation is saught by requiring that (REF ) admit nontrivial integrable solutions.", "Equation (REF ) is a particular case of the integral system obtained when the surface conductivity is anisotropic; see Section REF .", "The derivation of (REF ) and its extension is described in Section REF .", "In Section , we use the Fourier transform in $x$ in order to state the respective matrix Riemann-Hilbert problem." ], [ "EP dispersion relation for isotropic sheet", "Without loss of generality, assume that $\\Re \\,q>0$ .", "By (REF ) the EP dispersion relation is $\\exp \\left\\lbrace \\left[Q_+({\\rm i}q)+Q_-(-{\\rm i}q)\\right]-\\left[R_+({\\rm i}q)+R_-(-{\\rm i}q)\\right]\\right\\rbrace =-1~;$ for $\\Re \\,q<0$ simply replace $q$ by $-q$ in this relation.", "In the above, we define $Q_\\pm (\\xi )=\\pm \\frac{1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty \\frac{\\ln \\mathcal {P}_{\\rm TM}(\\xi ^{\\prime })}{\\xi ^{\\prime }-\\xi }\\,{\\rm d}\\xi ^{\\prime }~,\\quad R_\\pm (\\xi )=\\pm \\frac{1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty \\frac{\\ln \\mathcal {P}_{\\rm TE}(\\xi ^{\\prime })}{\\xi ^{\\prime }-\\xi }\\,{\\rm d}\\xi ^{\\prime }~,\\ \\pm \\Im \\,\\xi >0~;$ $\\mathcal {P}_{\\rm TM}(\\xi )=1-\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}(k_{\\rm eff}^2-\\xi ^2)\\widehat{K}(\\xi ;q)~,\\quad \\mathcal {P}_{\\rm TE}(\\xi ;q)=1-{\\rm i}\\omega \\mu \\sigma \\widehat{K}(\\xi ;q)~.$ Here, $\\widehat{K}(\\xi ;q)$ is the Fourier transform of kernel $K(x;q)$ ; $\\widehat{K}(\\xi ;q)=({\\rm i}/2)(k_{\\rm eff}^2-\\xi ^2)^{-1/2}$ with $\\Im \\sqrt{k_{\\rm eff}^2-\\xi ^2}>0$ for wave decay in $|z|$ .", "The derivation of (REF ) is provided in Section ." ], [ "Electric field tangential to plane of sheet near and away from edge", "Suppose that $q$ satisfies (REF ).", "By using the ensuing Fourier integrals for the components of $E_\\parallel (x,0)$ , we show that $\\mathbf {e}_x\\cdot E_\\parallel (x,0)$ is singular at the edge, viz., $\\mathbf {e}_x\\cdot E_\\parallel (x,0)=\\mathcal {O}(\\sqrt{k_0 x})\\ \\mbox{as}\\ x\\downarrow 0\\quad \\mbox{and}\\quad \\mathbf {e}_x\\cdot E_\\parallel (x,0)=\\mathcal {O}((k_0 x)^{-1/2})\\ \\mbox{as}\\ x\\uparrow 0~;$ whereas $\\mathbf {e}_y\\cdot E_\\parallel (x,0)$ is continuous and finite at the edge.", "For the derivations see Section REF .", "For the far field on the sheet (as $x\\rightarrow +\\infty $ for $z=0$ ) we write $E_\\parallel =E_\\parallel ^{\\rm sp}+E_\\parallel ^{\\rm rad}$ ; $E_\\parallel ^{\\rm sp}$ amounts to a 2D bulk SP as the residue contribution to the Fourier integrals from a zero of $\\mathcal {P}_{\\rm TM}(\\xi )$ or $\\mathcal {P}_{\\rm TE}(\\xi )$ , whereas $E_\\parallel ^{\\rm rad}$ is the branch cut contribution.", "We derive asymptotic formulas for $E_\\parallel ^{\\rm rad}(x,0)$ and exact formulas for $E_\\parallel ^{\\rm sp}(x,0)$ for $x>0$ .", "For example, we find that $|E_\\parallel ^{\\rm rad}(x,0)|=\\mathcal {O}\\biggl (\\frac{e^{-\\sqrt{q^2-k_0^2}\\,x}}{(\\sqrt{q^2-k_0^2} x)^{3/2}}\\biggr )\\quad \\mbox{as}\\ |\\sqrt{q^2-k_0^2}\\,x|\\rightarrow +\\infty ~,$ keeping $q/k_0$ and $\\omega \\mu \\sigma /k_0$ fixed.", "For details, see Section REF .", "In a similar vein, we have $|E_\\parallel ^{\\rm sp}(x,0)|=\\mathcal {O}\\bigl (e^{{\\rm i}k_{\\rm sp}x}\\bigr )$ where, for a lossless ambient medium ($k_0>0$ ), $k_{\\rm sp}$ is the zero in the upper half $\\xi $ -plane of $\\mathcal {P}_{\\rm TM}(\\xi )$ if $\\Im \\,\\sigma >0$ , or $\\mathcal {P}_{\\rm TE}(\\xi )$ if $\\Im \\,\\sigma <0$ ; see Section REF ." ], [ "Approximation for EP dispersion relation at low frequency", "If $|\\omega \\mu \\sigma (\\omega )/k_0|\\gg 1$ along with $\\Im \\,\\sigma (\\omega )>0$ , we show that (REF ) yields the approximation $\\frac{q-k_0}{k_0}\\sim \\frac{\\epsilon ^2}{2\\pi ^2}\\mathcal {A}(\\epsilon )^2\\quad \\mbox{where}\\quad e^{\\mathcal {A}(\\epsilon )}=\\frac{2e\\pi }{\\epsilon ^2\\mathcal {A}(\\epsilon )}~;\\quad \\epsilon =\\frac{{\\rm i}\\,2k_0}{\\omega \\mu \\sigma }\\quad (|\\epsilon |\\ll 1)~.$ For details, see Section .", "In view of the semi-classical Drude model[10] for $\\sigma (\\omega )$ , the above asymptotic formula indicates how $q$ approaches $k_0$ at low enough frequency, $\\omega $ ." ], [ "EP dispersion relation in nonretarded regime", "In the nonretarded frequency regime, when $|\\omega \\mu \\sigma (\\omega )/k_0|\\ll 1$ with $\\Im \\,\\sigma (\\omega )>0$ ,[9], [10] the EP dispersion relation can be derived by the quasi-electrostatic approach.", "[19], [23], [25] By carrying out an asymptotic expansion for exact result (REF ), we derive the formula $q\\sim {\\rm i}\\eta _0\\,\\frac{2k_0^2}{\\omega \\mu \\sigma }\\biggl \\lbrace 1-\\eta _1\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2\\biggr \\rbrace ~.$ In the above, $\\eta _0$ is a numerical factor ($\\eta _0\\simeq 1.217$ ) that amounts to the result of the quasi-electrostatic approximation;[23] and $\\eta _1$ is a positive constant ($\\eta _1\\simeq 0.416$ ) that signifies the leading-order correction due to retardation; see Section ." ], [ "Extension of EP theory to two coplanar conducting sheets", "Consider the coplanar sheets described by the sets $\\Sigma ^L=\\lbrace (x,y,z)\\in \\mathbb {R}^3\\,:\\,z=0,\\, x<0\\rbrace $ and $\\Sigma ^R=\\lbrace (x,y,z)\\in \\mathbb {R}^3\\,:\\,z=0,\\, x>0\\rbrace $ , which lie in an isotropic homogeneous medium.", "Suppose that their scalar, spatially constant conductivities are $\\sigma ^L$ and $\\sigma ^R$ , respectively ($\\sigma ^{L}\\ne \\sigma ^{R}$ and $\\sigma ^L\\sigma ^R\\ne 0$ ).", "The electric field tangential to the sheets in $\\Sigma =\\Sigma ^L\\cup \\Sigma ^R$ satisfies $E_\\parallel (x,0) = \\frac{{\\rm i}\\omega \\mu }{k_0^2}\\begin{pmatrix}\\displaystyle \\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2 & \\quad \\displaystyle {\\rm i}q \\frac{{\\rm d}}{{\\rm d}x} \\\\\\displaystyle {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} & k_{\\rm eff}^2 \\end{pmatrix}\\int _{-\\infty }^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\sigma (x^{\\prime })\\,E_\\parallel (x^{\\prime },0) \\quad \\mbox{all}\\ x\\ \\mbox{in}\\ \\mathbb {R}~,$ where $\\sigma (x)=\\sigma ^L+\\vartheta (x) (\\sigma ^R-\\sigma ^L)$ ; $\\vartheta (x)=1$ if $x>0$ and $\\vartheta (x)=0$ if $x<0$ (see Section ).", "We show that the equation for $E_\\parallel (x,0)$ admits nontrivial integrable solutions if $q$ obeys (REF ) with definitions (REF ) for $Q_\\pm (\\xi )$ and $R_{\\pm }(\\xi )$ .", "However, in the latter formulas one should make the replacement $\\mathcal {P}_{\\rm \\varpi }(\\xi )\\rightarrow \\mathcal {P}_{\\varpi }^R(\\xi )/\\mathcal {P}_{\\varpi }^L(\\xi )$ where $\\mathcal {P}_{\\varpi }^{\\ell }(\\xi )$ is defined by (REF ) with $\\sigma \\rightarrow \\sigma ^{\\ell }$ ($\\varpi ={\\rm TM}, {\\rm TE}$ and $\\ell =R, L$ ).", "See Section  for some details." ], [ "Boundary value problem and integral equations", "In this section, we formulate a boundary value problem for time harmonic Maxwell's equations in the geometry with a semi-infinite conducting sheet in an unbounded, isotropic and homogeneous medium.", "We also derive a system of integral equations for the electric field tangential to the plane of the sheet.", "Our formulation includes nonhomogeneous and anisotropic sheets with local surface conductivities; a generalization is provided in Ref.", "MMSLL-preprint." ], [ "Geometry and boundary value problem", "The geometry of the problem is depicted in Fig.", "REF .", "This consists of: a semi-infinite conducting sheet, $\\Sigma =\\lbrace (x,y,z)\\in \\mathbb {R}^3: x> 0~,\\ z=0\\rbrace $ , in the $xy$ -plane; and the surrounding unbounded, homogeneous and isotropic medium.", "The sheet, $\\Sigma $ , has a local and in principle tensor-valued surface conductivity, $\\sigma ^\\Sigma $ , which may depend on coordinates $x,y$ and the frequency, $\\omega $ .", "Thus, allowing $\\sigma ^\\Sigma $ to act on vectors in $\\mathbb {R}^3$ , we use the matrix representation $\\sigma ^\\Sigma =\\begin{pmatrix} \\sigma _{xx} & \\sigma _{xy} & 0\\\\\\sigma _{yx} & \\sigma _{yy} & 0\\\\0 & 0 & 0\\end{pmatrix}~,$ where matrix elements $\\sigma _{\\ell m}$ ($\\ell , m=x, y$ ) are in principle complex-valued functions of $x$ , $y$ and $\\omega $ .", "(However, in Section REF $\\sigma ^\\Sigma $ is not allowed to depend on $y$ ).", "The ambient space has a constant dielectric permittivity, $\\varepsilon $ , and a constant magnetic permeability, $\\mu $ .", "For non-active 2D materials, the Hermitian part of this $\\sigma ^\\Sigma $ must be positive semidefinite; also, $\\sigma ^\\Sigma $ must obey the Onsager reciprocity relations.", "[44], [45] We note in passing that by causality $\\sigma ^\\Sigma (\\omega )$ must be analytic in the upper $\\omega $ -plane (for $\\Im \\,\\omega >0$ ), if $\\omega $ becomes complex.", "Figure: Geometry of the problem.", "A semi-infinite conducting sheet, Σ\\Sigma , lies in the xyxy-plane for x>0x>0, and has the local and in principle tensor-valued surface conductivity σ Σ \\sigma ^\\Sigma .", "The sheet is surrounded by a homogeneous and isotropic medium of wave number k 0 =ωεμk_0=\\omega \\sqrt{\\varepsilon \\mu }, where ε\\varepsilon is the dielectric permittivity and μ\\mu is the magnetic permeability.The curl laws of the time harmonic Maxwell equations outside the sheet $\\Sigma $ read $\\nabla \\times E={\\rm i}\\omega B~,\\ \\nabla \\times (\\mu ^{-1} B)=-{\\rm i}\\omega \\varepsilon E+J_e\\quad \\mbox{in}\\ \\mathbb {R}^3\\setminus \\overline{\\Sigma }~;\\ \\overline{\\Sigma }:=\\lbrace (x,y,z): x\\ge 0, z=0\\rbrace ~.$ Here, $E$ and $B$ are the electric and magnetic fields, respectively; and $J_e$ is the compactly supported current density of an external source.", "On $\\Sigma $ , we impose the boundary conditions [46] $\\left[\\mathbf {e}_z\\times E\\right]_\\Sigma =0~,\\quad \\left[\\mathbf {e}_z\\times B\\right]_\\Sigma =\\mu \\,\\mathfrak {J}\\quad \\mbox{on}\\ \\Sigma ~.$ Here, $[Q]_\\Sigma :=Q(x,y,z=0^+)-Q(x,y,z=0^-)$ for $x>0$ , which denotes the jump of $Q(x,y,z)$ across $\\Sigma $ ; and $\\mathfrak {J}$ on $\\Sigma $ is the vector-valued surface flux induced on the sheet, viz., $\\mathfrak {J}(x,y)=\\sigma ^\\Sigma \\, E(x,y,z)=(\\sigma _{xx}E_x+\\sigma _{xy}E_y)\\mathbf {e}_x+(\\sigma _{yx}E_x+\\sigma _{yy}E_y)\\mathbf {e}_y\\quad \\mbox{for}\\ z=0~,\\ x>0~.$ We set $\\mathfrak {J}(x,y)\\equiv 0$ if $x<0$ .", "More generally, $\\mathfrak {J}$ can be a linear functional of $E$ at $z=0$ .", "[25] We alert the reader that (REF ) and (REF ) introduce the volume current density, $J_e$ , as distinct from the induced surface current density, $\\mathfrak {J}$ , on $\\Sigma $ .", "This distinction is justified if the domain of Maxwell's equations is $\\mathbb {R}^3\\setminus \\overline{\\Sigma }$ ; and highlights the different physical origins of the two current densities.", "An alternate yet mathematically equivalent view, which we adopt for convenience in Section REF , is to extend the domain of Maxwell's laws to the whole Euclidean space and include $\\mathfrak {J}$ in their source term by treating it as a distribution (delta function in $z$ ).", "We now discuss a suitable far-field condition.", "[47], [48], [49] To determine the EP dispersion relation, we will set $J_e=0$ and solve the ensuing homogeneous boundary value problem by assuming that the solution is a wave, the EP, that travels along and remains localized near the $y$ -axis (Section REF ).", "In this setting, the imposition of an outgoing wave in the $\\pm z$ -directions in addition to having an exponentially decaying and outgoing wave in the $y$ -direction may yield a solution that is exponentially increasing with $|z|$ , similarly to the problem of the infinitely long microstrip.", "[49] We will therefore consider solutions that decay exponentially in the directions perpendicular to the sheet.", "An implication of this assumption is outlined in the end of Section ." ], [ "Edge-plasmon polariton and integral equations for tangential electric field", "Next, we derive integral equations for $E_x$ and $E_y$ at $z=0$ , by introducing the EP as a particular solution.", "We assume that the surface conductivity, $\\sigma ^\\Sigma $ , is independent of $y$ and the fields are traveling waves in the $y$ -direction; the related wave number is to be determined.", "We invoke the vector potential, ${A}^{\\text{sc}}$ , of the scattered field $(E^{\\rm sc}, B^{\\rm sc})$ in the Lorenz gauge.", "The ${A}^{\\text{sc}}$ of course satisfies $B^{\\rm sc}=\\nabla \\times {A}^{\\text{sc}}$ outside $\\overline{\\Sigma }$ .", "Our derivation of integral equations for the electric field here is akin to the derivation of the Pocklington integral equation for electric currents on thin cylindrical antennas in uniform media.", "[50] Our integral formalism is a particular case of the “electric field integral equation” approach in electromagnetics.", "[51] To account for the EP, we consider fields of the form $F(x,y,z)=e^{{\\rm i}q y} \\widetilde{F}(x,z;q)$ and replace $\\nabla $ by $(\\partial _x,{\\rm i}q,\\partial _z)$ , where $q$ is a complex wave number to be determined and $F=E,\\,B, {A}^{\\text{sc}}, \\mathfrak {J}, J_e$ .", "Now drop the tildes from all respective variables, which depend on $x$ or $z$ , for ease of notation.", "This procedure amounts to taking the Fourier transform of Maxwell's equations and the boundary conditions with respect to $y$ (where $q$ is the `dual variable').", "By (REF ) and (REF ), ${A}^{\\text{sc}}$ is due to the electron flow on the sheet and, thus, obeys the following nonhomogeneous Helmholtz equation on $\\mathbb {R}^2$ : $(\\Delta _{x,z}+k_0^2-q^2){A}^{\\text{sc}}(x,z)=-\\mu \\, \\mathfrak {J}(x)\\,\\delta (z)\\ \\mbox{for\\ all}\\ (x,z)\\ \\mbox{in}\\ \\mathbb {R}^2~,$ where $\\delta (z)$ is the Dirac delta function and $\\Delta _{x,z}$ is the 2D Laplacian ($\\Delta _{x,z}=\\partial ^2/\\partial x^2+\\partial ^2/\\partial z^2$ ).", "The vector potential ${A}^{\\text{sc}}(x,z)$ is given in terms of the surface current, $\\mathfrak {J}(x)$ , by [47] ${A}^{\\text{sc}}(x,z)=\\mu \\int _{\\mathbb {R}} G(x,z;x^{\\prime },0)\\, \\mathfrak {J}(x^{\\prime })\\,\\,{\\rm d}x^{\\prime }~\\quad \\mbox{in}\\ \\mathbb {R}^2\\setminus \\overline{\\Sigma }_2~,$ where $\\overline{\\Sigma }_2:=\\lbrace (x,z)\\in \\mathbb {R}^2: x\\ge 0, z=0\\rbrace $ and $G(x,z;x^{\\prime },z^{\\prime })$ is given by $G(x,z;x^{\\prime },z^{\\prime })=\\textstyle {\\frac{{\\rm i}}{4}}H_0^{(1)}\\left(k_{\\rm eff}\\sqrt{(x-x^{\\prime })^2+(z-z^{\\prime })^2}\\right)~,\\quad k_{\\rm eff}:=\\sqrt{k_0^2-q^2}~,\\ \\Im \\,k_{\\rm eff}>0~,$ by use of the first-kind Hankel function, $H_0^{(1)}(w)$ , of the zeroth order.", "[52] This $G$ comes from the (retarded) Green function for the scalar Helmholtz equation, with effective wave number $k_{\\rm eff}$ .", "Note that by (REF ) ${A}^{\\text{sc}}(x,z)$ is continuous everywhere, [48] and equals ${A}^{\\text{sc}}=A_x^{\\rm sc} \\mathbf {e}_x+A_y^{\\rm sc} \\mathbf {e}_y$ where each component $A_\\ell ^{\\rm sc}=\\mathbf {e}_\\ell \\cdot {A}^{\\text{sc}}$ is determined by $\\mathbf {e}_\\ell \\cdot \\mathfrak {J}$ at $z=0$ ($\\ell =x,\\,y$ ).", "We compute $B_x^{\\rm sc}(x,z)&=-\\frac{\\partial A_y^{\\rm sc}}{\\partial z}~,\\quad B_y^{\\rm sc}(x,z)=\\frac{\\partial A_x^{\\rm sc}}{\\partial z}~,\\quad B_z^{\\rm sc}(x,z)=\\frac{\\partial A_y^{\\rm sc}}{\\partial x}-{\\rm i}q A_x^{\\rm sc}\\quad \\mbox{in}\\ \\mathbb {R}^2\\setminus \\overline{\\Sigma }_2~.$ Hence, by the Ampère-Maxwell law from (REF ) we find the field components (defined in $\\mathbb {R}^2\\setminus \\overline{\\Sigma }_2$ ) $E_x^{\\rm sc}(x,z)&=\\frac{{\\rm i}\\omega }{k_0^2}\\left\\lbrace \\left(\\frac{\\partial ^2}{\\partial x^2}+k_0^2\\right)A_x^{\\rm sc}+{\\rm i}q\\frac{\\partial A_y^{\\rm sc}}{\\partial x}\\right\\rbrace ~,\\ E_y^{\\rm sc}(x,z)=\\frac{{\\rm i}\\omega }{k_0^2}\\left({\\rm i}q\\frac{\\partial A_x^{\\rm sc}}{\\partial x}+k_{\\rm eff}^2 A_y^{\\rm sc}\\right)~,$ which are continuous across the half line $\\Sigma _2:=\\lbrace (x,z)\\in \\mathbb {R}^2: x>0, z=0\\rbrace $ , the projection of the physical sheet on the $xz$ -plane.", "Thus, $E_x^{\\rm sc}$ and $E_y^{\\rm sc}$ obey the first condition in (REF ).", "One can verify that ($E^{\\rm sc}$ , $B^{\\rm sc}$ ) satisfies Faraday's law in (REF ) and the second condition in (REF ).", "To obtain the desired integral equations for $E_x$ and $E_y$ , we use (REF ).", "Thus, we find $E_x^{\\rm sc}(x,z)&=\\frac{{\\rm i}\\omega \\mu }{k_0^2}\\left\\lbrace \\left(\\frac{\\partial ^2}{\\partial x^2}+k_0^2\\right)\\int _0^\\infty G(x,z;x^{\\prime },0)\\,\\left[\\sigma _{xx}E_x(x^{\\prime },0)+\\sigma _{xy}E_y(x^{\\prime },0)\\right]\\,{\\rm d}x^{\\prime }\\right.\\\\&\\qquad \\left.", "+{\\rm i}q\\frac{\\partial }{\\partial x}\\int _0^\\infty G(x,z;x^{\\prime },0)\\,\\left[\\sigma _{yx}E_x(x^{\\prime },0)+\\sigma _{yy} E_y(x^{\\prime },0)\\right]\\,{\\rm d}x^{\\prime }\\right\\rbrace ~,\\\\E_y^{\\rm sc}(x,z)&=\\frac{{\\rm i}\\omega \\mu }{k_0^2}\\left\\lbrace {\\rm i}q\\frac{\\partial }{\\partial x} \\int _0^\\infty G(x,z;x^{\\prime },0)\\,\\left[\\sigma _{xx}E_x(x^{\\prime },0)+\\sigma _{xy}E_y(x^{\\prime },0)\\right]\\,{\\rm d}x^{\\prime }\\right.", "\\\\&\\qquad \\left.", "+k_{\\rm eff}^2 \\int _0^\\infty G(x,z;x^{\\prime },0)\\,\\left[\\sigma _{yx}E_x(x^{\\prime },0)+\\sigma _{yy}E_y(x^{\\prime },0)\\right]\\,{\\rm d}x^{\\prime }\\right\\rbrace \\quad \\mbox{in}\\ \\mathbb {R}^2\\setminus \\overline{\\Sigma }_2~.$ In these expressions, we take the limit $z\\rightarrow 0$ for $x\\ne 0$ .", "[48] In the absence of any external source, when $J_e\\equiv 0$ , we have $(E_x^{\\rm sc}, E_y^{\\rm sc})=(E_x, E_y)$ .", "For ease of notation, define $u(x):=E_x(x,0)~,\\ v(x):=E_y(x,0)~,\\ K(x;q):=G(x,0;0,0)=({\\rm i}/4)H_0^{(1)}(k_{\\rm eff}|x|)~.$ The resulting system of (homogeneous) integral equations reads as $u(x)&=\\frac{{\\rm i}\\omega \\mu }{k_0^2}\\left\\lbrace \\left(\\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2\\right)\\int _0^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\,\\left[\\sigma _{xx}u(x^{\\prime })+\\sigma _{xy}v(x^{\\prime })\\right]\\right.", "\\\\&\\qquad \\left.", "+{\\rm i}q\\frac{{\\rm d}}{{\\rm d}x}\\int _0^\\infty {\\rm d}x^{\\prime }\\,K(x-x^{\\prime };q)\\,\\left[\\sigma _{yx}u(x^{\\prime })+\\sigma _{yy} v(x^{\\prime })\\right] \\right\\rbrace ~, \\\\v(x)&=\\frac{{\\rm i}\\omega \\mu }{k_0^2}\\left\\lbrace {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} \\int _0^\\infty {\\rm d}x^{\\prime }\\,K(x-x^{\\prime };q)\\,\\left[\\sigma _{xx}u(x^{\\prime })+\\sigma _{xy}v(x^{\\prime })\\right] \\right.", "\\\\&\\qquad \\left.", "+k_{\\rm eff}^2 \\int _0^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\,\\left[\\sigma _{yx}u(x^{\\prime })+\\sigma _{yy}v(x^{\\prime })\\right]\\right\\rbrace ~,\\quad \\mbox{all}\\ x\\ \\mbox{in}\\ \\mathbb {R}\\setminus \\lbrace 0\\rbrace ~.", "$ Let us now formally extend the domain of these equations to the whole $\\mathbb {R}$ .", "We have $\\begin{pmatrix} u(x) \\\\v(x)\\end{pmatrix} = \\frac{{\\rm i}\\omega \\mu }{k_0^2}\\begin{pmatrix}\\displaystyle \\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2 & \\quad \\displaystyle {\\rm i}q \\frac{{\\rm d}}{{\\rm d}x} \\\\\\displaystyle {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} & k_{\\rm eff}^2 \\end{pmatrix}\\int _0^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\,\\sigma ^\\Sigma _2\\,\\begin{pmatrix} u(x^{\\prime }) \\\\v(x^{\\prime })\\end{pmatrix} \\quad \\mbox{all}\\ x\\ \\mbox{in}\\ \\mathbb {R}~, $ where $\\sigma _2^\\Sigma :=\\begin{pmatrix}\\sigma _{xx} & \\sigma _{xy} \\\\\\sigma _{yx} & \\sigma _{yy}\\end{pmatrix}~.", "$ Note that this $\\sigma _2^\\Sigma $ may depend on the coordinate $x^{\\prime }$ .", "For a generalization of (REF ) to 2D materials in which the induced surface current density is a linear functional of $(u(x), v(x))$ , see Ref. MMSLL-preprint.", "The problem is to find $q$ so that matrix equation (REF ) admits nontrivial solutions.", "The right-hand side of (REF ) involves the field components tangential to the physical sheet, for $x^{\\prime }>0$ under the integral sign.", "Strictly speaking, (REF ) yields a system of integral equations for $u(x)$ and $v(x)$ by restriction of both sides of this equation to $x>0$ .", "By solving these equations for an isotropic sheet, we will verify that any nontrivial, admissible electric field component $u(x)$ , which is normal to the edge, is singular and discontinuous at $x=0$ (Section ).", "In contrast, $v(x)$ turns out to be continuous at $x=0$ (Section ).", "We can state a more precise definition of the EP; cf.", "Refs. Fetter1985,VolkovMikhailov1988.", "Definition 1 (Edge plasmon-polariton).", "The EP amounts to nontrivial integrable solutions $(u,v)$ and corresponding wave number, $q$ , of (REF ) ($u,\\,v\\in L^1(\\mathbb {R})$ ).", "The EP dispersion relation describes how the $q$ of this solution is related to the angular frequency, $\\omega $ .", "An assumption underlying Definition 1 is that nontrivial integrable solutions $u(x)$ and $v(x)$ of (REF ), and the corresponding $q$ 's, exist for some range of frequencies $\\omega $ , given some meaningful model for the surface conductivity $\\sigma _2^\\Sigma $ .", "We will construct such solutions by the Wiener-Hopf method for the simplified model with $\\sigma _2^\\Sigma =\\sigma I_2$ where $I_2$ is the $2\\times 2$ unit matrix and $\\sigma $ is a spatially constant but $\\omega $ -dependent scalar quantity (Sections  and )." ], [ "Homogeneous sheet: Coupled functional equations", "In this section, we reduce (REF ) to a system of functional equations on the real line for Fourier-transformed fields via the Wiener-Hopf method, [26], [29] if $\\sigma _{xx}~,\\ \\sigma _{xy}~,\\ \\sigma _{yx}~,\\ \\sigma _{yy}\\ \\mbox{are\\ spatially\\ constant}.", "$ Equation (REF ) is recast to the system $\\begin{pmatrix} u(x) \\\\v(x)\\end{pmatrix} = \\frac{{\\rm i}\\omega \\mu }{k_0^2}\\begin{pmatrix}\\displaystyle \\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2 & \\quad \\displaystyle {\\rm i}q \\frac{{\\rm d}}{{\\rm d}x} \\\\\\displaystyle {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} & k_{\\rm eff}^2 \\end{pmatrix}\\sigma ^\\Sigma _2\\,\\int _{-\\infty }^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime };q)\\,\\begin{pmatrix} u_{>}(x^{\\prime }) \\\\v_{>}(x^{\\prime })\\end{pmatrix} \\quad x\\ \\mbox{in}\\ \\mathbb {R}~, $ where $\\begin{pmatrix} u_{>}(x) \\\\v_{>}(x)\\end{pmatrix}:= \\begin{pmatrix} u(x) \\\\v(x)\\end{pmatrix}\\quad \\mbox{if}\\ x>0~,\\quad \\begin{pmatrix} u_{>}(x) \\\\v_{>}(x)\\end{pmatrix}\\equiv 0\\quad \\mbox{if}\\ x<0~.$ Furthermore, we define the functions $u_{<}(x)$ and $v_{<}(x)$ via the relations $u(x)=u_{>}(x)+u_{<}(x)$ and $v(x)=v_{>}(x)+v_{<}(x)$ for all $x$ in $\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ .", "The Fourier transform of $f(x)$ , with $f\\in L^1(\\mathbb {R})$ , is $\\widehat{f}(\\xi )= \\int _{-\\infty }^{\\infty } {\\rm d}x\\, f(x) e^{-{\\rm i}\\xi x}~.$ Hence, if $f(x)\\equiv 0$ for $x<0$ then $\\widehat{f}(\\xi )$ is analytic in the lower half $\\xi $ -plane, $\\mathbb {C}_-$ ; whereas $\\widehat{f}(\\xi )$ is analytic in the upper half plane, $\\mathbb {C}_+$ , if $f(x)\\equiv 0$ for $x>0$ .", "[26], [28], [53] The application of the Fourier transform in $x$ to (REF ) yields the system $\\begin{pmatrix} \\widehat{u}_+(\\xi ) \\\\\\widehat{v}_+(\\xi )\\end{pmatrix} +\\begin{pmatrix} \\widehat{u}_-(\\xi ) \\\\\\widehat{v}_-(\\xi )\\end{pmatrix}= \\frac{{\\rm i}\\omega \\mu }{k_0^2}\\, \\widehat{K}(\\xi ;q)\\begin{pmatrix}\\displaystyle k_0^2-\\xi ^2 & \\ \\displaystyle ({\\rm i}q)({\\rm i}\\xi ) \\\\\\displaystyle ({\\rm i}q)({\\rm i}\\xi ) & \\ k_0^2-q^2 \\end{pmatrix}\\sigma ^\\Sigma _2\\,\\begin{pmatrix} \\widehat{u}_{-}(\\xi ) \\\\\\widehat{v}_{-}(\\xi )\\end{pmatrix}\\quad \\mbox{all}\\ \\xi \\ \\mbox{in}\\ \\mathbb {R}~, $ which describes two coupled functional equations on the real line.", "In the above, we have $\\widehat{K}(\\xi ;q)=\\int _{-\\infty }^\\infty {\\rm d}x\\,K(x;q) \\, e^{-{\\rm i}\\xi x}=\\frac{{\\rm i}}{2}(k_{\\rm eff}^2-\\xi ^2)^{-1/2}~,\\quad k_{\\rm eff}=\\sqrt{k_0^2-q^2}~,$ where $\\Im \\sqrt{k_{\\rm eff}^2-\\xi ^2}>0$ since we impose decay of $\\widehat{G}(\\xi ,z;0,0)$ with $\\Im \\,k_{\\rm eff}>0$ as $|z|\\rightarrow \\infty $ ; see (REF ).", "With this choice of the top Riemann sheet $\\widehat{K}(\\xi ;q)$ is an even function of $\\xi $ .", "Note that the requisite branch cuts, which emanate from $\\xi =\\pm k_{\\rm eff}=\\pm \\sqrt{k_0^2-q^2}$ ($\\Im \\,k_{\\rm eff}>0$ ), lie in $\\mathbb {C}_\\pm $ and are infinite and symmetric with respect to the origin.", "We also define $\\begin{pmatrix} \\widehat{u}_{\\pm }(\\xi ) \\\\\\widehat{v}_{\\pm }(\\xi )\\end{pmatrix}= \\int _{-\\infty }^{\\infty }{\\rm d}x\\ \\begin{pmatrix}u_{< \\atop >}(x) \\\\v_{< \\atop >}(x)\\end{pmatrix}e^{-{\\rm i}\\xi x}~.$ Of course, $\\widehat{u}_\\pm (\\xi )$ and $\\widehat{v}_\\pm (\\xi )$ depend on $q$ ; for ease of notation, we suppress this dependence.", "Two comments are in order.", "First, $\\widehat{u}_\\pm (\\xi )$ and $\\widehat{v}_\\pm (\\xi )$ for real $\\xi $ are viewed as limits of the corresponding analytic functions as $\\xi $ approaches the real axis from $\\mathbb {C}_+$ or $\\mathbb {C}_-$ .", "Thus, (REF ) expresses a Riemann-Hilbert problem on the real line.", "This type of problem, and the respective matrix Wiener-Hopf integral equation associated with it, can be solved explicitly, with the solution in simple closed form, only in a limited number of cases; see, e.g., Refs. GohbergKrein1960,WuWu1963,Abrahams1997.", "We will solve (REF ) explicitly for the special case with a scalar constant conductivity, i.e., if $\\sigma _2^\\Sigma =\\sigma I_2$ where $\\sigma $ is a scalar constant in $x$ and $y$ and $I_2={\\rm diag}(1,1)$ (Section ).", "Second, recall that we impose $\\Im \\sqrt{k_{\\rm eff}^2-\\xi ^2}>0$ with $\\Im \\,k_{\\rm eff}>0$ in the $\\xi $ -plane.", "Suppose for a moment that $\\Re \\,q>0$ and $\\Im \\,q>0$ , i.e., the EP is an outgoing and decaying wave in the positive $y$ -direction; then, $\\Im \\,k_{\\rm eff}^2=\\Im (k_0^2-q^2)<0$ if the ambient medium is lossless ($k_0>0$ ).", "Hence, the condition $\\Re \\,k_{\\rm eff}<0$ must be satisfied, given that $\\Im \\,k_{\\rm eff}>0$ .", "By the prescribed choice of the branch cut for $\\sqrt{k_{\\rm eff}^2-\\xi ^2}$ and the respective integration path in the $\\xi $ -plane, we conclude that $\\Re \\sqrt{k_{\\rm eff}^2-\\xi ^2}<0$ ; cf.", "Ref. TTWu1957.", "The sign reversal of $\\Re \\,q$ , i.e., the mapping $q\\mapsto -q^*$ , causes the sign change of $\\Re \\sqrt{k_{\\rm eff}^2-\\xi ^2}$ ." ], [ "Edge plasmon on isotropic homogeneous sheet", "In this section, we restrict attention to the case with an isotropic and homogeneous conducting sheet.", "Hence, we set $\\sigma _2^\\Sigma =\\sigma \\,I_2~, $ where $\\sigma $ is a scalar function of $\\omega $ with $\\Re \\,\\sigma (\\omega )\\ge 0$ .", "We will explicitly solve (REF ) via a suitable transformation of $(\\widehat{u}_\\pm (\\xi ), \\widehat{v}_\\pm (\\xi ))$ and subsequent factorizations in the $\\xi $ -plane.", "[26], [40] Equation (REF ) is recast to the system $\\begin{pmatrix} \\widehat{u}_+(\\xi ) \\\\\\widehat{v}_+(\\xi )\\end{pmatrix} +\\begin{pmatrix} \\widehat{u}_-(\\xi ) \\\\\\widehat{v}_-(\\xi )\\end{pmatrix}= \\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}\\, \\widehat{K}(\\xi ;q)\\begin{pmatrix}\\displaystyle k_0^2-\\xi ^2 & \\ \\displaystyle ({\\rm i}q)({\\rm i}\\xi ) \\\\\\displaystyle ({\\rm i}q)({\\rm i}\\xi ) & \\ k_0^2-q^2 \\end{pmatrix}\\begin{pmatrix} \\widehat{u}_{-}(\\xi ) \\\\\\widehat{v}_{-}(\\xi )\\end{pmatrix}\\qquad (\\mbox{all\\ real}\\ \\xi )~.", "$ Now define the matrix $\\Lambda (\\xi ;q):= \\begin{pmatrix}\\displaystyle 1-\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}(k_0^2-\\xi ^2)\\widehat{K}(\\xi ;q) & \\ \\displaystyle -\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}({\\rm i}q)({\\rm i}\\xi ) \\widehat{K}(\\xi ;q) \\\\\\displaystyle -\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}({\\rm i}q)({\\rm i}\\xi )\\widehat{K}(\\xi ; q) & \\ \\displaystyle 1-\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}(k_0^2-q^2) \\widehat{K}(\\xi ;q) \\end{pmatrix}~.$ Accordingly, the functional equations under consideration are expressed by $\\Lambda (\\xi ; q)\\begin{pmatrix} \\widehat{u}_{-}(\\xi ) \\\\\\widehat{v}_{-}(\\xi )\\end{pmatrix}+\\begin{pmatrix} \\widehat{u}_{+}(\\xi ) \\\\\\widehat{v}_{+}(\\xi )\\end{pmatrix}=0\\qquad (\\mbox{all\\ real}\\ \\xi )~.", "$" ], [ "Linear transformation and explicit solution", "The key observation is that in the present setting we can explicitly define a matrix valued function, $\\mathcal {T}(\\xi ;q)$ , such that the transformed vector valued function $\\begin{pmatrix}U(\\xi ) \\\\V(\\xi )\\end{pmatrix}:= \\mathcal {T}(\\xi ;q)\\begin{pmatrix}\\widehat{u}(\\xi )\\\\\\widehat{v}(\\xi )\\end{pmatrix}$ has the following properties: (i) ($U_s(\\xi ), V_s(\\xi ))^T=\\mathcal {T}(\\xi ;q) (\\widehat{u}_s(\\xi ),\\widehat{v}_s(\\xi ))^T$ where the superscript $T$ denotes transposition and $s=\\pm $ ; and (ii) the components $U_s(\\xi )$ and $V_s(\\xi )$ separately satisfy two (decoupled) functional equations on the real axis.", "The first property [item (i)] directly follows from (REF ) if each matrix element in $\\mathcal {T}(\\xi ;q)$ is an entire function of $\\xi $ .", "To this end, we diagonalize $\\Lambda (\\xi ;q)$ .", "Consider an invertible matrix $\\mathcal {S}$ such that $\\Lambda (\\xi ;q)=\\mathcal {S}(\\xi ;q)\\, {\\rm diag}(\\mathcal {P}_1(\\xi ;q),\\mathcal {P}_2(\\xi ;q))\\,\\mathcal {S}^{-1}(\\xi ;q)\\qquad (\\xi \\in \\mathbb {C})~,$ where $\\mathcal {P}_j(\\xi ;q)$ are eigenvalues of $\\Lambda (\\xi ;q)$ ($j=1,\\,2$ ).", "The associated eigenvalues satisfy $&\\mathcal {P}^2-\\left\\lbrace 2-\\frac{i\\omega \\mu \\sigma }{k_0^2}(2k_0^2-q^2-\\xi ^2)\\widehat{K}(\\xi ;q)\\right\\rbrace \\mathcal {P}+1-\\frac{i\\omega \\mu \\sigma }{k_0^2}(2k_0^2-q^2-\\xi ^2)\\widehat{K}(\\xi ;q)\\\\&\\qquad + \\left(\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0}\\right)^2 (k_{\\rm eff}^2-\\xi ^2) \\,\\widehat{K}(\\xi ;q)^2=0~,$ which has the distinct solutions $\\mathcal {P}_1=\\mathcal {P}_{\\rm TM}(\\xi ;q):=1-\\frac{{\\rm i}\\omega \\mu \\sigma }{k_0^2}(k_{\\rm eff}^2-\\xi ^2)\\widehat{K}(\\xi ; q)~,\\quad \\mathcal {P}_2=\\mathcal {P}_{\\rm TE}(\\xi ;q):=1-{\\rm i}\\omega \\mu \\sigma \\widehat{K}(\\xi ; q)$ where $k_{\\rm eff}=\\sqrt{k_0^2-q^2}$ and $\\widehat{K}(\\xi ;q)$ is defined by (REF ).", "Note that $\\mathcal {P}_{\\rm TM}(\\pm {\\rm i}q;q)=\\mathcal {P}_{\\rm TE}(\\pm {\\rm i}q;q)$ .", "We will show how the contributions from $\\mathcal {P}_{\\rm TM}$ and $\\mathcal {P}_{\\rm TE}$ enter the EP dispersion relation (Section REF ).", "In regard to these eigenvalues, $\\mathcal {P}_{\\rm TM}$ corresponds to TM polarization while $\\mathcal {P}_{\\rm TE}$ amounts to TE polarization.", "This terminology is motivated as follows: The roots $\\xi $ of $\\mathcal {P}_{\\rm TM}(\\xi ;0)=0$ or $\\mathcal {P}_{\\rm TE}(\\xi ;0)=0$ provide the propagation constants in the $x$ -direction for the TM- or TE-polarized SP on the respective infinite 2D conducting material with $q=0$ .", "[8], [9], [54] Alternatively, by replacing $\\xi $ by $\\sqrt{q_x^2+q_y^2}$ in these roots, where $q_\\ell $ is the wave number in the $\\ell $ -direction ($\\ell =x,y$ ), and solving for $\\omega (q_x,q_y)$ one recovers the continuum energy spectrum of the TM- or TE-polarized SP on the infinite sheet.", "[9], [5] The roots $\\xi $ for each case are present in the top Riemann sheet under suitable conditions on the phase of $\\sigma $ (see Section REF ).", "[8], [9], [54], [55] By an elementary calculation, eigenvectors of $\\Lambda (\\xi ;q)$ are given by $\\begin{pmatrix}{\\rm i}\\xi \\cr {\\rm i}q\\end{pmatrix}\\ \\mbox{for}\\ \\mathcal {P}=\\mathcal {P}_{\\rm TM}\\quad \\mbox{and}\\quad \\begin{pmatrix}{\\rm i}q \\cr -{\\rm i}\\xi \\end{pmatrix}\\ \\mbox{for}\\ \\mathcal {P}=\\mathcal {P}_{\\rm TE}~,$ which depend on the material parameters through $q$ if the latter satisfies a dispersion relation.", "Hence, the matrix $\\mathcal {S}$ can be taken to be equal to $\\mathcal {S}(\\xi ;q)=\\begin{pmatrix}{\\rm i}\\xi & {\\rm i}q \\cr {\\rm i}q & -{\\rm i}\\xi \\end{pmatrix}~,$ which is an entire matrix valued function of $\\xi $ , and invertible for all complex $\\xi $ with $\\xi \\ne \\pm {\\rm i}q$ .", "Once we compute $\\mathcal {S}^{-1}=-(\\xi ^2+q^2)^{-1}\\mathcal {S}$ , we write $\\Lambda (\\xi ;q)=-\\frac{1}{q^2+\\xi ^2}\\mathcal {S}(\\xi ;q)\\begin{pmatrix}\\mathcal {P}_{\\rm TM}(\\xi ; q) & 0 \\cr 0 & \\mathcal {P}_{\\rm TE}(\\xi ;q)\\end{pmatrix}\\mathcal {S}(\\xi ;q)~.$ Accordingly, by (REF ) we obtain the expression $\\begin{pmatrix}\\mathcal {P}_{\\rm TM}(\\xi ; q) & 0 \\cr 0 & \\mathcal {P}_{\\rm TE}(\\xi ;q)\\end{pmatrix}\\mathcal {S}(\\xi ;q)\\begin{pmatrix}\\widehat{u}_-(\\xi )\\cr \\widehat{v}_-(\\xi )\\end{pmatrix}+\\mathcal {S}(\\xi ;q)\\begin{pmatrix}\\widehat{u}_+(\\xi )\\cr \\widehat{v}_+(\\xi )\\end{pmatrix}=0\\quad (\\mbox{all\\ real}\\ \\xi )~.$ Thus, by recourse to (REF ) we can set $\\mathcal {T}(\\xi ;q)=\\mathcal {S}(\\xi ;q)=\\begin{pmatrix}{\\rm i}\\xi & {\\rm i}q \\cr {\\rm i}q & -{\\rm i}\\xi \\end{pmatrix}~.$ This choice implies the transformation $(\\widehat{u},\\widehat{v})\\mapsto (U, V)$ with $U(\\xi )={\\rm i}\\xi \\, \\widehat{u}(\\xi )+{\\rm i}q\\, \\widehat{v}(\\xi )~,\\quad V(\\xi )={\\rm i}q\\,\\widehat{u}(\\xi )-{\\rm i}\\xi \\,\\widehat{v}(\\xi )~.$ Evidently, $(U_\\pm (\\xi ), V_\\pm (\\xi ))^T$ may result from the application of $\\mathcal {T}(\\xi ;q)$ to $(\\widehat{u}_\\pm (\\xi ), \\widehat{v}_\\pm (\\xi ))^T$ .", "Remark 1 (On the transformation for $\\widehat{u}$ and $\\widehat{v}$ ).", "Equations (REF ) represent the Fourier tranforms with respect to $x$ of $-\\partial E_z(x,z)/\\partial z=\\partial E_x/\\partial x+{\\rm i}q\\, E_y$ and $-i\\omega B_z(x,z)={\\rm i}q E_x(x,z)-\\partial E_y/\\partial x$ at $z=0$ by omission of any boundary terms for $E_x(x,0)$ and $E_y(x,0)$ (as $x\\rightarrow 0$ ).", "This absence of boundary terms is consistent with the presence of a non-integrable singularity of $\\partial E_x/\\partial x$ and the continuity of $E_y(x,z)$ as $x\\rightarrow 0$ at $z=0$ (Section ).", "We return to the task of computing $\\widehat{u}(\\xi )$ and $\\widehat{v}(\\xi )$ .", "The functions $U_\\pm (\\xi )$ and $V_\\pm (\\xi )$ obey $&\\mathcal {P}_{\\rm TM}(\\xi ;q)U_-(\\xi )+U_+(\\xi )=0~,\\\\&\\mathcal {P}_{\\rm TE}(\\xi ;q)V_-(\\xi )+V_+(\\xi )=0\\quad \\mbox{for\\ all}\\ \\xi \\ \\mbox{in}\\ \\mathbb {R}~.$ Hence, loosely speaking, the contributions from the TE- and TM-polarizations are now decoupled.", "Our goal is to solve (REF ) explicitly (Section REF ); and then account for transformation (REF ) in order to obtain $q$ as well as the corresponding nontrivial $\\widehat{u}_\\pm (\\xi )$ and $\\widehat{v}_\\pm (\\xi )$ .", "We should alert the reader that the approach of matrix diagonalization, which we apply above, is tailored to the present isotropic model of the surface conductivity.", "This approach is in principle not suitable for a strictly anisotropic conductivity in functional equations ().", "This limitation can be attributed to the ensuing analytic structure of the matrix $\\mathcal {S}(\\xi ;q)$ ." ], [ "Derivation of EP dispersion relation", "Let us assume that for all admissible $q$ the functions $\\mathcal {P}_{\\rm TM}(\\xi ;q)$ and $\\mathcal {P}_{\\rm TE}(\\xi ;q)$ satisfy $\\mathcal {P}_{\\rm TM}(\\xi ;q)\\ne 0\\ \\mbox{and}\\ \\mathcal {P}_{\\rm TE}(\\xi ;q)\\ne 0\\quad \\mbox{for\\ all\\ real}\\ \\xi ~.$ Hence, the functions $\\ln \\mathcal {P}_{\\rm TM}(\\xi )$ and $\\ln \\mathcal {P}_{\\rm TE}(\\xi )$ , which we invoke below, are analytic in a vicinity of the real axis in the $\\xi $ -plane.", "The above conditions imply that the respective bulk SPs, for fixed $q$ , do not have real propagation constants in the first Riemann sheet ($\\Im \\sqrt{k_0^2-q^2-\\xi ^2}>0$ ).", "To simplify the notation, we henceforth suppress the $q$ -dependence in quantities such as $\\mathcal {P}_{\\rm TM}$ and $\\mathcal {P}_{\\rm TE}$ .", "In order to solve (REF ) we need to carry out factorizations of $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ , i.e., determine `split functions' $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ such that [26] $Q(\\xi ):=\\ln \\mathcal {P}_{\\rm TM}(\\xi )=Q_+(\\xi )+Q_-(\\xi )~,\\quad R(\\xi ):=\\ln \\mathcal {P}_{\\rm TE}(\\xi )=R_+(\\xi )+R_-(\\xi )~,$ which is a classic problem in complex analysis.", "The EP dispersion relation will be expressed in terms of functions $Q_\\pm $ and $R_\\pm $ .", "Note that $Q(\\xi )$ and $R(\\xi )$ are even functions in the top Riemann sheet.", "It is useful to introduce the (vector-valued) index, $\\nu $ , for functional equations (REF ).", "This $\\nu $ expresses the indices associated with $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ on the real axis, viz., [26], [29] $\\nu :=\\frac{1}{2\\pi {\\rm i}}\\lim _{M\\rightarrow +\\infty }\\int _{-M}^M\\begin{pmatrix}\\lbrace {\\mathcal {P}^{\\prime }_{\\rm TM}}(\\xi )/\\mathcal {P}_{\\rm TM}(\\xi )\\rbrace \\cr \\lbrace {\\mathcal {P}^{\\prime }_{\\rm TE}}(\\xi )/\\mathcal {P}_{\\rm TE}(\\xi )\\rbrace \\end{pmatrix} \\,{\\rm d}\\xi =\\frac{1}{2\\pi }\\lim _{M\\rightarrow +\\infty }\\begin{pmatrix}\\arg \\mathcal {P}_{\\rm TM}(\\xi ) \\cr \\arg \\mathcal {P}_{\\rm TE}(\\xi )\\end{pmatrix}\\Biggl |_{\\xi =-M}^M~,$ where the prime here denotes differentiation with respect to the Fourier variable $\\xi $ .", "The components of this $\\nu $ express the changes of the values for $(2\\pi {\\rm i})^{-1}\\ln \\mathcal {P}_{\\rm TM}(\\xi )$ and $(2\\pi {\\rm i})^{-1}\\ln \\mathcal {P}_{\\rm TE}(\\xi )$ as $\\xi $ moves between the extremities of the real axis.", "Thus, each component of $\\nu $ is the winding number with respect to the origin of a contour, $C_0^{\\varpi }$ , in the complex $\\mathcal {P}_\\varpi $ -plane under the mapping $\\xi \\mapsto \\mathcal {P}_\\varpi (\\xi )$ which maps the real axis to $C_0^\\varpi $ ($\\varpi ={\\rm TM}$ or ${\\rm TE}$ ).", "Because $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ are even functions of $\\xi $ , we can assert that $\\nu =0$ which implies that splitting (REF ) makes sense and can be carried out directly via the Cauchy integral formula.", "[26], [55] In contrast, for certain strictly anisotropic conducting sheets, the index for the underlying Wiener-Hopf integral equations in the quasi-electrostatic approach may be nonzero, which implies distinct possibilities regarding the existence, or lack thereof, of the EP.", "[23], [25] This material anisotropy lies beyond the scope of the present paper.", "Therefore, we can directly apply the Cauchy integral formula and obtain [26], [55] $Q_\\pm (\\xi )&=\\pm \\frac{1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty \\frac{Q(\\xi ^{\\prime })}{\\xi ^{\\prime }-\\xi }\\ {\\rm d}\\xi ^{\\prime }=\\pm \\frac{\\xi }{{\\rm i}\\pi }\\int _0^\\infty \\frac{Q(\\xi ^{\\prime })}{{\\xi ^{\\prime }}^2-\\xi ^2}\\ {\\rm d}\\xi ^{\\prime }~, \\\\R_\\pm (\\xi )&=\\pm \\frac{1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty \\frac{R(\\xi ^{\\prime })}{\\xi ^{\\prime }-\\xi }\\ {\\rm d}\\xi ^{\\prime }=\\pm \\frac{\\xi }{{\\rm i}\\pi }\\int _{-\\infty }^\\infty \\frac{R(\\xi ^{\\prime })}{{\\xi ^{\\prime }}^2-\\xi ^2}\\ {\\rm d}\\xi ^{\\prime }\\quad (\\pm \\Im \\,\\xi >0)~, $ in view of definitions (REF ).", "Equations (REF ) then read $e^{Q_-(\\xi )}U_-(\\xi )=-e^{-Q_+(\\xi )}U_+(\\xi )~,\\quad e^{R_-(\\xi )}V_-(\\xi )=-e^{-R_+(\\xi )}V_+(\\xi )\\qquad \\mbox{for\\ all}\\ \\xi \\ \\mbox{in}\\ \\mathbb {R}~.$ By analytic continuation of each side of the above equations to complex $\\xi $ , in $\\mathbb {C}_+$ or $\\mathbb {C}_-$ , we infer that there exist entire functions $\\mathcal {E}_j(\\xi )$ ($j=1,\\,2$ ) such that [26] $& e^{Q_-(\\xi )}U_-(\\xi )=-e^{-Q_+(\\xi )}U_+(\\xi )=\\mathcal {E}_1(\\xi )~,\\\\& e^{R_-(\\xi )}V_-(\\xi )=-e^{-R_+(\\xi )}V_+(\\xi )=\\mathcal {E}_2(\\xi )\\qquad \\mbox{for\\ all}\\ \\xi \\ \\mbox{in}\\ \\mathbb {C}~.$ Each of these $\\mathcal {E}_j(\\xi )$ can be determined by examination of $Q_\\pm (\\xi )$ , $R_\\pm (\\xi )$ , $U_\\pm (\\xi )$ and $V_\\pm (\\xi )$ as $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_+$ or $\\mathbb {C}_-$ .", "It is compelling to consider only polynomials as candidates for $\\mathcal {E}_j(\\xi )$ .", "Let us now discuss in detail the issue of determining $\\mathcal {E}_j(\\xi )$ .", "Recall that the electric-field components $E_x(x,0)$ and $E_y(x,0)$ are assumed to be integrable on $\\mathbb {R}$ .", "Hence, $\\widehat{u}_\\pm (\\xi )\\rightarrow 0$ and $\\widehat{v}_\\pm (\\xi )\\rightarrow 0$ as $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_\\pm $ .", "[28], [53] By transformation (REF ), we infer that $U_\\pm (\\xi ),\\,V_\\pm (\\xi )\\ \\mbox{cannot\\ grow\\ as\\ fast\\ as}\\ \\xi $ in the limit $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_\\pm $ .", "To express this behavior, we write $|U_\\pm (\\xi )|< \\mathcal {O}(\\xi )$ and $|V_\\pm (\\xi )|<\\mathcal {O}(\\xi )$ as $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_\\pm $ .", "Now consider the asymptotics for $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ when $|\\xi |$ is large; see the Appendix.", "We can assert that $e^{Q_\\pm (\\xi )}=\\mathcal {O}(\\sqrt{\\xi })\\ \\mbox{and}\\ e^{R_\\pm (\\xi )}\\rightarrow 1\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_\\pm ~.$ These estimates imply that $|e^{Q_-(\\xi )}U_-(\\xi )|< \\mathcal {O}(\\xi \\sqrt{\\xi })\\ \\mbox{and}\\ |e^{-Q_+(\\xi )}U_+(\\xi )|<\\mathcal {O}(\\sqrt{\\xi })\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty $ in $\\mathbb {C}_-$ or $\\mathbb {C}_+$ , respectively.", "In a similar vein, we have $|e^{\\mp R_\\pm (\\xi )}V_\\pm (\\xi )|<\\mathcal {O}(\\xi )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_\\pm ~.$ Hence, we find that the entire functions $\\mathcal {E}_1(\\xi )$ and $\\mathcal {E}_2(\\xi )$ satisfy $\\mathcal {E}_1(\\xi )<\\mathcal {O}(\\sqrt{\\xi })\\ \\mbox{and}\\ \\mathcal {E}_2(\\xi )<\\mathcal {O}(\\xi )\\ \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}~.$ Thus, resorting to Liouville's theorem, we conclude that $\\mathcal {E}_1(\\xi )=C_1={\\rm const}.\\quad \\mbox{and}\\quad \\mathcal {E}_2(\\xi )=C_2={\\rm const}.\\qquad \\mbox{for\\ all}\\ \\xi \\in \\mathbb {C}~.$ These constants, $C_1$ and $C_2$ , have units of electric field and are both arbitrary so far.", "We proceed to determine $\\widehat{u}_\\pm (\\xi )$ and $\\widehat{v}_\\pm (\\xi )$ in terms of $C_1$ and $C_2$ , and then obtain the EP dispersion relation.", "Equations (REF ) and (REF ) lead to $U_\\pm (\\xi )=\\mp C_1 e^{\\pm Q_\\pm (\\xi )}~,\\quad V_\\pm (\\xi )=\\mp C_2 e^{\\pm R_\\pm (\\xi )}~.$ In view of transformation (REF ), we readily obtain the formulas $\\widehat{u}_-(\\xi )&=-\\frac{{\\rm i}\\xi \\, U_-(\\xi )+{\\rm i}q\\, V_-(\\xi )}{q^2+\\xi ^2}=-\\frac{{\\rm i}\\xi \\,C_1 e^{-Q_-(\\xi )}+{\\rm i}q\\,C_2 e^{-R_-(\\xi )}}{q^2+\\xi ^2}~,\\\\\\widehat{v}_-(\\xi )&=-\\frac{{\\rm i}q\\,U_-(\\xi )-{\\rm i}\\xi \\,V_-(\\xi )}{q^2+\\xi ^2}=-\\frac{{\\rm i}q\\,C_1 e^{-Q_-(\\xi )}-{\\rm i}\\xi \\,C_2e^{-R_-(\\xi )}}{q^2+\\xi ^2}~, $ for the fields $u_>(x)$ and $v_>(x)$ , along with the formulas $\\widehat{u}_+(\\xi )&=-\\frac{{\\rm i}\\xi \\, U_+(\\xi )+{\\rm i}q\\, V_+(\\xi )}{q^2+\\xi ^2}=\\frac{{\\rm i}\\xi \\,C_1 e^{Q_+(\\xi )}+{\\rm i}q\\,C_2 e^{R_+(\\xi )}}{q^2+\\xi ^2}~,\\\\\\widehat{v}_+(\\xi )&=-\\frac{{\\rm i}q\\,U_+(\\xi )-{\\rm i}\\xi \\,V_+(\\xi )}{q^2+\\xi ^2}=\\frac{{\\rm i}q\\,C_1 e^{Q_+(\\xi )}-{\\rm i}\\xi \\,C_2e^{R_+(\\xi )}}{q^2+\\xi ^2}~, $ in regard to $u_<(x)$ and $v_<(x)$ .", "Notice the appearance of the factor $(\\xi ^2+q^2)^{-1}$ .", "Now define ${\\rm sg}(q):=\\left\\lbrace \\begin{array}{lr}1 & \\ \\mbox{if}\\ \\Re \\,q>0~,\\cr -1 & \\ \\mbox{if}\\ \\Re \\,q<0~,\\end{array}\\right.$ which is the signum function for $\\Re \\,q$ .", "Since $\\widehat{u}_-(\\xi )$ and $\\widehat{v}_-(\\xi )$ are analytic in $\\mathbb {C}_-$ , by (REF ) we impose the conditions that ${\\rm i}\\xi \\,C_1 e^{-Q_-(\\xi )}+{\\rm i}q\\,C_2 e^{-R_-(\\xi )}=0$ and ${\\rm i}q\\,C_1 e^{-Q_-(\\xi )}-{\\rm i}\\xi \\,C_2e^{-R_-(\\xi )}=0$ at $\\xi =-{\\rm i}q\\,{\\rm sg}(q)$ , which entail the relation $C_1 e^{-Q_-(-{\\rm i}q\\,{\\rm sg}(q))} +{\\rm i}\\, {\\rm sg}(q)\\,C_2 e^{-R_-(-{\\rm i}q\\,{\\rm sg}(q))}=0\\quad \\mbox{if}\\ \\Re \\, q\\ne 0~.$ Another condition should be dictated at $\\xi ={\\rm i}q\\, {\\rm sg}(q)$ by use of $Q_+$ and $R_+$ .", "By (REF ) we require that ${\\rm i}\\xi C_1 e^{Q_+(\\xi )}+{\\rm i}q C_2 e^{R_+(\\xi )}$ and ${\\rm i}q C_1 e^{Q_+(\\xi )}-{\\rm i}\\xi C_2 e^{R_+(\\xi )}$ vanish at $\\xi ={\\rm i}q\\,{\\rm sg}(q)$ .", "Thus, $C_1e^{Q_+({\\rm i}q\\, {\\rm sg}(q))}-{\\rm i}\\, {\\rm sg}(q) C_2 e^{R_+({\\rm i}q\\, {\\rm sg}(q))}=0~, \\ \\Re \\, q\\ne 0~.", "$ Equations (REF ) form a linear system for $(C_1, C_2)$ .", "For nontrivial solutions of this system, we require that $e^{R_+({\\rm i}q\\,{\\rm sg}(q))-Q_-(-{\\rm i}q\\,{\\rm sg}(q))}+e^{Q_+({\\rm i}q\\,{\\rm sg}(q))-R_-(-{\\rm i}q\\,{\\rm sg}(q))}=0~,$ which is recast to the expression $\\lbrace Q_+({\\rm i}q\\,{\\rm sg}(q))+Q_-(-{\\rm i}q\\, {\\rm sg}(q))\\rbrace -\\lbrace R_+({\\rm i}q\\,{\\rm sg}(q))+R_-(-{\\rm i}q\\,{\\rm sg}(q))\\rbrace ={\\rm i}(2l+1)\\pi $ for any $l$ in $\\mathbb {Z}$ .", "Equations (REF ) form our core result.", "Recall that $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ are defined by (REF ).", "By virtue of (REF ) and (REF ), the constants $C_1$ and $C_2$ are interrelated, as expected.", "Remark 2.", "Dispersion relation (REF ) or (REF ) exhibits reflection symmetry with respect to $q$ , i.e., it is invariant under the replacement $q\\rightarrow -q$ , as anticipated for the case with an isotropic surface conductivity.", "If $\\sigma ^*(\\omega )=-\\sigma (\\omega )$ and the ambient medium is lossless, we can verify that if $q(\\omega )$ is a solution of  (REF ) so is $q^*(\\omega )$ ; thus, if $q(\\omega )$ is unique for $\\Re \\,q(\\omega )>0$ or $\\Re \\,q(\\omega )<0$ , with fixed $\\omega $ , this $q(\\omega )$ must be real.", "The integer $l$ that appears in (REF ) deserves some attention.", "Remark 3.", "Because $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ are analytic and single valued, only one value of the integer $l$ is relevant in dispersion relation (REF ); cf.", "Refs.", "VolkovMikhailov1988,MMSLL-preprint for a similar discussion.", "This $l$ should be chosen in conjunction with the branch for the logarithm in the integrals for $Q_\\pm $ and $R_\\pm $ .", "Of course, relation (REF ) should furnish physically anticipated results.", "For example, $q$ approaches the known quasi-electrostatic limit if $|q|\\gg |k_0|$ and $\\Im \\,\\sigma >0$ (Section  and Ref.", "VolkovMikhailov1988); also, $q$ should approach $k_0$ at low enough frequencies and thus yield a gapless energy spectrum $\\omega (q)$ of the EP in the dissipationless case, if $q$ is real (Section  and Ref. VolkovMikhailov1988).", "We choose to set $l=0$ which implies that the branch of the logarithm $w=\\ln \\mathcal {P}_\\varpi (\\xi )$ ($\\varpi ={\\rm TM}, {\\rm TE}$ ) in the integrals for $Q_\\pm $ and $R_\\pm $ is such that $-\\pi <\\Im \\,w\\le \\pi $ , when $\\xi $ lies in the top Riemann sheet (see Sections  and ).", "Remark 4.", "Equations (REF ) express the combined effect of TM and TE polarizations via the terms $Q_\\pm (\\pm {\\rm i}q{\\rm sg}(q))$ and $R_\\pm (\\pm {\\rm i}q{\\rm sg}(q))$ , respectively.", "In the nonretarded frequency regime, the $R_\\pm $ terms become relatively small (see Section  for details)." ], [ "Tangential electric field and bulk surface plasmons", "In this section, we compute the electric field tangential to the sheet.", "The EP wave number, $q$ , satisfies dispersion relation (REF ).", "For definiteness, we henceforth assume that $\\Re \\,q>0\\ \\mbox{and}\\ \\Im \\,q\\ge 0~.$ First, by (REF )–(REF ) we obtain the Fourier transforms $\\widehat{u}_-(\\xi )=-C_1 \\left[{\\rm i}\\xi \\, e^{-Q_-(\\xi )}-q\\, e^{-Q_-(-{\\rm i}q)} e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right] (q^2+\\xi ^2)^{-1}~, $ $\\widehat{v}_-(\\xi )=-C_1 \\left[ {\\rm i}q\\, e^{-Q_-(\\xi )}+\\xi \\, e^{-Q_-(-{\\rm i}q)} e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right] (q^2+\\xi ^2)^{-1}~;$ and $\\widehat{u}_+(\\xi )=C_1 \\left[ {\\rm i}\\xi \\,e^{Q_+(\\xi )}+q\\,e^{Q_+({\\rm i}q)} e^{R_+(\\xi )-R_+({\\rm i}q)}\\right] (q^2+\\xi ^2)^{-1}~,$ $\\widehat{v}_+(\\xi )= C_1 \\left[ {\\rm i}q\\,e^{Q_+(\\xi )}-\\xi e^{Q_+({\\rm i}q)} e^{R_+(\\xi )-R_+({\\rm i}q)}\\right] (q^2+\\xi ^2)^{-1}~.$ These functions are analytic at $\\xi =\\pm {\\rm i}q$ .", "The inverse Fourier transforms are $E_x(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty }^\\infty {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}\\xi \\, e^{-Q_-(\\xi )}-q\\, e^{-Q_-(-{\\rm i}q)} e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right]~, \\\\E_y(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty }^\\infty {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}q \\, e^{-Q_-(\\xi )}+\\xi \\, e^{-Q_-(-{\\rm i}q)}e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right]\\quad x>0~; $ and $E_x(x,0)&=\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty }^\\infty {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}\\xi \\, e^{Q_+(\\xi )}+q\\,e^{Q_+({\\rm i}q)} e^{R_+(\\xi )-R_+({\\rm i}q)}\\right]~, \\\\E_y(x,0)&=\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty }^\\infty {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}q \\, e^{Q_+(\\xi )}-\\xi \\,e^{Q_+({\\rm i}q)}e^{R_+(\\xi )-R_+({\\rm i}q)}\\right]\\quad x<0~.", "$ The task now is to approximately evaluate the above integrals for fixed $q$ in the following regimes: (i) $|qx|\\ll 1$ , close to the edge (Section REF ); and (ii) for sufficiently large $|qx|$ if $x>0$ (Section REF ).", "We describe two types of plausibly emerging SPs, which for fixed $q$ and $\\omega $ have distinct propagation constants in the $x$ -direction, on the sheet away from the edge.", "For large $|qx|$ , our calculation indicates the localization of the EP on the sheet near the material edge." ], [ "Tangential electric field near the edge, $|qx|\\ll 1$", "Consider $x>0$ , for points on the sheet.", "In (), we shift the integration path in the lower half $\\xi $ -plane, keeping in mind that the integrands are analytic at $\\xi =-{\\rm i}q$ , and write $E_x(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}\\xi \\, e^{-Q_-(\\xi )}-q\\, e^{-Q_-(-{\\rm i}q)} e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right]~, \\\\E_y(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}q \\, e^{-Q_-(\\xi )}+\\xi \\, e^{-Q_-(-{\\rm i}q)}e^{R_-(-{\\rm i}q)-R_-(\\xi )}\\right]\\quad x>0~,$ for a positive constant $\\delta _1$ with $\\delta _1\\gg |q|$ and $\\delta _1 x \\ll 1$ .", "Thus, the factor $e^{{\\rm i}\\xi x}$ in each integrand has a magnitude close to unity.", "From the Appendix, we use the asymptotic formulas $e^{-Q_-(\\xi )}=\\mathfrak {Q}(\\xi )[1+o(1)]\\quad \\mbox{and}\\quad e^{-R_-(\\xi )}=1+o(1)\\ \\mbox{as}\\ \\xi \\rightarrow \\infty ~,$ where $\\mathfrak {Q}(\\xi ):= \\left(-\\frac{\\omega \\mu \\sigma \\xi }{2k_0^2}\\right)^{-1/2}$ which has a branch cut emanating from the origin in $\\mathbb {C}_+$ .", "The component of the electric field parallel to the edge on the sheet approaches the limit $\\lim _{x\\downarrow 0}E_y(x,0)&= -\\frac{C_1}{2\\pi {\\rm i}}\\left[{\\rm i}q\\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\,(q^2+\\xi ^2)^{-1}\\,e^{-Q_-(\\xi )} \\right.", "\\\\& \\quad \\left.", "+e^{-Q_-(-{\\rm i}q)+R_-(-{\\rm i}q)}\\lim _{x\\downarrow 0}\\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\,e^{{\\rm i}\\xi x} (q^2+\\xi ^2)^{-1}\\xi e^{-R_-(\\xi )} \\right]~.$ Since $e^{-Q_-(\\xi )}=\\mathcal {O}(\\xi ^{-1/2})$ as $\\xi \\rightarrow \\infty $ , we infer that the first one of the above integrals converges.", "In fact, we see that this integral vanishes by closing the integration path through a large semicircle in $\\mathbb {C}_-$ .", "The second integral is evaluated via the approximations $q^2+\\xi ^2\\sim \\xi ^2$ and $e^{-R_-(\\xi )}\\sim 1$ since $|\\xi |\\gg |q|$ .", "Hence, at the edge $E_y(x,0)$ on the sheet has the finite value $\\lim _{x\\downarrow 0}E_y(x,0)=:E_y(0^+,0)=-C_1 e^{-Q_-(-{\\rm i}q)+R_-(-{\\rm i}q)}~, $ where $q$ satisfies (REF ); cf.", "Ref.", "VolkovMikhailov1988 in the context of the quasi-electrostatic approach.", "It can be shown that the correction to this leading-order term for $E_y(x,0)$ is of the order of $|k_0 x|$ .", "In a similar vein, we can address $E_x(x,0)$ , the component of the electric field on the sheet normal to the edge.", "Without further ado, we compute (with $\\delta _1\\gg |q|$ ) $E_x(x,0)&\\sim -\\frac{C_1}{2\\pi {\\rm i}}\\left[\\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{i\\xi x}}{q^2+\\xi ^2}\\,({\\rm i}\\xi )\\,\\mathfrak {Q}(\\xi )-q e^{-Q_-(-{\\rm i}q)+R_-(-{\\rm i}q)}\\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{-R_-(\\xi )}}{q^2+\\xi ^2}\\right]\\\\&= -\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{i\\xi x}}{q^2+\\xi ^2}\\,({\\rm i}\\xi )\\,\\mathfrak {Q}(\\xi )\\sim -\\frac{C_1}{2\\pi }\\int _{-\\infty -{\\rm i}\\delta _1}^{+\\infty -{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{i\\xi x}}{\\xi }\\,\\mathfrak {Q}(\\xi )~.$ By applying integration by parts once and wrapping the integration contour around the positive imaginary axis in the $\\xi $ -plane, we obtain $E_x(x,0)\\sim 2C_1 \\left(\\frac{2{\\rm i}\\,k_0}{\\pi \\omega \\mu \\sigma }\\right)^{1/2}\\sqrt{k_0 x}~,\\quad |qx|\\ll 1~,\\ x>0~.$ Thus, the surface current normal to the edge vanishes, as it happens also for line currents at the ends of cylindrical antennas with a delta-function voltage generator.", "[50] For a similar result in the scattering of waves from conducting films, see equation (39) in Ref. MML2017.", "Consider $x<0$ , if the observation point lies at $z=0$ outside the sheet.", "By (), we have $E_x(x,0)&=\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}\\xi \\, e^{Q_+(\\xi )}+q\\,e^{Q_+({\\rm i}q)} e^{R_+(\\xi )-R_+({\\rm i}q)}\\right]~, \\\\E_y(x,0)&=\\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2} \\left[{\\rm i}q \\, e^{Q_+(\\xi )}-\\xi \\,e^{Q_+({\\rm i}q)}e^{R_+(\\xi )-R_+({\\rm i}q)}\\right]\\quad x<0~,$ where $\\delta _1\\gg |q|$ and $\\delta _1 |x|\\ll 1$ .", "We will also need the following formulas (see Appendix): $e^{Q_+(\\xi )}=\\mathfrak {Q}(-\\xi )^{-1}[1+o(1)]\\quad \\mbox{and}\\quad e^{R_+(\\xi )}=1+o(1)\\ \\mbox{as}\\ \\xi \\rightarrow \\infty ~,$ noting that $\\mathfrak {Q}(-\\xi )$ has a branch cut emanating from the origin in $\\mathbb {C}_-$ .", "For $|qx|\\ll 1$ , we therefore compute $\\lim _{x\\uparrow 0}E_y(x,0)&=:E_y(0-,0)= \\frac{C_1}{2\\pi {\\rm i}}\\left[{\\rm i}q\\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{Q_+(\\xi )}}{q^2+\\xi ^2}\\right.", "\\\\& \\quad \\left.", "-e^{Q_+({\\rm i}q)-R_+({\\rm i}q)}\\lim _{x\\uparrow 0}\\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1}{\\rm d}\\xi \\ e^{i\\xi x}\\,\\frac{\\xi }{q^2+\\xi ^2}\\,e^{R_+(\\xi )}\\right]\\\\&= -\\frac{C_1}{2\\pi {\\rm i}}\\, e^{Q_+({\\rm i}q)-R_+({\\rm i}q)}\\lim _{x\\uparrow 0}\\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1}{\\rm d}\\xi \\ e^{i\\xi x}\\,\\frac{\\xi }{q^2+\\xi ^2}\\,e^{R_+(\\xi )}\\\\&= C_1\\, e^{Q_+({\\rm i}q)-R_+({\\rm i}q)}~.$ In the above, the integral of the first line is convergent; in fact, this integral vanishes.", "In the integrand of the remaining integral, we use the approximations $q^2+\\xi ^2\\sim \\xi ^2$ and $e^{R_+(\\xi )}\\sim 1$ .", "By dispersion relation (REF ) and limit (REF ), we conclude that $E_y(x,0)$ is continuous across the edge, viz., $E_y(0^-,0)=E_y(0^+,0)~.", "$ On the other hand, the $x$ -component of the electric field at $z=0$ outside the sheet is $E_x(x,0)&\\sim \\frac{C_1}{2\\pi {\\rm i}} \\left[ \\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2}\\,({\\rm i}\\xi ) \\, \\mathfrak {Q}(-\\xi )^{-1} +q\\, e^{Q_+({\\rm i}q)-R_+({\\rm i}q)}\\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1}{\\rm d}\\xi \\ \\frac{e^{R_+(\\xi )}}{q^2+\\xi ^2}\\right] \\\\&= \\frac{C_1}{2\\pi {\\rm i}} \\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2}\\,({\\rm i}\\xi ) \\, \\mathfrak {Q}(-\\xi )^{-1} \\sim \\frac{C_1}{2\\pi } \\int _{-\\infty +{\\rm i}\\delta _1}^{+\\infty +{\\rm i}\\delta _1} {\\rm d}\\xi \\ \\frac{e^{{\\rm i}\\xi x}}{\\xi }\\, \\mathfrak {Q}(-\\xi )^{-1} \\\\&= C_1 \\left(\\frac{\\omega \\mu \\sigma }{2\\pi {\\rm i}\\,k_0}\\right)^{1/2} \\frac{1}{\\sqrt{k_0 |x|}}~,\\quad |qx|\\ll 1,\\ x<0~.$ Thus, $\\partial E_x(x,z)/\\partial x$ indeed has a non-integrable singularity as $x\\uparrow 0$ at $z=0$ (see Remark 1)." ], [ "Far field: Two types of bulk SPs in the direction normal to the edge", "Next, we describe the bulk SPs in the $x$ -direction with recourse to the Fourier integrals for $E_x(x,0)$ and $E_y(x,0)$ .", "By (REF ) and (), for $x>0$ we use the integral representations $E_x(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty {\\rm d}\\xi \\,\\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2}\\left\\lbrace \\frac{{\\rm i}\\xi }{\\mathcal {P}_{\\rm TM}(\\xi )} e^{Q_+(\\xi )}-e^{R_-(-{\\rm i}q)-Q_-(-{\\rm i}q)}\\frac{q}{\\mathcal {P}_{\\rm TE}(\\xi )} e^{R_+(\\xi )}\\right\\rbrace ~,$ $E_y(x,0)&=-\\frac{C_1}{2\\pi {\\rm i}}\\int _{-\\infty }^\\infty {\\rm d}\\xi \\,\\frac{e^{{\\rm i}\\xi x}}{q^2+\\xi ^2}\\left\\lbrace \\frac{{\\rm i}q}{\\mathcal {P}_{\\rm TM}(\\xi )} e^{Q_+(\\xi )}+e^{R_-(-{\\rm i}q)-Q_-(-{\\rm i}q)}\\frac{\\xi }{\\mathcal {P}_{\\rm TE}(\\xi )} e^{R_+(\\xi )}\\right\\rbrace ~,$ where $q$ solves (REF ), and $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ are defined by (REF ).", "Note that the integrands are analytic at $\\xi ={\\rm i}q$ .", "Definition 2 (2D bulk SPs).", "Consider the electric field tangential to the sheet.", "For every $q$ solving (REF ) with given $\\omega $ , the 2D bulk SPs in the positive $x$ -direction are identified with waves that arise from (REF ) as residues of the integrands from the zeros of $\\mathcal {P}_{\\rm TM}(\\xi )$ or $\\mathcal {P}_{\\rm TE}(\\xi )$ in the upper half $\\xi $ -plane of the top Riemann sheet ($\\Re \\sqrt{\\xi ^2+q^2-k_0^2}>0$ ).", "If the ambient medium is lossless ($k_0>0$ ) we can characterize these waves as follows.", "If $\\Im \\,\\sigma (\\omega )>0$ , only the zeros $\\xi =\\pm k_{\\rm sp}^{\\rm e}$ ($k_{\\rm sp}^{\\rm e}\\in \\mathbb {C}_+$ ) of $\\mathcal {P}_{\\rm TM}(\\xi )$ are present in the top Riemann sheet; see (REF ) below.", "In this case, only $k_{\\rm sp}^{\\rm e}$ contributes to the residues, which amounts to a TM-like bulk SP.", "[9], [54], [55] Similarly, if $\\Im \\,\\sigma (\\omega )<0$ only the zero $\\xi =k_{\\rm sp}^{\\rm m}\\in \\mathbb {C}_+$ of $\\mathcal {P}_{\\rm TE}(\\xi )$ contributes to the residues; see () below.", "This case signifies a TE-like bulk SP.", "[9], [54], [55] Definition 2 does not explain how these 2D SPs can be separated from other contributions to the Fourier integrals for $E_x(x,0)$ and $E_y(x,0)$ .", "We address this issue in a simplified way.", "By (REF ) we proceed to calculate $E_x(x,0)$ and $E_y(x,0)$ by contour integration in the far field, for sufficiently large $|\\sqrt{q^2-k_0^2}\\,x|$ , and thus indicate the emergence of bulk SPs as possibly distinct contributions.", "By closing the path in the upper half $\\xi $ -plane, we write $E_\\ell (x,0)=E_\\ell ^{\\rm sp}(x,0)+E_\\ell ^{\\rm rad}(x,0)\\qquad (\\ell =x,\\,y)~,$ where $E_\\ell ^{\\rm sp}$ is the residue contribution, which amounts to a bulk SP in the $x$ -direction (Definition 2), and $E_\\ell ^{\\rm rad}$ is the contribution from the branch cut emanating from the point ${\\rm i}\\sqrt{q^2-k_0^2}$ ($\\Re \\sqrt{q^2-k_0^2}>0$ ).", "We refer to the latter contribution as the `radiation field'.", "[55] In this simplified treatment, we focus on large enough distances from the edge so that the relevant pole contribution is sufficiently separated from the branch point contribution.", "First, we consider $E_\\ell ^{\\rm sp}(x,0)$ ($\\ell =x,\\,y$ ).", "After some algebra, for $k_0>0$ we obtain $\\frac{E_x^{\\rm sp}(x,0)}{C_1}=\\left\\lbrace \\begin{array}{lr}{\\displaystyle {\\rm i}\\biggl [1-\\left(\\frac{\\omega \\mu \\sigma }{2k_0}\\right)^2\\biggr ]^{-1} e^{Q_+(k_{\\rm sp}^{\\rm e})} e^{ik_{\\rm sp}^{\\rm e}x}}~,& \\Im \\sigma >0~,\\\\{\\displaystyle \\frac{q}{k_{\\rm sp}^{\\rm m}}\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2\\biggl [1-\\left(\\frac{\\omega \\mu \\sigma }{2k_0}\\right)^2\\biggr ]^{-1} e^{R_+(k_{\\rm sp}^{\\rm m})-R_+({\\rm i}q)+Q_+({\\rm i}q)} e^{ik_{\\rm sp}^{\\rm m}x}}~, & \\Im \\sigma <0~;\\end{array} \\right.$ $\\frac{E_y^{\\rm sp}(x,0)}{C_1}=\\left\\lbrace \\begin{array}{lr}{\\displaystyle \\frac{{\\rm i}q}{k_{\\rm sp}^{\\rm e}}\\biggl [1-\\left(\\frac{\\omega \\mu \\sigma }{2k_0}\\right)^2\\biggr ]^{-1} e^{Q_+(k_{\\rm sp}^{\\rm e})} e^{ik_{\\rm sp}^{\\rm e}x}}~,& \\Im \\sigma >0~, \\\\{\\displaystyle -\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2\\biggl [1-\\left(\\frac{\\omega \\mu \\sigma }{2k_0}\\right)^2\\biggr ]^{-1} e^{R_+(k_{\\rm sp}^{\\rm m})-R_+({\\rm i}q)+Q_+({\\rm i}q)} e^{ik_{\\rm sp}^{\\rm m}x}}~, & \\Im \\sigma <0~.\\end{array} \\right.$ In the above, from the zeros of $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ we define the wave numbers $k_{\\rm sp}^{\\rm e}&={\\rm i}\\sqrt{q^2-\\biggl (\\frac{{\\rm i}2k_0^2}{\\omega \\mu \\sigma }\\biggr )^2-k_0^2}\\quad \\mbox{if}\\ \\Im \\,\\sigma >0\\ (\\mbox{TM})~, \\\\k_{\\rm sp}^{\\rm m}&={\\rm i}\\sqrt{q^2-\\biggl (\\frac{\\omega \\mu \\sigma }{2{\\rm i}}\\biggr )^2-k_0^2}\\quad (\\Im \\,k_{\\rm sp}^{\\rm e, m}>0)\\quad \\mbox{if}\\ \\Im \\,\\sigma <0\\ (\\mbox{TE})~,$ so that $k_{\\rm sp}^{\\rm e}$ or $k_{\\rm sp}^{\\rm m}$ lies in the top Riemann sheet, respectively.", "[54], [55] Remark 5.", "In the nonretarded frequency regime (Section ), if $|\\omega \\mu \\sigma /k_0|\\ll 1$ and $\\Im \\,\\sigma (\\omega )>0$ , the $q$ that solves (REF ) for fixed $\\omega $ is given by $q\\sim \\eta _0\\, ({\\rm i}2k_0^2/(\\omega \\mu \\sigma ))$ with $\\eta _0>1$ ;[19], [23] thus, the TM-like SP is significantly damped.", "In this regime, we approximate $k_{\\rm sp}^{\\rm e}\\sim {\\rm i}\\sqrt{q^2-[{\\rm i}2k_0^2/(\\omega \\mu \\sigma )]^2}$ .", "Note that this approximation can be used in the exponential factor for $E_x^{\\rm sp}(x,0)$ and $E_y^{\\rm sp}(x,0)$ with a small error if $|\\omega \\mu \\sigma |x\\ll 1$ , along with $|\\omega \\mu \\sigma /k_0|\\ll 1$ .", "Next, we calculate the contributions, $E_\\ell ^{\\rm rad}(x,0)$ , along the branch cut ($\\ell =x,\\,y$ ).", "Suppose that $\\sqrt{q^2-k_0^2}>0$ .", "By the change of variable $\\xi \\mapsto \\varsigma $ with $\\xi ={\\rm i}\\sqrt{q^2-k_0^2}(1+\\varsigma )$ and $\\varsigma >0$ , we express the requisite integrals as $E_x^{\\rm rad}(x,0)&= -\\frac{C_1}{2\\pi }\\biggl \\lbrace \\frac{\\omega \\mu \\sigma }{k_0}\\frac{\\sqrt{q^2-k_0^2}}{k_0}\\int _0^\\infty {\\rm d}\\varsigma \\ \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x\\varsigma } e^{Q_+({\\rm i}\\sqrt{q^2-k_0^2}(1+\\varsigma ))}}{(1+\\varsigma )^2-\\frac{q^2}{q^2-k_0^2}}\\frac{(1+\\varsigma )\\sqrt{\\varsigma (2+\\varsigma )}}{1-\\bigl (\\frac{\\omega \\mu \\sigma }{2k_0}\\bigr )^2\\frac{q^2-k_0^2}{k_0^2}\\varsigma (2+\\varsigma )}\\\\&\\mbox{} +4e^{-R_+({\\rm i}q)+Q_+({\\rm i}q)}\\frac{q}{\\omega \\mu \\sigma }\\int _0^\\infty {\\rm d}\\varsigma \\ \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x\\varsigma }}{(1+\\varsigma )^2-\\frac{q^2}{q^2-k_0^2}}\\frac{e^{R_+({\\rm i}\\sqrt{q^2-k_0^2}(1+\\varsigma ))}}{1-\\bigl (\\frac{2}{\\omega \\mu \\sigma }\\bigr )^2(q^2-k_0^2)\\varsigma (2+\\varsigma )} \\sqrt{\\varsigma (2+\\varsigma )}\\biggr \\rbrace \\\\&\\mbox{}\\qquad \\times e^{-\\sqrt{q^2-k_0^2}\\,x}$ and $E_y^{\\rm rad}(x,0)&= \\frac{{\\rm i}C_1}{2\\pi }\\biggl \\lbrace \\frac{q}{k_0}\\frac{\\omega \\mu \\sigma }{k_0}\\int _0^\\infty {\\rm d}\\varsigma \\ \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x\\varsigma } e^{Q_+({\\rm i}\\sqrt{q^2-k_0^2}(1+\\varsigma ))}}{(1+\\varsigma )^2-\\frac{q^2}{q^2-k_0^2}}\\frac{\\sqrt{\\varsigma (2+\\varsigma )}}{1-\\bigl (\\frac{\\omega \\mu \\sigma }{2k_0}\\bigr )^2\\frac{q^2-k_0^2}{k_0^2}\\varsigma (2+\\varsigma )}\\\\&\\mbox{} +4e^{-R_+({\\rm i}q)+Q_+({\\rm i}q)}\\frac{\\sqrt{q^2-k_0^2}}{\\omega \\mu \\sigma }\\int _0^\\infty {\\rm d}\\varsigma \\ \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x\\varsigma }}{(1+\\varsigma )^2-\\frac{q^2}{q^2-k_0^2}}\\frac{e^{R_+({\\rm i}\\sqrt{q^2-k_0^2}(1+\\varsigma ))} (1+\\varsigma ) \\sqrt{\\varsigma (2+\\varsigma )}}{1-\\bigl (\\frac{2}{\\omega \\mu \\sigma }\\bigr )^2(q^2-k_0^2)\\varsigma (2+\\varsigma )}\\biggr \\rbrace \\\\&\\mbox{}\\qquad \\times e^{-\\sqrt{q^2-k_0^2}\\,x}~.$ In the far field, when $|\\sqrt{q^2-k_0^2}\\,x|\\gg 1$ with $\\biggl |\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2\\frac{q^2-k_0^2}{k_0^2}\\frac{1}{\\sqrt{q^2-k_0^2}x}\\biggr |\\ll 1\\quad \\mbox{and}\\quad \\biggl |\\frac{q^2-k_0^2}{(\\omega \\mu \\sigma )^2}\\frac{1}{\\sqrt{q^2-k_0^2}\\,x}\\biggr |\\ll 1,$ the major contribution to integration in the above branch cut integrals comes from the endpoint, $\\varsigma =0$ .", "Accordingly, we evaluate $E_x^{\\rm rad}(x,0)&\\sim \\frac{C_1}{\\sqrt{2\\pi }}\\frac{q^2-k_0^2}{k_0^2}\\biggl \\lbrace \\frac{\\omega \\mu \\sigma }{2k_0}\\frac{\\sqrt{q^2-k_0^2}}{k_0} e^{Q_+({\\rm i}\\sqrt{q^2-k_0^2})}+2\\frac{q}{\\omega \\mu \\sigma }e^{-R_+({\\rm i}q)+Q_+({\\rm i}q)}e^{R_+({\\rm i}\\sqrt{q^2-k_0^2})}\\biggr \\rbrace \\\\&\\mbox{}\\quad \\times \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x}}{(\\sqrt{q^2-k_0^2} x)^{3/2}}~,$ $E_y^{\\rm rad}(x,0)&\\sim -\\frac{{\\rm i}C_1}{\\sqrt{2\\pi }}\\frac{q^2-k_0^2}{k_0^2}\\biggl \\lbrace \\frac{\\omega \\mu \\sigma }{2k_0}\\frac{q}{k_0} e^{Q_+({\\rm i}\\sqrt{q^2-k_0^2})}+2\\frac{\\sqrt{q^2-k_0^2}}{\\omega \\mu \\sigma }e^{-R_+({\\rm i}q)+Q_+({\\rm i}q)}e^{R_+({\\rm i}\\sqrt{q^2-k_0^2})}\\biggr \\rbrace \\\\&\\mbox{}\\quad \\times \\frac{e^{-\\sqrt{q^2-k_0^2}\\,x}}{(\\sqrt{q^2-k_0^2} x)^{3/2}}\\qquad (x>0)~.$ The above far-field formulas for $E_x^{\\rm rad}(x,0)$ and $E_y^{\\rm rad}(x,0)$ can be analytically continued to complex $\\sqrt{q^2-k_0^2}$ with $\\Re \\sqrt{q^2-k_0^2}>0$ .", "Remark 6.", "By the formulas for $E_\\ell ^{\\rm rad}(x,0)$ ($\\ell =x, y$ ), this contribution may decay rapidly with $x$ if $|q|\\gg k_0$ .", "This can occur in the nonretarded frequency regime (see Section ), where $\\Im \\,\\sigma >0$ and $q\\simeq \\eta _0 ({\\rm i}2k_0^2/(\\omega \\mu \\sigma ))$ with $\\eta _0>1$ .", "[23] By inspection of the simplified formulas for the TM-like bulk SP, $E_\\ell ^{\\rm sp}$ , and the radiation field $E_\\ell ^{\\rm rad}$ , we expect that, in the nonretarded frequency regime, the SP contribution can be dominant over the radiation field.", "Hence, the EP electric field tangential to the sheet can be localized near the edge on the 2D material.", "A more accurate study of the electric field would involve the derivation of asymptotic formulas for the requisite Fourier integrals in an intermediate regime of distances from the edge, between the near and far fields.", "In addition, the $q(\\omega )$ must be numerically computed from dispersion relation (REF ) for various material parameters and frequencies of interest.", "These tasks will be the subject of future work.", "[43]" ], [ "On the low-frequency EP dispersion relation", "In this section, we derive an asymptotic formula for the $q$ that obeys (REF ) if $\\biggl |\\frac{\\omega \\mu \\sigma (\\omega )}{k_0}\\biggr |\\gg 1\\quad \\mbox{and}\\quad \\Im \\,\\sigma (\\omega )> 0\\qquad (k_0=\\omega \\sqrt{\\mu \\varepsilon })~.$ One way to motivate these conditions is to invoke the Drude model for doped single-layer graphene, which is expected to be accurate for small enough plasmon energies.", "[10] By this model, $\\sigma (\\omega )={\\rm i}[e^2 v_F \\sqrt{n_s}/(\\sqrt{\\pi }\\hbar )](\\omega +{\\rm i}/\\tau _e)^{-1}$ ; $e$ is the electron charge, $v_F$ is the Fermi velocity, $\\tau _e$ is the relaxation time of microscopic collisions, $n_s$ is the electron surface density, and $\\hbar $ is Planck's constant, while the interband transitions are neglected in the calculation of this $\\sigma (\\omega )$ .", "[56] Hence, within this model, the conditions of (REF ) are obeyed if $\\tau _e^{-1}\\ll \\omega \\ll \\omega _p\\quad \\mbox{where}\\ \\omega _p=\\biggl |Z_0\\frac{e^2\\sqrt{n_s}v_F}{\\sqrt{\\pi }\\hbar }\\biggl |~,\\quad Z_0= \\sqrt{\\frac{\\mu }{\\varepsilon }}~;$ $Z_0$ is the characteristic impedance of the (unbounded) ambient medium.", "For given $\\omega $ , the conditions in (REF ) call for large enough relaxation time, $\\tau _e$ , and surface density, $n_s$ .", "We expect that $q/k_0=\\mathcal {O}(1)$ with $|q|> k_0$ in this regime.", "Our task here is to refine this anticipated result.", "Note that the model for $\\sigma (\\omega )$ can be improved by consideration of both the intraband and interband transitions in the linear-response quantum theory for $\\sigma $ .", "[56] First, we convert (REF ) for the EP dispersion to a more explicit expression with $\\Re \\,q>0$ .", "Consider integral formulas (REF ) for $Q_\\pm (\\xi )$ .", "By changing the integration variable, $\\xi ^{\\prime }$ , according to $\\xi ^{\\prime }=q\\,\\varsigma $ , we can alternatively write (REF ), with $l=0$ , as $I(q):=&\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\left\\lbrace \\ln \\left[1+\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0}\\frac{q}{k_0}\\left(\\varsigma ^2+1-k_0^2/q^2\\right)^{1/2}\\right] \\right.", "\\\\& \\qquad \\left.", "-\\ln \\left[1-\\frac{{\\rm i}\\omega \\mu \\sigma }{2q}\\left(\\varsigma ^2+1-k_0^2/q^2\\right)^{-1/2}\\right]\\right\\rbrace = {\\rm i}\\pi ~;$ $\\Re \\sqrt{q^2(\\varsigma ^2+1)-k_0^2}>0~.$ The last condition defines the top Riemann sheet in the $\\varsigma $ -plane for the integrand in (REF ).", "By use of (REF ), we notice that in the present frequency regime we have $I(q)&=\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\,\\biggl \\lbrace \\ln \\biggl (e^{{\\rm i}\\pi }\\biggl (-\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0}\\biggr )\\biggr )-\\ln \\bigg (-\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0}\\biggr )\\biggr \\rbrace +\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\, \\ln \\biggl (\\frac{\\varsigma ^2+\\bar{q}^2}{1-\\bar{q}^2}\\biggr )\\\\&\\qquad +\\mathcal {O}(\\epsilon \\ln \\epsilon )\\\\&={\\rm i}\\pi +\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\, \\ln \\biggl (\\frac{\\varsigma ^2+\\bar{q}^2}{1-\\bar{q}^2}\\biggr ) +\\mathcal {O}(\\epsilon \\ln \\epsilon )~;\\quad \\epsilon ={\\rm i}\\frac{2k_0}{\\omega \\mu \\sigma }~,\\quad \\bar{q}^2=1-\\frac{k_0^2}{q^2}\\quad (|\\bar{q}|<1)$ with $|\\epsilon |\\ll 1$ .", "In the above, we defined the branch of the logarithm, $w=\\ln (\\cdot )$ , by $-\\pi <\\Im \\,w\\le \\pi $ ; accordingly, $\\bar{q}\\rightarrow 0$ as $\\epsilon \\rightarrow 0$ , when $q$ approaches $k_0$ (see Remark 3).", "More generally, we may define $(2l_0-1)\\pi <\\Im \\,w\\le (2l_0+1)\\pi $ for some $l_0\\in \\mathbb {Z}$ while we set $l=l_0$ in (REF ).", "We proceed to describe in some detail the asymptotics for (REF ).", "By invoking the identity $& \\ln \\biggl (1-\\epsilon ^{-1}\\sqrt{\\frac{\\varsigma ^2+\\bar{q}^2}{1-\\bar{q}^2}}\\biggr )-\\ln \\biggl (1+\\epsilon ^{-1}\\sqrt{\\frac{1-\\bar{q}^2}{\\varsigma ^2+\\bar{q}^2}}\\biggr )={\\rm i}\\pi +2\\ln \\biggl (\\sqrt{\\frac{\\varsigma ^2+\\bar{q}^2}{1-\\bar{q}^2}}\\biggr )\\\\&\\mbox{}\\quad +\\ln \\biggl (1-\\epsilon \\sqrt{\\frac{1-\\bar{q}^2}{\\varsigma ^2+\\bar{q}^2}}\\biggr )-\\ln \\biggl (1 +\\epsilon \\frac{\\varsigma }{\\sqrt{1-\\bar{q}^2}} +\\epsilon \\frac{\\sqrt{\\varsigma ^2+\\bar{q}^2}-\\varsigma }{\\sqrt{1-\\bar{q}^2}}\\biggr )~,$ and treating the term $\\epsilon (\\sqrt{\\varsigma ^2+\\bar{q}^2}-\\varsigma )/\\sqrt{1-\\bar{q}^2}$ as a perturbation in the last logarithm, we approximate (REF ) by the relation $I_1(\\bar{q})-\\bar{\\epsilon }I_2(\\bar{q})-I_3(\\bar{\\epsilon })\\sim \\frac{\\pi }{2}\\ln (1-\\bar{q}^2)~,\\quad \\bar{\\epsilon }=\\frac{\\epsilon }{\\sqrt{1-\\bar{q}^2}}~,$ where $I_1(\\chi )&=\\int _{0}^\\infty {\\rm d}\\varsigma \\ \\frac{\\ln (\\varsigma ^2+\\chi ^2)}{1+\\varsigma ^2}~,\\quad I_2(\\chi )=\\int _0^\\infty {\\rm d}\\varsigma \\,\\biggl (\\frac{1}{\\sqrt{\\varsigma ^2+\\chi ^2}}-\\frac{\\varsigma }{1+\\varsigma ^2}\\biggr )~,\\\\I_3(\\chi )&=\\int _0^\\infty {\\rm d}\\varsigma \\ \\frac{\\ln (1+\\chi \\varsigma )}{1+\\varsigma ^2}~.$ The first two integrals can be computed directly.", "For $I_1(\\chi )$ , by contour integration we obtain $I_1(\\chi )=\\pi \\ln (1+\\chi )= \\pi \\left(\\chi -\\textstyle {\\frac{1}{2}}\\chi ^2\\right)+\\mathcal {O}(\\chi ^3)~,\\quad |\\chi |\\ll 1~.$ The integral $I_2(\\chi )$ is expressed as $I_2(\\chi )&=\\lim _{M\\rightarrow \\infty }\\biggl \\lbrace \\int _0^M\\frac{{\\rm d}\\varsigma }{\\sqrt{\\varsigma ^2+\\chi ^2}}-\\int _0^M{\\rm d}\\varsigma \\,\\frac{\\varsigma }{1+\\varsigma ^2}\\biggr \\rbrace \\\\&=\\lim _{M\\rightarrow \\infty }\\biggl \\lbrace \\ln \\biggl (\\frac{M}{\\chi }+\\sqrt{1+\\frac{M^2}{\\chi ^2}}\\biggr )-\\textstyle {\\frac{1}{2}}\\ln (1+M^2)\\biggr \\rbrace =\\ln (2/\\chi )~.$ In order to obtain an asymptotic expansion for $I_3(\\chi )$ as $\\chi \\rightarrow 0$ we use the Mellin transform technique.", "[57] The idea is to compute the Mellin transform, $\\widetilde{I}_3(s)$ , of $I_3(\\chi )$ , and then employ its inversion formula; the desired asymptotic expansion for $I_3(\\chi )$ comes from the residues at poles of ${\\widetilde{I}}_3(s)$ in the $s$ -plane with $\\Re \\,s\\ge \\alpha $ , for some suitable real $\\alpha $ .", "For $\\chi >0$ , define $\\widetilde{I}_3(s)=\\int _0^\\infty {\\rm d}\\chi \\, I_3(\\chi )\\chi ^{-s}=\\frac{1}{2}\\frac{\\Gamma (s)}{(s-1)^2}\\Gamma (2-s)\\,\\Gamma \\left(\\frac{s}{2}\\right)\\,\\Gamma \\left(-\\frac{s}{2}+1\\right)~,\\quad 1<\\Re \\,s<2=\\alpha $ so that this integral converges, where $\\Gamma (\\zeta )$ is the gamma function.", "[58] Here, we interchanged the order of integration (in $\\chi $ and $\\varsigma $ ) and used a known integral for the beta function, $B(\\zeta _1,\\zeta _2)=\\Gamma (\\zeta _1)\\Gamma (\\zeta _2)/\\Gamma (\\zeta _1+\\zeta _2)$ .", "[58] Consider the inversion formula $I_3(\\chi )=\\frac{1}{2\\pi {\\rm i}}\\int _{c_1-{\\rm i}\\infty }^{c_1+{\\rm i}\\infty } {\\rm d}s\\ \\chi ^{s-1} \\widetilde{I}_3(s)~,\\qquad 1<c_1<2=\\alpha ~,$ and shift the integration path to the right, i.e., into the region of the $s$ -plane with $\\Re \\,s\\ge \\alpha =2$ , noticing that $\\widetilde{I}_3(s)$ has poles at the integers $s=n$ (with $n\\ge 2$ ) in this region.", "By applying the residue theorem at the double pole $s=2$ and the simple pole $s=3$ , we find $I_3(\\chi )= \\chi (1-\\ln \\chi )+\\frac{\\pi }{4}\\chi ^2+\\mathcal {O}(\\chi ^3\\ln \\chi )\\qquad \\mbox{as}\\ \\chi \\rightarrow 0~.$ This expansion can be analytically continued to complex $\\chi $ with $\\Re \\,\\chi \\ge 0$ .", "Consequently, after some algebra, dispersion relation (REF ) is reduced to the formula $q\\sim k_0\\biggl \\lbrace 1+\\frac{1}{2\\pi ^2}\\epsilon ^2\\,\\mathcal {A}(\\epsilon )^2\\biggr \\rbrace ~,\\quad \\epsilon =\\frac{{\\rm i}\\,2k_0}{\\omega \\mu \\sigma }~,$ where for simplicity we neglected terms $o(\\epsilon ^2)$ on the right-hand side.", "In the above, the function $\\mathcal {A}(\\epsilon )$ amounts to logarithmic corrections and solves the equation $e^{\\mathcal {A}}=\\frac{2e\\pi }{\\epsilon ^2 \\mathcal {A}}~.$ Note that an expansion for $\\mathcal {A}(\\epsilon )$ can be formally constructed via the iterative scheme $\\mathcal {A}^{(n+1)}(\\epsilon )=\\ln \\biggl (\\frac{2e\\pi }{\\epsilon ^2}\\biggr )-\\ln \\mathcal {A}^{(n)}(\\epsilon )~,\\quad \\mathcal {A}^{(0)}(\\epsilon )=\\ln \\frac{2 e\\pi }{\\epsilon ^2}\\quad (n=0,\\,1,\\,\\ldots )~.$ Finally, one can verify that the result furnished by () does not violate the conditions $\\mathcal {P}_{\\rm TM}(\\xi ;q)\\ne 0$ and $\\mathcal {P}_{\\rm TE}(\\xi ;q)\\ne 0$ for all real $\\xi $ , which are assumed for the application of the underlying Wiener-Hopf factorization in Section REF ." ], [ "On the nonretarded frequency regime", "In this section, we simplify (REF ) under the conditions $\\biggl |\\frac{\\omega \\mu \\sigma (\\omega )}{k_0}\\biggr |\\ll 1\\ \\mbox{and}\\ \\Im \\,\\sigma (\\omega )>0~,$ which signify the nonretarded frequency regime in the context of our isotropic conductivity model.", "[5], [9] We show how the quasi-electrostatic approximation of previous works [19], [23] can be refined.", "In fact, we derive a correction to this approximation which indicates the role of the TE polarization through the relatively small $R_\\pm (\\pm {\\rm i}q{\\rm sg}(q))$ terms in (REF ).", "In this regime, we expect to have[23], [17] $\\eta (q)=-\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0}\\frac{q}{k_0}=-\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0}\\frac{1}{\\delta }=\\mathcal {O}(1)~,\\quad \\delta =\\frac{k_0}{q}~;$ thus, $|\\delta |\\ll 1$ ($|q|\\gg |k_0|$ ).", "The definition of $\\eta (q)$ is inspired by the quasi-electrostatic approach of Ref.", "VolkovMikhailov1988 where it is found that $\\eta \\simeq 1.217$ .", "We assume that $\\Re \\,q>0$ .", "First, we expand in $\\delta $ the integral pertaining to $\\mathcal {P}_{\\rm TE}$ for fixed $\\eta $ .", "By () we have $R_+({\\rm i}q)+R_-(-{\\rm i}q)&=\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\ln \\biggl [1+\\eta \\delta ^2\\,(\\varsigma ^2+1-\\delta ^2)^{-1/2}\\biggr ]\\\\&\\sim \\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\ln \\biggl [1+\\eta \\delta ^2\\,(1+\\varsigma ^2)^{-1/2}\\biggl (1+\\frac{\\delta ^2}{2}\\frac{1}{1+\\varsigma ^2}\\biggr )\\biggr ]\\\\&\\sim \\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\biggl \\lbrace \\eta \\delta ^2\\,(1+\\varsigma ^2)^{-1/2}\\biggl (1+\\frac{\\delta ^2}{2}\\frac{1}{1+\\varsigma ^2}\\biggr )-\\frac{1}{2}\\frac{(\\eta \\delta ^2)^2}{1+\\varsigma ^2}\\biggr \\rbrace \\\\&=\\frac{2}{\\pi }\\eta \\delta ^2\\biggl [1+\\biggl (\\frac{1}{3}- \\frac{\\pi }{8}\\eta \\biggr )\\delta ^2\\biggr ]~,\\quad |\\delta |\\ll 1~.$ In contrast, the integral pertaining to $\\mathcal {P}_{\\rm TM}$ is $Q_+({\\rm i}q)+Q_-(-{\\rm i}q)=\\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\ln \\left(1-\\eta \\sqrt{\\varsigma ^2+1-\\delta ^2}\\right)=\\mathcal {O}(1)~,$ which is dominant over $R_+({\\rm i}q)+R_-(-{\\rm i}q)$ .", "By expanding in $\\delta $ for fixed $\\eta =\\eta (q)$ , we find $&Q_+({\\rm i}q)+Q_-(-{\\rm i}q)\\sim \\frac{2}{\\pi }\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\,\\ln \\biggl \\lbrace 1-\\eta \\sqrt{1+\\varsigma ^2}\\biggl [1-\\frac{\\delta ^2}{2}\\frac{1}{1+\\varsigma ^2}-\\frac{\\delta ^4}{8}\\frac{1}{(1+\\varsigma ^2)^2}\\biggr ]\\biggr \\rbrace \\\\&\\mbox{} \\quad \\sim \\frac{2}{\\pi }\\biggl \\lbrace \\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ \\ln \\left(1-\\eta \\sqrt{1+\\varsigma ^2}\\right)-\\frac{1}{2}\\eta \\delta ^2\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{(1+\\varsigma ^2)^{3/2}}\\ \\left(\\eta \\sqrt{1+\\varsigma ^2}-1\\right)^{-1}\\\\&\\mbox{} \\quad -\\frac{1}{8}\\eta \\delta ^4\\biggl [\\eta \\int _0^\\infty \\frac{{\\rm d}\\varsigma }{(1+\\varsigma ^2)^2}\\,(\\eta \\sqrt{1+\\varsigma ^2}-1)^{-2} +\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{(1+\\varsigma ^2)^{5/2}}\\ \\left(\\eta \\sqrt{1+\\varsigma ^2}-1\\right)^{-1}\\biggr ]\\biggr \\rbrace ~.$ By neglecting terms $\\mathcal {O}(\\delta ^4)$ , we thus approximate dispersion relation (REF ) with $l=0$ by $2\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ln \\left(\\eta \\sqrt{1+\\varsigma ^2}-1\\right)\\sim \\eta \\delta ^2\\left\\lbrace \\eta \\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2} \\left(\\eta \\sqrt{1+\\varsigma ^2}-1\\right)^{-1}+1\\right\\rbrace ~,$ where the term ${\\rm i}\\pi =\\ln (-1)$ from (REF ) was combined with the logarithm from $Q_+({\\rm i}q)+Q_-(-{\\rm i}q)$ , resulting in the reversal of the sign of its argument.", "Note that $\\eta \\delta =-{\\rm i}\\omega \\mu \\sigma /(2k_0)$ .", "Thus, the solution, $q$ , of (REF ) is expressed via the expansion $\\eta (q)\\sim \\eta _0 \\biggl \\lbrace 1-\\eta _1\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2\\biggr \\rbrace ~,\\quad \\eta _j=\\mathcal {O}(1)\\quad (j=0,\\,1)~;\\quad \\biggl |\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr |\\ll 1~.$ The coefficients $\\eta _j$ are determined below.", "The substitution of the above expansion into (REF ) along with a dominant balance argument yield the desired equations for $\\eta _j$ , viz., $\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2}\\ln \\left(\\eta _0\\sqrt{1+\\varsigma ^2}-1\\right)=0~,$ $2\\eta _0\\eta _1 \\int _0^\\infty \\frac{{\\rm d}\\varsigma }{\\sqrt{1+\\varsigma ^2}}\\biggl (\\eta _0\\sqrt{1+\\varsigma ^2}-1\\biggr )^{-1}=\\int _0^\\infty \\frac{{\\rm d}\\varsigma }{1+\\varsigma ^2} \\biggl (\\eta _0\\sqrt{1+\\varsigma ^2}-1\\biggr )^{-1}+\\eta _0^{-1}~.$ Equation (REF ) is in agreement with the corresponding result in the quasi-electrostatic limit derived in Ref.", "VolkovMikhailov1988 and gives $\\eta _0\\simeq 1.217$ ; cf.", "their equation (39) and the subsequent relation in view of the change of variable $\\varsigma \\mapsto x$ with $\\varsigma =\\cot x$ here.", "On the other hand, after some algebra (REF ) entails $\\eta _1=\\frac{1}{2} \\left\\lbrace 1-\\sqrt{1-\\eta _0^{-2}}\\frac{\\displaystyle \\frac{\\pi }{2}-\\eta _0^{-1}}{\\displaystyle \\frac{\\pi }{2}+\\arcsin (\\eta _0^{-1})}\\right\\rbrace \\quad \\left(0< \\arcsin w<\\frac{\\pi }{2}\\ \\mbox{if}\\ 0<w<1\\right)~.$ The numerical evaluation of this coefficient yields $\\eta _1\\simeq 0.416$ by use of $\\eta _0\\simeq 1.217$ .", "According to our expansion for $\\eta (q)$ above, the EP wave number is furnished by $q= \\eta _0\\,\\frac{{\\rm i}2k_0^2}{\\omega \\mu \\sigma }\\biggl \\lbrace 1-\\eta _1\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^2+\\mathcal {O}\\biggl (\\biggl (\\frac{\\omega \\mu \\sigma }{2k_0}\\biggr )^4\\biggr )\\biggr \\rbrace ~.$ Evidently, the leading-order correction term, which is of the relative order of $(\\omega \\mu \\sigma /k_0)^2$ , comes from contributions of both TM and TE polarizations, i.e., from both the $Q_\\pm $ and $R_\\pm $ terms in dispersion relation (REF ).", "By virtue of (REF ), it is of some interest to describe $q(\\omega )$ when $\\sigma (\\omega )$ is given by the Drude model which is relevant to doped single-layer graphene for small enough plasmonic energies (see Section ).", "[9], [10] By use of the formula $\\sigma (\\omega )={\\rm i}(\\mathcal {D}/\\pi )\\,(\\omega +{\\rm i}/\\tau _e)^{-1}$ (beginning of Section ), where the dimensional parameter $\\mathcal {D}$ is the Drude weight,[10] we obtain $\\Re \\,q(\\omega )&\\sim \\frac{2\\eta _0\\varepsilon \\pi }{\\mathcal {D}}\\omega ^2 \\biggl \\lbrace 1+\\frac{\\eta _1}{4\\pi ^2}\\frac{(Z_0 \\mathcal {D})^2}{\\omega ^2+\\tau _e^{-2}}\\biggr \\rbrace \\sim \\frac{2\\eta _0\\varepsilon \\pi }{\\mathcal {D}} \\biggl \\lbrace \\omega ^2+\\frac{\\eta _1}{4\\pi ^2}(Z_0 \\mathcal {D})^2\\biggr \\rbrace ~,\\\\\\Im \\,q(\\omega )&\\sim \\frac{2\\eta _0\\varepsilon \\pi }{\\mathcal {D}} \\omega \\tau _e^{-1}\\biggl \\lbrace 1-\\frac{\\eta _1}{4\\pi ^2}\\frac{(Z_0 \\mathcal {D})^2}{\\omega ^2+\\tau _e^{-2}}\\biggr \\rbrace \\sim \\frac{2\\eta _0\\varepsilon \\pi }{\\mathcal {D}} \\omega \\tau _e^{-1}\\biggl \\lbrace 1-\\frac{\\eta _1}{4\\pi ^2}\\frac{(Z_0 \\mathcal {D})^2}{\\omega ^2}\\biggr \\rbrace ~.$ Here, the formulas on the rightmost-hand side come from applying the condition $\\omega \\tau _e\\gg 1$ ." ], [ "Extension: Two coplanar conducting sheets", "In this section, we extend our formalism to the setting with two coplanar, semi-infinite sheets of distinct isotropic and homogeneous conductivities.", "Consider the `left' sheet $\\Sigma ^L=\\lbrace (x,y,z)\\in \\mathbb {R}^3\\,:\\,z=0,\\, x<0\\rbrace $ and the `right' sheet $\\Sigma ^R=\\lbrace (x,y,z)\\in \\mathbb {R}^3\\,:\\,z=0,\\, x>0\\rbrace $ that have scalar, spatially constant surface conductivities $\\sigma ^L(\\omega )$ and $\\sigma ^R(\\omega )$ , respectively ($\\sigma ^{L}\\ne \\sigma ^{R}$ and $\\sigma ^L\\sigma ^R\\ne 0$ ).", "We formulate and solve a system of Wiener-Hopf integral equations for the electric field tangential to the plane of the sheets on $\\Sigma =\\Sigma ^L\\cup \\Sigma ^R$ in order to derive the dispersion relation for the EP that propagates along the $y$ -axis.", "The surface current density is $\\mathfrak {J}(x,y)=e^{{\\rm i}q y}\\sigma (x) \\lbrace E_x(x,z) \\mathbf {e}_x+E_y(x,z)\\mathbf {e}_y\\rbrace \\bigl |_{z=0}$ where $\\sigma (x)=\\sigma ^L+ \\vartheta (x) (\\sigma ^R-\\sigma ^L)\\qquad (\\sigma ^R\\ne \\sigma ^L)~;$ the Heaviside step function $\\vartheta (x)$ is defined by $\\vartheta (x)=1$ if $x>0$ and $\\vartheta (x)=0$ if $x<0$ .", "By using the vector potential in the Lorenz gauge (Section REF ),[47] we obtain the system $\\begin{pmatrix} u(x) \\\\v(x)\\end{pmatrix} = \\frac{{\\rm i}\\omega \\mu }{k_0^2}\\begin{pmatrix}\\displaystyle \\frac{{\\rm d}^2}{{\\rm d}x^2}+k_0^2 & \\quad \\displaystyle {\\rm i}q \\frac{{\\rm d}}{{\\rm d}x} \\\\\\displaystyle {\\rm i}q\\frac{{\\rm d}}{{\\rm d}x} & k_{\\rm eff}^2 \\end{pmatrix}\\int _{-\\infty }^\\infty {\\rm d}x^{\\prime }\\, K(x-x^{\\prime })\\sigma (x^{\\prime })\\,\\begin{pmatrix} u(x^{\\prime }) \\\\v(x^{\\prime })\\end{pmatrix} \\quad x\\ \\mbox{in}\\ \\mathbb {R}~,$ where $u(x)=E_x(x,0)$ and $v(x)=E_y(x,0)$ .", "By applying the Fourier transform with respect to $x$ , we obtain the functional equations [cf.", "(REF )] $\\Lambda ^R(\\xi )\\begin{pmatrix} \\widehat{u}_{-}(\\xi ) \\\\\\widehat{v}_{-}(\\xi )\\end{pmatrix}+\\Lambda ^L(\\xi )\\begin{pmatrix} \\widehat{u}_{+}(\\xi ) \\\\\\widehat{v}_{+}(\\xi )\\end{pmatrix}=0\\qquad (\\mbox{all\\ real}\\ \\xi )~, $ where $\\Lambda ^{\\ell }(\\xi ):= \\begin{pmatrix}\\displaystyle 1-\\frac{{\\rm i}\\omega \\mu \\sigma ^{\\ell }}{k_0^2}(k_0^2-\\xi ^2)\\widehat{K}(\\xi ;q) & \\ \\displaystyle -\\frac{{\\rm i}\\omega \\mu \\sigma ^{\\ell }}{k_0^2}({\\rm i}q)({\\rm i}\\xi ) \\widehat{K}(\\xi ;q) \\\\\\displaystyle -\\frac{{\\rm i}\\omega \\mu \\sigma ^{\\ell }}{k_0^2}({\\rm i}q)({\\rm i}\\xi )\\widehat{K}(\\xi ; q) & \\ \\displaystyle 1-\\frac{{\\rm i}\\omega \\mu \\sigma ^{\\ell }}{k_0^2}(k_0^2-q^2) \\widehat{K}(\\xi ;q) \\end{pmatrix}~;\\quad \\ell =R,\\,L~.$ In the spirit of our analysis for a single sheet (Section REF ), we diagonalize the matrices $\\Lambda ^{\\ell }(\\xi )$ .", "Their eigenvalues are [cf.", "(REF )] $\\mathcal {P}_{\\rm TM}^{\\ell }(\\xi )=1-\\frac{{\\rm i}\\omega \\mu \\sigma ^{\\ell }}{k_0^2}(k_{\\rm eff}^2-\\xi ^2)\\widehat{K}(\\xi )~,\\quad \\mathcal {P}_{\\rm TE}^{\\ell }(\\xi )=1-{\\rm i}\\omega \\mu \\sigma ^{\\ell }\\widehat{K}(\\xi )\\qquad (\\ell =R,\\,L)~.$ Recall that $\\widehat{K}(\\xi )=({\\rm i}/2)(k_{\\rm eff}^2-\\xi ^2)^{-1/2}$ where $k_{\\rm eff}=\\sqrt{k_0^2-q^2}$ ; $\\Im \\sqrt{k_{\\rm eff}^2-\\xi ^2}>0$ for the first Riemann sheet.", "Accordingly, functional equations (REF ) become $\\begin{pmatrix}\\mathcal {P}_{\\rm TM}^L(\\xi ) & 0 \\\\0 & \\mathcal {P}_{\\rm TE}^L(\\xi )\\end{pmatrix}\\mathcal {S}(\\xi )\\begin{pmatrix}\\widehat{u}_+(\\xi ) \\\\\\widehat{v}_+(\\xi )\\end{pmatrix}+\\begin{pmatrix}\\mathcal {P}_{\\rm TM}^R(\\xi ) & 0 \\\\0 & \\mathcal {P}_{\\rm TE}^R(\\xi )\\end{pmatrix}\\mathcal {S}(\\xi )\\begin{pmatrix}\\widehat{u}_-(\\xi ) \\\\\\widehat{v}_-(\\xi )\\end{pmatrix}=0~,$ for real $\\xi $ , where the matrix $\\mathcal {S}(\\xi )$ is defined by (REF ).", "Hence, we recover and apply transformation (REF ) where $(\\widehat{u}, \\widehat{v})\\mapsto (U,V)$ ; the functions $U_\\pm (\\xi )$ and $V_\\pm (\\xi )$ satisfy the equations $& \\mathcal {P}_{\\rm TM}^L(\\xi )U_+(\\xi )+\\mathcal {P}_{\\rm TM}^R(\\xi )U_-(\\xi )=0~,\\\\&\\mathcal {P}_{\\rm TE}^L(\\xi )V_+(\\xi )+\\mathcal {P}_{\\rm TE}^R(\\xi )V_-(\\xi )=0\\qquad \\mbox{all\\ real}\\ \\xi ~.$ These equations form an extension of (REF ) to the geometry of two coplanar sheets.", "Following the procedure of Section REF , we assume that $\\mathcal {P}_{\\rm TM}^{\\ell }(\\xi )\\ne 0\\ \\mbox{and}\\ \\mathcal {P}_{\\rm TE}^{\\ell }(\\xi )\\ne 0\\quad \\mbox{for\\ all\\ real}\\ \\xi \\qquad (\\ell =R,\\,L)~.$ By defining the functions $\\mathcal {P}_{\\rm TM}(\\xi )= \\frac{\\mathcal {P}_{\\rm TM}^R(\\xi )}{\\mathcal {P}_{\\rm TM}^L(\\xi )}~,\\qquad \\mathcal {P}_{\\rm TE}(\\xi )= \\frac{\\mathcal {P}_{\\rm TE}^R(\\xi )}{\\mathcal {P}_{\\rm TE}^L(\\xi )}$ with $Q(\\xi )=\\ln \\mathcal {P}_{\\rm TM}(\\xi )$ and $R(\\xi )=\\ln \\mathcal {P}_{\\rm TE}(\\xi )$ , we carry out the splittings indicated in (REF ); the logarithmic functions here are such that $Q(\\xi )=\\ln \\mathcal {P}_{\\rm TM}^R(\\xi )-\\ln \\mathcal {P}_{\\rm TM}^L(\\xi )$ and $R(\\xi )=\\ln \\mathcal {P}_{\\rm TE}^R(\\xi )-\\ln \\mathcal {P}_{\\rm TE}^L(\\xi )$ when $\\xi $ lies in the top Riemann sheet (cf.", "Remark 3).", "Because in the present setting the indices associated with $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ on the real axis are zero, i.e., $\\nu =0$ as in (REF ), the split functions $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ are given by integrals (REF ) under (REF ).", "Note that $e^{Q_\\pm (\\xi )}=\\mathcal {O}(1)$ and $e^{R_\\pm (\\xi )}\\rightarrow 1$ as $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_\\pm $ ; see Appendix.", "The Wiener-Hopf method furnishes the entire functions $\\mathcal {E}_1(\\xi )=C_1={\\rm const.", "}$ and $\\mathcal {E}_2(\\xi )=C_2={\\rm const.", "}$ , as in the case with a single conducting sheet (Section REF ).", "Some intermediate steps are slightly different because of the asymptotics for $e^{Q_\\pm (\\xi )}$ in the setting with two sheets (see Appendix).", "We omit any further details about how to obtain $\\mathcal {E}_1(\\xi )$ and $\\mathcal {E}_2(\\xi )$ here.", "Consequently, we obtain the formulas $U_\\pm (\\xi )=\\mp C_1 e^{\\pm Q_\\pm (\\xi )}$ and $V_\\pm (\\xi )=\\mp C_2 e^{\\pm R_+(\\xi )}$ where $C_1$ and $C_2$ are arbitrary constants, which in turn yield (REF ) and (REF ) for $\\widehat{u}_\\pm (\\xi )$ and $\\widehat{v}_\\pm (\\xi )$ .", "By the analyticity of $u_-(\\xi )$ and $v_-(\\xi )$ at $\\xi =-{\\rm i}q{\\rm sg}(q)$ , and the analyticity of $u_+(\\xi )$ and $v_+(\\xi )$ at $\\xi ={\\rm i}q{\\rm sg}(q)$ , we subsequently derive relations (REF )." ], [ "Conclusion and discussion", "In this paper, by using the theory of the Wiener-Hopf integral equations we derived the dispersion relation for the edge plasmon-polariton that propagates along the straight edge of a semi-infinite, planar conducting sheet.", "The sheet lies in a uniform isotropic medium.", "Our treatment takes into account retardation effects, in the sense that, given a spatially homogeneous scalar conductivity of the 2D material as a function of frequency, the underlying boundary value of Maxwell's equations is solved exactly.", "Thus, we avoid the restrictive assumptions of the quasi-electrostatic approximation.", "Our formalism was directly extended to the geometry with two semi-infinite, coplanar conducting sheets.", "In our formal analysis, the existence of the EP dispersion relation on the isotropic sheet is connected to the notion of zero index in Krein's theory.", "[26] In the setting of the dissipationless Drude model for the surface conductivity,[10] for example, this zero index mathematically expresses the property that, for every (real) EP wave number $q$ , the corresponding EP frequency, or energy, $\\omega (q)$ is smaller than the energy of the 2D bulk SP of the same wave number.", "Thus, the character of this EP remains intact in the isotropic setting, in contrast to the situation with a strictly anisotropic conductivity, e.g., in the presence of a static magnetic field, where a branch of $\\omega (q)$ may cross the respective dispersion curve of the 2D bulk SP.", "[23] This latter possibility is studied in some generality, yet within the quasi-electrostatic approach, elsewhere.", "[25], [43] The EP dispersion relation derived here expresses the simultaneous presence of distinct polarization effects.", "To be more precise, the effect of the TM polarization, which alone provides the fine scale of the bulk SP in the nonretarded frequency regime, is accompanied by a contribution that amounts to the TE polarization.", "In this framework, we were able to smoothly connect two non-overlapping asymptotic regimes: (i) the low-frequency limit, in which the EP wave number, $q$ , approaches the free-space propagation constant, $k_0$ , and thus $q/\\omega \\sim {\\rm const.", "}$ ; and (ii) the nonretarded frequency regime, where $q$ is much larger in magnitude than $k_0$ and $q/\\omega ^2 \\sim {\\rm const.", "}$ In each of these regimes, we derived corrections to the anticipated, leading-order formulas for $q(\\omega )$ by invoking the semi-classical Drude model.", "Our work has limitations and leaves several open questions.", "Two noteworthy issues are the stability of the EP under perturbations of the edge and the semi-infinite character of the sheet geometry.", "As a next step, it is tempting to analyze the EP dispersion in microstrips, which may be more closely related to the actual experimental setups.", "[13], [16] This setting calls for developing approximate solution schemes for the related integral equations for the electric field.", "Since we addressed only isotropic and homogeneous surface conductivities, it is natural to investigate how to analyze anisotropic or nonhomogeneous sheets with nonlocalities.", "[25] In this context, a possibility is to couple the full Maxwell equations with linearized models of viscous electron flow in the hydrodynamic regime,[59] where the viscosity and compressibility induce nonlocal effects in the effective conductivity tensor within linear response theory; moreover, the edge as a boundary of the viscous 2D electron system necessarily affects the form of the conductivity tensor." ], [ "ACKNOWLEDGMENTS", "The author is indebted to Vera Andreeva, Tony Low, Alex Levchenko, Andy Lucas, Mitchell Luskin, Matthias Maier, Marco Polini, Tobias Stauber, and Tai Tsun Wu for useful discussions.", "The author also acknowledges: partial support by the MURI Award No.", "W911NF-14-1-0247 of the Army Research Office (ARO) and Grant No.", "1517162 of the Division of Mathematical Sciences (DMS) of the NSF; the support by a Research and Scholarship Award from the Graduate School, University of Maryland in the spring of 2019; and the support of the Institute for Mathematics and its Applications (NSF Grant DMS-1440471) at the University of Minnesota for several visits.", "*" ], [ "On asymptotic expansions for $Q_\\pm (\\xi )$ and {{formula:75f70dfb-343a-41c9-bcca-54dc1b316229}} as {{formula:ec9e0580-0a82-4e89-9040-992fc8c3acdf}}", "In this appendix, we sketch the derivations of asymptotic formulas for the split functions $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ as $\\xi \\rightarrow \\infty $ in $\\mathbb {C}_\\pm $ (see Sections  and ).", "For analogous asymptotic expansions, see Refs.", "MML2017,MMSLL-preprint." ], [ "Single conducting sheet", "Consider formulas (REF ) for $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ with the functions $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ introduced in (REF ).", "We express the associated integrals in the forms $Q_\\pm (\\xi )=\\pm \\frac{1}{{\\rm i}\\pi }\\int _0^{\\infty e^{-{\\rm i}\\arg \\xi }} {\\rm d}\\varsigma \\ \\frac{Q(\\xi \\varsigma )}{\\varsigma ^2-1}~,\\quad R_\\pm (\\xi )=\\pm \\frac{1}{{\\rm i}\\pi }\\int _0^{\\infty e^{-{\\rm i}\\arg \\xi }}{\\rm d}\\varsigma \\ \\frac{R(\\xi \\varsigma )}{\\varsigma ^2-1}\\quad (\\pm \\Im \\,\\xi >0)~,$ where $Q(\\zeta )=\\ln \\biggl (1+\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0^2}\\sqrt{\\zeta ^2-k_{\\rm eff}^2}\\biggr )~,\\quad R(\\zeta )=\\ln \\biggl (1-\\frac{{\\rm i}\\omega \\mu \\sigma }{2}\\frac{1}{\\sqrt{\\zeta ^2-k_{\\rm eff}^2}}\\biggr )~;\\quad \\Re \\sqrt{\\zeta ^2-k_{\\rm eff}^2}>0~.$ First, let us focus on $Q_+(\\xi )$ .", "The numerator in the corresponding integrand is expressed as $Q(\\xi \\varsigma )=\\ln \\biggl (\\frac{{\\rm i}\\omega \\mu \\sigma }{2k_0^2}\\xi \\varsigma \\biggr )+Q_1(\\xi \\varsigma )~;\\quad Q_1(\\zeta )=\\ln \\biggl (\\sqrt{1-\\frac{k_{\\rm eff}^2}{\\zeta ^2}}+\\frac{2k_0^2}{{\\rm i}\\omega \\mu \\sigma }\\frac{1}{\\zeta }\\biggr )~.$ Notice that $Q_1(\\zeta )=\\mathcal {O}(\\zeta ^{-1})$ as $\\zeta \\rightarrow \\infty $ .", "Thus, by substitution of this $Q(\\xi \\varsigma )$ into the integral for $Q_+(\\xi )$ and exact evaluation of the contribution of the first term, we obtain [55], [25] $Q_+(\\xi )=\\frac{1}{2}\\ln \\biggl (\\frac{\\omega \\mu \\sigma \\xi }{2k_0^2}\\biggr )+\\mathcal {O}\\biggl (\\frac{1+\\ln \\xi }{\\xi }\\biggr )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_+~;$ the correction term can be systematically derived via the Mellin transform technique.", "[57] In the above asymptotic formula for $Q_+(\\xi )$ , the branch cut for the logarithm can lie in the lower half $\\xi $ -plane or the negative real axis.", "By symmetry, we have $Q_-(\\xi )=\\frac{1}{2}\\ln \\biggl (-\\frac{\\omega \\mu \\sigma \\xi }{2k_0^2}\\biggr )+\\mathcal {O}\\biggl (\\frac{1+\\ln \\xi }{\\xi }\\biggr )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_-~,$ where the branch cut for the logarithm can lie in the upper half $\\xi $ -plane or the negative real axis.", "To reconcile the last two asymptotic formulas for $Q_+(\\xi )$ and $Q_-(\\xi )$ , we take the branch cut for each logarithm along the negative real axis.", "Accordingly, we verify that $Q_+(\\xi )+Q_-(\\xi )\\sim \\ln \\biggl (\\frac{{\\rm i}\\omega \\mu \\sigma \\xi }{2k_0^2}\\biggr )\\sim Q(\\xi )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\quad \\mbox{in}\\ \\mathbb {C}_+\\ \\mbox{and}\\ \\mathbb {C}_-~.$ We now turn our attention to $R_+(\\xi )$ .", "We write $R(\\xi \\varsigma )=\\ln \\biggl (1-\\frac{{\\rm i}\\omega \\mu \\sigma }{2\\xi \\varsigma }\\biggr )+R_1(\\xi \\varsigma )~;\\quad R_1(\\zeta )=\\ln \\biggl \\lbrace 1-\\frac{{\\rm i}\\omega \\mu \\sigma }{2\\zeta }\\frac{(1-k_{\\rm eff}^2/\\zeta ^2)^{-1/2}-1}{1-{\\rm i}\\omega \\mu \\sigma /(2\\zeta )}\\biggr \\rbrace ~,$ where $R_1(\\zeta )=\\mathcal {O}(\\zeta ^{-3})$ as $\\zeta \\rightarrow \\infty $ .", "The substitution of the above expression for $R(\\xi \\varsigma )$ into the integral for $R_+(\\xi )$ yields $R_+(\\xi )=\\frac{1}{\\pi } \\frac{\\omega \\mu \\sigma }{2\\xi }\\,\\ln \\biggl (\\frac{2\\xi }{\\omega \\mu \\sigma }\\biggr )+\\mathcal {O}(1/\\xi )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_+~.$ In the last formula, the logarithm comes from the first term shown in (REF ); while the $\\mathcal {O}(1/\\xi )$ correction term is attributed to both the first and second terms appearing in (REF ).", "Similarly, we have $R_-(\\xi )=-\\frac{1}{\\pi } \\frac{\\omega \\mu \\sigma }{2\\xi }\\,\\ln \\biggl (-\\frac{2\\xi }{\\omega \\mu \\sigma }\\biggr )+\\mathcal {O}(1/\\xi )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_-~.$ We note in passing that $R_+(\\xi )+R_-(\\xi )=\\mathcal {O}(1/\\xi )$ as $\\xi \\rightarrow \\infty $ , as expected because the sum of $R_+(\\xi )$ and $R_-(\\xi )$ should be exactly equal to $R(\\xi )$ ." ], [ "Two coplanar conducting sheets", "Consider integral formulas (REF ) for $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ where the functions $\\mathcal {P}_{\\rm TM}(\\xi )$ and $\\mathcal {P}_{\\rm TE}(\\xi )$ are now defined by (REF ) (Section ).", "The EP is assumed to propagate along the joint boundary of two coplanar sheets of distinct, scalar surface conductivities $\\sigma ^R$ and $\\sigma ^L$ with $\\sigma ^R\\ne \\sigma ^L$ and $\\sigma ^R\\sigma ^L\\ne 0$ .", "For this geometry, we have $Q(\\zeta )=\\ln \\frac{\\mathcal {P}_{\\rm TM}^R(\\zeta )}{\\mathcal {P}_{\\rm TM}^L(\\zeta )}=\\ln \\mathcal {P}_{\\rm TM}^R(\\zeta )-\\ln \\mathcal {P}_{\\rm TM}^L(\\zeta )=\\ln \\biggl (\\frac{\\sigma ^R}{\\sigma ^L}\\biggr )+\\mathcal {O}(1/\\zeta )\\quad \\mbox{as}\\ \\zeta \\rightarrow \\infty $ and $R(\\zeta )=\\ln \\frac{\\mathcal {P}_{\\rm TE}^R(\\zeta )}{\\mathcal {P}_{\\rm TE}^L(\\zeta )} =\\ln \\mathcal {P}_{\\rm TE}^R(\\zeta )-\\ln \\mathcal {P}_{\\rm TE}^L(\\zeta )=\\mathcal {O}(1/\\zeta )\\quad \\mbox{as}\\ \\zeta \\rightarrow \\infty ~,$ in the appropriately chosen branch of the logarithm, $w=\\ln \\mathcal {P}_\\varpi ^\\ell $ ($\\varpi ={\\rm TM}, {\\rm TE}$ and $\\ell =R, L$ ).", "By inspection of the resulting integrals for $Q_\\pm (\\xi )$ and $R_\\pm (\\xi )$ here we realize that their treatment for a single sheet in Section REF of this Appendix can be directly applied to the present setting of two sheets.", "Without further ado, in regard to $R_\\pm (\\xi )$ we can assert that $R_\\pm (\\xi )=\\pm \\frac{1}{\\pi } \\frac{\\omega \\mu }{2\\xi }\\biggl \\lbrace \\sigma ^R\\ln \\biggl (\\pm \\frac{2\\xi }{\\omega \\mu \\sigma ^R}\\biggr )-\\sigma ^L\\ln \\biggl (\\pm \\frac{2\\xi }{\\omega \\mu \\sigma ^L}\\biggr )\\biggr \\rbrace +\\mathcal {O}(1/\\xi )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_\\pm ~;$ thus, $R_\\pm (\\xi )=o(1)$ .", "On the other hand, in regard to the asymptotics for $Q_\\pm (\\xi )$ we find $Q_\\pm (\\xi )=\\frac{1}{2}\\ln \\biggl (\\frac{\\sigma ^R}{\\sigma ^L}\\biggr )+\\mathcal {O}\\biggl (\\frac{1+\\ln \\,\\xi }{\\xi }\\biggr )\\quad \\mbox{as}\\ \\xi \\rightarrow \\infty \\ \\mbox{in}\\ \\mathbb {C}_\\pm ~,$ with $\\sigma ^R\\ne \\sigma ^L$ and $\\sigma ^L\\sigma ^R\\ne 0$ ." ] ]

[ [ "Joint Stochastic Approximation and Its Application to Learning Discrete\n Latent Variable Models" ], [ "Abstract Although with progress in introducing auxiliary amortized inference models, learning discrete latent variable models is still challenging.", "In this paper, we show that the annoying difficulty of obtaining reliable stochastic gradients for the inference model and the drawback of indirectly optimizing the target log-likelihood can be gracefully addressed in a new method based on stochastic approximation (SA) theory of the Robbins-Monro type.", "Specifically, we propose to directly maximize the target log-likelihood and simultaneously minimize the inclusive divergence between the posterior and the inference model.", "The resulting learning algorithm is called joint SA (JSA).", "To the best of our knowledge, JSA represents the first method that couples an SA version of the EM (expectation-maximization) algorithm (SAEM) with an adaptive MCMC procedure.", "Experiments on several benchmark generative modeling and structured prediction tasks show that JSA consistently outperforms recent competitive algorithms, with faster convergence, better final likelihoods, and lower variance of gradient estimates." ], [ "INTRODUCTION", "A wide range of machine learning tasks involves observed and unobserved data.", "Latent variable models explain observations as part of a partially observed system and usually express a joint distribution $p_\\theta (x,h)$ over observation $x$ and its unobserved counterpart $h$ , with parameter $\\theta $ .", "Models with discrete latent variables are broadly applied, including mixture modeling, unsupervised learning [30], [27], structured output prediction [32], [28] and so on.", "Currently variational methods are widely used for learning latent variable models, especially those parameterized using neural networks.", "In such methods, an auxiliary amortized inference model $q_\\phi (h|x)$ with parameter $\\phi $ is introduced to approximate the posterior $p_\\theta (h|x)$ [18], [35], and some bound of the marginal log-likelihood, used as a surrogate objective, is optimized over both $\\theta $ and $\\phi $ .", "Two well-known bounds are the evidence lower bound (ELBO) [16] and the multi-sample importance-weighted (IW) lower bound [5].", "Though with progress (as reviewed in Section ), a difficulty in variational learning of discrete latent variable models is to obtain reliable (unbiased, low-variance) Monte-Carlo estimates of the gradient of the bound with respect to (w.r.t.)", "$\\phi $since the gradient of the ELBO w.r.t.", "$\\theta $ usually can be reliably estimated.. Additionally, a common drawback in many existing methods for learning latent-variable models is that they indirectly optimize some bound of the target marginal log-likelihood.", "This leaves an uncontrolled gap between the marginal log-likelihood and the bound, depending on the expressiveness of the inference model.", "There are hard efforts to develop more expressive but increasingly complex inference models to reduce the gap [39], [34], [25], [17]Notably, some methods are mainly applied to continuous latent variables, e.g.", "it is challenging to directly apply normalizing flows to discrete random variables, though recently there are some effort [49]..", "But it is highly desirable that we can eliminate the effect of the gap on model learning or ideally directly optimize the marginal log-likelihood, without increasing the complexity of the inference model.", "In this paper, we show that the annoying difficulty of obtaining reliable stochastic gradients for the inference model and the drawback of indirectly optimizing the target log-likelihood can be gracefully addressed in a new method based on stochastic approximation (SA) theory of the Robbins-Monro type [36].", "These two seemingly unrelated issues are inherently related to our choice that we optimizes the ELBO or the similar IW lower bound.", "Specifically, we propose to directly maximize w.r.t.", "$\\theta $ the marginal log-likelihood“directly” is in the sense that we set the marginal log-likelihood as the objective function in stochastic optimization.", "and simultaneously minimize w.r.t.", "$\\phi $ the inclusive divergence$KL[p_\\theta ||q_\\phi ] \\triangleq \\int p_\\theta \\log \\left( p_\\theta / q_\\phi \\right)$ $KL[p_\\theta (h|x)||q_\\phi (h|x)]$ between the posterior and the inference model, and fortunately, we can use the SA framework to solve the joint optimization problem.", "The key is to recast the two gradients as expectations and equate them to zero; then the equations can be solved by applying the SA algorithm, in which the inference model serves as an adaptive proposal for constructing the Markov Chain Monte Carlo (MCMC) sampler.", "The resulting learning algorithm is called joint SA (JSA), as it couples SA-based model learning and SA-based adaptive MCMC and jointly finds the two sets of unknown parameters ($\\theta $ and $\\phi $ ).", "It is worthwhile to recall that there is an another class of methods in learning latent variable models for maximum likelihood (ML), even prior to the recent development of variational learning for approximate ML, which consists of the expectation-maximization (EM) algorithm [8] and its extensions.", "Interestingly, we show that the JSA method amounts to coupling an SA version of EM (SAEM) [7], [20] with an adaptive MCMC procedure.", "This represents a new extension among the various stochastic versions of EM in the literature.", "This revealing of the connection between JSA and SAEM is important for us to appreciate the new JSA method.", "The JSA learning algorithm can handle both continuous and discrete latent variables.", "The application of JSA in the continuous case is not pursued in the present work, and we leave it as an avenue of future exploration.", "In this paper, we mainly present experimental results for learning discrete latent variable models with Bernoulli and categorical variables, consisting of stochastic layers or neural network layers.", "Our results on several benchmark generative modeling and structured prediction tasks demonstrate that JSA consistently outperforms recent competitive algorithms, with faster convergence, better final likelihoods, and lower variance of gradient estimates." ], [ "STOCHASTIC APPROXIMATION (SA)", "[tb] The general stochastic approximation (SA) algorithm $t=1,2,\\cdots $ Monte Carlo sampling: Draw a sample $z^{(t)}$ with a Markov transition kernel $K_{\\lambda ^{(t-1)}}(z^{(t-1)},\\cdot )$ , which starts with $z^{(t-1)}$ and admits $p_{\\lambda ^{(t-1)}}(\\cdot )$ as the invariant distribution.", "SA updating: Set $\\lambda ^{(t)} = \\lambda ^{(t-1)} + \\gamma _t F_{\\lambda ^{(t-1)}}(z^{(t)})$ , where $\\gamma _t$ is the learning rate.", "Stochastic approximation methods are an important family of iterative stochastic optimization algorithms, introduced in [36] and extensively studied [3], [6].", "Basically, stochastic approximation provides a mathematical framework for stochastically solving a root finding problem, which has the form of expectations being equal to zeros.", "Suppose that the objective is to find the solution $\\lambda ^*$ of $f(\\lambda ) = 0$ with $f(\\lambda ) = E_{z \\sim p_\\lambda (\\cdot ) } [ F_\\lambda (z) ],$ where $\\lambda $ is a $d$ -dimensional parameter vector, and $z$ is an observation from a probability distribution $p_\\lambda (\\cdot )$ depending on $\\lambda $ .", "$F_\\lambda (z) \\in R^d $ is a function of $z$ , providing $d$ -dimensional stochastic measurements of the so-called mean-field function $f(\\lambda )$ .", "Intuitively, we solve a system of simultaneous equations, $f(\\lambda ) = 0$ , which consists of $d$ constraints, for determining $d$ -dimensional $\\lambda $ .", "Given some initialization $\\lambda ^{(0)}$ and $z^{(0)}$ , a general SA algorithm iterates Monte Carlo sampling and parameter updating, as shown in Algorithm REF .", "The convergence of SA has been established under conditions [3], [2], [41], including a few technical requirements for the mean-field function $f(\\lambda )$ , the Markov transition kernel $K_{\\lambda ^{(t-1)}}(z^{(t-1)},\\cdot )$ and the learning rates.", "Particularly, when $f(\\lambda )$ corresponds to the gradient of some objective function, then $\\lambda ^{(t)}$ will converge to local optimum, driven by stochastic gradients $F_\\lambda (z)$ .", "To speed up convergence, during each SA iteration, it is possible to generate a set of multiple observations $z$ by performing the Markov transition repeatedly and then use the average of the corresponding values of $F_\\lambda (z)$ for updating $\\lambda $ , which is known as SA with multiple moves [44], as shown in Algorithm in Appendix.", "Remarkably, Algorithm REF shows stochastic approximation with Markovian perturbations [3].", "It is more general than the non-Markovian SA which requires exact sampling $z^{(t)} \\sim p_{\\lambda ^{(t-1)}}(\\cdot )$ at each iteration and in some tasks can hardly be realized.", "In non-Markovian SA, we check that $F_\\lambda (z)$ is unbiased estimates of $f(\\lambda )$ , while in SA with Markovian perturbations, we check the ergodicity property of the Markov transition kernel." ], [ "VARIATIONAL LEARNING METHODS", "Here we briefly review the variational methods, recently developed for learning latent variable models [18], [35].", "Consider a latent variable model $p_\\theta (x,h)$ for observation $x$ and latent variable $h$ , with parameter $\\theta $ .", "Instead of directly maximizing the marginal log-likelihood $\\log p_\\theta (x)$ for the above latent variable model, variational methods maximize the variational lower bound (also known as ELBO), after introducing an auxiliary amortized inference$q_\\phi (h|x)$ uses a single, global set of parameters $\\phi $ over the entire training set, which is called amortized inference.", "model $q_\\phi (h|x)$ : $\\begin{split}ELBO(\\theta ,\\phi ;x) &\\triangleq E_{q_\\phi (h|x)}\\log \\frac{p_\\theta (x,h)}{q_\\phi (h|x)}\\\\&=\\log p_\\theta (x) - KL\\left[ q_\\phi (h|x) || p_\\theta (h|x) \\right]\\end{split}$ It is known that maximizing the ELBO w.r.t.", "${\\phi }$ amounts to minimize the exclusive KL-divergence $KL[q_{{\\phi }}({h}|{x})|| p_{{\\theta }}({h}|{x})]$ , which has the annoying effect of high variance in estimating gradients mentioned before." ], [ "DEVELOPING JSA", "Consider a latent variable model $p_\\theta (x,h)$ for observation $x$ and latent variable $h$ , with parameter $\\theta $ .", "Like in variational methods, we also jointly train the target model $p_\\theta (x,h)$ together with an auxiliary amortized inference model $q_\\phi (h|x)$ .", "The difference is that we propose to directly maximize w.r.t.", "$\\theta $ the marginal log-likelihood and simultaneously minimizes w.r.t.", "$\\phi $ the inclusive KL divergence $KL(p_\\theta (h|x)||q_\\phi (h|x))$ between the posterior and the inference model, pooled over the empirical dataset: $\\left\\lbrace \\begin{split}& \\min _{\\theta } KL\\left[ \\tilde{p}(x) || p_\\theta (x) \\right] \\\\& \\min _{\\phi } E_{\\tilde{p}(x)} KL\\left[ p_\\theta (h|x)|| q_\\phi (h|x) \\right] \\\\\\end{split}\\right.$ where $\\tilde{p}(x) \\triangleq \\frac{1}{n} \\sum _{i=1}^{n} \\delta (x-x_i)$ denotes the empirical distribution for a training dataset consisting of $n$ independent and identically distributed (IID) data points $\\left\\lbrace x_1, \\cdots , x_n \\right\\rbrace $ .", "In such a way, we pursue direct maximum likelihood estimation of $\\theta $ and at the same time avoid the high-variance gradient estimate w.r.t.", "$\\phi $ .", "Fortunately, we can use the SA framework to solve the joint optimization problem Eq.", "(REF ) as described below.", "First, it can be shown that the gradients for optimizing the two objectives in Eq.", "(REF ) can be derived as followsThe first equation in Eq.", "(REF ) directly follows from the Fisher identity Eq.", "(REF ).", ": $\\left\\lbrace \\begin{split}g_\\theta \\triangleq -\\nabla _\\theta KL[ \\tilde{p}&(x) || p_\\theta (x) ]\\\\&=E_{\\tilde{p}(x) p_\\theta (h|x)}\\left[\\nabla _\\theta logp_\\theta (x,h)\\right] \\\\g_\\phi \\triangleq -\\nabla _\\phi E_{\\tilde{p}(x)} &KL\\left[ p_\\theta (h|x)|| q_\\phi (h|x) \\right]\\\\&=E_{\\tilde{p}(x) p_\\theta (h|x)}\\left[ \\nabla _\\phi logq_\\phi (h|x)\\right]\\end{split}\\right.$ By the following Proposition REF , Eq.", "(REF ) can be recast in the expectation form of Eq.", "(REF ).", "The optimization problem can then be solved by setting the gradients to zeros and applying the SA algorithm to finding the root for the resulting system of simultaneous equations.", "Proposition 1 The gradients w.r.t.", "$\\theta $ and $\\phi $ as in Eq.", "(REF ) can be recast in the expectation form of Eq.", "(REF ) (i.e.", "as expectation of stochastic gradients), by letting $\\lambda \\triangleq (\\theta , \\phi )^T$ , $z \\triangleq (\\kappa , h_1,\\cdots ,h_n)^T$ , $p_\\lambda (z) \\triangleq \\frac{1}{n} \\prod _{i=1}^{n} p_\\theta (h_i|x_i)$ , $f(\\lambda ) \\triangleq (g_\\theta , g_\\phi )^T$ , and $F_\\lambda (z) \\triangleq \\left( \\begin{array}{c}\\sum _{i=1}^{n} \\delta (\\kappa =i) \\nabla _\\theta logp_\\theta (x_i,h_i) \\\\\\sum _{i=1}^{n} \\delta (\\kappa =i) \\nabla _\\phi logq_\\phi (h_i|x_i)\\end{array} \\right).$ In order to avoid visiting all $h_1,\\cdots ,h_n$ at every SA iteration (to be explained below), we introduce an index variable $\\kappa $ which is uniformly distributed over $1,\\cdots ,n$ .", "$\\delta (\\kappa =i)$ denotes the indicator of $\\kappa $ taking $i$ .", "This can be readily seen by rewriting Eq.", "(REF ) as: $ \\left\\lbrace \\begin{split}& g_\\theta = \\frac{1}{n} \\sum _{i=1}^{n} E_{p_\\theta (h_i|x_i)}\\left[\\nabla _\\theta logp_\\theta (x_i,h_i)\\right]\\\\& g_\\phi = \\frac{1}{n} \\sum _{i=1}^{n} E_{p_\\theta (h_i|x_i)}\\left[ \\nabla _\\phi logq_\\phi (h_i|x_i)\\right]\\end{split}\\right.$ and applying $E [\\delta (\\kappa =i)] = \\frac{1}{n}$ and the independence between $\\kappa , h_1,\\cdots ,h_n$ .", "Recall that as defined in Proposition REF , $z$ consists of $n+1$ independent components, $\\kappa $ , and $h_1,\\cdots ,h_n$ .", "At iteration $t$ , the SA algorithm need to draw sample $z^{(t)} \\sim p_{\\lambda ^{(t-1)}}(z)$ either directly or through a Markov transition kernel, and then to update parameters based on stochastic gradients $F_{\\lambda ^{(t-1)}}(z^{(t)})$ calculated using this sample $z^{(t)}$ .", "The introduction of $\\kappa $ allows us to calculate the stochastic gradients for only one $h_i$ , indexed by $\\kappa $ , instead of calculating over the full batch of all $n$ data points (as originally suggested by Eq.", "(REF )).", "Minibatching (i.e.", "drawing a subset of data points) can be achieved by running SA with multiple moves, once the SA procedure with only one $h_i$ is established (see Appendix for details).", "The introduction of $\\kappa $ also allows us to avoid drawing a new full set of $h_1,\\cdots ,h_n$ in sampling $z^{(t)} \\sim p_{\\lambda ^{(t-1)}}(z)$ at every SA iteration.", "Suppose that we can construct a base transition kernel for each $h_i$ , denoted by $B_{\\lambda ,i}(h_i^{(t-1)},\\cdot )$ , which admits $p_\\theta (h_i|x_i)$ as the invariant distribution, $i=1,\\cdots ,n$ .", "Specifically, we can first draw $\\kappa $ uniformly over $1,\\cdots ,n$ , and then draw $h_\\kappa $ by applying $B_{\\lambda ,\\kappa }(h_\\kappa ^{(t-1)},\\cdot )$ and leave the remaining components $h_1,\\cdots ,h_{\\kappa -1}, h_{\\kappa +1},\\cdots ,h_n$ unchanged.", "This is a special instanceWe just sample according to the marginal distribution of $h_i, i=1,\\cdots ,n$ , instead of the conditional distribution of $h_i$ , since they are independent.", "of random-scan Metropolis-within-Gibbs sampler [9].", "For designing the base transition kernel $B_{\\lambda ,i}(h_i^{(t-1)},\\cdot )$ , $i=1,\\cdots ,n$ , we propose to utilize the auxiliary inference model $q_\\phi (h_i|x_i)$ as a proposal and use the Metropolis independence sampler (MIS) [22], targeting $p_\\theta (h_i|x_i)$ .", "Given current sample $h_i^{(t-1)}$ , MIS works as follows: Propose $h_i \\sim q_\\phi (h_i|x_i)$ ; Accept $h_i^{(t)}=h_i$ with prob.", "$ min\\left\\lbrace 1, \\frac{w(h_i)}{w( h_i^{(t-1)})} \\right\\rbrace $ , where $w(h_i) = \\frac{p_\\theta (h_i|x_i)}{q_\\phi (h_i|x_i)}$ is the importance sampling weight.", "Remarkably, we can cancel out the intractable $p_{\\theta }(x_i)$ which appears in both the numerator $w(h_i)$ and denominator $w(h^{(t-1)}_i)$ .", "The Metropolis-Hastings (MH) ratio is then calculated as $\\frac{p_\\theta (h_i,x_i)}{q_\\phi (h_i|x_i)} / \\frac{p_\\theta (h^{(t-1)}_i,x_i)}{q_\\phi (h^{(t-1)}_i|x_i)}$ .", "Finally, the JSA algorithm is summarized in Algorithm REF .", "[tb] The JSA algorithm Monte Carlo sampling: Draw $\\kappa $ over $1,\\cdots ,n$ , pick the data point $x_\\kappa $ along with the cached $h^{(old)}_\\kappa $ , and use MIS to draw $h_\\kappa $ ; SA updating: Update $\\theta $ by ascending: $\\nabla _\\theta logp_\\theta (x_\\kappa ,h_\\kappa )$ ; Update $\\phi $ by ascending: $\\nabla _\\phi logq_\\phi (h_\\kappa |x_\\kappa )$ ; convergence" ], [ "CONNECTING JSA TO MCMC-SAEM", "In this section, we reveal that the JSA algorithm amounts to coupling an SA version of EM (SAEM) [7], [20] with an adaptive MCMC procedure.", "To simplify discussion, we consider learning with a single training data point.", "Learning with a set of IID training data points can be similarly analyzed.", "The EM algorithm is an iterative method to find maximum likelihood estimates of parameters for latent variable models (or in statistical terminology, from incomplete data).", "At iteration $t$ , the E-step calculates the Q-function $Q(\\theta |\\theta ^{(t-1)}) = E_{p_{\\theta ^{(t-1)}}(h|x)}\\left[ \\nabla _\\theta logp_\\theta (x,h)\\right]$ and the M-step updates $\\theta $ by maximizing $Q(\\theta |\\theta ^{(t-1)})$ over $\\theta $ or performing gradient ascent over $\\theta $ when a closed-form solution is not available.", "In the E-step, when the expectation in $Q(\\theta |\\theta ^{(t-1)})$ cannot be tractably evaluated, SAEM has been developed [7].", "SAEM can be readily obtained from the Fisher identity (see Appendix for proof), $\\nabla _\\theta log p_\\theta (x) = E_{p_\\theta (h|x)}\\left[ \\nabla _\\theta logp_\\theta (x,h)\\right],$ which again can be viewed as in the expectation form of Eq.", "(REF ) (i.e.", "as expectation of stochastic gradients).", "So we can apply the SA algorithm, which yields SAEM.", "It can be seen that the Monte Carlo sampling step in SA fills the missing values for latent variables through sampling $h^{\\prime } \\sim p_\\theta (h|x)$ , which is analogous to the E-step in EM.", "The SA updating step performs gradient ascent over $\\theta $ using $\\nabla _\\theta logp_\\theta (x,h^{\\prime })$ , analogous to the M-step in EM.", "When exact sampling from $p_{\\theta ^{(t-1)}}(h|x)$ is difficult, an MCMC-SAEM algorithm has been developed [20].", "MCMC-SAEM draws a sample of the latent $h$ by applying a Markov transition kernel which admits $p_{\\theta ^{(t-1)}}(h|x)$ as the invariant distribution.", "Given $\\theta ^{(t-1)}$ , the MCMC step in classic MCMC-SAEM is non-adaptive in the sense that the proposal of the transition kernel is fixed.", "In contrast, in JSA, the auxiliary amortized inference model $q_\\phi (h|x)$ acts like an adaptive proposal, adjusted from past realizations of the Markov chain, so that the Markov transition kernel is adapted." ], [ "RELATED WORK", "Novelty.", "MCMC-SAEM [20] and adaptive MCMC [1] have been separately developed in the SA framework.", "To the best of our knowledge, JSA represents the first method that couples MCMC-SAEM with adaptive MCMC, and encapsulate them through a joint SA procedure.", "The model learning of $\\theta $ and the proposal tuning through $\\phi $ are coupled, evolving together to converge (see Appendix for convergence of JSA).", "This coupling has important implications for reducing computational complexity (see below) and improving the performance of MCMC-SAEM with adaptive proposals.", "Depending on the objective functions used in joint training of the latent variable model $p_\\theta $ and the auxiliary inference model $q_\\phi $ , JSA is also distinctive among existing methods.", "A class of methods relies on a single objective function, either the variational lower bound (ELBO) or the IW lower bound, for optimizing both $p_\\theta $ and $q_\\phi $ , but suffers from seeking reliable stochastic gradients for the inference model.", "The reweighted wake-sleep (RWS) algorithm uses the IW lower bound for optimizing $p_\\theta $ and the inclusive divergence for optimizing $q_\\phi $ [4].", "In contrast, JSA directly optimizes the marginal log-likelihood for $p_\\theta $ .", "Though both RWS and JSA use the inclusive divergence for optimizing $q_\\phi $ , the proposed samples from $q_\\phi (h|x)$ are always accepted in RWS, which clearly lacks an accept/reject mechanism by MCMC for convergence guarantees.", "Compared to the earlier work [46], this paper is a new development by introducing the random-scan sampler and a formal proof of convergence (Proposition REF and Appendix ).", "A similar independent work pointed out by one of the reviewers is called Markov score climbing (MSC) [29] (see Appendix for differences between JSA and MSC).", "Computational complexity.", "It should be stressed that though we use MCMC to draw samples from the posterior $p_\\theta (h|x)$ , we do not need to run the Markov chain for sufficiently long time to converge within one SA iterationTo understand this intuitively, first, note that the inference model is adapted to chase the posterior on the fly, so the proposed samples from $q_\\phi (h|x)$ are already good approximate samples for the posterior.", "Second, the chain could be viewed as running continuously across iterations and dynamically close to the slowly-changing stationary distribution..", "This is unlike in applications of MCMC solely for inference.", "In learning latent variable models, JSA only runs a few steps of transitions (the same as the number of samples drawn in variational methods) per parameter update and still have parameter convergence.", "Thus the training time complexity of JSA is close to that of variational methods.", "One potential problem with JSA is that we need to cache the sample for latent $h$ per training data point in order to make a persistent Markov Chain.", "Thus JSA trades storage for quality of model learning.", "This might not be restrictive given today's increasingly large and cheap storage.", "Minimizing inclusive divergence  $KL[p_\\theta (h|x)||$ $q_\\phi (h|x)]$ w.r.t.", "$\\phi $ ensures that $q_\\phi (h|x)>0$ wherever $p_\\theta (h|x)>0$ , which is a basic restriction that makes $q_\\phi (h|x)$ a valid proposal for MH sampling of $p_\\theta (h|x)$ .", "The exclusive divergence is unsuitable for this restriction.", "Inclusive minimization also avoid the annoying difficulty of obtaining reliable stochastic gradients for $\\phi $ , which is suffered by minimizing the exclusive divergence.", "Learning discrete latent variable models.", "In order to obtain reliable (unbiased, low-variance) Monte-Carlo estimates of the gradient of the bound w.r.t.", "$\\phi $ , two well-known possibilities are continuous relaxation of discrete variable [26], [15], which gives low-variance but biased gradient estimates, and the REINFORCE method [31], which yields unbiased but high-variance gradient estimates.", "Different control variates - NVIL [27], VIMCO [28], MuProp [12], REBAR [43], RELAX [10], are developed to reduce the variance of REINFORCE.", "Other recent possibilities include a Rao-Blackwellization procedure [23], a finite difference approximation [24], a gradient reparameterization via Dirichlet noise [47] or sampling without replacement [19].", "The gap between the marginal log-likelihood and the ELBO is related to the mismatch between the inference model and the true posterior.", "Some studies develop more expressive but increasingly complex inference models, which can also be used in JSA.", "To reduce the gap without introducing any additional complexity to the inference model, there are methods to seek tighter bound, e.g.", "the IW bound in [5].", "But it is shown in [33] that using tighter bound of this form is detrimental to the process of learning the inference model.", "There are efforts to incorporate MCMC into variational inference to reduce the gap.", "[39] introduces some auxiliary variables from a Markov chain in defining a tighter ELBO, but with additional neural networks for forward and backward transition operators.", "[14] seeks ML estimate of $\\theta $ by performing additional Hamiltonian Monte Carlo (HMC) steps from the proposal given by $q_\\phi $ , but still estimate $\\phi $ by maximizing ELBO.", "This method only works with continuous latent variables by using HMC and would be limited since minimizing exclusive divergence encourages low entropy inference models which are not good for proposals.", "In contrast, JSA is not severely suffered by the mismatch, since although the mismatch affects the sampling efficiency of MIS and SA convergence rate, we still have theoretical convergence to ML estimate.", "Adaptive MCMC  is an active research area.", "A classic example is adaptive scaling of the variance of the Gaussian proposal in random-walk Metropolis [37].", "Recently, some adaptive MCMC algorithms are analyzed as SA procedures to study their ergodicity properties [1].", "These algorithms minimize the inclusive divergence between the target distribution and the proposal, but use a mixture of distributions from an exponential family as the adaptive proposal.", "The L2HMC [21] learns a parametric leapfrog operator to extend HMC mainly in continuous space, by maximizing expected squared jumped distance.", "The auxiliary variational MCMC [13] optimizes the proposal by minimizing exclusive divergence." ], [ "EXPERIMENTS", "In this section, we evaluate the effectiveness of the proposed JSA method in training models for generative modeling with Bernoulli and categorical variables and structured output prediction with Bernoulli variables." ], [ "BERNOULLI LATENT VARIABLES", "We start by applying various methods to training generative models with Bernoulli (or binary) latent variables, comparing JSA to REINFORCE [45], NVIL [27], RWS [4], Concrete [26]/Gumbel-Softmax [15], REBAR [43], VIMCO [28], and ARMARSM with Bernoulli latent variables reduces to ARM.", "[48].", "For JSA, RWS and VIMCO, we use particle-number $=2$ (i.e.", "computing gradient with 2 Monte Carlo samples during training), which yields their theoretical time complexity comparable to ARM.", "We follow the model setup in [26], which is also used in [48].", "For $q_\\phi (h|x)$ and $p_\\theta (x|h)$ , three different network architectures are examined, as summarized in Table REF in Appendix, including “Nonlinear” that has one stochastic but two Leaky-ReLU deterministic hidden layers, “Linear” that has one stochastic hidden layer, and “Linear two layers” that has two stochastic hidden layers.", "We use the widely used binarized MNIST [38] with the standard training/validation/testing partition (50000/10000/10000), making our experimental results directly comparable to previous results in [4], [48].", "During training, we use the Adam optimizer with learning rate $0.0003$ and minibatch size 50; during testing, we use 1,000 proposal samples to estimate the negative log-likelihood (NLL) for each data point in the test set, as in [28], [48].", "This setup is also used in section REF but with minibatch size 200.", "In all experiments in section REF , REF and REF , we run different methods on the training set, calculate the validation NLL for every 5 epochs, and report the test NLL when the validation NLL reaches its minimum within a certain number of epochsSpecifically, 1,000, 500 and 200 epochs are used for experiments in section REF , REF and REF respectively.", "(i.e.", "we apply early-stopping).", "For training with JSA, theoretically we need to cache a latent $h$ -sample for each training data point.", "Practically, our JSA method runs in two stages.", "In stage I, we run without caching, i.e.", "at each iteration, we accept the first proposed sample from $q_\\phi (h|x)$ as an initialization and then run MIS with multiple moves.", "After stage I, we switch to running JSA in its standard manner.", "The idea is that when initially the estimates of $\\theta $ and $\\phi $ are far from the root of Eq.", "(REF ), the sample may not be valuable enough to be cached and large randomness could force the estimates moving fast to a neighborhood around the root.", "This two-stage scheme yields fast learning while ensuring convergence.", "In this experiment, we use the first 600 epochs as stage I and the remaining 400 epochs as stage II.", "See [11], [42] for similar two-stage SA algorithms and more discussions.", "Table REF lists the testing NLL results and Figure REF plots training and testing NLL against training epochs and time.", "We observe that JSA significantly outperforms other methods with faster convergence and lower NLL for both training and test set.", "The training time (or complexity) for JSA is comparable to other methods.", "Notably, the NLL of JSA drops around epoch 600 when JSA switches from stage I (no caching) and stage II (caching).", "This drop indicates the effectiveness of our two-stage scheme and the empirical benefit of our theoretical development (keeping a persistent chain).", "Table: Test NLL of different methods with three different network architectures on generative modeling with Bernoulli variables on MNIST, where ** denotes the results reported in , and the others are obtained based on our implementation.The mean and standard deviation results are computed over five independent trials with different random seeds.Our Pytorch code and hyperparameters follow [48].", "Some differences are: [48] uses TensorFlow, runs up to 1,200 epochs, and monitors the validation NLL every one epoch.", "Our run saves time and should be no worse under their conditions.", "Figure: NLL vs training time for “Two layers”Table: Test NLL of different methods on generative modeling of categorical variables on MNIST.", "The mean and standard deviation results are computed over five independent trials with different random seeds.Figure: NLL vs training timeFigure: Log variance of gradient of different methods on generative modeling with categorical variables on MNIST, estimatedevery 50 (out of 500) epochs during training.", "Each method is computed 1000times with different latent samples for the first minibatch (the first 200 records) in the epoch.The left and right are variances of gradients w.r.t.", "θ\\theta and φ\\phi respectively.We report (the logarithm of) the sum of the variances per parameter.Table: Test NLL of different methods on structured output prediction with Bernoulli variables on MNIST, where “n” denotes particle-number.", "The mean and standard deviation results are computed over five independent trials with different random seeds." ], [ "CATEGORICAL LATENT VARIABLES", "In this experiment, we evaluate various methods for training generative models with categorical latent variables.", "Following the setting of [47], there are 20 latent variables, each with 10 categories, and the architectures are listed in Table REF in Appendix.", "We compare JSA with other methods that can handle categorical latent variables, including VIMCO [28], straight through Gumbel-Softmax (ST Gumbel-S.) [15] and ARSM [47].", "For JSA, ST Gumbel-Softmax and VIMCO, we use particle-number $=20$ , which yields their theoretical time complexity close to ARSM.", "JSA runs with the two-stage scheme, using the first 300 epochs as stage I and the remaining 200 epochs as stage II.", "We use the code from [47], and implement JSA and VIMCO, making the results directly comparable.", "We use a binarized MNIST dataset by thresholding each pixel value at 0.5, the same as in [47].", "Table REF lists the test NLL results and Figure REF plots training and testing NLL against training epochs and time.", "Similar to results with Bernoulli variables, JSA significantly outperforms other competitive methods.", "The training time (or complexity) for JSA is comparable to VIMCO and ST Gumbel-Softmax, and ARSM costs much longer time.", "Also, when JSA switches to stage II, a significant decrease of NLL is observed, which clearly shows the benefit of JSA with caching.", "Figure REF plots the gradient variances of different methods.", "The gradient variances w.r.t.", "$\\theta $ for different methods are close and generally smaller than the variances w.r.t.", "$\\phi $ .", "Notably, the gradient variance w.r.t.", "$\\phi $ from JSA is the smallest, which clearly validates the superiority of JSA." ], [ "STRUCTURED OUTPUT PREDICTION", "Structured prediction is another common benchmark task for latent variable models.", "The task is to model a complex observation $x$ given a context $c$ , i.e.", "model the conditional distribution $p(x|c)$ .", "We use a conditional latent variable model $p_\\theta (x,h|c)=p_\\theta (x|h,c)p_\\theta (h|c)$ , whose model part is similar to conditional VAE [40].", "Specifically, we examine the standard task of modeling the bottom half of a binarized MNIST digit (as $x$ ) from the top half (as $c$ ), based on the widely used binarized MNIST [38].", "The latent $h$ consists of 50 Bernoulli variables and the conditional prior $p_\\theta (h|c)$ feeds $c$ to two deterministic layers of 200 tanh units to parameterize factorized Bernoulli distributions of $h$ .", "For $p_\\theta (x|h,c)$ , we concatenate $h$ with $c$ and feed it to two deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs.", "For $q_\\phi (h|x,c)$ , we feed the whole MNIST digit to two deterministic layers of 200 tanh units to parameterize factorized Bernoulli distributions of $h$ .", "RWS, VIMCO, and JSA are conducted with the same models and training setting.", "During training, we use Adam optimizer with learning rate $0.0003$ and minibatch size 100.", "During testing, we use 1,000 samples from $q_\\phi (h|x,c)$ to estimate $-log p_\\theta (x|c)$ (NLL) by importance sampling for each data point.", "JSA runs with the two-stage scheme, using the first 60 epochs as stage I and afterwards as stage II.", "Table REF lists test NLL results against different number of particles (i.e.", "the number of Monte Carlo samples used to compute gradients during training).", "JSA performs the best consistently under different number of particles.", "Both JSA and RWS clearly benefit from using increasing number of particles; but VIMCO notOur result of VIMCO here is accordance with the analysis and result of IWAE in [33], which show that using the IW bound with increasing number of particles actually hurt model learning.", "IWAE and VIMCO are two such examples, with continuous and discrete latent variables respectively.." ], [ "CONCLUSION", "We introduce a new class of algorithms for learning latent variable models - JSA.", "It directly maximizes the marginal likelihood and simultaneously minimizes the inclusive divergence between the posterior and the inference model.", "JSA couples SA-based model learning and SA-based adaptive MCMC and jointly finds the two sets of unknown parameters for the target latent variable model and the auxiliary inference model.", "The inference model serves as an adaptive proposal for constructing the MCMC sampler.", "To our knowledge, JSA represents the first method that couples MCMC-SAEM with adaptive MCMC, and encapsulate them through a joint SA procedure.", "JSA provides a simple and principled way to handle both discrete and continuous latent variable models.", "In this paper, we mainly present experimental results for learning discrete latent variable models with Bernoulli and categorical variables, consisting of stochastic layers or neural network layers.", "Our results on several benchmark generative modeling and structured prediction tasks demonstrate that JSA consistently outperforms recent competitive algorithms, with faster convergence, better final likelihoods, and lower variance of gradient estimates.", "JSA has wide applicability and is easy to use without any additional parameters, once the target latent variable model and the auxiliary inference model are defined.", "The code for reproducing the results in this work is released at https://github.com/thu-spmi/JSA" ], [ "Acknowledgements", "Z. Ou is also affiliated with Beijing National Research Center for Information Science and Technology.", "This work was supported by NSFC Grant 61976122, China MOE-Mobile Grant MCM20170301.", "The authors would like to thank Zhiqiang Tan for helpful discussions." ], [ "On convergence of JSA", "The convergence of SA has been established under conditions [3], [2], [41], including a few technical requirements for the mean-field function $f(\\lambda )$ , the transition kernel $K_{\\lambda ^{(t-1)}}(z^{(t-1)},\\cdot )$ and the learning rates.", "We mainly rely on Theorem 5.5 in [2] to show the convergence of JSA.", "For the transition kernel in JSA, it is shown in [9] that the random-scan Metropolis-within-Gibbs sampler satisfies the $V$ -uniform ergodicity under some mild conditions.", "The $V$ -uniform ergodicity of the transition kernel is the key property for the transition kernel $K_{\\lambda }$ to satisfy the drift condition, which is an important condition used in [2] to establish the convergence of the SA algorithm.", "Specifically, we can apply Theorem 5.5 in Andrieu et al.", "(2005) to verify the conditions (A1) to (A4) to show JSA convergence.", "(A1) is the Lyapunov condition on $f(\\lambda )$ , which typically holds for stochastic optimization problems like in this paper, in which $f(\\lambda )$ is a gradient field for some bounded, real-valued and differentiable objective functions.", "(A2) and (A3) hold under the drift condition, which is satisfied by the transition kernel in JSA as outlined above.", "(A4) gives conditions on the learning rates, e.g.", "satisfying that $\\sum _{t=0}^\\infty \\gamma _t = \\infty $ and $\\sum _{t=0}^\\infty \\gamma _t^2 < \\infty $ ." ], [ "Proof of the Fisher identity", "Note that $E_{p_\\theta (h|x)}\\left[ \\nabla _\\theta logp_\\theta (h|x)\\right]=0$ , so we have $E_{p_\\theta (h|x)}\\left[ \\nabla _\\theta logp_\\theta (x,h)\\right] = E_{p_\\theta (h|x)}\\left[ \\nabla _\\theta logp_\\theta (x)+logp_\\theta (h|x)\\right] =\\nabla _\\theta log p_\\theta (x)$ ." ], [ "Additional comments about JSA", "Minibatching in JSA.", "In our experiments, we run JSA with multiple moves [44] to realize minibatching.", "Specifically, we draw a minibatch of data points, say, $x_{\\kappa _1},\\cdots , x_{\\kappa _m}$ , and for each $x_{\\kappa _j}, j=1,\\cdots ,m$ , we generate multiple $h$ -samples $h_{\\kappa _{j},k}, k=1,\\cdots ,\\textit {particle-number}$ .", "In our pytorch implementation, firstly, we generate proposals for all $h_{\\kappa _{j},k}, j=1,\\cdots ,m, k=1,\\cdots ,\\textit {particle-number}$ , according to $q_\\phi $ and calculate their importance weights.", "This can be easily organized in tensor operations.", "Then we perform accept/reject to obtain the $h$ -samples, which takes negligible computation.", "In this way, JSA can efficiently utilize modern tensor libraries.", "Comparison with [29].", "A similar independent work pointed out by one of the reviewers is called Markov score climbing (MSC) [29].", "It is interesting that MSC uses the conditional importance sampler (CIS), whereas JSA uses the random-scan sampler.", "We see two further differences.", "First, the random-scan sampler in JSA satisfies the $V$ -uniform ergodicity, which ensures that the transition kernel satisfies the drift condition for the SA convergence.", "It is not clear if the CIS sampler satisfies the assumptions listed in [29].", "Second, the model setups in [29] and our work are different for large-scale data.", "By using the random-scan sampler, JSA can support minibatching in our setup.", "Minibatching of MSC in the setup of [29] leads to systematic errors.", "Overfitting observed in Figure REF and REF .", "Similar to previous studies in optimizing discrete latent variable models (e.g.", "[43], [19]), we observe overfitting in Figure REF and REF , when the training process was deliberately prolonged in order to show the effects of different optimization methods for decreasing the training NLLs.", "A common approach (which is also used in our experiments) to address overfitting is monitoring NLL on validation data and apply early-stopping.", "Thus, for example in Figure REF a, the testing NLL for JSA should be read from the early-stopping point shortly after epoch 300.", "Different methods early-stop at different epochs, by monitoring the validation NLLs as described in the 3rd paragraph in section REF .", "For example, in training with categorical latent variables (Figure REF ), JSA, ARSM and VIMCO early-stop at epoch 320, 485 and 180 respectively.", "These early-stopping results are precisely the testing NLLs reported in Table 1 and 2, from which we can see that JSA significantly outperforms other competitive methods across different latent variable models." ], [ "Additional tables and figures", "[tb] SA with multiple moves $t=1,2,\\cdots $ Set $z^{(t,0)}=z^{(t-1,K)}$ .", "For $k$ from 1 to $K$ , generate $z^{(t,k)} \\sim K_{\\lambda ^{(t-1)}}(z^{(t, k-1)},\\cdot )$ , where $K_{\\lambda ^{(t-1)}}(z^{(t, k-1)},\\cdot )$ is a Markov transition kernel that admits $p_{\\lambda ^{(t-1)}}(\\cdot )$ as the invariant distribution.", "Set $\\lambda ^{(t)} = \\lambda ^{(t-1)} + \\gamma _t \\lbrace \\frac{1}{K} \\sum _{z\\in B^{(t)}} F_{\\lambda ^{(t-1)}}(z) \\rbrace $ , where $B^{(t)} = \\lbrace z^{(t,k)} | k = 1,\\cdots ,K \\rbrace $ .", "Table: The three different network architectures in generative modeling with Bernoulli variables on MNIST, which are the same as Table 1 in .", "The following symbols “→\\rightarrow ”, “]”, “)”, and “⇝\\leadsto ” represent deterministic linear transform, leaky rectified linear units (LeakyReLU) nonlinear activation, sigmoid nonlinear activation, and random sampling respectively.Table: Network architectures in generative modeling with categorical variables on MNIST, which are the same as in .", "The following symbols “→\\rightarrow ”, “]”, “)”, “⇝\\leadsto ” and “↪\\hookrightarrow ” represent deterministic linear transform, LeakyReLU nonlinear activation, sigmoid nonlinear activation, random sampling over Bernoulli distributions and random sampling over categorical distributions respectively.For sampling over categorical distributions, there are 20 categorical variables each with 10 categories, thus the 200 output units are divided into 20 groups each with 10 units.Then we apply softmax to each group, and obtain the categorical distribution of each variable.After obtaining random sample of each variable, we use one-hot encoding and concatenate them to obtain a 200 dimensional array.Figure: NLL vs training time for “Noninear”" ] ]

[ [ "Astrophysical Implications of Neutron Star Inspiral and Coalescence" ], [ "Abstract The first inspiral of two neutron stars observed in gravitational waves was remarkably close, allowing the kind of simultaneous gravitational wave and electromagnetic observation that had not been expected for several years.", "Their merger, followed by a gamma-ray burst and a kilonova, was observed across the spectral bands of electromagnetic telescopes.", "These GW and electromagnetic observations have led to dramatic advances in understanding short gamma-ray bursts; determining the origin of the heaviest elements; and determining the maximum mass of neutron stars.", "From the imprint of tides on the gravitational waveforms and from observations of X-ray binaries, one can extract the radius and deformability of inspiraling neutron stars.", "Together, the radius, maximum mass, and causality constrain the neutron-star equation of state, and future constraints can come from observations of post-merger oscillations.", "We selectively review these results, filling in some of the physics with derivations and estimates." ], [ "Introduction", "Observing a 100-second train of gravitational waves (GWs) from inspiraling neutron stars[1] followed after 1.7 s seconds by a gamma-ray burst[2] immediately and incompletely resolved the 50-year old mystery of short gamma-ray bursts (sGRBs).", "Within 11 hours, afterglow light was seen from a host galaxy 130 Ly away, implying that the speed of gravitational waves agreed with the speed of light to a few seconds in 130 My, one part in $10^{15}$ .", "Over the next two years, spanning photon energies from $10^{-5}$ eV to 100 MeV, telescopes across the world monitored the post-merger light.", "These observations, together with numerical simulations of mergers, elucidate the nature of the afterglow and give compelling evidence that a substantial fraction of the universe's heaviest elements are forged by rapid neutron bombardment of nuclei ejected in the merger.", "They also imply that a massive post-merger neutron star briefly sustained itself against collapse.", "This in turn, leads to a new lower limit on the maximum neutron star mass and the most precise current estimate of its value.", "The primary observational constraints on the behavior of cold matter above nuclear density – on the neutron-star equation of state – are this maximum mass and measurements of neutron star masses and radii.", "In the late inspiral, tides alter the waveform; by comparing the GW170817 inspiral waveform to a template bank of waveforms developed in years of analytic and numerical studies, the Ligo/VIRGO collaboration (LVC), as well as subsequent authors, measured the masses and radii of the two neutron stars to within about 2 km.", "These and future GW measurements supplement electromagnetic observations of neutron stars in binary systems that have given precise measurements of mass and approximate measurements of radius.", "Observed neutron stars with masses above 2$M_\\odot $ and the strong evidence from GW170817 that the maximum mass is above 2.1$M_\\odot $ are close to ruling out the hypothesis that neutron stars are really strange quark stars.", "The evidence makes it significantly less likely that neutron stars have quark cores and reduces the likelihood of cores with hyperons;[3] the constraints on EOS parameters are not, however, stringent enough to rule out either alternative.", "[4] We selectively review the implications for physics of the inspiral and merger of two neutron stars.", "Papers on astrophysical implications of NS-NS merger typically use models and relations whose derivations must be tracked through the literature.", "In our discussion, we fill in some of the physics, with calculations and estimates.", "The plan of the paper is as follows.", "In Secs.", "and , we discuss the merger of binary neutron stars and key parts of the physics underlying gamma-ray bursts, kilonovae, and the creation of r-process elements.", "We turn in Sec.", "to the determination of NS radius and tidal deformability from binary inspiral and from electromagnetic observations; and in Sec.", "to inferring maximum NS mass from inspiral and postmerger observations and from X-ray binaries.", "Finally, in Sec.", ", we consider the implications for the EOS of neutron star matter of radius and maximum-mass observations and of future observations of post-merger oscillations of a hypermassive star prior to collapse.", "Among the recent, more detailed expositions are a Shibata-Hotokezaka review of merger and mass ejection,[5] reviews of kilonovae by Metzger[6], of gamma-ray bursts by Berger[7] and Mészáros,[8], and of observational and theoretical constraints on the neutron-star EOS by Lattimer.", "[9]" ], [ "Mergers of compact binaries and short gamma-ray bursts", "In the merger of a double neutron star (NS-NS) system, if the total mass $M$ is small enough to temporarily sustain itself against collapse, the system must rid itself of the difference between the orbital energy just prior to merger and the kinetic energy of a differentially rotating remnant.", "A rough estimate of the size of a $2.7M_\\odot $ remnant is obtained assuming an average density equal to nuclear density, $\\rho _n = 2.6\\times 10^{14}\\rm \\ g/cm^3$ : $ \\displaystyle R \\sim \\left[2.7 M_\\odot /(\\frac{4}{3}\\pi \\rho )\\right]^{1/3}\\sim 17 \\ \\rm km$ .", "(this roughly agrees with numerical simulations, when low-density material surrounding the remnant is included).", "The available energy is then of order the gravitational binding energy $E \\sim \\frac{1}{10}\\frac{GM^2}{R}\\sim 10^{53}\\left(\\frac{M}{2.7 M_\\odot }\\right)^2\\frac{15\\ \\rm km}{R}\\ \\rm erg,$ an energy per baryon of order 50 MeV.", "Almost all the energy of the merger is ultimately carried away by gravitational waves and by thermal neutrinos.", "These are produced by electron and positron capture, $e^+ + n\\rightarrow p+\\bar{\\nu }_e, \\ p+e^-\\rightarrow n + \\nu _e\\ $ , by pair annihilation, $e^+ +e^-\\rightarrow \\nu +\\bar{\\nu }$ , and by electrons interacting with the hot plasma $e^-\\rightarrow e^-+\\nu +\\bar{\\nu },$ the latter processes yielding all neutrino flavors.", "The remaining energy is in the kinetic energy of ejected material, in kinetic and thermal energy of the remnant, in a growing magnetic field, and in light.", "Because only about 1% of the available energy is emitted as light, the features of electromagnetic observations are sensitive to initial conditions, but even the gross behavior of the system depends on the initial masses and on the neutron-star equation of state.", "Most important is the total mass $M=m_1+m_2$ of the system.", "The nature and final fate of the merger depends on whether $M$ is larger or smaller than three critical masses: $M_{\\rm thres}$ is the threshold mass, the mass above which the merged stars – the merger remnant – promptly collapses to a black hole; it is the maximum mass that can be supported against collapse by pressure and differential rotation of the initial hot remnant.", "$M_{\\rm max,rot}$ is the maximum mass of a cold, uniformly rotating neutron star.", "$M_{\\rm max,rot}$ is about 20% larger than $M_{\\rm max, spherical}$ , the maximum mass of a cold, nonrotating star, which we will denote by $M_{\\rm max}$ for simplicity.", "If $M> M_{\\rm thres}$ , the system will promptly collapse to a black hole.", "The value of $M_{\\rm thres}$ depends on the neutron star equation of state (EOS), and as we discuss below, measuring $M_{\\rm thres}$ gives the strongest astrophysical constraint on the EOS at several times nuclear density.", "When $M>M_{\\rm thres}$ , the star collapses in a few dynamical times, where $t_{\\rm dynamical} \\sim \\sqrt{\\frac{R^3}{GM}}= 0.12 \\left(\\frac{R}{17 \\ \\rm km}\\right)^{3/2} \\left(\\frac{M}{2.7M_\\odot }\\right)^{-1/2} \\ \\rm ms.$ Even in this prompt collapse, some matter can remain in a disk, if the tidal torques are large enough.", "The height of tides raised on the less massive star $m_1$ is roughly $h\\sim \\frac{m_2}{m_1}R_1\\left(\\frac{R_1}{d}\\right)^3$ with $R_1$ the radius of $m_1$ and $d$ the distance between the stars.", "For approximately equal masses, the disk mass is negligible, but if $m_1/m_2 \\lesssim 0.8$ and the EOS is stiff below twice nuclear density (so that the radius of $m_1$ is not too small), tidal disruption can leave a disk with mass $\\gtrsim .001 M_\\odot $ .", "[5] Figure: The merger outcome depends on the system's total mass.", "M>M thres M>M_{\\rm thres} leads toprompt collapse with a small or negligible disk.", "M<M thres M< M_{\\rm thres} yields a massive hot, differentially rotatingneutron star.", "If M thres >M>M max , rot M_{\\rm thres}>M> M_{\\rm max, rot} this differentially rotating remnant collapsesto a black hole as viscosity and a magnetic field enforce uniform rotation.Figure adapted from Shibata and HotokezakaWhen $M<M_{\\rm thres}$ , the merger yields a hot, massive neutron star surrounded by a thick disk of material.", "The massive remnant is initially supported against collapse by differential rotation and degeneracy pressure (the pressure of the zero-temperature neutron star EOS), which still dominates thermal pressure in the dense core.", "At the intersection of the merging stars, velocity fields from each star are oppositely oriented, leading to a differentially rotating remnant and to an unstable boundary layer between fluids with different velocities.", "This is the Kelvin-Helmholtz instability, responsible, for example, for ocean waves – waves at the interface between air and water.", "Here the length scale is much shorter than the radius of the star, and the growth time much faster than dynamical.", "Differential rotation also leads to rapid magnetic field growth, in part through the magnetorotational instability.", "[10], [11] The ensuing gross behavior over the next several seconds is tied to a redistribution of angular momentum.", "A ring of fluid with mass $m$ , radius $r$ , and velocity $v$ has the same angular momentum $mvr$ as a more distant ring at radius $R$ and velocity $vr/R$ , but the energy of the more distant ring is smaller by the factor $r^2/R^2$ .", "Dissipation – here viscosity and magnetic turbulence – therefore transports angular momentum outward, driving the rotation law toward uniform rotation.", "The smaller amount of kinetic energy available in uniform rotation leads to the collapse that differential rotation had temporarily halted: A star is called hypermassive if its mass is in the range $M_{\\rm thres}> M > M_{\\rm max,rot}$ , below the threshold for prompt collapse, but too large to be supported by uniform rotation.", "[12] Finally, even if $M< M_{\\rm max,rot}$ , collapse remains a short-term threat, because the newly generated magnetic field will spin the star down, and radiation will cool its outer part.", "The spin-down time from a magnetic field of order $10^{15}$ G is a few minutes, and unless the remnant's mass is below the maximum mass $M_{\\rm max}$ of a cold, nonrotating star, the merging stars are fated to end as a black hole surrounded by a disk (see, e.g., [5] and references therein)." ], [ "Short gamma-ray bursts", "Beginning in the mid-sixties with observations of the Vela satellites, bursts of gamma rays were detected and subsequently identified with galaxies at cosmological distances.", "There are two primary classes: Long bursts, with duration generally greater than two seconds, occur in star-forming galaxies, and coincident supernovae are seen when the burst is close enough for the supernova to be detected.", "They are compellingly linked to the core-collapse of a massive star, ending as a black hole or possibly a magnetar (a neutron star with a magnetic field of order $10^{15}$ G).", "For short gamma-ray bursts, on the other hand, no associated supernova has been seen, and, when associated galaxies are seen, they often have only old stars, ruling out supernovae as the source.", "Because the bursts last less than 2 s, they must emerge from a volume that is not much more than 2 light-seconds across.", "The cosmological distance implies a luminosity comparable to the neutron-star binding energy seen in supernovae.", "With supernovae ruled out, the leading candidates have been NS-BH and NS-NS mergers[13], [14] (see, e.g., [8], [7] for a history and extensive references).", "The gamma-ray burst is emitted by a jet of relativistic particles that fly outward along the axis of rotation.", "The restriction of the burst to a narrow beam is associated with relativistic beaming and to collimation of the jet by pressure of the surrounding matter and perhaps by a toroidal magnetic field.", "[15], [16] A leading candidate for the engine that launches the jet is the Poynting flux from a magnetic field of strength above $10^{15}$ G that accelerates plasma near the axis to relativistic speeds.", "Driven by the Kelvin-Helmholtz instability and the magnetorotational instability (MRI) associated with differential rotation, the magnetic field grows dramatically on a dynamical timescale.", "Recent general relativistic magnetohydrodyamical simulations of NS-NS mergers with magnetic field growth are reported by Rezzolla et al.", "[17], Kiuchi et al., [18] and by Ruiz et al.", "[19], who find the emergence of a mildly relativistic jet.", "Because of its astrophysical importance, we briefly discuss the MRI and estimate its growth time, following Balbus and Hawley.[20].", "The post-merger star inherits a small magnetic field from its progenitors.", "Because the matter is hot and conducting, a displacement $\\xi $ of the fluid deforms the magnetic field.", "The energy per unit volume of the magnetic field is $\\frac{B^2}{8\\pi }$ , and bending the field lines to a curve with radius of curvature $R$ then gives a force per unit volume of order $B^2/R$ .", "Because a displacement $\\mathbf {\\xi }\\cos ({\\mathbf {k}\\cdot \\mathbf {x}})$ has radius of curvature $\\lambda ^2/\\xi $ , the restoring force per unit volume for a displacement perpendicular to $\\mathbf {B}$ , with $\\mathbf {k}$ along $\\mathbf {B}$ , is $ f \\sim \\xi B^2/\\lambda ^2$ .", "The way the instability works can be seen in a toy model, with the star replaced by a disk about a central mass and each particle in the disk moving with its Keplerian angular velocity $\\Omega = \\sqrt{GM/r^3}$ .", "A particle displaced outward by an amount $\\xi $ (keeping its initial angular momentum) moves in an elliptical orbit with the same frequency $\\Omega $ : In a frame moving with the original angular velocity $\\Omega $ , the particle oscillates about its original position with oscillation frequency (epicyclic frequency) $\\Omega $ , corresponding to a restoring force per unit volume $\\rho \\Omega ^2\\xi $ .", "Now consider the same model with a magnetic field perpendicular to the disk.", "When a fluid element is displaced outward, its angular velocity decreases; in the rotating frame it moves backward, and it drags the field lines backward.", "The deformed field pulls the fluid element forward, countering the decrease in its angular velocity.", "If the resulting angular velocity at $r+\\xi $ is larger than $\\Omega (r+\\xi )$ of the unperturbed disk, the fluid element's acceleration will be larger than the gravitational force on it and it will accelerate outward.", "The criterion for instability is then that the magnetic force per unit volume is larger than the restoring force per unit volume or, roughly,$\\displaystyle \\frac{B^2}{\\lambda ^2} \\gtrsim \\rho \\Omega ^2$ .", "Modes with wavelength of order $R$ are unstable on a dynamical time when $B^2\\sim \\rho \\Omega ^2 R^2$ .", "Thus, energy associated with differential rotation can flow into a magnetic field on a dynamical timescale.", "The instability criterion for a differentially rotating star has this form, as long as $d\\Omega /dr \\sim -\\Omega /R$ , and it is satisfied in the the outer part of the massive neutron star.", "[21] The inner part can retain two density maxima and a quadrupole velocity field from the merger until collapse; the MRI is not confined to the outer part of the star, but one cannot use the criterion for a differentially rotating star.", "Short wavelength magnetic turbulence and viscosity convert energy associated with differential rotation in the outer part of the remnant to heat and kinetic energy.", "The available energy is a fraction of the difference $\\Delta E$ between the initial differential rotation and a final uniform rotation, and the change in angular velocity is comparable to the angular velocity itself.", "Estimating the moment of inertia as $I\\sim \\frac{2}{5} MR^2$ , with $M=2.6 M_\\odot $ and $R=14$ km,Despite the fact that any spherical Newtonian star has moment of inertia smaller than $I=\\frac{2}{5} MR^2$ (the uniform-density case), the enhanced strength of relativistic gravity at high compactness increases the value of $I$ for the same $M$ and $R$ .", "Following Bejger & Haensel[22], Lattimer & Schutz[23] find $I\\approx 0.237 MR^2\\left[1+4.2 (M/M_\\odot )/(R/1\\rm km) + 90 (M/M_\\odot )^4/(R/1\\rm km)^4\\right]$ , to about 3%.", "we have $\\Delta E \\sim \\frac{1}{2} I\\Omega ^2 \\sim 10^{53} \\left(\\frac{I}{4\\times 10^{45}\\ \\rm g\\, cm^2}\\right)\\left(\\frac{\\Omega }{7000\\rm \\ s^{-1}}\\right)^2\\ \\rm erg.$ With $\\sim 10^{57}$ baryons per $M_\\odot $ , and $\\lesssim 0.05 M_\\odot $ in the ejecta this energy is over 1000 MeV/(ejecta baryon).", "Unless $M<M_{\\rm max,rot}$ , collapse leaves a large part of the rotational energy in the black hole.", "Most of the remainder is emitted as neutrinos.", "Part, less than $10^{51}$ erg, powers the ejection of a thick torus extending from the equatorial plane and moving with roughly the escape velocity, about $0.1$ c, or a kinetic energy of about 5 MeV per baryon.", "The part that is ordinarily observed, the gamma-ray burst itself, has a typical energy less than 2$\\times 10^{50}$ erg.", "[24], [7]This energy is computed after estimating the solid angle of the beam.", "Because the beam angle is often not known, an isotropic equivalent energy $E_{\\rm iso}$ , the larger energy of an isotropic emitter with the same observed flux, is common in the literature: Emitted energy $= E_{\\rm iso}\\times $ (solid angle of the beam)/$4\\pi $ .", "For SGRBs, the estimated average relation is[7] $E \\sim 0.015 E_{\\rm iso}$ .", "Remarkably, however, the gamma-rays are emitted by highly relativistic matter: As we review in the next paragraphs, there is compelling evidence that the gamma-rays are emitted by a jet whose matter has a Lorentz factor of order 20-100 at the time of observation.", "If the emitting particles were accelerated baryons, they would need an energy per baryon $m_n\\Gamma > 20,000$ MeV.", "For accelerated electrons or $e^+e^-$ pairs, the energy per particle $m_e\\Gamma $ instead exceeds a more manageable 10 MeV.", "A lower limit on $\\Gamma $ is set by the observation of gamma-ray energies above the threshold, $m_ec^2 = 511$ keV, for pair creation from $\\gamma +\\gamma \\rightarrow e^++e^-$ .", "We will soon see that, if this were the energy in the rest frame of the matter, the interaction of the photons with a high density of pairs would make the emitting region opaque.", "But the prompt emission of the gamma rays (and the fact that the spectrum is not a blackbody) means that the matter is nearly transparent, that their mean free path $\\ell $ is of order the radius $R$ of the emitting region.", "We can estimate $\\Gamma $ as follows.", "[25] For a total energy ${\\cal E}(m_ec^2)$ of light whose energy per photon is above $m_e c^2 = 511$ keV in the rest frame of the radiating matter, a fraction of order unity of the photons are converted to pairs, giving a total number of pairs $N\\sim {\\cal E}/m_e c^2$ and a number density $n = \\frac{\\cal E}{m_e c^2 R^3},$ with $R$ the radius of the emitting region.", "The scattering cross section of gamma-rays from electrons (and positrons) is the classical Thomson cross section $\\sigma _T = \\frac{8\\pi }{3}r_e^2 = 0.67\\times 10^{-24}\\ \\rm cm^2,$ where $r_e = e^2/(m_ec^2)$ is the classical electron radius.", "The mean free path, $ \\ell = \\frac{1}{n\\sigma _T}$ , is then $\\displaystyle \\ell \\sim \\frac{m_e c^2 R^3}{{\\cal E}\\sigma _T}$ , and the condition $\\ell \\gtrsim R$ is $\\frac{m_e c^2 R^2}{{\\cal E}\\sigma _T} \\gtrsim 1.$ If the matter is not highly relativistic, this is wildly inconsistent with observation: The burst is emitted in about a second from matter expanding outward with $v\\approx c$ , implying $R\\sim 3\\times 10^{10}$ cm.", "The total burst energy is ${\\cal E}_{\\rm observed}\\gtrsim 10^{48}$ erg.", "If the matter is not highly relativistic, this observed energy is comparable to the energy in the rest frame of the matter, and we have $\\frac{m_e c^2 R^2}{{\\cal E}\\sigma _T} \\approx 10^{-9} \\left(\\frac{R}{3\\times 10^5\\rm km}\\right)^2\\left(\\frac{\\cal E}{10^{48}}\\ \\rm erg\\right)^{-1}.$ That is, if $\\Gamma \\sim 1$ , the mean free path is less than $R$ by a factor of at least $10^9$ .", "If the matter is highly relativistic, however, the observed energy is much larger than the energy in the rest frame of the matter.", "The most obvious reason is relativistic beaming, but there is a second effect that is more important: Because photons are blue-shifted by the factor $\\sqrt{(1+v/c)/(1-v/c)}\\approx 2\\Gamma $ for $v\\approx c$ , the rest-frame energy peaks at a photon energy $E_p = E_{p,\\rm observed}/2\\Gamma $ , leaving far fewer photons with energy high enough for pair creation.", "Beyond the peak energy, the observed energy decreases, either as a power law or an exponential (see, e.g., a review by Nakar[26]).", "We obtain a conservative lower limit (an underestimate) with an exponential cutoff: With ${\\cal E}(E)$ the energy greater than $E$ , the cutoff has the formAs reviewed in Nakar[26] the range of SGRBs are fit to models of the form ${\\cal E}(E) = A E^{\\alpha +1} e^{-E/E_0}$ , with $\\alpha $ in the range $-2$ to $0.5$ (other models use broken power laws).", "We chose a representative value $\\alpha =-1$ , with $A=1$ to give a total energy of order $E_0$ .", "Observed peak energies range from 20 to 1000 keV.", "[27] $\\displaystyle {\\cal E}_{\\rm observed}(E) = {\\cal E}_{\\rm observed}(E_p) \\exp (-E/E_{p})$ , with $E_{p,\\rm observed}= 500$ keV a typical value.", "The energy distribution in the matter rest frame is then given by ${\\cal E}(E) = \\frac{{\\cal E}_{\\rm observed}(E_p)}{2\\Gamma } \\exp (-2\\Gamma E/E_p),$ and Eq.", "(REF ) for ${\\cal E} = {\\cal E}(m_ec^2) $ gives $\\ell > R$ for $(2\\Gamma )^{-1}e^{-2\\Gamma } < 10^{-9}$ , or $\\Gamma > 9$ .", "SGRBs have varying cutoffs and peak energies, and estimated values of $\\Gamma $ are larger than our deliberate underestimate, ranging from 20 to 100.", "Without the effect of the energy cutoff, relativistic beaming and blue-shifting alone enhance the total observed energy of the burst by a factor of $\\Gamma ^3$ .", "The factor of $10^{9}$ needed for $\\ell \\sim R$ would then require $\\Gamma \\gtrsim 10^3$ .", "As mentioned above, magnetic transfer of rotational energy in merger simulations has produced only mildly relativistic outflows.", "A competing central engine, neutrino pair annilation $\\nu +\\bar{\\nu }\\rightarrow e^+ + e^-$ followed by $e^\\pm $ annihilation to produce gamma rays, also appears to provide too little energy.", "[28], [29], [30], [31], [32], [33], [34] Following collapse, however, the spinning black hole will retain a magnetic field that can again communicate its rotational energy to the surrounding plasma (the Blandford-Znajek mechanism[35]) and that may provide the engine that turns the outflow into a highly relativistic jet.", "[36], [17], [18], [37] The Burst GRB170817A The gamma-ray burst associated with GW170817 was unusually weak, with luminosity less than $4\\times 10^{47}$ erg s$^{-1}$ , orders of magnitude below that of a typical sGRB[24] with known redshift.", "A debate over whether the jet had made it past the surrounding matter or was choked was finally resolved by direct observation[38] of “superluminal motion,” a source that moved a measured distance $d$ perpendicular to the line of sight in a measured time less than $d/c$ .", "Recall that, for a source moving at $v\\approx c$ at an angle $\\theta $ to the line of sight, light emitted after a time $t$ travels a distance shorter by $vt\\cos \\theta $ than light emitted at $t=0$ .", "The difference in arrival time is then $\\displaystyle t-vt\\cos \\theta /c$ , and the apparent velocity perpendicular to the line of sight is $\\displaystyle v_{\\rm app} = \\frac{v\\sin \\theta }{1-v\\cos \\theta /c}$ .", "This is large for small $\\theta $ and has maximum value $v_{\\rm app}=\\Gamma c$ at $\\theta =1/\\Gamma $ .", "The observed apparent velocity of the late-time jet implies $\\Gamma \\approx 4$ .", "This is consistent with an initial $\\Gamma $ of order 10, still below the previously observed SGRB range.", "The leading scenario describing the atypically weak event has a a jet that is surrounded by matter dense enough to shroud it, called a cocoon (following the early Blandford-Rees model of galactic jets[39], [40]).", "Only part of the jet breaks out[41], [42], [38], [43], [16], and it is seen at an angle away from the jet axis.", "Beyond the smaller gamma-ray luminosity, what leads to this scenario is a major difference between the subsequent behavior of GW170817 and previously analyzed SGRBs.", "In earlier events, as interactions with surrounding matter slow the highly relativistic initial jet, its average photon energy quickly decreases, moving from gamma-rays to X-rays.", "Measured by time in the rest-frame of the jet, X-rays are seen within two minutes of the initial burst, but it was 9 days after the GW17-817 merger before the first evidence of X-rays.", "[44] Their decline has been similarly much more gradual than that of a typical sGRB (see [45], [46] and references therein).", "The idiosyncratic behavior of this event suggests a previously unidentified class of NS-NS mergers.", "It would include both sGRBs like GRB170817A, in which a jet is partially blocked by a cocoon, and other mergers with choked jets that never break out from the surrounding matter.", "In the last year, von Kienlinet al.", "[47] searched archival GRB data for evidence of overlooked events, finding 12 candidates, including one noticed a year earlier with a measured redshift.", "[48], [49] In all of these events the gamma-ray burst is followed by a delayed, soft (lower-frequency) tail, whose spectrum resembles that of GRB 170817A.", "In the one case where a redshift is measured, GRB 150101B, the burst energy is $10^3$ larger than 170817, although still at the low end of sGRBs.", "An analysis by Nakar & Piran[43] finds that at most two of the candidates are consistent with the cocoon shock-breakout model.", "This is still consistent with a large unseen class of mergers, because events as weak as GRB 170817A would not have been seen at typical distances of measured sGRBs." ], [ "r-process nucleosyntheis and kilonovae", "The major media story following GW170817 was a “clash of neutron stars forges gold.”[50] We briefly review the formation of the heaviest elements from rapid neutron bombardment and the evidence that merger ejecta from NS-NS and perhaps NS-BH mergers are the sites of this nucleosynthesis." ], [ "r-process nucleosynthesis", "Nucleosynthesis in stellar cores proceeds in thermodynamic equilibrium, forging elements with successively greater binding energy per nucleon.", "As Fig.", "REF illustrates, the binding energy per nucleon trends upward from helium through the iron group (baryon number $A\\sim 50-60$ ), because larger nuclei have a smaller fraction of surface nucleons, nucleons with unsaturated bonds.", "But the binding energy per nucleon cannot exceed its value for full saturation, while the Coulomb repulsion per nucleon grows as $Z^2/A$ .", "Figure: Binding energy per nucleon peaks at the iron group.", "From OpenStax under Creative CommonsAfter the iron group, the increasing Coulomb energy reverses the trend, gradually reducing the binding energy per nucleon as $Z$ increases.", "The result is that only elements up to the iron group can form in thermodynamic equilibrium.", "Stable nuclei, however, extend up to lead (Z=82, A=208).", "They lie in a valley of stability, limited on the high-proton side by Coulomb repulsion and on the high-neutron side by the increased neutron Fermi energy, and these heavier elements are almost entirely formed by neutron bombardment.", "[51] What nuclei are reached depends on whether the time between neutron captures is longer or shorter than the time for unstable nuclei to decay back to stability, ordinarily by $\\beta $ -decay.", "Slow bombardment, the s-process, proceeds through stable nuclei until an unstable nucleus is reached that then decays back to stability before the next neutron capture.", "As $A$ increases, however, a smaller fraction of nuclei are stable.", "Not all stable nuclei can be reached by a path through the valley of stability, and, more often, along the available paths, capture cross sections limit the abundance of s-process elements.", "With rapid bombardment, the r-process, highly unstable neutron-rich nuclei are built before they have time to decay.", "After the bombardment stops, a series of decays take the nuclei back to stability.", "The s-process and r-process nuclides comprise overlapping sets.", "In Fig.", "REF , the solid zig-zag line shows a segment of the s-process through stable nuclides.", "Figure: Stable elements are represented by boxes, neutron number increasing to the right,proton number increasing vertically.", "Heavy solid lines show s-process paths.Arrows show beta-decay to stability of initial rr-process nuclides.", "Adapted from Kaeppler et al.The hashed area (labeled r-process band) shows the unstable nuclides formed by the initial rapid bombardment, and the dotted lines with arrows pointing back to stability show the result of $\\beta $ -decays back to stability.", "That mergers of NS-BH and NS-NS binaries might be a primary site for $r$ -process nucleosynthesis was first suggested by Lattimer and Schramm[53] and by Symbalisty and Schramm[54], respectively.", "Although the astrophysics community long favored supernovae, evidence for mergers and against supernovae gradually accumulated (see Lattimer[55] for a history of the debate).", "The strongest observational evidence is an uneven distribution of $r$ -process nuclides that implies they are synthesized in rare events: For example, of dwarf galaxies large enough to have hosted thousands of supernovae (with of order $10^6$ stars), only about 10% have $r$ -process elements, a percentage consistent with SGRB rates.", "[56] On the theoretical side, in increasingly sophisticated supernova and merger simulations, only the mergers yield significant amounts of heavy $r$ -process matter.", "The measured abundance of r-process nuclides peaks at magic numbers that correspond to nuclei with filled shells of neutrons, reached during rapid bombardment before the nuclei decay to stability.", "The three main peaks are at $A \\approx $ 80, 130, 195.", "Because the abundance distribution reflects the binding energy per nucleon of the pre-decay nuclides, it is robust, emerging from a variety of simulations,[57] when there is a wide enough range of neutron richness in the reacting matter.", "Some ejected matter must be highly neutron rich to reach the second and third peaks: To form elements from lanthanides (rare earth elements) through uranium, a ratio of protons to the total number of baryonsThis ratio is conventionally measured by the electron fraction $Y_e := $ (number density of electrons)/(number density of baryons).", "less than 0.25 is needed.", "[58], [59], [5] To reproduce the observed abundance of lighter r-process nuclides, the ejecta must also include a less neutron rich component, with the ratio proton/baryon $\\gtrsim 0.25$ , and that is seen in simulations of mergers with $M<M_{\\rm thres}$ , mergers in which a massive neutron star briefly supports itself against collapse.", "[59], [60], [61], [62], [6] In merger simulations, the first ejecta are neutron rich.", "If a high-mass neutron star is formed, the $e^\\pm $ production and subsequent weak interactions that lead to loss of most of the merger energy in neutrinos also deplete the number of neutrons: The higher density of neutrons means a higher rate of $n\\rightarrow p$ than $p\\rightarrow n$ conversions, from positron and neutrino capture.", "An additional and larger part of ejecta emerges as viscosity and a rapidly growing magnetic field convert differential rotation to uniform rotation, transporting angular momentum outward.", "Indirectly powering this second part of the ejection is the energy difference $ \\Delta E \\sim 10^{53}$ erg of Eq.", "(REF ) in the massive neutron star and the smaller energy of differential rotation in the surrounding disk: Heated by what liberated energy is not radiated as neutrinos, the expanding disk adds to the ejecta.", "Figure: Abundance of r-process elements shows peaks associated with neutron magic numbers (filled neutron shells)in bombarded nuclei prior to their decay to stability.", "From Hotokezaka et al.The LIGO-Virgo collaboration has seen one unambiguous NS-NS inspiral and coalescence (GW170817) and one additional event (GW190425),[64] whose total mass is $3.4^{+0.3}_{-0.1}M_\\odot $ .", "[64], with a possibility that one of the component stars has a mass as large as $2.5 M_\\odot $ .", "If both stars are neutron stars, this second event slightly elevates the estimated rate of NS-NS mergers, but it may belong to the class of prompt-collapse mergers, a class unlikely to contribute significantly to the r-process element abundance.", "We begin with an estimate that assumes only one relevant event has been observed.", "We then suppose GW190425 is the result of a BH-NS merger and estimate the contribution to $r$ -process matter from events of this kind.", "If, based on one event in a year of observation, we assume that a coalescence occurs within the observed distance at least every 5 years, we can estimate as follows an event rate that lies within the large error bars of the several recent estimates.", "The electromagnetic counterparts of GW170817 were identified with the galaxy NCG4993, whose 40 Mpc distance corresponds to a volume $V = \\frac{4}{3}\\pi (40\\ \\rm Mpc)^3 = 2.7\\times 10^5\\rm Mpc^3.$ The merger rate is commonly written in Gpc$^{-3}$ yr$^{-1}$ or in events per Milky-Way equivalent galaxy (MWEG) per year: $\\mbox{NS-NS merger rate}&\\sim \\frac{1}{(5\\ \\rm yr)(2.7\\times 10^{-4}\\rm Gpc^3)} \\sim 700\\ \\rm Gpc^{-3}yr^{-1}\\\\&\\sim 0.7\\times 10^{-4}\\rm MWEG^{-1}yr^{-1},$ where we have used a volume of about 100 Mpc$^3$ per MWEG.", "The LIGO-Virgo rate estimate (excluding the prompt-collapse class)[64] is the range $760^{+1740}_{-650}\\rm Gpc^{-3}yr^{-1}$ .", "(Fig.", "5 of Eldridge et.", "al.", "[65] gives a compendium of recent rate estimates for short gamma-ray bursts, which generally fall within this LIGO/Virgo range for NS-NS mergers).", "The observed local abundance by mass of r-process matter[66] is about $10^{-7}$ .", "A MWEG has total stellar mass $10^{11} M_\\odot $ , giving a mass of $\\sim 10^4 M_\\odot $ of r-process matter.", "From the properties of the electromagnetic counterpart, the ejecta mass for GW170817 has been estimated to be in the range of 0.03 to 0.06 $M_\\odot $[67], [68], [69], [70], [71], [72], [45], [73], [74], [75], [76], (at the high end of what is expected from numerical simulations).", "The r-process mass per MWEG in the lifetime of the universe is then of order $(10^{-4} \\mbox{mergers/yr/MWEG})(5\\times 10^{-2} M_\\odot /\\mbox{merger})(10^{10}\\rm yr) \\sim 5\\times 10^4 M_\\odot ,$ with uncertainties in the merger rate and in the r-process production per merger allowing a range from $5\\times 10^3 M_\\odot $ to $10^5 M_\\odot $ .", "If we use $0.01 M_\\odot $ as a lower bound on the r-process matter per merger, we have an upper limit on the NS-NS merger rate: Adopting $2\\times 10^4 M_\\odot $ as an upper limit on the amount of r-process matter in a MWEG immediately gives $\\mbox{NS-NS merger rate} < 2\\times 10^{-4} \\rm yr^{-1}\\ MWEG^{-1}\\approx 2000\\ Gpc^{-3}yr^{-1},$ below the upper edge of the LIGO/Virgo range and of the estimates summarized in Eldridge et al.[65].", "Suppose now that the event GW190425 is a BH-NS merger.", "In recent simulations of a merger of a 2 $M_\\odot $ black hole with neutron stars of radii from 11.6 to 13 km, Kyutoku et al.", "[77] find tidal disruption leads to a disk of mass 0.04 $M_\\odot $ to 0.1 $M_\\odot $ , of which 15%-30% is ejected, giving an $r$ -process mass $\\lesssim 0.03 M_\\odot $ (but see [78] for a larger estimate based on ealier simulations).", "Despite a distance to the event four times that to GW170817, the LIGO/Virgo paper estimates a comparable GW190425-like merger rate, $460_{-390}^{+1050}$ Gpc$^{-3}$ yr$^{-1}$ , and our previous steps then give a comparably large BH-NS contribution to r-process matter: $2\\times 10^4 M_\\odot \\ \\rm MWEG^{-1}$ .", "The uncertainty in ejecta mass, the very large uncertainty in event rate, the likelihood that this was not a NS-BH event, and the fact that one must add the unknown rate from NS-BH binaries with larger BH masses, all allow BH-NS events to be the dominant or a negligible contributor to $r$ -process matter." ], [ "Kilonovae", "Coined by Metzger et al.", "[79], the term kilonova refers to the peak brightness of the radioactively powered ejecta after a NS-NS or BH-NS merger: The peak brightness is about 1000 times that of of a classical nova (the explosive fusion of hydrogen to helium at the surface of an accreting white dwarf).", "The brightness and timescale are key signatures of the formation of r-process material whose radioactive decay powers the kilonova, and we give brief estimates here.", "Li and Paczynski[80] first suggested an observable optical glow from merger ejecta in 1998; see Metzger[6] for a review and references.", "For a typical merger – and, apparently, for GW170817 – the mass is in the range $M_{\\rm max,rot} < M < M_{\\rm thres}$ , meaning that a hypermassive neutron star forms, is supported by differential rotation for tens or hundreds of milliseconds (shorter for higher mass and/or a softer EOS),[81] and collapses to a black hole as viscosity and magnetic field windup enforce uniform rotation.", "The first, dynamical, ejecta emerge immediately, in less than about 10 ms, powered by tidal disruption, by shocks that convert some of the collision's enormous kinetic energy to heat;[5] matter at the interface between the merging stars is squeezed out by the collision and then swept out by the rotating double core.", "[82] The larger part of the ejecta, however, emerges later.", "As we will see, the time $t_p$ to peak brightness depends on whether or not the merger produces the heavier r-process elements, in particular the lanthanides.", "The time $t_p$ is much longer than the timescale of seconds for the larger part of the ejected matter, for neutrinos to carry away most of the merger energy, and for a gamma-ray burst to be launched.", "$t_p$ is long because the ejected matter is initially dense enough to be opaque – that is, the photon mean free path is short.", "It depends on whether or not lanthanides are present because the interaction cross section for resonant scattering or absorption grows with the number of available electron transitions: The lanthanides have a partly filled $f$ -shell, and that leads to a large number of available transitions.", "To estimate $t_p$ , note first that peak brightness occurs when photons finally diffuse through the ejecta with a speed $v_{\\rm diffusion}$ equal to the ejecta's expansion velocity, $v_{\\rm ej}$ .", "Because the ejecta move at close to the escape velocity from the massive neutron star, $v_{\\rm ej} \\sim 0.1 c$ .", "We will find $v_{\\rm diffusion}$ in terms of $t_p$ and solve $v_{\\rm ej} = v_{\\rm diffusion}$ for $t_p$ .", "We use $v_{\\rm ej}& = \\mbox{ejecta velocity} \\qquad \\qquad \\qquad t_p=\\mbox{time from merger to peak brightness}\\nonumber \\\\\\ell &= \\mbox{photon mean free path} \\quad \\quad \\ n= \\mbox{number density of ions}\\nonumber \\\\\\sigma &= \\mbox{interaction cross section}\\qquad M_{\\rm ej} = \\mbox{total ejecta mass} \\nonumber \\\\m_S & = \\mbox{average mass of ejecta ions}\\nonumber $ To find the diffusion speed, note that the time for a photon to diffuse through a radius $R$ is is the random walk time: If $N=$ number of collisions, we have $R=\\sqrt{N} \\ell $ , $ N = R^2/\\ell ^2$ , implying $t_{\\rm diffusion} = N\\ell / c = R^2/(\\ell c)$ and $v_{\\rm diffusion} \\sim \\frac{R}{t_{\\rm diffusion}} \\sim \\frac{\\ell }{R} c.$ The radius at peak brightness is $R=v_{\\rm ej} t_p$ .", "To find $\\ell $ in terms of $t_p$ , write $\\ell = 1/n\\sigma $ , where the number density is $n = \\rho /{m_S}$ , with $\\rho = M_{\\rm ej}/\\left(\\frac{4}{3}\\pi R^3\\right)$ .", "Then $\\Longrightarrow \\ \\frac{1}{\\ell }= n\\sigma = \\frac{\\sigma M_{\\rm ej}}{m_S\\left(\\frac{4}{3}\\pi R^3\\right)}.$ Using Eqs.", "(REF ) and (REF ), and replacing $R$ by $v_{\\rm ej}t_p$ , we have $t_p = \\frac{3}{4\\pi } \\frac{\\sigma M_{\\rm ej}}{m_S c\\, v_{\\rm ej}t_p } \\quad \\Longrightarrow \\ \\quad t_p = \\sqrt{\\frac{3}{4\\pi } \\frac{\\sigma M_{\\rm ej}}{m_S c\\,v_{\\rm ej}}}.$ The time $t_p$ then depends on the opacity, $\\kappa = \\sigma /m_S$ , of order 0.1 $\\rm cm^2/g$ for ejecta with only light elements.", "[83] For resonant interactions, the cross section $\\sigma $ is enhanced by the number of transitions, roughly proportional to $C^2$ , where the complexity $C$ is the number of ways to assign valence electrons to states within shells.", "[84] Because the lanthanides have a partially filled f-shell, with $2(2\\ell +1) = 14$ , lanthanide production from within the f-shell alone increases $\\sigma $ by a factor of more than 100, and the total number of relevant transitions is an order of magnitude larger.", "The lanthanides have $A\\sim 150$ , giving $\\kappa = \\sigma /m_S \\gtrsim 10\\ \\rm cm^2/g$ .", "Eq.", "(REF ) then gives $t_p = 7\\times 10^5 \\sqrt{\\frac{\\kappa }{10\\rm \\ cm^2/g}\\ \\frac{ M_{\\rm ej}}{0.01M_\\odot }\\ \\frac{0.1c}{v_{\\rm ej}}}\\quad \\rm s.$ Thus light escapes only after several days, when the spatial extent of the ejecta is of order $v_{\\rm ej} t_p \\sim 10^{10}$ km, vastly larger than the initial post-merger configuration, a 20-30 km torus.", "This long time is a signature of the presence of heavy r-process elements: The time is long only in the context of neutron star mergers.", "It is short compared to the timescale of supernova light curves, associated with much larger ejected envelopes that have little or no heavy $r$ -process matter.", "Without the heavier elements, elements with valence d-shell electrons (e.g., the iron group) dominate the opacity, and it is smaller by a factor of 10 or more.", "[85], [84] A second key feature comes from the fact that number of lanthanide transitions increases with energy, with the result that the light emerging at at peak emission is red.", "As mentioned earlier, unless the collapse is prompt, electron and neutrino captures deplete the neutrons in ejected matter, and this processed part of the ejecta is lanthanide poor, more transparent and, in particular, transparent to shorter wavelength light.", "Because the light emerges sooner, Metzger and Fernández[86] and Kasen et al.", "[87] predicted early blue emission, lasting about two days in mergers with hypermassive neutron stars, and evolving to the red and infrared spectrum.", "The GW170817 kilonova From ultraviolet through infrared, observations of the GW170817 remnant (labeled the astronomical transient AT2017gfo) strikingly confirmed the behavior of a radioactively-powered kilonova (see, e.g., [6] and references therein).", "A spectrum initially peaked in ultraviolet progressed through blue to red over about three days and then to infrared [88], [89], [90], supporting models whose ejecta had a range of neutron fractions.", "Subsequent observations strengthen the case: Watson et al.", "[91] identify spectral lines of Sr, a light neutron-capture element, providing direct evidence that the early blue light and its rapid fading is associated with decays in a lanthanide-poor component.", "And the apparent match to decays in lanthanide enriched matter of the slower progression through longer wavelengths was strengthened by infrared observations at 43 and 74 days.", "[92] In that time, the measured infrared luminosity dropped by a factor of 6, a decline consistent with decays from a small set of nuclides with half-lives of roughly 14 days.", "There are no light radioactive nuclei with decay times close to that, but it matches some of the more abundant heavy $r$ -process elements (e.g., $^{143}$ Pr, 13.6 d;$^{56}$ Eu, 15.2d).", "Because merger simulations that produce lanthanides ordinarily yield elements up to and beyond the third peak, it is likely that the GW170817 merger synthesized the full range of $r$ -process elements.", "Given the estimated $0.03$ to $0.06\\, M_\\odot $ of $r$ -process ejecta in this event, there is little doubt that mergers involving neutron stars contribute a major part of the universe's $r$ -process matter.", "There is, however, evidence that they are not the only $r$ -process site.", "An example is a high observed abundance of Eu in the early universe – too high for the early-time merger rate inferred from the event rate of observed SGRBs.", "[93]" ], [ "From inspiral waveform", "The radius and tidal deformability of neutron stars in binary systems can be inferred from the effect of tides on the inspiral waveform.", "[94], [95] (see, e.g., the Ligo/VIRGO analyis of neutron star properties from GW170817[96] for an extensive list of references).", "As a neutron star binary loses energy to gravitational waves, the separation of the binary continuously shrinks, with a chirp-like increase in both the amplitude and frequency of the emitted waves.", "In the late inspiral, tides alter the waveform in two ways at the same post-Newtonian order: Orbital energy is lost directly to work done in raising tides.", "Indirectly, by distorting the stars, tides increase the system's quadrupole moment and thereby increase the rate at which energy is radiated in gravitational waves.", "The system loses its orbital energy more rapidly, leading to coalescence at larger separation and lower frequency.", "For point-particle inspiral, because the orbital energy is $E_{\\rm orbit}=-GM^2/2d$ , with $d$ the distance between the stars, the energy loss to gravitational waves is is related to $d$ by $\\displaystyle \\frac{\\dot{E}_{\\rm GW}}{E_{\\rm orbit}} = \\frac{\\dot{d}}{d}$ .", "From the fact that the height of tides $h$ is of order $h\\sim R^4/d^3$ , it is easy to see that the rate of orbital energy loss to raising tides and to enhanced GW emission are each of the same order in $R/d$ .", "The tide raises a mass $\\delta M\\sim M h/R$ , doing work $E_{\\rm tide} \\sim GM\\delta Mh/R^2 \\sim GM^2 R^5/d^6$ , whence $\\frac{\\dot{E}_{\\rm tide}}{\\dot{E}_{GW}} \\sim \\frac{R^5}{d^5}.$ The quadrupole moment of each star is increased by $\\delta Q\\sim \\delta M R^2$ , and, because the tidal bulge corotates in line with the stars, the parallel axis theorem implies that the quadrupole moment of the binary system increases by $2\\delta Q$ .", "The quadrupole formula for the rate of gravitational wave emission, $\\dot{E}_{GW}\\propto \\dddot{Q}^2$ , then gives the enhancement $\\delta \\dot{E}_{GW}$ in energy loss $\\frac{\\delta \\dot{E}_{GW}}{\\dot{E}_{GW}} \\sim \\frac{\\delta Q}{Q}\\sim \\frac{\\delta M}{M}\\frac{R^2}{d^2} \\sim \\frac{R^5}{d^5}.$ When the orbital separation is large, the inspiral phase can be described by point-particle dynamics.", "A NS binary enters the sensitivity window of ground-based GW instruments, which currently ranges from a few tens of Hz to a few kHz (see, e.g.", "[97]), only a few tens of seconds before merging.", "The GW signal during the inspiral is characterized by the chirp mass, ${M}_\\mathrm {chirp}=\\frac{\\left(m_1 m_2 \\right)^{3/5}}{\\left(m_1+m_2\\right)^{1/5}}.$ The GW amplitude at lowest post-Newtonian order and ignoring tidal effects is $h(t) \\sim \\frac{1}{r} M_{\\rm chirp}^{5 / 3} f_{\\mathrm {GW}}^{2 / 3},$ (with factors of order unity omitted and $c=G=1$ ), where $r$ is the distance to the source and $ f_{\\mathrm {GW}}$ is the GW frequency.", "The latter is given by the first time-derivative of the phase $\\phi (t) = -2 (5M_{\\rm chirp})^{-5 / 8} (t_{\\rm merger} -t)^{5 / 8}+\\phi _0$ , where $t_{\\rm merger}$ is the time at merger and $\\phi _0$ is an initial phase.", "The mass ratio $q=m_1/m_2$ with $m_1\\le m_2$ also affects the dynamics and the waveform, but only at higher post-Newtonian (PN) order[98], [99].", "Mass-ratio effects thus become more pronounced in the last phase of the inspiral.", "Numerical simulations[100] show that contact between the two NSs occurs a few GW cycles prior to merger at frequency $f_{\\rm GW}^{\\rm contact} \\sim 700 (M/2.8MM_\\odot )$  Hz.", "Because tides change the rate at which the orbit loses energy at order $(R/d)^5$ , with $d$ the distance between stars, they are important in late inspiral[101], [99]; phase accumulated from the earlier inspiral, however, induces a time shift that corresponds to a larger phase difference in the later high-frequency cycles.", "[102] Since tidal effects influence the internal structure of neutron stars, the calculation of the relevant coupling constants needs to be carried out in full general relativity.", "In the rest frame of one of the neutron stars, the quadrupolar tidal field sourced by the companion is $ \\mathcal {E}_{i j}=R_{t i t j}$ , where $R_{\\mu \\alpha \\nu \\beta }$ is the Riemann tensor of the spacetime describing the companion only.", "The neutron star responds to the tidal disturbance by its companion, by adjusting its internal structure to a new equilibrium configuration.", "At asymptotically large distances from its center, this adjustment enters in the form of multipole moments $\\frac{1+g_{t t}}{2}=\\frac{G M}{r c^{2}}+\\frac{\\left(3 n^{i} n^{j}-\\delta ^{i j}\\right) Q_{i j}}{2 r^{3}}+\\mathcal {O}\\left(r^{-4}\\right)-\\frac{1}{2} n^{i} n^{j} \\mathcal {E}_{i j} r^{2}+\\mathcal {O}\\left(r^{3}\\right),$ where $Q_{i j}$ is the neutron star's mass quadrupole moment tensor and $n^{i}=x^{i} / r$ are components of a unit vector.", "The neutron star's response to the tidal perturbation can be described in terms of excitations of its oscillation modes, which are either resonantly excited (when the tidal forcing frequency coincides with the mode frequency) or are otherwise adiabatically driven.", "For example, the tidally induced quadrupole moment $Q_{ij}$ is a sum of contributions from all quadrupolar $l=2$ modes.", "Because the fundamental mode has the strongest tidal coupling, overtones are usually omitted in this sum.", "When the orbital separation is much larger than the neutron star radius, the induced quadrupole moment is linearly proportional to the tidal field: $Q_{i j}^{\\text{adiab }}=-\\lambda \\mathcal {E}_{ij},$ where $\\lambda $ is the tidal deformability parameter, related to the tidal Love number $k_2$ and the neutron star radius by $\\lambda =2 /(3 G) k_{2} R^{5}$ .", "It is customary to define a dimensionless tidal deformability, $\\Lambda =\\frac{ c^{10}}{G^{4} M^{5}}\\lambda ,$ which can be computed by solving a second-order ordinary differential equation, in addition to the well-known TOV equations for neutron star structure (see [103], [104], [105] for recent reviews).", "In the frequency domain, the GW signal for a binary neutron star coalescence is[106] $\\tilde{h}(f)=\\mathcal {A} f_{\\rm GW}^{-7 / 6} \\exp \\left[\\mathrm {i}\\left(\\psi _{\\text{point-mass }}+\\psi _{\\text{tidal }}\\right)\\right],$ where the amplitude $\\mathcal {A}$ includes both point-mass and matter effects.", "The tidal contribution to the frequency-domain GW phasing is[107] $\\psi _{\\text{tidal }}=\\frac{3}{128\\left(\\pi G M_{\\rm chirp} f_{\\rm GW} / c^{3}\\right)^{5 / 3}}\\left[-\\frac{39}{2} \\tilde{\\Lambda }\\left(\\pi G M f_{\\rm GW} / c^{3}\\right)^{10 / 3}\\right],$ and it depends on the effective (or weighted average) tidal deformability $\\tilde{\\Lambda }=\\frac{16 c^{10}}{13 G^{4}} \\frac{\\left(m_1+12 m_2\\right) m_1^{4} \\Lambda _{1}+\\left(m_{2}+12 m_{1}\\right) m_{2}^{4} \\Lambda _{2}}{M^{5}}$ which can be constrained through GW observations (for equal-mass binaries, $\\tilde{\\Lambda }= \\Lambda _1 = \\Lambda _2)$ .", "More elaborate models have been developed for matching theoretical waveforms of BNS mergers to GW observations (see[103], [108] and references therein).", "An accurate description in the high-frequency regime is obtained through the effective-one-body (EOB) formalism[109], [101], where the relative motion of the two stars is equivalent to the motion of a particle of mass equal to the reduced mass $\\mu =m_1m_2/M$ in an effective potential.", "The EOB two-body Hamiltonian for nonspinning binaries can be expanded as $H_{\\mathrm {EOB}} \\simeq M c^{2}+\\frac{\\mu }{2} p^{2}+\\frac{\\mu }{2}\\left(-\\frac{2 G M}{c^{2} d^{2}}+\\ldots -\\frac{\\kappa ^{T}}{d^{5}}\\right),$ where $p$ is momentum.", "The constant $\\kappa ^T$ encodes the effect of quadrupolar tidal interactions at leading order in $R/d$ and in a post-Newtonian expansion.", "It is equal to $\\kappa ^T = \\kappa _1 +\\kappa _2$ , where $\\kappa _{i}$ $(i=1,2)$ are the quadrupole tidal polarizability coupling constants for the individual stars, in a multipolar expansion of the tidal potential (see [108] for a recent review) and are equal to a function of mass times the corresponding tidal deformability $\\Lambda _{i}$ .", "Furthermore, phenomenological tidal models can be constructed by fitting EOB models to numerical relativity simulations[110], [111], [112].", "The chirp mass is measured with high accuracy from the inspiral signal and for GW170817 it was obtained as[1], [97] $M_{\\rm chirp} = 1.186(1)M_\\odot $ .", "With the current sensitivity, the total binary mass of sources at a few tens of Mpc can be obtained with an accuracy of order one percent [113], [114], [1].", "For GW170817 the total mass was $ M_\\mathrm {tot}=2.74^{+0.04}_{-0.01}~M_\\odot $ .", "By comparison, the mass ratio $q$ was poorly constrained to be between 0.7 and 1.", "The dimensionless tidal deformability was initially constrained to be $\\tilde{\\Lambda }< 800$ at the 90% confidence level, assuming a uniform prior[1], which corresponds to $\\kappa ^{\\rm T}<150$ .", "A subsequent improved analysis[97] gave a 90% highest posterior density interval of $\\widetilde{\\Lambda }=300_{-230}^{+420}$ .", "A somewhat different range for $\\tilde{\\Lambda }$ was recently obtained with other numerical-relativity calibrated waveforms[115].", "Constraints on $\\Lambda $ can be translated to constraints on radii.", "For example, for $1.4\\,M_\\odot $ models, the empirical relation $\\Lambda _{1.4}=2.88 \\times 10^{-6}(R_{1.4} / \\mathrm {km})^{7.5},$ was found[116], leading to $R_{1.4}<13.6$ km for $\\Lambda _{1.4}<800$ , taking into account the error bars in (REF ).", "The effective tidal deformability $\\tilde{\\Lambda }$ was also shown to correlate well with the radius of the primary star $R(m_1)$ , for a fixed chirp mass[117].", "The initial analysis of the GW170817 merger[118] resulted in the constraint of $R=11.9_{-1.4}^{+1.4}$ km (for both stars involved in the merger).", "A large number of other estimates of NS radii based on the observation of GW170817 (or in combination with multimessenger and/or experimental constraints) have appeared, see e.g.", "[119], [120], [121], [122], [123], [124], [125], [126], and references in the review articles.", "[127], [103], [128], [108], [104], [105].", "Setting EOS constraints through multiple detections has been considered in[129], [130], [131], [132], [133].", "Besides EOS constraints, EOS-insensitive relations are also useful in extracting source properties[134]." ], [ "From electromagnetic observations", "With varying accuracy and model dependence, a number of observational methods, have been used to measure neutron star radii (see, for example, summaries by Lattimer[9] and by Ozël and Freire[135]).", "Particularly promising is the recently launched NICER instrument, which looks at X-rays from pulsars.", "Charged particles that spiral around closed field lines of the rapidly rotating magnetic field collide with the surface at magnetic poles, creating X-ray emitting hot spots.", "By accurately modeling the images of a rotating star with regions of varying temperature, the NICER project can match the observed periodic variation in X-ray intensity to obtain a best fit to the temperature distribution, mass and radius.", "Preliminary results[136] for the millisecond pulsar J0030+0451 give equatorial radius $12.71^{+1.14}_{-1.19}$ km and mass $1.34^{+0.15}_{-0.16}M_\\odot $ , associated with a preferred hot-region model.", "Associated with an alternative temperature distribution[137] is an additional km uncertainty.", "NICER anticipates observing several pulsars with eventual accuracy in radius measurement to $\\sim 0.5$ km.", "A second method uses quiescent X-ray binaries, binary systems in which the neutron star accretes mass episodically from a companion and radiates steadily when it is not accreting.", "[9], [135], [138], [139] In this quiescent stage, assuming the radiation is thermal, the star behaves as a blackbody with intrinsic luminosity $A \\sigma T^4$ , with $A$ the star's surface area, and if the distance to the system is known, one can infer the radius.", "Two uncertainties are the amount of intervening interstellar matter to absorb the X-rays, and the composition of the outer atmosphere (by mistaking helium for hydrogen one underestimates the radius).", "A final related method uses the expansion of the neutron star's atmosphere in an X-ray burst, the explosive nuclear reaction at the star's surface that occurs when the amount of accreted matter reaches its critical mass.", "Again, one extracts the star's radius from a blackbody temperature, in this case after the ejected atmosphere has settled back to the surface (see [140], [141] and references therein)." ], [ "From post-merger", "The maximum mass of a non-rotating neutron star, $M_{\\rm max}$ , represents the ultimate constraint on the EOS, in the sense that it refers to the star in which the highest possible densities are sampled.", "From the observations of GW170817 and its electro-magnetic counterpart one can arrive at constraints on $M_{\\rm max}$ , that are, however, dependent on the assumption on makes about the fate of the remnant.", "By combining the total binary mass of GW170817 inferred from the GW signal with conservative upper limits on the energy in the GRB and and in the ejecta from EM observations, a relatively short-lived remnant is favoured, setting an upper limit of $M_{\\max } \\lesssim 2.17 M_{\\odot }(90 \\%) $ [142].", "Rezzolla et al.", "[143] argue that the remnant survived for a longer time and collapsed as a supermassive neutron star near the maximum mass supported by uniform rotation.", "Using an empirical relation between the maximum mass for nonrotating models and uniformly rotating models, they arrive at an upper limit of $M_{\\mathrm {max}} \\lesssim 2.16_{-0.15}^{+0.17}\\, M_{\\odot }$ (notice the large uncertainty).", "A longer-lived remnant surrounded by a torus was also considered in [81] and, in combination with the absence of optical counterparts from relativistic ejecta, the maximum mass was argued to be in the range of $2.15-2.25 M_{\\odot }$ .", "Under different assumptions (a BH is formed in a delayed collapse soon after the NSNS merger and the sGRB is triggered by collimated, magnetically confined, helical jet and powered by a magnetized disk) one can write[144] $ \\beta M_{\\mathrm {max}} \\approx M_{\\rm max,rot}\\, \\lesssim \\,M_{\\mathrm {GW170817}} \\approx 2.74 M_\\odot \\,\\lesssim \\, M_{\\text{thresh }} \\approx \\alpha M_{\\max },$ with $\\alpha \\approx 1.3-1.7$ from numerical simulations[145], [146] and $\\beta \\approx 1.2$ for microphysical EOS[147], [148], [149], [150].", "This implies an upper limit of $M_{\\max } \\lesssim 2.16 M_{\\odot } $ .", "A more conservative upper limit based on causality is obtained by considering the relations[151] $M_{\\max }^{\\mathrm {}}=4.8\\left(\\frac{2 \\times 10^{14} \\mathrm {g}/\\mathrm {cm}^{3}}{\\epsilon _{m} / c^{2}}\\right)^{1 / 2} M_{\\odot } \\ \\ \\mathrm {and} \\ \\ M_{\\rm max,rot }=6.1\\left(\\frac{2 \\times 10^{14} \\mathrm {g} / \\mathrm {cm}^{3}}{\\epsilon _{m} / c^{2}}\\right)^{1 / 2} M_{\\odot },$ where $\\epsilon _{m}$ is a matching energy density, above which the EOS is assumed to be at the causal limit (speed of sound equal to speed of light).", "Then, $\\beta \\approx 1.27$ and $M_{\\max } \\lesssim 2.28 M_\\odot $ .", "Similar considerations were used in.", "[152] A prompt collapse to a BH ($M_{\\rm tot}>M_{\\rm thres})$ is expected to be accompanied by systematically less massive and more neutron-rich ejecta, resulting in a less luminous and redder kilonova than for the case of a delayed collapse[153] (see[154] for an alternative method to infer the threshold mass).", "In this case, one can use the empirical relation[146] $M_{\\mathrm {thres}} \\approx \\left(-3.606 \\frac{G M_{\\mathrm {max}}}{c^{2} R_{1.6}}+2.38\\right) M_{\\mathrm {max}},$ where $R_{1.6}$ is the radius of a $1.6 M_\\odot $ star, to directly constrain $M_{\\rm max}$ .", "Combining a large number of future observations is expected to tighten the constraints on $M_{\\rm max}$[153] and other EOS properties[155]." ], [ "From binary systems", "The three largest accurately measured neutron star masses have values close to 2.0$M_\\odot $ .", "The stars, J1614-2230[156], J0348+0432[157], and, most recently J0740+6620[158], have, respectively, measured masses $1.97\\pm 0.04M_\\odot $ , $2.01\\pm 0.04$ , and $2.14\\pm 0.1 M_\\odot $ (the uncertainties represent 1-$\\sigma $ errors).", "Each is in a binary system, and each orbit is circular to one part in $10^6$ .", "Two of the systems are eclipsing binaries: Seen almost edge on, their orbital planes are nearly in the line of sight (angles $i$ between plane perpendicular to line of sight and plane of orbit are $89.17^\\circ $ for J1614+2230 and $i=87.35^\\circ $ for J0740+6620).", "Obscured by the complexity of the detailed analyses is an underlying simplicity that is exact for circular orbits in the line of sight: For Newtonian binaries (and these are nearly Newtonian), two equations among the three unknowns, $m_1$ , $m_2$ and the orbital radius $a$ , are Kepler's third law, $ a^3 = GM/\\Omega ^2$ and the expression for the orbital velocity $v_1 = r_1\\Omega $ of the neutron star in terms of its distance $ r_1 = a\\ m_2/M$ from the system's center of mass: $v_1 = \\frac{m_2}{M} a\\Omega $ .", "The period $P$ of the orbit is given by the periodic Doppler shift in the observed time interval between pulses, and the pulsar velocity $v_1$ relative to the system's center of mass can be measured from half the difference between the maximum blue- and red-shifts.", "The third measurement needed to determine the three variables $m_1,m_2,a$ is the Shapiro time delay, (see, for example [159]), the delay in light travel time along a light trajectory that grazes the companion with a distance $r_0$ of closest approach.", "With $t$ proper time measured on Earth and $D$ the distance from the pulsar to the Earth, the delay is, to lowest order in $M/r_0$ , $r_0/D$ and $r_0/a$ , $\\Delta t = \\frac{2Gm_2}{c^3}\\ln \\left(8aD/r_0^2\\right), \\vspace{-5.69054pt} $ where $r_0$ is the distance of closest approach of the light ray to the center of the companion, and $D$ the distance of the binary system to the Earth.", "The distance $D$ , however, is not accurately measured, and, in practice, one measures not the absolute time delay, but its change due to the change in $r_0$ as the pulsar orbits.", "Eq.", "(REF ) implies the difference is independent of $D$ : $\\Delta t(r_0^{\\prime })-\\Delta t(r_0) = \\frac{2Gm_2}{c^3}\\ln \\left(r_0^2/r_0^{\\prime 2}\\right),\\vspace{-5.69054pt} $ Because the ratio of impact parameters depends only on the observed angular pulsar positions, it can be directly measured.", "The measured time delay now determines $m_2$ , and the remaining two relations then give $a$ and the pulsar mass $m_1$ .", "Because the Shapiro time delay is small, of order $10^{-5}$ s, the analysis for the real system must include the comparable small corrections to a circular Newtonian orbit: post-Newtonian corrections as well as eccentricity and orbital inclination.", "As usual, orbital measurements rely on accurate timing of the extremely stable rotation of old neutron stars.", "In the remaining system, J0348+0432, a white dwarf and neutron star orbit in an plane far from the line of sight.", "In this case, the Shapiro time delay cannot be accurately measured; instead, the authors obtain $m_2$ from the mass-radius relation for low-mass white dwarfs, estimating the radius and surface gravity from hydrogen Balmer lines at the surface of the dwarf." ], [ "Neutron star EOS", "Above nuclear saturation density, $\\rho _n = 2.7\\times 10^{14} \\rm g/cm^3$ , and up to at least a few times nuclear density, the star consists primarily of neutrons with a small fraction of protons, electrons, and muons.", "The primary uncertainty in its composition is in the star's dense core.", "At high enough density, the Fermi energy of down quarks in compressed nucleons must exceed the rest mass of strange quarks, and a transition from nucleons to hyperons occurs; and at a presumably higher density, whose value in cold matter is similarly uncertain, the nucleons themselves dissolve, creating strange quark matter, comprising free up, down, and strange quarks.", "Should these critical densities be below the central density of the maximum-mass neutron star, the phase transitions to hyperons and/or strange quark matter will soften the equation of state making the star more compact and lowering the maximum mass.", "Because old neutron stars are cold, with thermal energy far below the Fermi energy, their matter satisfies a one-parameter EOS of the form $P=P(\\rho )$ , where $\\rho $ is the baryon mass density and $P$ the pressure of the star.", "The EOS determines the one-parameter family of neutron stars, with the star's mass a function of its radius or central density, and the relation can be inverted: The $M(R)$ curve can be inverted to give $P(\\rho )$ .", "[160].", "Although simultaneous measurements of mass and radius are currently restricted to stars whose mass is of order $1.4 M_\\odot $ , these, together with causality and the maximum neutron star mass, substantially restrict the universe of candidate EOSs.", "The radius of a 1.4$M_\\odot $ neutron star is closely correlated with the pressure at about twice nuclear density.[161].", "For EOSs whose maximum masses range from 2.5 $M_\\odot $ to $2.0 \\,M_\\odot $ , corresponding central densities range from about $2\\times 10^{15}$ to $2.9\\times 10^{15}$ g/cm$^3$ .", "The maximum mass is then approximately governed by the pressure at densities of order 7-$8 \\rho _n \\sim 2\\times 10^{15}$ g/cm$^3$ .", "[162].", "The diagram below, patterned after Özel & Freire[135] portrays the approximate relation.", "Figure: Maximum mass and the radius at about 1.4M ⊙ M_\\odot roughly correspond to the parts of theEOS curve sketched here.To systematize the observational constraints on the neutron star equation, Read et al.", "[163] introduce a parameterized equation of state above nuclear density, specifying the pressure at three fiducial densities with linear interpolation of $\\log p$ vs $\\log \\rho $ .", "Using these piecewise polytropic models, subsequent authors have used GW and electromagnetic observations, causality, and constraints from nuclear theory to constrain the $p(\\rho )$ curve.", "[162], [164], [165], [166], [167], [9] An alternative spectral representaion of the EOS due to Lindblom[160], [168] can give a more accurate map from observational constraints to the EOS[167] with a less obvious physical interpretation of the parameters in the spectral expansion." ], [ "Constraints from causality and minimum radius", "Causality, in the form $v_{\\rm sound}< c$ , where $v_{\\rm sound} = dp/d\\epsilon $ , with $\\epsilon $ the energy density.The propagation of signals along characteristics of relativistic fluid equation exceeds the speed of light unless this relation is satisfied for fluids obeying a one-parameter EOS or a for a stable star with a 2-parameter EOS of the form $p=p(\\rho ,s)$ , with $s$ the specific entropy.", "[170] significantly constrains the EOS above nuclear density.", "Because the upper limit on mass appears to exceed 2.1$M_\\odot $ , the EOS must be stiff at high density, and causality then requires a minimum pressure above a few times nuclear density.", "Because stiffer EOSs yield stars with larger radii, a large upper mass limit then sets a lower limit on the radius of neutron star.", "We now obtain that limit, beginning with the lower limit on the radius of the maximum mass star.", "Using the maximally soft EOS consistent with causality[171] and with a given $M_{\\rm max}$ , Haensel et al.", "[172] and Lattimer[173] obtain $ M_\\mathrm {max}\\le \\frac{1}{2.82}\\frac{c^2\\,R_\\mathrm {max}}{G},$ where $R_\\mathrm {max}$ is the radius of the maximum-mass nonrotating star.", "This implies that an EOS cannot become arbitrarily stiff (see also [151]).", "There is a tight empirical relation[146], analogous to Eq.", "(REF ), between $k=M_\\mathrm {thres}/M_\\mathrm {max}$ and $\\displaystyle C_\\mathrm {max}=\\frac{G\\,M_\\mathrm {max}}{c^2\\,R_\\mathrm {max}}$  , $M_\\mathrm {thres}=\\left( -3.38\\frac{G\\,M_\\mathrm {max}}{c^2\\,R_\\mathrm {max}}+2.43\\right)\\,M_\\mathrm {max}.$ Inserting the causality constraint Eq.", "(REF ) in Eq.", "(REF ) results in the constraint $M_\\mathrm {thres}\\le 0.436\\frac{c^2\\,R_\\mathrm {max}}{G}, \\ \\ \\ \\mathrm {or} \\ \\ \\ R_\\mathrm {max} \\ge 2.29\\frac{G\\ M_{\\rm thres}}{c^2}.$ Hence, a given measurement or estimate of $M_\\mathrm {thres}$ sets a lower bound on $R_\\mathrm {max}$ .", "As previously mentioned, electromagnetic observations of GW170817 give an ejecta mass between 0.03 and 0.06 $M_\\odot $ .", "Based on this mass range, it was suggested in [169] that the merger did not result in a prompt collapse, because direct BH formation implies significantly reduced mass ejection (see e.g.", "Fig.", "7 in [82]).", "This implies that the measured total binary mass of GW170817 is smaller than the threshold binary mass $M_\\mathrm {thres}$ for prompt BH formation and thus $M_\\mathrm {thres}> 2.74^{+0.04}_{-0.01}~M_\\odot $ .", "Using this condition in Eq.", "(REF ) results in a lower limit on $R_\\mathrm {max}$ .", "The detailed calculation and error analysis in [169] yields $R_\\mathrm {max}\\ge 9.60^{+0.14}_{-0.03}$  km.", "Following the same line of arguments, one can set a lower limit on the radius $R_{1.6}$ of a nonrotating 1.6 $M_\\odot $ NS.", "Replacing the EOS above 1.6 $M_\\odot $ with the causal limit EOS, yields $M_{\\max }=\\frac{1}{3.10} \\frac{c^{2} R_{1.6}}{G},$ and combining this with Eq.", "(REF ) one obtains[169] $R_{1.6}\\ge 10.68^{+0.15}_{-0.04}$  km.", "The minimal set of assumptions, Eq.", "(REF , REF ) thus set an absolute lower limit on NS radii that rules out very soft nuclear matter.", "The lower limits on $R_{\\rm max}$ and $R_{1.6}$ are shown in the left panel of Fig.", "REF on top of a set of mass-radius relations of a large sample of EOS.", "It is straightforward to convert the above radius constraints to a limit on the tidal deformability.", "A limit of $R_{1.6}>10.7$  km corresponds to a lower bound on the tidal deformability of a 1.4 $M_\\odot $ NS of about $\\Lambda _{1.4}>200$ (constraints on the tidal deformability that rely on the post-merger properties of GW170817, not all consistent which each other, were also derived in e.g.", "[174], [175], [176]).", "With future observations, an event that is identified as a prompt collapse will set an upper bound on $M_\\mathrm {thres}$ , which constrains the maximum mass and radii of nonrotating NSs from above.", "The resulting constraints from a hypothetical future detection are shown in Fig.", "REF (right panel).", "A large number of detections of the inspiral phase of binary neutron star mergers may also set constraints on $M_{\\rm max}$ [177].", "An extension of the empirical relation Eq.", "(REF ) as a more accurate, bilinear fit of the form $M_{\\max }\\left(M_{\\text{thres }}, \\tilde{\\Lambda }_{\\text{thres }}\\right)=0.632 M_{\\text{thres }}-0.002 \\tilde{\\Lambda }_{\\text{thres }}+0.802,$ where $\\tilde{\\Lambda }_{\\text{thres }}=\\tilde{\\Lambda }\\left(M_{\\text{thres }} / 2\\right)$ , was recently presented in[178], and a unique signature of strong phase transitions was found in the $\\tilde{\\Lambda }_{\\text{thres }}$ vs. $M_{\\text{thres}}$ parameter space." ], [ "Future constraints from post-merger NS oscillations", "A likely outcome of a NS merger is the formation of a metastable, differentially rotating NS remnant.", "The expected GW spectrum of such a case (in terms of the effective GW amplitude $h_{\\rm eff}=\\tilde{h}(f)\\cdot f$ , where $f$ is frequency and $\\tilde{h}(f)$ the Fourier transform), is shown in Fig.", "REF for a NS merger simulation of two stars with a mass of 1.35 $M_\\odot $ each, assuming the EOS DD2.", "[179], [180] (notice that these hydrodynamical simulations start only a few orbits before merging).", "The postmerger spectrum exhibits several distinct peaks in the kHz range [184], [185], [186], [187], [188], [189], [190], [183], [191], [192], [193], [194], [195], [196], [197], which originate from certain oscillation modes and other dynamical processes in the postmerger remnant.", "There is a dominant oscillation frequency $f_\\mathrm {peak}$ (also denoted as $f_2$ ), which typically has the highest signal-to-noise ratio (SNR) of all distinct postmerger features.", "In addition, there are secondary peaks, denoted as $f_\\mathrm {spiral}$ and $f_{2-0}$ , that are above the aLIGO/aVIRGO noise level and additional peaks at higher frequencies, which are less likely to be observed, even with third-generation detectors.", "Understanding the physical mechanisms generating these different features is essential for the detection and interpretation of postmerger GW signals.", "An effective method for analyzing oscillation modes of rotating stars, based on a Fourier extraction of their eigenfunctions from simulation data, was presented in [198] and applied to NS merger remnants in [184].", "Throughout the star, the Fourier spectrum exhibits a discrete dominant frequency $f_\\mathrm {peak}$ , with an extracted eigenfunction that has a clear $m=2$ quadrupole structure and no nodal lines in the radial direction.", "Thus, $f_{\\rm peak}$ is produced by the fundamental quadrupolar fluid oscillation mode of the post-merger remnant, a result confirmed by hydrodynamical simulations of the late-time remnant with the $m=2$ fundamental mode added as as perturbation[183].", "Figure: Empirical surfaces for the three main post-merger frequencies, as a function of M chirp M_{\\rm chirp} and R 1.6 R_{1.6}.", "The surfaces are shown only in regions where data points are available.", "Figurefrom Ref.The collision of the two stars also excites the fundamental quasi-radial mode of oscillation in the remnant, whose frequency we denote by $f_0$ .", "Since these oscillations are nearly spherically symmetric even in a rotating remnant, the quasi-radial mode produces only weak GW emission (typically at a frequency where the spectrum is still dominated by the inspiral phase).", "However, both the quadrupole and the quasi-radial oscillations have a large initial amplitude, and thus there exist non-linear couplings between them.", "At second order in the perturbation, the coupling of the two modes results in the appearance of daughter modes, here with quasi-linear combination frequencies[184] $f_{2\\pm 0}=f_\\mathrm {peak}\\pm f_0$ .", "At frequencies below $f_\\mathrm {peak}$ , there is at least one more pronounced secondary peak, which (when present) typically appears in between $f_\\mathrm {peak}-f_0$ and $f_\\mathrm {peak}$ , as shown in Fig.", "REF .", "This secondary peak, denoted by $f_{\\rm spiral}$ , is generated by the orbital motion of two bulges that form right after merging [188].", "This feature is strongest in equal-mass binaries, appearing as two small spiral arms.", "Matter in these bulges or spiral arms cannot follow the faster rotation of the quadrupole pattern of the inner core; instead, the antipodal bulges orbit the central remnant with a smaller orbital frequency.", "The structure is transient and dissolves within a few milliseconds.", "Notice that the $f_{2+0}$ frequency can also be present in the spectrum, but it is typically too weak to be detectable even with the Einstein Telescope (although this could become possible with dedicated high-frequency detectors).", "Figure: Spectral classification of the postmerger GW emission, as obtainedby a machine-learning algorithm, verifying the classification introducedin.", "Theclassification is shown in the mass vs. radius parameter space of nonrotatingneutron star models, constructed with various EOSs and masses.A clustering algorithm separates the models into three different types (shownas red squares for Type I, black ×\\times for Type II and blue circles forType III).", "Then, a supervised-learning classification algorithm locates theborders between the three different types in this parameter space.", "Figurefrom Ref.In the range of total masses $2.4 M_\\odot \\le M_{\\rm tot} \\le 3.0 M_\\odot $ , the secondary peaks appear in distinct frequency ranges[188]: $f_{\\rm peak} -1.3 {\\rm kHz} \\le f_{2-0} \\le f_{\\rm peak} - 0.9 {\\rm kHz} $ , while $f_{\\rm peak} - 0.9 {\\rm kHz} \\le f_{\\rm spiral} \\le f_{\\rm peak} - 0.5{\\rm kHz}$ .", "All three distinct post-merger GW frequencies can be described by empirical relations with small scatter.", "Recently, multivariate empirical relations were constructed, where each frequency is given as a function of the chirp mass $M_{\\rm chirp}$ and the radius of a $1.6\\,M_\\odot $ nonrotating star[199], see Fig.", "REF .", "Examining the post-merger GW spectrum for equal-mass (or nearly equal-mass) binaries, one can arrive at the following spectral classification[188]: Type I: The $f_{2-0}$ peak is the strongest secondary feature, while the $f_\\mathrm {spiral}$ peak is suppressed or hardly visible.", "This behavior is found for mergers with relatively high total binary masses and soft EOSs.", "Type II: Both secondary features $f_{2-0}$ and $f_\\mathrm {spiral}$ are clearly present and have roughly comparable strength.", "Type III: The $f_\\mathrm {spiral}$ peak is the strongest secondary feature, while the $f_{2-0}$ peak is either strongly suppressed or even absent.", "This behavior is found for mergers with relatively low total binary masses and stiff EOSs.", "This classification scheme was reproduced in Ref.", "[199] using machine-learning algorithms, see Fig.", "REF .", "For a given EOS, different spectral types may occur, depending on the total mass of the binary.", "For asymmetric mass ratios of $q\\sim 0.7$ the above classification scheme has to be modified (for example, in such asymmetric cases the $f_{\\rm spiral}$ secondary peak will be considerably weaker).", "In [97] an unmodelled data analysis search was performed to extract the postmerger GW emission.", "No signal was found, as is expected for the given distance of the event and the sensitivity of the instruments during the observations.", "Improving the sensitivity of detectors by a factor of a few, however, may allow the detection of postmerger gravitational-wave emission at a distance of few tens of Mpc [200], [194], [201], [202].", "With this sensitivity, the masses of the individual stars could be measured with a precision of a percent at the same distances [113], [114], [1], if the accuracy of mass measurements scales roughly with $(\\mathrm {SNR})^{-1}$ or $(\\mathrm {SNR})^{-1/2}$ .", "For binaries at larger distances, stacking a large number of individual signals may allow the detection of the post-merger phase with third-generation detectors.", "[203], [204] Figure: Multivariate empirical relation for Λ 1.6 \\Lambda _{1.6} as a functionof f peak f_{\\rm peak} and M chirp M_{\\rm chirp}.", "Figurefrom Ref.The peak frequency $f_\\mathrm {peak}$ of 1.35-1.35 $M_\\odot $ mergers shows a clear correlation with the radius $R_{1.35}$ of a nonrotating NS with 1.35 $M_\\odot $ (see Fig.", "4 in [205] and Fig.", "12 in [206]).", "Similar tight correlations exist for other fiducial masses (see Figs.", "9 to 12 in [206]).", "The tightest relation for a 1.35-1.35 $M_\\odot $ is with the radius $R_{1.6}$ .", "With $f_{\\mathrm {peak}}$ in kHz and $R_{1.6}$ in km, the relation has the form $f _ { \\mathrm { peak } } = \\left\\lbrace \\begin{array} { l l }{ - 0.2823 \\cdot R _ { 1.6 } + 6.284, } & { \\textrm { for } f_ { \\rm peak } < 2.8 \\mathrm { kHz } }, \\\\{ - 0.4667 \\cdot R _ { 1.6 } + 8.713, } & { \\textrm { for } f_ { \\rm peak } > 2.8 \\mathrm { kHz } }.", "\\end{array} \\right.$ For $R_{1.6}$ the maximum scatter of this relation is less than 200 m. For other fixed binary masses, e.g.", "1.2-1.2 $M_\\odot $ , 1.2-1.5 $M_\\odot $ or 1.5-1.5 $M_\\odot $ mergers, similar scalings between $f_\\mathrm {peak}$ and NS radii exist [206] and a single relation, scaled by the total mass, is [183] $f _ { \\rm peak } / M _ { \\mathrm { tot } } = 0.0157 \\cdot R _ { 1.6 } ^{ 2 } - 0.5495 \\cdot R _ { 1.6 } + 5.5030,$ (see [207] for a similar rescaling but with the tidal coupling constant).", "A multivariate extension[199] of the above empirical relations yields the radius of nonrotating neutron stars at a specified mass as a function of two observable variables, $R_{\\rm x}=R_{\\rm x}(f_{\\rm peak}, M_{\\rm chirp})$ , where $\\rm x$ stands for the value of the mass in solar masses ($f_{\\rm peak}$ obtained from the post-merger GW spectrum and $M_{\\rm chirp}$ obtained from the inspiral phase).", "As an example, for the case of $M = 1.6 M_\\odot $ the empirical relation for the radius is $\\begin{split}R_{1.6}= 35.442 -13.46 M_{\\mathrm {chirp}} -9.262 f_{\\mathrm {peak}}/M_{\\mathrm {chirp}}\\\\+3.118 M_{\\mathrm {chirp}}^2 +2.307 f_{\\mathrm {peak}} +0.758 \\left( f_{\\mathrm {peak}}/M_{\\mathrm {chirp}}\\right)^2,\\end{split}$ with a maximum residual of 0.654 km and $R^2=0.954$ .", "This relation was constructed using an extended set of simulations and available GW catalogues, that included equal and unequal mass binaries.", "Multivariate empirical relations can also be constructed for the dimensionless tidal deformability[199].", "For example, for $M = 1.6 M_\\odot $ one obtains $\\Lambda _{1.6} = 2417 + 770.2 M_{\\rm chirp} - 1841 f_{\\rm peak}+ 262.9 f_{\\rm peak}^2,$ with maximum residual of 99.85 and $R^2=0.964$ , see Fig.", "REF .", "Additional empirical relations for setting EOS constraints have been presented e.g.", "in Refs.", "[185], [207], [193], [208], [209].", "A one-armed instability was studied in Refs.", "[210], [211], [212], [213], [214] and the excitation of inertial modes (under the assumption of low viscosity in the remnant) was studied in Refs.", "[215], [216] The detection of phase transitions through GW observations of the post-merger phase was considered in[217], [218], [219], [220], [221], [222], [223].", "For the impact of magnetic fields on the post-merger phase, see the review[224] and references therein and for the effect of a possible strong turbulent viscosity, see the review[5] and references therein, as well as the recent study using a calibrated subgrid-scale turbulence model[225].", "Because of the high scientific return expected from observing the post-merger phase with gravitational waves, designs for new GW detectors, which will be dedicated to operate with enhanced sensitivity in the kHz regime, have been presented.", "[226], [227], [228] For a large number of sources at cosmological distances, the cumulative information on radius will be dominated by sources at $z\\sim 1$ .", "[229]" ], [ "Acknowledgments", "We are grateful to A. Bauswein for comments on the manuscript and to the referee for many corrections.", "N.S.", "is supported by the ARIS facility of GRNET in Athens (SIMGRAV, SIMDIFF and BNSMERGE allocations) and the Aristoteles Cluster at AUTh, as well as by the COST actions CA16214 (PHAROS), CA16104 (GWVerse), CA17137 (G2Net) and CA18108 (QG-MM)." ] ]

[ [ "Recent advances in the calculation of dynamical correlation functions" ], [ "Abstract We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems.", "Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties.", "The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed.", "For most of systems of interest approximation schemes must be used.", "The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system.", "Insights have been gained from theorems that were discovered with that method.", "For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system.", "The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains.", "Effects of disorder were considered in some of those systems.", "In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent.", "Another important approach is the numerically exact diagonalization method.", "It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit.", "In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations." ], [ "Introduction", "Dynamical correlation functions are central to the understanding of time-dependent properties of many-body systems.", "They appear ubiquitously in the formulation of the fluctuation-dissipation theory, where the response of a system to a weak external perturbation is cast in terms of a time-dependent relaxation function of the unperturbed system [1], [2].", "In this article, we are concerned with the recent calculations of such correlation functions.", "We shall cover two lines of approach, namely the method of recurrence relations and the method of exact diagonalization.", "The method of recurrence relations was developed in the early 1980s [3], [4], [5], [6], [7] following the ideas of the Mori-Zwanzig projection operator formalism [8], [9].", "Essentially one solves the Heisenberg equation of motion for an operator of an interacting system, from which one obtains dynamic correlation functions, a generalized Langevin equation, memory functions, etc.", "Review articles found in the literature cover the earlier developments [10], [11], [12].", "On the other hand, exact diagonalization methods have also been used in several areas of physics [13], [14], [15], [16], [17].", "In this method one numerically determines the eigenvalues and eigenfunctions of a given Hamiltonian of a finite system to find the dynamical correlations of interest.", "The main drawback is that one is bound by computer limitations and must deal with finite systems.", "In addition, being a numerical method, it does not provide any new general insight in the form of theorems, etc.", "Nevertheless, one can obtain surprisingly good results which can be readily extended to the thermodynamic limit.", "In a way, exact diagonalization complements the method of recurrence relations, especially when solutions become hard to obtain by analytic means.", "Other approaches can be found in Refs.", "[18], [19], [20], [21], [22], [23], [24]." ], [ "Dynamical correlation functions", "Consider a system of $N$ elements such as particles, spins, etc., governed by a time-independent Hamiltonian $H$ , in thermal equilibrium with a heat bath at temperature $T$ .", "For two dynamical variables $X$ and $Y$ of the system, the time-dependent correlation function is given by the average: $<Y(0)X(t)> \\, \\equiv (1/Z) \\, {\\rm Tr}\\, [ Y(0) X(t) \\exp (-\\beta H)],$ where $\\rm Tr [ \\dots ]$ denotes a trace over a complete set of states.", "Here, $\\beta = 1/k_BT$ is the inverse temperature, $Z\\, \\equiv {\\rm Tr}\\,\\exp (-\\beta H)$ is the canonical partition function, and $X(t)$ is a time-dependent operator in Heisenberg representation $X(t) = \\exp (iHt/\\hbar ) X \\exp (-iHt/\\hbar )$ , which satisfies: $i \\hbar \\frac{dX(t)}{dt} = [X(t),H], \\qquad X(0) = X,$ where $[X(t),H]$ is the quantum commutator.", "In a classical system, the operators are replaced by classical dynamic variables, the trace by integral over the phase space, and the commutators by Poisson brackets.", "For a given variable, the time-dependent correlation function $C(t)$ reads: $C(t) = \\frac{<X(0)X(t)>}{<X(0)X(0)>}.$ Its Fourier transform $S(\\omega )$ is called the spectral density, or frequency spectrum: $S(\\omega ) = \\int _{-\\infty }^{\\infty } C(t) \\exp (-i\\omega t)\\, dt.$ If we use the integral representation of the Dirac $\\delta $ -function: $\\delta (t) = \\frac{1}{2 \\pi } \\int _{-\\infty }^{\\infty } \\exp (- i\\omega t)\\, d\\omega ,$ then we obtain $C(t) = \\frac{1}{2 \\pi } \\int _{-\\infty }^{\\infty } S(\\omega ) \\exp (i \\omega t) \\, d\\omega .$ Since the Hamiltonian is time-independent, it follows that $C(t)$ in Eq.", "(REF ), has the property $<X(0)X(t)> \\, = \\, <~X(\\tau ) X(t + \\tau )>$ .", "If we take $\\tau = -t$ , then $ <~X(0)X(t)>\\,= \\, <X(-t)X(0)>$ .", "Also, it follows that $S(\\omega )$ is real.", "Due to the invariance of the trace under cyclic permutations, one finds that $S(-\\omega ) = \\exp (-\\beta \\hbar \\omega ) S(\\omega )$ .", "In the classical limit ($\\hbar = 0$ ) or at infinite temperature $\\beta = 1/k_BT = 0$ , it follows that $S(\\omega )$ is even in $\\omega $ .", "In general, the asymmetry in $S(\\omega )$ is a typical quantum feature, and is referred to as the detailed balance.", "Dynamical correlation functions appear in the relaxation function $R(t)$ from linear response theory [25], [2]: $R(t) = \\int _0^{\\beta } d\\lambda {<} \\exp (\\lambda H) Y \\exp (-\\lambda H)X(t) {>}$ $\\qquad \\qquad \\qquad \\qquad - \\beta <X><Y>$ where $<\\dots >$ is a canonical average, and $X$ and $Y$ are operators.", "Time-dependent correlation functions appear in the dynamical structure factor, are related to the inelastic neutron-scattering cross section, where the neutron energy changes upon the scattering process.", "For a system of interacting spins on a lattice, the dynamic structure factor reads: $S^{\\alpha }(q,\\omega ) = \\sum _n \\int _{-\\infty }^{\\infty } dt \\exp [i(qn-\\omega t)]<S_j^{\\alpha }(0)S_{j+n}^{\\alpha }(t)>.$ where $S$ are spin variables and the sum runs over all the lattice sites.", "In light scattering experiments, the scattered intensity is given by the differential cross section, proportional to: $I({\\mathbf {k}}, \\omega ) = \\int _{-\\infty }^{\\infty } dt \\exp (-i\\omega t)<A_k^{\\dagger }(0)A_k(t)>.$ where the form of operator $A$ is system dependent.", "It also depends on the the particular frequency of the incoming light." ], [ "The method of recurrence relations", "The time evolution of a Hermitian operator $A(t)$ is governed by the Heisenberg equation: $\\frac{dA(t)}{dt} = i {\\cal L} A(t),$ where: ${\\cal L} A(t) \\equiv HA(t) - A(t)H = [H, A(t)].$ Consider a time-independent and Hermitian Hamiltonian $H$ .", "From now on we will be using a system of units in which $\\hbar = 1$ .", "We seek a solution to Eq.", "(REF ) for $t \\ge 0$ , thus we set $A(t)=0$ for $t < 0$ .", "In the method of recurrence relations, the formal solution: $A(t) = \\exp (iHt)A\\exp (-iHt)$ is cast as an orthogonal expansion in a realized Hilbert space $\\cal S$ of $d$ dimensions.", "That Hilbert space $\\cal S$ is realized by the scalar product: $(X, Y) = \\beta ^{-1} \\int _0^{\\beta } d\\lambda \\, <X(\\lambda )Y> - <X><Y>.$ where $X$ , $Y$ $\\subset \\cal S$ , $\\beta $ is the inverse temperature, $X(\\lambda ) = \\exp (\\lambda H) X \\exp (-\\lambda H)$ , and $< \\dots >$ denotes a canonical ensemble average.", "Thus, the time evolution of $A(t)$ is written as: $A(t) = \\sum _{\\nu = 0}^{d-1} a_{\\nu }(t) f_{\\nu }.$ where $\\lbrace f_{\\nu }\\rbrace $ is a complete set of states in $\\cal S$ , while the time-dependence is carried out by the coefficients $a_{\\nu }(t)$ .", "The dimensionality $d$ of the realized Hilbert space $\\cal S$ is still unknown, but it will be determined later.", "If $d$ turns out to be finite, the solutions are oscillatory functions.", "However, in most interesting cases $d$ is infinite.", "The method of recurrence relations imposes constraints on which type of solutions are admissible.", "By choosing the basal vector $f_0 = A(0) = A$ , the remaining basis vectors are obtained following the Gram-Schmidt orthogonalization procedure, which is equivalent to the recurrence relation: $f_{\\nu + 1} = i{\\cal L}f_{\\nu } + \\Delta _{\\nu } f_{\\nu - 1}, \\qquad \\nu \\ge 0$ with $f_1 \\equiv 0$ , $\\Delta _0 \\equiv 0$ .", "The quantity $\\Delta _{\\nu }$ is defined as the ratio between the norms of consecutive basis vectors: $\\Delta _{\\nu } = \\frac{(f_{\\nu },f_{\\nu })}{(f_{\\nu - 1},f_{\\nu -1})} \\qquad \\nu \\ge 1.$ The $\\Delta $ 's are referred to as the recurrants whereas Eq.", "(REF ) is termed the first recurrence relation, or RRI.", "The time-dependent correlation function $C(t)$ is given by: $C(t) = \\frac{<A(0)A(t)>}{<A(0)A(0)>} = (f_0,A(t)) = a_0(t).$ The basal coefficient $a_0(t)$ is just the time-dependent correlation function.", "The time-dependent coefficients $a_{\\nu }(t)$ obey a second recurrence relation (RRII): $\\Delta _{\\nu + 1} a_{\\nu + 1}(t) = - \\dot{a}_{\\nu }(t) + {a}_{\\nu -1}(t) \\qquad \\nu \\ge 0,$ where $\\dot{a}_{\\nu }(t) = da_{\\nu }(t)/dt$ , and $a_{-1} \\equiv 0$ .", "It follows from Eq.", "(REF ) that the initial choice $f_0 = A(0)$ implies $a_{0}=1$ and $a_{\\nu }(0) = 0$ for $\\nu \\ge 1$ .", "Thus the complete time evolution of $A(t)$ is obtained by the two recurrence relations RRI and RRII.", "One should note that only in very few cases a closed analytic solution to a model can be found.", "More often, as in many-body problems, approximations are required.", "A generalized Langevin equation can be derived for $A(t)$  [3], [4], [5]: $\\frac{dA(t)}{dt} = \\int _0^t dt^{\\prime }\\,\\phi (t-t^{\\prime })A(t^{\\prime }) = F[t]$ where $\\phi $ is the memory function and $F[t]$ the random force.", "The random force is given as an expansion in the subspace of $\\cal S$ : $F[t] = \\sum _{\\nu =1}^{d-1} b_{\\nu }(t) f_{\\nu },$ where the coefficients $b_{\\nu }$ satisfy the convolution equation: $a_{\\nu }(t) = \\int _0^t dt^{\\prime } b_{\\nu }(t - t^{\\prime }) a_0(t^{\\prime }), \\qquad \\nu \\ge 1.$ The memory function $\\phi (t)$ is $\\phi (t) = \\Delta _1 b_1(t)$ .", "The remaining $b_{\\nu }$ 's, $b_2$ , $b_3$ , are the second memory function, the third memory function, $\\dots $ , etc.", "Consider now the Lapace transform $a_{\\nu }(z)$ of $a_{\\nu }(t)$ : $a_{\\nu }(z) = \\int _0^{\\infty } dt \\exp (-zt) a_{\\nu }(t), \\quad {\\rm Re}\\, z > 0.$ Then RRII can be transformed in the following way: $1 &&= a_0(z) + \\Delta _1 a_1(z), \\\\a_{\\nu - 1}(z) &&= a_{\\nu }(z) + \\Delta _{\\nu +1} a_{\\nu +1}(z), \\quad \\nu \\ge 1.$ These equations can be solved for $a_0(z)$ : $a_0(z) = 1/(z+ \\Delta _1/(z + \\Delta _2/z + \\dots \\Delta _{d-1}/z)),$ resulting in a continued fraction.", "As can be seen from Eq.", "(REF ) and the recurrence relation RRII, that the time-dependence actually depends on the recurrants $\\Delta _{\\nu }$ only.", "Therefore, the knowledge of all recurrants provides the necessary means to obtain the time correlation function.", "Moreover, the structure of RRII must be obeyed by time correlation functions.", "Thus, a pure exponential decay as well as special polynomials can be ruled out as solutions, since their recursion relations are not congruent to RRII.", "Also, from RRII one obtains $(da_0(t)/dt)|_0 = 0$ , which precludes a pure time exponential as well as other functions that do not have zero derivative at $t=0$ .", "The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [26], [27], [29], [28], harmonic oscillator chains [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], many-particle systems [43], [42], [40], [41], spin chains [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], plasmonic Dirac systems [60], [61], etc." ], [ "The method of exact diagonalization", "Given a system governed by a Hamiltonian $H$ , one wishes to numerically determine the time correlation function $C(t)$ , defined by: $C(t) = \\frac{<A(0)A(t)>}{<AA>},$ where $A(t) = \\exp (iHt) A \\exp (-iHt) $ , $\\hbar =1$ , and the brackets denote canonical averages.", "We consider here self-adjoint operators $A$ and the Hamiltonian $H$ .", "One numerically diagonalizes $H$ and then uses its eigenvalues $E_n$ and eigenvectors $|n>$ , $H|n> = E_n|n>$ , to calculate $C(t)$ in Eq.", "(REF ), where: $<AA> = \\frac{1}{Z}\\sum _n \\exp (-\\beta H) <n|A^2|n>,$ $<A(0)A(t)> = \\frac{1}{Z} \\sum _{n,m} e^{-\\beta E_n} e^{- i (E_n - E_m)} |<n|A|m>|^2,$ with the partition function $Z = \\sum _n \\exp (-\\beta E_n$ ).", "Notice that the time correlation function is normalized to unity at $t=0$ , that is, $C(0) = 1$ .", "Another quantity of interest is the moment $\\mu _k$ , also referred to as the frequency moments which can be obtained from the Taylor expansion of $C(t)$ about $t=0$ : $C(t) = \\sum _{k=0}^{\\infty } \\frac{(-1)^k}{ (2k)!}", "\\mu _{2k} t^{2k}.$ Since $C(0)=1$ , it follows that $\\mu _0 = 1$ .", "The moments are given by: $\\mu _{2k} = \\frac{1}{Z} {\\rm Tr}\\, [e^{-\\beta H} A {\\cal L}^{2k} A],$ where $\\cal L$ is the Liouville operator, Eq.", "(REF ).", "From the moments, one can use conversion formulas to obtain the recurrants $\\Delta $ 's of the method of recurrence relations from the frequency moments [11].", "Suppose the moments $\\mu _0=1$ and $\\mu _{2k}$ , $k= 1, \\dots , K$ are known.", "The first $K$ recurrants $\\Delta _{\\nu }$ are determined by the equations: $\\Delta _{\\nu } = \\mu _{2\\nu }^{(\\nu )}, \\qquad \\mu _{2k}^{(\\nu )} = \\frac{ \\mu _{2k}^{(\\nu - 1)}}{\\Delta _{\\nu - 1}} - \\frac{\\mu _{2k-2}^{(\\nu - 2)}}{\\Delta _{\\nu - 2}}, \\quad ,$ for $k = \\nu , \\nu + 1, \\dots , K$ and $\\nu = 1, 2, \\dots , K$ , with $\\mu _{2k}^{(0)}= \\mu _{2k}$ , $\\Delta _{-1} = \\Delta _0 = 1$ , $\\mu _{2k}^{(-1)} = 0.$ s For instance, if the first moments $\\mu _0 =1$ , $\\mu _2$ , $\\mu _4$ , $\\dots $ , $\\mu _{10}$ are given, the recurrences are obtained from Eq.", "(REF ): $&&\\Delta _1 = \\mu _2, \\nonumber \\\\&&\\Delta _2 = -\\mu _2 + \\mu _4/\\mu _2, \\nonumber \\\\&&\\Delta _3 = \\mu _4 (\\mu _4/\\mu _2 - \\mu _6/\\mu _4)/\\mu _2 (\\mu _2 - \\mu _4/\\mu _2), \\nonumber \\\\&&\\Delta _4 = - \\mu _4/\\mu _2 + \\mu _6/\\mu _4 \\nonumber \\\\&&- \\mu _4 (\\mu _4/\\mu _2 - \\mu _6/\\mu _4)/ \\mu _2 (\\mu _2 - \\mu _4/\\mu _2) \\nonumber \\\\&&+ \\mu _6 (\\mu _6/\\mu _4 - \\mu _8/\\mu _6)/ \\mu _4 (\\mu _4/\\mu _2 - \\mu _6/\\mu _4).$ Conversely, suppose one has the first $K$ known recurrants, $\\Delta _{\\nu }$ , $\\nu = 1, 2, \\dots , K$ , and $\\Delta _{-1} = \\Delta _0 = 1$ .", "Then, the moments $\\mu _{2 \\nu }^{(0)} = \\mu _{2 \\nu }$ are obtained from the following conversion formula: $\\mu _{2k}^{(\\nu - 1)} = \\Delta _{\\nu - 1} \\mu _{2k}^{(\\nu )} + \\frac{\\Delta _{\\nu -1}}{\\Delta _{\\nu - 2}} \\mu _{2k - 2}^{(\\nu - 2)},$ for $\\nu = k, k-1, \\dots , 1$ and $k = 1, 2, \\dots , K$ , with $\\mu _{2k}^{(-1)}~=~0$ .", "In case the first recurrants $\\Delta _1$ , $\\Delta _2$ , $\\dots $ , $\\Delta _4$ , are known, the moments $\\mu $ are found to be: $&&\\mu _0 = 1, \\nonumber \\\\&&\\mu _2 = \\Delta _1, \\nonumber \\\\&&\\mu _4 = \\Delta _1(\\Delta _1 + \\Delta _2), \\nonumber \\\\&&\\mu _6 = \\Delta _1((\\Delta _1 + \\Delta _2)^2 + \\Delta _1 \\Delta _3), \\nonumber \\\\&&\\mu _8 = \\Delta _1 \\left( (\\Delta _1+\\Delta _2)^2 + \\Delta _2 \\Delta _3 \\right) \\nonumber \\\\&& \\times \\left( \\Delta _1 + \\Delta _2 + \\frac{\\Delta _2 \\Delta _3}{\\Delta _1 + \\Delta _2}+ \\Delta _2 \\Delta _3 \\frac{\\Delta _3 + \\Delta _4 + \\frac{\\Delta _2 \\Delta _3}{\\Delta _1 + \\Delta _2}}{(\\Delta _1 + \\Delta _2)^2 + \\Delta _2 \\Delta _3} \\right).", "\\nonumber \\\\&&{ }$ Typically, the analytical determination of the recurrants becomes increasingly time consuming.", "In practice, only a few of them can be obtained to be used in an extrapolation scheme to obtain higher-order recurrants.", "Several extrapolation schemes have been used.", "One of the simplest is to set the unknown recurrants to zero, thus truncating the continued fraction for $a_0(z)$ , which leads to a finite number of poles in the complex plane [18].", "In other problems, it is most appropriate to introduce a Gaussian termination, that is, a sequence of recurrants $\\Delta {\\nu }$ that grow linearly with its index $\\nu $ , $\\Delta _{\\nu } = \\nu \\Delta $  [44], [11].", "Other extrapolation schemes are tailored to the problem at hand, especially if the recurrants are not expected to grow indefinitely." ], [ "Applications to interacting systems", "The dynamics of spin chains has attracted a great deal of attention in recent decades.", "Exact results for the longitudinal dynamics of the one dimensional XY model have been obtained with the Jordan-Wigner transformation [62].", "Later, exact results for the transverse time correlation functions of the XY and the transverse Ising chain were obtained at the high temperature limit by using different methods [63], [64], [44].", "A great deal of progress was achieved in the calculations of the dynamic correlation functions of spin models in one dimension.", "It was soon recognized that exact solutions using the method of recurrence relations were difficult to obtain, however a notable exception is the classical harmonic oscillator chain where the time correlation functions were obtained exactly [30].", "The problem of a mass impurity in the harmonic chain was solved later, and its dynamical correlation functions were found to have the same form as in the quantum electron gas in two dimensions, thus showing that unrelated quantities in these two models displayed the same dynamical behavior, that is, the have dynamic equivalence [65].", "It should be mentioned that harmonic oscillator chains have been the subject of a considerable amount of work with the method of recurrence relations [31], [32], [33], [34], [35], [36], [37], [38], [39].", "The method of recurrence relations provides important insights on how to proceed to obtain reliable approximate solutions.", "The cornerstone quantity in the dynamics is the recurrant, which is the only quantity that ultimately determines the dynamics of the model.", "Often it is only possible to determine a few of the recurrants analytically.", "The calculations become too lenghty so that one must stop at a given order.", "Thus, an extrapolation method must be devised for the higher order recurrants, which hopefully will have the essential ingredients to produce reliable time-dependent correlation functions for longer times as well as spectral densities with the expected behavior near the origin $\\omega =0$  [55], [66], [67], [42].", "The dynamics of the transverse Ising model in two dimensions was studied with the method of recurrence relations [68], [69], [71], [70].", "The dynamic structure factor of that model compares well to the experimental data of the compound LiTbF${}_4$  [72].", "The dynamics of spin ladders has also attracted interest from researchers.", "The dynamical correlation functions were obtained for a two-leg spin ladder with XY interaction along each leg and interchain Ising couplings in a random magnetic field.", "More recently, the dynamics of a ladder with Ising couplings in the legs and steps as well as four-spin plaquette interactions in a magnetic field [73] have been also investigated.", "The dynamical correlations of the Heisenberg model in one dimension have been have been a subject of great interest in the recent decades [13], [14], [15], [21], [74].", "The method of recurrence relations has been employed in various works [76], [75], [77], [78], [79], [80], [81].", "In spite of the progress made thus far, the long-time dynamics of the Heisenberg spin model is still an open problem.", "For instance, there is the standing problem on the power law exponent $\\alpha > 0$ ( $\\sim t^{-\\alpha }$  ) of the time correlation function as $t \\rightarrow \\infty $ .", "From the work of Fabricius et al.", "[15], we find that the time correlation functions of the Heisenberg model decay more slowly than that in the XY model, for which the exact solution is known, $C(t) \\sim t^{-1}$ for large $t$ .", "Thus we infer that the numerical evidence suggests that $\\alpha \\ge 1$ for the Heisenberg model.", "There has been a great deal of work that uses exact diagonalization to study the dynamics of spin systems [13], [14].", "Earlier works with the Heisenberg model used this technique.", "Later on, other systems were scrutinized by using exact diagonalization.", "One of those systems is the Ising model with four-spin interactions in a transverse field.", "The time correlation function was obtained for one dimensional and infinite temperature [82], where the Gaussian behavior shown in the usual transverse Ising model was ruled out.", "The effects of disorder on the dynamics of that model were obtained for the cases where the random variables are drawn from bimodal distributions of random couplings and fields [16], [83], [84].", "Dynamical correlation functions were also obtained for the system at finite temperatures, ranging from $T=0$ to $T= \\infty $  [85]." ], [ "Heisenberg model with Dzyaloshinskii-Moriya interactions", "The dynamical structure factor for a quantum spin Heisenberg chain with Dzyaloshinskii-Moriya (DM) interactions [86], [87] has been investigated by different approaches, such as spin wave theory [88], mean-field [89], and projection operator techniques [90].", "The dynamics of the related XY model with DM interactions was also studied by employing Jordan-Wigner fermions [91], [92], [93].", "The dynamical correlation functions of the spin-1/2 Heisenberg model with DM interactions in a transverse magnetic field was studied recently with the method of recurrence relations.", "The model Hamiltonian for a one-dimensional chain is given by: $H &&= - J \\sum _i (\\sigma _i^x \\sigma _{i+1}^x + \\sigma _i^y \\sigma _{i+1}^y+ \\sigma _i^z \\sigma _{i+1}^z) \\qquad \\qquad \\nonumber \\\\&& - D\\sum _i (\\sigma _i^x \\sigma _{i+1}^y - \\sigma _i^y \\sigma _{i+1}^x)- \\sum _i B_i \\sigma _i^x,$ where $J$ is the Heisenberg coupling, $D$ is the Dzyaloshinskii-Moriya interaction, and $B_i$ is a magnetic field perpendicular to the DM axis.", "The quantities $\\sigma _i^{x,y,z}$ are the usual Pauli operators.", "The effects of a uniform magnetic field $B_j = B$ on the dynamics are investigated in the infinite temperature limit [94].", "The purpose is to determine the time correlation function $C(t) = <\\sigma _j^z \\sigma _j^z(t)>$ and its associated spectral density $S(\\omega )$ .", "The first four recurrants are determined analytically and an extrapolation scheme is devised to obtain higher order recurrants.", "Such scheme must take into account what is already known from the solutions of related problems.", "One crucial point is to determine whether or not the extrapolated recurrants grow indefinitely.", "The time correlation function of the longitudinal spin component in the $XY$ chain is known exactly at $T= \\infty $ , $C(t) = J_0^2(t) \\sim t^{-1}$ asymptotically for large times, where $J_0$ is the Bessel function of first kind [62].", "In this case, the recurrants tend to a constant finite value as $\\nu \\rightarrow \\infty $ .", "There are numerical indications that the time correlation function of the Heisenberg model decays as a power law [15], which suggests that the extrapolated recurrants grow asymptotically to a finite value.", "For the Hamiltonian Eq.", "(REF ) the extrapolation is the following power-law: $\\Delta _{\\nu } = \\Delta _{\\infty } - \\frac{b}{\\nu ^{\\beta }}, \\quad \\nu \\ge n_c$ where $n_c$ is the order of the last exactly calculated recurrant.", "The limit value $\\Delta _{\\infty }$ is obtained by extrapolating the last two recurrants to the origin of $1/\\nu $ .", "The constants $b$ and $\\beta $ guarantee smooth behavior of the recurrants above and below $\\nu = n_c$ .", "Once the recurrants are obtained, the relaxation function and its spectral density can be readily obtained.", "For the special case without DM interaction, the result shows good agreement with the known results for the XY and Heisenberg models.", "The full calculation reveals that the effects of the external field are to produce stronger and more rapid oscillations in the relaxation functions, as well as a suppression of the central peak in the spectral density.", "In addition a peak centered at a well defined frequency appears, which is attributed to an enhancement of the collective mode of spins precessing about the external field.", "It should be noted that the method of recurrence relations was also used to study the dynamics of the XY model with DM interaction [58].", "The effects of disorder in a transverse magnetic field on the dynamical correlation functions are investigated with the bimodal distribution for $B_i$ : $\\rho (\\lbrace B_i\\rbrace ) = \\Pi _i [ (1-p) \\delta (B_i - B_A) + p \\delta (B_i - B_B)].$ The method of recurrence relations is then applied to obtain the dynamical correlation functions for a given realization of disorder [59].", "Next, the average over the random fields is performed by using the distribution Eq.", "(REF ).", "This is accomplished by defining the scalar product in the Hilbert space $\\cal S$ , so as to include an average over the random variables in addition to the thermal average.", "Four recurrants $\\Delta _{\\nu }$ are obtained, and an extrapolation is made for the remaining recurrants.", "In practice only the first dozen are needed to attain convergence.", "The time correlation function and its associated spectral density were obtained for $D=1$ and $B_A= 0$ and $B_B= 4$ in units of the Heisenberg coupling $J$ .", "When the probability $p$ is very small, a strong central mode appears, as well as a shoulder in the spectral density.", "As $p$ increases, there is supression of the central mode as well as the shoulder.", "On the other hand, for large values of the probability $p$ , a nonzero frequency peak appears, resulting from the precession of the spins around the magnetic field adding further suppression of the central peak.", "This central mode behavior versus collective dynamics, is a known feature of the dynamics of spin systems and they are in some sense universal.", "However, in the present case the appearence of a shoulder for small $p$ is an interesting novel feature." ], [ "Random transverse Ising model", "Consider the $s=1/2$ spin model in one dimension: $H = - \\frac{1}{2}\\sum _i J_i \\sigma _i^x \\sigma _{i+1}^x - \\frac{1}{2} \\sum _i B_i \\sigma _i^z,$ where $J_i$ and $B_i$ are exchange couplings and transverse fields, respectively.", "These couplings and fields are random variables drawn from distribution functions.", "The quantities $\\sigma _i^{\\alpha } (\\alpha = x,y,z)$ are Pauli matrices.", "The model is referred to as the random transverse Ising model (RTIM), and its dynamical correlation function in the infinite temperature limit has been investigated by using the method of recurrence relations [95].", "The time correlation $C(t)$ is defined by: $C(t) = \\overline{<\\sigma _j^x \\sigma _j^x(t)>},$ where the line indicates that an average over the random variables is performed after the statistical average $< \\dots >$ .", "The time evolution of $\\sigma _j^z(t)$ in a system governed by the Hamiltonian Eq.", "(REF ) is given as an expansion in a Hilbert space $\\cal S$ of $d$ dimensions, where $d$ is to be determined later: $\\sigma _j^x(t) = \\sum _{\\nu = 0}^{d-1} a_{\\nu }(t)f_{\\nu },$ where $f_{\\nu }$ are orthogonal vectors spanning $\\cal S$ .", "The time dependence is contained in the coefficients $a_{\\nu }(t)$ .", "The inner product in $\\cal S$ in the infinite temperature limit is defined in such a way that it encompasses both the thermal average in a realization of disorder and the average over the random variables: $(A, B) = \\overline{<A B^{\\dagger >}} - \\overline{<A><B^{\\dagger }>},$ where $A$ and $B$ are vectors in $\\cal S$ .", "This definition of scalar product ensures that the form of the recurrence relations in unchanged.", "The zeroth basis vector $f_0$ is chosen as the variable of interest, $f_0 = \\sigma _j^x$ .", "Thus, the zeroth-order coefficient $a_0(t)$ can be identified with the time-dependent correlation function of interest: $a_0(t) = (f_0,f_0) = \\overline{<\\sigma _j^x \\sigma _j^x(t)>} = C(t).$ The remaining basis vectors $f_{\\nu }$ , $\\nu = 1, 2, ..., d-1$ , are obtained from the recurrence relation RRI, Eq.", "(REF ).", "The first vectors are then: $&& \\qquad \\qquad \\qquad \\quad f_0 = \\sigma _j^x , \\nonumber \\\\&& \\qquad \\qquad \\qquad f_1 = B_j \\sigma _j^y , \\nonumber \\\\&& f_2 = (\\Delta _1 - B_j^2) \\sigma _j^x + B_j J_{j-1}\\sigma _{j-1}^x \\sigma _j^z+ B_j J_j \\sigma _j^z \\sigma _{j+1}^x, \\nonumber \\\\&& \\qquad f_3 = - B_j (J_{j-1}^2 + J_j^2 + B_j^2 -\\Delta _1 - \\Delta _2)\\sigma _j^y \\nonumber \\\\&& \\quad - 2 B_j J_{j-1} J_j \\sigma _{j-1}^x \\sigma _j^y \\sigma _{j+1}^x + B_{j-1}J_{j-1} \\sigma _{j-1}^y \\sigma _j^z \\nonumber \\\\&& \\qquad \\qquad \\quad + B_jB_{j+1} J_j \\sigma _j^z \\sigma _{j+1}^y,$ etc.", "The vectors $f_4$ , $f_5$ , $\\dots $ , $f_9$ were obtained analytically but not reported because of their length [95].", "However, they were used in all of the subsequent calculations.", "The first three recurrants are the following, $&& \\qquad \\qquad \\qquad \\qquad \\Delta _1 = \\overline{B_j^2}, \\nonumber \\\\&& \\qquad \\qquad \\Delta _2 = 2 \\overline{J_j^2} - \\overline{B_j^2} + \\overline{B_j^4}/\\overline{B_j^2}, \\nonumber \\\\&& \\Delta _3 = \\frac{\\overline{B_j^6}+ 2 \\overline{J_j^2}^2 \\, \\overline{B_j^2} + 2\\overline{J_j^4}\\overline{B_j^2} + 2 \\overline{J_j^2}\\,\\overline{ B_j^2}^2 - \\overline{B_j^4}^2/ \\overline{B_j^2}}{ 2 \\overline{J_j^2} \\overline{B_j^2} - \\overline{B_j^2}^2 + \\overline{B_j^4} }.", "\\nonumber \\\\{\\ \\ } &&{} {\\ \\ }$ Notice that the couplings and fields are site-dependent.", "There are two types of disorder considered in Ref.", "[95], random fields and random spin couplings.", "Each case is treated separately.", "In both cases a simple bimodal distribution is used for the random variable.", "The field $B_i$ (or the coupling $J_i$ ) can assume two distinct values, with probalities $q$ ($p$ ) and $1-q$ ($1-p)$ , respectively.", "The time correlaton function and the spectral density are then obtained numerically.", "For the pure cases, ($p=q=1$ ), two types of behavior emerge, depending on the relative strength between $J$ and $B$ .", "For $J>B$ , the dynamics is dominated by a central-mode behavior, whereas for $J<B$ a collective-mode is the prevailing dynamics.", "In the disordered cases, the dynamics is neither central-mode nor collective-mode type, but something in between those types of dynamics." ], [ "Transverse Ising model with next-to-nearest neighbors interactions", "Consider the transverse Ising model with an additional axial next-nearest-neighbor interaction (transverse ANNNI model) [17].", "The Hamiltonian for a chain with $L$ spins can be written as: $H = J_1 \\sum _{i=1}^{L}\\,\\sigma _{i}^{z}\\sigma _{i+1}^{z}- J_{2}\\sum _{i=1}^{L}\\,\\sigma _{i}^{z}\\sigma _{i+2}^{z}- B\\sum _{i=1}^{L}\\,\\sigma _{i}^{x}~,$ where $\\sigma _i^{\\alpha }$ are the usual spin-1/2 operators, $\\alpha = x, y, z$ .", "Periodic boundary conditions are imposed on this model, namely $\\sigma _{i+L}^{\\alpha } = \\sigma _{i}^{\\alpha }$ .", "Consider antiferromagnetic ($J_1 >0)$ Ising interactions.", "A competing ferromagnetic interaction is assumed for the next-nearest-interaction ($J_2 > 0$ ).", "The transverse magnetic field ($B$ ) induces the quantum fluctuations.", "In what follows we set $J_1 = 1$ as the unity of energy.", "In the absence of a transverse magnetic field and of thermal fluctuations ($T=0$ ) the ground-state properties of the model are exactly soluble and several phases are present.", "For $J_2 < 0.5$ the ground state is ordered ferromagnetically.", "For $J_2 > 0.5$ , a phase consisting of two up-spins followed by two down-spins is periodically formed.", "The phase is known as $<2,2>$ -phase or an anti-phase.", "For $J_2 = 0.5$ , the model has a multiphase point.", "The ground-state is highly degenerate with many phases of the type $<p,q>$ corresponding to a periodic phase with $p$ -up spins followed by $q$ -down spins, among other spin configurations.", "The number of degenerate states increases exponentially with the size of the system $L$ .", "In the case where $J_2 = 0$ and the magnetic field is switched on, the model becomes the Ising model in a transverse field which was exactly solved by Pfeuty [96].", "In this model, due to quantum fluctuations induced by the transverse magnetic field, a second order phase transition occurs at $B= 1$ , which separates a ferromagnetic phase at low magnetic fields from a paramagnetic phase at high magnetic fields.", "For the full Hamiltonian, Eq.", "(REF ), the competing interaction between the ferromagnetic and antiferromagnetic terms induces frustration in the magnetic ordering.", "This will give rise to a much richer variety of phases when either the transverse magnetic field or the spin-spin interations are varied, such as ferromagnetic or antiferromagnetic phases, disordered or paramagnetic phases, and floating phases [17].", "Such variety of phases in the ground-state could carry over their effects into the dynamics at the high temperature limit, like the known transverse Ising model.", "In this model, a signal of the ground-state transition is manifested in the Gaussian behavior at criticality of the dynamical correlation functions at $T=\\infty $  [44].", "The main quantity of interest is the time-dependent correlation function: $C(t) = <\\sigma _j^x(0) \\sigma _j^x(t) >,$ where $\\sigma _j^x(t) = e^{iHt} \\sigma _j^x e^{-iHt}$ and $< \\cal O>$ is a canonical average of the operator $\\cal O$ .", "The method of exact diagonalization will be employed to study the dynamics, however, the recurrants of the method of recurrence relations will also be obtained.", "The numerical calculations will be performed at the high-temperature limit, $T=\\infty $ , hence: $C(t) = \\frac{1}{2^L} {\\rm Tr}\\, (\\sigma _j^x e^{iHt} \\sigma _j^x e^{-iHt}).$ One of the properties of $C(t)$ is that it is real and an even function of the time $t$ .", "Therefore, the Taylor expansion about $t=0$ has only even powers of $t$ : $C(t) = \\sum _{k=0}^{\\infty } \\frac{(-1)^k}{(2k)!}", "\\mu _{2k}\\, t^{2k},$ where the frequency moments are expressed in terms of the trace over iterated commutators: $\\mu _{2k} = \\frac{1}{2^L} {\\rm Tr}\\, (\\sigma _j^x {\\cal L}^{2k} \\sigma _j^x),$ with ${\\cal L}$ defined such that: ${\\cal L}A = [H, A] = HA - AH,$ where $H$ is the Hamiltonian and $A$ an operator.", "Figure: Time correlation function of a tagged spin in the TI modelwhen B=J=1.0B = J =1.0 for some chain sizes LL, as indicated.Here and in the next figures J=1J = 1 is the energy unit.The exact solution is a Gaussian, which lies underneaththe L=13L=13 curve.The correlation function is calculated in the Lehman representation.", "First, we consider the energies $E_n$ and eigenstates $|n>$ of the Hamiltonian, obtained from the eigenvalue equation $H|n> = E_n |n>$ .", "Then, the correlation function takes the form: $C(t)&&= \\frac{1}{2^L}\\sum _{m,n}\\cos (E_{n}-E_{m})t |<n|\\sigma _{j}^{x} |m>|^{2}, \\nonumber \\\\&&= \\sum _{k=0}^{\\infty } \\frac{(-1)^k}{(2k)!}", "\\mu _{2k} t^{2k},$ where the moments $\\mu _{2k}$ are given by: $\\mu _{2k} = \\frac{1}{2^L}\\sum _{m,n}(E_{n}-E_{m})^{2k} | <n|\\sigma _{j}^{x} |m>|^{2}.$ The spectral density $S(\\omega )$ is simply the Fourier transform of $C(t)$ : $S(\\omega ) = \\int _{-\\infty }^{\\infty }\\,C(t)e^{-i\\omega t}dt.$ After using Eq.", "(REF ), the spectral density can be cast in the form: $S(\\omega ) = \\frac{\\pi }{2^L} \\sum _{m,n}|<n|\\sigma _j^x|m>|^2[\\delta (\\omega -\\epsilon _{nm}) + \\delta (\\omega +\\epsilon _{nm})],$ where $\\epsilon _{nm} \\equiv E_n - E_m$ .", "The Dirac $\\delta $ -function is approximated by a rectangular window of width $a$ and unit area, centered at the zeros of their arguments.", "The width $a$ , can be adjusted to reduce fluctuations.", "Another approach could be the use of histograms, such as in Ref. [85].", "However, the general shape of the spectral density $S(\\omega )$ is the same, although the rectangle approximation gives more accurate results.", "Therefore, both dynamical correlation functions $C(t)$ and $S(\\omega )$ can be calculated directly via exact diagonalization.", "Figure: Spectral density for the TI model (B=1B = 1 and J 2 =0J_2=0)and different chain sizes.", "The plots are the time Fourier transforms ofthe curves in Fig. .", "The curves for finite chains oscillatearound the exact Gaussian result of the infinite chain.As a case test, Guimarães et al.", "[17] consider $B=1$ and $J_2=0$ , the usual transverse Ising model (TIM) with dynamical correlation functions known exactly in the high temperature limit [64], [44].", "Figure REF shows their numerical results for the time correlation function for $B = 1$ and several lattice sizes.", "The results for $L=12, 13$ , agree very well with the exact result of the infinite system, $C(t) = \\exp (-2t^2)$ in the time interval of interest $0 \\le t \\le 10$ .", "Convergence toward the thermodynamic result increases as the system size grows.", "However, already for $L=13$ the numerical calculations reproduce the Gaussian behavior found by the exact calculation.", "The corresponding spectral density is shown in Fig.", "REF for different chain sizes.", "The Dirac $\\delta $ functions are approximated by a rectangle of unit area and width $a=0.1$ .", "That is the best value for the width $a$ to reduce the fluctuations due to finite-size effects.", "Those fluctuations decrease in amplitude as one considers larger system sizes.", "The frequency-dependent Gaussian of the exact result is already masked by the curve for $L=13$ .", "Therefore, the method works just fine with the transverse Ising model, and very likely will do so with the transverse ANNNI model.", "In the following, consider the representative cases $B=0.5$ , $1.0$ , and $2.0$ .", "These cases should cover the relevant possibilities for $B$ in the transverse ANNNI model.", "Consider first $B=0.5$ .", "The time correlation function $C(t)$ is shown in Fig.", "REF for different next-to-nearest neighbor couplings $J_2$ .", "There are pairs of curves for a given $J_2$ , dashed lines for $L=12$ and solid lines for $L=13$ .", "Those two lines agree very well with each other for the range of time $t$ displayed.", "The quantitative agreement between the $L=12$ and $L=13$ curves is an indication that within the accuracy used, the thermodynamic value has already been obtained.", "The features shown in Fig.", "REF are real and will not change in the thermodynamic limit.", "They possibly could be traced back to the rich ground-state phase diagram, however, a careful investigation is still necessary to clarify that point.", "In general, the decay of $C(t)$ with time is slower for larger $J_2$ .", "The corresponding spectral density is displayed in Fig.", "REF , calculated for $L=13$ .", "Other than the height near the origin $\\omega = 0$ , the remaining plots should not change essentially for larger chain sizes, or at the thermodynamic limit.", "The distinctive feature is the enhancement of the central mode as $J_2$ increases.", "Figure: Time-dependent correlation function for B=0.5B=0.5 andseveral values of the NNN coupling J 2 J_2.Figure: Spectral density for B=0.5B = 0.5 and several values of J 2 J_2.", "All the curves were obtainedfor chains with L=12L= 12 spins.The time correlation function for $B=1$ is depicted in Fig.", "REF for several $J_2$ .", "The curves shown are from $L=12$ and $L=13$ .", "Note that for $J_2=0$ the calculation reproduces the known Gaussian solution of the TI model.", "When $J_2=0.5$ , oscillations are present in $C(t)$ .", "For $J_2 \\ge 1$ the curves decay at much slower rate.", "A careful examination of the figures shows oscillations of relatively small amplitudes.", "The spectral density $S(\\omega )$ is shown in Fig.", "REF , where the calculations were done with $L=13$ .", "For $J_2=0$ the Gaussian of the TI model is reproduced.", "We observe an enhancement of the central model behavior as the values of $J_2$ are increased.", "Finally, consider the case where the transverse field is larger ($B=2$ ) than the Ising coupling.", "The time correlation function is shown in Fig.", "REF for several values of $J_2$ .", "The curves were obtained from a chain of size $L=13$ .", "For small values of $J_2$ the correlation function, $C(t)$ , shows oscillations typical of collective mode, such as that found in the TI model ($J_2 = 0$ ).", "As $J_2$ becomes larger, the amplitude of the oscillations decreases.", "For large enough $J_2$ , the system displays an enhancement of the central model.", "Figure REF depicts the corresponding spectral density $S(\\omega )$ for the values of $J_2$ used in the previous figure.", "For $J_2=0$ , the dynamics is dominated by the two-peak structure characteristic of collective mode.", "As $J_2$ increases, a reduction of the intensity of the peaks of the collective mode is observed in tandem with a growth of the central peak.", "For $J_2 \\ge 2 $ the dynamics seems to be dominated entirely by the central mode.", "The recurrants $\\Delta _{\\nu }$ of the method of recurrence relations are calculated numerically for $J_2=1$ and several values of $B$ .", "First the moments $\\mu $ are obtained by using Eq.", "(REF ).", "Next, use the conversion formulas Eq.", "(REF ).", "Table REF shows some numerical results for the recurrants when $B=1.0$ , and $J_2=1.0$ , obtained for $L= 11$ , 12, and 13.", "The rightmost column shows the extrapolated value of $\\Delta _{\\nu }$ for $L= \\infty $ .", "As can be seen, with relatively small chain sizes ($L \\le 13$ ), one can infer the thermodynamic value of the lower-order recurrants.", "Higher order recurrants are still obtained, but with lesser accuracy.", "The results for the thermodynamic estimates of the recurrants are shown in Fig.", "REF for $B=1.0$ and various values for $J_2$ .", "For $J_2=0$ the linear behavior that leads to Gaussian behavior is recovered [44].", "As $J_2$ increases, $\\Delta _{\\nu }$ increases at higher rates on the average and becomes rather erratic, therefore, it is difficult to predict a trend based on their behavior.", "Still the results shown are already the thermodynamic values, and it is very difficult to devise extrapolation schemes for the $\\Delta $ .", "Notwithstanding, such an endeavor will not uncover any new physics in regard to the dynamics of the transverse ANNNI model considered here.", "Figure: Time-dependent correlation function for B=1B = 1 and several valuesof J 2 J_2.", "The curves were obtained with L=12L=12 and 13.Figure: Spectral density for B=1B = 1 and several values of J 2 J_2, obtainedfor chains of length L=13L=13.Figure: Time-dependent correlation function for B=2.0B = 2.0 and various values of J 2 J_2.The chain size is L=13L=13.Figure: Spectral density for the case B=2B=2 and several values of J 2 J_2.The plots were obtained for L=12L=12.Table: Recurrants for the transverse ANNNI model, with B=1.0 B=1.0, J 2 =1.0J_2=1.0 andseveral chain sizes.", "The rightmost column is the extrapolation for the thermodynamiclimit (L=∞L=\\infty ).Figure: Recurrants of the infinite transverse ANNNI model with B=1.0B=1.0 and several values of J 2 J_2." ], [ "Summary and perspectives", "The dynamical correlation functions play a crucial role in the fluctuation-dissipation theorem and in the linear response theory.", "However, the calculation of those quantities is often a very complicated problem in itself.", "The method of recurrence relations is an exact procedure that allows one to obtain of time correlation functions, spectral densities, and dynamical structure factors.", "We have shown the main features of the method and the inherent difficulties one might encounter in an attempt to apply to a many-body problem.", "Another method that is showing great potential is exact diagonalization, a numerical method which relies mostly on computer capabilities.", "Nevertheless, the two methods can be used together, one complementing the other, to achieve progress in the calculation of dynamical correlation functions.", "Acknowledgements This work was supported by FAPERJ and PROPPI-UFF (Brazilian agencies)." ], [ "Conflict of Interest Statement", "The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest." ], [ "Author Contributions", "The authors J. F and O. F. A.", "B. declare that they planned and carried out the elaboration of this work with equal contribution from each author." ] ]

[ [ "Gamma-ray Blazar variability: New statistical methods of time-flux\n distributions" ], [ "Abstract Variable \\gama-ray emission from blazars, one of the most powerful classes of astronomical sources featuring relativistic jets, is a widely discussed topic.", "In this work, we present the results of a variability study of a sample of 20 blazars using \\gama-ray (0.1--300~GeV) observations from Fermi/LAT telescope.", "Using maximum likelihood estimation (MLE) methods, we find that the probability density functions that best describe the $\\gamma$-ray blazar flux distributions use the stable distribution family, which generalizes the Gaussian distribution.", "The results suggest that the average behavior of the \\gama-ray flux variability over this period can be characterized by log-stable distributions.", "For most of the sample sources, this estimate leads to standard log-normal distribution ($\\alpha=2$).", "However, a few sources clearly display heavy tail distributions (MLE leads to $\\alpha<2$), suggesting underlying multiplicative processes of infinite variance.", "Furthermore, the light curves were analyzed by employing novel non-stationarity and autocorrelation analyses.", "The former analysis allowed us to quantitatively evaluate non-stationarity in each source -- finding the forgetting rate (corresponding to decay time) maximizing the log-likelihood for the modeled evolution of the probability density functions.", "Additionally, evaluation of local variability allows us to detect local anomalies, suggesting a transient nature of some of the statistical properties of the light curves.", "With the autocorrelation analysis, we examined the lag dependence of the statistical behavior of all the $\\{(y_t,y_{t+l})\\}$ points, described by various mixed moments, allowing us to quantitatively evaluate multiple characteristic time scales and implying possible hidden periodic processes." ], [ "Introduction ", "A class of active galaxies that profusely shine in radio frequency is known as radio-loud galaxy.", "These kinds of galaxies often show presence of parsec-scale relativistic jets.", "If the jet is oriented towards the Earth, the relativistic effects become dominant such that the Doppler boosted non-thermal emission makes the sources remarkably brighter over a wide range of electromagnetic frequencies.", "The emission is found to be more pronounced in higher energy spectrum e.g.", "X-ray and $\\gamma $ -rays, and the objects could also be the sources neutrino emission flying through the inter-galactic medium[14], [15].", "In addition, the parsec scale jets seem to be most efficient cosmic particle accelerators, where the particles, mainly leptons, are accelerated several orders of rest-mass electron energies.", "As a result, a large hoard of accelerated high energy particles become source of incoherent synchrotron emission by decelerating into the ambient jet magnetic field, and thereby making the extended jet “visible\".", "These energetic particles might also up-scatter the surrounding synchrotron photons which they themselves produced [20], [21] or the low-energy electrons of the external origins e.g.", "from accretion disk ([9]), broad-line region ([27]), and dusty torus ([5]), resulting in a large output of MeV-TeV emission.", "Variability over minute to decade timescales is one of the characteristic, defining properties of blazars.", "Numerous studies on various energy bands on all timescales have been conducted over the years using all available ground and space based instruments [25], [19], [3].", "Particularly, in the $\\gamma $ -ray regime, the power spectral study has shown that statistical nature of the variability can well be described by a single power-law in the Fourier domain [1]; whereas in some sources, applying continuous regressive models, some breaks in the power spectra, possibly corresponding to characteristic timescales [26] were reported.", "Indeed, time domain analysis of blazars serves as one of the most important tools to unravel the physical process occurring at the innermost regions around the central engines.", "As an attempt to understand the phenomenon of multi-timescale, multi-frequency variability in the sources, several emission models have been invoked; some of the widely discussed models include various magnetohydrodynamic instabilities in the turbulent jets [4], [23] disk and the jets , shocks traveling down jets [24], [28], `jets-in-a jet' model [12] and effects of jet orientation or geometric models [16].", "In spite of the collaborative efforts of the researchers working in the instruments and observations, modeling and theory, the details of the processes shaping multi-timescale variability still remains debated.", "The importance of time domain analysis with a focus on constraining the nature of the variability possibly can not be exaggerated as variability studies provides us with an excellent tool to probe into the energetics of the supermassive black hole systems.", "In this work, we perform statistical analysis of decade long Fermi/LAT observations of 20 blazars that were presented in [1].", "The source names, their 3FGL catalog name, source classification, RA, Dec and red-shift are presented in column 1, 2, 3, 4, 5 and 6, respectively, of Table .", "In Section , the details of the analyses methods carried out on the $\\gamma $ -ray light curves are discussed.", "The results and the discussions are presented in Section .", "llllll The source sample of the Fermi/LAT blazars 500pt Source name 3FGL name Source class R.A. (J2000) Dec. (J2000) Red-shift W Comae 3FGL J1221.4+2814 BL Lac $12^h 21^m31.7^s$ $+28^d 13^m 59^s$ 0.102 PKS 1502+106 3FGLJ1504.4+1029 FSRQ $15^h 04^m25^s.0$ $+10^d 29^m 39^s$ 1.84 4C+38.41 3FGL J1635.2+3809 FSRQ $16^h35^m15.5^s$ $+38^d08^m04^s$ 1.813 BL Lac 3FGL J2202.7+4217 BL Lac $22^h02^m43.3^s$ $+42^d16^m40^s$ 0.068 3C 279 3FGL J1256.1-0547 FSRQ $12^h56^m11.1665^s$ $-05^d47^m21.523^s$ 0.536 CTA 102 3FGL J2232.5+1143 FSRQ $22^h32^m36.4^s$ $+11^d43^m51^s$ 1.037 4C +21.35 3FGLJ1224.9+2122 FSRQ $12^h24^m54.4^s$ $+21^d22^m46^s$ 0.432 Mrk 501 3FGL J1653.9+3945 BL Lac $16^h53^m52.2167^s$ $+39^d45^m36.609^s$ 0.0334 PKS 0454-234 3FGLJ0457.0-2324 BL Lac $04^h 57^m03.2^s$ $-23^d 24^m 52^s$ 1.003 1ES 1959+65 3FGL J2000.0+6509 BL Lac $19^h59^m59.8521^s$ $+65^d08^m54.652^s$ 0.048 PKS 1424-418 3FGLJ1427.9-4206 FSRQ $14^h27^m56.3^s$ $-42^d06^m19^s$ 1.522 PKS 2155-304 3FGL J2158.8-3013 BL Lac $21^h58^m52.0651^s$ $-30^d13^m32.118^s$ 0.116 S5 0716+714 3FGL J0721.9+7120 BL Lac $07^h21^m53.4^s$ $+71^d20^m36^s$ 0.3 3C 66A 3FGL J0222.6+4301 BL Lac $02^h22^m41.6^s$ $+43^d02^m35.5^s$ 0.444 Mrk 421 3FGLJ1104.4+3812 BL Lac $11^h04^m273^s$ $+38^d12^m32^s$ 0.03 ON +325 3FGL J1217.8+3007 BL Lac $12^h17^m52.1^s$ $+30^d07^m01^s$ 0.131 AO 0235+164 3FGL J0238.6+1636 BL Lac $02^h 38^m38.9^s$ $+16^d 36^m 59^s$ 0.94 PKS 1156 3FGL J1159.5+2914 BL Lac $11^h59^m31.8^s$ $+29^d14^m44^s$ 0.7247 3C 454.3 3FGL J2254.0+1608 FSRQ $22^h53^m57.7^s$ $+16^d08^m54^s$ 0.859 3C 273 3FGL J1229.1+0202 FSRQ $12^h29^m06.6997^s$ $+02^d03^m08.598^s$ 0.158" ], [ "Methodology and analysis ", "There is applied and extended methodology from [10] article: first normalize marginal distributions with a parametric distribution (log-stable here) as in copula theory [11], then model evolution of normalized variables, or joint distribution for autocorrelations, in both cases representing PDE in polynomial basis." ], [ "Normalization with log-stable distribution", "As in copula theory, in the discussed methodology it is convenient to first normalize marginal distributions to nearly uniform on $[0,1]$ range.", "For this purpose we need to approximate their density, preferably with some parametric family.", "This density represents averaged density over the entire time period ($\\approx 10$ years here).", "In non-stationarity analysis we will additionally search for evolution of probability density inside this period, as a correction to density used for normalization.", "In autocorrelation analysis, for pairs of values shifted by various lags, we will evaluate distortion from uniform joint distribution on $[0,1]^2$ .", "A standard assumption for parametric distribution of this type of data, suggested by central limit theorem for multiplicative processes, is log-normal distribution: Gaussian distribution for logarithmized values.", "To verify this assumption, there were tested two larger families containing Gaussian distribution: exponential power distributions $\\rho (x)\\sim \\exp (-|x|^\\kappa )$ and stable distributions containing also heavy $\\sim |x|^{-\\alpha -1}$ .", "The highest log-likelihoods were achieved by using stable distributions for logarithmized values, hence they were applied for normalization.", "Stable distribution [7] is defined by 4 parameters.", "As Gaussian distribution it has $\\mu \\in (-\\infty ,\\infty )$ location parameter, and $\\sigma \\in (0,\\infty )$ scale parameter.", "Additionally it has $\\alpha \\in (0,2]$ stability parameter.", "For $\\alpha =2$ we get standard Gaussian distribution, for $\\alpha =1$ we get Cauchy distribution with $1/x^2$ heavy tails.", "Generally for $\\alpha \\in (0,2)$ it has $|x|^{-\\alpha -1}$ heavy tails, leading to infinite variance.", "This family also has $\\beta \\in [-1,1]$ skewness parameter, which allows for some asymmetry of the distribution, however, its influence weakens while $\\alpha $ approaches 2, and for $\\alpha =2$ this parameter has no effect.", "Finally examples of probability distribution function (PDF) and cumulative distribution functions (CDF) for some parameters of stable distribution are presented in Fig.", "REF .", "As the name suggests, these distributions are stable as in the central limit theorem, this time in its generalized version [13].", "While addition of finite variance i.i.d.", "random variables asymptotically leads to Gaussian distribution, for infinite variance variables such summation usually leads to a stable distribution.", "Good agreement here suggests multiplicative process with finite variance for some sources ($\\alpha =2$ ), but infinite for the remaining ($\\alpha <2$ ).", "Figure: Visual evaluation of agreement of MLE stable distributions for the 20 time series - orange curve being equal blue diagonal would mean perfect agreement.", "Specifically, the original time series (x t )(x_t) was first logarithmized, then there was performed maximal likelihood estimation (α,β\\alpha ,\\beta parameters are written), then y t = CDF αβμσ (ln(x t ))y_t=\\textrm {CDF}_{\\alpha \\beta \\mu \\sigma }(\\ln (x_t)) sequence was calculated using cumulative distribution function (CDF) for the found parameters, then orange curves are sorted y t {y_t} values - ideally from uniform distribution which would give diagonal (blue).", "We can see that agreement is quite decent, the α=2\\alpha =2 cases correspond to just log-normal distribution.", "However, as seen in Fig.", ", some objects have clearly lower α\\alpha - denoting heavier tail.", "The β\\beta parameter denotes asymmetry and is limited to [-1,1][-1,1], what seems insufficient for a few sequences.In the presented analysis, the flux values $(x_t)$ were first logarithmized, then individually for each object there was performed maximum likelihood estimation (MLE) of parameters of stable distribution using Wolfram Mathematica software.", "To verify estimation of $\\alpha $ and evaluate its accuracy, there was also performed estimation with various fixed $\\alpha $ , which log-likelihoods are presented in Fig.", "REF .", "We can see that for some objects we should indeed assume $\\alpha <2$ heavy tails, especially: 3C 273, 3C 454.3, PKS 1156, AO 235+164, ON +325.", "Then there was performed normalization using cumulative distribution functions (CDF) of the found distributions: assuming a given sequence $(\\ln (x_t))$ has lead to $(\\alpha ,\\beta ,\\mu ,\\sigma )$ MLE parameters, there was calculated $y_t = \\textrm {CDF}_{\\alpha \\beta \\mu \\sigma } (\\ln (x_t))$ sequence, which would be from uniform distribution on $[0,1]$ if $(\\ln (x_t))$ were exactly from this stable distribution.", "Beside log-likelihood tests, there was also performed visual evaluation if such normalized variables $(y_t)$ are from nearly uniform distribution: by sorting them (empirical distribution) and comparing with diagonal - which would be obtained for uniform distribution.", "Figure REF presents such visual evaluation, where we can see a relatively good agreement, especially at the boundaries corresponding to tails.", "In this Figure there are also written $\\alpha $ and $\\beta $ parameters of the found stable distributions.", "For most of sequences we got $\\alpha =2$ , what means that indeed log-normal distribution has turned out the bast choice.", "However, a few last sequences (they were ordered by $\\alpha $ ) obtained lower $\\alpha $ in this ML estimation, suggesting heavier tails.", "Parameters of such ML estimation can be treated as features of objects e.g.", "for classification purposes, especially this $\\alpha $ parameter defining type of tail of distribution.", "The normalized sequences $(y_t)$ are later presented in Fig.", "REF as dots - where we can see that in horizontal direction they have nearly uniform distribution.", "However, local density evolves in vertical direction corresponding to time, what is considered in non-stationarity analysis." ], [ "Modelling non-stationarity with polynomials of evolving contribution", "After normalization, $(y_t)_{t=1..n}$ variables are from nearly uniform distributions.", "Here we would like to model distortion from this uniform distribution, like its evolution for non-stationarity analysis, by representing this density as polynomial and modelling its coefficients as discussed in [10].", "For this purpose we could model joint distribution of $\\lbrace (y_t,t)\\rbrace $ pairs, with times $t$ rescaled to $[0,1]$ range, and predict $\\rho (y|t)$ conditional distributions using polynomial model for their joint distributions.", "There were performed 10-fold cross-validation tests of log-likelihood for such approach, but has led to inferior evaluation than further adaptive approach, hence we will focus only on adaptive approach here, especially that it also provides evaluation of non-stationarity of the sequences.", "We would like to model distortion from uniform density on $[0,1]$ (for normalized variables) as linear combinations using some basis $\\lbrace f_j:j\\in B\\rbrace $ , $B^+=B\\backslash \\lbrace 0\\rbrace $ , $f_0=1$ : $\\rho (x) = \\sum _{j\\in B} a_j\\ f_j(x)= 1 +\\sum _{j\\in B^+} a_j\\ f_j(x)$ as discussed in [10], these coefficients have similar interpretations as moments: $a_1$ as expected value, $a_2$ as variance, $a_3$ as skewness, $a_4$ as kurtosis, etc.", "Using orthornomal family of functions $\\int _0^1 f_i(x) f_j(x) dx=\\delta _{ij}$ , mean-square error estimation is given by just averages of functions over the data sample $(y_i)_{i=1..n}$ : $a_j = \\frac{1}{n}\\sum _{i=1}^n f_j(x_i)$ There were tested various orthornormal families like trigonometric, and generally the best results were obtained for (rescaled Legendre) polynomials - $f_0,f_1,f_2,f_3,f_4,f_5$ are correspondingly: $ 1,\\sqrt{3}(2x-1), \\sqrt{5}(6x^2-6x+1), \\sqrt{7}(20x^3-30x^2+12x-1),(70x^4-140x^3+90x^2-20x+1)$ .", "For adaptivity we can replace average in (REF ) with exponential moving average for some $\\eta \\in (0,1)$ forgetting rate: $a_j(t+1)=\\eta \\, a_j(t) + (1-\\eta )f_j(x_t)=a_j(t) + (1-\\eta ) (f_j(x_t)-a_j(t))$ estimating density $\\rho _t(x)= \\sum _{j\\in B} a_j(t)\\ f_j(x)$ for a given time $t$ based only on previous values, with exponentially weakening weights $\\propto \\eta ^{\\Delta t}$ for value $\\Delta t$ time ago.", "There remains a difficult question of choosing this $\\eta $ rate, defining strength of updates, which generally could also evolve.", "To find the optimal $\\eta $ (fixed here), there was searched space of $\\eta =0,0.01,\\ldots ,1$ for fixed basis $B=\\lbrace 0,1,2,3,4\\rbrace $ , evaluating log-likelihood: average $\\ln (\\rho _t(y_t))$ for $\\rho _t(y)=\\sum _j a_j(t)\\, f_j(y)$ .", "However, the problem is that such $\\rho $ as polynomial sometimes gets below zero, hence we need to reinterpret such negative predicted densities as small positive, what is referred as calibration - there was used log-likelihood as average $\\ln (\\tilde{\\rho }_t(y_t))$ instead, where $\\tilde{\\rho }_t=\\max (\\rho _t,\\epsilon )/N$ and $N$ is normalization constant to integrate to 1, and $\\epsilon $ was arbitrarily chosen as 0.3 here.", "Such search for optimal $\\eta $ using log-likelihood evaluation is presented in Fig.", "REF .", "While in financial time series optimal $\\eta $ is usually close to 1, here it can be very far, suggesting strong non-stationarity - this e.g.", "optimal $\\eta $ can be treated as a feature characterizing non-stationarity of an object.", "The obtained log-likelihoods are relatively large: while stationary $\\rho =1$ density would have log-likelihood 0, here it can go up to $\\approx 0.8$ , corresponding to mean $\\exp (0.8)\\approx 2.2$ times localization in the $[0,1]$ range.", "Finally the predicted evolving densities using optimized $\\eta $ are presented in Fig.", "REF , together with $(y_t)$ points.", "Their values (horizontal direction) average to nearly uniform distribution, however, there are obvious clusters in their time evolution (vertical direction), exploited in discussed adaptive model - with predictions visualized as density.", "Fig.", "REF show time non-uniformity of evaluation of such prediction: $\\tilde{\\rho }_t(y_t)$ sequences (blue points), which smoothing (orange line) can be used to evaluate local variability.", "The discussed approach is optimized for fixed time difference between measured values, what is not exactly true for the analyzed data: we can see in Fig.", "REF that density is constant between succeeding observations.", "Varying time difference could be included e.g.", "by faster modification (lower $\\eta $ ) for longer time differences, but such attempts did not lead to essential improvement of evaluation, hence are not presented.", "We could also use separate $\\eta _j$ for each parameter $a_j$ and optimize them individually (also vary in time), modify basis size and $\\epsilon $ in calibration - there were performed such initial tests, but obtained improvement was nearly negligible, hence they are omitted here for simplicity.", "The presented results also did not include errors of values.", "They could be included e.g.", "by replacing values with discretized sets of values weighted with probabilities ($f_j(x)\\rightarrow \\sum \\textrm {Pr}(x) f_j(x)$ ), but its contribution was also nearly negligible here.", "Figure: Non-stationarity evaluation: while the found log-stable distribution is average over a long period, local probability distribution might evolve in time.", "For normalized variable yy there were found evolving in time a 1 ,a 2 ,a 3 ,a 4 a_1,a_2,a_3,a_4 coefficients using exponential moving average: a j (t+1)=ηa j (t)+(1-η)f j (y t )a_{j}(t+1)=\\eta \\, a_{j}(t)+(1-\\eta ) f_j(y_t).The plots show log-likelihood dependence from η\\eta , without prediction log-likelihood would be zero - we can see that in all but two cases (W Comae, and ON +325), we can essentially increase log-likelihood by adapting parameters.", "Moreover, while e.g.", "in financial time series usually η>0.99\\eta >0.99, here the optimal one can be much smaller - corresponding to extremely fast forgetting of these systems.", "Both the optimal η\\eta and corresponding log-likelihood can be used as features of the object describing non-stationarity for example for classification purposes." ], [ "Autocorrelation analysis", "There was also performed autocorrelation analysis for $(y_t)_{t=1..n}$ series using polynomial $(f_j)$ basis.", "We look at pairs $(y_t,y_{t+l})$ shifted by lag $l$ , up to maximal lag $m$ which was chosen here as $m=100$ : $P_l=\\lbrace (y_t,y_{t+l}): \\textrm { values in }t, t+l\\textrm { are available}\\rbrace \\qquad \\textrm {for}\\qquad l=1,\\ldots ,m$ The available data has regular time difference (7 days), however, some values are missing.", "Hence there were used all available pairs with chosen lag, such sets of pairs usually have size varying with lag (usually decreasing).", "If uncorrelated, thanks to normalization, these pairs would be from nearly uniform $\\rho =1$ joint distribution on $[0,1]^2$ .", "We would like to model distortion from this uniform distribution using polynomial basis - let us start with product basis of orthornormal polynomials $(f_j(y)\\cdot f_k(z))_{(j,k)\\in B}$ : $\\rho _l(y,z)=1+\\sum _{(j,k)\\in B^+} a_{jk}(l)\\ f_j(y)\\, f_k(z)$ thanks to orthonormality we can use MSE estimation as in (REF ): $a_{jk}(l)=\\frac{1}{|P_l|}\\sum _{(y,z)\\in P_l} f_j(y)\\,f_k(z)$ As $f_0=1$ , coefficients $a_{j0}$ describe marginal distributions of the first variable - averaged over the second variable.", "Coefficients $a_{0k}$ marginal distribution of the second variable.", "$a_{11}$ is approximately dependence between their expected values - has similar interpretation as correlation coefficient.", "Further $a_{kl}$ coefficients can be seen as higher mixed moments: describe dependence between $j$ -th moment of the first variable and $k$ -th moment of the second variable.", "Their direct interpretation is through $f_j(y)f_k(z)$ density, some of which are presented in third row of Fig.", "REF .", "Top row of Fig.", "REF contains examples of such pairs for “3C 66A\" sequence.", "Second row adds isolines of joint density modeled in discussed way - using $B=\\lbrace (j,k):j,k=0,\\ldots ,4\\rbrace $ polynomial basis.", "Third row presents $f_j(y)f_k(z)$ densities for some $(j,k)$ - correspondingly: $(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)$ .", "Fourth row shows lag $l$ dependence $a_{jk}(l)$ for these presented 6 coefficients.", "The last two rows try to improve the above arbitrarily chosen basis with feature extraction using PCA (principal component analysis) over lag $l$ .", "Specifically, for each object we have $|B|=25$ sequences for $m=100$ lags.", "Averaging over lags we can find $C_{j,k}$ covariance matrix $25\\times 25$ and look at its few eigenvectors corresponding to the highest eigenvalues: $C v = \\lambda v$ .", "Then define new basis $f_v = v\\cdot (f_{jk}:(j,k)\\in B)$ and corresponding sequence $a_v(l) = v\\cdot (a_{jk}(l):(j,k)\\in B)$ over lag $l$ .", "For 3 highest eigenvalues there are presented such $f_v$ densities and $a_v$ sequences: of size of contribution of $f_v$ to joint density of $(y_t,y_{t+l})$ .", "However, as discussed, these are highly non-stationary time series.", "Wanting to focus here on statistical dependencies of values shifted by $l$ , it would be beneficial to try to remove contribution of non-stationarity.", "There are many ways to realize it, for example we could use modelled evolved density from non-stationarity analysis for additional normalization, but such analysis would be model dependent.", "There was used simpler more unequivocal approach instead - subtracting contribution of (evolving) marginal distributions from mixing terms: $\\tilde{a}_{jk}=a_{jk}-a_{j0}\\,a_{0k}\\qquad \\textrm {for}\\quad k,j>0$ then we can analogously perform PCA on $\\tilde{a}_{jk}(l)$ , leading to bottom row in Fig.", "REF and then analogously for all 20 objects in Fig.", "REF and REF .", "This way we get a few lag dependencies for each object, hopefully nearly independent thanks to PCA.", "As we can see in Fig.", "REF , REF they are quite complicated, but often have clear minima-maxima structure, which could correspond to some characteristic time differences.", "Their deeper analysis might be quite complicated and is planned for future work, for example trying to fit it with such dependence for coupled pendulums.", "A first suggestion is that periodic processes should have alternating maxima/minima in fixed distances.", "Green lines show results of such manual attempts, to be improved in a future.", "The blazar $\\gamma $ -ray flux time series were fitted with general log-stable distribution parameterized by 4 parameters: location, variance, stability and asymmetry parameters.", "The maximum likelihood estimation for most of the sources result in $\\alpha $ =2, that indicates log-normal distribution consistent with our previous result in [1].", "This suggests that the observed variability is shaped by the multiplicative processes in contrast to additive processes.", "The log-normal flux distribution indicate strong disk-jet connection in the sources, in the sense that the imprints of disk related variability can make their way into the jet, propagate along and finally detected by the observer.", "The analysis also revealed $\\alpha <2$ heavy tails for some sources ($\\rho (x)\\sim |x|^{-\\alpha -1}$ ), again pointing to multiplicative process of infinite variance.", "This could be an important result that provides insights into the nature of $\\gamma $ -ray production in blazar, with an implication that the flux contributing to the higher end of the heavier tail of the PDF is probably contributed by large amplitude flaring events that could of distinguished origin compared to the origin of the lower amplitude fluxes.", "Another important result that the work has revealed is that, although the PDF is log-stable distribution over a long period, it displayed transient non-stationarity suggesting the processes linked to the origin of variability are fast forgetting.", "Such fast changing PFDs can be expected in the turbulent jet scenario [23].", "Novel auto-correlation analysis of lag dependence of multiple mixed moments shows complex minima/maxima structures; the timescales corresponding to the extremum, typically in the order of a few months, can be interpreted as some characteristic timescales associated with the jet processes.", "These timescales could be driven by the accretion disk related timescales e.g.", "dynamical, thermal, viscous timescales [8] in an AGN with a central black hole of mass in the order of$\\sim 10^{8}-10^{9} M_{\\odot }$ ; however the timescales could be altered by the jet Lorentz factors.", "Additionally, the observed timescales can also be linked to Rayleigh-Taylor and Kelvin-Helmholtz instabilities developing at a disk-magnetosphere interface [17], or non-thermal (e.g.", "synchrotron and inverse-Compton) cooling timescales of the accelerated charged particle in the jet.", "Apart from aperiodic timescales, various jet and accretion disk related instabilities can set up (quasi-) periodic oscillations as observed in the multi-frequency light curves of several blazars [2] GB acknowledges the financial support by the Narodowe Centrum Nauki (NCN) grant UMO-2017/26/D/ST9/01178.", "Figure: Using η\\eta forgetting rate maximizing log-likelihood from Fig.", ", there is presented calculated evolution of density of normalized variables yy.", "Horizontal direction is [0,1][0,1] range (with averaged nearly uniform density), vertical is time from bottom to top (2008-2018, synchronized for all objects).", "There are missing some values - leading to constant predicted densities between obtained values here.", "Observe that for some we can directly see oscillations.Figure: Blue points: ln(ρ ˜ t (y t ))\\ln (\\tilde{\\rho }_t(y_t)) in horizontal direction for evolution from Fig.", "in vertical direction.", "They average to log-likelihoods (maximal in Fig.", "), additionally allowing to evaluate local time variability.", "Orange lines present their smoothing with rate 0.9 exponential moving average, it can interpreted as local agreement with the found model, its rapid decreases can be interpreted as anomalies.Figure: Example of discussed autocorrelation analysis for \"3C 66A\" object after normalization to nearly uniform variables y t = CDF (ln(x t ))y_t=\\textrm {CDF}(\\ln (x_t)).", "Row 1: (y t ,y t+l )(y_t, y_{t+l}) pairs for various lags ll.", "Row 2: their modeled joint distribution as ρ(y,z)=1+∑ jk a jk f j (y)f k (z)\\rho (y,z)=1+\\sum _{jk} a_{j k}\\, f_{j}(y) f_k(z).", "Row 3: densities (orange - positive, blue - negative) of some used f j (y)f k (z)f_{j}(y)f_{k}(z) functions from basis of orthonormal polynomials - for (j,k)=11,12,21,22,33,44(j,k)=11,12,21,22,33,44.", "Row 4: their found corresponding coefficients a jk (l)a_{j k}(l) for various lags l=1,...,100l=1,\\ldots ,100.", "Row 5: sequences for basis found with principal component analysis (PCA) of all such sequences, densities of corresponding functions, and their eigenvalues.", "Row 6: as in row 5, but with earlier removal of contribution of marginal distributions a ˜ jk =a jk -a j0 a 0k \\tilde{a}_{jk}=a_{jk}-a_{j0}\\,a_{0k}, getting clearer signal only from dependencies between values shifted by lag ll.", "This way we get decorrelated multiple lag dependencies for each object, there was presented manual attempt of fitting alternating minima-maxima suggesting periodic process.Figure: Final plots for the first 10 sequences - PCA with removed marginals as in Row 6 of Fig.", ".", "Especially clear minima and maxima in lag dependence can be interpreted as characteristic times of given object, with statistical interpretations presented in corresponding perturbations to joint densities ρ(y t ,y t+l )≈1+∑ v a v (l)f v (y t ,y t+l )\\rho (y_t,y_{t+l})\\approx 1 +\\sum _v a_v(l) f_v(y_t,y_{t+l}).", "Green lattice present attempts to manually deduce periodic processes as alternating minima/maxima in fixed distances, to be improved in the future.Figure: Final plots for the last 10 sequences - PCA with removed marginals as in Row 6 of Fig.", "." ] ]

[ [ "Scale invariant regularity estimates for second order elliptic equations\n with lower order coefficients in optimal spaces" ], [ "Abstract We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\\rm div}(A\\nabla u+bu)+c\\nabla u+du=-{\\rm div}f+g$, assuming that $A$ is elliptic and bounded.", "In the setting of Lorentz spaces, under the assumptions $b,f\\in L^{n,1}$, $d,g\\in L^{\\frac{n}{2},1}$ and $c\\in L^{n,q}$ for $q\\leq\\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with \"good\" constants (that is, depending only on the norms of the coefficients) do not hold in general.", "On the other hand, assuming a necessary smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's estimate for subsolutions with \"good\" constants.", "We also show the reverse Moser estimate for nonnegative supersolutions with \"good\" constants, under no smallness assumptions when $q<\\infty$, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions.", "Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates." ], [ "Introduction", "In this article we are interested in local and global regularity for subsolutions and supersolutions to the equation $\\mathcal {L}u=-\\operatorname{div}f+g$ , in domains $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ , where $\\mathcal {L}$ is of the form $\\mathcal {L}u=-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du.$ In particular, we investigate the validity of the maximum principle, Moser's estimate, the Harnack inequality and continuity of solutions, in a scale invariant setting; that is, we want our estimates to not depend on the size of $\\Omega $ .", "We will also assume throughout this article that $n\\ge 3$ .", "In this work $A$ will be bounded and uniformly elliptic in $\\Omega $ : for some $\\lambda >0$ , $\\left<A(x)\\xi ,\\xi \\right>\\ge \\lambda \\Vert \\xi \\Vert ^2,\\quad \\forall x\\in \\Omega ,\\,\\,\\,\\forall \\xi \\in \\operatorname{\\mathbb {R}}^n.$ For the lower order coefficients and the terms on the right hand side, we consider Lorentz spaces that are scale invariant under the natural scaling for the equation.", "That is, we assume that $b,f\\in L^{n,1}(\\Omega ),\\quad c\\in L^{n,q}(\\Omega ),\\quad d,g\\in L^{\\frac{n}{2},1}(\\Omega ),\\quad q\\le \\infty .$ In the case that $q=\\infty $ , it is also necessary to assume that the norm of $c$ is small for our results to hold.", "As explained in Section , these assumptions are the optimal ones to imply our estimates in the setting of Lorentz spaces.", "Note also that there will be no size assumption on $\\Omega $ and no regularity assumption on $\\partial \\Omega $ .", "The main inspiration for this work comes from the local and global pointwise estimates for subsolutions to the fore mentioned operator in [28], where it is also assumed that $d\\ge \\operatorname{div}c$ in the sense of distributions.", "Focusing on the case when $c,d\\equiv 0$ for simplicity, and assuming that $b\\in L^{n,1}$ , a maximum principle for subsolutions to $-\\operatorname{div}(A\\nabla u+bu)\\le -\\operatorname{div}f+g$ is shown in [28], while a Moser type estimate is the context of [28].", "The main feature of these estimates is their scale invariance, with constants that depend only on the ellipticity of $A$ and the $L^{n,1}$ norm of $b$ , as well as the $L^{\\infty }$ norm of $A$ for the Moser estimate.", "Following this line of thought, it could be expected that the consideration of all the lower order coefficients in the definition of $\\mathcal {L}$ should yield the same type of scale invariant estimates, with constants being “good\"; that is, depending only on $n$ , $q$ , the ellipticity of $A$ , and the norms of the coefficients involved (as well as $\\Vert A\\Vert _{\\infty }$ in some cases).", "However, it turns out that this does not hold.", "In particular, if $B_1$ is the unit ball in $\\operatorname{\\mathbb {R}}^n$ , in Proposition REF we construct a bounded sequence $(d_N)$ in $L^{\\frac{n}{2},1}(B_1)$ and a sequence $(u_N)$ of nonnegative $W_0^{1,2}(B_1)$ solutions to the equation $-\\Delta u_N+d_Nu_N=0$ in $B_1$ , such that $\\Vert u_N\\Vert _{W_0^{1,2}(B_1)}\\le C,\\quad \\text{while}\\quad \\Vert u_N\\Vert _{L^{\\infty }(B_{1/2})}\\xrightarrow[N\\rightarrow \\infty ]{}\\infty .$ We also show in Remark REF that the equation $-\\Delta u-\\operatorname{div}(bu)+c\\nabla u=0$ has the same feature, which implies that the constants in Moser's local boundedness estimate, as well as the Harnack inequality, cannot be “good\" without any further assumptions.", "Since scale invariant estimates with “good\" constants do not hold in such generality, we first prove estimates where the constants are allowed to depend on the coefficients themselves.", "This is the context of the global bound in Proposition REF , where it is shown that, if $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ is a domain and $u\\in Y^{1,2}(\\Omega )$ (see (REF )) is a subsolution to $\\mathcal {L}u\\le -\\operatorname{div}f+g$ , then, for any $p>0$ , $\\sup _{\\Omega }u^+\\le C\\sup _{\\partial \\Omega }u^++C^{\\prime }\\left(\\int _{\\Omega }|u^+|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{n,1}+C\\Vert g\\Vert _{\\frac{n}{2},1},$ where $C$ is a “good\" constant, while $C^{\\prime }$ depends on the coefficients themselves and $p$ .", "Note the appearance of a constant in front of the term $\\sup _{\\partial \\Omega }u^+$ ; such a constant can be greater than 1, and this follows from the fact that constants are not necessarily subsolutions to our equation in the generality of our assumptions.", "Having proven the previous estimate, we then turn to show various scale invariant estimates with “good\" constants, assuming an extra condition on the lower order coefficients, which is necessary in view of the fore mentioned discussion.", "Such a condition is some type of smallness: in particular, we either assume that the norms of $b,d$ are small, or that the norms of $c,d$ are small.", "Under these smallness assumptions, we show in Propositions REF and REF that we can take $C^{\\prime }=0$ in (REF ), leading to a maximum principle, and the Moser estimate for subsolutions to $\\mathcal {L}u\\le -\\operatorname{div}f+g$ is shown in Propositions REF and REF ; that is, in the case when $b,d$ are small, or $c,d$ are small, then for any $p>0$ , $\\sup _{B_r}u\\le C\\left(_{B_{2r}}|u^+|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where the constant $C$ is “good\", and also depends on $p$ .", "In addition, the analogous estimate close to the boundary is deduced in Propositions REF and REF .", "On the other hand, somewhat surprisingly, we discover that even if the scale invariant Moser estimate with “good\" constants requires some type of smallness, it turns out that the scale invariant reverse Moser estimate with “good\" constants holds in the full generality of our initial assumptions.", "That is, in Proposition REF , we show that if $u\\in W^{1,2}(B_{2r})$ is a nonnegative supersolution to $\\mathcal {L}u\\ge -\\operatorname{div}f+g$ , and under no smallness assumptions (when $q<\\infty $ ), then for some $\\alpha =\\alpha _n$ , $\\left(_{B_r}u^{\\alpha }\\right)^{\\frac{1}{\\alpha }}\\le C\\inf _{B_{r/2}}u+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ is a “good\" constant.", "Moreover, the analogue of this estimate close to the boundary is deduced in Proposition REF .", "Then, the Harnack inequality (Theorems REF and REF ) and continuity of solutions (Theorems REF and REF ) are shown combining (REF ) and (REF ); for those, in order to obtain estimates with “good\" constants, it is again necessary to assume a smallness condition.", "Finally, having shown the previous estimates, we also obtain their analogues in the generality of our initial assumptions, with constants that depend on the coefficients themselves (Remarks REF , REF and REF ).", "As a special case, we remark that all the scale invariant estimates above hold, with “good\" constants, in the case of the operators $\\mathcal {L}_1u=-\\operatorname{div}(A\\nabla u)+c\\nabla u,\\qquad \\mathcal {L}_2u=-\\operatorname{div}(A\\nabla u+bu),$ under no smallness assumptions when $b\\in L^{n,1}$ and $c\\in L^{n,q}$ , $q<\\infty $ ." ], [ "The techniques", "The assumption that the coefficients $b,d$ lie in scale invariant spaces is reflected in the fact that the classical method of Moser iteration does not seem to work in this setting.", "More specifically, an assumption of the form $b\\in L^{n,q}$ for some $q>1$ does not necessarily guarantee pointwise upper bounds (see Remark REF ), and it is necessary to assume that $b\\in L^{n,1}$ in order to deduce these bounds.", "However, Moser's method does not seem to be “sensitive\" enough to distinguish between the cases $b\\in L^{n,1}$ and $b\\in L^{n,q}$ for $q>1$ .", "Thus, a procedure more closely related to Lorentz spaces has to be followed, and the first results in this article (Section ) are based on a symmetrization technique, leading to estimates for decreasing rearrangements.", "This technique involves a specific choice of test functions and has been used in the past by many authors, going back to Talenti's article [33]; here we use a slightly different choice, utilized by Cianchi and Mazya in [12].", "However, since all the lower order coefficients are present, our estimates are more complicated, and we have to rely on an argument using Grönwall's inequality (as in [5], for example) to give a bound on the decreasing rearrangement of our subsolution.", "On the other hand, the main drawback of the symmetrization technique is that it does not seem to work well when we combine it with cutoff functions; thus, we are not able to suitably modify it in order to directly show local estimates like (REF ).", "The idea to overcome this obstacle is to pass from small to large norms using a two-step procedure (in Section ), utilizing the maximum principle.", "Thus, relying on Moser's estimate for the operator $\\mathcal {L}_0u=-\\operatorname{div}(A\\nabla u)+c\\nabla u$ when the norm of $c$ is small, the first step is a perturbation argument based on the maximum principle that allows us to pass to the operator $\\mathcal {L}$ when all the lower order terms have small norms.", "Then, the second step is an induction argument relying on the maximum principle (similar to the proofs of [28]), which allows us to pass to arbitrary norms for $b$ or $c$ .", "To the best of our knowledge, the combination of the symmetrization technique with the fore mentioned argument in order to obtain local estimates has not appeared in the literature before (with the exception of [28], which used estimates on Green's function), and it is one of the novelties of this article.", "Since we do not obtain Moser's estimate (REF ) using test functions and Moser's iteration, in order to deduce the reverse Moser estimate (REF ) we transform supersolutions to subsolutions via exponentiation (in Section ).", "The advantage of this procedure is that, if the exponent is negative and close to 0, we obtain a subsolution to an equation with the coefficients $b,d$ being small, thus we can apply (REF ) to obtain a scale invariant estimate with “good\" constants, without any smallness assumptions (when $q<\\infty $ ).", "This estimate has negative exponents appearing on the left hand side, and we show (REF ) passing to positive exponents using an estimate for supersolutions and the John-Nirenberg inequality (as in [24]).", "One drawback of this technique is that we do not obtain the full range $\\alpha \\in (0,\\frac{n}{n-2})$ for the left hand side, as in [19], but this does not affect the proof of the Harnack inequality.", "Then, the Harnack inequality and continuity of solutions are deduced combining (REF ) and (REF ).", "Finally, the optimality of our assumptions is shown in Section .", "In particular, the sharpness of our spaces to guarantee some type of estimates (either having “good\" constants, or not) is shown, and the failure of scale invariant estimates with “good\" constants is exhibited by the construction in Proposition REF .", "The first fundamental contribution to regularity for equations with rough coefficients was made by De Giorgi [13] and Nash [26] and concerned Hölder continuity of solutions to the operator $-\\operatorname{div}(A\\nabla u)=0$ ; a different proof, based on the Harnack inequality, was later given by Moser in [24].", "The literature concerning this subject is vast, and we refer to the books by Ladyzhenskaya and Ural'tseva [23] and Gilbarg and Trudinger [19], as well as the references therein, for equations that also have lower order coefficients in $L^p$ .", "However, in these results, the norms of those spaces are not scale invariant under the natural scaling of the equation, so it is not possible to obtain scale invariant estimates without extra assumptions on the coefficients (like smallness, for example).", "One instance of a scale invariant setting where $b,d,f,g\\equiv 0$ and $c\\in L^n$ was later treated by Nazarov and Ural'tseva in [27].", "Another well studied case of coefficients is the class of Kato spaces.", "The first work on estimates for Schrödinger operators with the Laplacian and potentials in a suitable Kato class was by Aizenman and Simon in [2] using probabilistic techniques, which was later generalized (with nonprobabilistic techniques) by Chiarenza, Fabes and Garofalo in [11], allowing a second order part in divergence form.", "The case in [2] was also later treated using nonprobabilistic techniques by Simader [29] and Hinz and Kalf [20].", "In these works, $b,c\\equiv 0$ , while $d$ is assumed to belong to $K_n^{\\operatorname{loc}}(\\Omega )$ , which is comprised of all functions $d$ in $\\Omega $ such that $\\eta _{\\Omega _1,d}(r)\\rightarrow 0$ as $r\\rightarrow 0$ , for all $\\Omega _1$ compactly supported in $\\Omega $ , where $\\eta _{\\Omega ,d}(r)=\\sup _{x\\in \\operatorname{\\mathbb {R}}^n}\\int _{\\Omega \\cap B_r(x)}\\frac{|d(y)|}{|x-y|^{n-2}}\\,dy$ (or, in some works, the supremum is considered over $x\\in \\Omega $ ).", "Moreover, adding the drift term $c\\nabla u$ , regularity estimates for $c$ in a suitable Kato class were shown by Kurata in [22].", "From Hölder's inequality (see (REF )), if $d\\in L^{\\frac{n}{2},1}(\\Omega )$ , we have that $\\eta _{\\Omega ,d}(r)\\le C_n\\sup _{x\\in \\operatorname{\\mathbb {R}}^n}\\Vert d\\Vert _{L^{\\frac{n}{2},1}(\\Omega \\cap B_r(x))}\\xrightarrow[r\\rightarrow 0]{} 0,$ therefore $L^{\\frac{n}{2},1}(\\Omega )\\subseteq K_n^{\\operatorname{loc}}(\\Omega )$ ; that is, the class of Lorentz spaces we consider in this work is weaker than the Kato class.", "However, the constants in the results involving Kato classes depend on the rate of convergence of the function $\\eta $ defined above to 0, leading to different constants than the ones that we obtain in this article.", "More specifically, let $d\\in L^{\\frac{n}{2},1}(\\operatorname{\\mathbb {R}}^n)$ be supported in $B_1$ , and set $d_M(x)=M^2d(Mx)$ for $M>0$ .", "Then, we can show that $\\eta _{B_1,d_M}(Mr)=\\eta _{B_1,d}(r)$ , thus the functions $\\eta _{B_1,d_M},\\eta _{B_1,d}$ do not converge to 0 at the same rate.", "Hence, the estimates shown using techniques involving Kato spaces, and concerning subsolutions $u_M$ to $-\\Delta u_M+d_Mu_M\\le 0$ in $B_1$ , lead to constants that could blow up as $M\\rightarrow \\infty $ .", "On the other hand, the $L^{\\frac{n}{2},1}(B_1)$ norm of $d_M$ is bounded above uniformly in $M$ , hence the results we prove in this article are not direct consequences of their counterparts involving Kato classes.", "Finally, considering all the lower order terms, Mourgoglou in [25] shows regularity estimates when the coefficients $b,d$ belong to the scale invariant Dini type Kato-Stummel classes (see [25]), and also constructs Green's functions.", "However, the framework we consider in this article for the Moser estimate and Harnack's inequality, as well as our techniques, are different from the ones in [25].", "For example, focusing on the case when $c,d\\equiv 0$ , the coefficient $b$ in [25] is assumed to be such that $|b|^2\\in \\mathcal {K}_{{\\rm Dini},2}$ , which does not cover the case $b\\in L^{n,1}$ , since for any $\\alpha >1$ , the function $b(x)=x|x|^{-2}\\left(-\\ln |x|\\right)^{-a}$ is a member of $L^{n,1}(B_{1/e})$ , while $|b|^2\\notin \\mathcal {K}_{{\\rm Dini},2}(B_{1/e})$ .", "We conclude with a brief discussion on symmetrization techniques.", "Such a technique was used by Weinberger in [34] in order to show boundedness of solutions with vanishing trace to $-\\operatorname{div}(A\\nabla u)=-\\operatorname{div}f$ and $-\\operatorname{div}(A\\nabla u)=g$ , where $f\\in L^p$ and $g\\in L^{\\frac{p}{2}}$ , $p>n$ .", "Another well known technique consists of a use of test functions that leads to bounds for the derivative of the integral of $|\\nabla u|^2$ over superlevel sets of $u$ , where $u$ is a subsolution to $\\mathcal {L}u\\le -\\operatorname{div}f+g$ .", "This bound, combined with Talenti's inequality [33], gives an estimate for the derivative of the decreasing rearrangement of $u$ , leading to bounds for $u$ in various spaces and comparison results.", "This technique has been used by many authors in order to study regularity properties of solutions to second order pdes, some works being [3], [4], [5], [7], [14], [15], [6], [1], [9].", "However, as we mentioned above, to the best of our knowledge, no local boundedness results have been deduced using this method so far.", "We also mention that, in order to treat lower order coefficients, pseudo-rearrangements of functions are also considered the literature, which are derivatives of integrals over suitable sets $\\Omega (s)\\subseteq \\Omega $ (see, for example, [28]).", "On the contrary, in this work we avoid this procedure, and as we mentioned above we rely instead on a slightly different approach, inspired by [12].", "Acknowledgments We would like to thank Professors Carlos Kenig and Andrea Cianchi for useful conversations regarding some parts of this article." ], [ "Definitions", "If $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ is a domain, $W_0^{1,2}(\\Omega )$ will be the closure of $C_c^{\\infty }(\\Omega )$ under the $W^{1,2}$ norm, where $\\Vert u\\Vert _{W^{1,2}(\\Omega )}=\\Vert u\\Vert _{L^2(\\Omega )}+\\Vert \\nabla u\\Vert _{L^2(\\Omega )}.$ When $\\Omega $ has infinite measure, the space $W^{1,2}(\\Omega )$ is not well suited to the problems we consider.", "For this reason, we let $Y_0^{1,2}(\\Omega )$ be the closure of $C_c^{\\infty }(\\Omega )$ under the $Y^{1,2}$ norm, where $\\Vert u\\Vert _{Y^{1,2}(\\Omega )}=\\Vert u\\Vert _{L^{2^*}(\\Omega )}+\\Vert \\nabla u\\Vert _{L^2(\\Omega )},$ and $2^*=\\frac{2n}{n-2}$ is the Sobolev conjugate to 2.", "From the Sobolev inequality $\\Vert \\phi \\Vert _{L^{2^*}(\\Omega )}\\le C_n\\Vert \\nabla \\phi \\Vert _{L^2(\\Omega )},$ for all $\\phi \\in C_c^{\\infty }(\\Omega )$ , we have that $Y_0^{1,2}(\\Omega )=W_0^{1,2}(\\Omega )$ in the case $|\\Omega |<\\infty $ .", "We also set $Y^{1,2}(\\Omega )$ to be the space of weakly differentiable $u\\in L^{2^*}(\\Omega )$ , such that $\\nabla u\\in L^2(\\Omega )$ , with the $Y^{1,2}$ norm.", "If $u$ is a measurable function in $\\Omega $ , we define the distribution function $\\mu _u(t)=\\left|\\left\\lbrace x\\in \\Omega : |u(x)|>t\\right\\rbrace \\right|,\\quad t>0.$ If $u\\in L^p(\\Omega )$ for some $p\\ge 1$ , then $\\mu _u(t)<\\infty $ for any $t>0$ .", "Moreover, we define the decreasing rearrangement of $u$ by $u^*(\\tau )=\\inf \\lbrace t>0:\\mu _u(t)\\le \\tau \\rbrace ,$ as in [18].", "Then, $u^*$ is equimeasurable to $u$ : that is, $\\left|\\left\\lbrace x\\in \\Omega :|u(x)|>t\\right\\rbrace \\right|=\\left|\\left\\lbrace s>0:u^*(s)>t\\right\\rbrace \\right|\\,\\,\\,\\text{for all}\\,\\,\\,t>0.$ Given a function $f\\in L^p(\\Omega )$ , we consider its maximal function $\\mathcal {M}_{f}(\\tau )=\\frac{1}{\\tau }\\int _0^{\\tau }f^*(\\sigma )\\,d\\sigma ,\\quad \\tau >0.$ Let $p\\in (0,\\infty )$ and $q\\in (0,\\infty ]$ .", "If $f$ is a function defined in $\\Omega $ , we define the Lorentz seminorm $\\Vert f\\Vert _{L^{p,q}(\\Omega )}=\\left\\lbrace \\begin{array}{c l} \\displaystyle \\left(\\int _0^{\\infty }\\left(\\tau ^{\\frac{1}{p}}f^*(\\tau )\\right)^q\\frac{d\\tau }{\\tau }\\right)^{\\frac{1}{q}}, & q<\\infty \\\\ \\displaystyle \\sup _{\\tau >0}\\tau ^{\\frac{1}{p}}f^*(\\tau ),& q=\\infty ,\\end{array}\\right.$ as in [18].", "We say that $f\\in L^{p,q}(\\Omega )$ if $\\Vert f\\Vert _{L^{p,q}(\\Omega )}<\\infty $ .", "Then $\\Vert \\cdot \\Vert _{p,q}$ is indeed a seminorm, since $\\Vert f+g\\Vert _{p,q}\\le C_{p,q}\\Vert f\\Vert _{p,q}+C_{p,q}\\Vert g\\Vert _{p,q},$ from [18].", "In addition, from [18], Lorentz spaces increase if we increase the second index, with $\\Vert f\\Vert _{L^{p,r}}\\le C_{p,q,r}\\Vert f\\Vert _{L^{p,q}}\\,\\,\\,\\,\\text{for all}\\,\\,\\,0<p<\\infty ,\\,\\,0<q<r\\le \\infty .$ Hölder's inequality for Lorentz functions states that $\\Vert fg\\Vert _{L^{p,q}}\\le C_{p_1,q_1,p_2,q_2}\\Vert f\\Vert _{L^{p_1,q_1}}\\Vert g\\Vert _{L^{p_2,q_2}},$ whenever $0<p,p_1,p_2<\\infty $ and $0<q,q_1,q_2\\le \\infty $ satisfy the relations $\\frac{1}{p}=\\frac{1}{p_1}+\\frac{1}{p_2}$ , $\\frac{1}{q}=\\frac{1}{q_1}+\\frac{1}{q_2}$ (see [18]).", "If $p\\in (1,\\infty ]$ and $q\\in [1,\\infty )$ , then [32] implies that $\\Vert \\mathcal {M}_f\\Vert _{p,q}\\le C_p\\Vert f\\Vert _{p,q},$ where $\\mathcal {M}_f$ is the maximal function defined in (REF ).", "For a function $u\\in Y^{1,2}$ , we will say that $u\\le s$ on $\\partial \\Omega $ if $(u-s)^+=\\max \\lbrace u-s,0\\rbrace \\in Y_0^{1,2}(\\Omega )$ .", "Moreover, $\\sup _{\\partial \\Omega }u$ will be defined as the infimum of all $s\\in \\mathbb {R}$ such that $u\\le s$ on $\\partial \\Omega $ .", "We now turn to the definitions of subsolutions, supersolutions and solutions.", "For this, let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and let $A$ be bounded in $\\Omega $ , $b,c\\in L^{n,\\infty }(\\Omega )$ , $d\\in L^{\\frac{n}{2},\\infty }(\\Omega )$ and $f,g\\in L^1_{\\operatorname{loc}}(\\Omega )$ .", "If $\\mathcal {L}u=-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du$ , we say that $u\\in Y^{1,2}(\\Omega )$ is a solution to the equation $\\mathcal {L}u=-\\operatorname{div}f+g$ in $\\Omega $ , if $\\int _{\\Omega }A\\nabla u\\nabla \\phi +b\\nabla \\phi \\cdot u+c\\nabla u\\cdot \\phi +du\\phi =\\int _{\\Omega }f\\nabla \\phi +g\\phi ,\\,\\,\\,\\text{for all}\\,\\,\\,\\phi \\in C_c^{\\infty }(\\Omega ).$ Moreover, we say that $u\\in Y^{1,2}(\\Omega )$ is a subsolution to $\\mathcal {L}u\\le -\\operatorname{div}f+g$ in $\\Omega $ , if $\\int _{\\Omega }A\\nabla u\\nabla \\phi +b\\nabla \\phi \\cdot u+c\\nabla u\\cdot \\phi +du\\phi \\le \\int _{\\Omega }f\\nabla \\phi +g\\phi ,\\,\\,\\,\\text{for all}\\,\\,\\,\\phi \\in C_c^{\\infty }(\\Omega ),\\,\\phi \\ge 0.$ We also say that $u$ is a supersolution to $\\mathcal {L}u\\ge -\\operatorname{div}f+g$ , if $-u$ is a subsolution to $\\mathcal {L}(-u)\\le \\operatorname{div}f-g$ ." ], [ "Main lemmas", "We now discuss some lemmas that we will use in the sequel.", "We begin with the following estimate, in which we show that a function in $L^{n,q}$ for $q>1$ fails to be in $L^{n,1}$ by a logarithm, with constant as small as we want.", "This fact will be useful in the proof of Lemma REF .", "Lemma 2.1 Let $f\\in L^{n,q}(\\Omega )$ for some $q\\in (1,\\infty )$ .", "Then, for any $0<\\sigma _1<\\sigma _2<\\infty $ and $\\operatorname{\\varepsilon }>0$ , $\\int _{\\sigma _1}^{\\sigma _2}\\tau ^{\\frac{1}{n}-1}f^*(\\tau )\\,d\\tau \\le \\operatorname{\\varepsilon }\\ln \\frac{\\sigma _2}{\\sigma _1}+C\\Vert f\\Vert _{n,q}^q,$ where $C$ depends on $q$ and $\\operatorname{\\varepsilon }$ .", "Let $p\\in (1,\\infty )$ be the conjugate exponent to $q$ .", "Then, from Hölder's inequality and (REF ), $\\int _{\\sigma _1}^{\\sigma _2}\\tau ^{\\frac{1}{n}-1}f^*(\\tau )\\,d\\tau &=\\int _{\\sigma _1}^{\\sigma _2}\\tau ^{-\\frac{1}{p}}\\tau ^{\\frac{1}{n}-\\frac{1}{q}}f^*(\\tau )\\,d\\tau \\le \\left(\\int _{\\sigma _1}^{\\sigma _2}\\tau ^{-1}\\,d\\tau \\right)^{\\frac{1}{p}}\\left(\\int _{\\sigma _1}^{\\sigma _2}\\tau ^{\\frac{q}{n}-1}f^*(\\tau )^q\\,d\\tau \\right)^{\\frac{1}{q}}\\\\&\\le \\left(p\\operatorname{\\varepsilon }\\ln \\frac{\\sigma _2}{\\sigma _1}\\right)^{\\frac{1}{p}}\\cdot (p\\operatorname{\\varepsilon })^{-\\frac{1}{p}}\\Vert f\\Vert _{n,q}\\le \\operatorname{\\varepsilon }\\ln \\frac{\\sigma _2}{\\sigma _1}+\\frac{(p\\operatorname{\\varepsilon })^{-\\frac{q}{p}}}{q}\\Vert f\\Vert _{n,q}^q,$ where we also used Young's inequality for the last step.", "The following describes the behavior of the Lorentz seminorm on disjoint sets.", "Lemma 2.2 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a set, and let $X,Y$ be nonempty and disjoint subsets of $\\Omega $ .", "If $f\\in L^{p,q}(\\Omega )$ for some $p,q\\in [1,\\infty )$ , then $\\Vert f\\Vert _{L^{p,q}(\\Omega )}^r\\ge \\Vert f\\Vert _{L^{p,q}(X)}^r+\\Vert f\\Vert _{L^{p,q}(Y)}^r,\\quad r=\\max \\lbrace p,q\\rbrace .$ Let $\\mu $ , $\\mu _X$ , $\\mu _Y$ be the distribution functions of $f,f|_X$ and $f|_Y$ , respectively.", "As in [28], we have that $\\mu \\ge \\mu _X+\\mu _Y$ .", "Also, if $p\\ge q$ , then $\\frac{q}{p}\\le 1$ , hence the reverse Minkowski inequality shows that $\\left(\\int _0^{\\infty }(\\mu _X(t)+\\mu _Y(t))^{\\frac{q}{p}}s^{q-1}\\,ds\\right)^{\\frac{p}{q}}\\ge \\left(\\int _0^{\\infty }\\mu _X(t)^{\\frac{q}{p}}s^{q-1}\\,ds\\right)^{\\frac{p}{q}}+\\left(\\int _0^{\\infty }\\mu _Y(t)^{\\frac{q}{p}}s^{q-1}\\,ds\\right)^{\\frac{p}{q}}.$ On the other hand, if $q>p$ , then $\\frac{q}{p}>1$ , hence $a^{\\frac{q}{p}}+b^{\\frac{q}{p}}\\le (a+b)^{\\frac{q}{p}}$ for all $a,b>0$ .", "Therefore, $\\int _0^{\\infty }(\\mu _X(t)+\\mu _Y(t))^{\\frac{q}{p}}s^{q-1}\\,ds\\ge \\int _0^{\\infty }\\mu _X(t)^{\\frac{q}{p}}s^{q-1}\\,ds+\\int _0^{\\infty }\\mu _Y(t)^{\\frac{q}{p}}s^{q-1}\\,ds.$ Then, the proof follows from the expression for the $L^{p,q}$ seminorm in [18].", "The next lemma will be useful in order to reduce to the case $d=0$ .", "Lemma 2.3 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and $d\\in L^{\\frac{n}{2},1}(\\Omega )$ .", "Then there exists a weakly differentiable vector valued function $e\\in L^{n,1}(\\Omega )$ , with $\\operatorname{div}e=d$ in $\\Omega $ and $\\Vert e\\Vert _{L^{n,1}(\\Omega )}\\le C_n\\Vert d\\Vert _{L^{\\frac{n}{2},1}(\\Omega )}$ .", "Extend $d$ by 0 outside $\\Omega $ , and consider the Newtonian potential $v$ of $d$ ; that is, we set $w(x)=C_n\\int _{\\operatorname{\\mathbb {R}}^n}\\frac{d(y)}{|x-y|^{n-2}}\\,dy.$ From [19] we have that $w$ is twice weakly differentiable in $\\Omega $ , and $\\Delta w=d$ .", "Setting $e=\\nabla w$ , we have that $\\operatorname{div}e=d$ .", "Moreover, $|e(x)|=|\\nabla w(x)|\\le C_n\\int _{\\operatorname{\\mathbb {R}}^n}\\frac{|d(y)|}{|x-y|^{n-1}}\\,dy,$ and the estimate follows from the first part of [18].", "The next lemma shows that $u^*$ is locally absolutely continuous, when $u\\in Y^{1,2}$ .", "Lemma 2.4 Let $\\Omega $ be a domain and $u\\in Y_0^{1,2}(\\Omega )$ .", "Then $u^*$ is absolutely continuous in $(a,b)$ , for any $0<a<b<\\infty $ .", "Extending $u$ by 0 outside $\\Omega $ , we may assume that $u\\in Y^{1,2}(\\operatorname{\\mathbb {R}}^n)$ .", "Consider the function $u^*$ defined in [10] (this $u^*$ is not the same as the one in (REF )!", "), and the function $\\tilde{u}(|x|)=u^*(x)$ (as in [10]).", "Then, from the argument for the proof of [28], it is enough to show that $\\tilde{u}$ is locally absolutely continuous in $(0,\\infty )$ .", "To show this, note that the proof of [10] shows that $u^*\\in Y^{1,2}(\\operatorname{\\mathbb {R}}^n)$ whenever $u\\in Y^{1,2}(\\operatorname{\\mathbb {R}}^n)$ (since $Y^{1,2}(\\operatorname{\\mathbb {R}}^n)$ is reflexive, bounded sequences have subsequences that converge weakly, and the rest of the argument runs unchanged).", "Hence, $u^*\\in W^{1,2}_{\\operatorname{loc}}(\\operatorname{\\mathbb {R}}^n)$ , and combining with [10], we obtain that $\\tilde{u}$ is locally absolutely continuous in $(0,\\infty )$ , as in [10], which completes the proof.", "We now turn to the following decomposition, which in similar to [28].", "This will be useful in a change of variables that we will perform in Lemma REF , as well as in the proof of the estimate in Lemma REF .", "Lemma 2.5 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and let $u\\in Y_0^{1,2}(\\Omega )$ .", "Then we can write $(0,\\infty )=G_u\\cup D_u\\cup N_u,$ where the union is disjoint, such that the following hold.", "If $x\\in G_u$ , then $u^*$ is differentiable at $x$ , $\\mu _u$ is differentiable at $u^*(x)$ , and $(u^*)^{\\prime }(x)\\ne 0$ .", "Moreover, $\\mu _u(u^*(x))=x\\quad \\text{and}\\quad \\mu _u^{\\prime }(u^*(x))=\\frac{1}{u^*(x)},\\quad \\text{for all}\\quad x\\in G_u.$ If $x\\in D_u$ , then $u^*$ is differentiable at $x$ , with $(u^*)^{\\prime }(x)=0$ .", "$N_u$ is a null set.", "The proof is the same as the proof of [28], where we use continuity of $u^*$ shown in Lemma REF , instead of [28].", "We now turn to the following lemma, which is based on [12].", "As we mentioned in the introduction, the properties of the function $\\Psi $ defined below will be crucial in the proof of Lemma REF and, using this lemma, we avoid the construction of pseudo-rearrangements (as in [28].", "Lemma 2.6 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain and $u\\in Y_0^{1,2}(\\Omega )$ with $u\\ge 0$ .", "For any $f\\in L^1(\\Omega )$ , the function $R_{f,u}(\\tau )=\\int _{[u>u^*(\\tau )]}|f|$ is absolutely continuous in $(0,\\infty )$ , and if $\\Psi _{f,u}=R^{\\prime }_{f,u}\\ge 0$ is its derivative, then for any $p>1$ and $q\\ge 1$ , $\\Vert \\Psi _{f,u}\\Vert _{L^{p,q}(0,\\infty )}\\le C_{p,q}\\Vert f\\Vert _{L^{p,q}(\\Omega )}.$ Moreover, for almost every $\\tau >0$ , $(-u^*)^{\\prime }(\\tau )\\le C_n\\tau ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|\\nabla u|^2,u}(\\tau )}.$ Let $u^{\\circ }$ be the function defined in [12]; that is, we define $u^{\\circ }(\\tau )=\\sup \\lbrace t^{\\prime }:\\mu _u(t^{\\prime })\\ge \\tau \\rbrace ,$ where $\\mu _u$ coincides with our definition of the distribution function (REF ), since $u\\ge 0$ .", "We will show that $u^*=u^{\\circ }$ , so that $R_{f,u}$ coincides with the function in [12].", "Then, the proof of the same lemma (where for absolute continuity of $u^*$ , we will use Lemma REF ) will show absolute continuity of $R_{f,u}$ , and (REF ) will follow from [12] and (REF ) .", "Note first that, from the definitions, $u^*(\\tau )\\le u^{\\circ }(\\tau )$ for all $\\tau $ .", "If now $u^*(\\tau )<u^{\\circ }(\\tau )$ , then we can find $t<t^{\\prime }$ with $\\mu _u(t)\\le \\tau $ and $\\mu _u(t^{\\prime })\\ge \\tau $ .", "Since $\\mu _u$ is decreasing, this will imply that $\\mu _u(t^{\\prime })\\le \\mu _u(t)$ , hence $\\mu _u$ is equal to $\\tau $ in $[t,t^{\\prime }]$ , which is a contradiction with continuity of $u^*$ from Lemma REF .", "This shows that $u^{\\circ }=u^*$ , and completes the proof of the first part.", "To show estimate (REF ), set $T_u(t)=\\displaystyle \\int _{[u>t]}|\\nabla u|^2$ , and note that, from [33], $C_n\\le \\mu _u(t)^{\\frac{2}{n}-2}(-\\mu _u^{\\prime }(t))\\left(-\\frac{d}{dt}\\int _{[u>t]}|\\nabla u|^2\\right)=C_n\\mu _u(t)^{\\frac{2}{n}-2}(-\\mu _u^{\\prime }(t))(-T_u)^{\\prime }(t),$ for every $t\\in F$ , where $F\\subseteq (0,\\sup _{\\Omega }u)$ has full measure (this estimate is shown for $u\\in W_0^{1,2}(\\Omega )$ , but the same proof as in [33] gives the result for $u\\in Y_0^{1,2}(\\Omega )$ ).", "Consider now the splitting $(0,\\infty )=G_u\\cup D_u\\cup N_u$ in Lemma REF .", "We claim that $u^*(\\tau )\\in F$ for almost every $\\tau \\in G_u$ : if this is not the case, then there exists $G\\subseteq G_u$ , with positive measure, such that if $\\tau \\in G$ , then $u^*(\\tau )\\notin F$ .", "Then, the set $u^*(G)$ has measure zero and $u^*$ is differentiable at every point $\\tau \\in G$ , hence [31] shows that $(u^*)^{\\prime }(\\tau )=0$ for almost every $\\tau \\in G$ .", "However, $u^*(\\tau )\\ne 0$ for every $\\tau \\in G_u$ from Lemma REF , which is a contradiction with the fact that $G$ has positive measure.", "So, $u^*(\\tau )\\in F$ for almost every $\\tau \\in G_u$ , and for those $\\tau $ , plugging $u^*(\\tau )$ in (REF ), we obtain that $C_n\\le \\mu _u(u^*(\\tau ))^{\\frac{2}{n}-2}(-\\mu _u^{\\prime }(u^*(\\tau ))(-T_u)^{\\prime }(u^*(\\tau )),$ and using (REF ), we obtain that $(-u^*)^{\\prime }(\\tau )\\le C_n\\tau ^{\\frac{2}{n}-2}(-T_u)^{\\prime }(u^*(\\tau )),$ for almost every $\\tau \\in G$ .", "Moreover, $R_{|\\nabla u|^2,u}=T_u\\circ u^*$ , and since $T_u$ is differentiable at $u^*(\\tau )$ for almost every $\\tau \\in G_u$ , multiplying the last estimate with $(-u^*)^{\\prime }(\\tau )$ implies that $\\left((-u^*)^{\\prime }(\\tau )\\right)^2\\le C_n\\tau ^{\\frac{2}{n}-2}(-T_u)^{\\prime }(u^*(\\tau ))\\cdot (-u^*)^{\\prime }(\\tau )=C_n\\tau ^{\\frac{2}{n}-2}R_{|\\nabla u|^2,u}^{\\prime }(\\tau ),$ which shows that (REF ) holds for almost every $\\tau \\in G_u$ .", "On the other hand, $(u^*)^{\\prime }(\\tau )=0$ when $\\tau \\in D_u$ , so (REF ) also holds for almost every $\\tau \\in D_u$ .", "Since $N_u$ has measure zero, (REF ) holds almost everywhere in $(0,\\infty )$ , which completes the proof.", "Finally, the following is a Grönwall type lemma, which we prove in the setting that will appear in Lemma REF .", "The reason for this is that the function $g_2g_3$ will not necessarily be integrable close to 0, which turns out to be inconsequential.", "Lemma 2.7 Let $M>0$ , and suppose that $f,g_1,g_2,g_3$ are functions defined in $(0,M)$ , with $g_2,g_3\\ge 0$ .", "Assume that $g_2g_3$ is locally integrable in $(0,M)$ , $g_3f\\in L^1(0,M)$ and $\\exp \\left(-\\int _{\\tau _0}^{\\tau }g_2g_3\\right)g_1(\\tau )g_3(\\tau )\\in L^1(0,M),\\qquad \\exp \\left(\\int _{\\operatorname{\\varepsilon }}^{\\tau _0}g_2g_3\\right)\\int _0^{\\operatorname{\\varepsilon }}g_3f\\xrightarrow[\\operatorname{\\varepsilon }\\rightarrow 0]{}0,$ for some $\\tau _0\\in (0,M)$ .", "If $\\displaystyle f(\\tau )\\le g_1(\\tau )+g_2(\\tau )\\int _0^{\\tau }g_3f$ in $(0,M)$ , then, for every $\\tau \\in (0,M)$ , $f(\\tau )\\le g_1(\\tau )+g_2(\\tau )\\int _0^{\\tau }g_1(\\sigma )g_3(\\sigma )\\exp \\left(\\int _{\\sigma }^{\\tau }g_2(\\rho )g_3(\\rho )\\,d\\rho \\right)\\,d\\sigma .$ Define $G(\\tau )=\\int _0^{\\tau }g_3f$ and $H(\\tau )=\\int _{\\tau _0}^{\\tau }g_2g_3$ , then $G$ is absolutely continuous in $[0,M]$ and $H$ is locally absolutely continuous in $(0,M)$ .", "Then, we have that $(e^{-H}G)^{\\prime }=e^{-H}(G^{\\prime }-H^{\\prime }G)\\le e^{-H}g_1g_3$ , and since $e^{-H}G$ is absolutely continuous in $(\\operatorname{\\varepsilon },\\tau )$ for $0<\\operatorname{\\varepsilon }<\\tau <M$ , we integrate to obtain $e^{-H(\\tau )}G(\\tau )-e^{-H(\\operatorname{\\varepsilon })}G(\\operatorname{\\varepsilon })\\le \\int _{\\operatorname{\\varepsilon }}^{\\tau }e^{-H}g_1g_3.$ The proof is complete after letting $\\operatorname{\\varepsilon }\\rightarrow 0$ and plugging the last estimate in the original estimate for $f$ ." ], [ "The main estimate", "The following lemma is the main estimate that will lead to global boundedness for subsolutions.", "The test function we use comes from [12] and it is a slight modification of test functions that have been used in the literature before (see, for example, the references for the decreasing rearrangements technique in the introduction).", "Lemma 3.1 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain.", "Let $A$ be uniformly elliptic and bounded in $\\Omega $ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(\\Omega )$ and $g\\in L^{\\frac{n}{2},1}(\\Omega )$ .", "There exists $\\nu =\\nu _{n,\\lambda }$ such that, if $c=c_1+c_2\\in L^{n,\\infty }(\\Omega )$ with $c_1\\in L^{n,q}(\\Omega )$ for some $q<\\infty $ and $\\Vert c_2\\Vert _{n,\\infty }<\\nu $ , then for any subsolution $u\\in Y_0^{1,2}(\\Omega )$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u\\le -\\operatorname{div}f+g$ in $\\Omega $ , and any $\\tau \\in (0,1)$ , $\\begin{split}-v^{\\prime }(\\tau )\\le C_1\\tau ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|f|^2}(\\tau )}+C_1\\tau ^{\\frac{2}{n}-1}\\mathcal {M}_g(\\tau )+C_1e^{C_2\\Vert c_1\\Vert _{n,q}^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{1}{n}-\\frac{1}{2}}\\sqrt{\\Psi _{|f|^2}(\\sigma )\\Psi _{|c|^2}(\\sigma )}\\,d\\sigma \\\\+C_1e^{C_2\\Vert c_1\\Vert _{n,q}^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{2}{n}-\\frac{1}{2}}\\mathcal {M}_{g}(\\sigma )\\sqrt{\\Psi _{|c|^2}(\\sigma )}\\,d\\sigma \\\\+C_1v(\\tau )\\tau ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|b|^2}(\\tau )}+C_1e^{C_2\\Vert c_1\\Vert _{n,q}^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{1}{n}-\\frac{1}{2}}v(\\sigma )\\sqrt{\\Psi _{|b|^2}(\\sigma )\\Psi _{|c|^2}(\\sigma )}\\sigma ^{\\frac{1}{n}-\\frac{1}{2}}\\,d\\sigma ,\\end{split}$ where $C_1$ depends on $n,\\lambda $ , $C_2$ depends on $n,\\lambda ,q$ , and where $v=(u^+)^*$ is the decreasing rearrangement of $u^+$ , $\\mathcal {M}_g$ is as in (REF ), and $\\Psi _{|b|^2}=\\Psi _{|b|^2,u^+},\\Psi _{|c|^2}=\\Psi _{|c|^2,u^+}, \\Psi _{|f|^2}=\\Psi _{|f|^2,u^+}$ are defined in Lemma REF .", "Fix $\\tau ,h>0$ , and consider the test function $\\psi =\\left\\lbrace \\begin{array}{l l} 0, & 0\\le u^+\\le v(\\tau +h) \\\\ u-v(\\tau +h), & v(\\tau +h)< u^+\\le v(\\tau ) \\\\ v(\\tau )-v(\\tau +h) & u^+>v(\\tau ).\\end{array}\\right.$ Since $\\psi \\in W_0^{1,2}(\\Omega )$ and $\\psi \\ge 0$ , we can use it as a test function, and from ellipticity of $A$ , $\\lambda \\int _{[v(\\tau +h)<u\\le v(\\tau )]}|\\nabla u|^2\\le v(\\tau )\\int _{[v(\\tau +h)<u\\le v(\\tau )]}|b\\nabla u|+(v(\\tau )-v(\\tau +h))\\int _{[u>v(\\tau +h)]}|c\\nabla u|\\\\+\\int _{[v(\\tau +h)<u\\le v(\\tau )]}|f\\nabla u|+(v(\\tau )-v(\\tau +h))\\int _{[u>v(\\tau +h)]}|g|.$ Letting $\\Psi (\\tau )=\\Psi _{|\\nabla u|^2,u^+}$ (as in Lemma REF ), dividing by $h$ , using the Cauchy-Schwartz inequality and letting $h\\rightarrow 0$ , we obtain that $\\Psi (\\tau )\\le C_{\\lambda }v(\\tau )\\sqrt{\\Psi _{|b|^2}(\\tau )}\\sqrt{\\Psi (\\tau )}+C_{\\lambda }(-v^{\\prime })(\\tau )\\int _{[u>v(\\tau )]}|c\\nabla u|\\\\+C_{\\lambda }\\sqrt{\\Psi _{|f|^2}(\\tau )}\\sqrt{\\Psi (\\tau )}+C_{\\lambda }(-v^{\\prime }(\\tau ))\\int _{[u>v(\\tau )]}|g|,$ where we also used continuity of the functions $R_{|c\\nabla u|,u^+}$ and $R_{|g|,u^+}$ , from Lemma REF .", "Moreover, from absolute continuity of $R_{|c\\nabla u|, u^+}$ and the Cauchy-Schwartz inequality, we obtain $\\int _{[u>v(\\tau )]}|c\\nabla u|=\\int _0^{\\tau }\\Psi _{|c\\nabla u|,u^+}\\le \\int _0^{\\tau }\\sqrt{\\Psi _{|c|^2}}\\sqrt{\\Psi }.$ Let now $\\mu $ be the distribution function of $u^+$ , and consider the decomposition $(0,\\infty )=G_{u^+}\\cup D_{u^+}\\cup N_{u^+}$ from Lemma REF .", "Then, for $\\tau \\in G_{u^+}$ , the Hardy-Littlewood inequality (see, for example, [8]) and (REF ) show that $(-v^{\\prime }(\\tau ))\\int _{[u>v(\\tau )]}|g|\\le (-v^{\\prime }(\\tau ))\\int _0^{\\mu (v(\\tau ))}g^*=(-v^{\\prime }(\\tau ))\\int _0^{\\tau }g^*=\\tau (-v^{\\prime }(\\tau ))\\mathcal {M}_g(\\tau ).$ On the other hand, if $\\tau \\in N_{u^+}$ , then $-v^{\\prime }(\\tau )=0$ , and since $N_{u^+}$ has measure 0, the last estimate holds almost everywhere.", "Hence, plugging the last estimate and (REF ) in (REF ), we obtain that $\\Psi (\\tau )\\le C_{\\lambda }v(\\tau )\\sqrt{\\Psi _{|b|^2}(\\tau )}\\sqrt{\\Psi (\\tau )}+C_{\\lambda }(-v^{\\prime })(\\tau )\\int _0^{\\tau }\\sqrt{\\Psi _{|c|^2}}\\sqrt{\\Psi }\\\\+C_{\\lambda }\\sqrt{\\Psi _{|f|^2}(\\tau )}\\sqrt{\\Psi (\\tau )}+C_{\\lambda }\\tau (-v^{\\prime }(\\tau ))\\mathcal {M}_g(\\tau ).$ Let $\\tau $ such that $\\Psi (\\tau )>0$ .", "Then, dividing the last estimate by $\\sqrt{\\Psi (\\tau )}$ and using (REF ), $\\sqrt{\\Psi (\\tau )}&\\le C_{\\lambda }v(\\tau )\\sqrt{\\Psi _{|b|^2}(\\tau )}+\\frac{C_{\\lambda }(-v^{\\prime })(\\tau )}{\\sqrt{\\Psi (\\tau )}}\\int _0^{\\tau }\\sqrt{\\Psi _{|c|^2}}\\sqrt{\\Psi }+C_{\\lambda }\\sqrt{\\Psi _{|f|^2}(\\tau )}+\\frac{C_{\\lambda }\\tau (-v^{\\prime }(\\tau ))\\mathcal {M}_g(\\tau )}{\\sqrt{\\Psi (\\tau )}}\\\\&\\le C\\sqrt{\\Psi _{|f|^2}(\\tau )}+C\\tau ^{\\frac{1}{n}}\\mathcal {M}_g(\\tau )+Cv(\\tau )\\sqrt{\\Psi _{|b|^2}(\\tau )}+C\\tau ^{\\frac{1}{n}-1}\\int _0^{\\tau }\\sqrt{\\Psi _{|c|^2}}\\sqrt{\\Psi },$ where $C=C_{n,\\lambda }$ .", "On the other hand, the last estimate holds also when $\\Psi (\\tau )=0$ , hence it holds for almost every $\\tau >0$ .", "Note now that, from subadditivity of $\\Psi $ and Lemma REF (since we can assume that $q>1$ ) for any $\\operatorname{\\varepsilon }>0$ , $\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c|^2}(\\rho )}\\,d\\rho &\\le \\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c_1|^2}(\\rho )}\\,d\\rho +\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c_2|^2}(\\rho )}\\,d\\rho \\\\&\\le \\operatorname{\\varepsilon }\\ln \\frac{\\tau }{\\sigma }+C_{q,\\operatorname{\\varepsilon }}\\left\\Vert \\sqrt{\\Psi _{|c_1|^2}}\\right\\Vert _{n,q}^q+\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\left\\Vert \\sqrt{\\Psi _{|c_2|^2}}\\right\\Vert _{n,\\infty }\\rho ^{-\\frac{1}{n}}\\,d\\rho \\\\&\\le \\operatorname{\\varepsilon }\\ln \\frac{\\tau }{\\sigma }+C_{n,q,\\operatorname{\\varepsilon }}\\Vert c_1\\Vert _{n,q}^q+C_n\\Vert c_2\\Vert _{n,\\infty }\\ln \\frac{\\tau }{\\sigma }.$ We choose $\\operatorname{\\varepsilon }=\\operatorname{\\varepsilon }_{n,\\lambda }$ and $\\nu _{n,\\lambda }$ such that $C\\operatorname{\\varepsilon }_{n,\\lambda }+CC_n\\nu _{n,\\lambda }\\le \\frac{1}{2}-\\frac{1}{n}$ ; then, we will have that $\\exp \\left(C\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c|^2}(\\rho )}\\,d\\rho \\right)\\le e^{C_2\\Vert c_1\\Vert _{n,q}^q}\\left(\\frac{\\tau }{\\sigma }\\right)^{\\frac{1}{2}-\\frac{1}{n}},$ where $C_2$ depends on $n,q$ and $\\lambda $ .", "Then, using that $v\\in L^{2^*}(0,\\infty )$ , (REF ) and Lemma REF , it is straightforward to check that the hypotheses of Grönwall's lemma (Lemma REF ) are satisfied, hence we obtain that $\\sqrt{\\Psi (\\tau )}\\le C\\sqrt{\\Psi _{|f|^2}(\\tau )}+C\\tau ^{\\frac{1}{n}}\\mathcal {M}_g(\\tau )+ Cv(\\tau )\\sqrt{\\Psi _{|b|^2}(\\tau )}\\\\+C\\tau ^{\\frac{1}{n}-1}\\int _0^{\\tau }\\sqrt{\\Psi _{|f|^2}(\\sigma )}\\sqrt{\\Psi _{|c|^2}(\\sigma )}\\exp \\left(C\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c|^2}(\\rho )}\\,d\\rho \\right)\\,d\\sigma \\\\+C\\tau ^{\\frac{1}{n}-1}\\int _0^{\\tau }\\sigma ^{\\frac{1}{n}}\\mathcal {M}_g(\\sigma )\\sqrt{\\Psi _{|c|^2}(\\sigma )}\\exp \\left(C\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c|^2}(\\rho )}\\,d\\rho \\right)\\,d\\sigma \\\\+C\\tau ^{\\frac{1}{n}-1}\\int _0^{\\tau }v(\\sigma )\\sqrt{\\Psi _{|b|^2}(\\sigma )}\\sqrt{\\Psi _{|c|^2}(\\sigma )}\\exp \\left(C\\int _{\\sigma }^{\\tau }\\rho ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|c|^2}(\\rho )}\\,d\\rho \\right)\\,d\\sigma ,$ where $C=C_{n,\\lambda }$ .", "Finally, using (REF ) to bound $\\sqrt{\\Psi }$ from below, and (REF ), the proof is complete." ], [ "The maximum principle", "Using Lemma REF , we now show global boundedness of subsolutions.", "Proposition 3.2 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain.", "Let $A$ be uniformly elliptic and bounded in $\\Omega $ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(\\Omega )$ , $d,g\\in L^{\\frac{n}{2},1}(\\Omega )$ , and suppose that $c=c_1+c_2\\in L^{n,\\infty }(\\Omega )$ with $c_1\\in L^{n,q}(\\Omega )$ for some $q<\\infty $ and $\\Vert c_2\\Vert _{n,\\infty }<\\nu $ , where $\\nu =\\nu _{n,\\lambda }$ appears in Lemma REF .", "There exists $\\tau _0\\in (0,\\infty )$ , depending on $b,c_1,c_2$ and $d$ such that, for any subsolution $u\\in Y^{1,2}(\\Omega )$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u\\le -\\operatorname{div}f+g$ we have that $\\sup _{\\Omega }u^+\\le C\\sup _{\\partial \\Omega }u^++Cv(\\tau _0)+C\\Vert f\\Vert _{n,1}+C\\Vert g\\Vert _{\\frac{n}{2},1},$ where $C$ depends on $n,q,\\lambda $ , $\\Vert b\\Vert _{n,1}, \\Vert c_1\\Vert _{n,q}$ and $\\Vert d\\Vert _{\\frac{n}{2},1}$ .", "In particular, for any $p>0$ , $\\sup _{\\Omega }u^+\\le C\\sup _{\\partial \\Omega }u^++C\\tau _0^{-1/p}\\left(\\int _{\\Omega }|u^+|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{n,1}+C\\Vert g\\Vert _{\\frac{n}{2},1}.$ If $s=\\sup _{\\partial \\Omega }u^+\\in (0,\\infty )$ , then for every $s^{\\prime }>s$ , $-\\operatorname{div}(A\\nabla (u-s^{\\prime })+b(u-s^{\\prime }))+c\\nabla (u-s^{\\prime })+d(u-s^{\\prime })\\le -\\operatorname{div}(f-s^{\\prime }b)+g-s^{\\prime }d,$ and $(u-s^{\\prime })^+\\in Y_0^{1,2}(\\Omega )$ ; hence, we can assume that $s=0$ , so $u^+\\in Y_0^{1,2}(\\Omega )$ .", "Consider the function $e$ from Lemma REF that solves the equation $\\operatorname{div}e=d$ in $\\operatorname{\\mathbb {R}}^n$ .", "Then, if we define $b^{\\prime }=b-e$ and $c^{\\prime }=c-e$ , $u$ is a subsolution to $-\\operatorname{div}(A\\nabla u+b^{\\prime }u)+c^{\\prime }\\nabla u\\le -\\operatorname{div}f+g,$ Set $c_1^{\\prime }=c_1-e$ , then $c^{\\prime }=c_1^{\\prime }+c_2$ .", "Let $C_1,C_2$ be the constants in Lemma REF , and denote $C_1e^{C_2\\Vert c_1^{\\prime }\\Vert _{n,q}^q}$ by $C_0$ .", "Moreover, set $\\begin{split}H(\\tau )=C_1\\tau ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|f|^2}(\\tau )}+C_1\\tau ^{\\frac{2}{n}-1}\\mathcal {M}_g(\\tau )+C_0\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{1}{n}-\\frac{1}{2}}\\sqrt{\\Psi _{|f|^2}(\\sigma )\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma \\\\+C_0\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{2}{n}-\\frac{1}{2}}\\mathcal {M}_g(\\sigma )\\sqrt{\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma .\\end{split}$ From Lemma REF , we have that $\\left\\Vert \\sqrt{\\Psi _{|f|^2}}\\right\\Vert _{n,1}\\le C_n\\Vert f\\Vert _{n,1},\\qquad \\left\\Vert \\sqrt{\\Psi _{|c^{\\prime }|^2}}\\right\\Vert _{n,\\infty }\\le C_n\\Vert c^{\\prime }\\Vert _{n,\\infty }.$ Then, since $\\frac{1}{n}-\\frac{3}{2}<-1$ , changing the order of integration and using (REF ) and (REF ), we have $\\int _0^{\\infty }\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{1}{n}-\\frac{1}{2}}\\sqrt{\\Psi _{|f|^2}(\\sigma )\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma \\,d\\tau &\\le C_n\\int _0^{\\infty }\\tau ^{\\frac{2}{n}-1}\\sqrt{\\Psi _{|f|^2}(\\sigma )\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma \\\\&\\le C_n\\Vert c^{\\prime }\\Vert _{n,\\infty }\\Vert f\\Vert _{n,1},$ and also, using (REF ), we obtain that $\\int _0^{\\infty }\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }\\sigma ^{\\frac{2}{n}-\\frac{1}{2}}\\mathcal {M}_{g}(\\sigma )\\sqrt{\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma &\\le C_n\\int _0^{\\infty }\\sigma ^{\\frac{3}{n}-1}\\mathcal {M}_{g}(\\sigma )\\sqrt{\\Psi _{|c^{\\prime }|^2}(\\sigma )}\\,d\\sigma \\,d\\tau \\\\&\\le C_n\\Vert c^{\\prime }\\Vert _{n,\\infty }\\Vert g\\Vert _{\\frac{n}{2},1}.$ The last two estimates and the definition of $H$ in (REF ) imply that $\\int _0^{\\infty }H\\le C(\\Vert c^{\\prime }\\Vert _{n,\\infty }+1)\\left(\\Vert f\\Vert _{n,1}+\\Vert g\\Vert _{\\frac{n}{2},1}\\right),$ where $C$ depends on $n,q,\\lambda $ and $\\Vert c_1^{\\prime }\\Vert _{n,q}$ .", "Set now $R(\\tau )=C_1\\tau ^{\\frac{1}{n}-1}\\sqrt{\\Psi _{|b^{\\prime }|^2}(\\tau )},\\qquad G(\\sigma )=C_1\\sigma ^{\\frac{2}{n}-1}\\sqrt{\\Psi _{|b^{\\prime }|^2}(\\sigma )\\Psi _{|c^{\\prime }|^2}(\\sigma )},$ Then, if $\\Vert c_1^{\\prime }\\Vert =\\Vert c_1^{\\prime }\\Vert _{n,q}$ , (REF ) shows that $-v^{\\prime }(\\tau )\\le H(\\tau )+R(\\tau )v(\\tau )+e^{C_2\\Vert c^{\\prime }_1\\Vert ^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma ,$ as long as $\\Vert c_2\\Vert _{n,\\infty }<\\nu _{n,\\lambda }$ .", "Since also $\\int _0^{\\infty }R\\le C_{n,\\lambda }\\Vert b^{\\prime }\\Vert _{n,1}$ from Lemma REF , we obtain that $\\begin{split}-\\left(e^{\\int _0^{\\tau }R}v\\right)^{\\prime }&=e^{\\int _0^{\\tau }R}\\left(-v^{\\prime }-Rv\\right)\\le e^{\\int _0^{\\tau }R}\\left(H(\\tau )+e^{C_2\\Vert c_1^{\\prime }\\Vert ^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma \\right)\\\\&\\le e^{C_{n,\\lambda }\\Vert b^{\\prime }\\Vert _{n,1}}H(\\tau )+ e^{C_{n,\\lambda }\\Vert b^{\\prime }\\Vert _{n,1}+C_2\\Vert c_1^{\\prime }\\Vert ^q}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\int _0^{\\tau }v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma .\\end{split}$ Set $B=\\exp \\left(C_{n,\\lambda }\\Vert b^{\\prime }\\Vert _{n,1}\\right)$ and $C^{\\prime }=\\exp \\left(C_{n,\\lambda }\\Vert b^{\\prime }\\Vert _{n,1}+C_2\\Vert c_1^{\\prime }\\Vert ^q\\right)$ , and let $\\tau _2>\\tau _1>0$ .", "Then $v$ is absolutely continuous in $(\\tau _1,\\tau _2)$ , from Lemma REF ; hence, integrating (REF ) in $(\\tau _1,\\tau _2)$ , we obtain that $\\begin{split}e^{\\int _0^{\\tau _1}R}v(\\tau _1)&\\le e^{\\int _0^{\\tau _2}R}v(\\tau _2)+B\\int _{\\tau _1}^{\\tau _2}H+C^{\\prime }\\int _{\\tau _1}^{\\tau _2}\\int _0^{\\tau }\\tau ^{\\frac{1}{n}-\\frac{3}{2}}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma d\\tau \\\\&\\le B\\left(v(\\tau _2)+\\Vert H\\Vert _1\\right)+C^{\\prime }\\int _{\\tau _1}^{\\tau _2}\\int _0^{\\tau }\\tau ^{\\frac{1}{n}-\\frac{3}{2}}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma d\\tau .\\end{split}$ Using Fubini's theorem, the last integral is equal to $\\int _0^{\\tau _1}\\int _{\\tau _1}^{\\tau _2}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\tau d\\sigma +\\int _{\\tau _1}^{\\tau _2}\\int _{\\sigma }^{\\tau _2}\\tau ^{\\frac{1}{n}-\\frac{3}{2}}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\tau d\\sigma \\\\\\le C_n\\tau _1^{\\frac{1}{n}-\\frac{1}{2}}\\int _0^{\\tau _1}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma +C_n\\int _{\\tau _1}^{\\tau _2}v(\\sigma )G(\\sigma )\\,d\\sigma ,$ therefore, plugging the last estimate in (REF ), and using that $v$ is decreasing, we obtain $v(\\tau _1)\\le B\\left(v(\\tau _2)+\\Vert H\\Vert _1\\right)+C^{\\prime }C_n\\tau _1^{\\frac{1}{n}-\\frac{1}{2}}\\int _0^{\\tau _1}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma +C^{\\prime }C_nv(\\tau _1)\\int _{\\tau _1}^{\\tau _2}G(\\sigma )\\,d\\sigma .$ Consider now $\\tau _0>0$ such that $\\int _0^{\\tau _0}G\\le \\frac{1}{2C^{\\prime }C_n};$ note that such $\\tau _0$ always exists, since $G\\in L^1(0,\\infty )$ from Lemma REF .", "Then, if $0<\\tau _1\\le \\tau _0$ , setting $\\tau _2=\\tau _0$ and plugging (REF ) in (REF ) we obtain that $v(\\tau _1)\\le 2B\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)+2C^{\\prime }C_n\\tau _1^{\\frac{1}{n}-\\frac{1}{2}}\\int _0^{\\tau _1}v(\\sigma )\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )\\,d\\sigma .$ Then, for $\\tau _1\\in (0,\\tau _0)$ , the hypotheses of Lemma REF are satisfied, and we obtain that $v(\\tau _1)&\\le 2B\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)+2C^{\\prime }C_n\\tau _1^{\\frac{1}{n}-\\frac{1}{2}}\\int _0^{\\tau _1}2B\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)\\sigma ^{\\frac{1}{2}-\\frac{1}{n}}G(\\sigma )e^{2C^{\\prime }C_n\\int _{\\sigma }^{\\tau _1}G}\\,d\\sigma \\\\&\\le 2B\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)+4C^{\\prime }C_nB\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)e^{2C^{\\prime }C_n\\Vert G\\Vert _1}\\int _0^{\\tau _1}G(\\sigma )\\,d\\sigma .$ This estimate holds for every $0<\\tau _1\\le \\tau _0$ , as long as (REF ) holds.", "Then, letting $\\tau _1\\rightarrow 0^+$ , and using the definition of $B$ and Lemma REF , we obtain that $\\lim _{\\tau _1\\rightarrow 0^+}v(\\tau _1)\\le 2B\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right)\\le \\exp \\left(C_{n,\\lambda }\\left(\\Vert b\\Vert _{n,1}+\\Vert d\\Vert _{\\frac{n}{2},1}\\right)\\right)\\left(v(\\tau _0)+\\Vert H\\Vert _1\\right),$ as long as $\\Vert c_2\\Vert _{n,\\infty }<\\nu _{n,\\lambda }$ and (REF ) hold.", "Combining with (REF ) then shows (REF ), and (REF ) follows from the fact that $v$ is decreasing and (REF ).", "As a corollary, we obtain the following maximum principle, which generalizes [28].", "From Remark REF , to have such an estimate with constants depending only on the norms of the coefficients for arbitrary $b\\in L^{n,1}$ requires that $c$ should have small norm; hence, we will assume that $c$ belongs to $L^{n,\\infty }$ and has small norm.", "Proposition 3.3 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain.", "Let $A$ be uniformly elliptic and bounded in $\\Omega $ , with ellipticity $\\lambda $ , and let $b,f\\in L^{n,1}(\\Omega )$ , $g\\in L^{\\frac{n}{2},1}(\\Omega )$ , with $\\Vert b\\Vert _{n,1}\\le M$ .", "There exists $\\beta =\\beta _{n,\\lambda ,M}>0$ such that, if $c\\in L^{n,\\infty }(\\Omega )$ and $d\\in L^{\\frac{n}{2},1}(\\Omega )$ with $\\Vert c\\Vert _{n,\\infty }<\\beta $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\beta $ , then for every subsolution $u\\in Y^{1,2}(\\Omega )$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le -\\operatorname{div}f+g$ in $\\Omega $ , we have that $\\sup _{\\Omega }u\\le C\\sup _{\\partial \\Omega }u^++C\\Vert f\\Vert _{n,1}+C\\Vert g\\Vert _{\\frac{n}{2},1},$ where $C$ depends on $n,\\lambda $ and $M$ .", "Assume that $\\Vert c\\Vert _{n,\\infty }<\\beta $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\beta $ , for $\\beta $ to be chosen later.", "Consider the $\\nu _{n,\\lambda }$ from Lemma REF , and take $c_1\\equiv 0$ and $q=1$ in Proposition REF .", "We will take $\\beta \\le \\nu _{n,\\lambda }$ , so it is enough to show that we can take $\\tau _0=\\infty $ in (REF ), since $\\lim _{\\tau \\rightarrow \\infty }v(\\tau )=0$ .", "Hence, from (REF ), and the definitions of $C^{\\prime }$ and $e$ from the proof of Proposition REF , it will be enough to have that $\\int _0^{\\infty }\\sigma ^{\\frac{2}{n}-1}\\sqrt{\\Psi _{|b-e|^2}(\\sigma )\\Psi _{|c-e|^2}(\\sigma )}\\,d\\sigma \\le C\\exp \\left(-C\\Vert b-e\\Vert _{n,1}-C\\Vert e\\Vert _{n,1}\\right),$ where $C$ depends on $n$ and $\\lambda $ only.", "We first bound the left hand side from above using Lemmas REF and REF , to obtain $\\int _0^{\\infty }\\sigma ^{\\frac{2}{n}-1}\\sqrt{\\Psi _{|b-e|^2}(\\sigma )\\Psi _{|c-e|^2}(\\sigma )}\\,d\\sigma &=\\left\\Vert \\sqrt{\\Psi _{|b-e|^2}\\Psi _{|c-e|^2}}\\right\\Vert _{\\frac{n}{2},1}\\le C\\left\\Vert \\sqrt{\\Psi _{|b-e|^2}}\\right\\Vert _{n,1}\\left\\Vert \\sqrt{\\Psi _{|c-e|^2}}\\right\\Vert _{n,\\infty }\\\\&\\le C\\Vert b-e\\Vert _{n,1}\\Vert c-e\\Vert _{n,\\infty }\\le C(M+\\beta )\\beta ,$ while $-C\\Vert b-e\\Vert _{n,1}-C\\Vert e\\Vert _{n,1}\\ge -C\\Vert b\\Vert _{n,1}-C\\Vert e\\Vert _{n,1}\\ge -CM-C\\beta .$ From the last two estimates, (REF ) will be satisfied as long as $C(M+\\beta )\\beta e^{CM+C\\beta }\\le 1.$ So, choosing $\\beta >0$ depending on $n,\\lambda $ and $M$ , such that the last estimate is satisfied and also $\\beta \\le \\nu _{n,\\lambda }$ completes the proof.", "In addition, we also obtain the following maximum principle, which concerns perturbations of the operator $-\\operatorname{div}(A\\nabla u)+c\\nabla u$ .", "Proposition 3.4 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain.", "Let $A$ be uniformly elliptic and bounded in $\\Omega $ , with ellipticity $\\lambda $ , and let $q<\\infty $ , $c=c_1+c_2\\in L^{n,\\infty }(\\Omega )$ , with $\\Vert c_2\\Vert _{n,\\infty }<\\nu $ and $\\Vert c_1\\Vert _{n,q}\\le M$ , where $\\nu =\\nu _{n,\\lambda }$ appears in Lemma REF .", "Assume also that $f\\in L^{n,1}(\\Omega )$ , $g\\in L^{\\frac{n}{2},1}(\\Omega )$ .", "There exists $\\gamma =\\gamma _{n,q,\\lambda ,M}>0$ such that, if $b\\in L^{n,1}(\\Omega )$ and $d\\in L^{\\frac{n}{2},1}(\\Omega )$ with $\\Vert b\\Vert _{n,1}<\\gamma $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\gamma $ , then for any subsolution $u\\in Y^{1,2}(\\Omega )$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le -\\operatorname{div}f+g$ in $\\Omega $ , we have $\\sup _{\\Omega }u\\le C\\sup _{\\partial \\Omega }u^++C\\Vert f\\Vert _{n,1}+C\\Vert g\\Vert _{\\frac{n}{2},1},$ where $C$ depends on $n,q,\\lambda $ and $M$ .", "As in the proof of Corollary REF , we will take $\\gamma \\le \\nu _{n,\\lambda }$ , and it will be enough to have that $\\int _0^{\\infty }\\sigma ^{\\frac{2}{n}-1}\\sqrt{\\Psi _{|b-e|^2}(\\sigma )\\Psi _{|c-e|^2}(\\sigma )}\\,d\\sigma \\le C\\exp \\left(-C\\Vert b-e\\Vert _{n,1}-C\\Vert c_1-e\\Vert _{n,q}^q\\right),$ whenever $\\Vert b\\Vert _{n,1}<\\gamma $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\gamma $ , and where $C=C_{n,q,\\lambda }$ .", "Then, a similar argument as in the proof of Proposition REF completes the proof." ], [ "The first step: all coefficients are small", "The first step to obtain the Moser estimate is via a coercivity assumption, which we now turn to.", "The following lemma is standard, and we only give a sketch of its proof.", "We will set $2_*=\\frac{2n}{n+2}$ .", "Lemma 4.1 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and $A$ be uniformly elliptic and bounded in $\\Omega $ , with ellipticity $\\lambda $ .", "There exists $\\theta =\\theta _{n,\\lambda }>0$ such that, if $b\\in L^{n,1}(\\Omega )$ , $c\\in L^{n,\\infty }(\\Omega )$ and $d\\in L^{\\frac{n}{2},1}(\\Omega )$ with $\\Vert b\\Vert _{n,1}\\le \\theta $ , $\\Vert c\\Vert _{n,\\infty }\\le \\theta $ and $\\Vert d\\Vert _{\\frac{n}{2},1}\\le \\theta $ , then the operator $\\mathcal {L}u=-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du$ is coercive, and every solution $v\\in W_0^{1,2}(\\Omega )$ to the equation $\\mathcal {L}u=-\\operatorname{div}F+G$ for $F\\in L^2(\\Omega )$ and $G\\in L^{2_*}(\\Omega )$ satisfies the estimate $\\Vert \\nabla v\\Vert _{L^2(B_{2r})}\\le C_{n,\\lambda }\\Vert F\\Vert _{L^2(\\Omega )}+C_{n,\\lambda }\\Vert G\\Vert _{L^{2_*}(\\Omega )}.$ Also, if $\\Omega =B_{2r}$ and $w\\in W^{1,2}(B_{2r})$ is a subsolution to $-\\operatorname{div}(A\\nabla w+bw)+c\\nabla w+dw\\le 0$ , then $\\int _{B_r}|\\nabla w|^2\\le \\frac{C}{r^2}\\int _{B_{2r}}|w^+|^2,$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "Moreover, for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u)+c\\nabla u\\le 0$ in $B_{2r}$ and $\\alpha \\in (1,2)$ , we have that $\\sup _{B_r}u\\le \\frac{C}{(\\alpha -1)^{n/2}}\\left(_{B_{\\alpha r}}|u^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "We first show (REF ), following the lines of the proof of [19]: if $\\phi $ is a smooth cutoff function, then using $u^+\\phi ^2$ as a test function, we obtain $\\begin{split}\\int _{B_{2r}}A\\nabla u^+\\nabla u^+\\cdot \\phi ^2&\\le -2\\int _{B_{2r}}A\\nabla u^+\\nabla \\phi \\cdot u^+\\phi -\\int _{B_{2r}}c\\nabla u^+\\cdot u^+\\phi ^2\\\\&\\le C\\Vert \\phi \\nabla u^+\\Vert _{L^2(B_{2r})}\\Vert u^+\\nabla \\phi \\Vert _{L^2(B_{2r})}+\\left\\Vert cu^+\\phi \\right\\Vert _{L^2(B_{2r})}\\Vert \\phi \\nabla u^+\\Vert _{L^2(B_{2r})}.\\end{split}$ Then, assuming that $\\Vert c\\Vert _{n,\\infty }\\le \\theta $ , for $\\theta $ to be chosen later, using Hölder's estimate (REF ) we have $\\left\\Vert cu^+\\phi \\right\\Vert _{L^2(B_{2r})}\\le C_n\\Vert c\\Vert _{n,\\infty }\\Vert u^+\\phi \\Vert _{L^{2^*,2}(B_{2r})}\\le C_n\\theta \\Vert u^+\\phi \\Vert _{L^{2^*,2}(B_{2r})},$ and combining with [28], we have $\\left\\Vert cu^+\\phi \\right\\Vert _{L^2(B_{2r})}\\le C_n\\theta \\Vert \\nabla (u^+\\phi )\\Vert _{L^2(B_{2r})}\\le C_n\\theta \\Vert \\phi \\nabla u^+\\Vert _{L^2(B_{2r})}+C_n\\theta \\Vert u^+\\nabla \\phi \\Vert _{L^2(B_{2r})}.$ So, choosing $\\theta $ such that $C_n\\theta <\\frac{\\lambda }{4}$ , and plugging in (REF ), we obtain that $\\int _{B_{2r}}|\\phi \\nabla u^+|^2\\le C\\int _{B_{2r}}|u^+\\nabla \\phi |^2,$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "This estimate corresponds to [19], and following the lines of the argument on [19] we obtain that $\\sup _{B_r}u\\le C\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "To complete the proof of (REF ) note that, for all $x\\in B_r$ , the last estimate shows that $\\sup _{B_{\\frac{\\alpha -1}{2}r}(x)}u\\le C\\left(_{B_{(\\alpha -1)r}(x)}|u^+|^2\\right)^{\\frac{1}{2}}\\le \\frac{C}{(\\alpha -1)^{n/2}r^{n/2}}\\left(\\int _{B_{\\alpha r}}|u^+|^2\\right)^{\\frac{1}{2}},$ since $B_{(\\alpha -1)r}(x)\\subseteq B_{\\alpha r}$ , and considering the supremum for $x\\in B_r$ shows (REF ).", "Finally, coercivity of $\\mathcal {L}$ , (REF ) and (REF ) follow via a combination of the procedure as in (REF ) and (REF ), where for (REF ) we use $v$ as a test function, and for (REF ) we use $w^+\\phi ^2$ as a test function.", "We now turn to local boundedness when all the lower order coefficients have small norms.", "Lemma 4.2 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "There exists $\\theta ^{\\prime }=\\theta ^{\\prime }_{n,\\lambda }>0$ such that, if $b\\in L^{n,1}(B_{2r})$ , $c\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le \\theta ^{\\prime }$ , $\\Vert c\\Vert _{n,\\infty }\\le \\theta ^{\\prime }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}\\le \\theta ^{\\prime }$ , then for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le 0$ , $\\sup _{B_r}u^+\\le C\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "Consider the $\\theta _{n,\\lambda }$ that appears in Lemma REF .", "We will take $\\theta ^{\\prime }\\le \\theta _{n,\\lambda }$ , so that the operator is coercive.", "Then, if $u$ is a subsolution to $\\mathcal {L}u\\le 0$ , the proof of [30] implies that $u^+$ is a subsolution to $\\mathcal {L}u^+\\le 0$ ; therefore, we can assume that $u\\ge 0$ .", "Assume first that $b,c,d$ are bounded in $B_{2r}$ , then [19] shows that $\\sup _{B_r}u<\\infty .$ Let $\\frac{1}{4}\\le \\eta <\\eta ^{\\prime }\\le \\frac{1}{2}$ .", "From coercivity of the operator $\\mathcal {L}_0u=-\\operatorname{div}(A\\nabla u)+c\\nabla u$ , and since $\\operatorname{div}(bu)-du\\in W^{-1,2}(B_{\\eta ^{\\prime }r})=\\left(W_0^{1,2}(B_{\\eta ^{\\prime }r})\\right)^*$ , the Lax-Milgram theorem shows that there exists $v\\in W_0^{1,2}(B_{\\eta ^{\\prime } r})$ such that $-\\operatorname{div}(A\\nabla v)+c\\nabla v=\\operatorname{div}(bu)-du.$ If $\\beta $ is as in Proposition REF , taking $\\theta ^{\\prime }\\le \\beta _{n,\\lambda ,\\theta _{n,\\lambda }}$ the same proposition shows that $\\sup _{B_{\\eta ^{\\prime } r}}v\\le C_{n,\\lambda }\\Vert bu\\Vert _{L^{n,1}(B_{\\eta ^{\\prime } r})}+C_{n,\\lambda }\\Vert du\\Vert _{L^{{\\frac{n}{2},1}}(B_{\\eta ^{\\prime } r})}\\le C_{n,\\lambda }\\theta ^{\\prime }\\sup _{B_{\\eta ^{\\prime } r}}u,$ since $u\\ge 0$ .", "In addition, from the Sobolev inequality, estimate (REF ) and the Hölder inequality, $\\Vert v\\Vert _{L^{2^*}(B_{\\eta ^{\\prime } r})}\\le C_n\\Vert \\nabla v\\Vert _{L^2(B_{\\eta ^{\\prime } r})}\\le C_{n,\\lambda }\\Vert bu\\Vert _{L^2(B_{\\eta ^{\\prime } r})}+C_{n,\\lambda }\\Vert du\\Vert _{L^{2_*}(B_{\\eta ^{\\prime } r})}\\le C_{n,\\lambda }\\Vert u\\Vert _{L^{2^*}(B_{\\eta ^{\\prime } r})}.$ Moreover, the function $w=u-v$ is a subsolution to $-\\operatorname{div}(A\\nabla w)+c\\nabla w\\le 0$ , so (REF ) implies that $\\sup _{B_{\\eta r}}w&\\le \\frac{C}{(\\frac{\\eta ^{\\prime }}{\\eta }-1)^{n/2}}\\left(_{B_{\\eta ^{\\prime }r}}|w^+|^2\\right)^{\\frac{1}{2}}\\le \\frac{C}{(\\eta ^{\\prime }-\\eta )^{n/2}}\\left(_{B_{\\eta ^{\\prime } r}}|u|^2\\right)^{\\frac{1}{2}}+\\frac{C}{(\\eta ^{\\prime }-\\eta )^{n/2}}\\left(_{B_{\\eta ^{\\prime } r}}|v|^2\\right)^{\\frac{1}{2}}\\\\&\\le \\frac{C}{(\\eta ^{\\prime }-\\eta )^{n/2}}\\left(_{B_{\\eta ^{\\prime } r}}|u|^{2^*}\\right)^{\\frac{1}{2^*}}\\le \\frac{C}{(\\eta ^{\\prime }-\\eta )^{n/2}}\\left(_{B_{r/2}}|u|^{2^*}\\right)^{\\frac{1}{2^*}},$ where we also used (REF ) for the penultimate estimate, and $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "Hence, the definition of $w$ , the last estimate and (REF ) show that $\\sup _{B_{\\eta r}}u\\le \\sup _{B_{\\eta r}}v+\\sup _{B_{\\eta r}}w\\le C_{n,\\lambda }\\theta ^{\\prime }\\sup _{B_{\\eta ^{\\prime }r}}u+\\frac{C}{(\\eta ^{\\prime }-\\eta )^{n/2}}\\left(_{B_{r/2}}|u|^{2^*}\\right)^{\\frac{1}{2^*}},$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "We now set $\\eta _N=\\frac{1}{2}-4^{-N}$ and apply the previous estimate for $\\eta =\\eta _N$ and $\\eta ^{\\prime }=\\eta _{N+1}$ .", "Then, $\\sup _{B_{\\eta _Nr}}u\\le C_{n,\\lambda }\\theta ^{\\prime }\\sup _{B_{\\eta _{N+1} r}}u+2^{nN}C\\left(_{B_{r/2}}|u|^{2^*}\\right)^{\\frac{1}{2^*}}.$ Inductively, this shows that, for any $N\\in \\mathbb {N}$ , $\\sup _{B_{\\eta _1r}}u\\le (C_{n,\\lambda }\\theta ^{\\prime })^N\\sup _{B_{\\eta _{N+1}r}}u+C\\sum _{i=1}^N(C_{n,\\lambda }\\theta ^{\\prime })^{i-1}2^{ni}\\cdot \\left(_{B_r}|u|^{2^*}\\right)^{\\frac{1}{2^*}}.$ We will consider $\\theta ^{\\prime }$ such that $C_{n,\\lambda }\\theta ^{\\prime }\\le \\frac{1}{2}$ .", "Then, letting $N\\rightarrow \\infty $ and using (REF ), we obtain that $\\sup _{B_{r/4}}u\\le C\\sum _{i=1}^{\\infty }\\left(2^nC_{n,\\lambda }\\theta ^{\\prime }\\right)^{i-1}\\cdot \\left(_{B_{r/2}}|u|^{2^*}\\right)^{\\frac{1}{2^*}},$ and choosing $\\theta ^{\\prime }$ that also satisfies $2^nC_{n,\\lambda }\\theta ^{\\prime }\\le \\frac{1}{2}$ shows that $\\sup _{B_{r/4}}u\\le C \\left(_{B_{r/2}}|u|^{2^*}\\right)^{\\frac{1}{2^*}}\\le C\\left(_{B_{2r}}|u|^2\\right)^{\\frac{1}{2}},$ where we used (REF ) and the Sobolev inequality for the last estimate, and where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "In the case that $b,c,d$ are not necessarily bounded, let $b^j$ be the coordinate functions of $b$ , and define $b_N$ having coordinate functions $b_N^j=b^j\\chi _{[|b^j|\\le N]}$ for $N\\in \\mathbb {N}$ ; define also similar approximations $c_N$ and $d_N$ for $c,d$ respectively.", "We then have that $\\Vert b_N\\Vert _{n,1}\\le \\Vert b\\Vert _{n,1}$ , and similarly for $c_N$ and $d_N$ .", "Since $\\theta ^{\\prime }\\le \\theta _{n,\\lambda }$ , from coercivity in Lemma REF and the Lax-Milgram theorem there exists $v_N\\in W_0^{1,2}(B_{r/2})$ that solves the equation $-\\operatorname{div}(A\\nabla v_N+b_Nv_N)+c_N\\nabla v_N+d_Nv_N=-\\operatorname{div}(A\\nabla u+b_Nu)+c_N\\nabla u+d_Nu$ in $B_{r/2}$ .", "Then, from (REF ), $\\Vert \\nabla v_n\\Vert _{L^2(B_{r/2})}\\le C\\Vert A\\nabla u+b_Nu\\Vert _{L^2(B_{r/2})}+C\\Vert c_N\\nabla u+d_Nu\\Vert _{L^{2_*}(B_{r/2})}\\le \\frac{C}{r}\\Vert u\\Vert _{L^2(B_{2r})},$ where we also used (REF ) and Hölder's inequality for the last estimate.", "So, $(v_N)$ is bounded in $W_0^{1,2}(B_{r/2})$ , hence from Rellich's theorem there exists a subsequence $(v_{N^{\\prime }})$ such that $v_{N^{\\prime }}\\rightarrow v_0\\,\\,\\,\\text{weakly in}\\,\\,\\,W_0^{1,2}(B_{r/2})\\,\\,\\,\\text{and strongly in}\\,\\,\\,L^{\\frac{n}{n-2}}(B_{r/2}),\\quad v_{N^{\\prime }}(x)\\rightarrow v_0(x)\\,\\,\\,\\forall \\,x\\in F,$ where $F\\subseteq B_{r/2}$ is a set with full measure.", "Note now that $w_N=u-v_N$ is a solution to $-\\operatorname{div}(A\\nabla w_N+b_Nw_N)+c_N\\nabla w_N+d_Nw_N=0$ in $B_{r/2}$ , and $b_N,c_N$ and $d_N$ are bounded, so (REF ) (where $B_{2r}$ is replaced by $B_{r/2}$ ) is applicable to $w^+$ ; therefore, for $x\\in F_N$ , where $F_N\\subseteq B_{r/16}$ has full measure, $w_N^+(x)\\le \\sup _{B_{r/16}}w_N^+\\le C\\left(_{B_{r/2}}|w_N^+|^2\\right)^{\\frac{1}{2}}\\le C\\left(_{B_{r/2}}u^2\\right)^{\\frac{1}{2}}+C\\left(_{B_{r/2}}v_N^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "Therefore, for all $x\\in F_N$ , $u(x)=v_N(x)+w_N(x)\\le v_N(x)+C\\left(_{B_{r/2}}u^2\\right)^{\\frac{1}{2}}+C\\left(_{B_{r/2}}v_N^2\\right)^{\\frac{1}{2}}\\le v_N(x)+C\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}},$ where we used the Sobolev inequality and (REF ) for the last estimate.", "Let now $F_0=F\\cap \\bigcap _{N=1}^{\\infty }F_N$ , then $F_0\\subseteq B_{r/16}$ has full measure, and if $x\\in F_0$ , then letting $N^{\\prime }\\rightarrow \\infty $ in the previous estimate, (REF ) implies that $u(x)\\le \\limsup _{N^{\\prime }\\rightarrow \\infty }v_{N^{\\prime }}(x)+C\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}}=v_0(x)+C\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}},$ for all $x\\in F_0$ .", "Finally, note that $v_{N^{\\prime }}$ is a subsolution to $-\\operatorname{div}(A\\nabla v_N+b_Nv_N)+c_N\\nabla v_N+d_Nv_N\\le -\\operatorname{div}((b_N-b)u)+(c_N-c)\\nabla u+(d_N-d)u$ in $B_{r/2}$ , and since $b_{N^{\\prime }}\\rightarrow b_N$ and $c_{N^{\\prime }}\\rightarrow c_N$ strongly in $L^2(B_{r/2})$ , while $d_{N^{\\prime }}\\rightarrow d_N$ strongly in $L^{\\frac{n}{2}}(B_{r/2})$ , using (REF ) and the variational formulation of subsolutions (REF ) we obtain that $v_0$ is a $W_0^{1,2}(B_{2r})$ subsolution to $-\\operatorname{div}(A\\nabla v_0+bv_0)+c\\nabla v_0+dv_0\\le 0.$ Hence, since $\\theta ^{\\prime }\\le \\beta _{n,\\lambda ,\\theta _{n,\\lambda }}$ , Proposition REF implies that $v_0\\le 0$ in $B_{r/2}$ , and plugging in (REF ) and covering $B_r$ with balls of radius $r/16$ completes the proof." ], [ "The second step: $b$ or {{formula:4f2cc668-2264-4057-95e1-8da14a552c20}} have large norms", "We now turn to scale invariant estimates with “good\" constants when $d$ is small, and either $b$ or $c$ are small as well.", "We first consider the case of small $c$ and assume that the right hand side is identically 0, for simplicity; the terms on the right hand side will be added in Proposition REF .", "Lemma 4.3 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ , and $b\\in L^{n,1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le M$ .", "There exists $\\overline{\\theta }=\\overline{\\theta }_{n,\\lambda ,M}>0$ such that, if $c\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert c\\Vert _{n,\\infty }<\\overline{\\theta }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\overline{\\theta }$ , then for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le 0$ , we have $\\sup _{B_r}u\\le C\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "We will proceed by induction on $M$ .", "Consider the $\\theta _{n,\\lambda }^{\\prime }$ and the constant $C_0=C_{n,\\lambda ,\\Vert A\\Vert _{\\infty }}\\ge 1$ that appear in Lemma REF .", "In addition, for any integer $N\\ge 0$ , set $C^{\\prime }_{n,\\lambda ,N}=C_{n,\\lambda ,2^{N/n}\\theta ^{\\prime }_{n,\\lambda }}\\ge 1$ , where the last constant appears in Proposition REF .", "We claim that, if $\\Vert b\\Vert _{L^{n,1}(B_{2r})}\\le 2^{N/n}\\theta _{n,\\lambda }^{\\prime }$ , then there exists $\\overline{\\theta }_{n,\\lambda ,N}>0$ such that, if we have that $\\Vert c\\Vert _{L^{n,\\infty }(B_{2r})}<\\overline{\\theta }_{n,\\lambda ,N}$ and $\\Vert d\\Vert _{L^{\\frac{n}{2},1}(B_{2r})}<\\overline{\\theta }_{n,\\lambda ,N}$ , then $\\sup _{B_r}u\\le 8^{\\frac{nN}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,\\lambda ,i}\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}}.$ For $N=0$ , letting $\\overline{\\theta }_{n,\\lambda ,0}=\\theta ^{\\prime }_{n,\\lambda }$ , the previous estimate holds from Lemma REF .", "Assume now that this estimate holds for some integer $N\\ge 0$ , for some constant $\\overline{\\theta }_{n,\\lambda ,N}$ .", "From Proposition REF there exists $\\beta ^{\\prime }_{n,\\lambda ,N}=\\beta _{n,\\lambda ,2^{N/n}\\theta ^{\\prime }_{n,\\lambda }}>0$ such that, if $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ is a domain, $A^{\\prime }$ is elliptic in $\\Omega $ with ellipticity $\\lambda $ , $\\Vert b^{\\prime }\\Vert _{L^{n,1}(\\Omega )}\\le 2^{N/n}\\theta ^{\\prime }_{n,\\lambda }$ , $\\Vert c^{\\prime }\\Vert _{L^{n,\\infty }(\\Omega )}<\\beta ^{\\prime }_{n,\\lambda ,N}$ and $\\Vert d^{\\prime }\\Vert _{L^{\\frac{n}{2},1}(\\Omega )}<\\beta ^{\\prime }_{n,\\lambda ,N}$ , then for any subsolution $v\\in Y^{1,2}(\\Omega )$ to $-\\operatorname{div}(A^{\\prime }\\nabla v+b^{\\prime }v)+c^{\\prime }\\nabla v+d^{\\prime }v\\le 0$ in $\\Omega $ , we have that $\\sup _{\\Omega }v\\le C^{\\prime }_{n,\\lambda ,N}\\sup _{\\partial \\Omega }v^+.$ We then set $\\overline{\\theta }_{n,\\lambda ,{N+1}}=\\min \\lbrace \\overline{\\theta }_{n,\\lambda ,N},\\beta ^{\\prime }_{n,\\lambda ,N+1}\\rbrace $ , and assume that $\\Vert b\\Vert _{L^{n,1}(B_{2r})}\\le 2^{(N+1)/n}\\theta ^{\\prime }_{n,\\lambda },\\qquad \\Vert c\\Vert _{L^{n,\\infty }(B_{2r})}<\\overline{\\theta }_{n,\\lambda ,{N+1}},\\quad \\text{and}\\quad \\Vert d\\Vert _{L^{\\frac{n}{2},1}(B_{2r})}<\\overline{\\theta }_{n,\\lambda ,{N+1}}.$ We will show that, in this case, (REF ) holds for $N+1$ .", "To show this, we distinguish between two cases: $\\Vert b\\Vert _{L^{n,1}(B_{3r/2})}\\le 2^{N/n}\\theta ^{\\prime }_{n,\\lambda }$ , and $\\Vert b\\Vert _{L^{n,1}(B_{3r/2})}>2^{N/n}\\theta ^{\\prime }_{n,\\lambda }$ .", "In the first case, let $x\\in B_r$ .", "Then, since $\\overline{\\theta }_{n,\\lambda ,{N+1}}\\le \\overline{\\theta }_{n,\\lambda ,N}$ and $B_{r/2}(x)\\subseteq B_{3r/2}$ , we have that $\\Vert b\\Vert _{L^{n,1}(B_{r/2}(x))}\\le 2^{N/n}\\theta ^{\\prime }_{n,\\lambda },\\qquad \\Vert c\\Vert _{L^{n,\\infty }(B_{r/2}(x))}<\\overline{\\theta }_{n,\\lambda ,N},\\quad \\text{and}\\quad \\Vert d\\Vert _{L^{\\frac{n}{2},1}(B_{r/2}(x))}<\\overline{\\theta }_{n,\\lambda ,N}.$ Therefore, from (REF ) for $N$ (in the ball $B_{r/2}(x)$ instead of $B_{2r}$ ), we have $\\sup _{B_{r/4}(x)}u&\\le 8^{\\frac{nN}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,\\lambda ,i}\\left(_{B_{r/2}(x)}|u^+|^2\\right)^{\\frac{1}{2}}\\\\&\\le 8^{\\frac{nN}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,\\lambda ,i}2^n\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}}\\le 8^{\\frac{n(N+1)}{2}}C_0\\prod _{i=0}^{N+1}C^{\\prime }_{n,\\lambda ,i}\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}},$ where we used that $C^{\\prime }_{n,\\lambda ,N+1}\\ge 1$ for the last step.", "So, (REF ) holds for $N+1$ in this case.", "In the second case, let $y\\in \\partial B_{7r/4}$ .", "Then $B_{r/4}(y)\\subseteq B_{2r}\\setminus B_{3r/2}$ , therefore, from Lemma REF , $\\Vert b\\Vert _{L^{n,1}(B_{r/4}(y))}^n\\le \\Vert b\\Vert _{L^{n,1}(B_{2r})}^n-\\Vert b\\Vert _{L^{n,1}(B_{3r/2})}^n< 2^{N+1}(\\theta ^{\\prime }_{n,\\lambda })^n-2^N(\\theta ^{\\prime }_{n,\\lambda })^n=(2^{N/n}\\theta ^{\\prime }_{n,\\lambda })^n.$ Moreover, from (REF ), we have that $\\Vert c\\Vert _{L^{n,\\infty }(B_{r/4}(y))}<\\overline{\\theta }_{n,\\lambda ,N}$ and $\\Vert d\\Vert _{L^{\\frac{n}{2},1}(B_{r/4}(y))}<\\overline{\\theta }_{n,\\lambda ,N}$ , hence (REF ) for $N$ (in the ball $B_{r/4}(y)$ instead of $B_{2r}$ ) implies that $\\sup _{B_{r/8}(y)}u\\le 8^{\\frac{nN}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,\\lambda ,i}\\left(_{B_{r/4}(y)}|u^+|^2\\right)^{\\frac{1}{2}}\\le 8^{\\frac{n(N+1)}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,\\lambda ,i}\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}}.$ Then, the last estimate, (REF ) and Proposition REF show that $\\sup _{B_r}u\\le C^{\\prime }_{n,\\lambda ,N+1}\\sup _{\\partial B_{7r/4}}u\\le C_{n,\\lambda ,N+1}^{\\prime }\\cdot 8^{\\frac{n(N+1)}{2}}C_0\\prod _{i=1}^NC^{\\prime }_{n,\\lambda ,i}\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}},$ which shows that (REF ) for $N+1$ in this case as well.", "Therefore, (REF ) holds for any $N\\in \\mathbb {N}$ , which completes the proof.", "Finally, we show Moser's estimate allowing right hand sides to the equation, and considering also different $L^p$ norms on the right hand side of the estimate.", "Proposition 4.4 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $b\\in L^{n,1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le M$ , and $p>0$ , $f\\in L^{n,1}(B_{2r})$ , $g\\in L^{\\frac{n}{2},1}(B_{2r})$ .", "There exists $\\operatorname{\\varepsilon }=\\operatorname{\\varepsilon }_{n,\\lambda ,M}>0$ such that, if $c\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert c\\Vert _{n,\\infty }<\\operatorname{\\varepsilon }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\operatorname{\\varepsilon }$ , then for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le -\\operatorname{div}f+g$ , we have that $\\sup _{B_r}u\\le C\\left(_{B_{2r}}|u^+|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ depends on $n,p,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Consider the $\\beta _{n,\\lambda ,M}$ from Proposition REF .", "If $\\Vert c\\Vert _{n,\\infty }<\\beta _{n,\\lambda ,M}$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\beta _{n,\\lambda ,M}$ , any solution $u\\in W_0^{1,2}(B_{2r})$ to the equation $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du=0$ in $B_{2r}$ should be identically 0, from Proposition REF .", "Hence, adding a term of the form $+Lu$ to the operator, for some large $L>0$ depending only on $n,\\lambda ,M$ , the operator becomes coercive, and a combination of the Lax-Milgram theorem and the Fredholm alternative (as in [16], for example) show that there exists a unique $v\\in W_0^{1,2}(B_{2r})$ such that $-\\operatorname{div}(A\\nabla v+bv)+c\\nabla v+dv=-\\operatorname{div}f+g,$ in $B_{2r}$ .", "Then, Proposition REF implies that $\\sup _{B_{2r}}|v|\\le C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ depends on $n,\\lambda $ and $M$ .", "Consider now the $\\overline{\\theta }_{n,\\lambda ,M}$ from Lemma REF and set $\\operatorname{\\varepsilon }=\\min \\lbrace \\beta _{n,\\lambda ,M},\\overline{\\theta }_{n,\\lambda ,M}\\rbrace $ .", "Then, assuming that $\\Vert c\\Vert _{n,\\infty }<\\operatorname{\\varepsilon }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\operatorname{\\varepsilon }$ , since $w=u-v$ is a subsolution to $-\\operatorname{div}(A\\nabla w+bw)+c\\nabla w+dw\\le 0$ , (REF ) implies that $\\sup _{B_r}w\\le C\\left(_{B_{2r}}|w^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Then, (REF ) for $p=2$ follows adding (REF ) and (REF ).", "Finally, in the case $p\\ge 2$ , (REF ) follows from Hölder's inequality, while in the case $p\\in (0,2)$ , the proof follows from the argument on [17].", "We now turn to the case when $c\\in L^{n,q}$ with $q<\\infty $ is allowed to have large norm.", "Lemma 4.5 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $q<\\infty $ and $c_1\\in L^{n,q}(B_{2r})$ with $\\Vert c_1\\Vert _{n,q}\\le M$ .", "There exist $\\xi =\\xi _{n,\\lambda }>0$ and $\\zeta =\\zeta _{n,q,\\lambda ,M}>0$ such that, if $b\\in L^{n,1}(B_{2r})$ , $c_2\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}<\\zeta $ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\zeta $ , then for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+(c_1+c_2)\\nabla u+du\\le 0$ , we have that $\\sup _{B_{r/4}}u\\le C\\left(_{B_{2r}}|u^+|^2\\right)^{\\frac{1}{2}},$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Let $C_n\\ge 1$ be such that $\\Vert h_1+h_2\\Vert _{n,\\infty }\\le C_n\\Vert h_1\\Vert _{n,\\infty }+C_n\\Vert h_2\\Vert _{n,\\infty }$ for all $h_1,h_2\\in L^{n,\\infty }$ (from (REF )), and $C_{n,q}\\ge 1$ be such that $\\Vert h\\Vert _{n,\\infty }\\le C_{n,q}\\Vert h\\Vert _{n,q}$ for all $h\\in L^{n,q}$ (from (REF )).", "Set $\\xi _{n,\\lambda }=\\frac{1}{2C_n}\\min \\left\\lbrace \\nu _{n,\\lambda },\\theta ^{\\prime }_{n,\\lambda }\\right\\rbrace >0,$ where $\\nu _{n,\\lambda }$ and $\\theta ^{\\prime }_{n,\\lambda }$ appear in Proposition REF and Lemma REF , respectively.", "For $N\\ge 0$ , set also $C^{\\prime }_{n,q,\\lambda ,N}=C_{n,q,\\lambda ,2^{N/q}C_{n,q}^{-1}\\xi _{n,\\lambda }}>1$ , where the last constant appears in Proposition REF , and consider the constant $C_0=C_{n,\\lambda ,\\Vert A\\Vert _{\\infty }}\\ge 1$ that appears in Lemma REF .", "We claim that, for any integer $N\\ge 0$ , if $\\Vert c_1\\Vert _{n,q}\\le 2^{N/q}C_{n,q}^{-1}\\xi _{n,\\lambda }$ , then there exists $\\zeta _{n,q,\\lambda ,N}$ such that, if $\\Vert b\\Vert _{n,1}<\\zeta _{n,q,\\lambda ,N}$ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi _{n,\\lambda }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\zeta _{n,q,\\lambda ,N}$ , then $\\sup _{B_{r/4}}u\\le 8^{\\frac{nN}{2}}C_0\\prod _{i=0}^NC^{\\prime }_{n,q,\\lambda ,i}\\left(_{B_{2r}}u^2\\right)^{\\frac{1}{2}}.$ For $N=0$ we can take $\\zeta _{n,q,\\lambda ,0}=\\xi _{n,\\lambda }$ , since we then have that $\\Vert c\\Vert _{n,\\infty }\\le C_nC_{n,q}\\Vert c_1\\Vert _{n,q}+C_n\\Vert c_2\\Vert _{n,\\infty }\\le 2C_n\\xi _{n,\\lambda }\\le \\theta ^{\\prime }_{n,\\lambda },$ and also $\\Vert b\\Vert _{n,1}\\le \\theta ^{\\prime }_{n,\\lambda }$ , $\\Vert d\\Vert _{\\frac{n}{2},1}\\le \\theta ^{\\prime }_{n,\\lambda }$ , therefore (REF ) for $N=0$ holds from Lemma REF .", "Assume now that (REF ) holds for some $N\\ge 0$ , and set $\\zeta _{n,q,\\lambda ,{N+1}}=\\min \\lbrace \\zeta _{n,q,\\lambda ,N},\\gamma ^{\\prime }_{n,q,\\lambda ,N+1}\\rbrace $ , where $\\gamma ^{\\prime }_{n,q,\\lambda ,N}=\\gamma _{n,q,\\lambda ,2^{N/q}C_{n,q}^{-1}\\xi _{n,\\lambda }}$ , and the $\\gamma $ appears in Proposition REF .", "We then continue as in the proof of the Lemma REF , using Lemma REF for $q>n$ and Proposition REF instead of Proposition REF ; this shows that (REF ) holds for $N+1$ if $\\Vert c_1\\Vert _{n,q}\\le 2^{(N+1)/q}C_{n,q}^{-1}\\xi _{n,\\lambda }$ , as long as $\\Vert b\\Vert _{n,1}<\\zeta _{n,q,\\lambda ,N+1}$ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi _{n,\\lambda }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\zeta _{n,q,\\lambda ,N+1}$ , and this completes the proof.", "Finally, we add right hand sides and allow different $L^p$ norms.", "Proposition 4.6 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ , and $q<\\infty $ , $c_1\\in L^{n,q}(B_{2r})$ with $\\Vert c_1\\Vert _{n,q}\\le M$ .", "Let also $p>0$ and $f\\in L^{n,1}(B_{2r})$ , $g\\in L^{\\frac{n}{2},1}(B_{2r})$ .", "There exist $\\xi =\\xi _{n,\\lambda }>0$ and $\\delta =\\delta _{n,q,\\lambda ,M}>0$ such that, if $b\\in L^{n,1}(B_{2r})$ , $c_2\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}<\\delta $ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\delta $ , then for any subsolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+(c_1+c_2)\\nabla u+du\\le -\\operatorname{div}f+g$ , we have that $\\sup _{B_r}u\\le C\\left(_{B_{2r}}|u^+|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ depends on $n,p,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "The proof is similar to the proof of Proposition REF , using Proposition REF instead of Proposition REF and Lemma REF instead of Lemma REF .", "Remark 4.7 Note that the analogue of Propositions REF and REF will hold under no smallness assumptions for $b,d$ or $c,d$ (when $c\\in L^{n,q}$ , $q<\\infty $ ), but then the constants depend on $b,d$ or $c,d$ and not just on their norms.", "This can be achieved considering $r^{\\prime }>0$ small enough, so that the norms of $b,d$ or $c,d$ are small enough in all balls of radius $2r^{\\prime }$ that are subsets of $B_{2r}$ , and after covering $B_r$ with balls of radius $r^{\\prime }$ ." ], [ "Estimates on the boundary", "We now turn to local boundedness close to the boundary.", "We will follow the same process as in the case of local boundedness in the interior.", "The following are the analogues of (REF ) and (REF ) close to the boundary; the proof is similar to the one of Lemma REF (as in [19]) and it is omitted.", "Lemma 4.8 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain and $B_{2r}\\subseteq \\operatorname{\\mathbb {R}}^n$ be a ball.", "Let also $A$ be uniformly elliptic and bounded in $\\Omega \\cap B_{2r}$ , with ellipticity $\\lambda $ .", "There exists $\\theta =\\theta _{n,\\lambda }>0$ such that, if $b\\in L^{n,1}(\\Omega \\cap B_{2r})$ , $c\\in L^{n,\\infty }(\\Omega \\cap B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le \\theta $ , $\\Vert c\\Vert _{n,\\infty }\\le \\theta $ and $\\Vert d\\Vert _{\\frac{n}{2},1}\\le \\theta $ , then, if $w\\in W^{1,2}(\\Omega \\cap B_{2r})$ is a subsolution to $-\\operatorname{div}(A\\nabla w+bw)+c\\nabla w+dw\\le 0$ with $w\\le 0$ on $\\partial \\Omega \\cap B_{2r}$ , we have that $\\int _{\\Omega \\cap B_r}|\\nabla w|^2\\le \\frac{C}{r^2}\\int _{\\Omega \\cap B_{2r}}|w^+|^2,$ where $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "Moreover, for any subsolution $u\\in W^{1,2}(\\Omega \\cap B_{2r})$ to $-\\operatorname{div}(A\\nabla u)+c\\nabla u\\le 0$ in $\\Omega \\cap B_{2r}$ and any $\\alpha \\in (1,2)$ , we have that $\\sup _{\\Omega \\cap B_r}u\\le \\frac{C}{(\\alpha -1)^{n/2}}\\left(_{B_{\\alpha r}}v^2\\right)^{\\frac{1}{2}},$ where $v=u^+\\chi _{\\Omega \\cap B_{2r}}$ , and $C$ depends on $n,\\lambda $ and $\\Vert A\\Vert _{\\infty }$ .", "To show local boundedness close to the boundary, we will need the following definition from [19]: if $u$ is a function in $\\Omega $ and $\\partial \\Omega \\cap B_{2r}\\ne \\emptyset $ , we define $s_u=\\sup _{\\partial \\Omega \\cap B_{2r}}u^+,\\qquad \\tilde{u}(x)=\\left\\lbrace \\begin{array}{l l}\\sup \\lbrace u(x),s_u\\rbrace , & x\\in B_{2r}\\cap \\Omega \\\\s_u, & x\\in B_{2r}\\setminus \\Omega \\end{array}\\right.$ where the supremum over $\\partial \\Omega \\cap B_{2r}$ is defined as on [19].", "The following proposition concerns the case of large $b$ .", "Proposition 4.9 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and $B_{2r}$ be a ball of radius $2r$ .", "Let also $A$ be uniformly elliptic and bounded in $\\Omega \\cap B_{2r}$ , with ellipticity $\\lambda $ , $b\\in L^{n,1}(\\Omega \\cap B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le M$ , and $p>0$ , $f\\in L^{n,1}(\\Omega \\cap B_{2r})$ , $g\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ .", "There exists $\\operatorname{\\varepsilon }=\\operatorname{\\varepsilon }_{n,\\lambda ,M}>0$ such that, if $c\\in L^{n,\\infty }(\\Omega \\cap B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ with $\\Vert c\\Vert _{n,\\infty }<\\operatorname{\\varepsilon }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\operatorname{\\varepsilon }$ , then for any subsolution $u\\in W^{1,2}(\\Omega \\cap B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\le -\\operatorname{div}f+g$ , we have that $\\sup _{\\Omega \\cap B_r}\\tilde{u}\\le C\\left(_{B_{2r}}|\\tilde{u}|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{L^{n,1}(\\Omega \\cap B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})},$ where $\\tilde{u}$ is defined in (REF ), and where $C$ depends on $n,p,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Subtracting a constant from $u$ , and since $\\tilde{u}\\ge s_u$ in $B_{2r}$ , we can reduce to the case when $u\\le 0$ on $\\partial \\Omega \\cap B_{2r}$ (that is, $s_u=0$ ).", "Then, based on Lemma REF and [19] instead of Lemma REF and [19], respectively, we can show the analogue of Lemma REF , replacing all the balls by their intersections with $\\Omega $ , for subsolutions $u\\in W^{1,2}(\\Omega \\cap B_{2r})$ with $u\\le 0$ on $\\partial \\Omega \\cap B_{2r}$ .", "We then continue with a similar argument as in the proofs of Lemma REF and Proposition REF , replacing all the balls by their intersections with $\\Omega $ .", "Finally, using a similar argument to the above, and going through the arguments of the proofs of Lemma REF and Proposition REF , we obtain the following estimate close to the boundary, in the case that $c$ is large.", "Proposition 4.10 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, and $B_{2r}$ be a ball of radius $2r$ .", "Let also $A$ be uniformly elliptic and bounded in $\\Omega \\cap B_{2r}$ , with ellipticity $\\lambda $ , and consider $q<\\infty $ and $c_1\\in L^{n,q}(\\Omega \\cap B_{2r})$ with $\\Vert c_1\\Vert _{n,q}\\le M$ .", "Let also $p>0$ , $f\\in L^{n,1}(\\Omega \\cap B_{2r})$ , and $g\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ .", "There exist $\\xi =\\xi _{n,\\lambda }>0$ and $\\delta =\\delta _{n,q,\\lambda ,M}>0$ such that, if $b\\in L^{n,1}(\\Omega \\cap B_{2r})$ , $c_2\\in L^{n,\\infty }(\\Omega \\cap B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ with $\\Vert b\\Vert _{n,1}<\\delta $ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\delta $ , then for any subsolution $u\\in W^{1,2}(\\Omega \\cap B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+(c_1+c_2)\\nabla u+du\\le -\\operatorname{div}f+g$ , we have that $\\sup _{\\Omega \\cap B_r}\\tilde{u}\\le C\\left(_{B_{2r}}|\\tilde{u}|^p\\right)^{\\frac{1}{p}}+C\\Vert f\\Vert _{L^{n,1}(\\Omega \\cap B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})},$ where $\\tilde{u}$ is defined in (REF ), and where $C$ depends on $n,p,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Remark 4.11 As in Remark REF , the analogues of Propositions REF and REF will hold under no smallness assumptions for $b,d$ or $c,d$ (when $c\\in L^{n,q}$ , $q<\\infty $ ), with constants depending on $b,d$ or $c,d$ and not just on their norms." ], [ "The lower bound", "In order to deduce the Harnack inequality, we will consider negative powers of positive supersolutions to transform them to subsolutions of suitable operators, where the coefficients $b,d$ will be small.", "This is the context of the following lemma.", "Lemma 5.1 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, $b,c,f\\in L^{n,\\infty }(\\Omega )$ and $d,g\\in L^{\\frac{n}{2},\\infty }(\\Omega )$ .", "Let also $u\\in W^{1,2}(\\Omega )$ be a supersolution to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\ge -\\operatorname{div}f+g$ with $\\inf _{\\Omega }u>0$ , and consider the function $v=u+\\Vert f\\Vert _{L^{n,1}(\\Omega )}+\\Vert g\\Vert _{L^{\\frac{n}{2},1}(\\Omega )}$ .", "Then, for any $k<0$ , $v^k$ is a $W^{1,2}(\\Omega )$ subsolution to $-\\operatorname{div}\\left(A\\nabla (v^k)+\\frac{k(bu-f)}{v}v^k\\right)+\\left(\\frac{(k-1)(bu-f)}{v}+c\\right)\\nabla (v^k)+\\frac{k(du-g)}{v}v^k\\le 0.$ We compute $-\\operatorname{div}(A\\nabla (v^k))=-\\operatorname{div}(A\\nabla v\\cdot kv^{k-1})=-k\\operatorname{div}(A\\nabla v)v^{k-1}-k(k-1)A\\nabla v\\nabla v\\cdot v^{k-2}.$ From ellipticity of $A$ we have that $A\\nabla v\\nabla v\\ge 0$ .", "Since also $k<0$ , the last identity shows that $-\\operatorname{div}(A\\nabla (v^k))\\le -\\operatorname{div}(A\\nabla u)\\cdot kv^{k-1}$ .", "Since $k<0$ , $v^{k-1}>0$ and $u$ is a supersolution, we have $-\\operatorname{div}(A\\nabla (v^k))\\le (\\operatorname{div}(bu)-c\\nabla u-du-\\operatorname{div}f+g)kv^{k-1},$ and the proof is complete after a straightforward computation.", "The next lemma bridges the gap between $L^p$ averages for positive and negative $p$ .", "Lemma 5.2 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ , and $b,c\\in L^{n,\\infty }(B_{2r})$ , $d\\in L^{\\frac{n}{2},\\infty }(B_{2r})$ .", "Let also $u\\in W^{1,2}(B_{2r})$ be a supersolution to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\ge 0$ in $B_{2r}$ , with $\\inf _{B_{2r}}u>0$ .", "Then there exists a constant $a=a_n$ such that $_{B_r}u^a_{B_r}u^{-a}\\le C,$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $\\Vert b\\Vert _{n,\\infty }$ , $\\Vert c\\Vert _{n,\\infty }$ and $\\Vert d\\Vert _{\\frac{n}{2},\\infty }$ .", "We use the test function from [24] (see also [19]): let $B_{2s}$ be a ball of radius $2s$ , contained in $B_{2r}$ .", "If $\\phi \\ge 0$ be a smooth cutoff supported in $B_{2s}$ , with $\\phi \\equiv 1$ in $B_s$ and $|\\nabla \\phi |\\le \\frac{C}{s}$ , then the function $\\phi ^2u^{-1}$ is nonnegative and belongs to $W_0^{1,2}(B_{2s})$ .", "Hence, using it as a test function, we obtain that $\\int _{B_{2s}}\\left(A\\nabla u\\frac{2\\phi \\nabla \\phi }{u}-A\\nabla u\\frac{\\phi ^2\\nabla u}{u^2}+b\\frac{2\\phi \\nabla \\phi }{u}u-b\\frac{\\phi ^2\\nabla u}{u^2}u+c\\nabla u\\frac{\\phi ^2}{u}+du\\frac{\\phi ^2}{u}\\right)\\ge 0,$ hence $\\int _{B_{2s}}A\\nabla u\\frac{\\nabla u}{u^2}\\phi ^2\\le \\int _{B_{2r}}\\left(A\\nabla u\\frac{2\\phi \\nabla \\phi }{u}+2b\\nabla \\phi \\cdot \\phi -b\\frac{\\nabla u}{u}\\phi ^2+c\\nabla u\\frac{\\phi ^2}{u}+d\\phi ^2\\right).$ Using ellipticity of $A$ , the Cauchy-Schwartz inequality, and Cauchy's inequality with $\\operatorname{\\varepsilon }$ , we obtain $\\int _{B_{2s}}\\frac{|\\nabla u|^2}{u^2}\\phi ^2&\\le C\\int _{B_{2s}}\\left(|\\nabla \\phi |^2+|b\\nabla \\phi |\\phi +(|b|^2+|c|^2+|d|)\\phi ^2\\right)\\\\&\\le Cs^{n-2}+Cs^{-1}\\Vert b\\Vert _{n,\\infty }\\Vert 1\\Vert _{L^{\\frac{n}{n-1},1}(B_{2s})}+C\\left\\Vert |b|^2+|c|^2+|d|\\right\\Vert _{\\frac{n}{2},\\infty }\\Vert 1\\Vert _{L^{\\frac{n}{n-2},1}(B_{2s})}\\\\&\\le Cs^{n-2},$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $\\Vert b\\Vert _{n,\\infty }$ , $\\Vert c\\Vert _{n,\\infty }$ and $\\Vert d\\Vert _{\\frac{n}{2},\\infty }$ , and where we used (REF ) for the second estimate.", "The proof is complete using the Poincaré inequality and the John-Nirenberg inequality, as on [24].", "The next bound is a reverse Moser estimate for supersolutions.", "Surprisingly, if we assume that the coefficient $c$ belongs to $L^{n,q}$ for some $q<\\infty $ , then we obtain a scale invariant estimate with “good\" constants under no smallness assumption on the coefficients.", "As mentioned before, for the Moser estimate in Propositions REF and REF , such a bound cannot hold with “good\" constants under these assumptions.", "Proposition 5.3 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(B_{2r})$ , $c_1\\in L^{n,q}(B_{2r})$ for some $q<\\infty $ , and $d,g\\in L^{\\frac{n}{2},1}(B_{2r})$ , with $\\Vert b\\Vert _{n,1}\\le M_b$ , $\\Vert c_1\\Vert _{n,q}\\le M_c$ and $\\Vert d\\Vert _{\\frac{n}{2},1}\\le M_d$ .", "There exist $a=a_n>0$ and $\\xi =\\xi _{n,\\lambda }>0$ such that, if $c_2\\in L^{n,\\infty }(B_{2r})$ with $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ , then for any nonnegative supersolution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+(c_1+c_2)\\nabla u+du\\ge -\\operatorname{div}f+g$ , we have that $\\left(_{B_r}u^a\\right)^{\\frac{1}{a}}\\le C\\inf _{B_{r/2}}u+C\\Vert f\\Vert _{L^{n,1}(\\Omega \\cap B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})},$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $M_b,M_c$ and $M_d$ .", "Adding a constant $\\delta >0$ to $u$ , we may assume that $\\inf _{B_{2r}}u>0$ ; the general case will follow by letting $\\delta \\rightarrow 0$ .", "Set $v=u+\\Vert f\\Vert _{L^{n,1}(B_{2r})}+\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})}$ , then $v$ is a supersolution to $-\\operatorname{div}\\left(A\\nabla v+\\frac{bu-f}{v}v\\right)+c\\nabla v+\\frac{du-g}{v}v\\ge 0,$ with $\\left\\Vert \\frac{bu-f}{v}\\right\\Vert _{n,1}\\le C_n\\Vert b\\Vert _{n,1}+C_n,\\qquad \\left\\Vert \\frac{du-g}{v}\\right\\Vert _{\\frac{n}{2},1}\\le C_n\\Vert d\\Vert _{\\frac{n}{2},1}+C_n.$ Then, since $\\inf _{B_{2r}}v>0$ , Lemma REF implies that there exists $a=a_n$ such that $_{B_r}v^a_{B_r}v^{-a}\\le C,$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $M_b,M_c$ and $M_d$ .", "For $k\\in (-1,0)$ to be chosen later, $v^k$ is a $W^{1,2}(B_{2r})$ subsolution to (REF ) for $c=c_1+c_2$ , and $\\left\\Vert \\frac{(k-1)(bu-f)}{v}+c_1\\right\\Vert _{n,q}\\le C_{n,q}(1-k)\\left\\Vert \\frac{bu}{v}\\right\\Vert _{n,q}+C_{n,q}(1-k)\\left\\Vert \\frac{f}{v}\\right\\Vert _{n,q}+C_{n,q}\\Vert c_1\\Vert _{n,q}\\le M,$ where $M$ depends on $n,q,M_b$ and $M_c$ .", "Then, for the $\\xi _{n,\\lambda }$ and the $\\delta _{n,q,\\lambda ,M}>0$ from Proposition REF and (REF ), if $\\left\\Vert \\frac{k(bu-f)}{v}\\right\\Vert _{L^{n,1}(B_r)}<\\delta _{n,q,\\lambda ,M},\\quad \\Vert c_2\\Vert _{L^{n,\\infty }(B_r)}<\\xi _{n,\\lambda },\\quad \\left\\Vert \\frac{k(du-g)}{v}\\right\\Vert _{L^{\\frac{n}{2},1}(B_r)}<\\delta _{n,q,\\lambda ,M},$ then $v^k$ satisfies the estimate $\\sup _{B_{r/2}}v^k\\le C_{B_r}v^k,$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "It is true that (REF ) holds for some $k\\in (-a,0)$ , depending on $n,q,\\lambda ,M_b,M_c$ and $M_d$ ; hence, for this $k$ , $\\left(_{B_r}v^k\\right)^{\\frac{1}{k}}\\le C(\\sup _{B_{r/2}}v^k)^{\\frac{1}{k}}=C\\inf _{B_{r/2}}v,$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty },M_b,M_c$ and $M_d$ .", "Since $-\\frac{a}{k}>1$ , Hölder's inequality implies that $_{B_r}v^k\\le \\left(_{B_r}v^{-a}\\right)^{-\\frac{k}{a}}\\Rightarrow \\left(_{B_r}v^k\\right)^{\\frac{1}{k}}\\ge \\left(_{B_r}v^{-a}\\right)^{-\\frac{1}{a}}\\ge C\\left(_{B_r}v^a\\right)^{\\frac{1}{a}},$ where we used (REF ) for the last step.", "Then, plugging the last estimate in (REF ), and using the definition of $v$ , the proof is complete." ], [ "Estimates on the boundary", "We now consider the analogue of Proposition REF close to the boundary.", "We will need the analogue of the definition of $\\tilde{u}$ in (REF ), from [19]: if $u\\ge 0$ is a function in $\\Omega $ and $\\partial \\Omega \\cap B_{2r}\\ne \\emptyset $ , we define $m_u=\\inf _{\\partial \\Omega \\cap B_{2r}}u,\\qquad \\bar{u}(x)=\\left\\lbrace \\begin{array}{l l}\\inf \\lbrace u(x),m_u\\rbrace , & x\\in B_{2r}\\cap \\Omega \\\\m_u, & x\\in B_{2r}\\setminus \\Omega \\end{array}\\right..$ The following is the analogue of Lemma REF close to the boundary.", "Lemma 5.4 Let $A$ be uniformly elliptic and bounded in $\\Omega \\cap B_{2r}$ , with ellipticity $\\lambda $ , and $b,c\\in L^{n,\\infty }(\\Omega \\cap B_{2r})$ , $d\\in L^{\\frac{n}{2},\\infty }(\\Omega \\cap B_{2r})$ .", "Let also $u\\in W^{1,2}(B_{2r})$ be a nonnegative supersolution to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du\\ge 0$ in $B_{2r}$ , and consider the function $\\bar{u}$ from (REF ).", "If $\\inf _{\\Omega \\cap B_{2r}}u>0$ and $m_u>0$ , then there exists a constant $a=a_n$ such that $_{B_r}\\bar{u}^a_{B_r}\\bar{u}^{-a}\\le C,$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $\\Vert b\\Vert _{n,\\infty }$ , $\\Vert c\\Vert _{n,\\infty }$ and $\\Vert d\\Vert _{\\frac{n}{2},\\infty }$ .", "As in the proof of [19], set $v=\\bar{u}^{-1}-m_u^{-1}\\in W^{1,2}(\\Omega \\cap B_{2r})$ , which is nonnegative in $\\Omega \\cap B_{2r}$ and vanishes on $\\partial \\Omega \\cap B_{2r}$ .", "Then, considering the test function $v\\phi ^2$ , where $\\phi $ is a suitable cutoff function, and using that $v>0$ if and only if $\\bar{u}=u$ , the proof follows by an argument as in the proof of Lemma REF .", "Using the previous lemma, we can show the following estimate.", "Proposition 5.5 Let $\\Omega \\subseteq \\operatorname{\\mathbb {R}}^n$ be a domain, $B_{2r}$ be a ball of radius $2r$ , and let $A$ be uniformly elliptic and bounded in $\\Omega \\cap B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(\\Omega \\cap B_{2r})$ , $c_1\\in L^{n,q}(\\Omega \\cap B_{2r})$ for some $q<\\infty $ , and $d,g\\in L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})$ , with $\\Vert b\\Vert _{n,1}\\le M_b$ , $\\Vert c_1\\Vert _{n,q}\\le M_c$ and $\\Vert d\\Vert _{\\frac{n}{2},1}\\le M_d$ .", "There exist $a=a_n>0$ and $\\xi =\\xi _{n,\\lambda }>0$ such that, if $c_2\\in L^{n,\\infty }(\\Omega \\cap B_{2r})$ with $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ , then for any nonnegative supersolution $u\\in W^{1,2}(\\Omega \\cap B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+(c_1+c_2)\\nabla u+du\\ge -\\operatorname{div}f+g$ , we have that $\\left(_{B_r}\\bar{u}^a\\right)^{\\frac{1}{a}}\\le C\\inf _{B_{r/2}}\\bar{u}+C\\Vert f\\Vert _{L^{n,1}(\\Omega \\cap B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(\\Omega \\cap B_{2r})},$ where $\\bar{u}$ is defined in (REF ), and where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ , $M_b,M_c$ and $M_d$ .", "As in the proof of Proposition REF , we can assume that $\\inf _{\\Omega \\cap B_{2r}}u>0$ , $m_u>0$ , and $f,g\\equiv 0$ .", "Let $a=a_n$ be as in Lemma REF .", "Then, Lemma REF and Proposition REF show that, for suitable $k\\in (-a,0)$ , if $w_k=u^k$ and $\\tilde{w_k}$ is as in (REF ), we have that $\\sup _{\\Omega \\cap B_{r/2}}\\tilde{w_k}\\le C_{B_r}\\tilde{w_k}\\le C\\left(_{B_r}\\tilde{w_k}^{-\\frac{a}{k}}\\right)^{-\\frac{k}{a}}.$ Since $\\tilde{w_k}=\\bar{u}^k$ , the proof is complete using also Lemma REF ." ], [ "The Harnack inequality, and local continuity", "We now show the Harnack inequality in the cases when $b,d$ are small, or when $c,d$ are small.", "Theorem 5.6 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le M$ , and $g\\in L^{\\frac{n}{2},1}(B_{2r})$ .", "There exists $\\operatorname{\\varepsilon }_{n,\\lambda ,M}>0$ such that, if $c\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert c\\Vert _{n,\\infty }<\\operatorname{\\varepsilon }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\operatorname{\\varepsilon }$ , then for any nonnegative solution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du =-\\operatorname{div}f+g$ , we have that $\\sup _{B_r}u\\le C\\inf _{B_r}u+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "The proof is a combination of Proposition REF (choosing $p=a_n$ in (REF ), as in Proposition REF ), and Proposition REF , (considering $q=n$ and $c_1\\equiv 0$ ), after also covering $B_r$ with balls of radius $r/4$ .", "Theorem 5.7 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ , and $q<\\infty $ , $c_1\\in L^{n,q}(B_{2r})$ with $\\Vert c_1\\Vert _{n,q}\\le M$ .", "Let also $f\\in L^{n,1}(B_{2r})$ , $g\\in L^{\\frac{n}{2},1}(B_{2r})$ .", "There exist $\\xi =\\xi _{n,\\lambda }>0$ and $\\delta =\\delta _{n,q,\\lambda ,M}>0$ such that, if $b\\in L^{n,1}(B_{2r})$ , $c_2\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}<\\delta $ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\delta $ , then for any nonnegative solution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du=-\\operatorname{div}f+g$ , we have that $\\sup _{B_r}u\\le C\\inf _{B_r}u+C\\Vert f\\Vert _{L^{n,1}(B_{2r})}+C\\Vert g\\Vert _{L^{\\frac{n}{2},1}(B_{2r})},$ where $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "The proof follows by a combination of Propositions REF and REF .", "We now turn to local continuity of solutions.", "For the following theorem, for $\\rho \\le 2r$ , we set $Q_{b,d}(\\rho )=\\sup \\left\\lbrace \\Vert b\\Vert _{L^{n,1}(B^{\\prime }_{\\rho })}+\\Vert d\\Vert _{L^{\\frac{n}{2},1}(B^{\\prime }_{\\rho })}:B^{\\prime }_{\\rho }\\subseteq B_{2r}\\right\\rbrace ,$ where $B_{\\rho }^{\\prime }$ runs over all the balls of radius $\\rho $ that are subsets of $B_{2r}$ .", "Also, we will follow the argument on [19].", "Theorem 5.8 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ .", "Let also $b,f\\in L^{n,1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}\\le M$ , $g\\in L^{\\frac{n}{2},1}(B_{2r})$ , and $\\mu \\in (0,1)$ .", "For every $\\mu \\in (0,1)$ , there exists $\\operatorname{\\varepsilon }=\\operatorname{\\varepsilon }_{n,\\lambda ,M}>0$ and $\\alpha =\\alpha _{n,\\lambda ,\\Vert A\\Vert _{\\infty },M,\\mu }\\in (0,1)$ such that, if $c\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert c\\Vert _{n,\\infty }<\\operatorname{\\varepsilon }$ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\operatorname{\\varepsilon }$ , then for any solution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du =-\\operatorname{div}f+g$ , we have that $|u(x)-u(y)|\\le C\\left(\\frac{|x-y|^{\\alpha }}{r^{\\alpha }}+Q_{b,d}(|x-y|^{\\mu }r^{1-\\mu })\\right)\\left(_{B_{2r}}|u|+Q_{f,g}(2r)\\right)+CQ_{f,g}(|x-y|^{\\mu }r^{1-\\mu }),$ for any $x,y\\in B_r$ , where $Q$ is defined in (REF ) and $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Let $\\rho \\in (0,r]$ , and set $M(\\rho )=\\sup _{B_{\\rho }}u$ , $m(\\rho )=\\inf _{B_{\\rho }}u$ .", "Then $v_1=M(\\rho )-u$ is nonnegative in $B_{\\rho }$ , and solves the equation $-\\operatorname{div}(A\\nabla v_1+bv_1)+c\\nabla v_1+dv_1=-\\operatorname{div}(M(\\rho )b-f)+(M(\\rho ) d-g)$ in $B_{\\rho }$ .", "Hence, from Theorem REF , (REF ) and (REF ), we obtain that $\\begin{split}M(\\rho )-m\\left(\\frac{\\rho }{2}\\right)&=\\sup _{B_{\\rho /2}}v_1\\le C\\inf _{B_{\\rho /2}}v_1+C\\Vert M(\\rho )b-f\\Vert _{L^{n,1}(B_{\\rho })}+C\\Vert M(\\rho ) d-g\\Vert _{L^{\\frac{n}{2},1}(B_{\\rho })}\\\\&=C\\left(M(\\rho )-M\\left(\\frac{\\rho }{2}\\right)\\right)+C\\sup _{B_r}|u|\\cdot Q_{b,d}(\\rho )+CQ_{f,g}(\\rho ),\\end{split}$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Moreover, $v_2=u-m(\\rho )$ is nonnegative in $B_{\\rho }$ , and solves the equation $-\\operatorname{div}(A\\nabla v_2+bv_2)+c\\nabla v_2+dv_2=-\\operatorname{div}(f-m(\\rho )b)+(g-m(\\rho )d)$ in $B_{\\rho }$ .", "Hence, from Theorem REF , as in (REF ), $M\\left(\\frac{\\rho }{2}\\right)-m(\\rho )\\le C\\left(m\\left(\\frac{\\rho }{2}\\right)-m(\\rho )\\right)+C\\sup _{B_r}|u|\\cdot Q_{b,d}(\\rho )+CQ_{f,g}(\\rho ).$ Adding (REF ) and (REF ) and defining $\\omega (\\rho )=M(\\rho )-m(\\rho )$ , we obtain that $\\omega \\left(\\frac{\\rho }{2}\\right)\\le \\theta _0\\omega (\\rho )+C\\sup _{B_r}|u|\\cdot Q_{b,d}(\\rho )+CQ_{f,g}(\\rho ),$ where $\\theta _0=\\frac{C-1}{C+1}\\in (0,1)$ .", "Then, [19] shows that, for $\\rho \\le r$ , $\\omega (\\rho )\\le C\\frac{\\rho ^{\\alpha }}{r^{\\alpha }}\\omega (r)+C\\sup _{B_r}|u|\\cdot Q_{b,d}(\\rho ^{\\mu }r^{1-\\mu })+CQ_{f,g}(\\rho ^{\\mu }r^{1-\\mu }),$ where $C$ depends on $n,\\lambda ,\\Vert A\\Vert _{\\infty },M$ , and $\\alpha =\\alpha _{n,\\lambda ,\\Vert A\\Vert _{\\infty },M,\\mu }$ .", "We then bound $\\sup _{B_r}|u|$ using Proposition REF (applied to $u$ and $-u$ , for $p=1$ ), which completes the proof.", "Finally, based on Proposition REF and Theorem REF , we obtain the following theorem when $b,d$ are small.", "Theorem 5.9 Let $A$ be uniformly elliptic and bounded in $B_{2r}$ , with ellipticity $\\lambda $ , and $q<\\infty $ , $c_1\\in L^{n,q}(B_{2r})$ with $\\Vert c_1\\Vert _{n,q}\\le M$ .", "Let also $f\\in L^{n,1}(B_{2r})$ , $g\\in L^{\\frac{n}{2},1}(B_{2r})$ .", "For every $\\mu \\in (0,1)$ , there exist $\\xi =\\xi _{n,\\lambda }>0$ , $\\delta =\\delta _{n,q,\\lambda ,M}>0$ and $\\alpha =\\alpha _{n,\\lambda ,\\Vert A\\Vert _{\\infty },M,\\mu }$ such that, if $b\\in L^{n,1}(B_{2r})$ , $c_2\\in L^{n,\\infty }(B_{2r})$ and $d\\in L^{\\frac{n}{2},1}(B_{2r})$ with $\\Vert b\\Vert _{n,1}<\\delta $ , $\\Vert c_2\\Vert _{n,\\infty }<\\xi $ and $\\Vert d\\Vert _{\\frac{n}{2},1}<\\delta $ , then for any solution $u\\in W^{1,2}(B_{2r})$ to $-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u+du=-\\operatorname{div}f+g$ , we have that $|u(x)-u(y)|\\le C\\left(\\frac{|x-y|^{\\alpha }}{r^{\\alpha }}+Q_{b,d}(|x-y|^{\\mu }r^{1-\\mu })\\right)\\cdot \\left(_{B_{2r}}|u|+Q_{f,g}(2r)\\right)+CQ_{f,g}(|x-y|^{\\mu }r^{1-\\mu }),$ for any $x,y\\in B_r$ , where $Q$ is defined in (REF ) and $C$ depends on $n,q,\\lambda ,\\Vert A\\Vert _{\\infty }$ and $M$ .", "Remark 5.10 As in Remarks REF and REF , the analogues of Theorems REF - REF will hold under no smallness assumptions for $b,d$ and $c,d$ (when $c\\in L^{n,q}$ , $q<\\infty $ ), but then the constants depend on $b,d$ or $c,d$ and not just on their norms." ], [ "Optimality of the assumptions", "We now turn to showing that our assumptions are optimal in order to deduce the estimates we have shown so far, in the setting of Lorentz spaces.", "We first show optimality for $b$ and $d$ .", "Remark 6.1 Considering the operators $\\mathcal {L}_1u=-\\Delta u-\\operatorname{div}(bu)$ and $\\mathcal {L}_2u=-\\Delta u+du$ , an assumption of the form $b\\in L^{n,q}$ , $d\\in L^{\\frac{n}{2},q}$ for some $q>1$ , with $\\Vert b\\Vert _{n,q}$ , $\\Vert d\\Vert _{\\frac{n}{2},1}$ being as small as we want, is not enough to guarantee the pointwise bounds in the maximum principle and Moser's estimate.", "Indeed, as in Lemma [21], set $u_{\\delta }(x)=\\left(-\\ln |x|\\right)^{\\delta }$ and $b_{\\delta }(x)=-\\frac{\\delta x}{|x|^2\\ln |x|}$ .", "Then, for $\\delta \\in (-1,1)$ , $b\\in L^{n,q}(B_{1/e})$ for all $q>1$ , $u_{\\delta }\\in W^{1,2}(B_{1/e})$ , and $u_{\\delta }$ solves the equation $-\\Delta u-\\operatorname{div}(b_{\\delta }u_{\\delta })=0$ in $B_{1/e}$ .", "However, $v_{\\delta }\\equiv 1$ on $\\partial B_{1/e}$ , and $v_{\\delta }\\rightarrow \\infty $ as $|x|\\rightarrow 0$ for $\\delta >0$ , so the assumption $b\\in L^{n,1}$ is optimal for the maximum principle and the Moser estimate.", "Note that $u_{\\delta }$ also solves the equation $-\\Delta u_{\\delta }+d_{\\delta }u_{\\delta }=0,\\qquad d_{\\delta }(x)=\\frac{\\delta (\\delta -1)}{|x|^2\\ln ^2|x|}+\\frac{\\delta (n-2)}{|x|^2\\ln |x|},$ and $d_{\\delta }\\in L^{\\frac{n}{2},q}(B_{1/e})$ for every $q>1$ ; hence, the assumption $d\\in L^{\\frac{n}{2},1}$ is again optimal.", "The same functions $b_{\\delta }$ and $d_{\\delta }$ serve as counterexamples to show optimality for the spaces of $b,d$ in the reverse Moser estimate.", "In particular, considering $\\delta <0$ , we have that $u_{\\delta }(0)=0$ , while $u_{\\delta }$ does not identically vanish close to 0, therefore the reverse Moser estimate cannot hold.", "We now turn to optimality for smallness of $c$ , when $c\\in L^{n,\\infty }$ .", "Remark 6.2 In the case of the operator $\\mathcal {L}_0u=-\\Delta u+c\\nabla u$ with $c\\in L^{n,\\infty }$ , smallness in norm is a necessary condition, in order to obtain all the estimates we have considered.", "Indeed, if $u(x)=-\\ln |x|-1$ , then $u\\in W_0^{1,2}(B_{1/e})$ , and $u$ solves the equation $-\\Delta u+c\\nabla u=0,\\qquad c=\\frac{(2-n)x}{|x|^2}\\in L^{n,\\infty }(B_{1/e}).$ However, $u$ is not bounded in $B_{1/e}$ , so the maximum principle, as well as Moser's and Harnack's estimates fail.", "On the other hand, the function $v(x)=(-\\ln |x|)^{-1}\\in W_0^{1,2}(B_{1/e})$ solves the equation $-\\Delta v+c^{\\prime }\\nabla v=0,\\qquad c^{\\prime }=\\frac{(n-2)x}{|x|^2}-\\frac{2x}{|x|^2\\ln |x|}\\in L^{n,\\infty }(B_{1/e}),$ with $v(0)=0$ and $v$ not identically vanishing close to 0, therefore smallness for $c\\in L^{n,\\infty }$ in the reverse Harnack estimate is necessary.", "Finally, we show the optimality of the assumption that either $b,d$ should be small, or $c,d$ should be small, so that in the maximum principle, as well as Moser's and Harnack's estimates, the constants depend only on the norms of the coefficients.", "The fact that $d$ should be small is based on the following construction.", "Proposition 6.3 There exists a bounded sequence $(d_N)$ in $L^{\\frac{n}{2},1}(B_1)$ and a sequence $(u_N)$ of nonnegative $W_0^{1,2}(B_1)\\cap C(\\overline{B})$ functions such that, for all $N\\in \\mathbb {N}$ , $u_N$ is a solution to the equation $-\\Delta u_N+d_Nu_N=0$ in $B_1$ , and $\\Vert u_N\\Vert _{W_0^{1,2}(B_1)}\\le C,\\quad \\text{while}\\quad u_N(0)\\xrightarrow[N\\rightarrow \\infty ]{}\\infty .$ We define $v(r)=\\left\\lbrace \\begin{array}{l l}\\frac{n}{2}+\\left(1-\\frac{n}{2}\\right)r^2, & 0<r\\le 1 \\\\r^{2-n}, & r>1.\\end{array}\\right.$ Set $u(x)=v(|x|)$ , then it is straightforward to check that $u$ is radially decreasing, $u\\ge 1$ in $B_1$ , $u\\le \\frac{n}{2}$ in $\\operatorname{\\mathbb {R}}^n$ , and $u\\in Y^{1,2}(\\operatorname{\\mathbb {R}}^n)\\cap C^1(\\operatorname{\\mathbb {R}}^n)$ .", "Then, the function $d=n(2-n)u^{-1}\\chi _{B_1}$ is bounded and supported in $B_1$ , and $u$ is a solution to the equation $-\\Delta u+du=0$ in $\\operatorname{\\mathbb {R}}^n$ .", "We now let $N\\in \\mathbb {N}$ with $N\\ge 2$ , and set $B_N$ to be the ball of radius $N$ , centered at 0.", "We will modify $u$ to be a $W_0^{1,2}(B_N)$ solution to a slightly different equation: for this, set $w_N=u-v(N)$ , and also $d_N=\\frac{du}{u-v(N)}.$ Since $d$ is supported in $B_1$ , $d_N$ is well defined.", "Note also that $w_N\\in W_0^{1,2}(B_N)$ , and $w_N$ is a solution to the equation $-\\Delta w_N+d_Nw_N=0$ in $B_N$ .", "Moreover, since $d$ is supported in $B_1$ , $u\\ge 1$ in $B_1$ and $v$ is decreasing, we have that $\\Vert d_N\\Vert _{L^{\\frac{n}{2},1}(B_N)}\\le C_n\\Vert d_N\\Vert _{L^{\\infty }(B_1)}\\le C_n\\frac{\\Vert d\\Vert _{L^{\\infty }(B_1)}\\Vert u\\Vert _{L^{\\infty }(B_1)}}{1-v(N)}\\le C_n.$ Let now $\\tilde{d}_N(x)=N^2d_N(Nx)$ and $\\tilde{w}_N(x)=w_N(Nx)$ , for $x\\in B_1$ .", "Then $\\tilde{w}_N\\in W_0^{1,2}(B_1)$ , $(\\tilde{d}_N)$ is bounded in $L^{\\frac{n}{2},1}(B_1)$ , and $\\tilde{w}_N$ is a solution to the equation $-\\Delta \\tilde{w}_N+\\tilde{d}_N\\tilde{w}_N=0$ in $B_1$ .", "Moreover, $\\tilde{w}_N(0)\\ge C_n$ , while $\\int _{B_1}|\\nabla \\tilde{w}_N|^2=N^{2-n}\\int _{B_N}|\\nabla w_N|^2= N^{2-n}\\int _{B_N}|\\nabla u|^2\\xrightarrow[N\\rightarrow \\infty ]{}0,$ since $\\nabla u\\in L^2(\\operatorname{\\mathbb {R}}^n)$ .", "Hence, considering the function $\\frac{\\tilde{w}_N}{\\Vert \\nabla \\tilde{w}_N\\Vert _{L^2(B_1)}}$ completes the proof.", "Remark 6.4 If $d_N,u_N$ are as in Proposition REF , then using the functions $e_N$ from Lemma REF that solve the equation $\\operatorname{div}e_N=d_N$ in $B_1$ , we have that $-\\operatorname{div}(\\nabla u_N-e_Nu)-e_N\\nabla u_N=0.$ So, for the operator $\\mathcal {L}u=-\\operatorname{div}(A\\nabla u+bu)+c\\nabla u$ , if both $b,c$ are allowed to be large, then the conclusion of Proposition REF shows that the constants in the maximum principle, as well as Moser's and Harnack's estimates, cannot depend only on the norms of the coefficients.", "Table: NO_CAPTION" ] ]

[ [ "Testing the cosmic curvature at high redshifts: the combination of LSST\n strong lensing systems and quasars as new standard candles" ], [ "Abstract The cosmic curvature, a fundamental parameter for cosmology could hold deep clues to inflation and cosmic origins.", "We propose an improved model-independent method to constrain the cosmic curvature by combining the constructed Hubble diagram of high-redshift quasars with galactic-scale strong lensing systems expected to be seen by the forthcoming LSST survey.", "More specifically, the most recent quasar data are used as a new type of standard candles in the range $0.036<z<5.100$, whose luminosity distances can be directly derived from the non-linear relation between X-ray and UV luminosities.", "Compared with other methods, the proposed one involving the quasar data achieves constraints with higher precision ($\\Delta \\Omega_k\\sim 10^{-2}$) at high redshifts ($z\\sim 5.0$).", "We also investigate the influence of lens mass distribution in the framework of three types of lens models extensively used in strong lensing studies (SIS model, power-law spherical model, and extended power-law lens model), finding the strong correlation between the cosmic curvature and the lens model parameters.", "When the power-law mass density profile is assumed, the most stringent constraint on the cosmic curvature $\\Omega_k$ can be obtained.", "Therefore, the issue of mass density profile in the early-type galaxies is still a critical one that needs to be investigated further." ], [ "Introduction", "The cosmic curvature is one of the fundamental issues in modern cosmology, which determines the evolution and structure of our Universe.", "Specifically, the spatial properties of our universe is not only closely connected with many important problems such as the properties of dark energy (DE) [28], [38], but also influences our understanding of inflationary models [45], [87], the most popular theories describing the evolution of early universe.", "More importantly, any detection of nonzero spatial curvature ($\\Omega _k\\ne 0$ ) would have significant consequences on the well-known FLRW metric, which has been investigated in many recent studies [33], [22].", "Therefore, precise measurements of spatial curvature allowing to better understand this degeneracy will have far-reaching consequences.", "The recent Planck 2018 results imposed very strong constraints on the curvature parameter, $\\Omega _k=0.001\\pm 0.002$ , based on cosmic microwave background (CMB) anisotropy measurements [1].", "However, it should be stressed here that the curvature inferred from CMB anisotropy data is obtained by assuming some specific dark energy model (the non-flat $\\Lambda $ CDM model).", "Therefore, it is necessary to consider different geometrical methods to derive model-independent measurements of the spatial curvature.", "Following this direction, great efforts have been made in the recent studies [9], [49], [91], [89], [68], with the combination of the well-known cosmic chronometers (which provide the expansion rate of the Universe [29]) and the observations of supernovae Ia (SNe Ia) (which provide the luminosity distances at different redshifts [83]).", "Later, such test has been implemented with updated observations of intermediate-luminosity radio quasars [65], the angular sizes of which could provide a new type of standard rulers at higher redshifts [23].", "Recently, [69] proposed a model-independent way to obtain constraints on the curvature, with strong gravitational lensing (SGL) data in the framework of the distance sum rule (DSR) in the FLRW metric [84], [33], [62].", "In the framework of strong gravitational lensing (SGL) [14], [15], the light can be bent by the gravity of massive body (at redshift $z_l$ ), which could produce multiple images for the distant sources (at redhsift $z_s$ ).", "Supplemented with the observations of the lens central velocity dispersion, the Einstein radius measurement [8], [13], [16] will enable a precise determination of the source-lens/lens distance ratio $d_{ls}/d_s$ [10], [11], [12] for individual strong lensing system.", "In addition, one should also estimate distances at redshifts $z_l$ and $z_s$ from different astrophysical probes covering these redshifts such as SNe Ia or Hubble parameters from cosmic chronometers [29], [28], [76], [49].", "The advantage of this method is that it is purely geometrical and the curvature can be constrained directly by observational data, without any pre-assumptions concerning the cosmological model and the FLRW metric [22].", "Such methodology has been first implemented with a SGL subsample from the Sloan Lens ACS Survey (SLACS) [8], which favors a spatially closed universe with the final results that the spatial curvature parameter could be constrained to $-1.22< \\Omega _k <0.63$ (95% C.L.)", "[70].", "More recently, several other studies have been carried out with enlarged galactic-scale SGL sample [16], as well as updated observations of SNe Ia data and radio quasars as distance indicators [18], [19], which furthermore confirmed the robustness of such consistency test as a practical measurement of the cosmic curvature [94], [65], [98].", "For instance, it has been demonstrated in a recent analysis [65] that 120 intermediate-luminosity radio quasars calibrated as standard rulers ($z\\sim 2.76$ ), in combination with 118 galactic-scale strong lensing systems, could provide an improved constraint on cosmic curvature $\\Omega _k<0.136$ .", "However, it should be pointed out that, the previous results still suffer from the sample size of available SGL data [70] and the reshift limitation of distance indicators [98].", "In the framework of the DSR, the purpose of this study is to assess the constraints on the spatial curvature, which could be achieved by confronting the currently largest standard candle quasar sample with the largest compilation of SGL observations expected from the forthcoming surveys.", "Specifically, the Large Synoptic Survey Telescope (LSST) is expected to discover $\\sim 10^5$ galaxy-scale lenses [61], [86], with the corresponding source redshift reaching $z\\sim 6$ .", "In this paper, we also take advantage of the recently compiled sample of quasar data set comprising 1598 quasars covering the redshift range of $0.036<z<5.100$ [73].", "Luminosity distances of these new type of standard candles are inferred from the recent method developed by [6], based on the relation between the UV and X-ray luminosities of high-redshift quasars.", "This paper is organized as follows.", "In Sec.", "2 and 3, we will briefly introduce the methodology, strong gravitational lensing models, as well as the the observational and simulated data in this analysis.", "In Sec.", "4, we show the forecasted constraints on the cosmic curvature.", "Finally, conclusions and discussions are summarized in Sec.", "5.", "On the assumption of cosmological principle, one always turn to the the FLRW metric to describe space-time of the Universe, which has the following form (in units where $c=1$ ): $ds^2=-dt^2+a^2(t)(\\frac{1}{1-kr^2}dr^2+r^2d\\Omega ^2).$ Here $k$ is a constant ($k=+1$ , $-$ 1, and 0 correspond to closed, open, and flat universe) associated with the curvature parameter as $\\Omega _k=-k/a_0^2 H_0^2$ , where $H_0$ denotes the Hubble constant.", "Let us introduce dimensionless comoving distances $d_l\\equiv d(0,z_l)$ , $d_s\\equiv d(0, z_s)$ and $d_{ls}\\equiv d(z_l, z_s)$ .", "For a galactic-scale strong lensing system, the dimensionless comoving distance $d(z)$ between the lensing galaxy (at redshift $z_l$ ) and the background source (at redshift $z_s$ ) is given by $\\nonumber d(z_l, z_s)&=& (1+z_s)H_0 D_A(z_l,z_s)\\\\&=&\\frac{1}{\\sqrt{|\\Omega _k|}}f\\left(\\sqrt{|\\Omega _k|}\\int ^{z_s}_{z_l}\\frac{H_0dz^{\\prime }}{H(z^{\\prime })} \\right),$ where $f(x)=\\left\\lbrace \\begin{array}{lll}\\sin (x)\\qquad \\,\\ \\Omega _k&<0, \\\\x\\qquad \\qquad \\,\\ \\Omega _k&=0, \\\\\\sinh (x)\\qquad \\Omega _k&>0.", "\\\\\\end{array}\\right.$ In the framework of FLRW metric, these distances are related via the distance sum rule [5], [29] $d_{ls}={d_s}\\sqrt{1+\\Omega _kd_l^2}-{d_l}\\sqrt{1+\\Omega _kd_s^2}.$ Note that in terms of dimensionless comoving distances DSR will reduce to an additivity relation $d_s =d_l+d_{ls}$ in the flat universe ($\\Omega _k=0$ ).", "The source/lens distance ratios $d_{ls}/d_s=D^{A}_{ls}/D^{A}_s$ can be assessed from the observations of multiple images in SGL systems [16].", "Meanwhile, if the two other two dimensionless distances $d_l$ and $d_s$ can be obtained from observations, the measurement of $\\Omega _k$ could be directly obtained [70].", "In this paper we will use the distance ratios $d_{ls}/d_s$ from the simulated SGL sample representative of the data obtainable from the forthcoming LSST survey [31], while the distances on cosmological scales ($d_l$ and $d_s$ ) will be inferred from the recent multiple measurements of 1598 quasars calibrated as standard candles [73]." ], [ "Strong gravitational lensing - distance ratio", "With the increasing number of detected SGL systems, strong gravitational lensing has become an important astrophysical tool to derive cosmological information from individual lensing galaxies, with both high-resolution imaging and spectroscopic observations.", "In this paper, we will focus on a method that can be traced back to [36] and furthermore extended in recent analysis [8], [13], [16], [26] based on different SGL samples.", "Specially, by combining the observations of SGL and stellar dynamics in elliptical galaxies, one could naturally measure the distance ratio $d_{ls}/d_s$ , based on the measurements of Einstein radius ($\\theta _E$ ) and the central velocity dispersion ($\\sigma _{lens}$ ) of the lens galaxies.", "The efficiency of such methodology lies in its ability to put constraints on the dynamic properties of dark energy [48], [53], the speed of light at cosmological scales [21], [25], and the validity of the General Relativity at galactic scale [20], [32].", "However, it should be stressed that cosmological application of SGL requires a better knowledge of the density profiles of early-type galaxies [17], [43], [32], the quantitative effect which has been assessed with updated galactic-scale strong lensing sample [26].", "Therefore, to describe the structure of the lens we will consider three types of models, which has been extensively investigated in the literature [65], [98].", "(I) Singular Isothermal Sphere (SIS) model: For the simplest SIS model, the distance ratio is given by [47].", "$ \\frac{d_{ls}}{d_s}=\\frac{c^2\\theta _E}{4\\pi \\sigma _{SIS}^2}=\\frac{c^2\\theta _E}{4\\pi \\sigma _{0}^2f_E^2},$ where $\\sigma _{SIS}$ is the dispersion velocity due to SIS lens mass distribution and $c$ the speed of light.", "In this analysis we also introduce a free parameter $f_E$ to quantify the difference between the SIS velocity dispersion ($\\sigma _{SIS}$ ) and the observed velocity dispersion of stars ($\\sigma _{0}$ ), as well as other possible systematic effects (see [60], [13] for more details).", "(II) Power-law model: Motivated by recent studies supporting non-negligible deviation from SIS for the slopes of density profiles of individual galaxies [47], [44], [79], we choose to generalize the SIS model to a spherically symmetric power-law mass distribution ($\\rho \\sim r^{-\\gamma }$ ).", "So the distance ratio for a power-law lens model can be written as [74], [47], [7] $ \\frac{d_{ls}}{d_s}=\\frac{c^2\\theta _E}{4\\pi \\sigma _{ap}^2}\\left(\\frac{\\theta _{ap}}{\\theta _E}\\right)^{2-\\gamma }f^{-1}(\\gamma ),$ where $f(\\gamma )$ is a certain function of the radial mass profile slope (see e.g.", "[16] for details), while the luminosity averaged line-of-sight velocity dispersion $\\sigma _{ap}$ can be measured inside the circular aperture of the angular radius $\\theta _{ap}$ .", "Note that SIS lens model corresponds to $\\alpha =2$ .", "(III) Extended power-law model: Considering the possible difference between the luminosity density profile ($\\nu (r)\\sim r^{-\\delta }$ ) and the total-mass (i.e.", "luminous plus dark-matter) density profile ($\\rho (r)\\sim r^{-\\alpha }$ ), one may solve the radial Jeans equation in spherical coordinate system to derive the dynamical mass inside the aperture radius [46].", "Therefore, the distance ratio – in the framework of this complicated lens profile – will be straightforwardly obtained, through the combination of dynamical mass and lens mass within the Einstein radius [26] $\\nonumber \\frac{d_{\\rm ls}}{d_{\\rm s}}&=& \\left(\\frac{c^2}{4\\sigma _{ap}^2}\\theta _{\\rm E}\\right)\\frac{2(3-\\delta )}{\\sqrt{\\pi }(\\xi -2 \\beta )(3-\\xi )} \\left( \\frac{\\theta _{\\rm ap}}{\\theta _{\\rm E}}\\right)^{2-\\alpha }\\\\&\\times &\\left[\\frac{\\lambda (\\xi )-\\beta \\lambda (\\xi +2)}{\\lambda (\\alpha )\\lambda (\\delta )}\\right]~,$ where $\\xi =\\alpha +\\delta -2$ , $\\lambda (x)=\\Gamma (\\frac{x-1}{2})/\\Gamma (\\frac{x}{2})$ .", "Note that $\\delta =\\alpha $ denotes that the shape of the luminosity density follows that of the total mass density, i.e., the power-law lens model.", "Moreover, in this model, a new parameter $\\beta (r) = 1 -{\\sigma ^2_t} / {\\sigma ^2_r}$ is included to quantify the anisotropy of stellar velocity.", "We assume that it follows a Gaussian distribution of $\\beta =0.18\\pm 0.13$ suggested by recent observations of several nearby early-type galaxies [37], [75]." ], [ "Distance calibration from high-redshift quasars", "In the past decades, great efforts have been made in investigating the “redshift - luminosity distance\" relation in quasars for the purpose of cosmological studies, based on different relations involving the quasar luminosity [3], [90], [88].", "In particular, the non-linear relation between the X-ray and UV luminosities of quasars looked very promising [2].", "However, suffering from the extreme variability and a wide range of luminosity, it still remains controversial whether quasars can be classified as \"true\" standard (or standardizable) candles in the Universe.", "Meanwhile, it should be pointed out that high scatter in the observed relations or the limitation of poor statistics remain the major uncertainties in most of these methods.", "Attempting to use these quasars by virtue of the non-linear relation between the X-ray and UV luminosities, one is usually faced with the challenge of large dispersions and observational biases.", "A key step forward was recently made by [73], who gradually refined the selection technique and flux measurements, which provided a suitable subsample of quasars (with an intrinsic dispersion smaller than 0.15 dex) to measure the luminosity distance.", "Following the approach described in [72], there exits a relation between the luminosities in the X-rays ($L_X$ ) and UV band ($L_{UV}$ ) $\\log (L_X)=\\hat{\\gamma }\\log (L_{UV})+\\beta ^{\\prime },$ where $\\hat{\\gamma }$ and $\\beta ^{\\prime }$ denote the slope parameter and the intercept.", "Combing Eq.", "(8) with the well-known expression of $L=F\\times 4\\pi D_L^2$ , the luminosity distance can be rewritten as $\\log (D_L)=\\frac{1}{2-2\\hat{\\gamma }}\\times [\\hat{\\gamma }\\log (F_{UV}) - \\log (F_X) + \\hat{\\beta }],$ a function of the respective fluxes ($F$ ), the slope parameter ($\\hat{\\gamma }$ ) and the normalization constant ($\\hat{\\beta }=\\beta ^{\\prime }+(\\hat{\\gamma }-1)\\log _{10} 4\\pi $ ).", "Therefore, from theoretical point of view, the luminosity distance can be directly determined from the measurements of the fluxes of $F_X$ and $F_{UV}$ , with a reliable knowledge of the dispersion $\\delta $ in this relation and the value for the two parameters ($\\hat{\\gamma }$ , $\\hat{\\beta }$ ) characterizing the $L_X - L_{UV}$ relation.", "However, it has been established that the $L_X - L_{UV}$ relation was characterized by a high dispersion.", "Through the analysis of different quasar samples with multiple observations available, previous works derived a consistent value for the slope parameter ($\\hat{\\gamma }=0.599\\pm 0.027$ ) and the intrinsic dispersion of the relation ($0.35\\sim 0.40$ dex) [54], [96].", "It was found in subsequent analysis quantifying the observational effects [55] that the magnitude of the intrinsic dispersion can be eventually decreased to the level of $<0.15$ dex.", "They identified a subsample of quasars without the major contributions from uncertainties in the measurement of the (2keV) X-ray flux, absorption in the spectrum in the UV and in the X-ray wavelength ranges, variability of the source and non-simultaneity of the observation in the UV and X-ray bands, inclination effects affecting the intrinsic emission of the accretion disc, and the selection effects due to the Eddington bias [73].", "Besides the lower dispersion in the relation, the reliability and effectiveness of the method strongly depend on the lack of evolution of the relation with redshift [6].", "Finally, [73] produced a final, high-quality catalog of 1598 quasars, by applying several filters (X-ray absorption, dust-reddening effects, observational contaminants in the UV, Eddington bias) to the parent sample from the Sloan Digital Sky Survey (SDSS) quasar catalogues [77], [63] and the XMM-Newton Serendipitous Source Catalogue [71].", "The final results indicated that such refined selection of the sources could effectively mitigate the large dispersion in the $L_X -L_{UV}$ relation, with a tractable amount of scatter avoiding possible contaminants and unknown systematics (see [73] for more details).", "More importantly, the similar analysis has supported the non-evolution of $L_X - L_{UV}$ relation with the redshift, which is supported by the subsequent study involving the intercept parameter $\\hat{\\beta } = 8.24\\pm 0.01$ , the slope parameter $\\hat{\\gamma }=0.633\\pm 0.002$ , and smaller dispersion $\\hat{\\delta }=0.24$ in a new, larger quasar sample [73].", "Therefore, with the gradually refined selection technique and flux measurements, as well as the elimination of systematic errors caused by various aspects, their discovery has a major implication: based on a Hubble diagram of quasars, new measurements of the expansion rate of the Universe could be obtained in the range of $0.036<z<5.10$ .", "Figure: Scatter plot of the flux measurements of 1598 quasars.Figure: Fractional uncertainty of the Einstein radius(Δθ E /θ E \\Delta \\theta _E/\\theta _E) determination as a function of theEinstein radius (θ E \\theta _E)(left panel) and the correspondinghistogram plot (right panel), based on the SL2S sample with HSTimaging and HST+CFHT imaging." ], [ "The observational quasar data", "In this paper, we turn the improved “clean\" sample including 1598 quasars, with reliable measurements of intrinsic X-ray and UV emissions assembled in [73].", "The flux measurements concerning X-ray and UV emissions with the final sample is shown in Fig. 1.", "Possible cosmological application of these standard candles has recently been discussed in the literature [58].", "More recently, the multiple measurements of high-redshift quasars have been used for testing the cosmic distance duality relation (CDDR), based on the relation between the UV and X-ray luminosities of quasars, combined with the VLBI observations of compact structure in radio quasars [97].", "According to the Eq.", "(6), we would be able to derive luminosity distances $D_L(z)$ and hence dimensionless co-moving distances of the lens $d_l$ and the source $d_s$ for each SGL system, from UV and X-ray fluxes of the quasars whose redshifts are equal to $z_l$ and $z_s$ , respectively.", "However, there are two potential problems to be solved.", "First is a high intrinsic scatter ($\\hat{\\delta }$ ) in the quasars sample, based on the UV and X-ray flux measurements.", "Second is that $\\hat{\\beta }$ and $\\hat{\\gamma }$ parameters are unknown.", "Fortunately, [58] used the quasars sample to achieve cosmological test without any external calibrator, treating the slope $\\hat{\\gamma }$ , the intercept $\\hat{\\beta }$ , and the intrinsic scatter $\\hat{\\delta }$ as free parameters to be fit.", "It was revealed that the quasar data can be self-calibrated under such individual optimization within a specified cosmology.", "For example in $\\Lambda $ CDM model one obtains: $\\hat{\\gamma }=0.639\\pm 0.005$ , $\\hat{\\beta }=7.02\\pm 0.012$ , $\\hat{\\delta }=0.231\\pm 0.0004$ , and $\\Omega _m=0.31\\pm 0.05$ .", "In this analysis, we will not confine ourselves to any specific cosmology, but instead we reconstruct the dimensionless co-moving distance function $d(z)$ , modeled as a polynomial expansion in $z$ or logarithmic polynomial expansion of $\\log (1+z)$ [70], [51], [50].", "For the first case, the dimensionless angular diameter distance is parameterized by a third-order polynomial function of redshift $d(z)=z+a_1z^2+a_2z^3,$ with the initial conditions of $d(0)=0$ and $d^{^{\\prime }}(0)=1$ .", "For the second case, we perform empirical fit to the quasar measurements, based on a third-order logarithmic polynomial of $d(z)=ln(10)(x+b_1x^2+b_2x^3),$ with $x=\\log (1+z)$ .", "Note that in the above two parameterizations, ($a_1$ , $a_2$ ) and ($b_1$ , $b_2$ ) represent two sets of constant parameters that need to be optimized and determined by flux measurements data in X-ray and UV emissions.", "Meanwhile, the logarithmic parametrization, benefiting from a more rapid convergence at high redshifts with respect to the standard linear parametrization, has proved to be a more reasonable approximation at high redshifts [73].", "Figure: Left panel: Fractional uncertainty of the velocitydispersion (Δσ v /σ v \\Delta \\sigma _v/\\sigma _v) as a function of the lenssurface brightness (BB) for the SLACS sample, with the best-fittedcorrelation function denoted as the red solid line.", "Right panel: Thedistribution of the velocity dispersion uncertainty for thesimulated SGL sample.Figure: The scatter plot of the simulated lensing systemsbased on the standard polynomial model (left panel) and logarithmicpolynomial model (right panel), with the gradient color denoting thevalue of the Einstein radius.In order to reconstruct the profile of dimensionless co-moving distance $d(z)$ function up to the redshifts $z=5.0$ , we make use of the publicly available code called the emcee https://pypi.python.org/pypi/emcee [34], to obtain the best-fit values and the corresponding $1\\sigma $ uncertainties of relevant parameters ($a_1$ , $a_2$ , $b_1$ and $b_2$ in our case).", "It is worth stressing here that one may worry that the cosmographic expansions are only valid at low redshift.", "However, the recent analysis of high-redshift Hubble diagram indicated that these relations are valid beyond $z\\sim 4$ , although fitting a log polynomial cosmography may hide certain features of the quasar data [95].", "Meanwhile, our results demonstrate that a third-order polynomial function adopted in [73] is sufficient enough to expand the luminosity distance, since the inclusion of higher orders in the polynomial expansion have negligible effect on the final reconstruction results.", "The chi-square $\\chi ^2$ objective function we minimized is defined as $\\chi ^2 = \\sum _{i=1}^{1598} \\frac{[\\log (F_{X,i})-\\Psi _{th} ([F_{UV}]_i;D_L[z_i])]^2}{\\sigma _{F_{X,i}}^2+\\hat{\\delta }^2},$ where $\\hat{\\delta }$ represents the global intrinsic dispersion, the $\\sigma _{F_{X,i}}$ denotes the i-th measurement error of flux $F_{X,i}$ in X-ray waveband.", "The function $\\Psi _{th}$ is defined as $\\Psi _{th}=\\hat{\\beta }+\\hat{\\gamma }\\log (F_{UV,i})+2(\\hat{\\gamma }-1)\\log (D_L(z_i)),$ in terms of the measured fluxes ($F_{X,i}$ , $F_{UV,i}$ ) and the luminosity distance $D_L(z)=c/H_0(1+z)d(z)$ .", "It should be pointed out that the measurement error of the flux in UV band is ignored in this analysis since $\\sigma _{F_{UV,i}}$ is insignificant comparing with $\\sigma _{F_{X,i}}$ and $\\hat{\\delta }$ .", "Meanwhile, we have also assumed a fiducial value for the Hubble constant $H_0=67.4$ km s$^{-1}$ Mpc$^{-1}$ , based on the results obtained from Planck 2018 data (TT, TE, EE+lowE+lensing) [1].", "For the first case, the best-fit quasar parameters and the 68% C.L.", "are determined as $\\hat{\\gamma }=0.613\\pm 0.011$ , $\\hat{\\beta }=7.970\\pm 0.312$ , and $\\hat{\\delta }=0.230\\pm 0.003$ .", "The corresponding results will change to $\\hat{\\gamma }=0.616\\pm 0.011$ , $\\hat{\\beta }=7.530\\pm 0.283$ , and $\\hat{\\delta }=0.230\\pm 0.003$ for the second case." ], [ "The simulated SGL data from LSST", "It is broadly reckoned that the future wide-area and deep surveys, such as the Large Synoptic Survey Telescope [86] and the Dark Energy Survey (DES) [35] will revolutionize the strong lensing science, by increasing the number of known galactic lenses by orders of magnitude.", "More specifically, the forthcoming photometric LSST survey will discover $\\sim 10^5$ strong gravitational lenses [31], the cosmological application of which has become the focus of the forecasted yields of LSST in the near future [20], [21], [56], [25].", "Based on the publicly available simulation programs github.com/tcollett/LensPop explicitly described in [31], we simulate a realistic population of strong lensing systems with early-type galaxies acting as lenses.", "The singular isothermal sphere (SIS) is adopted to model the mass distributions of the lensing galaxies, the number density of which is characterized by the velocity dispersion function (VDF) from the measurements of SDSS Data Release 5 [27].", "Now one important issue should be emphasized in our simulations.", "In order to achieve our $\\Omega _k$ test with the combination of strong lensing and stellar dynamics, valuable additional information such as spectroscopic redshifts ($z_l$ and $z_s$ ) and spectroscopic velocity dispersion ($\\sigma _{ap}$ ) are necessary.", "Since these dedicated observations and substantial follow-up efforts for a sample of $10^5$ SGL systems are expensive, it is more realistic to focus only on a particular well-selected subset of LSST lenses, as was proposed in the recent discussion of multi-object and single-object spectroscopy to enhance Dark Energy Science from LSST [42], [57].", "Therefore, in our analysis the final SGL sample is restricted to 5000 elliptical galaxies with the velocity dispersion of 200 km/s $<\\sigma _{ap} <$ 300 km/s, following the recent investigation of medium-mass lenses to minimize the possible discrepancy between Einstein mass and dynamical mass for the SIS model [17].", "The final simulated results show that the distributions of velocity dispersions and Einstein radii are very similar to those of the current SL2S sample [80].", "Figure: Redshift distribution of quasars used to assess distancesand SGL systems from future LSST survey.Concerning the uncertainty budget, LSST could provide high-quality (sub-arcsecond) imaging data in general, especially in the $g$ -band.", "However, in order to extract the full potential of LSST, obtaining high-resolution images for the lensing systems could also require additional imaging data from space-based facilities (HST), with detailed follow-up of individual SGL systems.", "Meanwhile, the participation of other ground-based facilities makes it possible to derive additional spectroscopic information, i.e., lens redshifts, source redshifts, and velocity dispersion measurements for individual lenses.", "In this analysis, different strategies will be applied to cope with the fractional uncertainty of the Einstein radius and stellar velocity dispersion, considering the possible correlations between the observational precision and other intrinsic properties of the lensing system (such as the mass or the brightness of the lens).", "For the uncertainty of the Einstein radius, we turn to 32 SGL systems recently detected by Strong Lensing Legacy Survey (SL2S), with Canada¨CFrance¨CHawaii Telescope (CFHT) near-infrared ground-based images or Hubble Space Telescope (HST) imaging data [80].", "The HST imaging data were taken with the Advanced Camera for Surveys (ACS; filters: F814W/F606W; exposure time: 800/400s), Wide Field and Planetary Camera 2 (WFPC2; filter: F606W; exposure time: 1200s), and Wide Field Camera 3 (WFC3; filters: F600LP/F475X; exposure time: 720s), which have been observed with HST as part of programs 10876, 11289 (PI: J. P. Kneib) and 11588 (PI: R. Gavazzi).", "In addition to space-based photometry, the NIR images for some of the SL2S lenses were observed with the instrument WIRCam [64] in the $K_s$ , $J$ and $H$ bands.", "We refer the reader to [80] for more detailed information of the CFHT observations for each target (exposure time, etc.).", "The scatter and histogram plots of the fractional uncertainty of Einstein radius are respectively shown in Fig.", "2, concerning the full sample with HST+CFHT imaging and the sub-sample with HST imaging.", "Not surprisingly, most of the lenses with high-precision Einstein radius measurements are derived from systems with HST data.", "Focusing on the full catalogue of SGL systems, one can clearly see a possible correlation between the fractional uncertainty of the Einstein radius and $\\theta _E$ , i.e.", "the lenses with smaller Einstein radii would suffer from large $\\theta _E$ uncertainty, as reported previously in the previous strong lensing analysis.", "Meanwhile, for the full sample with HST+CFHT imaging data (i.e., CFHT image when HST image is not available), the fractional uncertainty of the Einstein radius is taken at the level of 8%, 5% and 3% (the mean uncertainty within each certain $\\theta _E$ bin) for small Einstein radii lenses ($0.5\"<\\theta _E<1.0\"$ ), intermediate Einstein radii lenses ($1\"\\le \\theta _E<1.5\"$ ), and large Einstein radii lenses ($\\theta _E\\ge 1.5$ ).", "Such error strategy will be implemented in the simulations of our LSST lens sample.", "Meanwhile, in the optimistic case, i.e.", "when all of the LSST lenses considered in this work will be observed with HST-like image quality, it is reasonable to take the fractional uncertainty of the Einstein radius at a level of 3% [41].", "For the uncertainty of the velocity dispersion, we turn to 70 SGL systems observed in the Sloan Lens ACS survey (SLACS) [8] and quantitatively analyze its correlation with the lens surface brightness in the $i$ -band.", "The population of strong lenses is dominated by galaxies with velocity dispersion of $\\sigma _{ap}\\sim 230$ km/s (median value), while the Einstein radius distribution is characterized by the median value of $\\theta _E=1.10\"$ .", "Such restricted SLACS lens sample, which falls within the velocity dispersion criterion applied in this analysis (200 km/s $<\\sigma _{ap} <$ 300 km/s), is a good representative sample of what the future LSST survey might yield.", "As can be clearly seen from the results shown in Fig.", "3, strong evidence of anti-correlation between these two quantities is revealed in our analysis.", "Using the best-fitted correlation function derived from the current SGL sample, we obtain in Fig.", "3 the distribution of velocity dispersion uncertainty for the lenses discoverable in forthcoming LSST survey, which is well consistent with the previous strategy of assigning an overall error of 5% on $\\sigma _{ap}$ [16], [98].", "It should be pointed out that LSST will discover a number of fainter, smaller-separation lenses where it is not clear that the same level of precision can be reached.", "Therefore, two selection criteria of the Einstein radius and the $i$ -band magnitude are applied to our particular well-selected subset of LSST lenses ($\\theta _E>0.5\"$ and $m_i<22$ ).", "In this paper, we generate two SGL samples using the standard polynomial and logarithmic polynomial cosmographic reconstructions (taking the best fitted parameters of these reconstructions).", "The scatter plots of the simulated lensing systems based on standard polynomial and logarithmic polynomial cosmographic reconstructions are shown in Fig. 4.", "For a good comparison, Fig.", "5 illustrates the redshift coverage of the current quasar sample and simulated SGL sample, which demonstrates the perfect consistency between the redshift range of high-$z$ quasars and LSST lensing systems.", "Table: Constraints on the cosmic curvature and lens profileparameters for three types of lens models, in the framework ofstandard polynomial and logarithmic polynomial cosmographicreconstructions." ], [ "Results and discussion", "The constraints on the cosmic curvature, based on the simulated SGL systems supplemented with the constructed Hubble diagram of high-redshift quasars, are obtained by maximizing the likelihood ${\\cal L} \\sim \\exp {(-\\chi ^2 / 2)}$ .", "In our analysis, $\\chi ^2$ is constructed as $\\chi ^2(\\textbf {p},\\Omega _k)=\\sum _{i=1}^{N} \\frac{\\left({\\cal D}_{th}({z}_i;\\Omega _k)- {\\cal D}_{obs}({z}_i;\\textbf {p})\\right)^2}{\\sigma _{\\cal D}(z_i)^2},$ where ${\\cal D} = d_{ls}/d_s$ and $N$ is the number of the data points.", "The theoretical distance ratio $\\mathcal {D}_{th}$ dependent on $\\Omega _k$ is given by Eq.", "(REF ), while its observational counterpart is dependent on the lens model adopted Eq.", "(REF ), (REF ) and (REF ).", "Free parameters in these lens model are collectively denoted as $\\textbf {p}$ , and $\\sigma _D$ stands for the uncertainty of the distance ratio expressed as $\\sigma _D^2=\\sigma _{SGL}^{2}+\\sigma _{QSO}^2$ .", "Note that the statistical error of SGL ($\\sigma _{SGL}$ ) is propagated from the measurement uncertainties of the Einstein radius and velocity dispersion, while $\\sigma _{QSO}$ depends on the uncertainties of $d(z)$ function (polynomial and log-ploynomial parameterized distance) reconstructed from the quasars.", "As previously mentioned, the aim of this work is to estimate the cosmic curvature by combining the constructed Hubble diagram of high-redshift quasars with galactic-scale strong lensing systems expected to be seen by the forthcoming LSST survey.", "Therefore, our analysis will be performed on two different reconstruction schemes: the standard polynomial cosmographic reconstruction and the logarithmic polynomial reconstruction.", "The numerical results for the cosmic curvature $\\Omega _k$ and lens model parameters are summarized in Table 1, with the marginalized distributions with 1$\\sigma $ and 2$\\sigma $ contours shown in Fig. 7.", "Figure: The 2-D regions and 1-D marginalized distribution with the1-σ\\sigma and 2-σ\\sigma contours of all parameters from thestandard polynomial (blue dotted line) and the logarithmicpolynomial (green solid line) cosmographic reconstruction, in theframework of three lens models: SIS (left), power-law profile(middle), and extended power-law profile (right), respectively.Let us start our analysis with the standard polynomial cosmographic reconstruction and consider three lens mass density profiles: SIS, power-law model, and extended power-law model.", "For the simplest SIS model, the numerical and graphical results are respectively presented in Table 1 and Fig.", "7, with the best-fitted values for the parameters: $\\Omega _k=0.002\\pm 0.035$ and $f_E=1.000\\pm 0.002$ .", "On the one hand, one may clearly see the degeneracy between the cosmic curvature and the lens model parameters, a tendency revealed and extensively studied in the previous works [98].", "The best-fitted value of $f_E$ is exactly what one could expect knowing how the SGL data were simulated, i.e.", "the SIS velocity dispersion is equal (up to some noise added) to the observed velocity dispersion reported in mock catalog.", "This supports reliability of our procedure.", "On the other hand, a spatially flat universe is supported at much higher confidence levels ($\\Delta \\Omega _k \\sim 10^{-2}$ ), compared with the previous results obtained in [94], [65] by applying the above procedure to different available SGL subsamples.", "In the framework of power-law mass density profile, one can derive constraint on the cosmic curvature as $\\Omega _k=-0.007\\pm 0.029$ , with the best-fitted lens parameter and the corresponding 1$\\sigma $ uncertainty: $\\gamma =2.000\\pm 0.012$ .", "In addition, it is worth noting that when the fractional uncertainty of the Einstein radius is reduced to the level of 3% (with HST imaging), the resulting constraint on the cosmic curvature becomes $\\Delta \\Omega _k=0.028$ .", "Therefore, the estimation of the spatial curvature is more sensitive to the measurements of lens velocity dispersions, which indicates the importance of deriving additional spectroscopic information for individual lenses.", "In the case of extended power-law lens model, we get the weakest fits on the cosmic curvature in the three types of lens models, with the best-fit value and the marginalized 1$\\sigma $ uncertainty $\\Omega _k=0.003\\pm 0.045$ .", "Whereas, our analysis also yield improved constraints on the the total-density and luminosity density profiles, $\\alpha =2.000\\pm 0.014$ and $\\delta =2.171\\pm 0.035$ .", "Compared with the profile of the total mass, the density of luminous baryoic mass has exhibited slight different distribution in early-type galaxies, i.e, the stellar mass profile in the inner region of massive lensing galaxies could fall off steeply than that of the total mass.", "Such tendency, which has been revealed and studied in detail in [17], might helpfully contribute to the understanding of the presence of dark matter, which is differently spatially extended than luminous baryons in early-type galaxies.", "More importantly, besides the different degree of degeneracy between the lens model parameters, our analysis also reveals the strong correlation between $\\Omega _k$ and the parameters characterizing the lens mass profiles.", "Therefore, the large covariances of $\\Omega _k$ with the power-law parameters seen in Fig.", "6 motivates the future use of auxiliary data to improve constraints on the galaxy structure parameters.", "Now, the question is: What is the average $\\alpha $ , $\\delta $ and their intrinsic scatter for the overall population of early-type galaxies?", "One can use high-cadence, high-resolution and multi-filter imaging of the resolved lensed images, to put accurate constrains on the density profiles of galaxies [81], [85], [30], [92], with the newly developed state-of-the-art lens modeling techniques and kinematic modeling methods [82].", "More specifically, the joint lensing and dynamical studies of the SL2S lens sample have demonstrated that the precision of 5% could be obtained for the total-mass density slope inside the Einstein radius [74], [80] Note that the constraints on the mass density slope could be improved to the level of 1%, with precise time delay measurements for the quasar-galaxy strong lensing systems [93].", "Hence, the LSST lenses should be technically supported by dedicated follow-up imaging of the lensed images, possibly performed with more frequent visits on Hubble telescope and smaller ground-based telescopes.", "Meanwhile, observations of the lens galaxy spectra are also needed in order to obtain the kinematic velocity dispersions, which could be satisfied by Adaptive optics (AO) IFU spectroscopy on 8-40m-class telescopes.", "Other possible solutions to this issue can simultaneously satisfy all of these needs, focusing on the combination of AO imaging with slit spectroscopy [42].", "Another important issue is the choice of the $D_L(z)$ function reconstructed from current quasar sample that served for the $\\Omega _k$ estimation.", "Therefore, we perform a similar analysis with the logarithmic polynomial reconstruction and obtained the constraints in the parameter space of $\\Omega _k$ and ($f_E, \\gamma ,\\alpha , \\delta $ ) for three cases of mass density profiles.", "The results are also shown in Fig.", "6 and Table 1.", "Comparing constraints based on the two different reconstructions, we see that confidence regions of different parameters (cosmic curvature and lens model parameters) are well overlapped with each other; hence our results and discussions presented above are robust.", "The strong degeneracies between the cosmic curvature parameter and the lens model parameters are also present as illustrated in Fig. 8.", "More interestingly, compared with the standard polynomial reconstruction, the advantage of the logarithmic polynomial reconstruction is that it could provide more stringent constraints on the cosmic curvature: $\\Omega _k=-0.001\\pm 0.030$ , $\\Omega _k=-0.007\\pm 0.016$ and $\\Omega _k=0.002\\pm 0.031$ , respectively in the framework of three lens mass density profiles (SIS model, power-law spherical model, and extended power-law model).", "Our results indicate that logarithmic parametrization is a more reasonable approximation of theoretical values up to high redshift.", "Such findings, which highlight the importance of choosing a reliable $D_L(z)$ parametrization to better investigate the spatial properties in the early universe, have also been noted and discussed in the previous works [73], [58].", "It should be noted that, even though we focus on the simulated data of SGL systems trying to assess the performance of the method in the future, the reconstructed distances are obtained from the real data.", "Hence, the best-fitted values of $\\Omega _k$ somehow reflect what is supported by the observational data.", "Finally, an accurate reconstruction of cosmic curvature with redshift can greatly contribute to our understanding of the inflation models and fundamental physics.", "In order to address this issue, we divide the full SGL sample into five groups with $\\Delta z_s=1.0$ (based on the source redshifts) and obtain the constraints on $\\Omega _k$ in the framework of SIS model.", "The first subsample has 400 SGL with source redshifts $z_s<1.0$ , the second subsample has 2000 SGL with $1.0<z_s<2.0$ , the third subsample has 1800 SGL with $2.0<z_s<3.0$ , the fourth subsample has 600 SGL with $3.0<z_s<4.0$ , and the fifth subsample contains 200 SGL with source redshifts $4.0<z_s<5.0$ .", "The corresponding results are shown in Fig.", "7, with the $d(z)$ function (polynomial and log-ploynomial parameterized distance) reconstructed from the full quasar sample.", "Compared with the previous analysis performed to test cosmic curvature with different tests involving other popular astrophysical probes including SNe Ia [94] and compact radio quasars [65], it is suggested that our technique, i.e., using luminosity distance of quasars directly derived from the non-linear relation between X-ray and UV luminosities, will considerably improve such direct measurement of the spatial curvature in the early universe ($z\\sim 5.0$ ).", "Figure: Determination of cosmic curvature with five subsamples0<z<1.00<z<1.0, 1.0<z<2.01.0<z<2.0, 2.0<z<3.02.0<z<3.0, 3.0<z<4.03.0<z<4.0 and 4.0<z<5.04.0<z<5.0based on the source redshifts of SGL sample characterized by the SISlens model." ], [ "Conclusions", "In this paper, we re-estimate which precision can be achieved for the cosmic curvature in the near future, on the basis of the distance sum rule in the well-known FLRW metric [70].", "For the purpose, we focus on the simulated data of SGL systems expected to be detected by LSST, combined with the recently assembled catalog of 1598 high-quality quasars calibrated as standard candles.", "It is demonstrated that in the framework of such cosmological-model-independent way, the quasars have better coverage of redshift in SGL systems at high redshifts, which makes it possible to study the spatial properties in the early universe.", "Our main conclusions are summarized as follows: Based on the future measurements of a particular well-selected subset of 5000 LSST lenses (with source redshifts $z\\sim 5.0$ ), the final results show that the the cosmic curvature could be estimated with the precision of $\\Delta \\Omega _k \\sim 10^{-2}$ , which is comparable to that derived from the Planck CMB power spectra (TT, TE, EE+lowP) [1].", "It should be pointed out, even though the simulated data of SGL systems are used in our analysis , the reconstructed distances are obtained from the currently compiled quasar sample.", "In particular, no assumption on the cosmic curvature is made the simulation of the LSST lens sample, i.e., two SGL samples using the standard polynomial and logarithmic polynomial cosmographic reconstructions.", "Therefore, the best-fitted values of $\\Omega _k$ somehow reflect what is supported by the real observational data.", "In our analysis, three types of models, which has been extensively investigated in the literature is considered to describe the structure of the lens.", "Specially, one may obtain the most stringent fits on the cosmic curvature in the power-law lens model, while our $\\Omega _k$ estimation will be strongly affected by the complicated extended power law model (considering the possible difference between the luminosity density profile ($\\nu (r)\\sim r^{-\\delta }$ ) and the total-mass).", "Furthermore, our analysis also reveals the strong correlation between the cosmic curvature ($\\Omega _k$ ) and parameters characterizing the mass profile of lens galaxies ($f_E$ , $\\gamma $ , and $\\alpha $ , $\\delta $ ), which motivates the future investigation of lens density profiles through the combination of state-of-the-art lens modeling techniques and kinematic modeling methods [82].", "There are several sources of systematics that remain to be discussed and addressed in the future analysis.", "The first one is related to the galaxy structure parameters, especially those characterizing the stellar distribution in the lensing galaxies.", "In this paper, we adopted a power-law profile in the spherical Jeans equation, with the aim of connecting the observed velocity dispersion to the dynamical mass.", "However, many modern lens models have considered a two-component model that is the sum of a Sersic-like profile (fit to the stellar light distribution) and a NFW profile (fit to the dark matter distribution) [59].", "The luminosity distribution of spherical galaxies could also be well described by the well-known Hernquist profile, whose behavior follows an inner slope of $r^{-1}$ at small radii and $r^{-4}$ at large radii [40].", "Enlightened by the most recent studies trying to quantify how cosmological constraints are altered by different luminosity density profiles [56], such effect will contribute to the scatter in our cosmic-curvature test.", "This also highlights the importance of auxiliary data in improving constraints on the luminosity density profile, i.e., more high-quality integral field unit (IFU) data are needed to further improve the method in view of upcoming surveys [4].", "Our results indicate that, properly calibrated UV - X-ray relation in quasars has a great potential of becoming an important and precise distance estimator in cosmology.", "Based on the two cosmographic reconstructions of $D_L(z)$ function, our findings also highlight the importance of choosing a reliable reconstruction schemes in order to better investigate the nature of space-time geometry at high redshifts.", "This conclusion is also confirmed by the the reconstruction of cosmic curvature with the source redshift $z_s$ , with accurate observations and spectral characterization of quasars observed by SDSS [77], [63] and XMM [71].", "Finally, this paper seeks to highlights the potential of LSST, which is expected to find extraordinary numbers of new transients every night [78].", "For instance, one should recall that other promising settings for SGL systems have been proposed, for example, galactic-scale strong gravitational lensing systems with Type Ia supernovae [39], [21], [24] and gravitational waves (GWs) as background sources [52], [22], [66], [67].", "Benefit from LSST's wide-field of view and sensitivity, these upcoming improvements on the precision of cosmic curvature estimation will be very helpful for revealing the physical mechanism of cosmic acceleration, or the nature of cosmic origins." ], [ "Acknowledgments", "We are grateful to the referee for useful comments, which allowed to improve our paper substantially.", "This work was supported by National Key R&D Program of China No.", "2017YFA0402600; the National Natural Science Foundation of China under Grants Nos.", "11690023, and 11633001; Beijing Talents Fund of Organization Department of Beijing Municipal Committee of the CPC; the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University; and the Opening Project of Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences.", "M.B.", "was supported by the Key Foreign Expert Program for the Central Universities No.", "X2018002.", "This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.", "This work was partially supported by a grant from the Simons Foundation.", "M.B.", "is grateful for this support.", "He is also grateful for support from Polish Ministry of Science and Higher Education through the grant DIR/WK/2018/12." ] ]

[ [ "Detection and Classification of Internal Faults in Power Transformers\n using Tree-based Classifiers" ], [ "Abstract This paper proposes a Decision Tree (DT) based detection and classification of internal faults in a power transformer.", "The faults are simulated in Power System Computer Aided Design (PSCAD)/ Electromagnetic Transients including DC (EMTDC) by varying the fault resistance, fault inception angle, and percentage of winding under fault.", "A series of features are extracted from the differential currents in phases a, b, and c belonging to the time, and frequency domains.", "Out of these, three features are selected to distinguish the internal faults from the magnetizing inrush and another three to classify faults in the primary and secondary of the transformer.", "DT, Random Forest (RF), and Gradient Boost (GB) classifiers are used to determine the fault types.", "The results show that DT detects faults with 100\\% accuracy and the GB classifier performed the best among the three classifiers while classifying the internal faults." ], [ "Introduction", "Power transformers are an integral part of any Power network.", "They are expensive and once damaged their repairs are time-consuming.", "Thus, their protection is vital for reliable and stable operation of the power system.", "Transformer-protective relays are often tested for their dependability, stability, and speed of operation under different operating conditions.", "The protective relays should operate in cases of faults and avoid tripping the circuit breakers when there is no fault.", "Many relays having different characteristics such as the current differential, Buchholz, Volts/Hz, over current, etc.", "protect the transformers in case of internal conditions (phase (ph)-phase faults, phase-g faults, inter-turn faults, over fluxing).", "Although differential protection has been the primary protection for many years, it suffers from traditional challenges like magnetizing inrush current, saturation of core, Current Transformer (CT) ratio mismatch, external fault with CT saturation, etc.", "Percentage differential relay with second harmonic restraint [1] which detects magnetizing inrush may fail because of the lower content of second harmonics in inrush currents of modern transformers [2] and presence of higher second harmonics in internal faults with CT saturation and with distributed and series compensated lines [3].", "Researchers have proposed different intelligent techniques to distinguish internal faults and magnetizing inrush and classify internal faults in power transformers in the past.", "Tripathy et al.", "used the Probabilistic Neural Network (PNN) to differentiate different conditions in transformer operation [4].", "Genetic algorithm-based two parallel Artificial Neural Network (ANN) was used by Balaga et al.", "to detect and classify faults [5].", "Bigdeli et al.", "[6] proposed Support Vector Machine (SVM) based identification of different transformer winding faults (mechanical and short circuits).", "Patel et al.", "[7] reported that the Relevance Vector Machine (RVM) performs better than PNN and SVM for fault detection and classification.", "Wavelet-based protection and fault classification in an Indirect Symmetrical Phase Shift transformer (ISPST) was used by Bhasker et al.", "[8] [9].", "Shin et al.", "[10] used a fuzzy-based method to overcome mal-operations and enhance fault detection sensitivities of conventional differential relays.", "Segatto et al.", "[11] proposed two different subroutines for transformer protection based on ANN.", "Shah et al.", "[12] used Discrete Wavelet Transforms (DWT) and SVM based differential protection.", "Barbosa et al.", "[13] used Clarke’s transformation and Fuzzy logic based technique to generate a trip signal in case of an internal fault.", "RF classifier was used to discriminate internal faults and inrush in [14].", "Ensemble-based learning was used to classify 40 internal faults in the ISPST and the performance was compared with ANN and SVM in [15].", "In [16] DT based algorithms were used to discriminate the internal faults and other transient disturbances in an interconnected system with ISPST and power transformers.", "DWT and ANN were used for the detection and classification of internal faults in a two-winding three-phase transformer in [17].", "Applicability of seven machine learning algorithms with different sets of features as inputs is used to detect and locate faults in [18].", "Avoiding mal-operation of differential relays during magnetizing inrush and mis-operation during internal faults by correct detection of faults and inrush ensures the security and dependability of the protection system.", "Moreover, the classification of the internal faults may provide information about the faulty side of the transformer and may help in the evaluation of the amount of repair and maintenance needed.", "In a similar direction as in the above-cited publications, this study attempts to use the power of machine learning algorithms to segregate faults from inrush and then determine the type of internal faults in power transformers.", "The main contributions of this paper are: 11,088 cases for 11 different internal faults on the primary and secondary sides (1,008 cases for each) and 720 cases of magnetizing inrush are simulated in the transformer.", "A series of features belonging to the time, frequency, and time-frequency domains are extracted.", "DT distinguishes faults and inrush with an accuracy of 100% with Sample entropy as feature.", "Gradient Boost Classifier gives an accuracy of 95.4% for classifying the internal faults with change quantile and absolute energy as features.", "The rest of the paper is organized as follows.", "Section II illustrates the modeling and simulation of internal faults and magnetizing inrush in PSCAD/EMTDC.", "Section III describes the extraction and selection of features to differentiate faults from inrush and then classify the faults.", "Section IV and V consist of the classification framework and results respectively.", "Section VI concludes the paper.", "Figure: Transformer model showing the ac source, power transformer, multiple-run, faults, transmission line, and load components in PSCAD/EMTDCFigure: Typical 3-phase differential currents from left to right, top to bottom for (1) a-g, (2) b-g, (3) c-g, (4) ab-g, (5) ac-g, (6) bc-g, (7) 3-phase-g, (8) ab, (9) ac, (10) bc, and (11) 3-phase internal faults in the primary of the transformer" ], [ "System Modeling", "PSCAD/EMTDC version 4.2 is used for the modeling and simulation of the internal faults and magnetizing inrush in the power transformer.", "Figure REF shows the model consisting of the ac source, transmission line, power transformer, multi-run component, faults component, and a 3-phase load working at 60Hz.", "The source is rated at 400kV, the transformer is rated at 315 MVA and 400kV/230kV, the transmission line is rated at 230kV with 100km length, and the load is at 240 MW and 180 MVAR.", "Generally, the operation of a power transformer can be categorized in normal operation, internal fault, external fault, overexcitation, and magnetizing inrush or sympathetic inrush conditions.", "This paper specifically focuses on the detection and classification of the internal faults in the transformer.", "Winding phase-g faults (a-g, b-g, c-g), winding phase-phase-g faults (ab-g, ac-g, bc-g), winding phase-phase faults (ab, ac, bc), 3-phase and 3-phase-g faults are simulated using the multi-run component.", "The simulation run-time, fault inception time, and fault duration time are 0.2 secs, 0.1 secs, and 0.05 secs (3 cycles) respectively.", "The internal faults are simulated on the primary and secondary sides of the transformer.", "Different parameters of the transformer are varied to get data for training and testing.", "The fault inception angle is varied from 0 to 345 in steps of 15, the values of fault resistance are: 1, 5, and 10 ohms, and the percentage of winding under fault is varied from 20% to 80% in steps of 10% in the primary and secondary sides under load and no-load conditions.", "Consequently, 11,088 cases of differential currents for internal faults are generated with 1,008 cases for each of the 11 different internal faults.", "Figure REF shows the 3-phase differential currents of the 11 different internal faults.", "The magnetizing inrush currents are obtained by varying the switching angle from 0 to 345 in steps of 15 with residual flux of $\\pm 80\\%, \\pm 40\\%, 0\\% $ in phases a, b, and c for load and no-load conditions.", "The incoming transformer is connected to the source at 0.1s.", "As a result, 720 cases of magnetizing inrush are simulated.", "Figure REF shows the 3-phase differential currents of a typical magnetizing inrush condition for a residual flux of 0%.", "Figure: 3-phase magnetizing inrush differential currents" ], [ "Extraction and Selection of Features", "The differential currents used to extract the features are time-series signals which can be differentiated in many ways.", "The similarity between time series can be established at data-level using Euclidean or Dynamic Time Warping (DTW) [19] distance measures.", "The differential currents from a distinct fault type can also be differentiated from rest using features (e.g., mean, standard deviation (std), frequency, entropy, skewness, kurtosis, or wavelet coefficients) and the distance between the features [20] [21].", "Wang et al.", "[22] extracted features from trend, seasonality, periodicity, serial correlation, skewness, kurtosis, chaos, non-linearity, and self-similarity.", "Wirth et al.", "[23] further extended this approach to multivariate time series signals.", "Kumar et al.", "[24] used a variety of features from time and frequency domains to distinguish binary classes.", "Here, features are extracted from the 3-phase differential currents considering time-domain (e.g., minimum, maximum, median, number of peaks, mean, skewness, number of mean crossings, quantiles, and absolute energy), information-theoretic (sample entropy, approximate entropy, and binned entropy), and coefficients of auto-regression, discrete wavelets, and fast Fourier transforms.", "A number of features are extracted from the a,b, and c phase differential currents.", "More information on these features can be found in [25].", "Sample entropy computed from each of the three phases is used to distinguish an internal fault from an inrush.", "To classify the internal faults three most relevant features are then obtained by comparing the relative importance of the features by using Random Forest in Scikit Learn.", "The three features are a-phase change quantile, b-phase absolute energy, and a-phase change quantile.", "Change quantile calculates the average absolute value of consecutive changes of the time series inside two constant values qh and ql.", "Absolute energy is the sum of the squares of all the data points of the time series.", "Change quantile and absolute energy are expressed mathematically as given by equation (1) & equation (2) respectively.", "Figure: Relative importance of the top three features.$ Change\\ quantile = \\frac{1}{n}\\cdot {\\sum _{a=1}^{n-1} |S_{a+1} - S_{a}| }$ $Abs.\\ energy = \\sum _{a=1}^{n} S_a^{2}$ where, n is the total number of data points in the differential current for phase a, b, and c considered, S represents phase a, b, and c differential currents.", "The relative importance of the three selected features are shown in Figure REF .", "These features are used as the input to the classifiers.", "The importance of a feature (f) at node j is calculated by optimising the objective function ${I(_j,_f)}$ $I(_j,_f) = w_j\\cdot I(D_p)-\\frac{N_l}{N_p}\\cdot I(D_l)-\\frac{N_r}{N_p}\\cdot I(D_r)$ where, f is the feature to perform the split, $w_j $ = number of samples that reach the node, divided by the total number of samples, $D_p$ , $D_l$ and $D_r$ are the dataset of the parent and child nodes, I is \"gini\" impurity measure, and ${N_p},{N_l}$ and ${N_r}$ are number of samples at the parent and child nodes.", "Kernel density estimation plot is a useful statistical tool to picture data shape.", "The shapes of the probability distribution of multiple continuous attributes for different classes can be visualized in the same plot.", "In this Parzen–Rosenblatt window method [26] is used to estimate the underlying probability density of the three selected features for the seven different internal faults.", "Figure REF shows the kernel density estimation plots for the chosen features in phase a, b, and c. The Gaussian kernel function is used for approximation of the univariate features with a bandwidth of 0.1 for average change quantile in phases a and c, and bandwidth of 0.001 for absolute energy in phase b.", "Figure: Kernel Density Estimate plots showing the probability distribution of the selected features for the seven internal faults" ], [ "Fault Detection & Classification", "Percentage restraint differential protection compares the operating current and restraining current and thus differentiates external faults and normal operating conditions from the internal faults.", "Figure REF shows the detection and classification framework in use.", "The work in this paper is applicable for internal faults, magnetizing inrush, and normal operation in the power transformer.", "A DT trained on sample entropy of one-cycle of 3-phase differential currents detects an internal fault or an inrush.", "It sends a trip signal in case an internal fault is detected and computes change quantile and absolute entropy from the 3-phase differential currents.", "DT, RFC, and GB are trained on these features to classify the internal faults into seven different types of faults.", "The training of the classifiers is carried out on 4/5th and testing on 1/5th of the data and grid search is used to find the best hyperparameters.", "Three different classifiers are used for training and testing.", "The first classifier used is the Decision Tree (DT) [27] [28].", "DT classifier works on the principle of splitting the data on the basis of one of the available features which gives the largest Information Gain (IG).", "The splitting is repeated at every node till the child nodes are pure or they belong to the same class.", "IG is the difference between the impurity of the parent node and the child node.", "The impurity of the child node decides the (IG).", "DT is easier to interpret, can be trained quickly and, can model a high degree of nonlinearity in the relationship between the target and the predictor variables [29].", "The second classifier is Random Forest (RF).", "RF classifiers are a collection of decision trees that use majority vote of all the decision trees to make predictions.", "The trees are constructed by choosing random samples from the total training sample with replacement [30].", "The n_estimators hyperparameter which denotes the number of trees is the important parameter to be tuned.", "\"Grid Search\" is used to tune the number of trees in the RF classifier in this case.", "The third classifier used is the Gradient Boost (GB) [31].", "GB is also a collection of decision trees like RF.", "Unlike RF, in GB the trees are added in an iterative manner where each tree learns from the mistakes of previous trees.", "Thus, the learning rate becomes an important hyperparameter in GB.", "Higher learning rate and more number of trees increase the complexity of the model.", "Table: Classification Results for Decision TreeTable: Classification Results for Random ForestTable: Classification Results for Gradient Boost" ], [ "Fault detection", "Sample entropy computed from one-cycle (167 samples) of 3-phase differential currents is used to detect an internal fault or an inrush.", "DT distinguishes internal faults from magnetizing inrush with 100% accuracy.", "Since the number of cases for faults(11088) and inrush (720) is not balanced Synthetic Minority Over-Sampling Technique (SMOTE) [32] is used to create minority synthetic data and NearMiss algorithm is used for under-sampling the majority class." ], [ "Fault classification", "At first, attempts were made to classify the internal faults into 11 classes.", "But, it was observed that line to line to ground faults were misclassified as line to line faults and 3-phase faults as 3-phase-g.", "So, the line to line faults and line to line to ground faults are merged in one class.", "For instance fault types ab and ab-g form one class.", "Similarly, 3-phase and 3 phase-g faults form one class.", "After merging these internal fault types, the resultant number of classes became 7.", "The first three classes a-g, b-g, and c-g consist of 1008 samples, and the rest of the classes consist of 2016 samples.", "The misclassification for DT, RF and GB classifiers between different types of internal faults are reported in Table REF , Table REF , and Table REF respectively.", "The hyperparameters used for the Decision Tree Classifier are: criterion = 'gini', min_samples_leaf = 1, and min_samples_split = 2.", "The training and testing accuracies obtained are 93.65% and 93.6% respectively.", "The best hyperparameters obtained using \"grid search\" for the Random Forest Classifier are: criterion = 'gini', n_estimators = 595, min_samples_leaf = 1, and min_samples_split = 2.", "The training and testing accuracies obtained in this case are 93.61% and 95.1% respectively.", "For Gradient Boost Classifier the best hyperparameters obtained using \"grid search\" are: learning_rate = 0.1, max_depth = 10, n_estimators = 10000, criterion = friedman_mse, min_samples_leaf = 1, and min_samples_split = 2.", "The training and testing accuracies obtained for GB are 93.95% and 95.4% respectively.", "The default hyperparameter values are used for the rest of the hyperparameters for all three classifiers.", "Pre-processing, extraction of features, and selection of important features are done in Python 3.7 and MATLAB 2019, and the DT, RFC, and GB classifiers are built in Python 3.7 using Scikit-learn machine learning library [33].", "Intel Core i7-6560U CPU @ 2.20 GHz having 8 GB RAM is used to perform the simulations and for training the classifiers." ], [ "Conclusion", "Firstly, magnetizing inrush is distinguished from the internal faults in a power transformer in one cycle with DT and secondly, the three DT based classifiers are trained to classify the internal faults by using the most relevant features obtained from the differential currents in phases a, b, and c in this study.", "The random forest feature selection was used to reduce the number of features.", "The DT distinguishes faults from inrush currents without a single error.", "The GB classifier achieved the best accuracy of 95.4% for fault classification whereas, the DT has the lowest accuracy of 93.6% among the three classifiers.", "In this paper, only the internal faults and magnetizing inrush in the power transformer were considered.", "In the future, over-excitation, sympathetic inrush, turn-to-turn faults, and inter-winding faults can be considered for detailed analysis." ] ]

[ [ "Quantum self-learning Monte Carlo with quantum Fourier transform sampler" ], [ "Abstract The self-learning Metropolis-Hastings algorithm is a powerful Monte Carlo method that, with the help of machine learning, adaptively generates an easy-to-sample probability distribution for approximating a given hard-to-sample distribution.", "This paper provides a new self-learning Monte Carlo method that utilizes a quantum computer to output a proposal distribution.", "In particular, we show a novel subclass of this general scheme based on the quantum Fourier transform circuit; this sampler is classically simulable while having a certain advantage over conventional methods.", "The performance of this \"quantum inspired\" algorithm is demonstrated by some numerical simulations." ], [ "Introduction", "Monte Carlo (MC) simulation is a powerful statistical method that is generically applicable to compute statistical quantities of a given system by sampling random variables from its underlying probability distribution.", "A particularly efficient method is the Metropolis-Hastings (MH) algorithm [1]; this realizes a fast sampling from a target distribution via an appropriate acceptance/rejection filtering of the samples generated from an alternative proposal distribution which is easier to sample compared to the target one.", "Therefore, the most important task in this algorithm is to specify an appropriate proposal distribution satisfying the following three conditions; (i) it must be easy to sample, (ii) the corresponding probability can be effectively computed for judging acceptance/rejection of the sample, and (iii) it is rich in representation, meaning that it may lie near the target distribution.", "This is a long-standing challenging problem, but the recent rapid progress of machine learning enables us to take a circumventing pathway for the issue, the self-learning MC [2], [3], [4], [5], which introduces a parametrized proposal distribution and updates the parameters during the sampling so that it is going to mimic the target one.", "Despite of its potential power thanks to the aid of machine learning, this approach has been demonstrated only with a few physical models; for example in Ref.", "[2], for a target hard-to-sample Ising distribution, a parametrized proposal Ising distribution was applied to demonstrate the effectiveness of self-learning MC approach.", "To expand the scope of self-learning MC, we need a systematic method to design a parametrized proposal distribution that is generically applicable to a wide class of target distribution.", "Now we turn our attention to quantum regime, with the hope that the quantum computing might provide us an effective means to attack the above-mentioned problem.", "In fact the so-called quantum supremacy holds for sampling problems; that is, a (non-universal) quantum computer can generate a probability distribution which is hard to sample via any classical (i.e., non-quantum) computer.", "Especially, the Boson sampling [6] and Instantaneous Quantum Polynomial time computations [7], [8] are well known, the former of which is now even within reach of experimental demonstration [9], [10].", "Moreover, a recent trend is to extend this idea to quantum learning supremacy [11], meaning that a quantum circuit is trained to learn a given target distribution faster than any classical computer.", "With the above background in mind, in this paper we study a new type of self-learning MC that uses quantum computing to generate a proposal distribution.", "In fact this scheme satisfies the above-described three conditions.", "First, (i) is already fulfilled as an intrinsic nature of quantum computers.", "Second, it is well known that the task (ii) can be effectively executed using the amplitude estimation algorithm [12], [13], which is in fact twice as fast as the classical correspondence.", "Lastly the above-described fact, the expressive power of quantum computers for generating complex probability distributions, might enable us to satisfy (iii) and mimic a target distribution which is essentially hard to sample via any classical means.", "To realize the learning scheme in a quantum system, we take the variational method, meaning that a parametrized quantum circuit is trained so that its output probability distribution approaches to the target distribution.", "This schematic itself is also employed in the quantum generative modeling [14], [15], but application to MH might be of more practical use for the following reason.", "That is, unlike the generative modeling problem, MH need not generate a proposal distribution that is very close to the target, but rather it requires only a relatively high acceptance ratio and accordingly less demanding quantum computers.", "Of course the most difficult part is to design a parametrized quantum circuit which may successfully generate a suitable proposal distribution.", "Therefore, for the purpose of demonstrating the proof-of-concept, in this work we consider a special type of quantum circuit composed of the quantum Fourier transform (QFT), where the parameters to be learned are assigned corresponding to only the low-frequencies components.", "In fact, thanks to the rich expressive power of the Fourier transform in representing or approximating various functions, the proposed QFT sampler is expected to satisfy the condition (iii), in addition to (i) and (ii).", "We also emphasize that this QFT sampler or its variant (e.g., with different parametrization to cover the high-frequencies components) is a new type of circuit ansatz in the quantum variational method and might be applicable to other problems such as the quantum generative modeling.", "Now we state our bonus theorem; the proposed QFT sampler can be efficiently simulated with a classical means, using the iterative measurement technique [16].", "This is exactly the direction to explore a classical algorithm that fully makes use of quantum feature, i.e., a quantum-inspired algorithm such as [17].", "Actually we will show that this quantum-inspired sampler has a certain advantage over some conventional methods, in addition to the clear merit that the system with e.g., hundreds of qubits is simulable.", "We use the following notations: for a complex column vector $\\mathbf {a}$ , the symbols $\\mathbf {a}^\\dagger $ and $\\mathbf {a}^\\top $ represent its complex conjugate transpose and transpose vectors, respectively.", "Also $\\mathbf {a}^*$ denotes the elementwise complex column vector of $\\mathbf {a}$ .", "Hence $\\mathbf {a}^\\dagger = (\\mathbf {a}^*)^\\top $ .", "Let $p({\\mathbf {x}})$ and $q({\\mathbf {x}})$ be target and proposal probability distributions, respectively.", "To get a sample from $p({\\mathbf {x}})$ , the MH algorithm instead samples from $q({\\mathbf {x}})$ and accepts the result with valid probability determined by the detailed balance conditions.", "More specifically, assume that we have last accepted a sample ${\\mathbf {r}}$ and now obtain a sample $\\tilde{\\mathbf {r}}$ generated from the proposal distribution $q({\\mathbf {x}})$ .", "Then, this sample $\\tilde{\\mathbf {r}}$ is accepted with probability $A({\\mathbf {r}}, \\tilde{\\mathbf {r}})= {\\rm min}\\left\\lbrace 1, ~ \\frac{p(\\tilde{\\mathbf {r}})q({\\mathbf {r}})}{p({\\mathbf {r}})q(\\tilde{\\mathbf {r}})} \\right\\rbrace ,$ which is called the acceptance ratio.", "Note that the value $p(\\mathbf {r})$ is assumed to be easily computable for a given $\\mathbf {r}$ , while its sampling is hard.", "This procedure is repeated until the number of samples becomes enough large; then these accepted samples are governed by the target distribution $p({\\mathbf {x}})$ due to the detailed balance conditions.", "Note that, if $q({\\mathbf {x}})=p({\\mathbf {x}})$ , then the acceptance ratio is always exactly 1, which is maximally efficient; but of course this does not happen because $p({\\mathbf {x}})$ is hard to sample while $q({\\mathbf {x}})$ is assumed to be relatively easy to sample.", "In the context of self-learning MC, a parametric model of the proposal distribution $q({\\mathbf {x}};{\\mathbf {\\theta }})$ is considered, with ${\\mathbf {\\theta }}$ the vector of parameters.", "The self-learning MC aims to learn the parameters so that $q({\\mathbf {x}};{\\mathbf {\\theta }})$ well approximates the target $p({\\mathbf {x}})$ ." ], [ "General form of the quantum self-learning MH algorithm", "Here we describe the quantum sampler executing the self-learning MH algorithm, in the general setting.", "First, for the initial state $|{g^N}\\rangle =|{g}\\rangle ^{\\otimes N}$ with $|{g}\\rangle =[1,0]^\\top $ a qubit state, we apply the parametric unitary gate $U({\\mathbf {\\theta }})$ : $|{\\Psi (\\mathbf {\\theta })}\\rangle = U({\\mathbf {\\theta }}) |{g^N}\\rangle ,$ where $\\mathbf {\\theta }\\in {\\mathbb {C}}^m$ are the parameters to be tuned.", "The reason of taking the complex-valued parameters will be made clear in the next section when specializing to the QFT circuit.", "This state is measured in the computational basis, defining the probability distribution $q(\\mathbf {x};{\\mathbf {\\theta }})= | \\langle {\\mathbf {x}}|{\\Psi (\\mathbf {\\theta })}\\rangle |^2= | \\langle {\\mathbf {x}}|U({\\mathbf {\\theta }}) |{g^N}\\rangle |^2,$ where $\\mathbf {x}$ is the multi-dimensional random variable represented by binaries (an example is given in Appendix A).", "Equation (REF ) is the proposal distribution of our MH algorithm.", "Hence, our goal is to update $\\mathbf {\\theta }$ so that $q(\\mathbf {x};{\\mathbf {\\theta }})$ is going to well approximate the target distribution $p({\\mathbf {x}})$ .", "The learning process for updating $\\mathbf {\\theta }$ is executed via the standard gradient descent method of a loss function, as described below.", "First, for a fixed $\\mathbf {\\theta }$ , we obtain samples ${\\mathbf {r}_1, \\ldots , \\mathbf {r}_B}$ from $q(\\mathbf {x};{\\mathbf {\\theta }})$ by the computational-basis measurement.", "These samples are filtered according to the acceptance probability (REF ), thus producing samples governed by $p(\\mathbf {x})$ .", "Note now that, for a given $\\mathbf {r}$ , the value $q(\\mathbf {r}) = | \\langle {\\mathbf {r}}|{\\Psi (\\mathbf {\\theta })}\\rangle |^2$ must be effectively computed to do this filtering process (recall that the value of $p(\\mathbf {r})$ is assumed to be easily obtained); this computability indeed depends on the structure of $U({\\mathbf {\\theta }})$ , but in general we could apply the quantum amplitude estimation algorithm [12], [13] to reduce the cost for executing this task.", "The samples ${\\mathbf {r}_1, \\ldots , \\mathbf {r}_B}$ are also used to calculate the gradient descent vector of the loss function $L(\\mathbf {\\theta })$ for updating $\\mathbf {\\theta }$ .", "Noting that $L(\\mathbf {\\theta })$ is real while $\\mathbf {\\theta }$ is complex, its infinitesimal change with respect to $\\mathbf {\\theta }$ is given by $\\delta L = \\Big ( \\frac{\\partial L}{\\partial \\mathbf {\\theta }} \\Big )^\\top \\delta \\mathbf {\\theta }+ \\Big ( \\frac{\\partial L}{\\partial \\mathbf {\\theta }^*} \\Big )^\\top \\delta \\mathbf {\\theta }^*= \\Big ( \\frac{\\partial L}{\\partial \\mathbf {\\theta }} \\Big )^\\top \\delta \\mathbf {\\theta }+ \\Big ( \\frac{\\partial L}{\\partial \\mathbf {\\theta }} \\Big )^\\dagger \\delta \\mathbf {\\theta }^*.$ The gradient descent vector for updating the parameter from $\\mathbf {\\theta }$ to $\\mathbf {\\theta }^{\\prime } = \\mathbf {\\theta }+ \\delta \\mathbf {\\theta }$ is thus given by $\\mathbf {\\theta }^{\\prime } = \\mathbf {\\theta }- \\alpha \\Big (\\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}\\Big )^*,$ where $\\alpha >0$ is the learning coefficient; in fact then $\\delta L=-2\\alpha \\Vert \\partial L/\\partial \\mathbf {\\theta }\\Vert ^2 \\le 0$ .", "In this work, the loss function is set to the following cross entropy between $q(\\mathbf {x};\\mathbf {\\theta })$ and $p(\\mathbf {x})$ : $L(\\mathbf {\\theta }) = - \\sum _{\\mathbf {x}} p(\\mathbf {x}) \\log { q(\\mathbf {x};\\mathbf {\\theta }) },$ which is a standard measure for quantify the similarity of two distributions.", "The gradient vector of $L(\\mathbf {\\theta })$ can be computed as follows: $& & \\hspace*{-21.00006pt}\\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}= - \\sum _{\\mathbf {x}}\\frac{p(\\mathbf {x})}{q(\\mathbf {x};\\mathbf {\\theta })}\\frac{\\partial q(\\mathbf {x}; \\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}\\nonumber \\\\ & & \\hspace*{10.0pt}= - \\sum _{\\mathbf {x}} q(\\mathbf {x};\\mathbf {\\theta })\\frac{p(\\mathbf {x})}{q(\\mathbf {x};\\mathbf {\\theta })^2}\\frac{\\partial q(\\mathbf {x}; \\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}\\nonumber \\\\ & & \\hspace*{10.0pt}= \\lim _{n\\rightarrow \\infty } - \\frac{1}{n}\\sum _{i=1}^n\\frac{p(\\mathbf {r}_i)}{q(\\mathbf {r}_i;\\mathbf {\\theta })^2}\\frac{\\partial q(\\mathbf {r}_i; \\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}\\nonumber \\\\ & & \\hspace*{10.0pt}\\simeq - \\frac{1}{B} \\sum _{i=1}^B\\frac{p(\\mathbf {r}_i)}{q(\\mathbf {r}_i;\\mathbf {\\theta })^2}\\frac{\\partial q(\\mathbf {r}_i; \\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}.$ Note that $\\frac{\\partial q(\\mathbf {r}; \\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}= \\left[ |\\langle {\\mathbf {r}}| \\frac{\\partial U(\\mathbf {\\theta })}{\\partial \\theta _1} |{g^N}\\rangle |^2,\\ldots ,|\\langle {\\mathbf {r}}| \\frac{\\partial U(\\mathbf {\\theta })}{\\partial \\theta _m} |{g^N}\\rangle |^2 \\right]$ can be directly estimated when $U(\\mathbf {\\theta })$ is composed of Pauli operators with the parameters $\\lbrace \\theta _i\\rbrace $ corresponding to the rotation angles [18].", "Here we discuss the notable feature and possible quantum advantage of the quantum sampler for the MH algorithm, by referring to the three conditions mentioned in Sec. .", "First, the condition (i) is indeed satisfied because now the sampler is a quantum device that physically produces each measurement result ${\\mathbf {r}}$ according to the proposal probability distribution $q({\\mathbf {x}};{\\mathbf {\\theta }})$ , only in a few micro second in the case of superconducting devices.", "As for the condition (ii), it is in principle possible to effectively compute the probability $q({\\mathbf {r}};{\\mathbf {\\theta }})$ for a given ${\\mathbf {r}}$ , as discussed above in this subsection.", "Lastly for the condition (iii), $q({\\mathbf {x}};{\\mathbf {\\theta }})$ might be able to represent a wide class of probability distribution, which is even hard to sample via any classical means as mentioned in Sec. .", "Realization of this possible quantum advantage of course needs a clever designing of the ansatz $U(\\mathbf {\\theta })$ ." ], [ "The quantum Fourier transform sampler", "To show the proof of principle of the quantum sampler for self-learning MH algorithm, here we consider a special class of circuit composed of QFT, called the QFT sampler.", "Importantly, as will be shown, the QFT sampler is classically simulable, while it might have an advantage over classical algorithms." ], [ "1-dimensional QFT sampler", "We begin with the 1-dimensional QFT sampler; extension to the multi-dimensional case is discussed in the next subsection.", "As illustrated in Fig.REF , this sampler is composed of the QFT operation applied to a $N$ -qubits input state $|{\\rm in}\\rangle = |{{\\psi }({\\mathbf {\\theta }})}\\rangle \\otimes |{g^{N-M}}\\rangle $ where $|{{\\psi }({\\mathbf {\\theta }})}\\rangle =\\theta _0 |{0}\\rangle + \\theta _1 |{1}\\rangle + \\cdots + \\theta _{2^M-1} |{2^M-1}\\rangle ,$ where $\\lbrace |{0}\\rangle , |{1}\\rangle , \\ldots , |{2^M-1}\\rangle \\rbrace $ is the set of computational basis states in $({\\mathbb {C}}^2)^{\\otimes M}={\\mathbb {C}}^{2^M}$ , e.g., $|{0}\\rangle =|{g^M}\\rangle $ .", "That is, the first $M$ -qubits state contains the parameters ${\\mathbf {\\theta }}=[\\theta _0, \\cdots , \\theta _{2^M-1}]^\\top \\in {\\mathbb {C}}^{2^M}$ , while the residual $N-M$ qubits are set to $|{g}\\rangle $ states.", "Note that, if ${\\mathbf {\\theta }} \\in {\\mathbb {R}}^{2^M}$ , then the proposal distribution (REF ) below is limited to an even function, and thus ${\\mathbf {\\theta }}$ must take complex numbers.", "The output of QFT is given by $|{\\Psi (\\mathbf {\\theta })}\\rangle =U_{\\rm QFT}|{{\\rm in}}\\rangle $ , where $U_{\\rm QFT}$ is the QFT unitary operator whose matrix representation is given by $\\langle {k}|U_{\\rm QFT}|{j}\\rangle = \\frac{1}{\\sqrt{2^N}}e^{i2\\pi kj/2^N}.$ Now the measurement on $|{\\Psi (\\mathbf {\\theta })}\\rangle $ in the computational basis yields the probability distribution $q_{\\rm QFT}(x;{\\mathbf {\\theta }}) = |\\langle {x}|{\\Psi (\\mathbf {\\theta })}\\rangle |^2= | \\langle {x}|U_{\\rm QFT}|{{\\rm in}}\\rangle |^2,$ where the random variable $x$ is represented with binaries, i.e., $x \\in \\lbrace 0,1,\\cdots , 2^N-1\\rbrace $ .", "Again, the task of self-learning MH is to update $\\mathbf {\\theta }$ so that the proposal distribution $q_{\\rm QFT}(x;{\\mathbf {\\theta }})$ may approach to the target distribution $p(x)$ .", "To compute the gradient vector (REF ), let us express $\\langle {x}|U_{\\rm QFT}|{\\rm in}\\rangle $ as $\\langle {x}|U_{\\rm QFT}|{\\rm in}\\rangle = {\\mathbf {u}}_x^\\top {\\mathbf {\\theta }},$ where ${\\mathbf {u}}_x$ is the vector composed of the first $2^M$ elements of the $x$ -th row vector of $U_{\\rm QFT}$ ; hence the $j$ th element of ${\\mathbf {u}}_x$ is given by $({\\mathbf {u}}_{x})_j = \\frac{1}{\\sqrt{2^N}}e^{i2\\pi xj/2^N} (j=0, 1, \\cdots , 2^M-1).$ Then the partial derivative of $q_{\\rm QFT}(x;\\mathbf {\\theta }) = ({\\mathbf {u}}_x^\\top {\\mathbf {\\theta }})^*({\\mathbf {u}}_x^\\top {\\mathbf {\\theta }})= ({\\mathbf {u}}_x^\\dagger {\\mathbf {\\theta }}^*)({\\mathbf {u}}_x^\\top {\\mathbf {\\theta }})$ with respect to $\\mathbf {\\theta }$ is given by $\\frac{\\partial q_{\\rm QFT}(x;\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}= ({\\mathbf {u}}_x^\\dagger {\\mathbf {\\theta }}^*) {\\mathbf {u}}_x.$ Hence, from Eq.", "(REF ), the gradient vector of the loss function $L(\\mathbf {\\theta })$ can be calculated as follows: $& & \\hspace*{-21.00006pt}\\Big ( \\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }} \\Big )^*\\simeq -\\frac{1}{B} \\sum _{i=1}^B\\frac{p(r_i)}{q_{\\rm QFT}(r_i;\\mathbf {\\theta })^2}({\\mathbf {u}}_{r_i}^\\top {\\mathbf {\\theta }}){\\mathbf {u}}_{r_i}^*,$ where $\\lbrace r_1,r_2, \\cdots , r_B\\rbrace $ are samples taken from the proposal distribution $q_{\\rm QFT}(x;\\mathbf {\\theta })$ .", "Recall that we filter these samples via the rule (REF ) and thereby obtain samples that are subjected to the target distribution $p(x)$ .", "In this work, instead of the standard gradient update (REF ), we take the following momentum gradient descent [19]; $\\mathbf {\\theta }^{\\prime } = \\mathbf {\\theta }- \\alpha \\mathbf {m}^{\\prime }, ~~~\\mathbf {m}^{\\prime } = \\mu \\mathbf {m} + (1-\\mu )\\Big (\\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }}\\Big )^*,$ where $\\alpha $ and $\\mu $ are the learning coefficients.", "The updated vector $\\mathbf {\\theta }^{\\prime }$ is normalized and substituted into Eq.", "(REF ) for the next learning stage.", "Note that Eq.", "(REF ) is identical to the standard gradient descent when $\\mu =0$ .", "Here let us discuss how the basic conditions (i)-(iii) are reasonably fulfilled by the QFT sampler.", "First, as mentioned before, the QFT sampler enables us to obtain samples enough fast, thanks to the intrinsic feature of quantum devices, and thus it satisfies the condition (i).", "Also the probability $q_{\\rm QFT}(r;{\\mathbf {\\theta }})$ with given $r$ is obtained via just calculating the inner product (REF ) which costs of the order $O(2^M)$ , thereby the condition (ii) is satisfied when $2^M$ is a classically tractable number.", "As for the condition (iii), noting that the parameterized ket vector $|{\\psi (\\mathbf {\\theta })}\\rangle $ serves as the low-frequency components of the shape of the proposal distribution $q_{\\rm QFT}(x;{\\mathbf {\\theta }})$ , the QFT sampler may be able to well approximate the shape of the target $p(x)$ in view of the rich expressive power of Fourier decomposition.", "Now, one might think that this QFT sampler is still out of reach, even in the case of medium-size quantum devices with e.g., $N=100$ qubits.", "However, remarkably, the QFT sampler can be realized in a classical digital computer as long as $N^2$ and $2^M$ are classically tractable numbers; that is, in this regime, this is a “quantum-inspired\" algorithm that can deal with even a random variable on $2^N=2^{100}$ discrete elements.", "The trick relies on the use of the adaptive measurement technique [16], which enables us to sample only by applying $O(2^M + N)$ operations in a classical computer; see Appendix B for a detailed explanation.", "This means that, therefore, the QFT sampler fulfills the condition (i), even as a classical computer.", "Finally, we discuss a possible advantage of our QFT sampler, within a regime of classical sampling method.", "As a classical Fourier-based proposal distribution, one might think to employ the fast Fourier transform (FFT).", "However, to deal with a variable with $2^N$ discretized elements, FFT needs ${\\cal O}(N2^N)$ operations, while QFT can realize the same operation only with ${\\cal O}(N^2)$ gates.", "As is well known, this does not mean a quantum advantage in the typical application scene such as signal processing, because all the amplitude of the QFT-transformed state cannot be effectively determined [20].", "On the other hand, the presented scheme only requires sampling and thus determining $\\lbrace q_{\\rm QFT}(r_i;{\\mathbf {\\theta }}) \\rbrace _{i=1,\\ldots , B}$ rather than all the elements $\\lbrace q_{\\rm QFT}(x;{\\mathbf {\\theta }}) \\rbrace _{x=0,\\ldots , 2^N-1}$ .", "Hence we could say that the developed quantum-inspired algorithm has a solid computational advantage over the known classical algorithm, in the problem of determining a Fourier-based proposal distribution for the MH algorithm." ], [ "Multistage QFT sampler for multi-dimensional distributions", "The QFT sampler discussed above is able to sample only from a 1-dimensional distribution.", "To extend the scheme to the case of multi-dimensional distributions, we here develop a multistage QFT sampler, which is composed of single QFT samplers with their parameters updated via machine learning, as illustrated in Fig.", "REF .", "First, to see the idea, let us consider the case where the $D$ random variables $x_1, \\cdots , x_D$ are independent with each other and subjected to the $D$ -dimensional independent target distribution $p({\\mathbf {x}})$ .", "In this case, the following proposal distribution might work: $q({\\mathbf {x}} ; \\mathbf {\\theta }) = \\prod _{k=1}^{D} q_{\\rm QFT}(x_k ; \\mathbf {\\theta }_k),$ where $\\mathbf {x}=[x_1, \\cdots , x_D]^\\top $ is the $D$ -dimensional random variable represented by binaries $x_k \\in \\lbrace 0, 1,\\cdots , 2^N-1\\rbrace $ .", "Here $q_{\\rm QFT}(x_k ; \\mathbf {\\theta }_k)$ is a 1-dimensional QFT sampler parametrized by the $2^M$ -dimensional complex vector $\\mathbf {\\theta }_k$ ; we summarize these vectors to ${\\mathbf {\\theta }}=[\\mathbf {\\theta }_1^\\top , \\cdots , \\mathbf {\\theta }_D^\\top ]^\\top $ .", "Now based on Eq.", "(REF ) we construct the multistage QFT sampler that can deal with a non-independent multi-dimensional proposal distribution.", "The point is that, as illustrated in Fig.", "REF , the parameter vector $\\mathbf {\\theta }_k$ specifying the $k$ -th 1-dimensional QFT sampler in Eq.", "(REF ) is replaced by a vector of parametrized functions of random variables up to the $(k-1)$ -th stage, i.e., ${\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )$ , whose control parameter $\\mathbf {\\theta }_k$ is to be repeatedly modified through the learning process.", "Hence the proposal distribution is given by $q({\\mathbf {x}};\\mathbf {\\theta })= \\prod _{k=1}^{D} q_{\\rm QFT}\\left( x_k ; {\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k ) \\right).$ Note that this is simply a representation of the joint probability distribution over the multi-dimensional random variables $\\mathbf {x}=[x_1, \\cdots , x_D]^\\top $ via the series of conditional probabilities.", "The gradient of the cross entropy (REF ) is derived in the same way as the 1-dimensional case; $\\Big (\\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }_k}\\Big )^*\\simeq -\\frac{1}{B} \\sum _{i=1}^B \\frac{p(\\mathbf {r}_i)}{q({\\mathbf {r}}_i;\\mathbf {\\theta })}\\frac{({\\mathbf {u}}_{r_k}^\\top {{\\mathbf {f}}_k}){\\mathbf {u}}_{r_k}^*}{ q_{\\rm QFT}(r_k ; {\\mathbf {f}}_k) }\\frac{ \\partial {\\mathbf {f}}_k }{ \\partial \\mathbf {\\theta }_k },$ where $\\lbrace \\mathbf {r}_1, \\cdots , \\mathbf {r}_B \\rbrace $ are samples produced from the proposal distribution $q({\\mathbf {x}};\\mathbf {\\theta })$ .", "Note that each sample $\\mathbf {r}=[r_1, \\cdots , r_D]^\\top $ is formed from $r_k$ produced from the $k$ -th 1-dimensional QFT sampler, as shown in Fig.", "REF .", "Also, as in the 1-dimensional case (REF ), the vector ${\\mathbf {u}}_{x_k}$ is defined through $\\langle {x_k}|U_{\\rm QFT}|{\\rm in}\\rangle = {\\mathbf {u}}_{x_k}^\\top {\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )$ .", "This gradient vector (REF ) is used to update each parameter vector $\\mathbf {\\theta }_k$ using the momentum gradient descent (REF ), which is now of the form $\\mathbf {\\theta }_k^{\\prime } = \\mathbf {\\theta }_k - \\alpha \\mathbf {m}_k^{\\prime }, ~~~\\mathbf {m}_k^{\\prime } = \\mu \\mathbf {m}_k + (1-\\mu )\\Big (\\frac{\\partial L(\\mathbf {\\theta })}{\\partial \\mathbf {\\theta }_k}\\Big )^*,$ where the learning coefficients $(\\alpha , \\mu )$ do not depend on $k$ for simplicity.", "This eventually constitutes the total gradient descent vector minimizing the loss function (REF ) and accordingly move the proposal distribution (REF ) toward the target $p(\\mathbf {x})$ .", "Also recall that we use Eq.", "(REF ) to filter $\\lbrace \\mathbf {r}_1, \\cdots , \\mathbf {r}_B \\rbrace $ to obtain samples that are subjected to $p(\\mathbf {x})$ .", "In this work we examine the following four models of the parameterized function ${\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )$ .", "Identity (Id) model: ${\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )= {\\rm Norm}\\left( \\mathbf {\\theta }_k \\right)= \\frac{\\mathbf {\\theta }_k}{ \\Vert \\mathbf {\\theta }_k \\Vert },$ where $\\Vert \\mathbf {\\theta }\\Vert =\\sqrt{ \\mathbf {\\theta }^\\dagger \\mathbf {\\theta }}$ .", "The ${\\rm Norm}$ function ensures the normalization of $|{{\\mathbf {\\psi }}({\\mathbf {\\theta }})}\\rangle $ .", "This leads to the independent proposal distribution (REF ).", "Linear Basis Linear Regression (LBLR) model: $& & \\hspace*{-6.00006pt}{\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )\\nonumber \\\\ & & \\hspace*{-5.0pt}= {\\rm Norm}\\left( {\\mathbf {w}}_1^{(k)} x_1 + \\cdots + {\\mathbf {w}}_{k-1}^{(k)} x_{k-1} + \\mathbf {b}^{(k)} \\right),\\nonumber $ where in this case the parameters to be learned are the collection of $2^M$ -dimensional complex vectors, $\\mathbf {\\theta }_k = \\lbrace \\mathbf {w}_1^{(k)}, \\mathbf {w}_2^{(k)}, \\cdots , \\mathbf {w}_{k-1}^{(k)}, \\mathbf {b}^{(k)} \\rbrace $ .", "This model outputs a linear combination of their input argument.", "Nonlinear Basis Linear Regression (NBLR) Model: $& & \\hspace*{-6.00006pt}{\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )\\nonumber \\\\ & & \\hspace*{0.0pt}= {\\rm Norm}\\left( \\sum _{j=1}^J {\\mathbf {w}}_{j}^{(k)} \\phi _j^{(k)}(x_1, \\cdots , x_{k-1})+ \\mathbf {b}^{(k)} \\right),\\nonumber $ where $\\mathbf {\\theta }_k = \\lbrace \\mathbf {w}_1^{(k)},\\mathbf {w}_2^{(k)}, \\cdots , \\mathbf {w}_J^{(k)}, \\mathbf {b}^{(k)} \\rbrace $ are the collection of $2^M$ -dimensional complex vectors to be learned, and $\\lbrace \\phi _j^{(k)}(\\cdot )\\rbrace $ are fixed nonlinear basis functions.", "Different set of nonlinear basis functions should be chosen for a different target distribution.", "Neural Network (NN) model: $& & \\hspace*{-6.00006pt}{\\mathbf {f}}_k( x_1, \\cdots , x_{k-1} ; \\mathbf {\\theta }_k )= {\\rm Norm}\\big ( h(\\mathbf {y}^{(k)}) \\big ),\\nonumber \\\\ & & \\hspace*{-5.0pt}\\mathbf {y}^{(k)} = g_4^{(k)} \\circ {\\rm Sg} \\circ g_3^{(k)} \\circ {\\rm Sg} \\circ g_2^{(k)} \\circ {\\rm Sg} \\circ g_1^{(k)} (\\mathbf {x}),\\nonumber $ where $\\circ $ denotes $g_1\\circ g_2(\\mathbf {x})=g_1(g_2(\\mathbf {x}))$ .", "This is a fully connected 4-layers neural network with input $\\mathbf {x}=(x_1, \\cdots , x_{k-1})$ , shown in Fig.", "REF .", "Here $g_j^{(k)}(\\mathbf {x}) = {\\mathbf {W}}_j^{(k)} \\mathbf {x} + \\mathbf {b}_j^{(k)}$ is a linear transformation between the layers; $({\\mathbf {W}}_2^{(k)}, {\\mathbf {W}}_3^{(k)})$ are $256\\times 256$ real parameter matrices, and $({\\mathbf {W}}_1^{(k)}, {\\mathbf {W}}_4^{(k)})$ are $256\\times (k-1)$ and $2^{M+1} \\times 256$ real parameter matrices, respectively; also $({\\mathbf {b}}_1^{(k)}, {\\mathbf {b}}_2^{(k)}, {\\mathbf {b}}_3^{(k)})$ are 256 dimensional real parameter vectors, and ${\\mathbf {b}}_4^{(k)}$ is a $2^{M+1}$ dimensional real parameter vector.", "Hence $\\mathbf {\\theta }_k = (\\mathbf {W}_1^{(k)}, \\cdots , \\mathbf {W}_4^{(k)}, \\mathbf {b}_1^{(k)}, \\cdots , \\mathbf {b}_4^{(k)})$ are the parameters to be optimized.", "${\\rm Sg}(\\mathbf {x})$ is the sigmoid function whose $\\ell $ -th component is defined as $1/(1+e^{-x_\\ell })$ .", "Finally, for the output vector $\\mathbf {y}=[\\mathbf {y}_1^\\top , \\mathbf {y}_2^\\top ]^\\top $ where $\\mathbf {y}_1$ and $\\mathbf {y}_2$ are the $2^M$ real vectors, the function $h$ acts on it and produces a complex vector $h(\\mathbf {y})=\\mathbf {y}_1 + i\\mathbf {y}_2$ .", "Figure: Fully connected neural network." ], [ "Simulation results", "In this section we numerically examine the performance of the QFT sampler for several target distributions.", "For this purpose, the following criteria are used for the evaluation.", "First, the cross entropy (REF ) is employed to see if the proposal distribution approaches toward the target.", "Note that in general the cross entropy does not decrease to zero even in the case when the proposal distribution coincides with the target.", "Hence, to evaluate the distance between the two probability distributions, we use the following Wasserstein distance: ${\\rm W}(P,Q) = \\sup _{\\Vert f \\Vert _{L} \\le 1}\\Big \\lbrace \\mathbf {E}_{x \\sim P} [f(x)] - \\mathbf {E}_{x \\sim Q} [f(x)] \\Big \\rbrace ,$ where the supremum is taken over $f$ contained in the set of functions satisfying the 1-Lipschitz constraint.", "Note that here the Kullback Leibler divergence is not used, because it is applicable only when the supports of the two distributions coincide.", "We also use the average acceptance ratio to evaluate the efficiency of the sampler.", "Mathematically it is defined as the expectation value of Eq.", "(REF ), but in the simulation we simply take the ratio of the number of accepted samples to the number of all samples generated." ], [ "1-dimensional case", "First we discuss the performance of the 1-dimensional QFT sampler.", "The QFT sampler is composed of $N=10$ qubits wherein $M=4$ qubits are used for parametrization; see Fig.", "REF .", "The total learning step is 40000, where in each step $B=32$ samples are used to compute the gradient vector (REF ).", "The learning coefficients in the momentum update rule (REF ) are chosen as $\\alpha =0.01$ and $\\mu =0.9$ .", "In the upper panels of Fig.", "REF , five types of target distributions (red broken lines) and snapshots of proposal distributions in each 1000 steps (black solid lines) as well as the convergent distributions (blue solid lines) are plotted.", "In the lower three panels in each subfigures, CE (Cross entropy), WA (Wasserstein distance), and AC (Average acceptance ratio) are plotted, demonstrating that in each case the proposal distribution $q_{\\rm QFT}(x;\\mathbf {\\theta })$ is approaching to the target $p(x)$ , and accordingly AC increases.", "Note that in the case D, a wave-like structure still remains in the convergent proposal distribution, because of the absence of qubits corresponding to the high-frequency components.", "However, as mentioned in Sec.", ", this is not a serious issue in the framework of MH, which does not require a precise approximation of the target distribution via the proposal one but only needs a proposal distribution realizing relatively high acceptance ratio.", "In fact the lower panel of D in Fig.", "REF demonstrates about 30$\\%$ improvement in the acceptance ratio." ], [ "2-dimensional case", "Next we study several 2-dimensional (i.e., $D=2$ ) target distributions, which are shown in the top right panels from A to G in Fig.", "REF .", "In this case, the proposal distribution (REF ) is of the form $q({\\mathbf {x}};\\mathbf {\\theta })= q_{\\rm QFT}(x_2, {\\mathbf {f}}_2( x_1 ; \\mathbf {\\theta }_2 ))q_{\\rm QFT}(x_1, {\\mathbf {f}}_1( \\mathbf {\\theta }_1 )),$ where ${\\mathbf {f}}_1( \\mathbf {\\theta }_1 )$ is set to be ${\\mathbf {f}}_1( \\mathbf {\\theta }_1 )={\\rm Norm}(\\mathbf {\\theta }_1)$ .", "As for ${\\mathbf {f}}_2( x_1 ; \\mathbf {\\theta }_2 )$ , we study the four models described in Sec.", "REF ; i.e., Id, LBLR, NBLR, and NN models.", "The learning coefficients of the momentum gradient method are set to $\\alpha =0.01$ and $\\mu =0.9$ for all the cases, except that $\\alpha =0.001$ is chosen in the NN model for the cases from A to F. Also the two 1-dimensional QFT samplers in Eq.", "(REF ) are both composed of $N=10$ and $M=4$ qubits, for the cases from A to F, while $N=10$ and $M=5$ for the case G. The basis functions in the NBLR model ${\\mathbf {f}}_2( x_1 ; \\mathbf {\\theta }_2 )$ are chosen as follows; $\\phi _1(x)=\\bar{x}, ~~\\phi _2(x)=\\bar{x}^2, ~~\\phi _3(x)=\\bar{x}^3, ~~\\phi _4(x)=\\sqrt{|\\bar{x}|},$ where $\\bar{x}$ is defined as $\\bar{x}=2x/(2^N-1)-1$ .", "The total learning step is 40,000 for the cases from A to F and 400,000 for the case G. Finally, for all cases, $B=32$ samples are taken to compute the gradient in each step.", "With this setting, the proposal distributions at the final learning step corresponding to the four models (Id, LBLR, NBLR, and NN models from the left to right) are shown in the top panel of Fig.", "REF , and the change of the figure of merits (CE, WA, and AC) are also provided in the bottom panel.", "First, as expected, the Id model can only approximate the distribution having no correlation in the space dimension, i.e., the case of C. On the other hand, the LBLR model acquires a distribution close to the target, for the cases B, D, E, F, in addition to C. The figure of merits, CE, WA, and AC, also reflect this fact.", "(The blue and orange lines in the cases A and C almost coincide.)", "It is notable that the simple LBLR model greatly improves the performance of the Id model.", "To further handle the cases of A and G, some nonlinearities need to be introduced, as demonstrated by the NBLR and NN models.", "In particular, for all the cases from A to G, the NMLR model shows almost the same level of performance as the NN model, which is also supported by the figure of merits.", "Considering the fact that there are some jumps in WA and CE in the NN model, and the fact that the NN model costs a lot in the learning process, our conclusion is that the NMLR is the most efficient model in our case-study." ], [ "Application to a molecular simulation", "The last case-study is focused on the stochastic dynamics of two atoms obeying the Lennard-Jones (LJ) potential field, which is often employed in the field of molecular simulation.", "This problem requires sampling from the Boltzmann distribution $p(\\mathbf {r}_1, \\mathbf {r}_2) \\propto {\\rm exp} \\Big \\lbrace -\\beta {\\rm LJ} (\\Vert \\mathbf {r}_1 - \\mathbf {r}_2 \\Vert ) \\Big \\rbrace ,$ where ${\\rm LJ}(a) = a^{-12} - a^{-6}$ and $\\beta =0.1$ is the inverse temperature.", "$\\mathbf {r}_1 = [r_{1x}, r_{1y}, r_{1z}]^\\top $ and $\\mathbf {r}_2 = [r_{2x}, r_{2y}, r_{2z}]^\\top $ are the vectors of position variables of each atom.", "We apply the 6-stages QFT sampler to generate a proposal distribution for approximating this target distribution.", "The QFT sampler is configured by conditioning the samples in the order $r_{1x} \\rightarrow r_{2x} \\rightarrow r_{1y} \\rightarrow r_{2y} \\rightarrow r_{1z} \\rightarrow r_{2z}$ ; for instance, $q_{\\rm QFT}(r_{1y}, {\\mathbf {f}}_{1y}( r_{1x}, r_{2x}; \\mathbf {\\theta }_{1y} ))$ is conditioned on $(r_{1x}, r_{2x})$ .", "The output of the QFT sampler, $r_i \\in \\lbrace 0,1, \\cdots ,2^N - 1\\rbrace $ , is rescaled to $r_i/0.7(2^N - 1)$ .", "We again employ the four models (Id, LBLR, NBLR, and NN models) as the conditioning functions, where ${\\mathbf {f}}_1( \\mathbf {\\theta }_1 )$ is set to be ${\\mathbf {f}}_1( \\mathbf {\\theta }_1 )={\\rm Norm}(\\mathbf {\\theta }_1)$ .", "Each QFT sampler is composed of $N=10$ and $M=4$ qubits, and the total learning step is 10000, whereas $B=1024$ samples are used to compute the gradient in each step.", "The learning coefficient is $\\alpha =10$ for the Id, LBLR, and NBLR models, while $\\alpha =0.001$ for the NN case; in each case the momentum method with $\\mu =0.9$ is employed to update the parameters.", "The basis functions $\\lbrace \\phi _j^{(k)}(x_1, x_2, ..., x_{k-1}) \\rbrace $ for constructing the NBLR model are set to all the coefficients (except for the constant) of the third-order polynomial function $(1 + \\bar{x}_1 + \\bar{x}_2 + \\cdots + \\bar{x}_{k-1})^3$ with $\\bar{x}= 2x/(2^N - 1) - 1$ .", "Figure REF shows the change of CE and AC; WA was not calculated due to its heavy computational cost.", "We then find that, similar to the 2-dimensional case, the NBLR and NN models succeed in improving both CE and AC, although more learning steps are necessary compared to the previous cases.", "Note also that the Id and LBLR models are almost not updated, implying the validity to introduce nonlinearities in the model of QFT sampler." ], [ "Conclusion", "This paper provided a new self-learning Metropolis-Hastings algorithm based on quantum computing and an important subclass of this sampler that uses the QFT.", "This QFT sampler is shown to be classically simulable, and the effectiveness of this quantum inspired method is supported by several numerical simulations.", "There are a lot of rooms to be investigated more, such as the choice of the optimizer and the conditioning function for constructing the multistage QFT sampler.", "In particular, in extending to the case of multi-dimensional distribution, a completely different schematic than the proposed multistage QFT sampler could be considered.", "Although the QFT sampler offers a certain advantage over some classical sampling schemes as discussed at the end of Sec.", "III A, of course, this quantum inspired algorithm is not what fully makes use of the true power of quantum computation.", "The real goal is to establish a genuine quantum sampler, which may provide a faster sampling with higher acceptance ratio than any conventional classical method.", "Such a direction is in fact found in the literature [21], [22], which are though far beyond the reach of current available devices.", "We hope that the present work might be a first step for bridging this gap.", "This work was supported by IPA MITOU Target Program.", "KF and NY are supported by the MEXT Quantum Leap Flagship Program Grant Number JPMXS0118067394 and JPMXS0118067285, respectively." ], [ "Quantum circuit producing a multi-dimensional probability distribution", "The multi-dimensional probability distribution of random variables $\\mathbf {x}=(x_1, \\cdots , x_n)$ is simply obtained by measuring a state $|{\\Psi }\\rangle \\in \\otimes _{i=1}^n {\\cal H}_i$ in the computational basis $|{\\mathbf {x}}\\rangle =\\otimes _{i=1}^n |{x_i}\\rangle $ , i.e., $p(\\mathbf {x})=|\\langle {\\mathbf {x}}|{\\Psi }\\rangle |^2$ .", "The following is an example of 2-dimensional probability distribution such that the joint probability is explicitly represented via the conditional probability.", "Let $|{\\phi _1}\\rangle \\otimes |{\\phi _2}\\rangle $ be an initial state on ${\\cal H}_1\\otimes {\\cal H}_2$ .", "We consider a quantum circuit of the form $\\mathbf {U}=\\sum _x |{x}\\rangle \\langle {x}|\\otimes U_x$ , where $|{x}\\rangle $ is a computational basis state of ${\\cal H}_1$ , and $U_x$ is a unitary operator conditioned on the value $x$ ; that is, $\\mathbf {U}$ is a sum of controlled unitaries.", "Then the output probability distribution of the measurement result on the state $|{\\Psi }\\rangle =\\mathbf {U} |{\\phi _1}\\rangle \\otimes |{\\phi _2}\\rangle $ , in the computational basis $|{\\mathbf {x}}\\rangle =|{x_1}\\rangle |{x_2}\\rangle $ , is given by $p(\\mathbf {x})= |\\langle {x_1}|\\langle {x_2}| \\mathbf {U} |{\\phi _1}\\rangle |{\\phi _2}\\rangle |^2= |\\langle {x_1}|{\\phi _1}\\rangle |^2 |\\langle {x_2}|U_{x_1}|{\\phi _1}\\rangle |^2.$ Thus, in this case the joint probability $p(\\mathbf {x}) = p(x_1, x_2)$ is explicitly given as a product of the conditional probability $p(x_2 \\, | \\, x_1)=|\\langle {x_2}|U_{x_1}|{\\phi _1}\\rangle |^2$ and $p(x_1)=|\\langle {x_1}|{\\phi _1}\\rangle |^2$ ." ], [ "Classical simulation of the QFT sampler via adaptive measurement", "It is known that the a variety class of quantum circuits composed of the QFT can be simulated on a classical computer [16].", "This fact is indeed applied to our case, as described here.", "The point is twofold; one is that the input state to QFT is now of the form $|{{\\rm in}}\\rangle = |{\\psi (\\mathbf {\\theta })}\\rangle \\otimes |{g^{N-M}}\\rangle = [\\theta _0, \\ldots , \\theta _{2^M-1}, 0, \\ldots , 0]^\\top ,$ where $|{\\psi ({\\mathbf {\\theta }})}\\rangle = [\\theta _0, \\ldots , \\theta _{2^M-1}]^\\top $ is a relatively small quantum state whose entries can be efficiently determined.", "The other is that the circuit is terminated with the QFT part, meaning that the QFT is immediately followed by the measurement process.", "Here we explain a detailed classical algorithm for simulating our circuit in the case of $N=4$ and $M=2$ ; hence $|{{\\rm in}}\\rangle = |{\\psi ({\\mathbf {\\theta }})}\\rangle \\otimes |{g}\\rangle \\otimes |{g}\\rangle $ with $|{\\psi ({\\mathbf {\\theta }})}\\rangle $ a two-qubits state.", "In this case, the original quantum circuit to implement the QFT sampler is shown in Fig.", "REF A.", "The main body of this circuit is QFT, which consists of the Hadamard gates denoted by $H$ and the controlled-$R_n$ gates which rotate the target qubit by acting $R_n = \\exp {[-i2\\pi \\sigma _z 2^{-(n+1)}]}$ on it iff the control qubit is $|{e}\\rangle =[0, 1]^\\top $ .", "The output of QFT is measured in the computational basis, producing the binary output sequence $b_1 b_2 b_3 b_4$ with $b_i\\in \\lbrace 0,1\\rbrace $ .", "To classically simulate this circuit, we utilize the fact that, if the controlled rotation gate is immediately followed by a measurement on a control qubit, this process is replaced with the following feedforward type operation; that is, the control qubit is first measured, and, if the measurement result is “1\" the rotating operation acts on the target qubit.", "Repeating this replacement one by one from right to left in the circuit shown in Fig.", "REF A, we end up Fig.", "REF B.", "That is, all controlled-rotation gates are replaced with the 1-qubit rotation gate which depends on the antecedent measurement results.", "Figure: (A) Original circuit implementation of the QFT sampler on a quantum computer.", "(B) Equivalent circuit implementation of the QFT sampler, in which all the controlledrotation gates are replaced with the feedforward operations.The solid and dashed lines denote quantum and classical operations, respectively.", "(C) Equivalent circuit representation of the circuit B, emphasizing that the change ofthe first 2-qubits (possibly entangled) state |ψ(θ)〉|{\\psi ({\\mathbf {\\theta }})}\\rangle and the bottomtwo |g〉|{g}\\rangle states can be separately and adaptively tracked.Using this classically controlled implementation, the QFT sampler can be simulated on a classical computer as described below.", "In the example considered here, the input $|{{\\rm in}}\\rangle $ is divided into at least three unentangled states $\\lbrace |{\\psi ({\\mathbf {\\theta }})}\\rangle , |{g}\\rangle , |{g}\\rangle \\rbrace $ .", "Hence, these three parts can be tracked separately and adaptively, as shown in Fig.", "REF C. In general, the change of states of the QFT sampler can be computed by repeatedly multiplicating $2^M$ -dimensional matrices on $|{\\psi ({\\mathbf {\\theta }})}\\rangle $ and 2-dimensional matrices on $|{g}\\rangle $ in the adaptive manner, leading that the total computational cost is of the order ${\\mathcal {O}}(2^M + N)$ .", "Therefore, the QFT sampler can be classically simulated as long as $2^M$ is a classically tractable number." ] ]

[ [ "Dimension-Free Bounds on Chasing Convex Functions" ], [ "Abstract We consider the problem of chasing convex functions, where functions arrive over time.", "The player takes actions after seeing the function, and the goal is to achieve a small function cost for these actions, as well as a small cost for moving between actions.", "While the general problem requires a polynomial dependence on the dimension, we show how to get dimension-independent bounds for well-behaved functions.", "In particular, we consider the case where the convex functions are $\\kappa$-well-conditioned, and give an algorithm that achieves an $O(\\sqrt \\kappa)$-competitiveness.", "Moreover, when the functions are supported on $k$-dimensional affine subspaces--e.g., when the function are the indicators of some affine subspaces--we get $O(\\min(k, \\sqrt{k \\log T}))$-competitive algorithms for request sequences of length $T$.", "We also show some lower bounds, that well-conditioned functions require $\\Omega(\\kappa^{1/3})$-competitiveness, and $k$-dimensional functions require $\\Omega(\\sqrt{k})$-competitiveness." ], [ "Introduction", "We consider the convex function chasing (CFC) problem defined by [15], and independently studied under the name smooth online convex optimization (SOCO) by [23], [25].", "In this problem, an online player is faced with a sequence of convex functions over time, and has to choose a good sequence of responses to incur small function costs while also minimizing the movement cost for switching between actions.", "Formally, the player starts at some initial default action $x_0$ , which is usually modeled as a point in $\\mathbb {R}^d$ .", "Convex functions $f_1, f_2, \\dots $ arrive online, one by one.", "Upon seeing the function $f_t: \\mathbb {R}^d\\rightarrow \\mathbb {R}^+$ , the player must choose an action $x_t$ .", "The cost incurred by this action is $ \\Vert x_t - x_{t-1}\\Vert _2 + f_t(x_t), $ the former Euclidean distance term being the movement or switching cost between the previous action $x_{t-1}$ and the current action $x_t$ , and the latter function value term being the hit cost at this new action $x_t$ .", "(he problem can be defined for general metric spaces; in this paper we study the Euclidean case.)", "Given some sequence of functions $\\sigma = f_1, f_2, \\ldots , f_T$ , the online player's total cost for the associated sequence $X = (x_1, x_2, \\ldots , x_T)$ is $\\mathsf {cost}(X, \\sigma ) :=\\sum _{t=1}^T \\Big ( \\Vert x_t - x_{t-1}\\Vert _2 + f_t(x_t) \\Big ).", "$ The competitive ratio for this player is $\\max _{\\sigma } \\frac{\\mathsf {cost}(ALG(\\sigma ), \\sigma )}{\\min _Y \\mathsf {cost}(Y,\\sigma )}$ , the worst-case ratio of the cost of sequence of the player when given request sequence $\\sigma $ , to the cost of the optimal (dynamic) player for it (which is allowed to change its actions but has to also pay for its movement cost).", "The goal is to give an online algorithm that has a small competitive ratio.", "The CFC/SOCO problem is usually studied in the setting where the action space is all of $\\mathbb {R}^d$ .", "We consider the generalized setting where the action space is any convex set $K\\subseteq \\mathbb {R}^d$ .", "Formally, the set $K$ is fixed before the arrival of $f_1$ , and each action $x_t$ must be chosen from $K$ .", "The CFC/SOCO problem captures many other problems arising in sequential decision making.", "For instance, it can be used to model problems in “right-sizing” data centers, charging electric cars, online logistic regression, speech animation, and control; see, e.g., works by [23], [30], [21], [17], [18] and the references therein.", "In all these problems, the action $x_t$ of the player captures the state of the system (e.g., of a fleet of cars, or of machines in a datacenter), and there are costs associated both with taking actions at each timestep, and with changing actions between timesteps.", "The CFC/SOCO problem models the challenge of trading off these two costs against each other.", "One special case of CFC/SOCO is the convex body chasing problem, where the convex functions are indicators of convex sets in $\\mathbb {R}^d$ .", "This special case itself captures the continuous versions of problems in online optimization that face similar tensions between taking near-optimal actions and minimizing movement: e.g., metrical task systems studied by [12], [7], paging and $k$ -server (see [6], [9] for recent progress), and many others.", "Given its broad expressive power, it is unsurprising that the competitiveness of CFC/SOCO depends on the dimension $d$ of the space.", "Indeed, [15] showed a lower bound of $\\sqrt{d}$ on the competitive ratio for convex body chasing, and hence for CFC/SOCO as well.", "However, it was difficult to prove results about the upper bounds: Friedman and Linial gave a constant-competitive algorithm for body chasing for the case $d=2$ , and the function chasing problem was optimally solved for $d=1$ by [8], but the general problem remained open for any higher dimensions.", "The logjam was broken in results by [5], [1] for some special cases, using ideas from convex optimization.", "After intense activity since then, algorithms with competitive ratio $O(\\min (d, \\sqrt{d \\log T}))$ were given for the general CFC/SOCO problem by [4], [28].", "These results qualitatively settle the question in the worst case—the competitive ratio is polynomial in $d$ —although quantitative questions about the exponent for $d$ remain.", "However, this polynomial dependence on the dimension $d$ can be very pessimistic, especially in cases when the convex functions have more structure.", "In these well-behaved settings, we may hope to get better results and thereby escape this curse of dimensionality.", "This motivates our work in this paper: we consider two such settings, and give dimension-independent guarantees for them." ], [ "Well-Conditioned Functions.", "The first setting we consider is when the functions $f_t$ are all well-conditioned convex functions.", "Recall that a convex function has condition number $\\kappa $ if it is $\\alpha $ -strongly-convex and $\\beta $ -smooth for some constants $\\alpha ,\\beta > 0$ such that $\\frac{\\beta }{\\alpha }= \\kappa $ .", "Moreover, we are given a convex set $K$ , and each point $x_t$ we return must belong to $K$ .", "(We call this the constrained CFC/SOCO problem; while constraints can normally be built into the convex functions, it may destroy the well-conditionedness in our setting, and hence we consider it separately.)", "Our first main result is the following: [Upper Bound: Well-Conditioned Functions]theoremOBDmain There is an $O(\\sqrt{\\kappa })$ -competitive algorithm for constrained CFC/SOCO problem, where the functions have condition number at most $\\kappa $ .", "Observe that the competitiveness does not depend on $d$ , the dimension of the space.", "Moreover, the functions can have very different coefficients of smoothness and strong convexity, as long as their ratio is bounded by $\\kappa $ .", "In fact, we give two algorithms.", "Our first algorithm is a direct generalization of the greedy-like Move Towards Minimizer algorithm of [8].", "While it only achieves a competitiveness of $O(\\kappa )$ , it is simpler and works for a more general class of functions (which we called “well-centered”), as well as for all $\\ell _p$ norms.", "Our second algorithm is a constrained version of the Online Balanced Descent algorithm of [14], and achieves the competitive ratio claimed in thm:main.", "We then show a lower bound in the same ballpark: [Lower Bound: Well-Conditioned Functions]theoremLBDmain Any algorithm for chasing convex functions with condition number at most $\\kappa $ must have competitive ratio at least $\\Omega (\\kappa ^{1/3})$ .", "It remains an intriguing question to close the gap between the upper bound of $O(\\sqrt{\\kappa })$ from thm:main and the lower bound of $\\Omega (\\kappa ^{1/3})$ from thm:main-lbd.", "Since we show that $O(\\kappa )$ and $O(\\sqrt{\\kappa })$ are respectively tight bounds on the competitiveness of the two algorithms mentioned above, closing the gap will require changing the algorithm." ], [ "Chasing Low-Dimensional Functions.", "The second case is when the functions are supported on low-dimensional subspaces of $\\mathbb {R}^d$ .", "One such special case is when the functions are indicators of $k$ -dimensional affine subspaces; this problem is referred to as chasing subspaces.", "If $k=0$ we are chasing points, and the problem becomes trivial.", "[15] gave a constant-competitive algorithm for the first non-trivial case, that of $k=1$ or line chasing.", "[3] simplified and improved this result, and also gave an $2^{O(d)}$ -competitive algorithm for chasing general affine subspaces.", "Currently, the best bound even for 2-dimensional affine subspaces—i.e., planes—is $O(d)$ , using the results for general CFC/SOCO.", "[Upper Bound: Low-Dimensional Chasing]theoremSCCmain There is an $O(\\min (k, \\sqrt{k \\log T}))$ -competitive algorithm for chasing convex functions supported on affine subspaces of dimension at most $k$ .", "The idea behind thm:main-sub is to perform a certain kind of dimension reduction: we show that any instance of chasing $k$ -dimensional functions can be embedded into an $(2k+1)$ -dimensional instance, without changing the optimal solutions.", "Moreover, this embedding can be done online, and hence can be used to extend any $g(d)$ -competitive algorithm for CFC/SOCO into a $g(2k+1)$ -competitive algorithm for $k$ -dimensional functions." ], [ "Related Work", "There has been prior work on dimension-independent bounds for other classes of convex functions.", "The Online Balanced Descent ($\\mathsf {OBD}$ ) algorithm of [14] is $\\alpha $ -competitive on Euclidean metrics if each function $f_t$ is $\\alpha $ -locally-polyhedral (i.e., it grows at least linearly as we go away from the minimizer).", "Subsequent works of [19], [18] consider squared Euclidean distances and give algorithms with dimension-independent competitiveness of $\\min (3+O(1/\\alpha ),O(\\sqrt{\\alpha }))$ for $\\alpha $ -strongly convex functions.", "The requirement of squared Euclidean distances in these latter works is crucial for their results: we show in fct:lbd that no online algorithm can have dimension-independent competitiveness for non-squared Euclidean distances if the functions are only $\\alpha $ -strongly convex (or only $\\beta $ -smooth).", "Observe that our algorithms do not depend on the actual value of the strong convexity coefficient $\\alpha $ , only on the ratio between it and the smoothness coefficient $\\beta $ —so the functions $f_t$ may have very different $\\alpha _t, \\beta _t$ values, and these $\\alpha _t$ may even be arbitrarily close to zero.", "A related problem is the notion of regret minimization, which considers the additive gap of the algorithm's cost (REF ) with respect to the best static action $x^*$ instead of the multiplicative gap with respect to the best dynamic sequence of actions.", "The notions of competitive ratio and regret are known to be inherently in conflict: [2] showed that algorithms minimizing regret must have poor competitive ratios in the worst-case.", "Despite this negative result, many ideas do flow from one setting to the other.", "These is a vast body of work where the algorithm is allowed to move for free: see, e.g., books by [13], [20], [29] for many algorithmic ideas.", "This includes bounds comparing to the static optimum, and also to a dynamic optimum with a bounded movement cost [31], [10], [26], [11].", "Motivated by convergence and generalization bounds for learning algorithms, the path length of gradient methods have been studied by [27], [16].", "Results for CFC/SOCO also imply path-length bounds by giving the same function repeatedly: the difference is that these papers focus on a specific algorithm (e.g., gradient flow/descent), whereas we design problem-specific algorithms ($\\mathsf {M2M}$ or $\\mathsf {COBD}$ ).", "The CFC/SOCO problem has been considered in the case with predictions or lookahead: e.g., when the next $w$ functions are available to the algorithm.", "For example, [23], [24] explore the value of predictions in the context of data-server management, and provide constant-competitive algorithms.", "For more recent work see, e.g., [22] and the references therein." ], [ "Definitions and Notation", "We consider settings where the convex functions $f_t$ are non-negative and differentiable.", "Given constants $\\alpha ,\\beta >0$ , a differentiable function $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}$ is $\\alpha $ -strongly-convex with respect to the norm $\\Vert \\cdot \\Vert $ if for all $x,y\\in \\mathbb {R}^d$ , $f(y) - f(x) - \\langle \\nabla f(x), y-x \\rangle \\ge \\frac{\\alpha }{2}\\Vert x-y\\Vert ^2,$ and $\\beta $ -smooth if for all $x,y\\in \\mathbb {R}^d$ , $f(y) - f(x) - \\langle \\nabla f(x), y-x \\rangle \\le \\frac{\\beta }{2}\\Vert x-y\\Vert ^2.", "$ A function $f$ is $\\kappa $ -well-conditioned if there is a constant $\\alpha >0$ for which $f$ is both $\\alpha $ -strongly-convex and $\\alpha \\kappa $ -smooth.", "Of course, we focus on the Euclidean $\\ell _2$ norm (except briefly in §), and hence $\\Vert \\cdot \\Vert $ denotes $\\Vert \\cdot \\Vert _2$ unless otherwise specified.", "In the following, we assume that all our functions $f$ satisfy the zero-minimum property: i.e., that $\\min _y f(y) = 0$ .", "Else we can consider the function $g(x) = f(x) - \\min _y f(y)$ instead: this is also non-negative valued, with the same smoothness and strong convexity as $f$ .", "Moreover, the competitive ratio can only increase when we go from $f$ to $g$ , since the hit costs of both the algorithm and the optimum decrease by the same additive amount." ], [ "Cost.", "Consider a sequence $\\sigma = f_1, \\ldots ,f_T$ of functions.", "If the algorithm moves from $x_{t-1}$ to $x_t$ upon seeing function $f_t$ , the hit cost is $f_t(x_t)$ , and the movement cost is $\\Vert x_t - x_{t-1}\\Vert $ .", "Given a sequence $X = (x_1,\\dots , x_T)$ and a time $t$ , define $\\mathsf {cost}_t(X,\\sigma ) := \\Vert x_t - x_{t-1}\\Vert + f_t(x_t)$ to be the total cost (i.e., the sum of the hit and movement costs) incurred at time $t$ .", "When the algorithm and request sequence $\\sigma $ are clear from context, let $X_{ALG} = (x_1,x_2, \\dots , x_T)$ denote the sequence of points that the algorithm plays on $\\sigma $ .", "Moreover, denote the offline optimal sequence of points by $Y_{OPT} = (y_1,y_2,\\dots , y_T)$ .", "For brevity, we omit mention of $\\sigma $ and let $\\mathsf {cost}_t(ALG) := \\mathsf {cost}_t(X_{ALG}, \\sigma )$ and $\\mathsf {cost}_t(OPT) := \\mathsf {cost}_t(Y_{OPT},\\sigma )$ ." ], [ "Potentials and Amortized Analysis.", "Given a potential $\\Phi _t$ associated with time $t$ , denote $\\Delta _t \\Phi := \\Phi _t - \\Phi _{t-1}$ .", "Hence, for all the amortized analysis proofs in this paper, the goal is to show $ \\mathsf {cost}_t(ALG) + a\\cdot \\Delta _t \\Phi \\le b\\cdot \\mathsf {cost}_t(OPT) $ for suitable parameters $a$ and $b$ .", "Indeed, summing this over all times gives $ \\text{(total cost of }ALG) + a(\\Phi _T - \\Phi _0) \\le b \\cdot \\text{(total cost of }OPT).", "$ Now if $\\Phi _0 \\le \\Phi _T$ , which is the case for all our potentials, we get that the cost of the algorithm is at most $b$ times the optimal cost, and hence the algorithm is $b$ -competitive." ], [ "Deterministic versus Randomized Algorithms.", "We only consider deterministic algorithms.", "This is without loss of generality by the observation in [8]: given a randomized algorithm which plays the random point $X_t$ at each time $t$ , instead consider deterministically playing the “average” point $\\mu _t := \\mathbb {E}[X_t]$ .", "This does not increase either the movement or the hit cost, due to Jensen's inequality and the convexity of the functions $f_t$ and the norm $\\Vert \\cdot \\Vert $ ." ], [ "Algorithms", "We now give two algorithms for convex function chasing: §REF contains the simpler Move Towards Minimizer algorithm that achieves an $O(\\kappa )$ -competitiveness for $\\kappa $ -well-conditioned functions, and a more general class of well-centered functions (defined in sec:well-centered).", "Then §REF contains the Constrained Online Balanced Descent algorithm that achieves the $O(\\sqrt{\\kappa })$ -competitiveness claimed in thm:main." ], [ "Move Towards Minimizer: $O(\\kappa )$ -Competitiveness", "The Move Towards Minimizer ($\\mathsf {M2M}$ ) algorithm was defined in [8].", "The M2M Algorithm.", "Suppose we are at position $x_{t-1}$ and receive the function $f_t$ .", "Let $x^*_t := \\arg \\min _x f_t(x)$ denote the minimizer of $f_t$ .", "Consider the line segment with endpoints $x_{t-1}$ and $x^*_t$ , and let $x_t$ be the unique point on this segment with $\\Vert x_t - x_{t-1}\\Vert = f_t(x_t)-f_t(x_t^*)$ .Such a point is always unique when $f_t$ is strictly convex.", "The point $x_t$ is the one played by the algorithm.", "The intuition behind this algorithm is that one of two things happens: either the optimal algorithm $OPT$ is at a point $y_t$ near $x_t^*$ , in which case we make progress by getting closer to $OPT$ .", "Otherwise, the optimal algorithm is far away from $x_t^*$ , in which case the hit cost of $OPT$ is large relative to the hit cost of $ALG$ .", "Figure: The 𝖬2𝖬\\mathsf {M2M} Algorithm in dimension d=1d=1.As noted in §REF , we assume that $f_t(x_t^*)=0$ , hence $\\mathsf {M2M}$ plays a point $x_t$ such that $\\Vert x_t - x_{t-1}\\Vert = f_t(x_t)$ .", "Observe that the total cost incurred by the algorithm at time $t$ is $ \\mathsf {cost}_t(ALG) = f_t(x_t) + \\Vert x_t - x_{t-1}\\Vert = 2f_t(x_t) =2\\Vert x_t - x_{t-1}\\Vert .", "$" ], [ "The Analysis", "The proof of competitiveness for $\\mathsf {M2M}$ is via a potential function argument.", "The potential function captures the distance between the algorithm's point $x_t$ and the optimal point $y_t$ .", "Specifically, fix an optimal solution playing the sequence of points $Y_{OPT}=(y_1, \\dots , y_T)$ , and define $ \\Phi _t:= \\Vert x_t - y_t\\Vert .", "$ Observe that $\\Phi _0 = 0$ and $\\Phi _t \\ge 0$ .", "Theorem 2.1 With $c := 4+4\\sqrt{2}$ , for each $t$ , $\\mathsf {cost}_t(ALG) + 2\\sqrt{2} \\cdot \\Delta _t \\Phi \\le c\\cdot \\kappa \\cdot \\mathsf {cost}_t(OPT).", "$ Hence, the $\\mathsf {M2M}$ algorithm is $c \\kappa $ -competitive.", "The main technical work is in the following lemma, which will be used to establish the two cases in the analysis.", "Referring to Figure REF , imagine the minimizer for $f_t$ as being at the origin, the point $y$ as being the location of $OPT$ , and the points $x$ and $\\gamma x$ as being the old and new position of $ALG$ .", "Intuitively, this lemma says that either $ALG$ 's motion in the direction of the origin significantly reduces the potential, or $OPT$ is far from the origin and hence has significant hit cost.", "Lemma 2.2 (Structure Lemma) Given any scalar $\\gamma \\in [0,1]$ and any two vectors $x,y\\in \\mathbb {R}^d$ , at least one of the following holds: $ \\Vert y-\\gamma x\\Vert - \\Vert y-x\\Vert \\le -\\tfrac{1}{\\sqrt{2}} \\Vert x-\\gamma x\\Vert $ .", "$\\Vert y\\Vert \\ge \\tfrac{1}{\\sqrt{2}} \\Vert \\gamma x\\Vert $ .", "Let $\\theta $ be the angle between $x$ and $y-\\gamma x$ as in fig:structure-lemma.", "If $\\theta < \\frac{\\pi }{2}$ , then $\\Vert y\\Vert \\ge \\Vert \\gamma x\\Vert $ , and hence condition (ii) is satisfied.", "So let $\\theta \\in [\\frac{\\pi }{2}, \\pi ]$ ; using Figure REF observe that $\\Vert y\\Vert \\ge \\sin (\\theta )\\cdot \\Vert \\gamma x\\Vert .$ Figure: The Proof of Lemma .Suppose condition (i) does not hold.", "Then $ \\Vert y-x\\Vert < \\tfrac{1}{\\sqrt{2}} \\Vert (1-\\gamma ) x\\Vert + \\Vert y-\\gamma x\\Vert .", "$ Since both sides are non-negative, we can square to get $\\Vert y-x\\Vert ^2&< \\frac{1}{2}(1-\\gamma )^2 \\Vert x\\Vert ^2 + \\sqrt{2}\\cdot \\Vert (1-\\gamma )x\\Vert \\cdot \\Vert y-\\gamma x\\Vert + \\Vert y-\\gamma x\\Vert ^2\\\\\\Rightarrow \\Vert y-x\\Vert ^2 - \\Vert y-\\gamma x\\Vert ^2&< \\frac{1}{2}(1-\\gamma )^2 \\Vert x\\Vert ^2 + \\sqrt{2} (1-\\gamma )\\cdot \\Vert x\\Vert \\cdot \\Vert y-\\gamma x\\Vert .$ The law of cosines gives $\\Vert y-x\\Vert ^2 - \\Vert y-\\gamma x\\Vert ^2= (1-\\gamma )^2\\Vert x\\Vert ^2 - 2(1-\\gamma ) \\cos (\\theta )\\cdot \\Vert x\\Vert \\cdot \\Vert y-\\gamma x\\Vert .$ Substituting and simplifying, $ \\frac{1}{2} (1-\\gamma )\\Vert x\\Vert < (\\sqrt{2} +2\\cos (\\theta )) \\Vert y-\\gamma x\\Vert .$ As the LHS is non-negative, $\\cos (\\theta ) > -\\frac{1}{\\sqrt{2}}$ .", "Since $\\theta \\ge \\frac{\\pi }{2}$ , it follows that $\\sin (\\theta ) > \\frac{1}{\\sqrt{2}}$ .", "Now, (REF ) implies that $\\Vert y\\Vert \\ge \\sin (\\theta )\\cdot \\Vert \\gamma x\\Vert \\ge \\frac{1}{\\sqrt{2}}\\Vert \\gamma x\\Vert $ .", "[Proof of thm:k-competitive] First, the change in potential can be bounded as $ \\Delta _t \\Phi = \\Vert x_t - y_t \\Vert - \\Vert x_{t-1} - y_{t-1} \\Vert \\le \\Vert x_t - y_t \\Vert - \\Big ( \\Vert x_{t-1} - y_t \\Vert - \\Vert y_t - y_{t-1} \\Vert \\Big ).", "$ The resulting term $\\Vert y_t - y_{t-1} \\Vert $ can be charged to the movement cost of $OPT$ , and hence it suffices to show that $\\mathsf {cost}_t(ALG) + 2\\sqrt{2}\\cdot \\widetilde{\\Delta }_t \\Phi \\le (4+4\\sqrt{2})\\kappa \\cdot f_t(y_t), $ where $\\widetilde{\\Delta }_t \\Phi := \\Vert x_t - y_t \\Vert - \\Vert x_{t-1} - y_t \\Vert $ denotes the change in potential due to the movement of $ALG$ .", "Recall that $x^*_t$ was the minimizer of the function $f_t$ .", "The claim is translation invariant, so assume $x^*_t = 0$ .", "This implies that $x_t = \\gamma x_{t-1}$ for some $\\gamma \\in (0,1)$ .", "Lemma REF applied to $y = y_t$ , $x = x_{t-1}$ and $\\gamma $ , guarantees that one of the following holds: $\\widetilde{\\Delta }_t \\Phi = \\Vert x_t - y_t \\Vert - \\Vert x_{t-1} - y_t \\Vert \\le -\\tfrac{1}{\\sqrt{2}} \\Vert x_t - x_{t-1}\\Vert $ .", "$\\Vert y_t\\Vert \\ge \\tfrac{1}{\\sqrt{2}} \\Vert x_t\\Vert $ .", "Case I: Suppose $\\widetilde{\\Delta }_t \\Phi \\le -\\tfrac{1}{\\sqrt{2}} \\Vert x_t - x_{t-1}\\Vert $ .", "Since $\\mathsf {cost}_t(ALG) \\le 2\\Vert x_t-x_{t-1}\\Vert $ , $\\mathsf {cost}_t(ALG) + 2\\sqrt{2}\\cdot \\widetilde{\\Delta }_t \\Phi &\\le 2\\Vert x_t-x_{t-1}\\Vert - 2\\Vert x_t-x_{t-1}\\Vert = 0 \\\\&\\le (4+4\\sqrt{2})\\kappa \\cdot f_t(y_t).$ This proves (REF ).", "Case II: Suppose that $\\Vert y_t\\Vert \\ge \\tfrac{1}{\\sqrt{2}} \\Vert x_t\\Vert $ .", "By the well-conditioned assumption on $f_t$ (say, $f_t$ is $\\alpha _t$ -strongly-convex and $\\alpha _t \\kappa $ smooth) and the assumption that 0 is the minimizer of $f_t$ , we have $f_t(x_t)\\le \\frac{\\alpha _t \\kappa }{2} \\Vert x_t\\Vert ^2\\le \\alpha _t\\kappa \\Vert y_t\\Vert ^2\\le 2 \\kappa \\cdot f_t(y_t).$ By the triangle inequality and choice of $x_t$ such that $f_t(x_t) = \\Vert x_t - x_{t-1}\\Vert $ we have $\\widetilde{\\Delta }_t \\Phi =\\Vert x_t - y_t \\Vert - \\Vert x_{t-1} - y_t \\Vert \\le \\Vert x_t - x_{t-1}\\Vert = f_t(x_t).$ Using $\\mathsf {cost}_t(ALG) = 2f_t(x_t)$ , $\\mathsf {cost}_t(ALG) + 2\\sqrt{2}\\cdot \\widetilde{\\Delta }_t \\Phi &\\le 2f_t(x_t) + 2\\sqrt{2} f_t(x_t)\\\\&\\stackrel{(\\ref {eq:k-competitive-2})}{\\le } (4+4\\sqrt{2}) \\kappa \\cdot f_t(y_t).$ This proves (REF ) and hence the bound (REF ) on the amortized cost.", "Now summing (REF ) over all times $t$ , and using that $\\Phi _t \\ge 0 = \\Phi _0$ , proves the competitiveness.", "We extend Theorem REF to the constrained setting (by a modified algorithm); see §.", "We also extend the result to general norms by replacing Lemma REF by Lemma REF ; details appear in §.", "Moreover, the analysis of $\\mathsf {M2M}$ is tight: in Proposition REF we show an instance for which the $\\mathsf {M2M}$ algorithm has $\\Omega (\\kappa )$ -competitiveness." ], [ "Well-Centered Functions", "The proof of thm:k-competitive did not require the full strength of the well-conditioned assumption.", "In fact, it only required that each function $f_t$ is $\\kappa $ -well-conditioned “from the perspective of its minimizer $x_t^*$ ”, namely that there is a constant $\\alpha $ such that for all $x\\in \\mathbb {R}^d$ , $\\frac{\\alpha }{2}\\Vert x-x_t^*\\Vert ^2 \\le f_t(x) \\le \\frac{\\kappa \\alpha }{2} \\Vert x-x_t^*\\Vert ^2.", "$ Motivated by this observation, we define a somewhat more general class of functions for which the $\\mathsf {M2M}$ algorithm is competitive.", "Definition 1 Fix scalars $\\kappa ,\\gamma \\ge 1$ .", "A convex function $f: \\mathbb {R}^d\\rightarrow \\mathbb {R}^+$ with minimizer $x^*$ is $(\\kappa ,\\gamma )$ -well-centered if there is a constant $\\alpha > 0$ such that for all $x\\in \\mathbb {R}^d$ , $\\frac{\\alpha }{2}\\Vert x-x^*\\Vert ^\\gamma \\le f(x) \\le \\frac{\\alpha \\kappa }{2}\\Vert x-x^*\\Vert ^\\gamma .", "$ We can now give a more general result.", "Proposition 2.3 If each function $f_t$ is $(\\kappa ,\\gamma )$ -well centered, then with $c = 2 + 2\\sqrt{2}$ , $\\mathsf {cost}_t(ALG) + 2\\sqrt{2}\\cdot \\Delta _t\\Phi \\le c\\cdot 2^{\\gamma /2}\\kappa \\cdot \\mathsf {cost}_t(OPT).$ Hence, the $\\mathsf {M2M}$ algorithm is $c2^{\\gamma /2}\\kappa $ -competitive.", "Consider the proof of thm:k-competitive and replace (REF ) by $ f_t(x_t)\\le \\frac{\\alpha _t \\kappa }{2} \\Vert x_t\\Vert ^\\gamma \\le \\frac{\\alpha _t\\kappa }{2} \\Vert y_t\\Vert ^\\gamma \\cdot 2^{\\gamma / 2}\\le 2^{\\gamma / 2} \\kappa \\cdot f_t(y_t).$ The rest of the proof remains unchanged." ], [ "Constrained Online Balanced Descent: $O(\\sqrt{\\kappa })$ -Competitiveness", "The move-to-minimizer algorithm can pay a lot in one timestep if the function decreases slowly in the direction of the minimizer but decreases quickly in a different direction.", "In the unconstrained setting, the Online Balanced Descent algorithm addresses this by moving to a point $x_t$ such that $\\Vert x_t - x_{t-1}\\Vert = f_t(x_t)$ , except it chooses the point $x_t$ to minimize $f_t(x_t)$ .", "It therefore minimizes the instantaneous cost $\\mathsf {cost}_t(ALG)$ among all algorithms that balance the movement and hit costs.", "This algorithm can be viewed geometrically as projecting the point $x_{t-1}$ onto a level set of the function $f_t$ ; see fig:obd-vs-mtm.", "Figure: The 𝖮𝖡𝖣\\mathsf {OBD} Algorithm and the comparison to 𝖬2𝖬\\mathsf {M2M}.The point x t-1 x_{t-1} and the function f t f_t with minimizer x t * x_t^* are given.𝖮𝖡𝖣\\mathsf {OBD} plays the point x t x_t and 𝖬2𝖬\\mathsf {M2M} plays the point x ˜ t \\tilde{x}_t.In the constrained setting, it may be the case that $\\Vert x_t - x_{t-1}\\Vert < f_t(x_t)$ for all feasible points.", "Accordingly, the Constrained Online Balanced Descent ($\\mathsf {COBD}$ ) algorithm moves to a point $x_t$ that minimizes $f_t(x_t)$ subject to $\\Vert x_t - x_{t-1}\\Vert \\le f_t(x_t)$ .", "Formally, suppose that each $f_t$ is $\\alpha _t$ -strongly convex and $\\beta _t := \\kappa \\alpha _t$ -smooth, and let $x_t^*$ be the (global) minimizer of $f_t$ , which may lie outside $K$ .", "As before, we assume that $f_t(x_t^*) = 0$ .", "The Constrained OBD Algorithm.", "Let $x_t$ be the solution to the (nonconvex) program $\\min \\lbrace f_t(x) \\mid \\Vert x-x_{t-1}\\Vert \\le f_t(x), x\\in K\\rbrace $ .", "Move to the point $x_t$ .", "(Regarding efficient implementation of $\\mathsf {COBD}$ , see Remark REF .)", "As with $\\mathsf {M2M}$ , the choice of $x_t$ such that $\\Vert x_t-x_{t-1}\\Vert \\le f_t(x_t)$ implies that $ \\mathsf {cost}_t(ALG) = f_t(x_t) + \\Vert x_t - x_{t-1}\\Vert \\le 2f_t(x_t).$" ], [ "The Analysis.", "Again, consider the potential function: $\\Phi _t := \\Vert x_t - y_t \\Vert $ where $x_t$ is the point controlled by the $\\mathsf {COBD}$ algorithm, and $y_t$ is the point controlled by the optimum algorithm.", "We first prove two useful lemmas.", "The first lemma is a general statement about $\\beta $ -smooth functions that is independent of the algorithm.", "Lemma 2.4 Let convex function $f$ be $\\beta $ -smooth.", "Let $x^*$ be the global minimizer of $f$ , and suppose $f(x^*) = 0$ (as discussed in §REF ).", "Then for all $x\\in \\mathbb {R}^d$ , $\\Vert \\nabla f(x) \\Vert \\le \\sqrt{2\\beta f(x)}.$ The proof follows [13].", "Define $z := x -\\frac{1}{\\beta } \\nabla f(x)$ .", "Then $f(x) &\\ge f(x) - f(z) \\\\&\\ge {\\langle \\nabla f(x), x-z \\rangle } - \\frac{\\beta }{2}\\Vert x-z\\Vert ^2 \\\\&= {\\langle \\nabla f(x), \\frac{1}{\\beta } \\nabla f(x) \\rangle } - \\frac{1}{2\\beta } \\Vert \\nabla f(x)\\Vert ^2 = \\frac{1}{2\\beta } \\Vert \\nabla f(x)\\Vert ^2.$ The conclusion follows.", "The second lemma is specifically about $\\mathsf {COBD}$ .", "Lemma 2.5 For each $t \\ge 1$ , there is a constant $\\lambda \\ge 0$ and a vector $n$ in the normal cone to $K$ at $x_t$ such that $x_{t-1} - x_t = \\lambda \\nabla f_t(x_t) + n$ .", "Let $r = \\Vert x_t - x_{t-1}\\Vert $ .", "We claim that $x_t$ is the solution to the following convex program: $\\min \\quad & f_t(x)\\\\\\text{s.t.}", "\\quad & \\Vert x - x_{t-1}\\Vert ^2 \\le r^2\\\\& x\\in K$ Given this claim, the KKT conditions imply that there is a constant $\\gamma \\ge 0$ such that $\\nabla f_t(x) + \\gamma (x_t - x_{t-1})$ is in the normal cone to $K$ at $x_t$ and the result follows.", "We now prove the claim.", "Assume for a contradiction that the solution to this program is a point $z\\ne x_t$ .", "We have $f_t(z)< f_t(x_t)$ .", "Since $z\\in K$ and $x_t$ is the optimal solution to the nonconvex program $\\min \\lbrace f_t(x) \\mid \\Vert x-x_{t-1}\\Vert \\le f_t(x), x\\in K\\rbrace $ , we have $f(z) < \\Vert z-x_{t-1}\\Vert $ .", "But considering the line segment with endpoints $z$ and $x_{t-1}$ , the intermediate value theorem implies that there is a point $z^{\\prime }$ on this segment such that $f(z^{\\prime }) = \\Vert z^{\\prime }-x_{t-1}\\Vert $ .", "This point $z^{\\prime }$ is feasible for the nonconvex program and $f(z^{\\prime }) = \\Vert z^{\\prime }-x_{t-1}\\Vert < \\Vert z-x_{t-1}\\Vert = f(z) < f(x).$ This contradicts the choice of $x_t$ .", "The claim is proven, hence the proof of the lemma is complete.", "Remark 1 The convex program given in the proof can be used to to find $x_t$ efficiently.", "In particular, let $r^*$ denote the optimal value to the nonconvex program.", "For a given $r$ , if the solution to the convex program satisfies $f_t(x) < r$ , then $r^* < r$ .", "Otherwise, $r^* \\ge r$ .", "Noting that $0\\le f_t(x_t)\\le f_t(x_{t-1})$ , run a binary search to find $r^*$ beginning with $r = \\frac{1}{2} f_t(x_{t-1})$ .", "Theorem 2.6 With $c = 2\\sqrt{2\\kappa }$ , for each time $t$ it holds that $\\mathsf {cost}_t (ALG) + c\\cdot \\Delta _t \\Phi \\le 2(2+c)\\cdot \\mathsf {cost}_t(OPT).$ Hence, the $\\mathsf {COBD}$ algorithm is $2(2+c) = O(\\sqrt{\\kappa })$ -competitive.", "As in the proof of thm:k-competitive, it suffices to show that $\\mathsf {cost}_t(ALG) + c\\cdot \\widetilde{\\Delta }_t \\Phi \\le 2(2+c)\\cdot f_t(y_t),$ where $\\widetilde{\\Delta }_t \\Phi := \\Vert x_t - y_t\\Vert - \\Vert x_{t-1} - y_t\\Vert $ is the change in potential due to the movement of $ALG$ .", "There are two cases, depending on the value of $f_t(y_t)$ versus the value of $f_t(x_t)$ .", "In the first case, $f_t(y_t) \\ge \\frac{1}{2}f_t(x_t)$ .", "The triangle inequality bounds $\\widetilde{\\Delta }_t \\Phi = \\Vert x_t - y_t \\Vert - \\Vert x_{t-1} - y_t \\Vert \\le \\Vert x_t - x_{t-1}\\Vert \\le f_t(x_t)$ .", "Also using $\\mathsf {cost}_t(ALG) \\le 2f_t(x_t)$ , we have $\\mathsf {cost}_t(ALG) + c\\cdot \\widetilde{\\Delta }_t \\Phi \\le 2f_t(x_t) + c f_t(x_t)\\le 2(2+c)\\cdot f_t(y_t).$ In the other case, $f_t(y_t) \\le \\frac{1}{2} f_t(x_t)$ .", "Note that this implies that $x_t$ is not the minimizer of $f_t$ on the set $K$ .", "Any move in the direction of the minimizer gives a point in $K$ with lower hit cost, but this point cannot be feasible for the nonconvex program.", "Therefore, at the point $x_t$ , the constraint relating the hit cost to the movement cost is satisfied with equality: $\\Vert x_t-x_{t-1}\\Vert = f_t(x_t)$ .", "Let $\\theta _t$ be the angle formed by the vectors $\\nabla f_t(x_t)$ and $y_t-x_t$ ; see fig:obd-analysis.", "We now have $- \\langle \\nabla f_t(x_t), y_t-x_t\\rangle &\\ge f_t(x_t) - f_t(y_t) + \\frac{\\alpha _t}{2}\\Vert x_t-y_t\\Vert ^2 \\\\& \\ge \\frac{1}{2} f_t(x_t) + \\frac{\\alpha _t}{2}\\Vert x_t-y_t\\Vert ^2 \\\\\\Rightarrow - \\cos \\theta _t&\\ge \\frac{\\frac{1}{2}(f_t(x_t) + \\alpha _t\\Vert x_t-y_t\\Vert ^2)}{\\Vert \\nabla f_t(x_t)\\Vert \\cdot \\Vert x_t-y_t\\Vert }\\\\&\\ge \\frac{\\frac{1}{2}(f_t(x_t) +\\alpha _t\\Vert x_t-y_t\\Vert ^2)}{\\sqrt{2\\alpha _t \\kappa \\; f_t(x_t)}\\cdot \\Vert x_t-y_t\\Vert }\\\\&\\ge \\frac{1}{\\sqrt{2 \\kappa }} $ By Lemma REF , we have $x_{t-1} - x_t = \\lambda \\nabla f_t(x_t) + n$ for some $n$ in the normal cone to $K$ at point $x_t$ .", "Since $y_t\\in K$ we have $\\langle n, y_t - x_t\\rangle \\le 0$ .", "This gives $ - \\langle x_{t-1} - x_t, y_t - x_t \\rangle = - \\langle \\lambda \\nabla f_t(x_t) + n, y_t - x_t \\rangle \\ge - \\lambda \\nabla \\langle f_t(x_t), y_t - x_t\\rangle $ Furthermore, we have $\\lambda \\nabla f_t(x_t) = (x_{t-1} - x_t) - n$ , and since $\\langle x_{t-1} - x_t, n \\rangle < 0$ we have $\\Vert x_{t-1} - x_t\\Vert \\le \\lambda \\Vert \\nabla f_t(x_t)\\Vert $ Let $\\varphi _t$ be the angle formed by the vectors $x_{t-1}-x_t$ and $y_t-x_t$ ; see fig:obd-analysis.", "Figure: Proof of Theorem , case whenf t (y t )≤1 2f t (x t )f_t(y_t) \\le \\frac{1}{2}f_t(x_t).", "B λ B_\\lambda is the sublevel set of f t f_t with x t x_t is on its boundary.Combining the previous three inequalities, $-\\sec \\varphi _t&= \\frac{\\Vert x_{t-1} - x_t\\Vert \\cdot \\Vert y_t - x_t\\Vert }{-\\langle x_{t-1}-x_t, y_t - x_t \\rangle }\\\\&\\le \\frac{\\lambda \\Vert \\nabla f_t(x_t)\\Vert \\cdot \\Vert y_t - x_t\\Vert }{-\\lambda \\langle \\nabla f_t, y_t - x_t\\rangle } \\\\&= -\\sec \\theta _t \\\\&\\le \\sqrt{2\\kappa } = \\frac{c}{2}$ Now the law of cosines gives: $\\Vert x_t - x_{t-1}\\Vert ^2 - 2\\Vert x_t - x_{t-1}\\Vert \\cdot \\Vert x_t-y_t\\Vert \\cos \\varphi _t= \\Vert x_{t-1}-y_t\\Vert ^2 - \\Vert x_t - y_t\\Vert ^2.$ Rearranging: $\\Vert x_t - x_{t-1}\\Vert &= \\left(\\frac{\\Vert x_{t-1} - y_t\\Vert + \\Vert x_t - y_t\\Vert }{\\Vert x_t - x_{t-1}\\Vert - 2\\Vert x_t-y_t\\Vert \\cos \\varphi _t} \\right)\\Big (\\Vert x_{t-1} - y_t\\Vert - \\Vert x_t - y_t\\Vert \\Big )\\\\&\\le \\left(\\frac{\\Vert x_t - x_{t-1}\\Vert + 2\\Vert x_t-y_t\\Vert }{\\Vert x_t - x_{t-1}\\Vert - 2\\Vert x_t-y_t\\Vert \\cos \\varphi _t}\\right) \\Big (\\Vert x_{t-1} - y_t\\Vert - \\Vert x_t - y_t\\Vert \\Big ) \\\\& \\le -(\\sec \\varphi _t) \\cdot \\Big (\\Vert x_{t-1} - y_t\\Vert - \\Vert x_t - y_t\\Vert \\Big ).$ To see the last inequality, recall that $-\\cos \\varphi _t > 0$ ; hence $\\frac{a+b}{a+b(-\\cos \\varphi _t)}\\le \\frac{a+b}{(a+b)(-\\cos \\varphi _t)} = -\\sec \\varphi _t$ .", "Using that $\\mathsf {cost}_t(ALG) = 2\\Vert x_t - x_{t-1}\\Vert $ , we can rewrite the inequality above as $\\mathsf {cost}_t(ALG)- (2\\sec \\varphi _t) \\cdot \\widetilde{\\Delta }_t \\Phi \\le 0.$ Finally, observe that since $y_t\\in B_{\\lambda _t}$ , we have $\\widetilde{\\Delta }_t \\Phi \\le 0$ .", "Using the fact that $-\\sec (\\varphi _t) \\le \\frac{c}{2}$ , $\\mathsf {cost}_t(ALG) + c\\widetilde{\\Delta }_t \\Phi \\le \\mathsf {cost}_t(ALG) - (2\\sec \\varphi _t) \\cdot \\widetilde{\\Delta }_t \\Phi \\le 0 \\le 2(2+c)\\cdot f_t(y_t).$ This completes the proof.", "Again, our analysis of $\\mathsf {COBD}$ is tight: In Proposition REF we show an instance for which the $\\mathsf {COBD}$ algorithm has $\\Omega (\\sqrt{\\kappa })$ -competitiveness, even in the unconstrained setting." ], [ "Chasing Low-Dimensional Functions", "In this section we prove thm:main-sub, our main result for chasing low-dimensional convex functions.", "We focus our attention to the case where the functions $f_t$ are indicators of some affine subspaces $K_t$ of dimension $k$ , i.e., $f_t(x) = 0$ for $x \\in K_t$ and $f_t(x) = \\infty $ otherwise.", "(The extension to the case where we have general convex functions supported on $k$ -dimensional affine subspaces follows the same arguments.)", "The main ingredient in the proof of chasing low-dimensional affine subspaces is the following dimension-reduction theorem: Theorem 3.1 Suppose there is an $g(d)$ -competitive algorithm for chasing convex bodies in $\\mathbb {R}^d$ , for each $d \\ge 1$ .", "Then for any $k \\le d$ , there is a $g(2k+1)$ -competitive algorithm to solve instances of chasing convex bodies in $\\mathbb {R}^d$ where each request lies in an affine subspace of dimension at most $k$ .", "In particular, thm:reduction implies that there is a $(2k+1)$ -competitive algorithm for chasing subspaces of dimension at most $k$ , and hence proves thm:main-sub.", "Suppose we have a chasing convex bodies instance $K_1, K_2, \\dots , K_T$ such that each $K_t$ lies in some $k$ -dimensional affine subspace.", "We construct another sequence $K_1^{\\prime }, \\dots , K_T^{\\prime }$ such that (a) there is a single $2k+1$ dimensional linear subspace $L$ that contains each $K^{\\prime }_t$ , and (b) there is a feasible point sequence $x_1,\\dots , x_T$ of cost $C$ for the initial instance if and only if there is a feasible point sequence $x^{\\prime }_1,\\dots , x^{\\prime }_T$ for the transformed instance with the same cost.", "We also show that the transformation from $K_t$ to $K_t^{\\prime }$ , and from $x_t^{\\prime }$ back to $x_t$ can be done online, resulting in the claimed algorithm.", "Let $\\text{span}(S)$ denote the affine span of the set $S\\subseteq \\mathbb {R}^d$ .", "Let $\\dim (A)$ denote the dimension of an affine subspace $A\\subseteq \\mathbb {R}^d$ .", "The construction is as follows: let $L$ be an arbitrary $(2k+1)$ -dimensional linear subspace of $\\mathbb {R}^d$ that contains $K_1$ .", "We construct online a sequence of affine isometries $R_1,\\dots , R_T$ such that for each $t > 1$ : $R_t(K_t)\\subseteq L$ .", "$\\Vert R_t(x_t) - R_{t-1}(x_{t-1}) \\Vert = \\Vert x_t - x_{t-1}\\Vert $ for any $x_{t-1}\\in K_{t-1}$ and $x_t\\in K_t$ .", "Setting $x_t^{\\prime } = R_t(x_t)$ then achieves the goals listed above.", "To get the affine isometry $R_t$ we proceed inductively: let $R_1$ be the identity map, and suppose we have constructed $R_{t-1}$ .", "Let $A_t:= \\text{span}(R_{t-1}(K_t) \\cup R_{t-1}(K_{t-1}))$ .", "Note that $\\dim (A_t) \\le 2k+1$ .", "Let $\\rho _t$ be an affine isometry that fixes $\\text{span}(R_{t-1}(K_{t-1}))$ and maps $\\text{span}(R_{t-1}(K_t))$ into $L$ .", "Now define $R_t = \\rho _t\\circ R_{t-1}$ .", "Property (i) holds by construction.", "Moreover, since $x_{t-1}\\in K_{t-1}$ , we have $R_t(x_{t-1}) = R_{t-1}(x_{t-1})$ .", "Furthermore, $R_t$ is an isometry and hence preserves distances.", "Thus, $\\Vert R_t(x_t) - R_{t-1}(x_{t-1}) \\Vert = \\Vert R_t(x_t) - R_t(x_{t-1}) \\Vert = \\Vert x_t - x_{t-1}\\Vert .", "$ This proves (ii).", "Note that $R(x_1,\\dots , x_T) := (R_1(x_1), \\dots , R_T(x_T))$ is a cost-preserving bijection between point sequences that are feasible for $\\lbrace K_t\\rbrace _t$ and $\\lbrace K_t^{\\prime }\\rbrace _t$ respectively.", "It now follows that the instances $\\lbrace K_t\\rbrace _t$ and $\\lbrace K_t^{\\prime }\\rbrace _t$ are equivalent in the sense that $OPT(K_1^{\\prime },\\dots , K_T^{\\prime }) = OPT(K_1,\\dots , K_T)$ , and an algorithm that plays points $x^{\\prime }_t\\in K^{\\prime }_t$ can be converted into an algorithm of equal cost that plays points $x_t\\in K_t$ by letting $x_t = R_t^{-1}(x^{\\prime }_t)$ .", "However, each of $K_1^{\\prime }, \\dots , K_T^{\\prime }$ is contained in the $(2k+1)$ dimensional subspace $L$ , and thus we get the $g(2k+1)$ -competitive algorithm.", "Using the results for CFC/SOCO, this immediately gives an $O(\\min (k, \\sqrt{k \\log T}))$ -competitive algorithm to chase convex bodies lying in $k$ -dimensional affine subspaces.", "Moreover, the lower bound of [15] immediately extends to show an $\\Omega (\\sqrt{k})$ lower bound for $k$ -dimensional subspaces.", "Finally, the proof for $k$ -dimensional functions follows the same argument, and is deferred for now." ], [ "Lower Bounds", "In this section, we show a lower bound of $\\Omega (\\kappa ^{1/3})$ on the competitive ratio of convex function chasing for $\\kappa $ -well-conditioned functions.", "We also show that our analyses of the $\\mathsf {M2M}$ and $\\mathsf {COBD}$ algorithms are tight: that they have competitiveness $\\Omega (\\kappa )$ and $\\Omega (\\sqrt{\\kappa })$ respectively.", "In both examples, we take $K=\\mathbb {R}^d$ to be the action space." ], [ "A Lower Bound of $\\Omega (\\kappa ^{1/3})$", "The idea of the lower bound is similar to the $\\Omega (\\sqrt{d})$ lower bound [15], which we now sketch.", "In this lower bound, the adversary eventually makes us move from the origin to some vertex $\\varepsilon = (\\varepsilon _1, \\varepsilon _2, \\ldots , \\varepsilon _d)$ of the hypercube $\\lbrace -1,1\\rbrace ^d$ .", "At time $t$ , the request $f_t$ forces us to move to the subspace $\\lbrace x \\mid x_i = \\varepsilon _i\\; \\forall i \\le t \\rbrace $ .", "Not knowing the remaining coordinate values, it is best for us to move along the coordinate directions and hence incur the $\\ell _1$ distance of $d$ .", "However the optimal solution can move from the origin to $\\varepsilon $ along the diagonal and incur the $\\ell _2$ distance of $\\sqrt{d}$ .", "Since the functions $f_t$ in this example are not well-conditioned, we approximate them by well-conditioned functions; however, this causes the candidate $OPT$ to also incur nonzero hit costs, leading to the lower bound of $\\Omega (\\kappa ^{1/3})$ when we balance the hit and movement costs.", "We begin with a lemma analyzing a general instance defined by several parameters, and then achieve multiple lower bounds by appropriate choice of the parameters.", "Lemma A.1 Fix a dimension $d$ , constants $\\gamma >0$ and $\\lambda \\ge \\mu \\ge 0$ .", "Given any algorithm $ALG$ for chasing convex functions, there is a request sequence $f_1, f_2, \\dots , f_d$ that satisfies: Each $f_t$ is $2\\mu $ -strongly-convex and $2\\lambda $ -smooth (hence $(\\lambda /\\mu )$ -well-conditioned.)", "$OPT \\le \\gamma (1+\\mu d^{3/2} \\gamma ) \\sqrt{d}$ .", "$ALG \\ge (\\gamma - \\frac{1}{4\\lambda })d$ .", "Consider the instance where at each time $t \\in \\lbrace 1, \\ldots , d\\rbrace $ , we pick a uniformly random value $\\varepsilon _t \\in \\lbrace -1,1\\rbrace $ , and set $f_t(x) = \\lambda \\sum _{i=1}^t (x_i-\\gamma \\varepsilon _i)^2 + \\mu \\sum _{i=t+1}^d x_i^2.$ One candidate for $OPT$ is to move to the point $\\gamma \\varepsilon := (\\gamma \\varepsilon _1, \\gamma \\varepsilon _2, \\ldots , \\gamma \\varepsilon _d)$ , and take all the functions at that point.", "The initial movement costs $\\gamma \\sqrt{d}$ , and the $t^{\\text{th}}$ timestep costs $f_t(\\gamma \\varepsilon ) = \\mu (d-t)\\gamma ^2$ .", "Hence, the total cost over the sequence is at most $\\gamma \\sqrt{d} + \\mu \\binom{d}{2}\\gamma ^2\\le \\gamma \\Big (1+\\mu d^{3/2} \\gamma \\Big ) \\sqrt{d}.$ Suppose the algorithm is at the point $\\mathbf {z} = (z_1, \\ldots , z_d)$ after timestep $t-1$ , and it moves to point $\\mathbf {z}^{\\prime } = (z^{\\prime }_1, \\ldots , z^{\\prime }_d)$ at the next timestep.", "Moreover, suppose the algorithm sets $z^{\\prime }_t=a$ when it sees $\\varepsilon _t = 1$ , and sets $z^{\\prime }_t = b$ if $\\varepsilon _t=-1$ .", "Then for timestep $t$ , the algorithm pays in expectation at least $&\\frac{1}{2} [\\lambda (a - \\gamma )^2 + |a - z_t|]+ \\frac{1}{2} [\\lambda (b+\\gamma )^2 + | b - z_t| ] \\\\&= \\frac{\\lambda }{2} \\left[(a^2 - 2\\gamma a + \\gamma ^2)+ (b^2 + 2\\gamma b + \\gamma ^2) \\right]+ \\frac{1}{2}[|a - z_t| + | b - z_t| ] \\\\&\\ge \\frac{\\lambda }{2} \\left[(a^2 - 2\\gamma a + \\gamma ^2)+ (b^2 + 2\\gamma b + \\gamma ^2) \\right] + \\frac{1}{2} (a -b) \\\\&= \\frac{\\lambda }{2}\\left[\\left(a^2 - \\left(2\\gamma -\\frac{1}{\\lambda }\\right) a + \\gamma ^2\\right)+ \\left(b^2 + \\left(2\\gamma -\\frac{1}{\\lambda }\\right) b + \\gamma ^2\\right) \\right] \\\\&\\ge \\gamma - \\frac{1}{4\\lambda }.$ The last inequality follows from choosing $a = \\gamma - 1/(2ł)$ and $b = \\gamma + 1/(2ł)$ to minimize the respective quadratics.", "Hence, in expectation, the algorithm pays at least $\\gamma - \\frac{1}{4\\lambda }$ at each time $t$ .", "Summing over all times, we get a lower bound of $(\\gamma - \\frac{1}{4\\lambda })d$ on the algorithm's cost.", "In particular, Lemma REF implies a competitive ratio of at least $\\left(\\frac{\\gamma - 1/(4\\lambda )}{\\gamma (1+\\mu d^{3/2} \\gamma )}\\right)\\sqrt{d}$ for chasing a class of functions that includes $f_1,\\dots , f_d$ .", "It is now a simple exercise in choosing constants to get a lower bound on the competitiveness of any algorithm for chasing $\\kappa $ -well-conditioned functions, $\\alpha $ -strongly-convex functions, and $\\beta $ -smooth functions.", "Proposition A.2 The competitive ratio of any algorithm for chasing convex functions with condition number $\\kappa $ is $\\Omega (\\kappa ^{1/3})$ .", "Moreover, the competitive ratio of any algorithm for chasing $\\alpha $ -strongly-convex (resp., $\\beta $ -smooth) functions is $\\Omega (\\sqrt{d})$ .", "For $\\kappa $ -strongly convex functions, apply Lemma REF with dimension $d = \\kappa ^{2/3}$ , constants $\\gamma = \\lambda = 1$ and $\\mu = \\kappa ^{-1} = d^{-3/2}$ .", "This shows a gap of $\\Omega (\\sqrt{d}) = \\Omega (\\kappa ^{1/3})$ .", "For $\\alpha $ -strongly convex functions, choose $\\mu = \\alpha /2$ , $\\gamma = 1/(d^{3/2} \\alpha )$ , and $\\lambda = 1/\\gamma = d^{3/2}\\alpha $ .", "Finally, for $\\beta $ -smooth functions, choose $\\lambda = \\beta /2$ , $\\gamma = 1/\\beta $ , and $\\mu =0$ ." ], [ "A Lower Bound Example for $\\mathsf {M2M}$", "We show that the $\\mathsf {M2M}$ algorithm is $\\Omega (\\kappa )$ -competitive, even in $\\mathbb {R}^2$ .", "The essential step of the proof is the following lemma, which shows that, in a given timestep, $ALG$ can be forced to pay $\\Omega (k)$ times as much as some algorithm $Y=(y_1,\\dots , y_t)$ (we think of $Y$ as a candidate for $OPT$ ) while at each step $t$ , $ALG$ does not move any closer to $y_t$ .", "Lemma A.3 Fix $\\kappa > 0$ .", "Suppose that $(x_1, \\dots , x_{t-1} )$ is defined by the $\\mathsf {M2M}$ algorithm and $Y=(y_1,\\dots , y_{t-1})$ is a point sequence such that $y_{t-1}\\ne x_{t-1}$ .", "Define the potential $\\Phi _s = \\Vert x_s - y_s\\Vert .$ Then there is a $\\kappa $ -well-conditioned function $f_t$ and a choice of $y_t$ such that $\\mathsf {cost}_t(ALG) \\ge \\Omega (1)\\cdot \\Phi _{t-1} \\ge \\Omega (k) \\cdot \\mathsf {cost}_t (Y)$ $\\Phi _t \\ge \\Phi _{t-1}$ , and hence $y_t \\ne x_t$ .", "Observe that if we modify an instance by an isometry the algorithm's sequence will also change by the same isometry.", "So we may assume that $x_{t-1} = (\\gamma , \\gamma ) $ and $y_{t-1}= (2\\gamma ,0)$ , for some $\\gamma > 0$ .", "(See Figure REF .)", "Define $f_t(x) = \\frac{1}{4 \\gamma }\\left(\\frac{1}{\\kappa } \\cdot x_1^2 + x_2^2\\right).$ Note that $f_t$ is $\\kappa $ -well-conditioned.", "It is easily checked that $x_t = \\lambda x_{t-1}$ for some $\\lambda > \\frac{1}{2}$ (recall that $x_t$ is chosen to satisfy $f_t(x_t) = \\Vert x_t - x_{t-1}\\Vert $ ).", "Thus $ALG$ pays: $\\mathsf {cost}_t (ALG) = 2f_t(x_t) = 2\\lambda ^2 \\cdot f_t(x_{t-1}) \\ge \\Omega (1) \\cdot \\gamma = \\Omega (1)\\cdot \\Phi _{t-1}.$ Figure: Proof of Lemma  showing that𝖬2𝖬\\mathsf {M2M} is Ω(κ)\\Omega (\\kappa )-competitive.We choose $y_t = y_{t-1}$ so that the cost of $Y$ is: $\\mathsf {cost}_t(Y) = f_t(y_t) \\le O\\left(\\frac{1}{\\kappa }\\right) \\cdot \\gamma = O\\left(\\frac{1}{\\kappa }\\right) \\cdot \\Phi _{t-1}.$ Multiplying (REF ) by $\\Omega (\\kappa )$ and combining with (REF ) completes the proof of (i).", "The statement in (ii) follows from the fact that $x_t, x_{t-1}$ and $y_t$ form a right triangle with leg $\\Phi _{t-1}$ and hypotenuse $\\Phi _t$ .", "Proposition A.4 The $\\mathsf {M2M}$ algorithm is $\\Omega (\\kappa )$ competitive for chasing $\\kappa $ -well-conditioned functions.", "Suppose that before the first timestep, $y_0$ moves to $e_1$ and incurs cost 1.", "Now consider the instance given by repeatedly applying Lemma REF for $T$ timesteps.", "For each time $t$ , we have $\\Phi _t \\ge \\Phi _0$ .", "Thus, $\\mathsf {cost}_t(ALG) \\ge \\Omega (1) \\cdot \\Phi _{t-1} \\ge \\Omega (1)\\cdot \\Phi _0= \\Omega (1).$ Summing over all time, $ALG$ pays $\\mathsf {cost}(ALG) \\ge \\Omega (T)$ .", "Meanwhile, our candidate $OPT$ has paid at most $O(\\frac{1}{\\kappa }) \\cdot \\mathsf {cost}(ALG) + 1$ .", "The proof is completed by choosing $T \\ge \\Omega (\\kappa )$ ." ], [ "A Lower Bound Example for $\\mathsf {COBD}$", "We now give a lower bound for the $\\mathsf {COBD}$ algorithm.The lower bound example is valid even in the unconstrained setting, where $\\mathsf {COBD}$ and $\\mathsf {OBD}$ are the same algorithm.", "In the proof of Proposition REF we showed that the angle $\\theta _t$ between $y_t - x_t$ and $x_{t-1} - x_t$ satisfies $-\\sec (\\theta _t) \\le O(\\sqrt{\\kappa })$ .", "This bound corresponds directly determines to the competitiveness of $\\mathsf {COBD}$ .", "The essence of the lower bound is to give an example where $-\\sec (\\theta _t) \\ge \\Omega (\\sqrt{\\kappa })$ .", "Much like $\\mathsf {M2M}$ , the key to showing that $\\mathsf {COBD}$ is $\\Omega (\\sqrt{\\kappa })$ -competitive lies in constructing a single “bad timestep” that can be repeated until it dominates the competitive ratio.", "In the case of $\\mathsf {COBD}$ , this timestep allows us to convert the potential into cost to $ALG$ at a rate of $\\Omega (\\sqrt{\\kappa })$ .", "Lemma A.5 Fix $\\kappa \\ge 1$ .", "Suppose that $x_t$ is defined by the $\\mathsf {COBD}$ algorithm and that $Y=(y_1,\\dots , y_{t-1})$ is a point sequence such that $y_{t-1}\\ne x_{t-1}$ .", "Define the potential $\\Phi _s = \\Vert x_s - y_s\\Vert .$ Then there is a $\\kappa $ -well-conditioned function $f_t$ and a choice of $y_t$ such that $\\mathsf {cost}_t(ALG) \\ge \\Omega (\\frac{1}{\\sqrt{\\kappa }}) \\Phi _{t-1}$ .", "$\\mathsf {cost}_t(ALG) \\ge \\Omega (\\sqrt{\\kappa }) (-\\Delta _t \\Phi )$ .", "$\\mathsf {cost}_t(Y) = 0$ .", "Observe that modifying an instance by an isometry will modify the algorithm's sequence by the same isometry.", "After applying an appropriate isometry, we will define $f_t(x) = \\alpha (x_1^2 + \\kappa x_2^2)$ for some $\\alpha >0$ to be chosen later and $y_t = y_{t-1}$ .", "We claim that this can be done such that: $y_t = y_{t-1} = 0$ , $\\Vert x_t - x_{t-1}\\Vert = \\frac{1}{2\\sqrt{\\kappa }} \\Vert x_{t-1}-y_{t-1}\\Vert $ (which in turn is equal to $\\frac{1}{2\\sqrt{\\kappa }} \\Vert x_{t-1}\\Vert $ ).", "$x_t = \\gamma \\left[\\begin{matrix}\\sqrt{\\kappa }\\\\ 1\\end{matrix}\\right]$ for some $\\gamma >0$ , For any $\\alpha > 0$ , there is point a $x_\\alpha $ on the ray $\\left\\lbrace \\gamma \\left[\\begin{matrix}\\sqrt{\\kappa }\\\\ 1\\end{matrix}\\right]:\\gamma >0\\right\\rbrace $ such that $f_t(x_\\alpha ) = \\frac{1}{2\\sqrt{\\kappa }} \\Vert x_{t-1}-y_{t-1}\\Vert $ .", "Let $x^-_\\alpha := x_\\alpha + \\left(\\frac{1}{2\\sqrt{\\kappa }} \\Vert x_{t-1}-y_{t-1}\\Vert \\right)\\frac{\\nabla f_t(x_\\alpha )}{\\Vert \\nabla f_t(x_\\alpha )\\Vert }.$ Note that $x^-_\\alpha $ is defined so that applying $\\mathsf {COBD}$ to $x^-_\\alpha $ and $f_t$ outputs the point $x_\\alpha $ .", "Then $\\Vert x^-_\\alpha \\Vert $ increases continuously from $\\frac{1}{2\\sqrt{\\kappa }} \\Vert x_{t-1}-y_{t-1}\\Vert $ to $\\infty $ as $\\alpha $ ranges from 0 to $\\infty $ .", "Choose $\\alpha $ such that $\\Vert x^-_\\alpha \\Vert = \\Vert x_{t-1} - y_{t-1}\\Vert $ , and pick the isometry that maps $y_{t-1}$ to 0 and $x_{t-1}$ to $x^-_\\alpha $ .", "The claim follows.", "Now, (a) and (b) imply that $\\mathsf {cost}_t(ALG) = 2\\Vert x_t-x_{t-1}\\Vert = \\frac{1}{\\sqrt{\\kappa }}\\Vert x_{t-1}\\Vert = \\frac{1}{\\sqrt{\\kappa }} \\Phi _{t-1}.$ This proves $(i)$ .", "Furthermore, (b) and the triangle inequality give $\\Vert x_t\\Vert \\ge \\Vert x_{t-1}\\Vert - \\Vert x_{t-1} - x_t\\Vert = (2\\sqrt{\\kappa }- 1) \\cdot \\Vert x_{t-1}-x_t\\Vert \\ge \\sqrt{\\kappa }\\cdot \\Vert x_{t-1}-x_t\\Vert .$ There are $\\eta , \\nu > 0\\footnote {We omit the exact values (which depend on \\kappa and\\Vert x_{t-1}\\Vert ) as \\nu cancels out in the next step.", "}$ such that $x_{t-1} - x_t= \\eta \\nabla f_t(x_t)= \\nu \\left[\\begin{matrix}1 \\\\ \\sqrt{\\kappa } \\end{matrix}\\right].$ Letting $\\theta _t$ be the angle between $x_{t-1}-x_t$ and $y_t-x_t=-x_t$ (cf.", "fig:obd-analysis) we have $-\\cos (\\theta _t)= - \\frac{\\langle x_{t-1}-x_t, -x_t \\rangle }{\\Vert x_{t-1}-x_t\\Vert \\cdot \\Vert -x_t\\Vert }= \\frac{2\\sqrt{\\kappa }}{1+\\kappa }\\le \\frac{2}{\\sqrt{\\kappa }}.$ We now mirror the argument used in the proof of Theorem REF relating $\\mathsf {cost}_t(ALG)$ to $\\cos (\\theta _t)$ .", "$\\mathsf {cost}_t (ALG) &= 2\\Vert x_t-x_{t-1}\\Vert \\\\&= \\frac{\\Vert x_{t-1}\\Vert +\\Vert x_t\\Vert }{\\Vert x_{t-1}-x_t\\Vert - 2\\Vert x_t\\Vert \\cos (\\theta _t)}\\cdot (-\\Delta _t \\Phi ) \\\\&\\ge \\frac{\\Vert x_t\\Vert }{(1/\\sqrt{\\kappa }) \\Vert x_t\\Vert + (4/\\sqrt{\\kappa })\\cdot \\Vert x_t\\Vert }\\cdot (-\\Delta _t \\Phi ) \\\\&= \\frac{\\sqrt{\\kappa }}{5} (-\\Delta _t \\Phi ).$ Finally, observe that $\\mathsf {cost}_t(Y)= f_t(0) = 0$ .", "We can now get a lower bound on the competitiveness of $\\mathsf {COBD}$ .", "Proposition A.6 $\\mathsf {COBD}$ is $\\Omega (\\sqrt{\\kappa })$ competitive for chasing $\\kappa $ -well-conditioned functions.", "Suppose that before the first timestep, $y_0$ moves to $e_1$ and incurs cost 1.", "Now consider the instance given by repeatedly applying Lemma REF for $T$ timesteps.", "$\\mathsf {cost}(OPT)=1$ , so it remains to show that $\\mathsf {cost}(ALG) = \\Omega (\\sqrt{k})$ .", "Let $\\Phi _{min}:= \\min \\lbrace \\Phi _1, \\dots , \\Phi _T\\rbrace $ .", "Using $(i)$ and summing over all time we have $\\mathsf {cost}(ALG) \\ge \\frac{1}{\\sqrt{\\kappa }} \\sum _{t=0}^{T-1} \\Phi _t \\ge \\frac{T}{\\sqrt{\\kappa }} \\Phi _{min}.$ Using $(ii)$ and summing over all time (and using that $ALG$ incurs nonnegative cost at each step), $\\mathsf {cost}(ALG) \\ge \\Omega (\\sqrt{\\kappa })(\\Phi _0 - \\Phi _{min}) = \\Omega (\\sqrt{\\kappa })(1- \\Phi _{min})$ If $\\Phi _{min} \\ge \\frac{1}{2}$ then $\\mathsf {cost}(ALG) \\ge \\frac{T}{2\\sqrt{k}}$ by (REF ), else $\\Phi _{min} < \\frac{1}{2}$ and we have $\\mathsf {cost}(ALG) \\ge \\Omega (\\sqrt{k})$ by (REF ).", "Choosing $T = \\kappa $ completes the proof." ], [ "Constrained $\\mathsf {M2M}$", "We give a generalized version of the $\\mathsf {M2M}$ algorithm for the constrained setting where the action space $K\\subseteq \\mathbb {R}^d$ is an arbitrary convex set.", "This algorithm achieves the same $O(\\sqrt{\\kappa })$ -competitiveness respectively as in the unconstrained setting.", "The idea is to move towards $x_{K,t}^*$ , the minimizer of $f_t$ among feasible points, rather than the global minimizer.", "The proof of the algorithm's competitiveness proceeds similarly to the proof in the unconstrained setting.", "The difference is that it takes more care to show that $f(x_t) \\le O(\\kappa ) f(y_t)$ in Case II.", "The Constrained M2M Algorithm.", "Suppose we are at position $x_{t-1}$ and receive the function $f_t$ .", "Let $x^*_{K,t} := \\arg \\min _{x\\in K} f_t(x)$ denote the minimizer of $f_t$ among points in $K$ .", "Consider the line segment with endpoints $x_{t-1}$ and $x^*_{K,t}$ , and let $x_t$ be the unique point on this segment with $\\Vert x_t - x_{t-1}\\Vert = f_t(x_t)-f_t(x_{K,t}^*)$ .Such a point is always unique when $f_t$ is strictly convex.", "The point $x_t$ is the one played by the algorithm.", "Note that we assume that the global minimum value of $f_t$ is 0, as before.", "However, the minimum value of $f_t$ on the action space $K$ could be strictly positive.", "Proposition B.1 With $c = 25(2+2\\sqrt{2})$ , for each $t$ , $\\mathsf {cost}_t(ALG) + 2\\sqrt{2} \\cdot \\Delta _t \\Phi \\le c\\cdot \\kappa \\cdot \\mathsf {cost}_t(OPT).", "$ Hence, the constrained $\\mathsf {M2M}$ algorithm is $c \\kappa $ -competitive.", "As in the proof of Theorem REF , we begin by applying the structure lemma.", "This time, we use $x^*_{K,t}$ to be the origin.", "The proof of Case I is identical.", "Case II: Suppose that $\\Vert y_t - x^*_{K,t}\\Vert \\ge \\frac{1}{\\sqrt{2}} \\Vert x_t - x^*_{K,t}\\Vert $ .", "Let $x^*_t := \\arg \\min _{x} f_t(x)$ denote the global minimizer of $f_t$ .", "As before, we assume $f_t(x^*_t) = 0$ , and we translate such that $x^*_t = 0$ .", "We now show that $f_t(x_t) \\le 25\\kappa f_t(y_t)$ .", "If $f_t(x_t) \\le 25 \\kappa f_t(x^*_{K,t})$ , then since $f(y_t) \\ge f_t(x^*_{K,t})$ , we are done.", "So suppose that $f_t(x_t) > 25 \\kappa f_t(x^*_{K,t})$ .", "Now strong convexity and smoothness imply $\\Vert x_t\\Vert ^2 \\ge \\frac{2}{\\kappa \\alpha _t} f_t(x_t) \\ge 25\\cdot \\frac{2}{\\alpha _t} f(x^*_{K,t}) \\ge 25 \\Vert x^*_{K,t}\\Vert ^2.$ Thus $\\Vert x_t\\Vert \\ge 5 \\Vert x^*_{K,t}\\Vert $ .", "One application of the triangle inequality gives $\\Vert x_t - x^*_{K,t}\\Vert \\ge \\Vert x_t\\Vert - \\Vert x^*_{K,t}\\Vert \\ge 4\\Vert x^*_{K,t}\\Vert $ .", "Using the triangle inequality again, we get $\\Vert x_t\\Vert \\le \\Vert x_t-x^*_{K,t}\\Vert + \\Vert x^*_{K,t}\\Vert \\le \\frac{5}{4} \\Vert x-x^*_{K,t}\\Vert ,$ and $\\Vert y_t\\Vert \\ge \\Vert y-x^*_{K,t}\\Vert - \\Vert x^*_{K,t}\\Vert \\ge \\left(\\frac{1}{\\sqrt{2}} - \\frac{1}{4}\\right) \\Vert x-x^*_{K,t}\\Vert \\ge \\frac{1}{4} \\Vert x-x^*_{K,t}\\Vert $ Combining these two, we have $\\Vert y_t\\Vert \\ge \\Vert x_t\\Vert \\ge \\frac{1}{5}\\Vert x_t\\Vert $ Finally, we have $f_t(x_t)\\le \\frac{\\alpha _t \\kappa }{2} \\Vert x_t\\Vert ^2\\le \\frac{5\\alpha _t\\kappa }{2} \\Vert y_t\\Vert ^2\\le 25 \\kappa \\cdot f_t(y_t).$ We now proceed as in the proof of Theorem REF ." ], [ "A Structure Lemma for General Norms", "We can extend the $O(\\kappa )$ -competitiveness guarantee for $\\mathsf {M2M}$ for all norms, by replacing Lemma REF by the following Lemma REF in thm:k-competitive, and changing some of the constants in the latter accordingly.", "Lemma C.1 Fix an arbitrary norm $\\Vert \\cdot \\Vert $ on $\\mathbb {R}^d$ .", "Given any scalar $\\gamma \\in [0,1]$ and any two vectors $x,y\\in \\mathbb {R}^d$ , at least one of the following holds: $ \\Vert y-\\gamma x\\Vert - \\Vert y-x\\Vert \\le -\\frac{1}{2} \\Vert x-\\gamma x\\Vert $ .", "$\\Vert y\\Vert \\ge \\frac{1}{4}\\Vert \\gamma x\\Vert $ .", "As in the proof of Lemma REF we assume (ii) does not hold and show that (i) does.", "WLOG, let $\\Vert x\\Vert =1$ .", "Let $\\Vert \\cdot \\Vert _*$ denote the dual norm.", "Let $z_\\tau := \\nabla \\Vert \\tau x - y\\Vert = \\arg \\max _{\\Vert z\\Vert _*\\le 1}\\langle \\tau x - y, z\\rangle $ and note that $\\langle z_\\tau , \\tau x - y\\rangle =\\Vert \\tau x -y \\Vert $ .", "Then, $\\frac{d}{d\\tau } \\Vert \\tau x - y\\Vert &= \\left\\langle \\nabla \\Vert \\tau x - y\\Vert ,\\frac{d}{d\\tau }(\\tau x-y)\\right\\rangle \\\\&= \\left\\langle z_\\tau , x\\right\\rangle \\\\&= \\frac{\\langle z_\\tau , \\tau x - y \\rangle + \\langle z_\\tau , y \\rangle }{\\tau } \\\\&\\ge \\frac{\\Vert \\tau x - y\\Vert - \\Vert z_\\tau \\Vert _* \\Vert y\\Vert }{\\tau }\\\\&\\ge \\frac{(\\tau - \\Vert y\\Vert ) - 1\\cdot \\Vert y\\Vert }{\\tau }= 1 - \\frac{2\\Vert y\\Vert }{\\tau }.$ Given the bound $\\frac{d}{d\\tau } \\Vert \\tau x - y\\Vert \\ge 1 - \\frac{2\\Vert y\\Vert }{\\tau }$ we can say: $\\Vert y - \\gamma x\\Vert - \\Vert y-x\\Vert = - \\int _\\gamma ^1 \\frac{d}{d\\tau }\\big (\\Vert \\tau x - y\\Vert \\big ) \\, d\\tau \\le - \\int _\\gamma ^1 \\left( 1 -\\frac{2\\Vert y\\Vert }{\\tau } \\right)\\, d\\tau .", "$ Since by assumption condition (ii) does hold and $\\Vert x\\Vert = 1$ , we know that $\\Vert y\\Vert <\\frac{1}{4} \\Vert \\gamma x\\Vert = \\frac{1}{4} \\gamma $ .", "Hence $\\frac{2\\Vert y\\Vert }{\\tau } <\\frac{\\gamma /2}{\\tau } \\le 1/2$ for $\\tau \\ge \\gamma $ .", "The integrand in (REF ) is therefore at least half, and hence the result is at most $-\\frac{1}{2}(1-\\gamma )= -\\frac{1}{2}\\Vert x - \\gamma x\\Vert $ .", "Hence the proof." ] ]

[ [ "The first search for bosonic super-WIMPs with masses up to 1 MeV/c$^2$\n with GERDA" ], [ "Abstract We present the first search for bosonic super-WIMPs as keV-scale dark matter candidates performed with the GERDA experiment.", "GERDA is a neutrinoless double-beta decay experiment which operates high-purity germanium detectors enriched in $^{76}$Ge in an ultra-low background environment at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN in Italy.", "Searches were performed for pseudoscalar and vector particles in the mass region from 60 keV/c$^2$ to 1 MeV/c$^2$.", "No evidence for a dark matter signal was observed, and the most stringent constraints on the couplings of super-WIMPs with masses above 120 keV/c$^2$ have been set.", "As an example, at a mass of 150 keV/c$^2$ the most stringent direct limits on the dimensionless couplings of axion-like particles and dark photons to electrons of $g_{ae} < 3 \\cdot 10^{-12}$ and ${\\alpha'}/{\\alpha} < 6.5 \\cdot 10^{-24}$ at 90% credible interval, respectively, were obtained." ], [ "The first search for bosonic super-WIMPs with masses up to 1 MeV/c$^2$ with GERDA Gerda collaboration correspondence: [email protected] INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy INFN Laboratori Nazionali del Gran Sasso and Università degli Studi dell'Aquila, L'Aquila, Italy INFN Laboratori Nazionali del Sud, Catania, Italy Institute of Physics, Jagiellonian University, Cracow, Poland Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany Joint Institute for Nuclear Research, Dubna, Russia European Commission, JRC-Geel, Geel, Belgium Max-Planck-Institut für Kernphysik, Heidelberg, Germany Dipartimento di Fisica, Università Milano Bicocca, Milan, Italy INFN Milano Bicocca, Milan, Italy Dipartimento di Fisica, Università degli Studi di Milano and INFN Milano, Milan, Italy Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia National Research Centre “Kurchatov Institute”, Moscow, Russia Max-Planck-Institut für Physik, Munich, Germany Physik Department, Technische Universität München, Germany Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany Physik-Institut, Universität Zürich, Zurich, Switzerland M. Agostini Physik Department, Technische Universität München, Germany A.M. Bakalyarov National Research Centre “Kurchatov Institute”, Moscow, Russia M. Balata INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy I. Barabanov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia L. Baudis Physik-Institut, Universität Zürich, Zurich, Switzerland C. Bauer Max-Planck-Institut für Kernphysik, Heidelberg, Germany E. Bellotti Dipartimento di Fisica, Università Milano Bicocca, Milan, Italy INFN Milano Bicocca, Milan, Italy S. Belogurov [also at: ]NRNU MEPhI, Moscow, Russia Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia A. Bettini Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy L. Bezrukov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia D. Borowicz Joint Institute for Nuclear Research, Dubna, Russia E. Bossio Physik Department, Technische Universität München, Germany V. Bothe Max-Planck-Institut für Kernphysik, Heidelberg, Germany V. Brudanin Joint Institute for Nuclear Research, Dubna, Russia R. Brugnera Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy A. Caldwell Max-Planck-Institut für Physik, Munich, Germany C. Cattadori INFN Milano Bicocca, Milan, Italy A. Chernogorov Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia National Research Centre “Kurchatov Institute”, Moscow, Russia T. Comellato Physik Department, Technische Universität München, Germany V. D'Andrea INFN Laboratori Nazionali del Gran Sasso and Università degli Studi dell'Aquila, L'Aquila, Italy E.V.", "Demidova Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia N. Di Marco INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy E. Doroshkevich Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia V. Egorov deceased Joint Institute for Nuclear Research, Dubna, Russia F. Fischer Max-Planck-Institut für Physik, Munich, Germany M. Fomina Joint Institute for Nuclear Research, Dubna, Russia A. Gangapshev Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Max-Planck-Institut für Kernphysik, Heidelberg, Germany A. Garfagnini Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy C. Gooch Max-Planck-Institut für Physik, Munich, Germany P. Grabmayr Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany V. Gurentsov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia K. Gusev Joint Institute for Nuclear Research, Dubna, Russia National Research Centre “Kurchatov Institute”, Moscow, Russia Physik Department, Technische Universität München, Germany J. Hakenmüller Max-Planck-Institut für Kernphysik, Heidelberg, Germany S. Hemmer INFN Padova, Padua, Italy R. Hiller Physik-Institut, Universität Zürich, Zurich, Switzerland W. Hofmann Max-Planck-Institut für Kernphysik, Heidelberg, Germany M. Hult European Commission, JRC-Geel, Geel, Belgium L.V.", "Inzhechik [also at: ]Moscow Institute for Physics and Technology, Moscow, Russia Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia J. Janicskó Csáthy [presently at: ]Leibniz-Institut für Kristallzüchtung , Berlin, Germany Physik Department, Technische Universität München, Germany J. Jochum Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany M. Junker INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy V. Kazalov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Y. Kermaïdic Max-Planck-Institut für Kernphysik, Heidelberg, Germany H. Khushbakht Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany T. Kihm Max-Planck-Institut für Kernphysik, Heidelberg, Germany I.V.", "Kirpichnikov Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia A. Klimenko [also at: ]Dubna State University, Dubna, Russia Max-Planck-Institut für Kernphysik, Heidelberg, Germany Joint Institute for Nuclear Research, Dubna, Russia R. Kneißl Max-Planck-Institut für Physik, Munich, Germany K.T.", "Knöpfle Max-Planck-Institut für Kernphysik, Heidelberg, Germany O. Kochetov Joint Institute for Nuclear Research, Dubna, Russia V.N.", "Kornoukhov Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia P. Krause Physik Department, Technische Universität München, Germany V.V.", "Kuzminov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia M. Laubenstein INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy A. Lazzaro Physik Department, Technische Universität München, Germany M. Lindner Max-Planck-Institut für Kernphysik, Heidelberg, Germany I. Lippi INFN Padova, Padua, Italy A. Lubashevskiy Joint Institute for Nuclear Research, Dubna, Russia B. Lubsandorzhiev Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia G. Lutter European Commission, JRC-Geel, Geel, Belgium C. Macolino [presently at: ]LAL, CNRS/IN2P3, Université Paris-Saclay, Orsay, France INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy B. Majorovits Max-Planck-Institut für Physik, Munich, Germany W. Maneschg Max-Planck-Institut für Kernphysik, Heidelberg, Germany M. Miloradovic Physik-Institut, Universität Zürich, Zurich, Switzerland R. Mingazheva Physik-Institut, Universität Zürich, Zurich, Switzerland M. Misiaszek Institute of Physics, Jagiellonian University, Cracow, Poland P. Moseev Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia I. Nemchenok Joint Institute for Nuclear Research, Dubna, Russia K. Panas Institute of Physics, Jagiellonian University, Cracow, Poland L. Pandola INFN Laboratori Nazionali del Sud, Catania, Italy K. Pelczar Institute of Physics, Jagiellonian University, Cracow, Poland L. Pertoldi Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy P. Piseri Dipartimento di Fisica, Università degli Studi di Milano and INFN Milano, Milan, Italy A. Pullia Dipartimento di Fisica, Università degli Studi di Milano and INFN Milano, Milan, Italy C. Ransom Physik-Institut, Universität Zürich, Zurich, Switzerland L. Rauscher Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany S. Riboldi Dipartimento di Fisica, Università degli Studi di Milano and INFN Milano, Milan, Italy N. Rumyantseva National Research Centre “Kurchatov Institute”, Moscow, Russia Joint Institute for Nuclear Research, Dubna, Russia C. Sada Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy F. Salamida INFN Laboratori Nazionali del Gran Sasso and Università degli Studi dell'Aquila, L'Aquila, Italy S. Schönert Physik Department, Technische Universität München, Germany J. Schreiner Max-Planck-Institut für Kernphysik, Heidelberg, Germany M. Schütt Max-Planck-Institut für Kernphysik, Heidelberg, Germany A-K. Schütz Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany O. Schulz Max-Planck-Institut für Physik, Munich, Germany M. Schwarz Physik Department, Technische Universität München, Germany B. Schwingenheuer Max-Planck-Institut für Kernphysik, Heidelberg, Germany O. Selivanenko Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia E. Shevchik Joint Institute for Nuclear Research, Dubna, Russia M. Shirchenko Joint Institute for Nuclear Research, Dubna, Russia H. Simgen Max-Planck-Institut für Kernphysik, Heidelberg, Germany A. Smolnikov Max-Planck-Institut für Kernphysik, Heidelberg, Germany Joint Institute for Nuclear Research, Dubna, Russia D. Stukov National Research Centre “Kurchatov Institute”, Moscow, Russia A.A. Vasenko Institute for Theoretical and Experimental Physics, NRC “Kurchatov Institute”, Moscow, Russia A. Veresnikova Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia C. Vignoli INFN Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, Assergi, Italy K. von Sturm Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Padua, Italy INFN Padova, Padua, Italy T. Wester Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany C. Wiesinger Physik Department, Technische Universität München, Germany M. Wojcik Institute of Physics, Jagiellonian University, Cracow, Poland E. Yanovich Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia B. Zatschler Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany I. Zhitnikov Joint Institute for Nuclear Research, Dubna, Russia S.V.", "Zhukov National Research Centre “Kurchatov Institute”, Moscow, Russia D. Zinatulina Joint Institute for Nuclear Research, Dubna, Russia A. Zschocke Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany A.J.", "Zsigmond Max-Planck-Institut für Physik, Munich, Germany K. Zuber Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany G. Zuzel.", "Institute of Physics, Jagiellonian University, Cracow, Poland We present the first search for bosonic super-WIMPs as keV-scale dark matter candidates performed with the Gerda experiment.", "Gerda is a neutrinoless double-beta decay experiment which operates high-purity germanium detectors enriched in $^{76}$ Ge in an ultra-low background environment at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN in Italy.", "Searches were performed for pseudoscalar and vector particles in the mass region from 60 keV/c$^2$ to 1 MeV/c$^2$ .", "No evidence for a dark matter signal was observed, and the most stringent constraints on the couplings of super-WIMPs with masses above 120 keV/c$^2$ have been set.", "As an example, at a mass of 150 keV/c$^2$ the most stringent direct limits on the dimensionless couplings of axion-like particles and dark photons to electrons of $g_{ae} < \\text{3}\\cdot 10^{-12}$ and ${\\alpha ^{\\prime }}/{\\alpha } < \\text{6.5} \\cdot 10^{-24}$ at 90% credible interval, respectively, were obtained.", "95.35.+d,14.80.Mz,27.50.+e, 29.40.Wk The evidence for the existence of nonbaryonic dark matter (DM) in our Universe is overwhelming.", "In particular, recent measurements of temperature fluctuations in the cosmic microwave background radiation yield a 26.4% contribution of DM to the overall energy density in the $\\Lambda $ CDM model [1].", "However, all evidence is gravitational in nature, and the composition of this invisible form of matter is not known.", "Theoretical models for particle DM yield candidates with a wide range of masses and scattering cross sections with Standard Model (SM) particles [2], [3], [4].", "Among these, so-called bosonic super-weakly interacting massive particles (super-WIMPs) with masses at the keV-scale and ultra-weak couplings to the SM can be cosmologically viable and produce the required relic abundance [5], [6].", "Direct DM detection experiments, as well as experiments built to observe neutrinoless double-beta ($0\\nu \\beta \\beta $ ) decay can search for pseudoscalar (also known as axion-like particles, or ALPs) and vector (also known as dark photons) super-WIMPs via their absorption in detector materials in processes analogous to the photoelectric effect (also known as axioelectric effect in the case of axions).", "The ALP and dark photon energy is transferred to an electron, which deposits its energy in the detector.", "The expected signature is a full absorption peak in the energy spectrum, corresponding to the mass of the particle, given that these DM candidates have very small kinetic energies Most of the dark matter is cold and non-relativistic, hence $E\\simeq m_{DM}$.", "For ALPs the coupling to electrons is parameterised via the dimensionless coupling constant $g_{ae}$ [8], while for dark photons a kinetic mixing $\\alpha ^{\\prime }$ with strength $\\kappa $  [9] is introduced in analogy to the electromagnetic fine structure constant $\\alpha $ , such that ${\\alpha ^{\\prime }} = (e \\kappa )^2/4\\pi $ .", "The most stringent direct constraints on these couplings for particle masses at the keV-scale are set by experiments using liquid xenon (xmass, Pandax-II, lux, xenon100, [10], [11], [12], [13]), germanium crystals (Majorana Demonstrator, Supercdms, Edelweiss, cdex, [14], [15], [16], [17]) and calcium tungstate crystals (cresst-II [18]).", "Together, these experiments probe the super-WIMP mass region up to $\\sim $ 500 keV/c$^2$ .", "Here we describe a search for super-WIMP absorption in the germanium detectors operated by the Gerda collaboration, extending for the first time the mass region to 1 MeV/c$^2$ .", "At masses larger than twice the electron mass vector particles can decay into $(e^+, e^-)$ -pairs and their lifetime would be too short to account for the DM [6].", "The primary goal of Gerda is to search for the $0\\nu \\beta \\beta $ decay of $^{76}$ Ge, deploying High-Purity Germanium (HPGe) detectors enriched up to 87% in $^{76}$ Ge.", "The experiment is located underground at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN, Italy, at a depth of about 3500 meter water equivalent.", "The HPGe detector array is made of 7 enriched coaxial and 30 Broad Energy Germanium (BEGe) diodes, with average masses of 2.2 kg and 667 g, respectively, leading to a higher full absorption efficiency for the larger coaxial detectors.", "It is operated inside a 64 m$^3$ liquid argon (LAr) cryostat, which provides cooling and a high-purity, active shield against background radiation.", "The cryostat is inside a water tank instrumented with PMTs to detect Cherenkov light from muons passing through, and thus reduces the muon-induced background to negligible levels.", "A detailed description of the experiment can be found in [19], while the most recent $0\\nu \\beta \\beta $ decay results are presented in [20].", "Figure: The energy spectra of the BEGe and coaxial data sets, normalized by exposure.", "Only events with energies up to 1 MeV were considered in the analysis.", "The coaxial data set shows a significantly higher event rate (mainly from 39 ^{39}Ar decays) at energies below 500 keV due to the larger surface area of the signal readout electrodes .The dashed lines indicate the positions of the main known background gamma lines, also listed in Table .Due to its ultra-low background level [21] and excellent energy resolution ($\\sim $ 3.6 keV and $\\sim $ 3.0 keV full width at half maximum (FWHM) for coaxial and BEGe detectors at $Q_{\\beta \\beta }$ =2039 keV, respectively), the Gerda experiment is well-suited to search for other rare interactions, in particular for peak-like signatures as expected from bosonic super-WIMPs.", "Here we make the assumption that super-WIMPs constitute all of the DM in our galaxy, with a local density of 0.3 GeV/cm$^3$  [22].", "The absorption rate for dark photons and ALPs in an Earth-bound detector can be expressed as [5]: $R \\approx \\frac{4\\cdot 10^{23}}{A}\\frac{\\alpha ^{\\prime }}{\\alpha }\\left(\\frac{\\mathrm {[keV/c^2]}}{m_v}\\right)\\left(\\frac{\\sigma _{pe}}{\\mathrm {[b]}}\\right)\\mathrm {kg^{-1}d^{-1}}$ and $R \\approx \\frac{1.2\\cdot 10^{19}}{A}g_{ae}^{2}\\left(\\frac{m_a}{\\mathrm {[keV/c^2]}}\\right)\\left(\\frac{\\sigma _{pe}}{\\mathrm {[}{\\text{b}}]}\\right)\\mathrm {kg^{-1}d^{-1}}~,$ respectively.", "Here $g_{ae}$ and ${\\alpha ^{\\prime }}/{\\alpha }$ are the dimensionless coupling constants, A is the atomic mass of the absorber, $\\sigma _{pe}$ is the photoelectric cross section on the target material (germanium), and $m_v$ and $m_a$ are the DM particle masses.", "The linear versus inverse proportionality of the rate with the particle mass is due to the fact that rates scale as flux times cross section, where the cross section is proportional to $m_a^2$ and ${\\alpha ^{\\prime }}/{\\alpha }$ in the pseudoscalar and vector boson case, respectively [5].", "We perform the search for super-WIMPs in the (200 keV/c$^2$ - 1 MeV/c$^2$ ) mass range on data collected between December 2015 and April 2018, corresponding to 58.9 kg$\\cdot $ yr of exposure.", "The energy threshold of the HPGe detectors was lowered in October 2017 and enabled a search in the additional mass range of (60 keV/c$^2$ - 200 keV/c$^2$ ), corresponding to 14.6 kg$\\cdot $ yr of exposure accumulated until April 2018.", "The individual exposures for BEGe and coaxial detectors above (below) 200 keV are 30.8 (7.7) and 28.1 (6.9) kg$\\cdot $ yr, respectively.", "The lower energy bounds for our analysis were motivated by the energy thresholds of the Ge detectors and the shape of the background spectrum (dominated by $^{39}$ Ar decays) and the size of the fit window as explained in the following.", "In Gerda the energy reconstruction of events is performed through digital pulse processing [23].", "Events of non-physical origin such as discharges are rejected by a set of selection criteria based on waveform parameters (i.e., baseline, leading edge, and decay tail).", "The efficiency of these cuts for accepting signal events was estimated at $>$  98.7%.", "Since super-WIMPs would interact only once in a HPGe diode, events tagged in coincidence with the muon or LAr vetos, or observed in more than one germanium detector, were rejected as due to background interactions.", "We use the same set of cuts as in the Gerda main analysis for the $0\\nu \\beta \\beta $ decay [20], with the exception of the pulse shape discrimination cut, which had been tailored to the high-energy $0\\nu \\beta \\beta $ decay search.", "The muon and LAr veto accept signal events with efficiencies of 99.9% [24] and 97.5% [20], respectively.", "The total efficiency to observe a super-WIMP absorption in the HPGe diodes was determined as: $\\epsilon _\\text{tot} = \\epsilon _\\text{cuts}\\frac{1}{\\mathcal {E}} \\sum _i^{N_\\text{det}}\\mathcal {E}_\\text{i} \\cdot f_\\text{av,i}\\cdot \\epsilon _\\text{fep,i}~,$ where the efficiency of the event selection criteria $\\epsilon _\\text{cuts}$ and the exposure $\\mathcal {E}$ of each data set were taken into account.", "The index i runs over the individual detectors of that data set, containing $N_\\text{det}$ detectors, $\\mathcal {E}_i$ is the exposure, $f_\\text{av,i}$ the active mass fraction, and $\\epsilon _\\text{fep,i}$ the efficiency for detection of the full energy absorption of an electron emitted in the interaction.", "With the exception of $\\epsilon _\\text{fep,i}$ all parameters were identical to those in the analysis presented in [20].", "The full-energy absorption efficiency $\\epsilon _\\text{fep,i}$ accounts for partial energy losses, for example in a detector's dead layer.", "This efficiency was estimated for each detector at energies between 60 and 1000 keV with a Monte Carlo simulation of uniformly distributed electrons in the active volume of the detector using the MaGe framework [25].", "Table REF shows the average full energy absorption detection efficiencies $\\epsilon _\\text{fep}$ and the total efficiencies $\\epsilon _\\text{tot}$ at the lower and upper boundaries of the search region.", "At 60 keV, the full energy absorption was estimated as 99.5% for all detectors, while at 1000 keV it is 95.1% and 96.2% on average for BEGe and coaxial detectors, respectively.", "The energy dependence of the efficiency is caused by the photoabsorption cross section and the different size of the germanium diodes.", "The events which survived all selection criteria (with total efficiencies between 85.7% and 81.4%, see Table REF ) are shown in Fig.", "REF for the coaxial and BEGe detector data sets.", "Table: Detection efficiencies for the super-WIMP search.", "The average 〈ϵ fep 〉 \\langle \\epsilon _\\text{fep} \\rangle value for the detectors from one data set and the total efficiency ϵ tot \\epsilon _\\text{tot} are shown at 60 keV and 1000 keV for the two data sets.Table: Gamma lines accounted for in the background model for the super-WIMP search (branching ratio above 0.1%).The expected signal from super-WIMPs has been modeled with a Gaussian peak broadened by the energy resolution of the HPGe detectors.", "To estimate the potential signals from these particles we performed a binned Bayesian fit (with a 1 keV binning, while the systematic uncertainties on the energy scale are estimated at 0.2 keV) of the signal and a background model of the data.", "The fit was performed within a window of 24 keV in width, centered on the energy corresponding to the hypothetical mass of the particle and sliding with 1 keV step to examine each mass value.", "The total number of counts from signal and background was determined as follows: $R_\\text{tot}(E) = G_0(\\mathcal {N}_0, E_0, \\sigma _0) + F(E) + G_\\gamma ( \\mathcal {N}_\\gamma , E_\\gamma , \\sigma _\\gamma )~,$ where the Gaussian function $G_0$ models the peak signal of super-WIMPs at a fixed energy $E_0$ , corresponding to their mass.", "The Gaussian $G_\\gamma $ models the background gamma lines with energy $E_{\\gamma }$ listed in Table REF in case it is found within the sliding fit window.", "For more than one background gamma line the Eq.", "REF is modified accordingly to model all the peaks.", "$\\mathcal {N}_0$ and $\\mathcal {N}_{\\gamma }$ are the counts in the fitted signal and background peaks, respectively.", "The effective energy resolutions $\\sigma _0$ and $\\sigma _\\gamma $ of the detectors from the combined spectra are fixed to the values obtained from the regularly acquired calibration data, with systematic uncertainties around 0.1 keV [20].", "Finally, the polynomial fit function $F(E)$ describes the continuous background, and was chosen as a first- and second-order polynomial for energies above and below 120 keV, respectively.", "The higher order polynomial at lower energies is motivated by the curvature of the $^{39}$ Ar beta spectrum, see Fig.", "REF .", "At other energies, the spectrum has an approximately linear shape, and thus a first-order polynomial was judged sufficient.", "The Bayesian fit was performed with the BAT framework [26] using the Markov Chain Monte Carlo technique [27] to compute the marginalized posterior probability density function (PDF) given energy values of the data $\\mathbf {E}$ , $P(R_\\text{S}| \\mathbf {E})$ , where $R_S$ is the signal rate, i.e., the number of counts normalized by exposure.", "The probability for the signal count rate $P\\left(R_S, {\\theta } | \\mathbf {E}, M\\right)$ , given data $\\mathbf {E}$ and a model M, is described by Bayes' theorem as: $P\\left(R_S, {\\theta } | \\mathbf {E}, M\\right)=\\\\ \\frac{ P\\left(\\mathbf {E} | R_S, {\\theta }, M\\right)\\pi \\left(R_S\\right) \\pi ({\\theta })}{ \\int \\int P\\left(\\mathbf {E} | R_S, {\\theta }, M\\right)\\pi \\left(R_S\\right) \\pi ({\\theta }) d {\\theta } d R_S }.$ The denominator defines the overall probability of obtaining the observed data given a hypothetical signal.", "The numerator includes prior probabilities $\\pi $ for the signal count rate $R_S$ and for the nuisance parameters ${\\theta }$ (e.g., background shape) estimated before performing the fit.", "For ${\\theta }$ , flat priors were adopted, bound generously according to a preliminary fit with the Minuit algorithm [28].", "For the signal count rate, $R_S$ , the uniform (i.e., constant over the defined range) prior probability was constructed to be positive, with the upper bound defined by the total number of events in the signal region plus 10 times the expected Poisson fluctuations.", "The conditional probability $P\\left(\\mathbf {E} | R_S, {\\theta }, M\\right)$ is estimated according to the super-WIMP interaction model given by Eq.", "REF and Poisson fluctuations in the data.", "Figure: Best fit (red lines) and 68% uncertainty band (yellow bands) from marginalised posterior PDFs of the model parameters assuming a hypothetical signal at E 0 _0 in the BEGe data set.", "Top: Fit of a signal assumed at E 0 _0=520 keV; the excess is at a level of 2.6 σ\\sigma .", "A first order polynomial is used for the continuous background and a Gaussian for the background gamma line due to the decay of 85 ^{85}Kr.", "Bottom: Fit of a signal assumed at E 0 _0=87 keV, using a second order polynomial for the continuous background.The reported results were obtained from the combined fit of the BEGe and coaxial data sets.", "First, the BEGe data set was fit using a flat prior for the signal count rate R$_S$ , and the obtained posterior was used as a prior for the fit of the Coaxial data set.", "The results of the latter were then employed to evaluate the corresponding coupling constants of the super-WIMPs.", "The detection of the signal is ruled out when the significance of the best fit value for the count rate is less than 5 sigma, estimated as half of the 68% quantile of the posterior PDF.", "Additionally, if a fitted signal is in close proximity (within 5 $\\sigma $ of the energy resolution) to a known background gamma line, an upper limit was set irrespective of the mode, as uncertainties in the background rate do not allow to reliably claim an excess signal above gamma lines.", "An example for the fit using the model described by Eq.", "REF for two different background functions $F(E)$ is shown in Fig.", "REF .", "The obtained posterior PDFs do not show evidence for a signal in the energy range of the analysis.", "We thus set 90% credible interval (C.I.)", "upper limits on the signal count rate, corresponding to the 90% quantile of the posterior PDF $P(R_\\text{S}| \\mathbf {E})$ , accounting for the detection efficiencies according to Eq.", "REF .", "The 90% C.I.", "limits on the signal rate $R_S$ were converted into upper limits on the coupling strengths using Eqs.", "REF and REF .", "The results are presented in Fig.", "REF .", "We compare these to direct detection limits from cdex [17], Edelweiss-III [16], lux [12], the Majorana Demonstrator [14], Pandax-II [11], Supercdms [15], xenon100 [13] and xmass [10], as well as to indirect limits from horizontal branch and red giant stars [5].", "Above 120 keV/c$^2$ indirect $\\alpha ^{\\prime }/\\alpha $ limits from decays of vector-like particles into three photons ($V\\rightarrow 3\\gamma $ ) are significantly lower (ranging from 10$^{-12}$ at masses of 100 keV/c$^2$ to 10$^{-16}$ at 700 keV/c$^2$ ) than the available direct limits (not shown) [6].", "The improvement in sensitivity with respect to other crystal-based experiments is due to the much larger exposure in Gerda and the lower background rate over all of the search region.", "The weakening of our upper limits with increasing mass is primarily due to the steep decrease of the photoelectric cross section from about 45 barn at 100 keV to 0.085 barn at 1 MeV that overrules both the linear and inverse mass dependence in Eqs.", "REF and REF .", "The fluctuations in the upper limit curves are due to background fluctuations, where prominent peaks come from known gamma lines, shown in Table REF .", "Figure: Upper limits (at 90% C.I.)", "on the coupling strengths of pseudo-scalar (top) and vector (bottom) super-WIMPs.Only part of the data was acquired with a lower energy threshold, resulting in a lower exposure for data below 200 keV/c 2 ^2 and causing the step-like feature around this energy.", "Results from other experiments (see text) are also shown, together with indirect constraints from anomalous energy losses in horizontal branch (HB) and red giant (RG) stars (we refer to  for details).To summarise, in this Letter we demonstrated the capability of Gerda to search for other rare events besides the $0\\nu \\beta \\beta $ decay of $^{76}$ Ge.", "We performed a search for keV-scale DM in the form of bosonic super-WIMPs based on data with exposures of 58.9 kg$\\cdot $ yr and 14.6 kg$\\cdot $ yr in the mass ranges of (200 keV/c$^2$ -1 MeV/c$^2$ ) and (60 keV/c$^2$ -200 keV/c$^2$ ), respectively.", "Upper limits on the coupling strengths $g_{ae}$ and $\\alpha ^{\\prime }/\\alpha $ were obtained from a Bayesian fit of a background model and a potential peak-like signal to the measured data.", "Our limit is compatible with other direct searches in the mass range (60 keV/c$^2$ -120 keV/c$^2$ ) where the strongest limits were obtained by xenon-based DM experiments due to higher exposures and lower background rates in this low-energy region.", "Our search probes for the first time the mass region up to 1 MeV/c$^2$ and sets the best direct constraints on the couplings of super-WIMPs over a large mass range from (120 keV/c$^2$ -1 MeV/c$^2$ ).", "As an example, at a mass of 150 keV/c$^2$ the most stringent direct limits on the dimensionless couplings of axion-like particles and dark photons to electrons of $g_{ae} < \\text{3}\\cdot 10^{-12}$ and ${\\alpha ^{\\prime }}/{\\alpha } < \\text{6.5} \\cdot 10^{-24}$ (at 90% C.I.", "), respectively, were established.", "The limits are affected by the known background gamma lines, listed in Table REF , due to higher background rate at these energies.", "The sensitivity to new physics is expected to improve in the near future with the upcoming Legend-200 experiment.", "The experimental program aims to decrease the background rate and increase the number of HPGe detectors operated in an upgraded Gerda infrastructure at LNGS [29].", "The Gerda experiment is supported financially by the Swiss National Science Foundation (SNF), German Federal Ministry for Education and Research (BMBF), the German Research Foundation (DFG) via the Excellence Cluster Universe and the SFB1258, the Italian Istituto Nazionale di Fisica Nucleare (INFN), the Max Planck Society (MPG), the Polish National Science Centre (NCN), the Foundation for Polish Science (TEAM/ 2016-2/2017), and the Russian Foundation for Basic Research (RFBR).", "The institutions acknowledge also internal financial support.", "This project has received funding or support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreements No.", "690575 and No.", "674896, respectively.", "The Gerda collaboration thanks the directors and the staff of the LNGS for their continuous strong support of the Gerda experiment.", "tocchapterBibliography" ] ]

[ [ "Estimating the Prediction Performance of Spatial Models via Spatial\n k-Fold Cross Validation" ], [ "Abstract In machine learning one often assumes the data are independent when evaluating model performance.", "However, this rarely holds in practise.", "Geographic information data sets are an example where the data points have stronger dependencies among each other the closer they are geographically.", "This phenomenon known as spatial autocorrelation (SAC) causes the standard cross validation (CV) methods to produce optimistically biased prediction performance estimates for spatial models, which can result in increased costs and accidents in practical applications.", "To overcome this problem we propose a modified version of the CV method called spatial k-fold cross validation (SKCV), which provides a useful estimate for model prediction performance without optimistic bias due to SAC.", "We test SKCV with three real world cases involving open natural data showing that the estimates produced by the ordinary CV are up to 40% more optimistic than those of SKCV.", "Both regression and classification cases are considered in our experiments.", "In addition, we will show how the SKCV method can be applied as a criterion for selecting data sampling density for new research area." ], [ "Introduction", "An important step in machine learning applications is the evaluation of the prediction performance of a model in the task under consideration.", "For this one can use the k-fold cross validation (CV), which assumes the data are independent.", "Geographic information system (GIS) data sets represent an example where the independence assumption naturally does not hold due to the temporal or spatial autocorrelation (SAC).", "SAC and its effects on spatial data analysis has been extensively studied in spatial statistics literature, e.g.", "[24], [23].", "For example it has been shown that the failure to not account the effect of SAC in spatial data modeling can lead to over-complex model selection [21], [32].", "Generally speaking, natural data exhibits SAC because of the first law of geography and fundamental principle in geostatistical analysis according to Waldo Tobler [38]: \"Everything is related to everything else, but near things are more related than distant things\".", "In spatial statistics the degree of SAC of a data set can be measured using e.g.", "a semivariogram [13], Moran's I [26], Geary's C [18] or Getis's G [19].", "There are numerous applications involving spatial data which have problems caused by SAC in the data sets such as, natural resource detection, route selection, construction placement, natural disaster recognition, tree species detection, environmental monitoring etc.", "[2].", "Consider the example of harvesting operations in forestry where optimal route selections are of key importance.", "In order to minimize the risk of harvester sinking into the soil a route with the optimal carrying capacity is required.", "The route selection is based on predictions of soil types along the route which gives the harvester an estimate on the carrying capacity of the route.", "If the effect of SAC is not considered in the soil type predictions while estimating the model performance we might end up selecting a hazardous route.", "The reason for this is that the spatial model we are using gives over-optimistic prediction performance for soil types farther away from the harvester's current location.", "The model implicitly assumes that we have known soil types close to the predicted soil types which is not always the case.", "This fact must be taken into account in the model prediction performance evaluation in order to avoid over-optimistic estimation.", "An illustration of the considered example is shown in Figure REF .", "Figure: The forest harvesting example.", "The harvester driver needs to select an optimal route to target destination.", "Due to SAC it is to be expected that the prediction error increases the further away we make point predictions.", "The background map in the image (also in Figure ) is by the courtesy of OpenStreetMap.To counter the problems caused by SAC in spatial modeling one usually tries to incorporate SAC as an autocovariate factor into the prediction models themselves, e.g.", "autocovariate models, spatial eigenvector mapping, autoregressive models [10], [16], [6], [8], [7], [25], [14], [43].", "A review of such methods is well presented in [16].", "Other methods include spatial clustering and re-sampling techniques for countering SAC [20], [33], [11].", "Despite the vast literature of techniques for spatial prediction little attention is given for assessing the spatial prediction performance of a model via cross validation techniques.", "In [12] the author does not advocate CV for confirmatory data analysis because the independence assumption in the data samples is inherently not valid in geostatistical context.", "In this article, we propose a novel CV method called spatial k-fold cross validation (SKCV) for estimating prediction performance under SAC-based independence violations in the data.", "SKCV is also applicable for selecting grid sampling density for new research areas.", "More specifically, SKCV attempts to answer the following two questions: What is the prediction performance of a model at a certain geographical location when the closest data measurements used to train it lie at a given geographical distance?", "Conversely, if the prediction performance is required to be at least a given level, how dense data sampling grid should be used in the experiment area to achieve it?", "The question (2) is about the trade-off between the prediction performance and data collection costs.", "The SKCV method provides the model prediction performance as the function of geographical distance between the in-sample and out-of-sample data, and hence it indicates how close geographically training data has to be to the prediction area in order to achieve a required prediction performance.", "The idea in SKCV is to remove the optimistic bias due to SAC by omitting data samples from the training set, which are geographically too close to the test data.", "To evaluate how well SKCV answers the above questions, it is tested with three real world applications using public GIS-based data sets.", "The applications involve assessing the predictability of water permeability of soil and forest harvest track damage.", "Both regression and classification models were used in these experiments.", "The usability of the SKCV method for determining the needed sampling grid density is tested by measuring the difference between the performance of model constructed with a given grid density and the result predicted with SKCV.", "We will explain this comparison in more details in section 4.1.", "We wish to emphasize that we use SKCV in this manuscript for assessing the spatial prediction performance of a model and not for model complexity selection even though model complexity selection can also be applied with SKCV.", "In [32] the authors used a similar spatial cross validation method as SKCV for model variable selection.", "In their work, they compared a special case of SKCV method, the spatial leave-one-out method (SLOO) with Akaike information criterion (AIC) [1] as a criterion for model variable selection.", "It turned out that SAC caused the AIC to select biased variables, whereas SLOO prevented this.", "In [29], [30], [31] the SKCV method was called cross validation with a dead zone method.", "Related studies on spatial data analysis can also be found in the works of [5], [11], [34].", "In what follows, a formal description of the SKCV method will be given in Ch.", "2, followed by description of used data sets in Ch.", "3 and experimental analyses with three sample cases in Ch.", "4, and finally Ch.", "5 includes conclusions." ], [ "Spatial k-fold cross validation", "SKCV is a modification of the standard CV to overcome the biased prediction performance estimates of the model due to SAC of the data.", "The over-optimistic bias in the performance estimates is prevented by making sure that the training data set only contains data points that are at least a certain spatial or temporal distance away from the test data set.", "We will denote our data point as $\\textbf {d}_i = (\\textbf {x}_i, y_i, \\textbf {c}_i)$ , where $\\textbf {x}_i\\in \\mathbb {R}^n$ is a feature vector, $y_i\\in \\mathbb {R}$ a response value and $\\textbf {c}_i\\in \\mathbb {R}^2$ the geographical coordinate vector of $i$ th data point.", "The data set is denoted as $\\mathcal {D}=\\lbrace \\textbf {d}_1, \\textbf {d}_2, ..., \\textbf {d}_M\\rbrace $ .", "The value $r_\\delta \\in \\mathbb {R}^+$ is the so-called dead zone radius, which determines the data points to be eliminated from the training data set at each SKCV iteration.", "The set $\\mathcal {V}=\\lbrace \\mathcal {V}_1, ..., \\mathcal {V}_K\\rbrace $ is the set of cross validation folds, where each $\\mathcal {V}_p\\subset \\mathcal {D}$ and $\\mathcal {V}_p \\cap \\mathcal {V}_q = \\emptyset $ , when $p\\ne q$ and $\\bigcup _{p=1}^K \\mathcal {V}_p = \\mathcal {D}$ .", "The training of the model is performed by a learning algorithm $\\mathcal {A}$ .", "The vector $\\hat{\\textbf {y}}\\in \\mathbb {R}^M$ denotes the predicted response values by a prediction model $\\mathcal {F}$ .", "Note that the choice of $\\mathcal {F}$ does not affect the functionality of SKCV.", "We use the standard Euclidean distance $e$ to calculate the spatial distance between two data points $\\textbf {d}_i$ and $\\textbf {d}_j$ .", "A formal presentation of the SKCV method is given in Algorithm .", "When the number of folds $K$ equals the number of data points $M$ SKCV becomes SLOO.", "The SKCV algorithm is almost identical to normal CV with the exception of the reduction of the training set depicted in Figure REF and in line 2 of Algorithm .", "In particular, when $r_\\delta = 0$ SKCV reduces to normal CV.", "Spatial k-fold cross validation [1] $\\mathcal {V}, \\mathcal {D}, \\mathcal {A}, r_\\delta $ $\\hat{\\textbf {y}}$ $i \\leftarrow 1$ to $K$ $\\mathcal {H} \\leftarrow \\bigcup _{\\textbf {d}_k \\in \\mathcal {V}_i} \\left\\lbrace \\textbf {d}_j\\in \\mathcal {D}\\;|\\; e(\\textbf {c}_j, \\textbf {c}_k) \\le r_\\delta \\right\\rbrace $ Remove data points too close $\\mathcal {F} \\leftarrow \\mathcal {A}\\left(\\mathcal {D}\\setminus \\mathcal {H}\\right)$ Build model using reduced training set $\\textbf {d}_k \\in \\mathcal {V}_i$ $\\hat{\\textbf {y}}[k] \\leftarrow \\mathcal {F}\\left(\\textbf {x}_k, \\textbf {c}_k\\right)$ Make prediction $\\hat{\\textbf {y}}$ The predicted $\\hat{\\textbf {y}}$ Figure: Reduction of the training set in the SKCV procedure.", "The black and gray points correspond to test and training data points.", "The gray data points inside the perimeters of radius r δ r_\\delta are omitted from the training data, after which the test points are predicted using the remaining training data (i.e.", "the gray data points outside the perimeters).There are three issues one might consider with SKCV which we will address here.", "Firstly, since SKCV may involve removal of a large number of training data, this may introduce an extra pessimistic bias on the prediction performance not related to SAC.", "The size of this bias can be estimated via experiment in which one removes the same amount of randomly selected data from the training set on each CV round.", "Our experimental results in later sections confirm that the performance decrease observed by doing this is negligible compared to the one caused by SAC removal.", "Secondly, the above considered issue becomes far more severe if the number of SKCV folds $K$ is very small (say $K=2$ ).", "It could happen that most of the training data is removed because the combined dead zones of the test data points will have a large effective radius.", "This concern is application specific and the selection of the SKCV folds must be designed to suit the purposes of the application.", "For example with a sparse data set it would make a little practical sense to select the SKCV folds in such a way that all the training data is removed.", "For these reasons it is best to have $K=M$ , which corresponds to the SLOO case of SKCV if computational resources allow it.", "Thirdly, one could ask whether the prediction performance for a given $r_\\delta $ could be estimated by analyzing the prediction error obtained with, say, leave-one-out cross-validation simply as a function of the average distance to closest neighbors.", "While, this could be doable with data sets having both densely and scarcely measured areas, the data points in many available data sets tend to be much closer to each other than in the case we intend to simulate.", "For example, with a dense data set with a maximum distance of 3 m between a data point and its nearest neighbors, one can not simulate performing prediction for a data point having the closest measurements at least 25 meters away.", "Finally, let us consider the difference between spatial interpolation and regression.", "In the former, the only extra information available about the training data are their coordinates $\\textbf {c}$ , while in the latter one also has access to an additional information in the form of feature representation $\\textbf {x}$ .", "However, the SKCV algorithm works in a similar way in both cases, as it is independent on the type of information the learning algorithms use for training a model or what the model uses for predicting the responses for new points.", "Figure: Coverage of data on experimental cases 1-3.", "(a) case 1: 361201 data points, (b) case 2: 1691 data points, (c) case 3: 11795 data points.", "Blue areas correspond to areas where data was available.", "The axes correspond to locations in ETRS-TM35FIN coordinates in kilometers.", "The data set in case 1 is much more dense than data sets in cases 2 and 3." ], [ "Data sets", "The three experimental cases differ on the availability and resolution of the data sets.", "In Case 1 related to water permeability prediction data was available throughout the research area, with the exception of areas where there were obstacles (e.g.", "buildings or lakes).", "In case 2 also related to water permeability prediction, there were scattered field measurement data and in case 3 related to harvester track damage prediction the data set was clustered into several areas.", "These cases are typical of common types of spatial prediction applications.", "The availability of the data sets in three cases is illustrated in Figure REF .", "The data range from remote sensing data sets such as satellite and airborne imaging raster data to manually on-site collected samples of the soil [29], [30], [31], [40], [42], [39], [22].", "The formats of the data sets are TIFF-images and ASCII-files with different resolutions.", "A summary of the used data sets in the three cases is illustrated in Table REF .", "In the following paragraphs we briefly describe the used data sets.", "More detailed illustration of the data sets is given in the supplementary material.", "Digital elevation model data: we downloaded digital elevation model (DEM) data from the file service for open data by the National Land Survey of Finland (NLS).", "The DEM was made from airborne laser scanning data with grid size of 2 m. Several geomorphometric variables were derived from the NLS DEM in SAGA GIS environment.", "In our analysis we used the geomorphometric features: plan curvature, profile curvature, slope, topographic wetness index, flow area, aspect, diffuse insolation and direct insolation [42], [41], [40], [9], [35].", "These derived features are more efficient for prediction than raw height data alone.", "Multi-source national forest inventory data: a selected set of 43 features of the state of the Finnish forests in 20 m grid size are available as the Multi-Source National Forest Inventory (MS-NFI) by Natural Resources Institute Finland (LUKE).", "The MS-NFI data set is derived by interpolating field measured MS-NFI samples using inverse distance-weighted k-nearest neighbor method as the interpolation algorithm and Landsat imagery combined with DEM data as the basis of interpolation.", "Features include e.g.", "the biomass and volume of growing stock.", "The MS-NFI data features exhibit built-in dependencies which means the number of useful features is lower than 43.", "A detailed description of the MS-NFI is [39].", "Table: Summary of the used data sets in all experimental cases.", "Response value data sets are listed in emphasized form.", "Also the data format is shown either as TIFF-raster image or ASCII-vector file and grid resolution size in meters.Aerial gamma-ray spectroscopy data: the aerial gamma-ray flux of potassium (K) decay with the grid size of 50 m is provided by the Geological Survey of Finland (GTK).", "This data is related to e.g.", "the moisture dynamics, frost heaving [22] and density, porosity and grain size of the soil.", "High gamma-radiation indicates lower soil moisture and vice versa.", "Several statistical and textural features were derived from the gamma-ray data.", "These include: 3x3 windowed mean and standard deviation, Gabor filter features [17] and Local Binary Pattern features [28].", "Peatland data: the peatland data is provided by LUKE and uses topographic information provided by NLS.", "The peatland data is a binary raster mask of 1000 m grid size with values 0/1 corresponding to non-peatland/peatland areas.", "The peatland mask is derived from four NLS topographic database elements depicting different type of peatlands.", "The mask bit 1 refers to a spot where the location is mostly covered by peatland vegetation and the peat thickness exceeds 0.3 m over an local area of 1000 m$^2$ .", "Air-borne electromagnetic data: the air-borne electromagnetic (AEM) data was provided by the GTK.", "The apparent resistivity indicates the soil type factors, e.g.", "grain size distribution, water content and quality in the soil and cumulative weathering.", "Weather data: weather data on temperature (C) and rainfall (mm) for years 2011-2013 was provided by the Finnish Meteorological Institute (FMI).", "The grid size of the data set was 10 km.", "We used the mean temperature and rainfall of the last 30 days at each observation point of the response value.", "Stoniness data: stoniness was estimated by steel-rod sounding [37].", "The rod was pushed into the soil where the penetration depth and stone hits were recorded.", "Soil moisture data: gravimetric soil water content was measured from the samples by drying the soil samples and calculating the weight difference of dry and wet soil sample [4].", "Water permeability exponent data: water permeability indicates the nominal vertical speed of water through the soil sample.", "This feature was measured indirectly by observing the soil particle size distribution.", "The actual speed depends on inhomogenities (roots, rocks) and micro-cracks in the soil.", "The water permeability exponent is a logarithmic quantity $y$ derived from water permeability speed $v$ .", "Harvester track damage data: approximately 36 km of strip roads were traversed by Metsäteho Ltd. and visually assessed into damage classes by a forest operations expert.", "The soil damage classes used were: (1) No damage; (2) Slight damage; and (3) Damage.", "The original dataset required preprocessing by LUKE due to the inaccuracies in GPS-tracks.", "The strip road line segments were then converted to sample points used in the prediction process." ], [ "Experimental analysis with SKCV", "In this section the SKCV method is applied to three real world cases involving GIS-data making them suitable to illustrate the proposed method.", "In the first two cases the water permeability levels of boreal soil are predicted and in the final case the damage caused by movements of a forest harvester.", "The experiments provide useful results e.g.", "for forest industry where it is crucial to have accurate and optimistically unbiased prediction performance for soil conditions.", "It is estimated that forest industry in Finland alone has yearly costs of approximately 100M  caused by challenging trafficability conditions of the soil which increase time and fuel consumption and decrease the efficiency of timber harvesting operations [27], [36].", "These costs could be decreased by additional information on soil conditions, especially soil bearing capacity by utilizing public GIS-data.", "The research question (1) will be addressed in cases 1, 2, 3 (sections 4.1, 4.2, 4.3) and the research question (2) will be addressed in case 1 (section 4.1).", "In all experimental cases $k$ -nearest neighbor (kNN) algorithm was used as the prediction model $\\mathcal {F}$ and the predictor features $\\textbf {x}_i$ were z-score standardized.", "While there are many alternative prediction methods the choice does not have an effect on the presence of SAC in the data, therefore kNN was selected due to its simplicity.", "As a distance function that determines the nearest neighbors we use the Euclidean distance for the feature vectors $\\textbf {x}_i$ .", "Note that this is in contrast to the spatial distance $e$ used in SKCV.", "We implemented the analyses using $k$ -values of $\\lbrace 1,3,5,7,9,11,13,15\\rbrace $ for kNN.", "The general behavior of SKCV results was similar for all tested $k$ -values and for this reason we only report the results with $k=9$ .", "The performance measures used in the experiments were the standard root mean squared error (RMSE) for kNN-regression [3] and classification accuracy for kNN-classification.", "In cases 1 and 2 (regression) the predicted response value $\\hat{y}_i$ is defined as the average value of k-nearest neighbors and in case 3 (classification) the mode of the k-nearest neighbors.", "Figure: Calculation of the semivariogram.", "Data points within a distance range [m-t,m+t][m-t, m+t] are considered to be mm meters away from the center data point.The semivariograms and Moran's I statistics were calculated for the response variables $y_i$ in all experimental cases to confirm the presence of SAC in the data.", "In a semivariogram, a variable $X$ is spatially autocorrelated at a given distance range $[m-t, m+t]\\subset \\mathbb {R}^+$ with lag tolerance $t\\in \\mathbb {R}^+$ if its semivariogram value $\\gamma _t(m)\\in \\mathbb {R}^+$ is lower than the sill value of the variable $X$ [13].", "The lag tolerance $t$ gives us the maximum allowed deviation from $m\\in \\mathbb {R}^+$ when the distance between two data points is still considered to be $m$ meters (see Figure REF ).", "For example if $m=10$ meters and $t=1$ meter, then the semivariogram value $\\gamma _1(10)$ for a single data point $\\textbf {d}_i$ is calculated from the set $\\Gamma = \\lbrace \\textbf {d}_j \\in \\mathcal {D} \\;|\\; e(\\textbf {c}_j, \\textbf {c}_i) \\in [9, 11]\\rbrace $ .", "In other words, the data points in set $\\Gamma $ are considered to be 10 meters away from $\\textbf {d}_i$ .", "This is rarely exactly the case and hence we have to use the lag tolerance $t$ .", "The lag tolerance values in the experimental cases were selected to suite the resolution of the corresponding data.", "In the Moran's I autocorrelation plots we call baseline the 0 correlation." ], [ "CASE 1: Soil water permeability prediction based on soil type", "In this section we will consider the predictability of the soil water permeability levels based on the soil type.", "The response variable in this case is the water permeability exponent value $y\\in \\mathbb {R}$ , which is related both to the boreal soil type and to the water permeability itself.", "The exact relation between these two factors is presented in [29].", "Optimal harvesting routes avoid areas with small water permeability, where soil tends to stay moist and there is an elevated risk for ground damage and logistic problems.", "A reliable estimate of the water permeability distribution is needed when making routing decisions during the preliminary planning phase and during the harvest operations.", "The aim here is to increase the efficiency and minimize the harvesting costs.", "The target area is located in the municipality of Parkano, which is a part of the Pirkanmaa region of Western Finland.", "The size of the target area is approximately $144\\,\\text{km}^2$ (ETRS-TM35FIN coordinates at 278 kmE, 6882 kmN, zone 35).", "When considering all the features including the derived ones, we had a total of 49 predictor features in the data set.", "In the analysis of case 1 a total of 361201 data points were available.", "A summary of the data sets is illustrated in Table REF of section 3.", "In Figure REF depicting the semivariogram and Moran's I plot for the water permeability exponent $y$ we can see a clear presence of SAC.", "The predicted water permeability exponent $\\hat{y}_i$ for the $i$ th data point $\\textbf {d}_i=(\\textbf {x}_i, y_{i}, \\textbf {c}_i)$ using kNN-regression is defined as: $\\hat{y}_i = \\frac{1}{k}\\sum _{y \\in N_i}y,$ where $N_i$ is the set of water permeability exponent values $y$ of the $k$ -nearest neighbors of $\\textbf {d}_i$ .", "Figure: The semivariogram and Moran's I plot depicting the SAC of the water permeability exponent yy in case 1.", "(a) Semivariogram showing that γ(m)\\gamma (m) stays below the sill with t=10t=10 m. (b) Moran's I also revealing the presence of SAC in response value yy.Figure: Prediction performance estimates for 9NN using SKCV and SKCV-RLO in case 1.", "The curves are plotted with three spatial densities to illustrate how the spatial density of the data set affects the results.The estimated prediction performance for 9-nearest neighbor (9NN) using SKCV is illustrated in Figure REF , which answers to research question (1) with various distance values $r_\\delta $ .", "The spatial density in the results describes how many data points are in a given space, i.e.", "it describes the sparsity of the data set.", "From Figure REF we notice a clear rise in the prediction error (RMSE) when the distance between prediction point and training data increases.", "This was an expected result based on the SAC discovered in the semivariogram and Moran's I plots in Figure REF .", "With sparser data sets we notice the dead zone radius having a smaller effect on the results.", "To measure how much the SKCV's performance decrease along the increasing dead zone radius is caused only by the decreased size of the training set, we implement additional analysis which we refer to as SKCV random-leave-out (SKCV-RLO).", "SKCV-RLO is identical to the SKCV method (see Algorithm and Figure REF ) with the exception that instead of removing data points from the training set that are too close to the test data, i.e.", "inside the dead zone perimeter, we instead remove the same number of data points randomly from the training set as we would remove in SKCV.", "In Figure REF is illustrated the estimated prediction performance for 9NN using SKCV-RLO.", "On all spatial densities we notice SKCV-RLO being less sensitive to the number of data points removed from the training set giving more optimistic results than SKCV.", "This reinforces our claim that the prediction algorithm prefers to use data points which are geographically close to the prediction point and shows that random removal of training data points causes negligible change in prediction accuracy when compared with SAC-based data removal.", "Figure: Left: In SKCV the test point is always at least r δ r_\\delta meters away from training data.", "Right: In sample-generalize procedure we sample data points (the gray points) using a hexagonal grid and predict the rest of the area around the sampled points.", "The black point represents a prediction point where the distance to training data is maximum, i.e.", "r δ r_\\delta meters.Next, we focus our attention on research question (2), i.e.", "how densely we should sample data points from a new research area to achieve a given prediction level.", "Imagine that there are two distinct geographical areas which we refer to as areas $A$ and $B$ .", "In area $A$ , there exists a data set of measurements gathered from a certain subset of its coordinates but there are no measurements from area $B$ yet.", "The aim is to perform a number of measurements from area $B$ in order to construct a model for predicting the rest of the measurement values for every possible point in area $B$ .", "Performing measurements used to form a training set is expensive, and hence their number should be minimized under the constraint that at least a given prediction performance level is required.", "This trade-off between the number of training measurements and prediction performance is not known in advance and our hypothesis is that it can be estimated with SKCV on the existing data from area $A$ .", "Namely, if the prediction performance estimate provided by SKCV with dead zone radius $r_\\delta $ on area $A$ is as good or better than the required performance level, we hypothesize that we obtain as good prediction performance in area $B$ if we guarantee that the closest measurement points are at most at a distance of $r_\\delta $ from every point in area $B$ .", "Given this constraint, the number of measurement points in area $B$ is minimized via hexagonal sampling [15].", "To support our hypothesis (i.e.", "using SKCV to estimate the trade-off between number of measurement points and prediction performance) we use an auxiliary method called sample-generalize.", "In the sample-generalize procedure we firstly sample training data points hexagonally (e.g.", "measure their response variables) with sampling radius $r_\\delta $ , and secondly we use this data to train a model for predicting the responses from the rest of the area.", "Right side of Figure REF illustrates the sample-generalize procedure.", "Note that SKCV is inherently more pessimistic than sample-generalize since the prediction point is always at least $r_\\delta $ meters away from training data, whereas in sample-generalize the prediction point is always at most $r_\\delta $ meters away from training data (see Figure REF ).", "In order to inspect the goodness of SKCV as an estimator of the prediction performance of sample-generalize we implement a bias-variance analysis for nine smaller subareas formed using a 3x3 grid in the Parkano research area (see Figure REF ).", "We do this by firstly forming 72 $(A, B)$ area pairs (from 3x3 grid we get 9*8=72 area pairs, i.e.", "each smaller area has 8 pair possibilities) from the nine smaller subareas.", "Secondly, for each of the area pairs $(A,B)$ we calculate the prediction performance estimate with SKCV on area $A$ ($result_A$ ) and the prediction performance of sample-generalize on area $B$ ($result_B$ ) and then we take the difference of them ($result_A-result_B$ ).", "Lastly, we calculate the mean and standard deviation of the differences on the 72 area pairs.", "The resulting bias-variance plot is illustrated in Figure REF .", "From the plot we see that the SKCV estimates tend to be pessimistically biased on the range $r_\\delta \\in [0, 150]$ meters.", "In range $r_\\delta \\in [150, 340]$ meters the SKCV estimation is almost unbiased and in range $r_\\delta \\in [340, 400]$ meters it is optimistically biased.", "The results are pretty stable on all spatial densities for SKCV, the spatial density seems to shift the results simply by a constant value.", "Figure: (a) Division of a research area into nine smaller subareas using a 3x3 grid.", "Each smaller area is 16km 2 16\\,\\text{km}^2 in size and consists from approximately 40,000 data points.", "(b) Bias-variance (μ±σ)(\\mu \\pm \\sigma ) plot for the difference between the prediction performance estimate produced by SKCV and the actual prediction performance of sample-generalize of the 72 (9*8) area pairs.", "Solid curves represent the mean μ\\mu and dashed lines standard deviation σ\\sigma .", "Different colors represent different spatial densities for the data set in area AA where SKCV is implemented." ], [ "CASE 2: Soil water permeability prediction based on field measurements", "In this section we consider the predictability of forest soil water permeability based on field measurement data.", "The difference between the response variables in cases 1 and 2 is that in the case 1 the water permeability exponent $y$ is based on remote sensing data and in the case 2, $y$ is based on field measurements.", "Semivariogram and Moran's I plot for the response variable is presented in Figure REF which show clear SAC in the data.", "There is more variability in the SAC of case 2 than case 1 but we must note that the data set in case 2 was much smaller and more sparse.", "Figure: The semivariogram and Moran's I plot depicting the SAC of the response value of case 2.", "(a) Semivariogram with t=80t=80 m. (b) Moran's I.Figure: The SLOO and SLOO-RLO results in the Pomokaira analysis.", "The y-axis corresponds to the RMSE and x-axis to the length of dead zone radius r δ r_\\delta .The research area is located in Pomokaira, the northern part of the municipality of Sodankylä, which is a part of Finnish Lapland.", "The size of the target area is $18432 \\;\\text{km}^2$ .", "The center point of the rectangular target area is at ETRS-TM35FIN coordinates 7524 kmN, 488 kmE, zone 35.", "A total of 1691 data points were collected around the research area.", "The distances between the data points was much larger and they were not available from the entire research area when compared with the case 1 data set.", "102 feature variables were used for predicting the response value i.e.", "the water permeability exponent $y$ .", "The used data sets in case 2 are shown in Table REF .", "The response variable $y$ is predicted in exactly the same way as in case 1 using kNN-regression in Equation REF .", "Because the number of data points was significantly lower when compared with case 1 it was computationally feasible to implement SLOO and SLOO-RLO analyses on the data.", "The SLOO and SLOO-RLO results of case 2 are illustrated in Figure REF .", "The SLOO results show a clear drop in the prediction performance as the dead zone radius $r_\\delta $ is increased.", "A high optimistic bias is observed from the SLOO-RLO results when compared with SLOO.", "The SLOO results indicate that the prediction performance decreases radically after the distance between test and training data is approximately 40-50 meters.", "The effect of SAC can clearly be noted in these results." ], [ "CASE 3: Soil track damage classification", "In this case the goal is to assess the classification of forest harvester track damage.", "In other words, the task is to predict the damage that would occur to a soil point if a forest harvester drives through it.", "In particular, damage means the depression caused on the soil by the harvester.", "Track damage is affected by soil type, humidity, penetration resistance etc.", "The penetration resistance of soil is an important factor in forest harvesting operations which must be accounted for in order to prevent additional costs for harvesting.", "Peat areas for example cause challenging soil conditions for heavy machinery and extra carefulness is needed there.", "It is both expensive and laborious operations to get sunken forest harvesters out from peats.", "Therefore it is important to select harvesting routes which have the highest possible penetration resistance.", "As in cases 1 and 2, the semivariogram and Moran's I plot for the response variable of case 3 are presented in Figure REF , which also show a clear presence of SAC.", "Note that the track damage is an ordinal variable consisting from three classes and hence it was also possible to construct a variogram in this case.", "The research area consists from 13 different harvesting areas in Pieksämäki, a municipality located in the province of Eastern Finland 62$^{\\circ }$ 18'N 27$^{\\circ }$ 08'E.", "A total of 83 feature variables were used for classifying the soil damage.", "The sizes of the data sets collected from each of these areas ranged from hundreds of samples to thousands of samples.", "The total number of data points was 11795.", "As in cases 1 and 2 the used data sets in case 3 are shown in Table REF .", "In case 3 the predicted response value of $\\hat{y}_i$ (track damage class) is defined as the mode of set $N_i$ (kNN-classification), where $N_i$ is again the set of k-nearest neighbors of data point $\\textbf {d}_i$ .", "Figure: The semivariogram and Moran's I plot depicting the SAC of the response value of case 3.", "(a) Semivariogram with t=1t=1 m. (b) Moran's I.Figure: The SLOO and SLOO-RLO results in the Pieksämäki analysis.", "The y-axis corresponds to the fraction of successful classifications and x-axis to the length of dead zone radius r δ r_\\delta .The SLOO and SLOO-RLO analyses were conducted on each of the 13 harvest areas separately because the distances between the harvest areas were in worst cases dozens of kilometers.", "On each of these areas the SLOO and SLOO-RLO procedures were implemented and the results were averaged over all areas.", "Figure REF presents the SLOO and SLOO-RLO results for case 3.", "Similarly as in cases 1 and 2, the results in case 3 confirm the effect of SAC on prediction performance estimates.", "One can notice an exponential form decay in the SLOO results as a function of dead zone radius $r_\\delta $ whereas the SLOO-RLO results are almost unchanged as it was also in case 2.", "In the worst case we have approximately 40% difference in the results between SLOO and SLOO-RLO." ], [ "Conclusion", "Spatio-temporal autocorrelation is always present with GIS-based data sets and needs to be accounted for in machine learning approaches.", "As discussed above, traditional model performance criteria such as the CV method omit the consideration of the effect of SAC in the performance estimations with natural data sets.", "To account for the SAC in GIS-based data sets we demonstrated by the means of three experiments that the SKCV method can be used for estimating the prediction performance of spatial models without the optimistic bias due to SAC, while the ordinary CV can cause highly optimistically biased prediction performance estimates.", "We also showed that SKCV can be used as a data sampling density selection criterion for new research areas, which will result in reduced costs for data collection." ], [ "Acknowledgements", "We want to thank the Natural Resources Institute Finland (LUKE), Geological Survey of Finland (GTK), Natural Land Survey of Finland (NLS) and Finnish Meteorological Institute (FMI) for providing the data sets.", "This work was supported by the funding from the Academy of Finland (Grant 295336).", "The preprocessing of the data was partially funded by the Finnish Funding Agency for Innovation (Tekes)." ] ]

[ [ "Stability analysis for cosmological models in $f(T,B)$ gravity" ], [ "Abstract In this paper we study cosmological solutions of the $f(T,B)$ gravity using dynamical system analyses.", "For this purpose we consider cosmological viable functions of $f(T,B)$ that are capable of reproducing the dynamics of the Universe.", "We present three specific models of $f(T,B)$ gravity which have a general form of the solutions by writing the equations of motion as an autonomous system.", "Finally, we study its hyperbolic critical points and general trajectories in the phase space of the resulting dynamical variables which are compatible with the current late-time observations." ], [ "Introduction", "The $\\Lambda $ CDM cosmological model is supported by overwhelming observational evidence in describing the evolution of the Universe at all cosmological scales [1], [2] which is achieved by the inclusion of matter beyond the standard model of particle physics.", "This takes the form of dark matter which acts as a stabilizing agent for galactic structures [3], [4] and materializes as cold dark matter particles, while dark energy is represented by the cosmological constant [5], [6] and produces the measured late-time accelerated expansion [7], [8].", "However, despite great efforts, internal consistency problems persist with the cosmological constant [9], as well as a severe lack of direct observations of dark matter particles [10].", "On the other hand, the effectiveness of the $\\Lambda $ CDM model has also become an open problem in recent years.", "At its core, the $\\Lambda $ CDM model was convinced to describe Hubble data but the so-called $H_0$ tension problem calls this into question where the observational discrepancy between model independent measurements [11], [12] and predicted [13], [14] values of $H_0$ from the early-Universe appears to be growing [15].", "While measurements from the tip of the red giant branch (TRGB, Carnegie-Chicago Hubble Program) point to a lower $H_0$ tension, the issue may ultimately be resolved by future observations which may involve more exotic measuring techniques such as the use of gravitational wave astronomy [16], [17] with the LISA mission [18], [19].", "However, it may also be the case that physics beyond general relativity (GR) are at play here.", "The use of autonomous differential equations to investigate the cosmological dynamics of modified theories of gravity has been shown to be a powerful tool in elucidating cosmic evolution within these possible models of gravity [20].", "These analyses can reveal the underlying stability conditions of a theory from which it may be possible to constrain possible models on theoretical grounds alone.", "Theories beyond GR come in many different flavours [2], [21] where many are designed to impact the currently observed late-time cosmology dynamics.", "By and large, the main trust of these theories comes in the form of extended theories of gravity [22], [23], [21] which build on GR with correction factors that dominate for different phenomena.", "However, these are collectively all based on a common mechanism by which gravitation is expressed through the Levi-Civita connection, i.e.", "that gravity is communicated by means of the curvature of spacetime [1].", "In fact, it is the geometric connection which expresses gravity while the metric tensor quantifies the amount of deformation present [24].", "This is not the only choice where torsion has become an increasingly popular replacement and produced a number of well-motivated theories [25], [26], [27].", "Teleparallel Gravity (TG) collectively embodies the class of theories of gravity in which gravity is expressed through torsion through the teleparallel connection [28].", "This connection is torsion-ful while being curvatureless and satisfying the metricity condition.", "Naturally, all curvature quantities calculated with this connection will vanish irrespective of metric components.", "Indeed, the Einstein-Hilbert which is based on the Ricci scalar $\\accentset{\\circ }{R}$ (over-circles represent quantities calculated with the Levi-Civita connection) vanishes when calculated with the teleparallel connection, i.e.", "in general $R=0$ while $\\accentset{\\circ }{R}\\ne 0$ .", "Moreover, the identical dynamical equations can be arrived at in TG by replacing the Einstein-Hilbert action with it's so-called torsion scalar $T$ .", "By making this substitution, we produce the Teleparallel equivalent of General Relativity (TEGR), which differs from GR by a boundary term $B$ in its Lagrangian.", "The boundary term in TEGR consolidates the fourth-order contributions to many beyond GR theories.", "By extracting these contributions into a separate scalar, TEGR will have a meaningful and novel impact on extended theories and produce an impactful difference in the predictions of such theories.", "The most direct result of this fact will be that TG will produce a much broader plethora of theories in which dynamical equations are second order.", "This is totally different to the severely limited Lovelock theorem in curvature based theories [29].", "In fact, TG can produce a large landscape in which second-order field equations are produced [30], [31].", "TG also has a number of other attractive features such as its likeness to Yang-mills theory [25] offering a particle physics perspective to the theory, the possibility of it giving a definition to the gravitational energy-momentum tensor[32], [33], and that it does not necessitate the introduction of a Gibbons–Hawking–York boundary term to produce a well-defined Hamiltonian structure, among others.", "Taking the same reasoning as in $f(\\accentset{\\circ }{R})$ gravity [22], [23], [21], TEGR can be straightforwardly generalized to produce $f(T)$ gravity [34], [35], [36], [37], [38], [39].", "$f(T)$ gravity is generally second order due to the weakened Lovelock theorem in TG and has shown promise in several key observational tests [26], [40], [41], [42], [43], [44], [45], [46].", "However to fully embrace the TG generalization of $f(\\accentset{\\circ }{R})$ gravity, we must consider the $f(T,B)$ generalization of TEGR [47], [48], [49], [50], [51], [51], [52].", "In this scenario the second and fourth order contributions to $f(\\accentset{\\circ }{R})$ gravity are decoupled while this subclass becomes a particular limit of the arguments $T$ and $B$ , namely $f(\\accentset{\\circ }{R})=f(-T+B)$ .", "$f(T,B)$ gravity is an interesting theory of gravity and has shown promise in terms of solar system tests and the weak field regime [53], [54], [50], as well as cosmologically both in terms of its theoretical structure [47], [49], [51], [47] and confrontation with observational data [55].", "In this work, we explore the structure of $f(T,B)$ gravity through the dynamical systems approach in the cosmological context of a homogeneous and isotropic Universe using the Friedmann–Lemaître–Robertson–Walker metric (FLRW).", "This kind of system has been used to study higher-order modified teleparallel gravity that add a scalar field $\\phi $ depending on the boundary term [49], where the stability conditions for a number of exact solutions for an FLRW background are studied for a number of solutions such as de Sitter Universe and ideal gas solutions.", "In Ref.", "[56], a number of important reconstructions were presented together with their dynamical system evolution.", "This work is also interesting because they compare some of their results with Supernova type 1a data using a chi-square approach.", "Another interesting approach to determining and studying solutions is that of using Noether symmetries Ref.", "[57], which have [49] shown great promise in producing new cosmological solutions that admit more desirable cosmologies.", "On the other hand, in Ref.", "[51] the all important cosmological thermodynamics has been explored as well as the mater perturbations, which was complemented by background reconstructions of further cosmological solutions giving a rich literature of models together with Ref.[58].", "Ref.", "[59] then developed the $f(T,B)$ cosmology energy conditions which can give important information about regions of validity of the models.", "In the present study, the cosmic acceleration dynamics is reproduced only by a non-canonical $\\phi $ that mimics the $\\Lambda $ term.", "In our case, we introduce a $f(T,B)$ dark energy which is fluid-like in order to obtain a richer population of stability points that can be constrained by current observational surveys.", "We do this by first introducing briefly the technical details of $f(T,B)$ gravity in section  and then discussing its dynamical treatment in section .", "In section  we lay out the methodology of the analysis which includes the methods by which the analysis is conducted.", "The $f(T,B)$ gravity dynamical analysis is then realized in section  where the core results for each of the models is presented.", "Finally, we close in section  with a summary of our conclusions.", "In all that follows, Latin indices are used to refer to Minkowski space coordinates, while Greek indices refer to general manifold coordinates." ], [ "$f(T,B)$ cosmology", "We start by considering a flat homogeneous and isotropic FLRW metric in Cartesian coordinates with an absorbed lapse function ($N=1$ ) as (e.g through Ref.", "[1]) ${\\rm d}s^2=-{\\rm d}t^2+a(t)^2({\\rm d}x^2+{\\rm d}y^2+{\\rm d}z^2)\\,,$ where $a(t)$ is the scale factor.", "Also, we choose an arbitrary mapping over $\\tilde{f}(T,B) \\rightarrow -T + f(T,B)$ , which obeys the diffeomorphism invariance.", "As shown in [55], the choice of Lagrangian where $\\tilde{f}(T,B) \\rightarrow -T + f(T,B)$ represents an arbitrary Lagrangian over the torsion scalar and boundary term is diffomorphism invariant.", "In this proposal our choice of tetrad is given by $e^{a}_{\\phantom{a}\\mu }=\\mbox{diag}(1,a(t),a(t),a(t))\\,,$ which reproduces the metric in Eq.", "(REF ) and observes the symmetries of TG.", "In this spacetime, the torsion scalar can be given explicitly as $T = 6H^2\\,,$ while the boundary term is given by $B = 6\\left(3H^2+\\dot{H}\\right)\\,,$ which combine to produce the well known Ricci scalar of the flat FLRW metric $\\accentset{\\circ }{R}=-T+B=6\\left(\\dot{H}+2H^2\\right)$ (where again over-circles again represent quantities determined with the Levi-Civita connection).", "After the above considerations over the geometry, our field equations for a universe filled with a perfect fluid are $&&-3H^2\\left(3f_B + 2f_T\\right) + 3H\\dot{f}_B - 3\\dot{H} f_B + \\frac{1}{2}f = \\kappa ^2\\rho \\,, \\\\&&-\\left(3H^2+\\dot{H}\\right)\\left(3f_B + 2f_T\\right) - 2H\\dot{f}_T + \\ddot{f}_B + \\frac{1}{2}f = -\\kappa ^2 p\\,,$ where $\\rho $ and $p$ represent the energy density and pressure of a perfect fluid whose equation of state is $p= \\omega \\rho $ , respectively.", "These modified Friedmann equations show explicitly how a linear boundary contribution to the Lagrangian would act as a boundary term while other contributions of $B$ would contribute nontrivially to the dynamics of these equations.", "We can rewrite Eqs.", "(REF ,) by considering the modified TEGR components contained in the effective fluid contributions $3H^2 &=& \\kappa ^2 \\left(\\rho +\\rho _{\\text{eff}}\\right)\\,,\\\\3H^2+2 \\dot{H} &=& -\\kappa ^2\\left(p+p_{\\text{eff}}\\right)\\,,$ where $&&\\kappa ^2 \\rho _{\\text{eff}} = 3H^2\\left(3f_B + 2f_T\\right) - 3H\\dot{f}_B + 3\\dot{H}f_B - \\frac{1}{2}f\\,, \\\\&&\\kappa ^2 p_{\\text{eff}} = \\frac{1}{2}f-\\left(3H^2+\\dot{H}\\right)\\left(3f_B + 2f_T\\right)-2H\\dot{f}_T+\\ddot{f}_B\\,.$ The latter equation can be combined to obtain $2\\dot{H}=-\\kappa ^2\\left(\\rho + p + \\rho _{\\text{eff}} + p_{\\text{eff}}\\right)\\,.$ The effective fluid represents the modified part of the $f(T,B)$ Lagrangian which turns out to satisfy the conservation equation $\\dot{\\rho }_{\\text{eff}}+3H\\left(\\rho _{\\text{eff}}+p_{\\text{eff}}\\right) = 0\\,.$ For our purpose and in order to construct the dynamical system, the $f(T,B)$ Friedmann equations can be rewritten as $&&\\Omega + \\Omega _{\\text{eff}} = 1\\,, \\\\&& 3 + 2\\left(\\frac{H^{\\prime }}{H}\\right) = - \\frac{3f}{6H^2} + 9f_B +6f_T + 3\\left(\\frac{H^{\\prime }}{H}\\right)f_B+ 2\\left(\\frac{H^{\\prime }}{H}\\right)f_T + 2f^{\\prime }_T - \\left(\\frac{H^{\\prime }}{H}\\right) f^{\\prime }_B - f^{\\prime \\prime }_B\\,, $ where $\\Omega _{\\text{eff}} = 3f_B + 2f_T - f^{\\prime }_B - \\frac{f}{6H^2} + \\left(\\frac{H^{\\prime }}{H}\\right)f_B\\,,$ which each $i$ denotes the effective density parameter $\\Omega _i = \\kappa ^2 \\rho _i/3H^2$ .", "The prime $(\\prime )$ denotes derivatives with respect to $N=\\ln {a}$ , with a chain rule given by $d/dt = H (d/dN)$ .", "With the latter equations we can write the continuity equations for each fluid under the consideration $ && \\rho ^{\\prime } + 3(1 + \\omega )\\rho = 0, \\\\&& \\rho ^{\\prime }_{\\text{eff}} + 3(\\rho _{\\text{eff}} + p_{\\text{eff}}) = 0,$ where the effective fluid is related with the background cosmology derived from $f(T,B)$ gravity and $\\omega $ are related to the cold dark matter and non-relativistic fluids as matter contributions.", "This set of equations impose a condition over the form of the derivative $f^{\\prime }(T,B)$ .", "Using the Friedmann equations in Eq.", "(REF ) and Eq.", "(), we can directly write down the effective EoS for our $f(T,B)$ gravity as [55], [60] $\\omega _{\\mbox{eff}} &=& \\frac{p_{\\mbox{eff}}}{\\rho _{\\mbox{eff}}}\\\\&=& -1+\\frac{\\ddot{f}_B-3H\\dot{f}_B-2\\dot{H}f_T-2H\\dot{f}_T}{3H^2\\left(3f_B+2f_T\\right)-3H\\dot{f}_B+3\\dot{H}f_B-\\frac{1}{2}f}\\,, $ which can also be written as having a redshift dependence similar to $\\omega (z)$ .", "We can explicitly compute from Eq.", "(REF ) a dynamical equation in terms of the Hubble factor and its derivatives as $&& 6\\left(\\frac{H^{\\prime }}{H}\\right)f_B + 2\\left(\\frac{H^{\\prime }}{H}\\right)f_T + \\left(\\frac{H^{\\prime 2}}{H^2} + \\frac{H^{\\prime \\prime }}{H}\\right)f_B- \\frac{f^{\\prime }}{6H^2} = 0\\,.$ Notice that only the last term on the r.h.s contains information about the specific form of $f(T,B)$ theory (or in its derivative)." ], [ "$f(T,B)$ dynamical system structure", "To construct the dynamical autonomous system for our $f(T,B)$ cosmological model, we follow the approach outlined in Refs.", "[61], [62], [60].", "As a first step we introduce a set of conveniently specified variables which allow us to rewrite the evolution equations as an autonomous phase system.", "This set of equations will be subject to a generic constraint arising from our modified Friedmann equations in Eqs.", "(REF ,).", "For this system we propose to define the parameter [63] $\\lambda = \\frac{\\ddot{H}}{H^3} = \\frac{H^{\\prime 2}}{H^2} + \\frac{H^{\\prime \\prime }}{H}\\,.$ Notice that this expression depends explicitly on $N=\\ln {a}$ (time-dependence).", "It was discussed in the latter reference that for cases when $\\lambda $ = constant, some cosmological solutions can be recover, e.g.", "if $\\lambda =0$ , we can obtain a de Sitter/quasi de Sitter universe or if $\\lambda =9/2$ , a matter domination era can be derived.", "Since this ansatz shows cosmological viable scenarios as analogous to models with barotropic fluids, along the rest of this work we are going to consider $\\lambda $ = constant.", "Following this prescription, we can write our Friedmann evolution equations in term of dynamical variables $x \\equiv f_B\\,, \\quad y \\equiv f^{\\prime }_B\\,, \\quad z \\equiv \\frac{H^{\\prime }}{H} = \\frac{\\dot{H}}{H^2}\\,, \\quad w \\equiv -\\frac{f}{6H^2}\\,.$ From the latter definitions and the Friedmann evolution in Eq.", "(REF ) we can derive the constriction equation from the latter evolution as $\\Omega + 3x +2f_T - y +w +z x=1\\,, $ where $\\Omega $ is a parameter that depends on the other dynamical variables.", "Finally, we can write the autonomous system for this theory as $ &&z^{\\prime } = \\lambda - 2z^2\\,, \\\\&&x^{\\prime } = y, \\\\&&w^{\\prime } = -6zx - 2z f_T - \\lambda x - 2zw\\,, \\\\&&y^{\\prime } = 3w + (9 + 3z)x + f_T(6 + 2z) + 2 f^{\\prime }_T - zy\\nonumber \\\\ &&-3 - 2z\\,.", "$ To follow the constraint of the system in Eqs.", "(REF –REF ), we need to write $f_T$ as a dynamical variable or write it in terms of the described variables.", "This can be done by considering a specific form of $f(T,B)$ as we will show in section ()." ], [ "Stability methodology", "We can study our $f(T,B)$ autonomous system in Eqs.", "(REF ,,,) by performing stability analyses of the critical points, which can be investigated through linear perturbations around their critical values as $\\mathbf {x} = \\mathbf {x}_0 + \\mathbf {u}$ , where $\\mathbf {x}=(x,y,z,w)$ and $\\mathbf {u} = (\\delta x, \\delta y, \\delta z, \\delta w)$ .", "The equations of motion for each of our models can be written as $\\mathbf {x}^\\prime = \\mathbf {f}(\\mathbf {x})$ , which upon linearisation can be given by $\\mathbf {u}^\\prime = \\mathcal {M} \\mathbf {u}\\,, \\quad \\mathcal {M}_{ij}= \\left.", "\\frac{\\partial f_i}{\\partial x_j}\\right|_{\\mathbf {x}_\\ast }\\,,$ where $\\mathcal {M}$ is known as the linearisation matrix [20].", "The eigenvalues indicated by $\\omega $ of $\\mathcal {M}$ determine the stability (type) of the critical points, whereas the eigenvectors $\\mathbf {\\eta }$ of $\\mathcal {M}$ indicates the principal directions of the perturbations performed at linear level.", "As it is standard in the stability analysis, if $\\mathrm {Re}(\\omega ) < 0$ ($\\mathrm {Re}(\\omega ) > 0$ ) the critical point is called stable (unstable).", "More specific types of point will be indicated for each $f(T,B)$ scenario.", "For this case, we should consider perturbations of the four dynamical variables $(x,y,z,w)$ , keeping in mind that they are not all completely independent because they are bound together by the Friedmann constraint in Eq.", "(REF ).", "This dependence would then carry over to perturbative level.", "From Eq.", "(REF ) we notice that there is not an explicit dependency of $f(T,B)$ , therefore for the critical points following the above prescription $\\mathbf {x_*}$ / $\\mathbf {\\dot{x}} = \\mathbf {f}(\\mathbf {x_*})=0$ we require that $z = \\pm \\sqrt{\\frac{\\lambda }{2}}\\,, ~ \\quad ~y=0\\,.$ For each $f(T,B)$ cosmological case we will present the stability results, where we only consider the eigenvalues of the stability matrix $\\mathcal {M}$ for each of the critical points and for the perturbations that are compatible with Eq.", "(REF )." ], [ "Dynamical analyses for $f(T,B)$ models ", "In this work we consider three $f(T,B)$ scenarios.", "They were selected in order to obtain cosmological viable cases, in particular the late-time observed cosmic acceleration.", "The following models were studied in detail in Ref.", "[55], where cosmological constraints of each of them were found.", "In the following, we will focus on the corresponding parameter values which adapt to our autonomous system." ], [ "Stability analysis for General Taylor Expansion model", "The form for this model was presented in Ref.", "[50] as a general Taylor expansion of the $f(T,B)$ Lagrangian, given by $&& f(T,B) = f(T_0, B_0) + f_T(T_0,B_0) (T-T_0)+ f_B(T_0,B_0) (B-B_0) + \\frac{1}{2!", "}f_{TT}(T_0,B_0) (T-T_0)^2 \\nonumber \\\\&&+ \\frac{1}{2!", "}f_{BB}(T_0,B_0) (B-B_0)^2+ f_{TB}(T_0,B_0) (T-T_0)(B-B_0) + \\mathcal {O}(T^3,B^3)\\,,$ which gives the general Taylor expansion of the $f(T,B)$ Lagrangian about its Minkowski values for the torsion scalar $T$ and boundary term $B$ .", "We notice from here that we need to take into account beyond linear approximations since $B$ is a boundary term at linear order.", "Following the form for the FLRW tetrad in Eq.", "(REF ), where locally spacetime appears to be Minkowski with torsion scalar and boundary term values, we can consider $ T_0 = B_0 = 0\\,$ .", "Taking constants called $A_i$ , the Lagrangian can be rewritten as $f(T,B)\\simeq A_0+A_1 T + A_2 T^2 + A_3 B^2 + A_4 TB\\,,$ where the linear boundary term vanishes.", "We notice from this specific form that the first term can be seen as $A_0 \\approx \\Lambda $ , therefore we are dealing with a cosmic acceleration as a consequence of the $f(T,B)$ .", "Thus, the form of this model can be written in terms of the dynamical variables as $ f_T = -(3+z)x - 2w - A_1\\,,$ at linear order in torsion and with $A_0=0$ , i.e we are switching-off the cosmological constant.", "This can be done since an explicitly time-dependent factor appears and then a different approach has to be taken.", "The critical points for this model are $w= -A_1\\,, ~ \\quad ~x = \\frac{A_1 -1}{3 \\pm \\sqrt{\\frac{\\lambda }{2}}}\\,,$ which imply that the constriction evolution equation in Eq.", "(REF ) is now $\\Omega =0$ .", "According to these points we can compute the following eigenvalues for the system $\\omega _{1} &=&-3 \\mp \\sqrt{\\frac{\\lambda }{2}}\\,, \\quad \\omega _{2} =-3 \\mp 2\\sqrt{\\frac{\\lambda }{2}}\\,,\\nonumber \\\\\\omega _{3} &=&\\mp 4\\sqrt{\\frac{\\lambda }{2}}\\,, \\quad \\omega _{4} =\\pm 2\\sqrt{\\frac{\\lambda }{2}}\\,.$ Considering values as $\\lambda \\ne 0$ , we get that $Re(\\omega _3) = - Re(\\omega _4)\\ne 0$ , implying that for this system all the critical points are saddle-like for any value of $\\lambda $ .", "In Fig.", "(REF ), we show different views of the phase space of the dynamical system in Eq.", "(REF ) on 2-d surfaces.", "The solutions for the case are in agreement with the cosmological constraints found in [55].", "According to these results, our critical points behave as $A_{i}< A_{i+1}$ (which states for these values $A_0=0$ and $A_1=1$ ), show a quintessence behaviour and when $B$ dominates and $z\\approx 1$ .", "After that, a $\\Lambda $ CDM model behaviour is observed.", "Figure: Different views of the phase space of the dynamical system in Eq.", "() for λ≠0\\lambda \\ne 0.", "The system was reduced to a 2-d surface representing the different perspectives of its constraint.", "The arrows represent the direction of the velocity field and the trajectories reveal their stability properties as described for this model." ], [ "Stability analysis for Power Law model ", "If we consider a Lagrangian of separated power law style models for the torsion and boundary scalars, we can write a model like [49] $f(T,B) = b_0 B^k + t_0 T^m\\,.", "$ This is an interesting model since it was already been shown in Ref.", "[64] that for $m<0$ the Friedmann equations will be effected mostly in the accelerating late-time universe while for $m>0$ this impact will take place for the early universe, assuming no input from the boundary contribution.", "By incorporating the boundary term, this analysis will reveal an effect of $B$ on the combined evolution within $f(T,B)$ cosmology.", "The form for this model can be written in terms of the dynamical variables as: $f_T = -mw - \\frac{m}{k}(3+z)x\\,.$ The critical points for this scenario are $w = \\frac{k-m}{m(1-k)}\\,, ~ \\quad ~x = -\\frac{k}{m} \\Big (\\frac{m-1}{k-1}\\Big ) \\frac{1}{3 \\pm \\sqrt{\\frac{\\lambda }{2}}}\\,.$ For this case, the constriction (REF ) gives again $\\Omega =0$ .", "According to Eq.", "(REF ), we can analyse independently the positive and negative roots with $z = \\pm \\sqrt{\\frac{\\lambda }{2}}$ as follows." ], [ "Critical points.", "For this case, the eigenvalues derived from the stability matrix are $\\tiny \\omega _1 &=& -\\sqrt{2} \\sqrt{\\lambda }-3\\,, \\\\\\omega _2 &=& -2 \\sqrt{2} \\sqrt{\\lambda }\\,, \\\\\\omega _3 &=& -\\frac{1}{4 k}\\left(\\sqrt{\\alpha -\\beta +\\gamma }+k\\left(\\sqrt{2} \\sqrt{\\lambda } (1-2 m)-6\\right)+2 \\left(\\sqrt{2} \\sqrt{\\lambda }+6\\right) m \\right)\\,,\\\\\\omega _4 &=& \\frac{1}{4 k}\\left(\\sqrt{\\alpha -\\beta +\\gamma }+k\\left(\\sqrt{2} \\sqrt{\\lambda } (2 m-1)+6\\right)-2 \\left(\\sqrt{2} \\sqrt{\\lambda }+6\\right) m \\right)\\,,$ where $\\alpha &=& 2 k^2 \\left(\\lambda (2 m+1)^2+6 \\sqrt{2} \\sqrt{\\lambda } (6m-1)+18\\right)\\,, \\\\\\beta &=&8 k m \\left(\\lambda +6 \\sqrt{2} \\sqrt{\\lambda } (m+1)+2 \\lambda m+18\\right)\\,,\\\\\\gamma &=&8 \\left(\\lambda +6 \\sqrt{2} \\sqrt{\\lambda }+18\\right) m^2\\,,$ which under the conditions $Re(\\omega _1), Re(\\omega _2) < 0$ for any value of $\\lambda $ we get saddle/attractors points.", "Case $Re(\\omega _3) = 0$ .", "For the condition $Re(\\omega _3) = 0$ the critical regions areFrom this point, along the text we refer to the symbol $\\vee $ as or, $\\wedge $ as and.", "$k=1\\wedge 0<m<\\frac{1}{2}\\wedge 0<\\lambda <72 m^2-72 m+18\\,,$ $0<m<\\frac{1}{2}\\wedge 2 m<k\\le 1\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,,$ For the condition $Re(\\omega _4) = 0$ the critical regions are $k=1\\wedge \\left[\\left(0<m<\\frac{1}{2}\\wedge \\lambda >72 m^2-72 m+18\\right)\\vee \\left(m\\ge \\frac{1}{2}\\wedge \\lambda >0\\right)\\right]\\,,$ $0<m<\\frac{1}{2}\\wedge 2 m<k\\le 1\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,,$ Attractor regions.", "These cases can happen under the following conditions: $0 < m \\le \\frac{1}{2} \\wedge 0 < k < 1 \\wedge \\lambda > 18 \\,,$ $m > \\frac{1}{2} \\wedge 0 < k < 1 \\wedge \\lambda > 0 \\,,$" ], [ "Properties:", " If $m> k$ then, $w<-1/3$ ($b_0$ and $c_0$ fixed as positive).", "If $m< k$ then, we get $\\Lambda $ CDM.", "($b_0$ and $c_0$ fixed as positive).", "If $b_0 < t_0$ and vice versa, we get a crossover over the phantom divided-line ($w=-1$ ).", "We recover $\\Lambda $ CDM and late cosmic acceleration." ], [ "Critical points.", "For this case, the eigenvalues derived from the stability matrix are given by $\\tiny \\omega _1 &= \\sqrt{2} \\sqrt{\\lambda }-3\\,, \\\\\\omega _2 &= 2 \\sqrt{2} \\sqrt{\\lambda }\\,, \\\\\\omega _3 &= \\frac{1}{4 k}\\left(-\\sqrt{\\frac{\\gamma k^2}{4 m^2}+\\eta }\\right)+\\zeta \\,, \\\\\\omega _4 &= \\frac{1}{4 k}\\left(\\sqrt{\\frac{\\gamma k^2}{4 m^2}+\\eta }\\right)+\\zeta \\,.$ with $\\zeta &=& \\frac{1}{4 k}\\left(k \\left(\\sqrt{2} \\sqrt{\\lambda } (1-2 m)+6\\right)+2 \\left(\\sqrt{2} \\sqrt{\\lambda }-6\\right) m\\right)\\,,\\\\\\eta &=& 8 m^2\\left(\\lambda (k-1)^2+6 \\sqrt{2} \\sqrt{\\lambda } (k-1)+18\\right)- 8 m k\\left(3 \\sqrt{2} \\sqrt{\\lambda } (3 k-2)-k \\lambda +\\lambda +18\\right)\\,.$ Notice that according to the values of $\\omega _1$ and $\\omega _2$ , the critical point associated to the negative root case corresponds to a scenario where the universe has a contraction (accelerated) phase $\\lambda < 2$ ($\\lambda >2$ ), respectively, which represents a saddle point.", "On the other hand, according to the value of $\\omega _1$ , we notice that if $0 <\\lambda < \\frac{9}{2}$ , the critical point is hyperbolic and saddle-type.", "To simplify the analysed regions, we consider cases where $\\omega _1$ , or $\\omega _2$ , only have a non-vanishing real part.", "The next case to explore will be with a vanished real part (which correspond to a non-hyperbolic case).", "Conditions with $Re(\\omega _3) = 0$ .", "These regions are $0<m < 1\\wedge \\left((0<k<1\\wedge \\lambda =18)\\vee \\left(k=1\\wedge \\lambda >72 m^2-72 m+18\\right)\\vee (k>1\\wedge \\lambda =18)\\right),$ $k=1\\wedge \\left[\\left(m\\ge 1\\wedge \\lambda \\ge 72 m^2-72 m+18\\right)\\vee \\left(\\frac{3}{4}<m<1\\wedge \\lambda =72 m^2-72 m+18\\right)\\right]\\,,$ $0<k<1\\wedge \\left(m\\ge 1\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right),$ $1<k<2\\wedge \\left(\\frac{3 k}{2 k+2}<m\\le 1 \\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right) \\,,$ $k=2\\wedge m=1\\wedge \\frac{9}{2}<\\lambda \\le 18\\,,$ $k>2\\wedge 1\\le m<\\frac{3 k}{2 k+2}\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,,$ $k<0\\wedge \\left[(m>1\\wedge \\lambda =18)\\vee \\left(1\\le m\\le \\frac{3}{2}\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right)\\right]\\,,$ $m>\\frac{3}{2}\\wedge -\\frac{2 m}{2 m-3}<k<0\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,,$ Condition $Re(\\omega _4) = 0$ .", "These regions are: $k=1\\wedge \\frac{3}{4}<m\\le 1\\wedge \\frac{9}{2}<\\lambda \\le 72 m^2-72 m+18\\,,$ $\\textstyle &\\left[m\\ge 1\\wedge 0<k<1\\wedge \\left(\\lambda =18\\vee \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right)\\right]\\nonumber \\\\&\\vee \\left(m>1\\wedge k=1\\wedge \\frac{9}{2}<\\lambda <72 m^2-72 m+18\\right) \\nonumber \\\\&\\vee (k>1\\wedge \\lambda =18)\\,,$ $1<k<2\\wedge \\left[\\left(\\frac{3 k}{2 k+2}<m<1\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right)\\vee (m=1\\wedge \\lambda =18)\\right]\\,,$ $k=2\\wedge m=1\\wedge \\frac{9}{2}<\\lambda \\le 18\\,,$ $k>2\\wedge \\left(1\\le m<\\frac{3 k}{2 k+2}\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right),$ $k<0\\wedge \\left[(0<m<1\\wedge \\lambda =18)\\vee \\left(1\\le m\\le \\frac{3}{2}\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\right)\\right]\\,,$ $m>\\frac{3}{2}\\wedge -\\frac{2 m}{2 m-3}<k<0\\wedge \\lambda =\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,.$ Saddle regions.", "These regions are determine by the condition $Re(\\omega _3),Re(\\omega _4) >0$ , therefore $0<m<\\frac{1}{2} \\wedge \\left[ \\left(0 < k < 1 \\wedge \\lambda >18 \\right)\\vee \\left(k>1 \\wedge \\frac{9}{2}< \\lambda < 18\\right)\\right]\\,,$ $m=\\frac{1}{2} \\wedge \\left[ \\left( 0<k < 1 \\wedge \\lambda >18 \\right)\\vee \\left( k>1 \\wedge \\frac{9}{2}<\\lambda < 18\\right) \\right]\\,,$ $\\frac{1}{2} < m < 1 \\wedge 0 < k < 1 \\wedge \\lambda > 18\\,,$ $m \\ge 1 \\wedge \\left[\\left( 0 < k < \\frac{2m}{1+2m} \\wedge \\lambda > 18 \\right) \\vee \\left( k > \\frac{2m}{1+2m} \\wedge \\lambda > 18 \\right) \\right]\\,,$ $1<m<\\frac{3}{2}\\wedge k>-\\frac{2 m}{2 m-3}\\wedge \\frac{9}{2}<\\lambda <\\frac{18 (k-2 m)^2}{(-2 k m+k+2 m)^2}\\,,$ $k <0 \\wedge \\left[\\left( 0 < m < 1 \\wedge \\frac{9}{2} < \\lambda < 18 \\right) \\wedge \\left( m > 1 \\wedge \\frac{9}{2} < \\lambda < \\frac{18(k-2m)^2}{(k+2m -2km)^2}\\right) \\right]\\,.$ In Fig.", "(REF ) we show the dynamical phase space for this model.", "Figure: Different views of the phase space of the dynamical system in Eq.", "() for λ≠0\\lambda \\ne 0.", "The system was reduced to a 2-d surface representing the different perspectives of its constraint with x=-k mm-1 k-11 3+zx = -\\frac{k}{m} \\Big (\\frac{m-1}{k-1}\\Big ) \\frac{1}{3 + z}.", "The arrows represent the direction of the velocity field and the trajectories reveal their stability properties as described for this model." ], [ "Stability analysis for Mixed Power Law Model ", "In order to reproduce several important power law scale factors relevant for several cosmological epochs, in Ref.", "[49] a form of $f(T,B)$ given by $ f(T,B) = f_0 B^k T^m\\,,$ was presented, where the second and fourth order contributions will now be mixed, and $f_0,k,m$ are arbitrary constants.", "We can recover GR limit when the index powers vanish, i.e.", "when $k=0=m$ .", "For this case, the model can be written in terms of the dynamical variables through $f_T = -mw\\,.$ In comparison to the latter $f(T,B)$ scenarios, this case has the following particularity where $x = f_B = k f_0 B^{k-1} T^m = \\frac{k}{B}f = \\frac{k}{6(3H^2 + \\dot{H})}f = \\frac{f}{6H^2}\\frac{k}{3 +\\frac{\\dot{H}}{H^2}}=-\\frac{wk}{3+z}\\,,$ from which we can notice that $x$ is not an independent variable of the dynamical system.", "In the same way, when $y = x^{\\prime }$ we obtain directly that $y = y(w,z)$ .", "With these conditions, the autonomous system for this case can be reduced to a 2-d dynamical phase space $z^{\\prime } = \\lambda - 2 z^2\\,, \\quad w^{\\prime } = w\\left[\\frac{6z(k+m-1) + \\lambda k + 2z^2(m-1)}{3+z}\\right]\\,.$ The critical point of the latter system are $z = \\pm \\sqrt{\\frac{\\lambda }{2}}\\,, \\quad \\text{and} \\quad w=0\\,.$ Under these values, the constriction of the system is given by $\\Omega = 1$ .", "Again, we can consider the two roots as follow:" ], [ "Critical points", " Positive branch.", "This case are determinate by the condition $m>0$ , with eigenvalues $\\omega _1 = -2 \\sqrt{2} \\sqrt{\\lambda }\\,, \\\\\\omega _2 = (k+m-1)\\frac{6\\sqrt{\\frac{\\lambda }{2}} + \\lambda }{3 + \\sqrt{\\frac{\\lambda }{2}}}\\,.$ If $\\lambda >0$ , we obtain that the eigenvalues are real for any value of $\\lambda $ , $m$ and $k$ .", "On the other hand, $\\omega _2 = 0$ if $k=1-m$ , which represent a non-hyperbolic point.", "We obtain an attractor point if $k < 1-m$ , and saddle-like otherwise.", "Negative branch.", "The eigenvalues for this case are $\\omega _1 = 2 \\sqrt{2} \\sqrt{\\lambda }\\,, \\\\\\omega _2 = (k+m-1)\\frac{-6\\sqrt{\\frac{\\lambda }{2}} + \\lambda }{3 - \\sqrt{\\frac{\\lambda }{2}}}\\,.$ Again, both values are real.", "The critical point is repulsor-like if $k < 1-m$ and $\\lambda >0 \\wedge \\lambda \\ne 18$ .", "We obtain a saddle point if $k > 1-m$ and $\\lambda >0 \\wedge \\lambda \\ne 18$ .", "For $m>k$ or $m<k$ we get a phantom-like EoS ($w<-1$ ).", "Figure: Different views of the phase space of the dynamical system in Eq.().", "The system was reduced to a 2-d surface representing different perspectives.", "The arrows represent the direction of the velocity field and the trajectories reveal their stability properties as described for this model.This result reinforces our work in Ref.", "[55] where we find that for mixed power law models, that the equation of state does cross the phantom line ($\\omega =-1$ ) but preserves the quintessence behaviour up to high redshifts where the model limits to $\\Lambda $ CDM.", "For lower redshifts, both $m>k$ and $m<k$ scenarios mimick phantom energy.", "In fact, these two scenarios correspond to the critical points that we find in the above analysis." ], [ "Discussion", "Dynamical systems can reveal a lot of information about the cosmology of theories beyond GR, which may be difficult to study at background or using direct cosmological perturbation theory.", "In this work, we analysed dynamical systems within the $f(T,B)$ gravity context which was first studied in Refs., [65], [56] in a context of scalar field in order to study the implications on inflation solutions.", "TG offers an avenue to constructing theories which exhibit torsion rather than curvature by exchanging the Levi-Civita connection with its teleparallel connection analog.", "This produces a wide range of potential cosmological models since TG is naturally lower order and so produces novel manifestations of gravity in addition to those constructed through extensions to GR [22], [23], [21].", "$f(T,B)$ gravity is an interesting context within which to study dynamical systems since it is one of the rare higher order theories in TG that occurs naturally.", "Indeed, in section  we outline our strategy in terms of which dynamical variables will produce a suitable dynamical analysis of cosmological systems.", "$f(T,B)$ gravity acts as a TG generalisation of $f(\\accentset{\\circ }{R})$ gravity in that the second and fourth order contributions are decoupled from one another through the torsion scalar $T$ and boundary term $B$ with coincidence only for cases where $f(T,B)=f(-T+B)=f(\\accentset{\\circ }{R})$ .", "The effect of this point also plays out in the dynamical systems analysis where we also must take the dynamical variable defined $\\lambda $ in Eq.", "(REF ) which is directly analogous to the approach taken in [63].", "Indeed, the analysis where this parameter was probed against possible constant values was studied in [66].", "A feature we explore in this work through the methodology outlined in section .", "The core results of the work are presented in section  where the models (that are cosmological viable at background level) are analysed.", "We start by probing the general Taylor expansion model in Eq.", "(REF ) where the arbitrary function is expanded about the Minkowski values of the scalar arguments up to quadratic order (due to the linear form of $B$ being a boundary term).", "Here we find the critical points in Eq.", "(REF ) with system eigenvalues in Eq.", "(REF ).", "We find that for any nonvanishing constant value of $\\lambda $ , all the critical values are all saddle points.", "An interesting feature of this investigation is that the constraints found are consistent with Ref.", "[55] showing consistency in its confrontation with observations.", "The evolution for the various parameter combination is shown in Fig.", "(REF ).", "Afterward, we then study the power law model in section REF which considers a power law form for both scalar contributors.", "In this case, the determining factor is the $z$ variable which depends only on derivatives of the Hubble parameter as defined in Eq.", "(REF ).", "In this scenario, we find either attractors or saddle points for the positive branch and repellent or saddle points for the negative branch which is shown in Fig.", "(REF ).", "A similar picture also emerges for the mixed model investigated in section REF .", "Here, we again find the same variable to be the determining factor in the behaviour of the dynamics of the system.", "On the other hand, the system turns out to be relatively straightforward to analysis with clear cut results which tally with the general results of the power law model.", "These are shown in Fig.", "(REF ).", "The above results can be linked in a more straightforward manner if we consider directly the form for the Equation of State (EoS).", "According to the generic EoS reported in [55] and using our dynamical results, we obtain that for the first two cases (Taylor and Power law model) $\\omega _{\\text{eff}} = -1 \\mp \\frac{2}{3} \\sqrt{\\frac{\\lambda }{2}}\\,.$ Meanwhile, for the Mixed Power Law modelWe consider an approximate solution to avoid divergence in the critical points obtained for this case.", "$\\omega _{\\text{eff}} \\xrightarrow{} -1 \\mp \\frac{2}{3}(k + m)\\sqrt{\\frac{ \\lambda }{2}}\\,.$ Notice that we recover $\\Lambda $ CDM in the GR limit.", "In both EoS scenarios, we recover a $\\Lambda $ CDM model when $\\lambda $ vanishes, this can happen when we obtain de Sitter solutions as $H=\\text{constant}$ .", "Notice that we can rewrite the $z$ -variable from Eq.", "(REF ) using the definition of the second cosmographic parameter, the deceleration parameter ($q=-\\ddot{a}a/\\dot{a}^2$ ) as $\\dot{H}=-H^2 (q+1)$ , therefore $z=-(q+1)$ .", "In terms of $\\lambda $ , this latter parameter can be written as $q = -1 \\mp \\sqrt{\\frac{\\lambda }{2}}$ , which for $H=\\text{constant}$ we obtain that $q=-1$ .", "Also, we can rewrite the ansatz for $\\lambda $ given in Eq.", "(REF ) in terms of the third cosmographic parameter, the jerk ($\\dddot{a}/aH^3$ ) as $j = \\frac{\\ddot{H}}{H^3} -3q -2$ , which again in terms of $\\lambda $ is $j = \\lambda \\mp 3\\sqrt{\\frac{\\lambda }{2}} +1 $ .", "Notice that when $\\lambda =0$ , we recover the standard value $j=1$ .", "Another important feature of the analysis in this work is the role of couplings between the torsion tensor $T$ and the boundary term $B$ .", "As discussed in the introduction, these represent the second order and fourth order contributions to the field equations respectively, and in a particular combination, forming $f(\\accentset{\\circ }{R})=f(-T+B)$ gravity.", "In $f(\\accentset{\\circ }{R})$ gravity, these couplings do appear but in a very prescribed format.", "In the present case we allow for more novel models to develop.", "In particular in cases 1 and 3 of section , the coupling term in the Lagrangian plays an impactful role in the dynamics that ensue.", "This is an interesting property that should be investigated further.", "From the results obtained with this proposal, we notice that it will be interesting to study the behaviour of other $f(T,B)$ gravity models, which together with their confrontation with observational data, may open an avenue for producing other viable cosmological scenarios.", "Furthermore, the role of a varying $\\lambda $ is also an important future work which may better expose the dynamical behaviour of $f(T,B)$ gravity, since from the Taylor and Power Law cases we will require a non-autonomous system with $\\lambda \\ne \\text{const}$ .", "Another important higher-order extension to TG is $f(T,T_G)$ which may also have interesting properties.", "This study will be reported elsewhere.", "CE-R acknowledges the Royal Astronomical Society as FRAS 10147 and PAPIIT Project IA100220.", "CE-R and JLS would like to acknowledge networking support by the COST Action CA18108.", "JLS would also like to acknowledge funding support from Cosmology@MALTA which is supported by the University of Malta." ] ]

[ [ "Machine Learning Time Series Regressions with an Application to\n Nowcasting" ], [ "Abstract This paper introduces structured machine learning regressions for high-dimensional time series data potentially sampled at different frequencies.", "The sparse-group LASSO estimator can take advantage of such time series data structures and outperforms the unstructured LASSO.", "We establish oracle inequalities for the sparse-group LASSO estimator within a framework that allows for the mixing processes and recognizes that the financial and the macroeconomic data may have heavier than exponential tails.", "An empirical application to nowcasting US GDP growth indicates that the estimator performs favorably compared to other alternatives and that text data can be a useful addition to more traditional numerical data." ], [ "Introduction", "The statistical imprecision of quarterly gross domestic product (GDP) estimates, along with the fact that the first estimate is available with a delay of nearly a month, pose a significant challenge to policy makers, market participants, and other observers with an interest in monitoring the state of the economy in real time; see, e.g., for a recent discussion of macroeconomic data revision and publication delays.", "A term originated in meteorology, nowcasting pertains to the prediction of the present and very near future.", "Nowcasting is intrinsically a mixed frequency data problem as the object of interest is a low-frequency data series (e.g., quarterly GDP), whereas the real-time information (e.g., daily, weekly, or monthly) can be used to update the state, or to put it differently, to nowcast the low-frequency series of interest.", "Traditional methods used for nowcasting rely on dynamic factor models that treat the underlying low frequency series of interest as a latent process with high frequency data noisy observations.", "These models are naturally cast in a state-space form and inference can be performed using likelihood-based methods and Kalman filtering techniques; see for a recent survey.", "So far, nowcasting has mostly relied on the so-called standard macroeconomic data releases, one of the most prominent examples being the Employment Situation report released on the first Friday of every month by the US Bureau of Labor Statistics.", "This report includes the data on the nonfarm payroll employment, average hourly earnings, and other summary statistics of the labor market activity.", "Since most sectors of the economy move together over the business cycle, good news for the labor market is usually good news for the aggregate economy.", "In addition to the labor market data, the nowcasting models typically also rely on construction spending, (non-)manufacturing report, retail trade, price indices, etc., which we will call the traditional macroeconomic data.", "One prominent example of nowcast is produced by the Federal Reserve Bank of New York relying on a dynamic factor model with thirty-six predictors of different frequencies; see for more details.", "Thirty-six predictors of traditional macroeconomic series may be viewed as a small number compared to hundreds of other potentially available and useful nontraditional series.", "For instance, macroeconomists increasingly rely on nonstandard data such as textual analysis via machine learning, which means potentially hundreds of series.", "A textual analysis data set based on Wall Street Journal articles that has been recently made available features a taxonomy of 180 topics; see .", "Which topics are relevant?", "How should they be selected?", "constructs a daily business cycle index based on quarterly GDP growth and textual information contained in the daily business newspapers relying on a dynamic factor model where time-varying sparsity is enforced upon the factor loadings using a latent threshold mechanism.", "His work shows the feasibility of traditional state space setting, yet the challenges grow when we also start thinking about adding other potentially high-dimensional data sets, such as payment systems information or GPS tracking data.", "Studies for Canada (), Denmark (), India (), Italy (), Norway (), Portugal (), and the United States () find that payment transactions can help to nowcast and to forecast GDP and private consumption in the short term; see also for nowcasting unemployment rates with smartphone GPS data, among others.", "We could quickly reach numerical complexities involved with estimating high-dimensional state space models, making the dynamic factor model approach potentially computationally prohibitively complex and slow, although some alternatives to the Kalman filter exist for the large data environments; see e.g., and .", "In this paper, we study nowcasting a low-frequency series – focusing on the key example of US GDP growth – in a data-rich environment, where our data not only includes conventional high-frequency series but also nonstandard data generated by textual analysis of financial press articles.", "We find that our nowcasts are either superior to or at par with those posted by the Federal Reserve Bank of New York (henceforth NY Fed).", "This is the case when (a) we compare our approach with the NY Fed using the same data, or (b) when we compare our approach using an expanded high-dimensional data set.", "The former is a comparison of methods, whereas the latter pertains to the value of the additional (nonstandard) big data.", "To deal with such massive nontraditional data sets, instead of using the likelihood-based dynamic factor models, we rely on a different approach that involves machine learning methods based on the regularized empirical risk minimization principle and data sampled at different frequencies.", "We adopt the MIDAS (Mixed Data Sampling) projection approach which is more amenable to high-dimensional data environments.", "Our general framework also includes the standard same frequency time series regressions.", "Several novel contributions are required to achieve our goal.", "First, we argue that the high-dimensional mixed frequency time series regressions involve certain data structures that once taken into account should improve the performance of unrestricted estimators in small samples.", "These structures are represented by groups covering lagged dependent variables and groups of lags for a single (high-frequency) covariate.", "To that end, we leverage on the sparse-group LASSO (sg-LASSO) regularization that accommodates conveniently such structures; see .", "The attractive feature of the sg-LASSO estimator is that it allows us to combine effectively the approximately sparse and dense signals; see e.g., for a comprehensive treatment of high-dimensional dense time series regressions as well as for a complementary to ours Bayesian view of penalized MIDAS regressions.", "We recognize that the economic and financial time series data are persistent and often heavy-tailed, while the bulk of the machine learning methods assumes i.i.d.", "data and/or exponential tails for covariates and regression errors; see for a comprehensive review of high-dimensional econometrics with i.i.d.", "data.", "There have been several recent attempts to expand the asymptotic theory to settings involving time series dependent data, mostly for the LASSO estimator.", "For instance, and establish oracle inequalities for regressions with i.i.d.", "errors with sub-Gaussian tails; consider $\\beta $ -mixing series with exponential tails; , , and establish oracle inequalities for causal Bernoulli shifts with independent innovations and polynomial tails under the functional dependence measure of ; see also and for results on the adaptive LASSO based on the triplex tail inequality for mixingales of .", "Despite these efforts, there is no complete estimation and prediction theory for high-dimensional time series regressions under the assumptions comparable to the classical GMM and QML estimators.", "For instance, the best currently available results are too restrictive for the MIDAS projection model, which is typically an example of a causal Bernoulli shift with dependent innovations.", "Moreover, the mixing processes with polynomial tails that are especially relevant for the financial and macroeconomic time series have not been properly treated due to the fact that the sharp Fuk-Nagaev inequality was not available in the relevant literature until recently.", "The Fuk-Nagaev inequality, see , describes the concentration of sums of random variables with a mixture of the sub-Gaussian and the polynomial tails.", "It provides sharp estimates of tail probabilities unlike Markov's bound in conjunction with the Marcinkiewicz-Zygmund or Rosenthal's moment inequalities.", "This paper fills these gaps in the literature relying on the Fuk-Nagaev inequality for $\\tau $ -mixing processes of and establishes the nonasymptotic and asymptotic estimation and prediction properties of the sg-LASSO projections under weak tail conditions and potential misspecification.", "The class of $\\tau $ -mixing processes is fairly rich covering the $\\alpha $ -mixing processes, causal linear processes with infinitely many lags of $\\beta $ -mixing processes, and nonlinear Markov processes; see , for more details, as well as and for mixing properties of various processes encountered in time series econometrics.", "Our weak tail conditions require at least $4+\\epsilon $ finite moments for covariates, while the number of finite moments for the error process can be as low as $2+\\nu $ , provided that covariates are sufficiently integrable.", "From the theoretical point of view, we impose approximate sparsity, relaxing the assumption of exact sparsity of the projection coefficients and allowing for other forms of misspecification (see for further discussion on the topic of sparsity).", "Lastly, we cover the LASSO and the group LASSO as special cases.", "The rest of the paper is organized as follows.", "Section presents the setting of (potentially mixed frequency) high-dimensional time series regressions.", "Section characterizes nonasymptotic estimation and prediction accuracy of the sg-LASSO estimator for $\\tau $ -mixing processes with polynomial tails.", "We report on a Monte Carlo study in Section which provides further insights regarding the validity of our theoretical analysis in small sample settings typically encountered in empirical applications.", "Section covers the empirical application.", "Conclusions appear in Section ." ], [ "Notation:", "For a random variable $X\\in \\mathbf {R}$ , let $\\Vert X\\Vert _q=(\\mathbb {E}|X|^q)^{1/q}$ be its $L_q$ norm with $q\\ge 1$ .", "For $p\\in \\mathbf {N}$ , put $[p] = \\lbrace 1,2,\\dots ,p\\rbrace $ .", "For a vector $\\Delta \\in \\mathbf {R}^p$ and a subset $J\\subset [p]$ , let $\\Delta _J$ be a vector in $\\mathbf {R}^p$ with the same coordinates as $\\Delta $ on $J$ and zero coordinates on $J^c$ .", "Let $\\mathcal {G}$ be a partition of $[p]$ defining the group structure, which is assumed to be known to the econometrician.", "For a vector $\\beta \\in \\mathbf {R}^p$ , the sparse-group structure is described by a pair $(S_0,\\mathcal {G}_0)$ , where $S_0=\\lbrace j\\in [p]:\\;\\beta _j\\ne 0 \\rbrace $ and $\\mathcal {G}_0 = \\left\\lbrace G\\in \\mathcal {G}:\\; \\beta _{G} \\ne 0\\right\\rbrace $ are the support and respectively the group support of $\\beta $ .", "We also use $|S|$ to denote the cardinality of arbitrary set $S$ .", "For $b\\in \\mathbf {R}^p$ , its $\\ell _q$ norm is denoted as $|b|_q = \\left(\\sum _{j\\in [p]}|b_j|^q\\right)^{1/q}$ for $q\\in [1,\\infty )$ and $|b|_\\infty = \\max _{j\\in [p]}|b_j|$ for $q=\\infty $ .", "For $\\mathbf {u},\\mathbf {v}\\in \\mathbf {R}^T$ , the empirical inner product is defined as $\\langle \\mathbf {u},\\mathbf {v}\\rangle _T = T^{-1}\\sum _{t=1}^T u_tv_t$ with the induced empirical norm $\\Vert .\\Vert _T^2=\\langle .,.\\rangle _T=|.|_2^2/T$ .", "For a symmetric $p\\times p$ matrix $A$ , let $\\mathrm {vech}(A)\\in \\mathbf {R}^{p(p+1)/2}$ be its vectorization consisting of the lower triangular and the diagonal elements.", "For $a,b\\in \\mathbf {R}$ , we put $a\\vee b = \\max \\lbrace a,b\\rbrace $ and $a\\wedge b = \\min \\lbrace a,b\\rbrace $ .", "Lastly, we write $a_n\\lesssim b_n$ if there exists a (sufficiently large) absolute constant $C$ such that $a_n\\le C b_n$ for all $n\\ge 1$ and $a_n\\sim b_n$ if $a_n\\lesssim b_n$ and $b_n\\lesssim a_n$ ." ], [ "High-dimensional mixed frequency regressions", "Let $\\lbrace y_t:t\\in [T]\\rbrace $ be the target low frequency series observed at integer time points $t\\in [T]$ .", "Predictions of $y_t$ can involve its lags as well as a large set of covariates and lags thereof.", "In the interest of generality, but more importantly because of the empirical relevance we allow the covariates to be sampled at higher frequencies - with same frequency being a special case.", "More specifically, let there be $K$ covariates $\\lbrace x_{t-(j-1)/m,k},j\\in [m],t\\in [T],k\\in [K]\\rbrace $ possibly measured at some higher frequency with $m\\ge 1$ observations for every $t$ and consider the following regression model $\\phi (L)y_t = \\rho _0 + \\sum _{k=1}^K\\psi (L^{1/m}; \\beta _k)x_{t,k} + u_t,\\qquad t\\in [T],$ where $\\phi (L) = I - \\rho _1L - \\rho _2L^2 - \\dots - \\rho _JL^J$ is a low-frequency lag polynomial and $\\psi (L^{1/m};\\beta _k)x_{t,k} = 1/m \\sum _{j=1}^m\\beta _{j,k}x_{t-(j-1)/m,k}$ is a high-frequency lag polynomial.", "For $m$ = 1, we have a standard autoregressive distributed lag (ARDL) model, which is the workhorse regression model of the time series econometrics literature.", "Note that the polynomial $\\psi (L^{1/m}; \\beta _k)x_{t,k}$ involves the same $m$ number of high-frequency lags for each covariate $k\\in [K]$ , which is done for the sake of simplicity and can easily be relaxed; see Section .", "The ARDL-MIDAS model (using the terminology of ) features $J+1+m\\times K$ parameters.", "In the big data setting with a large number of covariates sampled at high-frequency, the total number of parameters may be large compared to the effective sample size or even exceed it.", "This leads to poor estimation and out-of-sample prediction accuracy in finite samples.", "For instance, with $m$ = 3 (quarterly/monthly setting) and 35 covariates at 4 lagged quarters, we need to estimate $m\\times K=420$ parameters.", "At the same time, say the post-WWII quarterly GDP growth series has less than 300 observations.", "The LASSO estimator, see , offers an appealing convex relaxation of a difficult nonconvex best subset selection problem.", "It allows increasing the precision of predictions via the selection of sparse and parsimonious models.", "In this paper, we focus on the structured sparsity with additional dimensionality reductions that aim to improve upon the unstructured LASSO estimator in the time series setting.", "First, we parameterize the high-frequency lag polynomial following the MIDAS regression or the distributed lag econometric literature (see ) as $\\psi (L^{1/m};\\beta _k)x_{t,k} = \\frac{1}{m}\\sum _{j=1}^m\\omega ((j-1)/m;\\beta _k)x_{t-(j-1)/m,k},$ where $\\beta _k$ is $L$ -dimensional vector of coefficients with $L\\le m$ and $\\omega :[0,1]\\times \\mathbf {R}^L\\rightarrow \\mathbf {R}$ is some weight function.", "Second, we approximate the weight function as $\\omega (u;\\beta _k) \\approx \\sum _{l=1}^L\\beta _{k,l}w_l(u),\\qquad u\\in [0,1],$ where $\\lbrace w_l:\\; l=1,\\dots ,L \\rbrace $ is a collection of functions, called the dictionary.", "The simplest example of the dictionary consists of algebraic power polynomials, also known as polynomials in the time series regression analysis literature.", "More generally, the dictionary may consist of arbitrary approximating functions, including the classical orthogonal bases of $L_2[0,1]$ ; see Online Appendix Section for more examples.", "Using orthogonal polynomials typically reduces the multicollinearity and leads to better finite sample performance.", "It is worth mentioning that the specification with dictionaries deviates from the standard MIDAS regressions and leads to a computationally attractive convex optimization problem, cf.", ".", "The size of the dictionary $L$ and the number of covariates $K$ can still be large and the approximate sparsity is a key assumption imposed throughout the paper.", "With the approximate sparsity, we recognize that assuming that most of the estimated coefficients are zero is overly restrictive and that the approximation error should be taken into account.", "For instance, the weight function may have an infinite series expansion, nonetheless, most can be captured by a relatively small number of orthogonal basis functions.", "Similarly, there can be a large number of economically relevant predictors, nonetheless, it might be sufficient to select only a smaller number of the most relevant ones to achieve good out-of-sample forecasting performance.", "Both model selection goals can be achieved with the LASSO estimator.", "However, the LASSO does not recognize that covariates at different (high-frequency) lags are temporally related.", "In the baseline model, all high-frequency lags (or approximating functions once we parameterize the lag polynomial) of a single covariate constitute a group.", "We can also assemble all lag dependent variables into a group.", "Other group structures could be considered, for instance combining various covariates into a single group, but we will work with the simplest group setting of the aforementioned baseline model.", "The sparse-group LASSO (sg-LASSO) allows us to incorporate such structure into the estimation procedure.", "In contrast to the group LASSO, see , the sg-LASSO promotes sparsity between and within groups, and allows us to capture the predictive information from each group, such as approximating functions from the dictionary or specific covariates from each group.", "Figure: Geometry of {β∈𝐑 2 :Ω(β)≤1}\\lbrace \\beta \\in \\mathbf {R}^2: \\Omega (\\beta )\\le 1 \\rbrace for different groupings and values of α\\alpha .To describe the estimation procedure, let $\\mathbf {y}$ = $(y_{1},\\dots ,y_T)^\\top ,$ be a vector of dependent variable and let $\\mathbf {X}$ = $(\\iota , \\mathbf {y}_{1},\\dots ,\\mathbf {y}_{J},Z_1W,\\dots ,Z_KW),$ be a design matrix, where $\\iota = (1,1,\\dots ,1)^\\top $ is a vector of ones, $\\mathbf {y}_{j} = (y_{1-j},\\dots , y_{T-j})^\\top $ , $Z_k = (x_{k,t-{(j-1)/m}})_{t\\in [T],j\\in [m]}$ is a $T\\times m$ matrix of the covariate $k\\in [K]$ , and $W=\\left(w_l\\left((j-1)/m\\right)/m\\right)_{j\\in [m],l\\in [L]}$ is an $m\\times L$ matrix of weights.", "In addition, put $\\beta $ = $(\\beta _0^\\top ,\\beta _1^\\top ,\\dots ,\\beta _K^\\top )^\\top $ , where $\\beta _0 = (\\rho _0,\\rho _1,\\dots ,\\rho _J)^\\top $ is a vector of parameters pertaining to the group consisting of the intercept and the autoregressive coefficients, and $\\beta _k\\in \\mathbf {R}^L$ denotes parameters of the high-frequency lag polynomial pertaining to the covariate $k\\ge 1$ .", "Then, the sparse-group LASSO estimator, denoted $\\hat{\\beta }$ , solves the penalized least-squares problem $\\min _{b\\in \\mathbf {R}^p}\\Vert \\mathbf {y} - \\mathbf {X}b\\Vert _T^2 + 2\\lambda \\Omega (b)$ with a penalty function that interpolates between the $\\ell _1$ LASSO penalty and the group LASSO penalty $\\Omega (b) = \\alpha |b|_1 + (1-\\alpha )\\Vert b\\Vert _{2,1},$ where $\\Vert b\\Vert _{2,1} = \\sum _{G\\in \\mathcal {G}}|b_G|_2$ is the group LASSO norm and $\\mathcal {G}$ is a group structure (partition of $[p]$ ) specified by the econometrician.", "Note that estimator in equation (REF ) is defined as a solution to the convex optimization problem and can be computed efficiently, e.g., using an appropriate coordinate descent algorithm; see .", "The amount of penalization in equation (REF ) is controlled by the regularization parameter $\\lambda >0$ while $\\alpha \\in [0,1]$ is a weight parameter that determines the relative importance of the sparsity and the group structure.", "Setting $\\alpha =1$ , we obtain the LASSO estimator while setting $\\alpha = 0$ , leads to the group LASSO estimator, which is reminiscent of the elastic net.", "In Figure REF we illustrate the geometry of the penalty function for different groupings and different values of $\\alpha $ covering (a) LASSO with $\\alpha $ = 1, (b) group LASSO with one group, $\\alpha $ = 0, and two sg-LASSO cases (c) one group and (d) two groups both with $\\alpha $ = 0.5.", "In practice, groups are defined by a particular problem and are specified by the econometrician, while $\\alpha $ can be fixed or selected jointly with $\\lambda $ in a data-driven way such as using the cross-validation." ], [ "High-dimensional regressions and $\\tau $ -mixing", "We focus on a generic high-dimensional linear projection model with a countable number of regressors $y_t = \\sum _{j=0}^\\infty x_{t,j}\\beta _j + u_t,\\qquad \\mathbb {E}[u_tx_{t,j}]=0,\\quad \\forall j\\ge 1,\\qquad t\\in \\mathbf {Z},$ where $x_{t,0}=1$ and $m_t \\triangleq \\sum _{j=0}^\\infty x_{t,j}\\beta _j$ is a well-defined random variable.", "In particular, to ensure that $y_t$ is a well-defined economic quantity, we need $\\beta _j\\downarrow 0$ sufficiently fast, which is a form of the approximate sparsity condition, see .", "This setting nests the high-dimensional ARDL-MIDAS projections described in the previous section and more generally may allow for other high-dimensional time series models.", "In practice, given a (large) number of covariates, lags thereof, as well as lags of the dependent variable, denoted $x_t\\in \\mathbf {R}^p$ , we would approximate $m_t$ with $x_t^\\top \\beta \\triangleq \\sum _{j=0}^px_{t,j}\\beta _j$ , where $p<\\infty $ and the regression coefficient $\\beta \\in \\mathbf {R}^p$ could be sparse.", "Importantly, our settings allows for the approximate sparsity as well as other forms of misspecification and the main result of the following section allows for $m_t\\ne x_t^\\top \\beta $ .", "Using the setting of equation (REF ), for a sample $(y_t,x_t)_{t=1}^T$ , write $\\mathbf {y} = \\mathbf {m} + \\mathbf {u},$ where $\\mathbf {y}=(y_1,\\dots ,y_T)^\\top $ , $\\mathbf {m} = (m_1,\\dots , m_T)^\\top $ , and $\\mathbf {u}=(u_1,\\dots ,u_T)^\\top $ .", "The approximation to $\\mathbf {m}$ is denoted $\\mathbf {X}\\beta $ , where $\\mathbf {X} = (x_1,\\dots ,x_T)^\\top $ is a $T\\times p$ matrix of covariates and $\\beta =(\\beta _1,\\dots ,\\beta _p)^\\top $ is a vector of unknown regression coefficients.", "We measure the time series dependence with $\\tau $ -mixing coefficients.", "For a $\\sigma $ -algebra $\\mathcal {M}$ and a random vector $\\xi \\in \\mathbf {R}^l$ , put $\\tau (\\mathcal {M},\\xi ) = \\bigg \\Vert \\sup _{f\\in \\mathrm {Lip}_1}\\left|\\mathbb {E}(f(\\xi )|\\mathcal {M}) - \\mathbb {E}(f(\\xi ))\\right|\\bigg \\Vert _1,$ where $\\mathrm {Lip}_1=\\left\\lbrace f:\\mathbf {R}^l\\rightarrow \\mathbf {R}:\\;|f(x)-f(y)|\\le |x-y|_1\\right\\rbrace $ is a set of 1-Lipschitz functions.", "Let $(\\xi _t)_{t\\in \\mathbf {Z}}$ be a stochastic process and let $\\mathcal {M}_t=\\sigma (\\xi _t,\\xi _{t-1},\\dots )$ be its canonical filtration.", "The $\\tau $ -mixing coefficient of $(\\xi _t)_{t\\in \\mathbf {Z}}$ is defined as $\\tau _k = \\sup _{j\\ge 1}\\frac{1}{j}\\sup _{t+k\\le t_1<\\dots <t_j}\\tau (\\mathcal {M}_t,(\\xi _{t_1},\\dots ,\\xi _{t_j})),\\qquad k\\ge 0.$ If $\\tau _k\\downarrow 0$ as $k\\rightarrow \\infty $ , then the process $(\\xi _t)_{t\\in \\mathbf {Z}}$ is called $\\tau $ -mixing.", "The $\\tau $ -mixing coefficients were introduced in as dependence measures weaker than mixing.", "Note that the commonly used $\\alpha $ - and $\\beta $ -mixing conditions are too restrictive for the linear projection model with an ARDL-MIDAS process.", "Indeed, a causal linear process with dependent innovations is not necessary $\\alpha $ -mixing; see also for an example of AR(1) process which is not $\\alpha $ -mixing.", "Roughly speaking, $\\tau $ -mixing processes are somewhere between mixingales and $\\alpha $ -mixing processes and can accommodate such counterexamples.", "At the same time, sharp Fuk-Nagaev inequalities are available for $\\tau $ -mixing processes which to the best of our knowledge is not the case for the mixingales or near-epoch dependent processes; see .", ", discuss how to verify the $\\tau $ -mixing property for causal Bernoulli shifts with dependent innovations and nonlinear Markov processes.", "It is also worth comparing the $\\tau $ -mixing coefficient to other weak dependence coefficients.", "Suppose that $(\\xi _t)_{t\\in \\mathbf {Z}}$ is a real-valued stationary process and let $\\gamma _k = \\Vert \\mathbb {E}(\\xi _{k}|\\mathcal {M}_0) - \\mathbb {E}(\\xi _{k})\\Vert _1$ be its $L_1$ mixingale coefficient.", "Then we clearly have $\\gamma _k\\le \\tau _k$ and it is known that $|\\mathrm {Cov}(\\xi _0,\\xi _k)| \\le \\int _0^{\\gamma _k}Q\\circ G(u)\\mathrm {d}u \\le \\int _0^{\\tau _k}Q\\circ G(u)\\mathrm {d}u \\le \\tau _k^\\frac{q-2}{q-1}\\Vert \\xi _0\\Vert _q^{q/(q-1)},$ where $Q$ is the generalized inverse of $x\\mapsto \\Pr (|\\xi _0|>x)$ and $G$ is the generalized inverse of $x\\mapsto \\int _0^x Q(u)\\mathrm {d}u$ ; see , Lemma A.1.1.", "Therefore, the $\\tau $ -mixing coefficient provides a sharp control of autocovariances similarly to the $L_1$ mixingale coefficients, which in turn can be used to ensure that the long-run variance of $(\\xi _t)_{t\\in \\mathbf {Z}}$ exists.", "The $\\tau $ -mixing coefficient is also bounded by the $\\alpha $ -mixing coefficient, denoted $\\alpha _k$ , as follows $\\tau _k \\le 2\\int _0^{2\\alpha _k}Q(u)\\mathrm {d}u \\le 2\\Vert \\xi _0\\Vert _q(2\\alpha _k)^{1/r},$ where the first inequality follows by , Lemma 7 and the second by Hölder's inequality with $q,r\\ge 1$ such that $q^{-1}+r^{-1}=1$ .", "It is worth mentioning that the mixing properties for various time series models in econometrics, including GARCH, stochastic volatility, or autoregressive conditional duration are well-known; see, e.g., , , ; see also for more examples and a comprehensive comparison of various weak dependence coefficients." ], [ "Estimation and prediction properties", "In this section, we introduce the main assumptions for the high-dimensional time series regressions and study the estimation and prediction properties of the sg-LASSO estimator covering the LASSO and the group LASSO estimators as special cases.", "The following assumption imposes some mild restrictions on the stochastic processes in the high-dimensional regression equation (REF ).", "Assumption 3.1 (Data) For every $j,k\\in [p]$ , the processes $(u_tx_{t,j,})_{t\\in \\mathbf {Z}}$ and $(x_{t,j}x_{t,k})_{t\\in \\mathbf {Z}}$ are stationary such that (i) $\\Vert u_0\\Vert _{q}<\\infty $ and $\\max _{j\\in [p]}\\Vert x_{0,j}\\Vert _{r}=O(1)$ for some constants $q>2r/(r-2)$ and $r>4$ ; (ii) the $\\tau $ -mixing coefficients are $\\tau _k \\le ck^{-a}$ and respectively $\\tilde{\\tau }_k\\le ck^{-b}$ for all $k\\ge 0$ and some $c>0$ , $a>(\\varsigma -1)/(\\varsigma -2)$ , $b>(r-2)/(r-4)$ , and $\\varsigma = qr/(q+r)$ .", "It is worth mentioning that the stationarity condition is not essential and can be relaxed to the existence of the limiting variance of partial sums at costs of heavier notations and proofs.", "Condition (i) requires that covariates have at least 4 finite moments, while the number of moments required for the error process can be as low as $2+\\epsilon $ , depending on the integrability of covariates.", "Therefore, (i) may allow for heavy-tailed distributions commonly encountered in financial and economic time series, e.g., asset returns and volatilities.", "Given the integrability in (i), (ii) requires that the $\\tau $ -mixing coefficients decrease to zero sufficiently fast; see Online Appendix, Section  for moments and $\\tau $ -mixing coefficients of ARDL-MIDAS.", "It is known that the $\\beta $ -mixing coefficients decrease geometrically fast, e.g., for geometrically ergodic Markov chains, in which case (ii) holds for every $a,b>0$ .", "Therefore, (ii) allows for relatively persistent processes.", "For the support $S_0$ and the group support $\\mathcal {G}_0$ of $\\beta $ , put $\\Omega _0(b) \\triangleq \\alpha |b_{S_0}|_1 + (1-\\alpha )\\sum _{G\\in \\mathcal {G}_0}|b_{G}|_2\\qquad \\text{and} \\qquad \\Omega _1(b) \\triangleq \\alpha |b_{S^c_0}|_1 + (1-\\alpha )\\sum _{G\\in \\mathcal {G}_0^c}|b_{G}|_2.$ For some $c_0>0$ , define $\\mathcal {C}(c_0) \\triangleq \\left\\lbrace \\Delta \\in \\mathbf {R}^p:\\; \\Omega _1(\\Delta )\\le c_0\\Omega _0(\\Delta ) \\right\\rbrace $ .", "The following assumption generalizes the restricted eigenvalue condition of to the sg-LASSO estimator and is imposed on the population covariance matrix $\\Sigma = \\mathbb {E}[\\mathbf {X}^\\top \\mathbf {X}/T]$ .", "Assumption 3.2 (Restricted eigenvalue) There exists a universal constant $\\gamma >0$ such that $\\Delta ^\\top \\Sigma \\Delta \\ge \\gamma \\sum _{G\\in \\mathcal {G}_0}|\\Delta _{G}|_2^2$ for all $\\Delta \\in \\mathcal {C}(c_0)$ , where $c_0 = (c+1)/(c-1)$ for some $c>1$ .", "Recall that if $\\Sigma $ is a positive definite matrix, then for all $\\Delta \\in \\mathbf {R}^p$ , we have $\\Delta ^\\top \\Sigma \\Delta \\ge \\gamma |\\Delta |_2^2$ , where $\\gamma $ is the smallest eigenvalue of $\\Sigma $ .", "Therefore, in this case Assumption REF is trivially satisfied because $|\\Delta |_2^2\\ge \\sum _{G\\in \\mathcal {G}_0}|\\Delta _{G}|_2^2$ .", "The positive definiteness of $\\Sigma $ is also known as a completeness condition and Assumption REF can be understood as its weak version; see and references therein.", "It is worth emphasizing that $\\gamma >0$ in Assumption REF is a universal constant independent of $p$ , which is the case, e.g., when $\\Sigma $ is a Toeplitz matrix or a spiked identity matrix.", "Alternatively, we could allow for $\\gamma \\downarrow 0$ as $p\\rightarrow \\infty $ , in which case the term $\\gamma ^{-1}$ would appear in our nonasymptotic bounds slowing down the speed of convergence, and we may interpret $\\gamma $ as a measure of ill-posedness in the spirit of econometrics literature on ill-posed inverse problems; see .", "The value of the regularization parameter is determined by the Fuk-Nagaev concentration inequality, appearing in the Online Appendix, see Theorem REF .", "Assumption 3.3 (Regularization) For some $\\delta \\in (0,1)$ $\\lambda \\sim \\left(\\frac{p}{\\delta T^{\\kappa -1}}\\right)^{1/\\kappa }\\vee \\sqrt{\\frac{\\log (8p/\\delta )}{T}},$ where $\\kappa = ((a+1)\\varsigma -1)/(a+\\varsigma -1)$ and $a,\\varsigma $ are as in Assumption REF .", "The regularization parameter in Assumption REF is determined by the persistence of the data, quantified by $a$ , and the tails, quantified by $\\varsigma = qr/(q+r)$ .", "This dependence is reflected in the dependence-tails exponent $\\kappa $ .", "The following result describes the nonasymptotic prediction and estimation bounds for the sg-LASSO estimator, see Online Appendix  for the proof.", "Theorem 3.1 Suppose that Assumptions REF , REF , and REF are satisfied.", "Then with probability at least $1 - \\delta - O(p^2(T^{1-\\mu }s_\\alpha ^\\mu + \\exp (-cT/s_\\alpha ^2)))$ $\\Vert \\mathbf {X}(\\hat{\\beta }- \\beta )\\Vert ^2_T \\lesssim s_\\alpha \\lambda ^2 + \\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2$ and $\\Omega (\\hat{\\beta }- \\beta ) \\lesssim s_\\alpha \\lambda + \\lambda ^{-1}\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 + \\sqrt{s_\\alpha }\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T$ for some $c>0$ , where $\\sqrt{s_\\alpha } = \\alpha \\sqrt{|S_0|} + (1-\\alpha )\\sqrt{|\\mathcal {G}_0|}$ and $\\mu = ((b + 1)r - 2)/(r+2(b-1))$ .", "Theorem REF provides nonasymptotic guarantees for the estimation and prediction with the sg-LASSO estimator reflecting potential misspecification.", "In the special case of the LASSO estimator ($\\alpha =1$ ), we obtain the counterpart to the result of for the LASSO estimator with i.i.d.", "data taking into account that we may have $m_t\\ne x_t^\\top \\beta $ .", "At another extreme, when $\\alpha =0$ , we obtain the nonasymptotic bounds for the group LASSO allowing for misspecification which to the best of our knowledge are new, cf.", "and .", "We call $s_\\alpha $ the effective sparsity constant.", "This constant reflects the benefits of the sparse-group structure for the sg-LASSO estimator that can not be deduced from the results currently available for the LASSO or the group LASSO.", "Remark 3.1 Since the $\\ell _1$ -norm is equivalent to the $\\Omega $ -norm whenever groups have fixed size, we deduce from Theorem REF that $|\\hat{\\beta }- \\beta |_1 \\lesssim s_\\alpha \\lambda + \\lambda ^{-1}\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 + \\sqrt{s_\\alpha }\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T.$ Next, we consider the asymptotic regime, in which the misspecification error vanishes when the sample size increases as described in the following assumption.", "Assumption 3.4 (i) $\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 = O_P\\left(s_\\alpha \\lambda ^2\\right)$ ; and (ii) $p^2T^{1-\\mu }s_\\alpha ^{\\mu }\\rightarrow 0$ and $p^2\\exp (-cT/s_\\alpha ^2)\\rightarrow 0$ .", "The following corollary is an immediate consequence of Theorem REF .", "Corollary 3.1 Suppose that Assumptions REF , REF , REF , and REF hold.", "Then $\\Vert \\mathbf {X}(\\hat{\\beta }- \\beta )\\Vert _T^2 = O_P\\left(\\frac{s_\\alpha p^{2/\\kappa }}{T^{2-2/\\kappa }}\\vee \\frac{s_\\alpha \\log p}{T}\\right)$ and $|\\hat{\\beta } - \\beta |_1 = O_P\\left(\\frac{s_\\alpha p^{1/\\kappa }}{T^{1-1/\\kappa }}\\vee s_\\alpha \\sqrt{\\frac{\\log p}{T}}\\right).$ If the effective sparsity constant $s_\\alpha $ is fixed, then $p=o(T^{\\kappa - 1})$ is a sufficient condition for the prediction and estimation errors to vanish, whenever $\\mu \\ge 2\\kappa -1$ .", "In this case Assumption REF (ii) is vacuous.", "More generally, $s_\\alpha $ is allowed to increase slowly with the sample size.", "Convergence rates in Corollary REF quantify the effect of tails and persistence of the data on the prediction and estimation accuracies of the sg-LASSO estimator.", "In particular, lighter tails and less persistence allow us to handle a larger number of covariates $p$ compared to the sample size $T$ .", "In particular $p$ can increase faster than $T$ , provided that $\\kappa >2$ .", "Remark 3.2 In the special case of the LASSO estimator with i.i.d.", "data, Corollary 4 of leads to the convergence rate of order $O_P\\left(\\frac{p^{1/\\varsigma }}{T^{1-1/\\varsigma }}\\vee \\sqrt{\\frac{\\log p}{T}}\\right)$ .", "If the $\\tau $ -mixing coefficients decrease geometrically fast (e.g., stationary AR(p)), then $\\kappa \\approx \\varsigma $ for a sufficiently large value of the dependence exponent $a$ , in which case the convergence rates in Corollary REF are close to the i.i.d.", "case.", "In this sense these rates depend sharply on the tails exponent $\\varsigma $ , and we can conclude that for geometrically decreasing $\\tau $ -mixing coefficients, the persistence of the data should not affect the convergence rates of the LASSO.", "Remark 3.3 In the special case of the LASSO estimator, if $(u_t)_{t\\in \\mathbf {Z}}$ and $(x_t)_{t\\in \\mathbf {Z}}$ are causal Bernoulli shifts with independent innovations and at least $q=r\\ge 8$ finite moments, one can deduce from , Lemma 5.1 and Corollary 5.1, the convergence rate of order $O_P\\left(\\frac{(p\\omega _T)^{1/\\varsigma }}{T^{1-1/\\varsigma }}\\vee \\sqrt{\\frac{\\log p}{T}}\\right)$ , where $\\omega _T=1$ (weakly dependent case) or $\\omega _T = T^{\\varsigma /2-1-a\\varsigma }\\uparrow \\infty $ (strongly dependent case), provided that the physical dependence coefficients are of size $O(k^{-a})$ .", "Note that for causal Bernoulli shifts with independent innovations, the physical dependence coefficients are not directly comparable to $\\tau $ -mixing coefficients; see , Remark 3.1 on p.32." ], [ "Monte Carlo experiments ", "We assess via simulations the out-of-sample predictive performance (forecasting and nowcasting), and the MIDAS weights recovery of the sg-LASSO with dictionaries.", "We benchmark the performance of our novel sg-LASSO setup against two alternatives: (a) unstructured, meaning standard, LASSO with MIDAS, and (b) unstructured LASSO with the unrestricted lag polynomial.", "The former allows us to assess the benefits of exploiting group structures, whereas the latter focuses on the advantages of using dictionaries in a high-dimensional setting." ], [ "Simulation Design ", "To assess the predictive performance and the MIDAS weight recovery, we simulate the data from the following DGP: $y_t = \\rho _1y_{t-1} + \\rho _2y_{t-2} + \\sum _{k=1}^K\\frac{1}{m}\\sum _{j=1}^m \\omega ((j-1)/m;\\beta _k)x_{t-(j-1)/m,k} + u_t,$ where $u_t\\sim _{i.i.d.", "}N(0,\\sigma _u^2)$ and the DGP for covariates $\\lbrace x_{k,t-(j-1)/m}:j\\in [m],k\\in [K]\\rbrace $ is specified below.", "This corresponds to a target of interest $y_t$ driven by two autoregressive lags augmented with high frequency series, hence, the DGP is an ARDL-MIDAS model.", "We set $\\sigma ^2_u=1$ , $\\rho _1=0.3$ , $\\rho _2=0.01$ , and take the number of relevant high frequency regressors $K$ = 3.", "In some scenarios we also decrease the signal-to-noise ratio by setting $\\sigma ^2_u$ = 5.", "We are interested in quarterly/monthly data, and use four quarters of data for the high frequency regressors so that $m$ = 12.", "We rely on a commonly used weighting scheme in the MIDAS literature, namely $\\omega (s;\\beta _k)$ for $k$ = 1, 2 and 3 are determined by beta densities respectively equal to $\\mathrm {Beta}(1,3),\\mathrm {Beta}(2,3)$ , and $\\mathrm {Beta}(2,2)$ ; see or , for further details.", "The high frequency regressors are generated as either one of the following: $K$ i.i.d.", "realizations of the univariate autoregressive (AR) process $x_h = \\rho x_{h-1}+ \\varepsilon _h,$ where $\\rho =0.2$ or $\\rho =0.7$ and either $\\varepsilon _h\\sim _{i.i.d.", "}N(0,\\sigma _\\varepsilon ^2)$ , $\\sigma _\\varepsilon ^2=1$ , or $\\varepsilon _h\\sim _{i.i.d.", "}\\text{student-}t(5)$ , where $h$ denotes the high-frequency sampling.", "Multivariate vector autoregressive (VAR) process $X_h = \\Phi X_{h-1} + \\mathbf {\\varepsilon }_h,$ where $\\varepsilon _{h}\\sim _{i.i.d.}", "N(0,I_K)$ and $\\Phi $ is a block diagonal matrix described below.", "For the AR simulation design, we initiate the processes as $x_0\\sim N\\left(0,\\sigma ^2/(1-\\rho ^2)\\right)$ and $y_0\\sim N\\left(0,\\sigma ^2(1-\\rho _2)/((1+\\rho _2)((1-\\rho _2)^2-\\rho _1^2))\\right).$ For the VAR, the initial value of $(y_t)$ is the same, while $X_0 \\sim N(0,I_K)$ .", "In all cases, the first 200 observations are treated as burn-in.", "In the estimation procedure, we add 7 noisy covariates which are generated in the same way as the relevant covariates and use 5 low-frequency lags.", "The empirical models use a dictionary which consists of Legendre polynomials up to degree $L = 10$ shifted to the $[0,1]$ interval with the MIDAS weight function approximated as in equation (REF ).", "The sample size is $T\\in \\lbrace 50, 100, 200\\rbrace ,$ and for all the experiments we use 5000 simulation replications.", "We assess the performance of different methods by modifying the assumptions on the error terms of the high-frequency process $\\mathbf {\\varepsilon }_h$ , considering multivariate high-frequency processes, changing the degree of Legendre polynomials $L$ , increasing the noise level of the low-frequency process $\\sigma ^2_u$ , using only half of the high-frequency lags in predictive regressions, and adding a larger number of noisy covariates.", "In the case of VAR high-frequency process, we set $\\Phi $ to be block-diagonal with the first $5\\times 5$ block having entries $0.15$ and the remaining $5\\times 5$ block(s) having entries $0.075$ .", "We estimate three different LASSO-type regression models.", "In the first model, we keep the weighting function unconstrained, and therefore we estimate 12 coefficients per high-frequency covariate using the unstructured LASSO estimator.", "We denote this model LASSO-U-MIDAS (inspired by the U-MIDAS of ).", "In the second model we use MIDAS weights together with the unstructured LASSO estimator; we call this model LASSO-MIDAS.", "In this case, we estimate $L+1$ number of coefficients per high-frequency covariate.", "The third model applies the sg-LASSO estimator together with MIDAS weights.", "Groups are defined as in Section ; each low-frequency lag and high-frequency covariate is a group, therefore, we have $K+5$ groups.", "We select the value of tuning parameters $\\lambda $ and $\\alpha $ using the 5-fold cross-validation, defining folds as adjacent blocks over the time dimension to take into account the time series dependence.", "This model is denoted sg-LASSO-MIDAS.", "For regressions with aggregated data, we consider: (a) Flow aggregation (FLOW): $x_{k,t}^A$ = $1/m\\sum _{j=1}^mx_{k,t-(j-1)/m}$ , (b) Stock aggregation (STOCK): $x_{k,t}^A$ = $x_{k,t}$ , and (c) Middle high-frequency lag (MIDDLE): single middle value of the high-frequency lag with ties solved in favor of the most recent observation (i.e., we take a single 6th lag if $m=12$ ).", "In these cases, the models are estimated using the OLS estimator, which is unfeasible when the number of covariates becomes equal to the sample size and we leave results blank in this case." ], [ "Simulation results", "Detailed results are reported in the Online Appendix.", "Tables REF –REF , cover the average mean squared forecast errors for one-step-ahead forecasts and nowcasts.", "The sg-LASSO with MIDAS weighting (sg-LASSO-MIDAS) outperforms all other methods in all simulation scenarios.", "Importantly, both sg-LASSO-MIDAS and unstructured LASSO-MIDAS with nonlinear weight function approximations perform much better than all other methods when the sample size is small ($T=50$ ).", "In this case, sg-LASSO-MIDAS yields the largest improvements over alternatives, in particular, with a large number of noisy covariates (bottom-right block).", "These findings are robust to increases in the persistence parameter of covariates $\\rho $ from 0.2 to 0.7.", "The LASSO without MIDAS weighting has typically large forecast errors.", "Comparing across simulation scenarios, all methods seem to perform worse with heavy-tailed or persistent covariates.", "In these cases, however, the impact on the sg-LASSO-MIDAS method is lesser compared to the other methods.", "This simulation evidence supports our theoretical results and findings in the empirical application.", "Lastly, forecasts using flow-aggregated covariates seem to perform better than other simple aggregation methods in all simulation scenarios, but significantly worse than the sg-LASSO-MIDAS.", "In Table REF –REF we report additional results for the estimation accuracy of the weight functions.", "In Figure REF –REF , we plot the estimated weight functions from several methods.", "The results indicate that the LASSO without MIDAS weighting can not accurately recover the weights in small samples and/or low signal-to-noise ratio scenarios.", "Using Legendre polynomials improves the performance substantially and the sg-LASSO seems to improve even more over the unstructured LASSO." ], [ "Nowcasting US GDP with macro, financial and textual news data ", "We nowcast US GDP with macroeconomic, financial, and textual news data.", "Details regarding the data sources appear in the Online Appendix Section .", "Regarding the macro data, we rely on 34 series used in the Federal Reserve Bank of New York nowcast model, discarding two series (\"PPI: Final demand\" and \"Merchant wholesalers: Inventories\") due to very short samples; see for more details regarding this data.", "For all macro data, we use real-time vintages, which effectively means that we take all macro series with a delay.", "For example, if we nowcast the first quarter of GDP one month before the quarter ends, we use data up to the end of February, and therefore all macro series with a delay of one month that enter the model are available up to the end of January.", "We use Legendre polynomials of degree three for all macro covariates to aggregate twelve lags of monthly macro data.", "In particular, let $x_{t+(h+1-j)/m,k}$ be $k^{{\\text{th}}}$ covariate at quarter $t$ with $m=3$ months per quarter and $h=2-1=1$ months into the quarter (2 months into the quarter minus 1 month due to publication delay), where $j = 1,2,\\dots ,12$ is the monthly lag.", "We then collect all lags in a vector $X_{t,k} = (x_{t+1/3,k}, x_{t+0/3,k},\\dots ,x_{t-10/3,k})^\\top $ and aggregate $X_{t,k}$ using a dictionary $W$ consisting of Legendre polynomials, so that $X_{t,k}W$ defines as a single group for the sg-LASSO estimator.", "In addition to macro and financial data, we also use the textual analysis data.", "We take 76 news attention series from and use Legendre polynomials of degree two to aggregate three monthly lags of each news attention series.", "Note that the news attention series are used without a publication delay, that is, for the one-month horizon, we take the series up to the end of the second month.", "Moreover, the news topic models involve rolling samples, avoiding look ahead biases when used in our nowcasts.", "We compute the predictions using a rolling window scheme.", "The first nowcast is for 2002 Q1, for which we use fifteen years (sixty quarters) of data, and the prediction is computed using 2002 January (2-month horizon) February (1-month), and March (end of the quarter) data.", "We calculate predictions until the sample is exhausted, which is 2017 Q2, the last date for which news attention data is available.", "As indicated above, we report results for the 2-month, 1-month, and the end-of-quarter horizons.", "Our target variable is the first release, i.e., the advance estimate of real GDP growth.", "We tune sg-LASSO-MIDAS regularization parameters $\\lambda $ and $\\alpha $ using 5-fold cross-validation, defining folds as adjacent blocks over the time dimension to take into account the time series nature of the data.", "Finally, we follow the literature on nowcasting real GDP and define our target variable to be the annualized growth rate.", "Let $x_{t,k}$ be the $k$ -th high-frequency covariate at time $t$ .", "The general ARDL-MIDAS predictive regression is $\\phi (L)y_{t+1} = \\mu + \\sum _{k=1}^K\\psi (L^{1/m}; \\beta _k)x_{t,k} + u_{t+1},\\qquad t=1,\\dots ,T,$ where $\\phi (L)$ is the low-frequency lag polynomial, $\\mu $ is the regression intercept, and $\\psi (L^{1/m};\\beta _k)x_{tk},k=1,\\dots ,K$ are lags of high-frequency covariates.", "Following Section , the high-frequency lag polynomial is defined as $\\psi (L^{1/m};\\beta _k)x_{t,k} = \\frac{1}{mq_k}\\sum _{j=1}^{mq_k}\\omega ((j-1)/mq_k;\\beta _k)x_{t+(h_k+1-j)/m,k},$ where for $k^{\\rm th}$ covariate, $h_k$ indicates the number of leading months of available data in the quarter $t$ , $q_k$ is the number of quarters of covariate lags, and we approximate the weight function $\\omega $ with the Legendre polynomial.", "For example, if $h_k=1$ and $q_k=4$ , then we have 1 month of data into a quarter and use $q_km=12$ monthly lags for a covariate $k$ .", "We benchmark our predictions against the simple AR(1) model, which is considered to be a reasonable starting point for short-term GDP growth predictions.", "We focus on predictions of our method, sg-LASSO-MIDAS, with and without financial data combined with series based on the textual analysis.", "One natural comparison is with the publicly available Federal Reserve Bank of New York, denoted NY Fed, model implied nowcasts.", "Table: Nowcast comparisons for models with macro and survey data only – Nowcast horizons are 2- and 1-month ahead, as well as the end of the quarter.", "Column Rel-RMSE reports root mean squared forecasts error relative to the AR(1) model.", "Column DM-stat-1 reports test statistic of all models relative to NY Fed nowcasts, while column DM-stat-2 reports the Diebold Mariano test statistic relative to sg-LASSO-MIDAS model.", "The last row reports the p-value of the average Superior Predictive Ability (aSPA) test, see , over the three horizons of sg-LASSO-MIDAS model compared to the NY Fed nowcasts.", "Out-of-sample period: 2002 Q1 to 2017 Q2.We adopt the following strategy.", "First, we focus on the same series that are used to calculate the NY Fed nowcasts.", "The purpose here is to compare models since the data inputs are the same.", "This means that we compare the performance of dynamic factor models (NY Fed) with that of machine learning regularized regression methods (sg-LASSO-MIDAS).", "Next, we expand the data set to see whether additional financial and textual news series can improve the nowcast performance.", "In Table REF , we report results based on real-time macro data used for the NY Fed model, see .", "The results show that the sg-LASSO-MIDAS performs much better than the NY Fed nowcasts at the longer, i.e.", "2-month, horizon.", "Our method significantly beats the benchmark AR(1) model for all the horizons, and the accuracy of the nowcasts improve with the horizon.", "Our end-of-quarter and 1-month horizon nowcasts are similar to the NY Fed ones, with the sg-LASSO-MIDAS being slightly better numerically but not statistically.", "We also report the average Superior Predictive Ability test of over all three horizons and the result reveals that the improvement of the sg-LASSO-MIDAS model versus the NY Fed nowcasts is significant at the 5% significance level.", "Table: Nowcast comparison table – Nowcast horizons are 2- and 1-month ahead, as well as the end of the quarter.", "Column Rel-RMSE reports root mean squared forecasts error relative to the AR(1) model.", "Column DM-stat-1 reports test statistic of all models relative to the NY FED nowcast, while column DM-stat-2 reports the Diebold Mariano test statistic relative to the sg-LASSO model.", "Columns DM-stat-3 and DM-stat-4 report the Diebold Mariano test statistic for the same models, but excludes the recession period.", "For the 1-month horizon, the last row SPF (median) reports test statistics for the same models comparing with the SPF median nowcasts.", "The last row reports the p-value of the average Superior Predictive Ability (aSPA) test, see , over the three horizons of sg-LASSO-MIDAS model compared to the NY Fed nowcasts, including (left) and excluding (right) financial crisis period.", "Out-of-sample period: 2002 Q1 to 2017 Q2.The comparison in Table REF does not fully exploit the potential of our methods, as it is easy to expand the data series beyond the small number used by the NY Fed nowcasting model.", "In Table REF we report results with additional sets of covariates which are financial series, advocated by , and textual analysis of news.", "In total, the models select from 118 series – 34 macro, 8 financial, and 76 news attention series.", "For the moment we focus only on the first three columns of the table.", "At the longer horizon of 2 months, the method seems to produce slightly worse nowcasts compared to the results reported in Table REF using only macro data.", "However, we find significant improvements in prediction quality for the shorter 1-month and end-of-quarter horizons.", "In particular, a significant increase in accuracy relative to NY Fed nowcasts appears at the 1-month horizon.", "We report again the average Superior Predictive Ability test of over all three horizons with the same result that the improvement of sg-LASSO-MIDAS versus the NY Fed nowcasts is significant at the 5% significance level.", "Lastly, we report results for several alternatives, namely, PCA-OLS, ridge, LASSO, and Elastic Net, using the unrestricted MIDAS scheme.", "Our approach produces more accurate nowcasts compared to these alternatives.", "Table REF also features an entry called SPF (median), where we report results for the median survey of professional nowcasts for the 1-month horizon, and analyze how the model-based nowcasts compare with the predictions using the publicly available Survey of Professional Forecasters maintained by the Federal Reserve Bank of Philadelphia.", "We find that the sg-LASSO-MIDAS model-based nowcasts are similar to the SPF-implied nowcasts.", "We also find that the NY Fed nowcasts are significantly worse than the SPF.", "Figure: Cumulative sum of loss differentials of sg-LASSO-MIDAS model nowcasts including financial and textual data compared with the New York Fed model for three nowcasting horizons: solid black line cumsfe for the 2-months horizon, dash-dotted black line - cumsfe for the 1-month horizon, and dotted line for the end-of-quarter nowcasts.", "The gray shaded area corresponds to the NBER recession period.In Figure REF we plot the cumulative sum of squared forecast error (CUMSFE) loss differential of sg-LASSO-MIDAS versus NY Fed nowcasts for the three horizons.", "The CUMSFE is computed as $\\text{CUMSFE}_{t,t+k} = \\sum _{q=t}^{t+k} e_{q,M1}^2 - e_{q,M2}^2$ for model $M1$ versus $M2.$ A positive value of $\\text{CUMSFE}_{t,t+k}$ means that the model $M1$ has larger squared forecast errors compared to model $M2$ up to $t+k,$ and negative values imply the opposite.", "In our case, $M1$ is the New York Fed prediction error, while $M2$ is the sg-LASSO-MIDAS model.", "We observe persistent gains for the 2- and 1-month horizons throughout the out-of-sample period.", "When comparing the sg-LASSO-MIDAS results with additional financial and textual news series versus those based on macro data only, we see a notable improvement at the 1-month horizon and a more modest one at the end-of-quarter horizons.", "In Figure REF , we plot the average CUMSFE for the 1-month and end-of-quarter horizons and observe that the largest gains of additional financial and textual news data are achieved during the financial crisis.", "The result in Figure REF prompts the question whether our results are mostly driven by this unusual period in our out-of-sample data.", "To assess this, we turn our attention again to the last two columns of Table REF reporting test statistics which exclude the financial crisis period.", "Compared to the tests previously discussed, we find that the results largely remain the same, but some alternatives seem to slightly improve (e.g.", "LASSO or Elastic Net).", "Note that this also implies that our method performs better during periods with heavy-tailed observations, such as the financial crisis.", "It should also be noted that overall there is a slight deterioration of the average Superior Predictive Ability test over all three horizons when we remove the financial crisis.", "Figure: Cumulative sum of loss differentials (CUMSFE) of sg-LASSO-MIDAS nowcasts when we include vs. when we exclude the additional financial and textual news data, averaged over 1-month and the end-of-quarter horizons.", "The gray shaded area corresponds to the NBER recession period.In Figure REF , we plot the fraction of selected covariates by the sg-LASSO-MIDAS model when we use the macro, financial, and textual analysis data.", "For each reference quarter, we compute the ratio of each group of variables relative to the total number of covariates.", "In each subfigure, we plot the three different horizons.", "For all horizons, the macro series are selected more often than financial and/or textual data.", "The number of selected series increases with the horizon, however, the pattern of denser macro series and sparser financial and textual series is visible for all three horizons.", "The results are in line with the literature – macro series tend to co-move, hence we see a denser pattern in the selection of such series, see e.g.", ".", "On the other hand, the alternative textual analysis data appear to be very sparse, yet still important for nowcasting accuracy, see also .", "Figure: The fraction of selected covariates attributed to macro (light gray), financial (dark gray), and textual (black) data for three monthly horizons." ], [ "Conclusion ", "This paper offers a new perspective on the high-dimensional time series regressions with data sampled at the same or mixed frequencies and contributes more broadly to the rapidly growing literature on the estimation, inference, forecasting, and nowcasting with regularized machine learning methods.", "The first contribution of the paper is to introduce the sparse-group LASSO estimator for high-dimensional time series regressions.", "An attractive feature of the estimator is that it recognizes time series data structures and allows us to perform the hierarchical model selection within and between groups.", "The classical LASSO and the group LASSO are covered as special cases.", "To recognize that the economic and financial time series have typically heavier than Gaussian tails, we use a new Fuk-Nagaev concentration inequality, from , valid for a large class of $\\tau $ -mixing processes, including $\\alpha $ -mixing processes commonly used in econometrics.", "Building on this inequality, we establish the nonasymptotic and asymptotic properties of the sparse-group LASSO estimator.", "Our empirical application provides new perspectives on applying machine learning methods to real-time forecasting, nowcasting, and monitoring with time series data, including unconventional data, sampled at different frequencies.", "To that end, we introduce a new class of MIDAS regressions with dictionaries linear in the parameters and based on orthogonal polynomials with lag selection performed by the sg-LASSO estimator.", "We find that the sg-LASSO outperforms the unstructured LASSO in small samples and conclude that incorporating specific data structures should be helpful in various applications.", "Our empirical results also show that the sg-LASSO-MIDAS using only macro data performs statistically better than NY Fed nowcasts at 2-month horizons and overall for the 1- and 2-month and end-of-quarter horizons.", "This is a comparison involving the same data and, therefore, pertains to models.", "This implies that machine learning models are, for this particular case, better than the state space dynamic factor models.", "When we add the financial data and the textual news data, a total of 118 series, we find significant improvements in prediction quality for the shorter 1-month and end-of-quarter horizons." ], [ "Acknowledgments", "We thank participants at the Financial Econometrics Conference at the TSE Toulouse, the JRC Big Data and Forecasting Conference, the Big Data and Machine Learning in Econometrics, Finance, and Statistics Conference at the University of Chicago, the Nontraditional Data, Machine Learning, and Natural Language Processing in Macroeconomics Conference at the Board of Governors, the AI Innovations Forum organized by SAS and the Kenan Institute, the 12th World Congress of the Econometric Society, and seminar participants at the Vanderbilt University, as well as Harold Chiang, Jianqing Fan, Jonathan Hill, Michele Lenza, and Dacheng Xiu for comments.", "We are also grateful to the referees and the editor whose comments helped us to improve our paper significantly.", "All remaining errors are ours.", "ONLINE APPENDIX" ], [ "Dictionaries ", "In this section, we review the choice of dictionaries for the MIDAS weight function.", "It is possible to construct dictionaries using arbitrary sets of functions, including a mix of algebraic polynomials, trigonometric polynomials, B-splines, Haar basis, or wavelets.", "In this paper, we mostly focus on dictionaries generated by orthogonalized algebraic polynomials, though it might be interesting to tailor the dictionary for each particular application.", "The attractiveness of algebraic polynomials comes from their ability to generate a variety of shapes with a relatively low number of parameters, which is especially desirable in low signal-to-noise environments.", "The general family of appropriate orthogonal algebraic polynomials is given by Jacobi polynomials that nest Legendre, Gegenbauer, and Chebychev's polynomials as a special case.", "Example A.1.1 (Jacobi polynomials) Applying the Gram-Schmidt orthogonalization to the power polynomials $\\lbrace 1,x,x^2,x^3,\\dots \\rbrace $ with respect to the measure $\\mathrm {d}\\mu (x) = (1-x)^{\\alpha }(1+x)^{\\beta }\\mathrm {d}x,\\qquad \\alpha ,\\beta >-1,$ on $[-1,1]$ , we obtain Jacobi polynomials.", "In practice Jacobi polynomials can be computed through the well-known tree-term recurrence relation for $n\\ge 0$ $P_{n+1}^{(\\alpha ,\\beta )}(x) = axP_{n}^{(\\alpha ,\\beta )}(x) + bP_{n}^{(\\alpha ,\\beta )}(x) - cP_{n-1}^{(\\alpha ,\\beta )}(x)$ with $a = (2n+\\alpha +\\beta +1)(2n+\\alpha +\\beta +2)/2(n+1)(n+\\alpha +\\beta +1)$ , $b=(2n+\\alpha +\\beta +1)(\\alpha ^2-\\beta ^2)/2(n+1)(n+\\alpha +\\beta +1)(2n+\\alpha +\\beta )$ , and $c = (\\alpha +n)(\\beta +n)(2n+\\alpha +\\beta +2)/(n+1)(n+\\alpha +\\beta +1)(2n+\\alpha +\\beta )$ .", "To obtain the orthogonal basis on $[0,1]$ , we shift Jacobi polynomials with affine bijection $x\\mapsto 2x-1$ .", "For $\\alpha =\\beta $ , we obtain Gegenbauer polynomials, for $\\alpha =\\beta =0$ , we obtain Legendre polynomials, while for $\\alpha =\\beta =-1/2$ or $\\alpha =\\beta =1/2$ , we obtain Chebychev's polynomials of two kinds.", "In the mixed frequency setting, non-orthogonalized polynomials, $\\lbrace 1,x,x^2,x^3,\\dots \\rbrace $ , are also called Almon polynomials.", "It is preferable to use orthogonal polynomials in practice due to reduced multicollinearity and better numerical properties.", "At the same time, orthogonal polynomials are available in Matlab, R, Python, and Julia packages.", "Legendre polynomials is our default recommendation, while other choices of $\\alpha $ and $\\beta $ are preferable if we want to accommodate MIDAS weights with other integrability/tail properties.", "We noted in the main body of the paper that the specification in equation (REF ) deviates from the standard MIDAS polynomial specification as it results in a linear regression model - a subtle but key innovation as it maps MIDAS regressions in the standard regression framework.", "Moreover, casting the MIDAS regressions in a linear regression framework renders the optimization problem convex, something only achieved by using the U-MIDAS of which does not recognize the mixed frequency data structure, unlike our sg-LASSO." ], [ "Proofs of main results", "Lemma A.2.1 Consider $\\Vert .\\Vert = \\alpha |.|_1 + (1-\\alpha )|.|_2$ , where $|.|_q$ is $\\ell _q$ norm on $\\mathbf {R}^p$ .", "Then the dual norm of $\\Vert .\\Vert $ , denoted $\\Vert .\\Vert ^*$ , satisfies $\\Vert z\\Vert ^* \\le \\alpha |z|_{1}^* + (1-\\alpha )|z|_{2}^*,\\qquad \\forall z\\in \\mathbf {R}^p,$ where $|.|_{1}^*$ is the dual norm of $|.|_1$ and $|.|_{2}^*$ is the dual norm of $|.|_2$ .", "Clearly, $\\Vert .\\Vert $ is a norm.", "By the convexity of $x\\mapsto x^{-1}$ on $(0,\\infty )$ $\\begin{aligned}\\Vert z\\Vert ^* & = \\sup _{b\\ne 0}\\frac{|\\langle z,b\\rangle |}{\\Vert b\\Vert } \\le \\sup _{b\\ne 0}\\left\\lbrace \\alpha \\frac{|\\langle z,b\\rangle |}{|b|_1} + (1-\\alpha )\\frac{|\\langle z,b\\rangle |}{|b|_2} \\right\\rbrace \\\\& \\le \\alpha \\sup _{b\\ne 0}\\frac{|\\langle z,b\\rangle |}{|b|_1} + (1-\\alpha )\\sup _{b\\ne 0}\\frac{|\\langle z,b\\rangle |}{|b|_2} \\\\& = \\alpha |z|_{1}^* + (1-\\alpha )|z|_{2}^*.\\end{aligned}$ [Proof of Theorem REF ] By Hölder's inequality for every $\\varsigma >0$ $\\max _{j\\in [p]}\\Vert u_0x_{0,j}\\Vert _\\varsigma \\le \\Vert u_0\\Vert _{\\varsigma q_1}\\max _{j\\in [p]}\\Vert x_{0,j}\\Vert _{\\varsigma q_2}$ with $q_1^{-1} + q_2^{-1}=1$ and $q_1,q_2\\ge 1$ .", "Therefore, under Assumption REF (i), $\\max _{j\\in [p]}\\Vert u_0x_{0,j}\\Vert _\\varsigma =O(1)$ with $\\varsigma =qr/(q+r)$ .", "Recall also that $\\mathbb {E}[u_tx_{t,j}]=0,\\forall j\\in [p]$ , see equation (REF ), which in conjunction with Assumption REF (ii) verifies conditions of Theorem REF and shows that there exists $C>0$ such that for every $\\delta \\in (0,1)$ $\\Pr \\left(\\left|\\frac{1}{T}\\sum _{t=1}^Tu_tX_t\\right|_\\infty \\le C\\left(\\frac{p}{\\delta T^{\\kappa -1}}\\right)^{1/\\kappa }\\vee \\sqrt{\\frac{\\log (8p/\\delta )}{T}}\\right) \\ge 1-\\delta .$ Let $G^* = \\max _{G\\in \\mathcal {G}}|G|$ be the size of the largest group in $\\mathcal {G}$ .", "Note that the sg-LASSO penalty $\\Omega $ is a norm.", "By Lemma REF , its dual norm satisfies $\\begin{aligned}\\Omega ^*(\\mathbf {X}^\\top \\mathbf {u}/T) & \\le \\alpha |\\mathbf {X}^\\top \\mathbf {u}/T|_\\infty + (1-\\alpha )\\max _{G\\in \\mathcal {G}}|(\\mathbf {X}^\\top \\mathbf {u})_G/T|_2 \\\\& \\le (\\alpha + (1-\\alpha )\\sqrt{G^*})|\\mathbf {X}^\\top \\mathbf {u}/T|_\\infty \\\\& \\le (\\alpha + (1-\\alpha )\\sqrt{G^*})C\\left(\\frac{p}{\\delta T^{\\kappa -1}}\\right)^{1/\\kappa }\\vee \\sqrt{\\frac{\\log (8p/\\delta )}{T}} \\\\& \\le \\lambda /c,\\end{aligned}$ where the first inequality follows since $|z|_1^*=|z|_\\infty $ and $\\left(\\sum _{G\\in \\mathcal {G}}|z_G|_2\\right)^* = \\max _{G\\in \\mathcal {G}}|z_G|_2$ , the second by elementary computations, the third by equation (REF ) with probability at least $1-\\delta $ for every $\\delta \\in (0,1)$ , and the last from the definition of $\\lambda $ in Assumption REF , where $c>1$ is as in Assumption REF .", "By Fermat's rule, the sg-LASSO satisfies $\\mathbf {X}^\\top (\\mathbf {X}\\hat{\\beta }- \\mathbf {y})/T + \\lambda z^* = 0$ for some $z^*\\in \\partial \\Omega (\\hat{\\beta })$ , where $\\partial \\Omega (\\hat{\\beta })$ is the subdifferential of $b\\mapsto \\Omega (b)$ at $\\hat{\\beta }$ .", "Taking the inner product with $\\beta -\\hat{\\beta }$ $\\begin{aligned}\\langle \\mathbf {X}^\\top (\\mathbf {y} - \\mathbf {X}\\hat{\\beta }),\\beta - \\hat{\\beta }\\rangle _T & = \\lambda \\langle z^*,\\beta - \\hat{\\beta }\\rangle \\le \\lambda \\left\\lbrace \\Omega (\\beta ) - \\Omega (\\hat{\\beta }) \\right\\rbrace ,\\end{aligned}$ where the inequality follows from the definition of the subdifferential.", "Using $\\mathbf {y}=\\mathbf {m} + \\mathbf {u}$ and rearranging this inequality $\\begin{aligned}\\Vert \\mathbf {X}(\\hat{\\beta }- \\beta )\\Vert ^2_T - \\lambda \\left\\lbrace \\Omega (\\beta ) - \\Omega (\\hat{\\beta }) \\right\\rbrace & \\le \\langle \\mathbf {X}^\\top \\mathbf {u},\\hat{\\beta }- \\beta \\rangle _T + \\langle \\mathbf {X}^\\top (\\mathbf {m}-\\mathbf {X}\\beta ),\\hat{\\beta }- \\beta \\rangle _T \\\\& \\le \\Omega ^*\\left(\\mathbf {X}^\\top \\mathbf {u}/T\\right)\\Omega (\\hat{\\beta }- \\beta ) + \\Vert \\mathbf {X}(\\hat{\\beta }- \\beta )\\Vert _T\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T \\\\& \\le c^{-1}\\lambda \\Omega (\\hat{\\beta }- \\beta ) + \\Vert \\mathbf {X}(\\hat{\\beta }- \\beta )\\Vert _T\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T.\\end{aligned}$ where the second line follows by the dual norm inequality and the last by $\\Omega ^*(\\mathbf {X}^\\top \\mathbf {u}/T)\\le \\lambda /c$ as shown in equation (REF ).", "Therefore, $\\begin{aligned}\\Vert \\mathbf {X}\\Delta \\Vert ^2_T & \\le c^{-1}\\lambda \\Omega (\\Delta ) + \\Vert \\mathbf {X}\\Delta \\Vert _T\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T + \\lambda \\left\\lbrace \\Omega (\\beta ) - \\Omega (\\hat{\\beta }) \\right\\rbrace \\\\& \\le (c^{-1}+1)\\lambda \\Omega (\\Delta ) + \\Vert \\mathbf {X}\\Delta \\Vert _T\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T\\end{aligned}$ with $\\Delta = \\hat{\\beta }- \\beta $ .", "Note that the sg-LASSO penalty can be decomposed as a sum of two seminorms $\\Omega (b) = \\Omega _0(b) + \\Omega _1(b),\\;\\forall b\\in \\mathbf {R}^p$ with $\\Omega _0(b) = \\alpha |b_{S_0}|_1 + (1-\\alpha )\\sum _{G\\in \\mathcal {G}_0}|b_{G}|_2\\qquad \\text{and} \\qquad \\Omega _1(b) =\\alpha |b_{S^c_0}|_1 + (1-\\alpha )\\sum _{G\\in \\mathcal {G}_0^c}|b_{G}|_2.$ Note also that $\\Omega _1(\\beta )=0$ and $\\Omega _1(\\hat{\\beta })=\\Omega _1(\\Delta )$ .", "Then by the triangle inequality $\\Omega (\\beta ) - \\Omega (\\hat{\\beta }) \\le \\Omega _0(\\Delta ) - \\Omega _1(\\Delta ).$ If $\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T \\le 2^{-1}\\Vert \\mathbf {X}\\Delta \\Vert _T$ , then it follows from the first inequality in equation (REF ) and equation (REF ) that $\\Vert \\mathbf {X}\\Delta \\Vert ^2_T \\le 2 c^{-1}\\lambda \\Omega (\\Delta ) + 2\\lambda \\left\\lbrace \\Omega _0(\\Delta ) - \\Omega _1(\\Delta ) \\right\\rbrace .$ Since the left side of this equation is positive, this shows that $\\Omega _1(\\Delta )\\le c_0\\Omega _0(\\Delta )$ with $c_0=(c+1)/(c-1)$ , and whence $\\Delta \\in \\mathcal {C}(c_0)$ , cf., Assumption REF .", "Then $\\begin{aligned}\\Omega (\\Delta ) & \\le (1+c_0)\\Omega _0(\\Delta ) \\\\& \\le (1+c_0)\\left(\\alpha \\sqrt{|S_0|}|\\Delta _{S_0}|_2 + (1-\\alpha )\\sqrt{|\\mathcal {G}_0|}\\sqrt{\\sum _{G\\in \\mathcal {G}_0}|\\Delta _G|_2^2}\\right) \\\\& \\le (1+c_0)\\sqrt{s_\\alpha }\\sqrt{\\sum _{G\\in \\mathcal {G}_0}|\\Delta _G|_2^2} \\\\& \\le (1+c_0)\\sqrt{s_\\alpha /\\gamma \\Delta ^\\top \\Sigma \\Delta }, \\end{aligned}$ where we use the Jensen's inequality, Assumption REF , and the definition of $\\sqrt{s_\\alpha }$ .", "Next, note that $\\begin{aligned}\\Delta ^\\top \\Sigma \\Delta & = \\Vert \\mathbf {X}\\Delta \\Vert ^2_T + \\Delta ^\\top (\\Sigma - \\hat{\\Sigma })\\Delta \\\\& \\le 2(c^{-1}+1)\\lambda \\Omega (\\Delta ) + \\Omega (\\Delta )\\Omega ^{*}\\left((\\hat{\\Sigma }- \\Sigma )\\Delta \\right) \\\\& \\le 2(c^{-1}+1)\\lambda \\Omega (\\Delta ) + \\Omega ^2(\\Delta )G^*|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty , \\end{aligned}$ where the first inequality follows from equation (REF ) and the dual norm inequality and the second by Lemma REF and elementary computations $\\begin{aligned}\\Omega ^*\\left((\\hat{\\Sigma }- \\Sigma )\\Delta \\right) & \\le \\alpha |(\\hat{\\Sigma }- \\Sigma )\\Delta |_\\infty + (1-\\alpha )\\max _{G\\in \\mathcal {G}}\\left|[(\\hat{\\Sigma }- \\Sigma )\\Delta ]_{G}\\right|_2 \\\\& \\le \\alpha |\\Delta |_1|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty + (1-\\alpha )\\sqrt{G^*}|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty |\\Delta |_1 \\\\& \\le G^*|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty \\Omega (\\Delta ).\\end{aligned}$ Combining the inequalities obtained in equations (REF and REF ) $\\begin{aligned}\\Omega (\\Delta ) & \\le (1+c_0)^2\\gamma ^{-1}s_\\alpha \\left\\lbrace 2(c^{-1}+1)\\lambda + G^*|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty \\Omega (\\Delta ) \\right\\rbrace \\\\& \\le 2(1+c_0)^2\\gamma ^{-1}s_\\alpha (c^{-1}+1)\\lambda + (1-A^{-1})\\Omega (\\Delta ),\\end{aligned}$ where the second line holds on the event $E \\triangleq \\lbrace |\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty \\le \\gamma /2G^*s_\\alpha (1+2c_0)^2 \\rbrace $ with $1 - A^{-1} = (1+c_0)^2/2(1+2c_0)^2<1$ .", "Therefore, inequalities in equation (REF and REF ) yield $\\begin{aligned}\\Omega (\\Delta ) & \\le \\frac{2A}{\\gamma }(1+c_0)^2(c^{-1}+1)s_\\alpha \\lambda \\\\\\Vert \\mathbf {X}\\Delta \\Vert ^2_T & \\le \\frac{4A}{\\gamma }(1+c_0)^2(c^{-1}+1)^2s_\\alpha \\lambda ^2.\\end{aligned}$ On the other hand, if $\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T > 2^{-1}\\Vert \\mathbf {X}\\Delta \\Vert _T$ , then $\\Vert \\mathbf {X}\\Delta \\Vert ^2_T \\le 4\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2.$ Therefore, on the event $E$ we always have $\\Vert \\mathbf {X}\\Delta \\Vert ^2_T \\le C_1s_\\alpha \\lambda ^2 + 4\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2$ with $C_1 = 4A\\gamma ^{-1}(1+c_0)^2(c^{-1}+1)^2$ .", "This proves the first claim of Theorem REF if we show that $\\Pr (E^c)\\le 2p(p+1)(c_1T^{1-\\mu }s_\\alpha ^{\\mu } + \\exp (-c_2T/s_\\alpha ^2)$ .", "To that end, by the Cauchy-Schwartz inequality under Assumptions REF (i) $\\max _{1\\le j\\le k\\le p}\\Vert x_{0,j}x_{0,k}\\Vert _{r/2} \\le \\max _{j\\in [p]}\\Vert x_{0,j}\\Vert _{r}^2 = O(1).$ This in conjunction with Assumption REF (ii) verifies assumptions of , Theorem 3.1 and shows that $\\begin{aligned}\\Pr (E^c) & = \\Pr \\left(\\left|\\frac{1}{T}\\sum _{t=1}^Tx_{t,j}x_{t,k} - \\mathbb {E}[x_{t,j}x_{t,k}]\\right|_\\infty > \\frac{\\gamma }{2G^*s_\\alpha (1+2c_0)^2}\\right) \\\\& \\le c_1T^{1-\\mu }s_\\alpha ^{\\mu }p(p+1) + 2p(p+1)\\exp \\left(-\\frac{c_2T^2}{s_\\alpha ^2B_T^2}\\right)\\end{aligned}$ for some $c_1,c_2>0$ and $B_T^2 = \\max _{j,k\\in [p]}\\sum _{t=1}^T\\sum _{l=1}^T|\\mathrm {Cov}(x_{t,j}x_{t,k}, x_{l,j}x_{l,k})|$ .", "Lastly, under Assumption REF , by , Lemma A.1.2, $B_T^2 = O(T)$ .", "To prove the second claim of Theorem REF , suppose first that $\\Delta \\in \\mathcal {C}(2c_0)$ .", "Then on the event $E$ $\\begin{aligned}\\Omega ^2(\\Delta ) & = (\\Omega _0(\\Delta ) + \\Omega _1(\\Delta ))^2 \\\\& \\le (1+2c_0)^2\\Omega _0^2(\\Delta ) \\\\& \\le (1+2c_0)^2\\Delta ^\\top \\Sigma \\Delta s_\\alpha /\\gamma \\\\& = (1+2c_0)^2\\left\\lbrace \\Vert \\mathbf {X}\\Delta \\Vert ^2_T + \\Delta ^\\top (\\Sigma -\\hat{\\Sigma })\\Delta \\right\\rbrace s_\\alpha /\\gamma \\\\& \\le (1+2c_0)^2\\left\\lbrace C_1s_\\alpha \\lambda ^2 + 4\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 + \\Omega ^2(\\Delta )G^*|\\mathrm {vech}(\\hat{\\Sigma }- \\Sigma )|_\\infty \\right\\rbrace s_\\alpha /\\gamma \\\\& \\le (1+2c_0)^2\\left\\lbrace C_1s_\\alpha \\lambda ^2 + 4\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2\\right\\rbrace s_\\alpha /\\gamma + \\frac{1}{2}\\Omega ^2(\\Delta ),\\end{aligned}$ where we use the inequality in equations (REF , REF , and REF ).", "Therefore, $\\Omega ^2(\\Delta ) \\le 2(1+2c_0)^2\\left\\lbrace C_1s_\\alpha \\lambda ^2 + 4\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2\\right\\rbrace s_\\alpha /\\gamma .$ On the other hand, if $\\Delta \\notin \\mathcal {C}(2c_0)$ , then $\\Delta \\notin \\mathcal {C}(c_0)$ , which as we have already shown implies $\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T>2^{-1}\\Vert \\mathbf {X}\\Delta \\Vert _T$ .", "In conjunction with equations (REF and REF ), this shows that $0\\le \\lambda c^{-1}\\Omega (\\Delta ) + 2\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 + \\lambda \\left\\lbrace \\Omega _0(\\Delta ) - \\Omega _1(\\Delta ) \\right\\rbrace ,$ and whence $\\begin{aligned}\\Omega _1(\\Delta ) & \\le c_0\\Omega _0(\\Delta ) + \\frac{2c}{\\lambda (c-1)}\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2 \\\\& \\le \\frac{1}{2}\\Omega _1(\\Delta ) + \\frac{2c}{\\lambda (c-1)}\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2.\\end{aligned}$ This shows that $\\begin{aligned}\\Omega (\\Delta ) & \\le (1+(2c_0)^{-1})\\Omega _1(\\Delta ) \\\\& \\le (1+(2c_0)^{-1})\\frac{4c}{\\lambda (c-1)}\\Vert \\mathbf {m} - \\mathbf {X}\\beta \\Vert _T^2.\\end{aligned}$ Combining this with the inequality in equation (REF ), we obtain the second claim of Theorem REF .", "The following result is proven in , see their Theorem A.1.", "Theorem A.1 Let $(\\xi _t)_{t\\in \\mathbf {Z}}$ be a centered stationary stochastic process in $\\mathbf {R}^p$ such that (i) for some $\\varsigma >2$ , $\\max _{j\\in [p]}\\Vert \\xi _{0,j}\\Vert _\\varsigma = O(1)$ ; (ii) for every $j\\in [p]$ , $\\tau $ -mixing coefficients of $\\xi _{t,j}$ satisfy $\\tau _k^{(j)}\\le ck^{-a}$ for some constants $c>0$ and $a>(\\varsigma -1)/(\\varsigma -2)$ .", "Then there exists $C>0$ such that for every $\\delta \\in (0,1)$ $\\Pr \\left(\\left|\\frac{1}{T}\\sum _{t=1}^T\\xi _t\\right|_\\infty \\le C\\left(\\frac{p}{\\delta T^{\\kappa -1}}\\right)^{1/\\kappa }\\vee \\sqrt{\\frac{\\log (8p/\\delta )}{T}} \\right) \\ge 1 - \\delta $ with $\\kappa = ((a+1)\\varsigma - 1)/(a+\\varsigma -1)$ ." ], [ "ARDL-MIDAS: moments and $\\tau $ -mixing coefficients", "The ARDL-MIDAS process $(y_t)_{t\\in \\mathbf {Z}}$ is defined as $\\phi (L)y_t = \\xi _t,$ where $\\phi (L) = I - \\rho _1L - \\rho _2L^2 - \\dots - \\rho _JL^J$ is a lag polynomial and $\\xi _t = \\sum _{j=0}^p x_{t,j}\\gamma _j + u_t$ .", "The process $(y_t)_{t\\in \\mathbf {Z}}$ is $\\tau $ -mixing and has finite moments of order $q\\ge 1$ as illustrated below.", "Assumption A.3.1 Suppose that $(\\xi _t)_{t\\in \\mathbf {Z}}$ is a stationary process such that (i) $\\Vert \\xi _t\\Vert _{q}<\\infty $ for some $q>1$ ; (ii) $\\beta $ -mixing coefficients satisfy $\\beta _k\\le Ca^{k}$ for some $a\\in (0,1)$ and $C>0$ ; and (iii) $\\phi (z)\\ne 0$ for all $z\\in \\mathbf {C}$ such that $|z|\\le 1$ .", "Note that by , (ii) holds if $(\\xi _t)_{t\\in \\mathbf {Z}}$ is a geometrically ergodic Markov process and that (iii) rules out the unit root process.", "Proposition A.3.1 Under Assumption REF , the ARDL-MIDAS process has moments of order $q>1$ and $\\tau $ -mixing coefficients $\\tau _k \\le C(a^{bk} + c^k)$ for some $c\\in (0,1),$ $C>0$ , and $b=1-1/q$ .", "Under (iii) we can invert the autoregressive lag polynomial and obtain $y_t = \\sum _{j=0}^\\infty \\psi _j\\xi _{t-j}$ for some $(\\psi _j)_{j=0}^\\infty \\in \\ell _1$ .", "Note that $(y_t)_{t\\in \\mathbf {Z}}$ has dependent innovations.", "Clearly, $(y_t)_{t\\in \\mathbf {Z}}$ is stationary provided that $(\\xi _t)_{t\\in \\mathbf {Z}}$ is stationary, which is the case by the virtue of Assumption REF .", "Next, since $\\Vert y_t\\Vert _q \\le \\sum _{j=0}^\\infty |\\psi _j|\\Vert \\xi _0\\Vert _q$ and $\\Vert \\xi _0\\Vert _q<\\infty $ under (i), we verify that $\\Vert y_t\\Vert _q<\\infty $ .", "Let $(\\xi _t^{\\prime })_{t\\in \\mathbf {Z}}$ be a stationary process distributed as $(\\xi _{t})_{t\\in \\mathbf {Z}}$ and independent of $(\\xi _t)_{t\\le 0}$ .", "Then by , Example 1, the $\\tau $ -mixing coefficients of $(y_t)_{t\\in \\mathbf {Z}}$ satisfy $\\begin{aligned}\\tau _k & \\le \\Vert \\xi _0 - \\xi _0^{\\prime }\\Vert _q\\sum _{j\\ge k}|\\psi _j| + 2\\sum _{j=0}^{k-1}|\\psi _j|\\int _0^{\\beta _{k-j}}Q_{\\xi _0}(u)\\mathrm {d}u \\\\& \\le 2\\Vert \\xi _0\\Vert _q\\sum _{j\\ge k}|\\psi _j| + 2\\Vert \\xi _0\\Vert _q\\sum _{j=0}^{k-1}|\\psi _j|\\beta _{k-j}^{1-1/q},\\end{aligned}$ where $(\\beta _k)_{k\\ge 1}$ are $\\beta $ -mixing coefficients of $(\\xi _t)_{t\\in \\mathbf {Z}}$ and the second line follows by Hölder's inequality.", ", p.85 shows that there exist $c\\in (0,1)$ and $K>0$ such that $|\\psi _j|\\le Kc^{j}$ .", "Therefore, $\\sum _{j\\ge k}|\\psi _j| = O(c^k)$ and under (ii) $\\sum _{j=0}^{k-1}|\\psi _j|\\beta _{k-j}^{1-1/q} \\le CK\\sum _{j=0}^{k-1}c^ja^{(k-j)(q-1)/q} \\le {\\left\\lbrace \\begin{array}{ll}CK\\frac{a^{k(q-1)/q} - c^k}{1 - ca^{(1-q)/q}} &\\text{if } c\\ne a^{(q-1)/q}, \\\\CKka^{k(q-1)/q} & \\text{otherwise}.\\end{array}\\right.", "}$ This proves the second statement." ], [ "Detailed description of data and models", "The standard macro variables are collected from Haver Analytics and ALFRED databases.", "ALFRED is a public data source for real-time data made available by the Federal Serve Bank of St. Louis; see the full list of the series with further details in Table REF .", "For series that are collected from the Haver Analytics database, we use as reported data, that is the first release is used for each data point.", "For the data that we collect from ALFRED, full data vintages are used.", "All the data is real-time, hence publication delays for each series are taken into consideration and we align each series accordingly.", "We use twelve monthly and four quarterly lags for each monthly and quarterly series respectively and apply Legendre aggregation with polynomial degree set to three.", "The groups are defined as lags of each series.", "On top of macro data, we add eight financial series which are collected from FRED database; the full list of the series appears in Table REF .", "These series are available in real time, hence no publication delays are needed in this case.", "We use three monthly lags and apply Legendre aggregation with polynomial degree set to two.", "As for macro, we group all lags of each series.", "Lastly, we add textual analysis covariates which are available at http://structureofnews.com/.", "The data is real time, i.e., topic models are estimated for each day and the monthly series are obtained by aggregating daily data; see for further details on the data construction.", "We use categories of series are potentially closely tied with economic activity, which are Banks, Economic Growth, Financial Markets, Government, Industry, International Affairs, Labor/income, and Oil & Mining.", "In total, we add 76 news attention series; the full list is available in Table REF .", "Three lags are used and Legendre aggregation of degree two is applied to each series.", "In this case, we group variables based on categories.", "To make the comparison with the NY Fed nowcasts as close as possible, we use 15 years (60 quarters) of the data and use rolling window estimation.", "The first nowcast is for the 2002 Q1 (first quarter that NY Fed publishes its historic nowcasts) and the effective sample size starts at 1988 Q1 (taking 15 years of data accounting for lags).", "We calculate predictions until the sample is exhausted, which is 2017 Q2, the last date for which news attention data is available.", "Real GDP growth rate data vintages are taken from ALFRED database.", "Some macro series start later than 1988 Q1, in which case we impute zero values.", "Lastly, we use four lags of real GDP growth rate in all models." ], [ "Alternative estimators", "We implemented the following alternative machine learning nowcasting methods.", "The first method is the PCA factor-augmented autoregression, where we estimate the first principal component of the data panel and use it together with four autoregressive lags.", "We denote this model PCA-OLS.", "We then consider three alternative penalty functions for the same linear model: ridge, LASSO, and Elastic Net.", "For these methods, we leave high-frequency lags unrestricted, and thus we call these methods the unrestricted MIDAS (U-MIDAS).", "As for the sg-LASSO-MIDAS model, we tune one- and two-dimensional regularization parameters via 5-fold cross-validation.", "Table: Data description table (macro data)– The Series column gives a time-series name, which is given in the second column Source.", "The column Units denotes the data transformation applied to a time-series.Table: Data description table (financial and uncertainty series) – The Series column gives a time-series name, which is given in the second column Source.", "The column Units denotes the data transformation applied to a time-series.", "[!htbp]r | cc Group Series 1 Banks Bank loans 2 Banks Credit ratings 3 Banks Financial crisis 4 Banks Mortgages 5 Banks Nonperforming loans 6 Banks Savings & loans 7 Economic Growth Economic growth 8 Economic Growth European sovereign debt 9 Economic Growth Federal Reserve 10 Economic Growth Macroeconomic data 11 Economic Growth Optimism 12 Economic Growth Product prices 13 Economic Growth Recession 14 Economic Growth Record high 15 Financial Markets Bear/bull market 16 Financial Markets Bond yields 17 Financial Markets Commodities 18 Financial Markets Currencies/metals 19 Financial Markets Exchanges/composites 20 Financial Markets International exchanges 21 Financial Markets IPOs 22 Financial Markets Options/VIX 23 Financial Markets Share payouts 24 Financial Markets Short sales 25 Financial Markets Small caps 26 Financial Markets Trading activity 27 Financial Markets Treasury bonds 28 Government Environment 29 Government National security 30 Government Political contributions 31 Government Private/public sector 32 Government Regulation 33 Government Safety administrations 34 Government State politics 35 Government Utilities 36 Government Watchdogs 37 Industry Cable 38 Industry Casinos 39 Industry Chemicals/paper 40 Industry Competition 41 Industry Couriers 42 Industry Credit cards 43 Industry Fast food 44 Industry Foods/consumer goods 45 Industry Insurance 46 Industry Luxury/beverages 47 Industry Revenue growth 48 Industry Small business 49 Industry Soft drinks 50 Industry Subsidiaries 51 Industry Tobacco 52 Industry Venture capital 53 International Affairs Canada/South Africa 54 International Affairs China 55 International Affairs France/Italy 56 International Affairs Germany 57 International Affairs Japan 58 International Affairs Latin America 59 International Affairs Russia 60 International Affairs Southeast Asia 61 International Affairs Trade agreements 62 International Affairs UK 63 Labor/income Executive pay 64 Labor/income Fees 65 Labor/income Government budgets 66 Labor/income Health insurance 67 Labor/income Job cuts 68 Labor/income Pensions 69 Labor/income Taxes 70 Labor/income Unions 71 Oil & Mining Agriculture 72 Oil & Mining Machinery 73 Oil & Mining Mining 74 Oil & Mining Oil drilling 75 Oil & Mining Oil market 76 Oil & Mining Steel Data description table (textual data) – The Group column is a group name of individual textual analysis series which appear in the column Series.", "Data is taken in levels." ] ]

[ [ "Two-Bar Charts Packing Problem" ], [ "Abstract We consider a Bar Charts Packing Problem (BCPP), in which it is necessary to pack bar charts (BCs) in a strip of minimum length.", "The problem is, on the one hand, a generalization of the Bin Packing Problem (BPP), and, on the other hand, a particular case of the Project Scheduling Problem with multidisciplinary jobs and one limited non-accumulative resource.", "Earlier, we proposed a polynomial algorithm that constructs the optimal package for a given order of non-increasing BCs.", "This result generalizes a similar result for BPP.", "For Two-Bar Charts Packing Problem (2-BCPP), when each BC consists of two bars, the algorithm we have proposed constructs a package in polynomial time, the length of which does not exceed $2\\ OPT+1$, where $OPT$ is the minimum possible length of the packing.", "As far as we know, this is the first guaranteed estimate for 2-BCPP.", "We also conducted a numerical experiment in which we compared the solutions built by our approximate algorithms with the optimal solutions built by the CPLEX package.", "The experimental results confirmed the high efficiency of the developed algorithms." ], [ "Introduction", "When solving the problem of optimizing the investment portfolio in the oil and gas sector, we faced the following problem [5].", "Suppose that the territory of the oil and gas field divided into clusters.", "For each cluster, a set of projects for its development is known.", "The project is characterized, in particular, by annual oil production.", "If we know the start year of the project, then we know the volumes of oil production in the first and all subsequent years of the project.", "The production schedule for each project can be represented in the form of a bar chart (BC), in which the bar's height corresponds to the volume of production in the corresponding year.", "It is required to determine the year of the launch of each project in such a way that the execution time of all projects is minimal, and the annual production volume from all cluster deposits does not exceed a predetermined value $D$ , which is defined, for example, by the pipeline capacity.", "Imagine a horizontal strip of height $D$ .", "Then, the problem described above comes down to finding packing of BCs in a part of the strip (a rectangle of height $D$ ) of minimum length.", "Moreover, when packing each BC, the bars corresponding to production volumes in different years can move vertically, but they are inextricably horizontal and cannot be rearranged.", "Fig.", "1 shows an example of feasible packing of three BCs, from which it follows that projects (a) and (b) start in the first year, project (c) starts in the fourth year, and all projects end in year 5.", "That is, the length of the strip (rectangle) into which all BCs are packed is 5.", "Figure: Feasible packing of BCs.We were unable to find any publications on BCs packing.", "Similar problems that were studied reasonably well are the Bin Packing Problem (BPP) [2], [4], [15], [16], [20], [25], [26] and the problem of packing rectangles in a strip [1], [3], [12], [13], [21], [23].", "In the classical BPP, a set $L$ of items, a size of each items, and a set of identical $D$ -size containers (bins) are specified.", "All items must be placed in a minimum number of bins.", "One of the well-known algorithms for packing items in containers is First Fit Decreasing (FFD).", "As part of this algorithm, objects are numbered in non-increasing order.", "Then, all items are scanned in order, and the current item is placed in the first suitable bin.", "In 1973, Johnson proved that the FFD algorithm uses no more than $11/9\\ OPT(L)+4$ containers [15].", "In 1985 Backer showed that the additive constant can be reduced to 3 [2].", "Yue in 1991 proved that $FFD(L)\\le 11/9\\ OPT (L)+1$ [25].", "Furthermore, in 1997 he improved the result to $FFD(L)\\le 11/9\\ OPT (L)+7/9$ together with Li [20].", "In 2007, Dósa found the tight boundary of the additive constant and gave an example when $FFD(L)=11/9\\ OPT(L)+6/9$ [4].", "A Modified First Fit Decreasing (MFFD) algorithm improves FFD by dividing items into groups by size and packing items from different groups separately.", "Johnson and Garey proposed this modification, and in 1985 they showed that $MFFD(L)\\le 71/60\\ OPT(L)+31/6$ [16].", "Subsequently, the result was improved by Yue and Zhang to $MFFD(L)\\le 71/60\\ OPT(L)+1$ [26].", "The BPP is a particular case of the problem of packing rectangles in a horizontal strip when all objects have the same width.", "Therefore, the problem of packing rectangles in a strip is NP-hard too.", "Moreover, if $P\\ne NP$ , then both problems are 3/2-inapproximable [24].", "Formally, the problem of tight packing of rectangles into a semi-infinite strip of height $D$ is as follows.", "For each rectangle $i\\in L$ , we know the width $w_i$ and the height $h_i$ .", "It is required to find the packing of the set of rectangles $L$ in a strip of minimum length.", "Rotation of rectangles is prohibited.", "The Bottom-Left algorithm proposed by Baker [1] arranges rectangles in descending order of height and has a ratio of 3.", "Coffman et al.", "[3] in 1980 proposed algorithms with ratio 3 and 2.7.", "Sleator [22] showed that his algorithm packs the rectangles into a strip whose length does not exceed $2\\ OPT(L)+w_{max}(L)/2$ , where $w_{max} (L)$ is the width of the widest rectangle in the set.", "Since $w_{max} (L) \\le OPT(L)$ , the algorithm guarantees a ratio of 2.5.", "This ratio was reduced by Schiermeyer [21] and Steinberg [23] to 2.", "Harren and van Stee [12] were the first who evaluated a ratio of less than 2.", "Their proposed algorithm has a ratio of 1.9396.", "The smallest estimate for the ratio known to date obtained by Harren et al.", "[13] in 2014, and equals $(5/3+\\varepsilon )OPT(L)$ , for any $\\varepsilon > 0$ .", "The problem under consideration is also a particular case of the project scheduling problem.", "Realy, each project consists of a sequence of jobs that needs to be done one after another without delay (no-wait).", "Each job has a unit duration and consumes the non-accumulative resource.", "It is required to find the start moment for each project in such a way that all projects are finished in minimum time, consuming together at most $D$ resource during each moment.", "Resource-limited scheduling has been the subject of many publications discussing renewable and nonrenewable or accumulative resources.", "An overview of the results can be found, for example, in [14], [18].", "For the case of an accumulative resource, exact and asymptotically exact algorithms have been developed [6], [7].", "In the case of a limited renewable resource, the scheduling problem is NP-hard, and polynomial algorithms with guaranteed accuracy estimates are not known for it.", "As a rule, heuristic algorithms are developed for its approximate solution, and a posteriori analysis is performed [8], [9], [14], [18].", "For example, in [7], [8], [9], [10], [11] multidisciplinary partially ordered jobs of arbitrary duration are considered that consume different amounts of a homogeneous resource at different time moments, and the authors developed approximate algorithms for solving the problem, and also performed a posteriori analysis, which showed a quite high efficiency.", "For comparison, the authors used a problem library PSPLIB [17].", "For some instances from the dataset J60 [8], [9], [10], and dataset J120 [11], the best-known solutions were improved.", "We were unable to find publications in which polynomial algorithms with guaranteed accuracy estimates proposed for such kind of project scheduling problem.", "The problem of packing BCs in the particular case when all BCs consist of one bar is a BPP.", "In this article, to construct an approximate solution to the problem, we developed a greedy algorithm ($GA$ ) and obtained some qualitative results for non-increasing BCs with an arbitrary number of bars.", "The main result of the article is the proof that, for arbitrary two-bar charts, algorithm $A$ , which uses the $GA$ as a procedure, builds a package whose length does not exceed $2\\ OPT+1$ , where $OPT$ is the minimum packing length.", "As far as we know, this is the first a priori estimate for the problem under consideration, which proves, in particular, that it belongs to the APX class.", "The rest of the article is organized as follows.", "Section 2 provides a statement of the packing problem for BCs with an arbitrary number of bars, as well as a statement in the case of two-bar charts (2-BCs) in the form of Boolean Linear Programming (BLP).", "Section 3 describes the greedy algorithm $GA$ for the densest packing of BCs in a unit-height strip.", "Some properties of the algorithm are also given there.", "Section 4 discusses the packing problem of 2-BCs.", "We proposed an approximate algorithm that uses the $GA$ algorithm as a procedure.", "At the preliminary stage of the algorithm, some BCs are combined into one BC, in which at least one bar has a height greater than 1/2.", "This section gives the main result of the article, which consists in proving that the developed algorithm builds a package whose length does not exceed $2\\ OPT + 1$ , where $OPT$ is the minimum length of a strip into which all 2-BCs can be packed.", "In Section 5, we described the results of a numerical experiment, which made it possible to conduct a posteriori analysis of the developed algorithm.", "To build the optimal solution to the BLP, we used the CPLEX package and compared the optimal solution with the solution built by our algorithms.", "In section 6, we summarize and outline directions for further research." ], [ "Formulation of the problem", "We have a horizontal stripe, the height of which, without loss of generality, equals 1 ($D=1$ ), and the set $S$ of BCs.", "Each BC $i\\in S$ consists of the sequence of $l_i$ bars ($\\max \\limits _{i\\in S}l_i=l$ ).", "Each bar $j$ in BC $i$ has a width that equals 1 and a height that equals $h^j_i\\in (0,1]$ ($H_i=\\max \\limits _{j=1,\\ldots ,l_i}h^j_i$ ).", "We introduce the Cartesian coordinate system so that the lower boundary of the strip coincides with the abscissa.", "Let us consider a part of the strip to the right of the origin 0, which we divide into the cells of width that equals 1, and number these cells with integers $1,2,\\ldots $ .", "BC $i$ is non-increasing (non-decreasing) if $h^j_i\\ge h^{j+1}_i$ ($h^j_i\\le h^{j+1}_i$ ) for all $j=1,\\ldots ,l_i-1$ .", "Packing is a function $p:S\\rightarrow Z^+$ , which associates with each BC $i$ an integer $p(i)$ corresponding to the cell number of the strip into which the first bar of BC $i$ falls.", "As a result of packing $p$ , bars from BC $i$ occupy the cells $p(i),p(i)+1,\\ldots ,p(i)+l_i-1$ .", "The packing is feasible if the sum of the heights of the bars that fall into one cell of the strip does not exceed 1.", "That is for each cell $k$ the inequality $\\sum \\limits _{i\\in S:p(i)\\le k\\le p(i)+l_i-1} h^{k-p(i)+1}_i \\le 1$ holds.", "The packing length $L(p)$ is the number of strip cells in which falls at least one bar.", "We can assume that any packing $p$ begins from the first cell, and in each cell from 1 to $L(p$ ), there is at least one bar.", "If this is not the case, then all or part of the package can be moved to the left.", "The Bar Charts Packing Problem (BCPP) is to build a feasible min-length package.", "Another measure of packing quality is density.", "It is a ratio of the sum of the bar's heights to the packing length.", "The density cannot be greater than 1, and the higher the density, the better the packing.", "Let us formulate the 2-BCPP problem, in which each BC has two bars, in the form of BLP.", "To do this, we introduce the following notation.", "Let the $i$ th 2-BC have the height of the first bar $a_i$ , and the second $b_i$ .", "We introduce the variables: $x_{ij}=\\left\\lbrace \\begin{array}{ll}1, & \\hbox{if the first bar of BC $i$ is in the cell $j$;} \\\\0, & \\hbox{else.", "}\\end{array}\\right.$ $y_j=\\left\\lbrace \\begin{array}{ll}1, & \\hbox{if the cell $j$ contains at least one bar;} \\\\0, & \\hbox{else.", "}\\end{array}\\right.$ Then 2-BCPP is written as follows.", "$\\sum \\limits _j y_j \\rightarrow \\min \\limits _{x_{ij},y_j\\in \\lbrace 0,1\\rbrace };$ $\\sum \\limits _j x_{ij} =1,\\ i\\in S;$ $\\sum \\limits _i a_ix_{ij} + \\sum \\limits _k b_kx_{k,j-1}\\le y_j,\\ \\forall j.$ Both problems BCPP and 2-BCPP are strongly NP-hard as the generalizations of the BPP [15].", "Moreover, these problems are $3/2$ -inapproximable [24]." ], [ "Greedy algorithm $G$", "First, we describe the version of the greedy algorithm (denote it by $G$ ), which builds an order-preserving package.", "That is, if the elements of the set $S$ are ordered, then the first bar of the $i$ th BC cannot be placed to the right of the first bar of the $j$ th BC if and only if $i<j$ .", "Let the elements of the set $S$ be arbitrarily numbered (ordered) by integers from 1 to $n=|S|$ .", "We denote the resulting ordered set by $P$ .", "In algorithm $G$ , the first BC in $P$ is placed starting from the first cell and excluded from $P$ .", "Then the following procedure is repeated for current list $P$ .", "For the first item in the $P$ , a cell is searched with the minimum number, not to the left of the cell containing the first bar of the previous BC, starting from which it can be placed preserving the feasibility of the packing.", "We exclude the first BC from the list $P$ .", "The process continues until $P\\ne \\emptyset $ .", "Algorithm $G$ constructs a feasible package for a specific permutation of BCs with $O(nl)$ time complexity.", "In [5], we proved the lemma, which can be rephrased for BCPP as follows.", "[5] If each BC is non-increasing, then for a given order of BCs, algorithm $G$ constructs the optimal order-preserving solution to the BCPP.", "The statement of the lemma generalizes the following statement for the BPP [19]: There always exists at least one ordering of items that allows first-fit to produce an optimal solution.", "Since the order does not matter for identical BCs, it is true the following If all BCs are equal non-increasing or non-decreasing, then the algorithm $G$ constructs a package of minimum length.", "In case of non-decreasing BCs, the optimal solution can be built by an algorithm similar to $G$ when the package is constructed from right to left.", "Since 2-BCs are either non-increasing or non-decreasing, the following also holds.", "If all 2-BCs are equal, then the algorithm $G$ constructs an optimal solution to the 2-BCPP." ], [ "2-BCPP", "In this section, we consider the problem in the case where each BC consists of two bars.", "To denote it, we use the 2-BCPP entry.", "Since 2-BCPP is a generalization of the BPP, let us try to use it.", "To do this, put each BC $i$ in a minimal 2-width rectangle that has a height equals $H_i = \\max \\lbrace a_i, b_i\\rbrace $ .", "As a result, we get a set of items $L$ , each of which $i$ has a width equals 2 and characterized only by its height $H_i$ .", "That is, we got BPP, for the solution of which we can use well-known approximate algorithms.", "Using the MFFD algorithm, it is possible to construct a package of items from $L$ using at most $71/60\\ OPT(L)+1$ bins, where $OPT(L)$ is the minimum number of containers for packing items from $L$ [26].", "This packing is feasible for 2-BCPP too.", "Since the height of the minimum bar in each rectangle containing BC can be arbitrarily small, the package constructed by the MFFD algorithm for 2-BCPP can have the density of 2 times less than the packing density of items from $L$ .", "Therefore, the strip length, in which the MFFD algorithm packs all 2-BCs, is limited to $142/60\\ OPT+2 \\approx 2.367\\ OPT+2$ , where $OPT$ is the minimum packing length of 2-BCs.", "Any BC $i\\in S$ fits a rectangle of width $l_i$ and height $H_i$ , and we can pack the resulting set of rectangles $R$ using the known algorithms.", "For example, an algorithm from [13] will construct (without rotation of the rectangles) a package whose length is not more than $(5/3 + \\varepsilon )\\ OPT(R)$ , where $OPT(R)$ is the minimum possible packing length of the rectangles of the set $R$ , and $\\varepsilon > 0$ .", "The resulting solution will be valid for BCPP too and will have a length at most $l (5/3+\\varepsilon )OPT$ , where $OPT$ is the optimal value of the BCPP objective function.", "If all BCs have the same width $l$ , then the MFFD algorithm for BPP will build a package for BCPP with a length of no more than $l(71/60\\ OPT+1)$ .", "Below for 2-BCPP we propose a greedy algorithm, which is somewhat different from $G$ .", "We denote it by $GA$ .", "Let, as before, the list $P$ be an ordered set of elements from $S$ .", "The first element in $P$ is placed in cells 1 and 2 and removed from $P$ .", "Let some BCs are packed and deleted from $P$ .", "Items deleted from $P$ do not move further.", "Then the typical procedure is performed, which consists of the following.", "For each BC in $P$ , we search the leftmost position that does not violate the feasibility of the packing.", "Among BCs that could be placed to the left of all, choose BC with a minimum number, fix its position in the package and delete it from $P$ .", "The algorithm stops when $P = \\emptyset $ .", "Algorithm $G$ builds an order-preserving package.", "As a result of the algorithm $GA$ , BCs with higher numbers can stand in the package to the left of BCs with lower numbers.", "Depending on the order of BCs in $P$ , the algorithm $GA$ builds different solutions.", "Further, we propose algorithm $A$ , using the $GA$ as a procedure, which builds a package of at most $2\\ OPT + 1$ length, where $OPT$ is the smallest possible packing length.", "Algorithm $A$ consists of three stages.", "The first stage is preparatory, and it consists of combining BCs so that, if it is possible, in each 2-BC, the height of at least one bar is greater than $1/2$ .", "For this, the pair of BCs $i$ and $j$ , for which $a_i, b_i, a_j, b_j \\le 1/2$ , are combined into one new BC of width 2 with the height of bars $a_i + a_j$ and $b_i + b_j$ .", "As a result, the set $S$ is transformed: two BCs $i$ and $j$ are removed from it, and one new BC is added.", "The procedure of combining BCs is repeated until $S$ contains the pairs of BCs with both bars no more than $1/2$ .", "As a result, for each BC, except possibly one, the maximum bar's height will be greater than $1/2$ (Fig.", "2$b$ ).", "Figure: Illustration of the operation of algorithm AA.", "a) a set SS; b) a set of combined BCs; c) a package constructed by algorithm AA.At the second stage of algorithm $A$ , we split the updated set $S$ into two subsets $S_1$ and $S_2$ .", "In $S_1$ we include non-increasing, and in $S_2$ non-decreasing BCs (Fig.", "2b).", "Without loss of generality, we assume that a small BC, for which both bars are at most $1/2$ , if it exists, is the last element of the set $S_1$ .", "We pack the set of elements in $S_1$ using the algorithm $GA$ (from left to right), and we pack the set of elements in $S_2$ by the analog of the algorithm $GA$ from right to left.", "Then we get two packages: left and right.", "At the third stage of algorithm $A$ , we shift the right package of the elements of the set $S_2$ to the left to the maximum so as not to violate the feasibility of the packing (Fig.", "2c).", "The time complexity of algorithm $A$ is $O(n^2)$ .", "The first stage of algorithm $A$ can be implemented with a time complexity of $O(n)$ as follows.", "Put $M=\\emptyset $ .", "Browse the BCs in numerical order.", "If the next BC has the first bar greater than the second and more than $1/2$ , then we put it in the set $S_1$ .", "If the next BC has a second bar greater than the first and more than $1/2$ , then we put it in the set $S_2$ .", "If both bars do not exceed $1/2$ and $M=\\emptyset $ , then put the current BC in $M$ and continue viewing.", "If both bars do not exceed $1/2$ and $M\\ne \\emptyset $ , then combine the current BC with BC in $M$ .", "If the resulting new combined BC has at least one bar greater than $1/2$ , then exclude it from $M$ and put it in $S_1$ if the first bar is larger than the second; otherwise we put it in $S_2$ .", "If, after combining the current BC with BC in $M$ , a new BC with both bars not exceeding $1/2$ is obtained, then we leave the new merged BC in $M$ and proceed to consider the next BC in $P$ .", "Thus, as a result of one scan of all BCs, we will form the sets $S_1$ and $S_2$ .", "At the second stage, we pack the elements of the sets $S_1$ and $S_2$ separately using the algorithm $GA$ .", "At each step of the algorithm $GA$ , one BC is added to the already constructed package.", "The complexity of the process of finding the best position for the current BC equals $O(n)$ .", "The number of steps of the algorithm $GA$ is at most $n$ .", "Therefore, the time complexity of the second stage of algorithm $A$ is $O(n^2)$ .", "The complexity of the third stage is $O(1)$ .", "The lemma is proved.", "Algorithm $A$ with time complexity $O(n^2)$ constructs a package for 2-BCPP whose length is at most $2\\ OPT+1$ , where $OPT$ is the minimum length of a strip into which all 2-BCs can be packed.", "We first consider the packing of the set $S_1$ .", "In this set, all BCs are not increasing, and the height of the first bar for all BCs, except, perhaps, the last BC, is greater than $1/2$ .", "Algorithm $GA$ , when packing next BC, shifts it as far as possible to the left.", "Let $k$ BCs are already packed, then the height of the last bar is $b_k$ .", "Consider the $(k+1)$ th BC.", "The following two cases are possible: $a_{k+1} + b_k \\le 1$ ; $a_{k+1} + b_k > 1$ .", "In the first case, we place the first bar of the $(k+1)$ th BC over the second bar of the $k$ th BC.", "In the second case, we put the $(k+1)$ th BC in the next two free cells.", "In both cases, after adding the $(k+1)$ th BC, the total packing density of the first non-empty cells, except, possibly, the last, is greater than $1/2$ .", "For the set $S_2$ , we carry out packing in a similar way from right to left.", "As a result, the packing density of non-empty cells without the first cell of the right package will be more than $1/2$ .", "Let us denote by $b$ the height of the last bar in the left package, and by $a$ the height of the first bar in the right package.", "After shifting the right package to the left to the maximum so that the total packing is feasible, we get one of the following cases.", "$a+b>1$ .", "Then the left and right packages are touching each other, and the density of the whole package is at least 1/2.", "$a+b\\le 1$ .", "Then the right bar of the left package and the left bar of the right package occupy the same cell (Fig.", "2$c$ ).", "In this case, $a+b$ can be at least 1/2 or at most 1/2, and then the density of the whole package is at least 1/2, excluding maybe one cell.", "In any way, we have that the package density of all cells, except, possibly, one, is more than $1/2$ .", "From this, we obtain the statement of the theorem.", "Algorithm $A$ constructs a package whose density is below bounded by $1/2$ , and this estimate is tight, which follows from the following instance.", "Let all BCs be equal with the height of the first bar $a_i = 1$ , and the second $b_i = \\varepsilon $ , $i\\in S$ .", "Then each BC in the optimal package will occupy two cells, and the density of such packing $(1+\\varepsilon )/2$ tends to $1/2$ when $\\varepsilon $ tends to 0.", "However, if instead of density, we compare the length of package constructed by the algorithm $GA$ with the minimum packing length, the difference will be less than two times.", "For the considered instance, for example, the ratio is 1, i.e., $GA$ constructs the optimal solution.", "Therefore, to obtain a more accurate estimate for the ratio, one needs to find a more accurate lower bound for the length of the optimal packing.", "After the first step of algorithm $A$ , several BCs are combined into one BC, which reduces the number of feasible packages.", "In addition, the length of the package constructed by the algorithm $GA$ substantially depends on the order of elements in the list $P$ .", "In the next section, we present the results of a numerical experiment in which the lengths of the package constructed by the algorithm $GA$ for different BCs ordering are compared with the minimal packing length." ], [ "Simulation", "For simulation, all the proposed algorithms were implemented in the Python programming language.", "In the numerical experiment, the input data waer generated randomly.", "As parameters $a_i$ , $b_i$ ($i=1,\\ldots ,n$ ) of the problem (1)-(3), random independent values were uniformly distributed in the segment $(0,1]$ .", "We treated the instances of different sizes $n\\in [10,1000]$ .", "For each value of $n$ , 100 different instances were generated.", "To build the optimal solution to BLP or to find the lower bound for the objective function, we used the IBM ILOG CPLEX 12.10 software package.", "The calculations were carried out on a computer Intel Core i7-3770 3.40GHz 16Gb RAM.", "We examined six different approximate algorithms: the algorithms $A$ , $GA$ , and their modifications.", "To evaluate the influence of the first stage of algorithm $A$ , we use algorithm $A1$ without the first stage.", "The quality of the solution built by algorithm depends significantly on the order of elements in the set $P$ .", "To evaluate the influence of the ordering, we implemented the algorithms $A$ _LO, $GA$ _LO, and $A1$ _LO.", "The abbreviation “LO” means that before packing, we order the BCs lexicographically in non-increasing order of bar's height and then apply the algorithms $A$ , $GA$ , and $A1$ , correspondently.", "Table 1 presents the results of a numerical experiment.", "One can see the benefits of the preliminary lexicographic ordering (LO) of the BCs.", "CPLEX operating time was limited to 20 seconds when $n<500$ , 40 seconds when $n=500$ , 120 seconds when $n=750$ and 300 seconds when $n=1000$ .", "For each size and each algorithm, the table shows the mean values $R_{av}$ and standard deviations $R_{sd}$ of $R$ which is defined as follows.", "If we know the optimal solution, then $R$ is the ratio.", "If CPLEX failed to find an optimal solution, then $R$ is the objective function of an approximate solution divided by the lower bound for objective function yielded by CPLEX during the allotted time.", "For $n\\le 75$ , CPLEX often builds only an approximate solution to the problem, which is tight enough (the average value of $R$ is about 1.11).", "However, when the size increases up to 1000, CPLEX builds the approximate solution significantly worse than the package build by the proposed approximate algorithms.", "We show the graphics of $R$ depending on $n$ for the algorithms CPLEX, $A$ , and $GA$ _LO in Fig.", "3.", "In this figure, we also showed the standard deviation from the mean values of $R$ for different algorithms.", "It is also important to note that for all $n$ , the running time of approximate algorithms did not exceed 1 second.", "For example, when $n=1000$ , the algorithm $A$ built solutions in 0.25 seconds.", "Table: Simulation results: mean values R av R_{av} and standard deviations R sd R_{sd} of RR.Figure: Dependence of RR on the dimension nn.Thus we can conclude that the algorithm $GA$ _LO (with lexicographic order of the BCs) is the best among all considered algorithms and, starting from $n=75$ , it builds solutions more accurately than CPLEX in 5 minutes.", "The algorithm $A1$ turned out to be better in most cases than the algorithm $A$ .", "We would also like to note that this experiment confirmed the significant influence of the ordering of BCs.", "All algorithms with preliminary lexicographic ordering turned out to be significantly more accurate than algorithms without preliminary ordering." ], [ "Conclusion", "We examined the problem BCPP of packing BCs in a strip of minimum length.", "For the particular case, when all BCs have two bars each, the polynomial algorithm $A$ developed by us builds a package whose length does not exceed $2\\ OPT+1$ , where $OPT$ is the minimum possible package length.", "As far as we know, this is the first guaranteed estimate for 2-BCPP, which proves, in particular, that it belongs to the APX class.", "We also conducted a numerical experiment in which we compared the solutions built by our approximate algorithms with the optimal solutions built by the CPLEX package.", "Based on the results of a numerical experiment, we conclude that the algorithm $GA$ _LO, which uses the greedy algorithm $GA$ to pack BCs lexicographically ordered in non-increasing order, significantly outperforms all the others.", "In particular, for the number of BCs $n=1000$ , it constructs a solution in less than 1 second, and the value of the objective function on this solution differs from the optimal value of the objective function by no more than 1.05 times (an average of 1.02 times).", "CPLEX in 5 minutes builds a solution on which the value of the objective function is, on average, 1.24 times worse than the optimal value of the objective function.", "On a larger dimension, CPLEX in 5 minutes does not produce a single feasible solution.", "If we increase the CPLEX operating time, the general trend will not change (see.", "Fig.", "3).", "In the future, we plan, firstly, to reduce the guaranteed estimate for the ratio, and, secondly, to consider BCs with a large number of bars." ] ]

[ [ "Local convergence of the FEM for the integral fractional Laplacian" ], [ "Abstract We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm.", "Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm." ], [ "Introduction", "It is well-known that the rate of convergence of the finite element method (FEM) applied to elliptic PDEs depends on the global regularity of the sought solution.", "However, if the quantity of interest is just the error on some subdomain, one could hope that - provided the solution is smoother locally - the error decays faster.", "This is indeed the case, and the proof of this observation goes back at least to the work [30].", "Since then, the local behavior of FEM approximations has been well understood and various refinements of the original arguments can be found in, e.g., [38], [17].", "In these works the locality of the differential operator is used to prove estimates of the form that the local error is bounded by a local best approximation and a local error in a weaker norm.", "Currently, models of anomalous diffusion are studied in various applications, which gives rise to fractional PDEs, i.e., fractional powers of elliptic operators.", "The numerical approximation of fractional PDEs by the finite element method, as studied here, is an active research field, and we mention, e.g., [29], [1], [3], [10], [6] for global a priori error analyses.", "For other numerical methods applied to fractional PDEs, we refer to [12], [9] for a semigroup approach, [34], [4] for techniques that exploit eigenfunction expansions, as well as the survey articles, [5], [27].", "In comparison to integer order elliptic operators, such as the Laplacian, dealing with the fractional version is much more challenging due to the non-local nature of fractional operators.", "In this regard, fractional operators are similar to the integral operators appearing in the boundary element method (BEM), [32].", "For the BEM, local error estimates and improved convergence results are available as well, see, e.g., [31], [37], [33], [18], which differ from the ones for the FEM in the way that the error contribution in the weaker norm – sometimes called 'slush term' in the literature – is in a global norm instead of a local norm due to the non-local nature of the appearing operators.", "In this article, we provide local error estimates for the FEM applied to the integral fractional Laplacian $(-\\Delta )^s$ for $s \\in (0,1)$ of the form $\\left\\Vert u-u_h \\right\\Vert _{H^1(\\Omega _0)} \\le C \\left(\\inf _{w_h\\in V_h}\\left\\Vert u-w_h \\right\\Vert _{H^1(\\Omega _1)}+ \\left\\Vert u-u_h \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right),$ where $\\Omega _0 \\subset \\subset \\Omega _1$ are open subsets of the computational domain, $V_h$ is the finite element space, $u$ denotes the exact solution of the fractional differential equation, and $u_h \\in V_h$ is its Galerkin approximation.", "A direct consequence of this estimate and a duality argument is that the FEM converges locally in the $H^1$ -norm with order $1-\\varepsilon $ for any $s \\in (0,1)$ and $\\varepsilon >0$ , provided the solution has $H^2$ -regularity locally, and the typical global regularity $u \\in H^{1/2+s-\\varepsilon }(\\Omega )$ .", "In contrast, global convergence in the $H^1$ -norm can only be expected for $s \\in (1/2,1)$ and then is limited to the rate $s-1/2-\\varepsilon $ , see [6].", "Recently and independently a local error analysis similar to ours was derived in [8] using different techniques.", "Our result differs from the estimates [8] in two ways: First, while [8] provides local estimates in the energy norm, we additionally study the stronger local $H^1$ -norm.", "Second, the slush term in [8] is in a different norm, the $L^2$ -norm, whereas we obtain the $H^{s-1/2}$ -norm.", "For $s<1/2$ this gives a stronger estimate, but for $s>1/2$ a weaker estimate.", "With our techniques the slush term could also be weakened to even weaker norms (such as the $L^2$ -norm for $s>1/2$ ), but we expressly chose the $H^{s-1/2}$ -norm in the slush term, as – using a duality argument – weaker norms would not give better convergence rates due to the limited regularity of the dual problem.", "The paper is structured as follows: Section  provides the model problem, the discretization by a lowest order Galerkin method and the main result, Theorem REF , which presents the local error estimate in the $H^1$ -norm, (REF ).", "If the solution is smoother locally, the improvement in the local convergence rates are stated in Corollary REF .", "Section  provides the proofs of the main results as well as the corresponding result for the energy norm.", "Finally, the numerical examples in Section  confirm the sharpness of the theoretical local convergence rates of our main result.", "Concerning notation: For bounded, open sets $\\omega \\subset \\mathbb {R}^d$ integer order Sobolev spaces $H^t(\\omega )$ , $t \\in \\mathbb {N}_0$ , are defined in the usual way.", "For $t \\in (0,1)$ , fractional Sobolev spaces are given in terms of the seminorm $|\\cdot |_{H^t(\\omega )}$ and the full norm $\\Vert \\cdot \\Vert _{H^t(\\omega )}$ by $|v|^2_{H^t(\\omega )} = \\int _{x \\in \\omega } \\int _{y \\in \\omega } \\frac{|v(x) - v(y)|^2}{\\left|x-y \\right|^{d+2t}}\\,dx\\,dy,\\qquad \\Vert v\\Vert ^2_{H^t(\\omega )} = \\Vert v\\Vert ^2_{L^2(\\omega )} + |v|^2_{H^t(\\omega )},$ where we denote the Euclidean distance in $\\mathbb {R}^d$ by $\\left|\\;\\cdot \\; \\right|$ .", "Moreover, for bounded Lipschitz domains $\\Omega \\subset \\mathbb {R}^d$ , we define the spaces $\\widetilde{H}^{t}(\\Omega ) := \\lbrace u \\in H^t(\\mathbb {R}^d) \\,: \\, u\\equiv 0 \\; \\text{on} \\; \\mathbb {R}^d \\backslash \\overline{\\Omega } \\rbrace $ of $H^t$ -functions with zero extension, equipped with the norm $\\left\\Vert v \\right\\Vert _{\\widetilde{H}^{t}(\\Omega )}^2 := \\left\\Vert v \\right\\Vert _{H^t(\\Omega )}^2 + \\left\\Vert v/\\rho ^t \\right\\Vert _{L^2(\\Omega )}^2,$ where $\\rho (x)$ is the distance of a point $x \\in \\Omega $ from the boundary $\\partial \\Omega $ .", "An equivalent norm is the full space norm $H^t(\\mathbb {R}^d)$ of the zero extension of $u$ .", "Throughout this work, we will frequently view functions in $\\widetilde{H}^t(\\Omega )$ as elements of $H^t(\\mathbb {R}^d)$ through the zero extension.", "For $t \\in (0,1)\\backslash \\lbrace \\frac{1}{2}\\rbrace $ , the norms $\\left\\Vert \\cdot \\right\\Vert _{\\widetilde{H}^{t}(\\Omega )}$ and $\\left\\Vert \\cdot \\right\\Vert _{H^{t}(\\Omega )}$ are equivalent, [21].", "Furthermore, for $t > 0$ we denote by $H^{-t}(\\Omega )$ the dual space of $\\widetilde{H}^t(\\Omega )$ .", "For $t \\in \\mathbb {R}$ , we denote by $H^t_{loc}(\\mathbb {R}^d)$ the distributions on $\\mathbb {R}^d$ whose restriction to any ball $B_R(0)$ is in $H^t(B_R(0))$ .", "As usual, we write $\\left< \\cdot ,\\cdot \\right>_{L^2(\\Omega )}$ for the duality pairing that extends the $L^2(\\Omega )$ inner product.", "We note that there are several different ways to define the fractional Laplacian $(-\\Delta )^s$ .", "A classical definition on the full space ${\\mathbb {R}}^d$ is in terms of the Fourier transformation ${\\mathcal {F}}$ , i.e., $({\\mathcal {F}} (-\\Delta )^s u)(\\xi ) = |\\xi |^{2s} ({\\mathcal {F}} u)(\\xi )$ .", "A consequence of this definition is the mapping property, (see, e.g., [5]) $(-\\Delta )^s: H^t(\\mathbb {R}^d) \\rightarrow H^{t-2s}(\\mathbb {R}^d), \\qquad t \\ge s,$ where the Sobolev spaces $H^t(\\mathbb {R}^d)$ , $t \\in \\mathbb {R}$ , are defined in terms of the Fourier transformation, [28].", "Alternative, equivalent definitions of $(-\\Delta )^s$ exist, e.g., via spectral, semi-group, or operator theory, [26] or via singular integrals.", "Specifically, the integral fractional Laplacian can alternatively be defined pointwise for sufficiently smooth functions $u$ as the principal value integral $(-\\Delta )^su(x) := C(d,s) \\; \\text{P.V.}", "\\int _{\\mathbb {R}^d}\\frac{u(x)-u(y)}{\\left|x-y \\right|^{d+2s}} \\, dy \\quad \\text{with} \\quad C(d,s):= - 2^{2s}\\frac{\\Gamma (s+d/2)}{\\pi ^{d/2}\\Gamma (-s)},$ where, $\\Gamma (\\cdot )$ denotes the Gamma function." ], [ "The model problem", "Let $\\Omega \\subset \\mathbb {R}^d$ be a bounded domain.", "We consider the fractional differential equation $(-\\Delta )^su &= f \\qquad \\text{in}\\, \\Omega , \\\\u &= 0 \\quad \\quad \\, \\text{in}\\, \\Omega ^c:=\\mathbb {R}^d \\backslash \\overline{\\Omega },$ where $s \\in (0,1)$ and $f \\in H^{-s}(\\Omega )$ is a given right-hand side.", "Equation (REF ) is understood as in weak form: Find $u \\in \\widetilde{H}^s(\\Omega )$ such that $a(u,v):= \\left< (-\\Delta )^s u,v \\right>_{L^2(\\mathbb {R}^d)} = \\left< f,v \\right>_{L^2(\\Omega )}\\qquad \\forall v \\in \\widetilde{H}^s(\\Omega ).$ The bilinear form $a$ has the alternative representation (cf., e.g., [26]) $a(u,v) =\\frac{C(d,s)}{2} \\int \\int _{\\mathbb {R}^d\\times \\mathbb {R}^d}\\frac{(u(x)-u(y))(v(x)-v(y))}{\\left|x-y \\right|^{d+2s}} \\, dx \\, dy\\qquad \\forall u,v \\in \\widetilde{H}^s(\\Omega ).$ Existence and uniqueness of $u \\in \\widetilde{H}^s(\\Omega )$ follow from the Lax–Milgram Lemma for any $f \\in H^{-s}(\\Omega )$ .", "The bilinear form $a$ induces an invertible operator $\\mathcal {A}: \\widetilde{H}^s(\\Omega ) \\rightarrow H^{-s}(\\Omega )$ .", "Our analysis hinges on the regularity pickup of certain dual problems.", "We formulate this as an assumption: Assumption 2.1 For the domain $\\Omega \\subset \\mathbb {R}^d$ and some $0< \\varepsilon < \\min \\lbrace s/2,1-s\\rbrace $ there holds the shift theorem for $\\mathcal {A}$ : $f \\in H^{1/2-s-\\varepsilon }(\\Omega ) \\quad \\Rightarrow \\quad u = \\mathcal {A}^{-1} f \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )$ together with the a priori estimate $\\left\\Vert u \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\le C \\left\\Vert f \\right\\Vert _{H^{1/2-s-\\varepsilon }(\\Omega )}.$ The constant $C>0$ depends only on $\\Omega $ , $d$ , $s$ , and $\\varepsilon $ .", "Remark 2.2 For smooth domains $\\Omega \\subset \\mathbb {R}^d$ with $\\partial \\Omega \\in C^\\infty $ , the shift theorem in Assumption REF holds for any $\\varepsilon >0$ , see, e.g., [22].", "For polygonal Lipschitz domains, which are of interest in applications, a similar shift theorem is mentioned in [7] as part of the forthcoming work [11]." ], [ "Discretization", "We consider a regular triangulation $h$ (in the sense of Ciarlet, [14]) of $\\Omega $ consisting of open simplices that is also $\\gamma $ -shape regular in the sense $\\max _{T \\in h} \\big ( \\operatornamewithlimits{diam}(T) / |T|^{1/d} \\big ) \\le \\gamma < \\infty .$ Here, $\\operatornamewithlimits{diam}(T):=\\sup _{x,y\\in T}|x-y| =:h_T$ denotes the Euclidean diameter of $T$ , whereas $|T|$ is the $d$ -dimensional Lebesgue volume.", "Additionally, we assume that $h$ is quasi-uniform with mesh width $h := \\max _{T \\in h} h_T$ .", "For an element $T \\in h$ , we define the element patch as $\\omega _T := \\operatornamewithlimits{interior}\\Big (\\bigcup _{T^{\\prime } \\in T)}\\overline{T^{\\prime }}\\Big ) \\quad \\text{with} \\quad T):=\\lbrace T^{\\prime }\\in \\ell \\,:\\, \\overline{T^{\\prime }}\\cap \\overline{T} \\ne \\emptyset \\rbrace .$ Similarly, for a function $\\eta \\in C_0^\\infty (\\mathbb {R}^d)$ , we write $\\omega _\\eta := \\operatornamewithlimits{interior}\\Big (\\bigcup _{T^{\\prime } \\in \\eta )}\\overline{T^{\\prime }}\\Big ) \\quad \\text{with} \\quad \\eta ):=\\lbrace T^{\\prime }\\in \\ell \\,:\\, \\overline{T^{\\prime }}\\cap \\operatornamewithlimits{supp}\\eta \\ne \\emptyset \\rbrace $ for all elements of the triangulation that have a non-empty intersection with the support of the function.", "For the discretization of (REF ), we consider the lowest order Galerkin method.", "More precisely, for $T \\in h$ , we denote the space of all affine functions on $T$ by $\\mathcal {P}^1(T)$ .", "The spaces of $h$ -piecewise affine and globally continuous functions are then defined as $S^{1,1}(h) := \\lbrace u \\in H^1(\\Omega ) \\,:\\, u|_{T} \\in \\mathcal {P}^1(T) \\text{ for all } T \\in h\\rbrace \\quad \\text{and} \\quad S^{1,1}_0(h) := S^{1,1}(h) \\cap H_0^1(\\Omega ).$ Using $S^{1,1}_0(h) \\subset \\widetilde{H}^s(\\Omega )$ as ansatz and test space, we seek a finite element solution $u_h \\in S^{1,1}_0(h)$ such that $a(u_h, v_h) = \\left< f,v_h \\right>_{L^2(\\Omega )}\\quad \\text{for all } v_h \\in S^{1,1}_0(h).$ The Lax–Milgram Lemma provides unique solvability of (REF )." ], [ "The main results", "The following theorem is the main result of this article.", "It estimates the local FEM-error in the $H^1$ -norm and the energy-norm by the local best approximation and a global error in a weaker norm.", "Theorem 2.3 Let $h$ be a quasi-uniform mesh and let Assumption REF be valid.", "Let $u$ solve (REF ) and $u_h$ be its Galerkin approximation solving (REF ).", "Let $\\Omega _0 \\subset \\Omega _{1/5} \\subset \\Omega _{2/5} \\subset \\Omega _{3/5} \\subset \\Omega _{4/5} \\subset \\Omega _1 \\subseteq \\Omega $ be given open sets such that $R:=\\operatornamewithlimits{dist}(\\Omega _0,\\partial \\Omega _1) > 0$ satisfies $\\frac{h}{R} \\le \\frac{1}{10}$ and $\\operatornamewithlimits{dist}(\\Omega _{j/5},\\partial \\Omega _{(j+1)/5}) = R/5$ for $j = 0,\\dots , 4$ .", "Let $\\eta , \\widetilde{\\eta } \\in C_0^\\infty (\\mathbb {R}^d)$ be cut-off functions satisfying $\\eta \\equiv 1$ on $\\Omega _0$ , $\\omega _\\eta \\subset \\Omega _{1/5}$ , $\\widetilde{\\eta }\\equiv 1$ on $\\Omega _{4/5}$ and $\\omega _{\\widetilde{\\eta }} \\subset \\Omega _1$ .", "Then $\\left\\Vert \\eta (u-u_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\le C \\left( \\operatornamewithlimits{inf}_{v_h \\in S^{1,1}_0(h)} \\left\\Vert \\widetilde{\\eta }(u- v_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} +\\left\\Vert u-u_h \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right),$ where the constant $C>0$ depends only on $\\Omega , \\Omega _0,\\Omega _1,R,d,s$ , and the $\\gamma $ -shape regularity of $h$ .", "Assume $u \\in H^1(\\Omega _1)$ .", "Then, $\\left\\Vert u-u_h \\right\\Vert _{H^1(\\Omega _0)} \\le C \\left( \\operatornamewithlimits{inf}_{v_h \\in S^{1,1}_0(h)} \\left\\Vert u- v_h \\right\\Vert _{H^1(\\Omega _1)} +\\left\\Vert u-u_h \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right),$ where the constant $C>0$ depends only on $\\Omega , \\Omega _0,\\Omega _1,R,d,s,\\varepsilon $ , and the $\\gamma $ -shape regularity of $h$ .", "Assuming additional regularity for the solution locally, the following corollary provides optimal rates for the local FEM-error.", "Corollary 2.4 With the assumptions of Theorem REF , let $\\Omega _1 \\subset \\Omega _2 \\subset \\Omega $ with $\\operatornamewithlimits{dist}(\\Omega _1,\\partial \\Omega _2) > 0$ .", "Let $\\varepsilon >0$ be such that Assumption REF holds.", "(i) Let $u \\in \\widetilde{H}^{s+\\alpha }(\\Omega )\\cap H^{s+\\beta }(\\Omega _2)$ with $0<\\alpha ,\\beta $ .", "Then, $\\left\\Vert \\eta (u-u_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\le C h^{\\min \\lbrace 1/2+\\alpha -\\varepsilon ,\\beta \\rbrace }.$ (ii) Let $u \\in \\widetilde{H}^{s+\\alpha }(\\Omega )\\cap H^{1+\\beta }(\\Omega _2)$ with $0<\\alpha ,\\beta $ .", "Then, $\\left\\Vert u-u_h \\right\\Vert _{H^1(\\Omega _0)} \\le C h^{\\min \\lbrace 1/2+\\alpha -\\varepsilon ,\\beta \\rbrace }.$ Here, the constants $C>0$ depend only on $\\Omega ,\\Omega _0,\\Omega _1,R,\\operatornamewithlimits{dist}(\\Omega _1,\\partial \\Omega _2),d,s,\\alpha ,\\beta ,$ the $\\gamma $ -shape regularity of $h$ , and $\\varepsilon $ .", "For sufficiently smooth right-hand sides $f$ , solutions of (REF ) can be expected to be in $H^{s+1/2-\\varepsilon }(\\Omega )$ for any $\\varepsilon >0$ (cf.", "the shift theorem of Assumption REF ), which gives $\\alpha = 1/2-\\varepsilon $ in Corollary REF .", "However, typically, solutions are smoother on any subdomain $\\Omega _2 \\subset \\Omega $ (cf.", "Lemma REF ) that satisfies $\\operatornamewithlimits{dist}(\\Omega _2, \\partial \\Omega )>0$ .", "For $u \\in H^{2}(\\Omega _2)$ , i.e., $\\beta = 1$ in the second statement of the Corollary REF , this leads to convergence of order $\\mathcal {O}(h^{1-2\\varepsilon })$ in the $H^1$ -norm locally.", "Remark 2.5 Corollary REF gives sharp local convergence results both in the $H^1$ -norm and the energy norm.", "For sufficiently high local regularity, both estimates give the same rate of convergence locally.", "This is due to the fact that in this case the slush term in Theorem REF dominates and both local error estimates employ the same slush term.", "However, weakening the norm of the slush term does not improve the convergence rates, since the duality arguments used (see the proof of Corollary REF ) to estimate the $H^{s-1/2}(\\Omega )$ -norm already exploits the maximal regularity of the dual problem available.", "We also mention that local estimates in the $L^2$ -norm are possible, but, for the same reason, the rate of convergence locally in $L^2$ for locally smooth solutions is not better than in the energy or $H^1$ -norm.", "We also refer to the numerical results in Section  for the sharpness of these observations." ], [ "The fractional Laplacian and the Caffarelli-Silvestre extension", "A key tool in the proof of a similar result for the BEM in [18] was the use of properties of the (single- or double-layer) potentials or, more precisely, a Caccioppoli type inequality.", "This interior regularity result allowed us to control derivatives of the potentials.", "For the fractional Laplacian a similar idea can be employed, where the role of the potential is replaced by the Caffarelli-Silvestre extension problem, cf., [15]: The fractional Laplacian can be understood as a Dirichlet-to-Neumann operator of a degenerate elliptic PDE on a half space in $\\mathbb {R}^{d+1}$ : Given $v \\in \\widetilde{H}^s(\\Omega )$ , let $U = U(x,\\mathcal {Y})$ solve for $s = 1-2\\alpha \\in (-1,1)$ $\\operatornamewithlimits{div}(\\mathcal {Y}^\\alpha \\nabla U) &= 0 \\;\\quad \\quad \\text{in} \\; \\mathbb {R}^d \\times (0,\\infty ), \\\\U(\\cdot ,0) & = v \\,\\quad \\quad \\text{in} \\; \\mathbb {R}^d.$ (The solution $U$ is unique by requiring $U$ to be in the Beppo-Levi space ${\\mathcal {B}}^1_\\alpha (\\mathbb {R}^d \\times \\mathbb {R}^+)$ introduced below.)", "Then, the fractional Laplacian can be recovered as the Neumann data of the extension problem in the sense of distributions, [15], [16]: $-d_s \\lim _{\\mathcal {Y}\\rightarrow 0^+} \\mathcal {Y}^\\alpha \\partial _\\mathcal {Y} U(x,\\mathcal {Y}) = (-\\Delta )^s v,\\qquad d_s = 2^{1-2s}\\left|\\Gamma (s) \\right|/\\Gamma (1-s).$ The natural Hilbert space for weak solutions of equation (REF ) is a weighted Sobolev space.", "For measurable subsets $\\omega \\subset \\mathbb {R}^d \\times \\mathbb {R}^+$ , we define the weighted $L^2$ -norm $\\left\\Vert U \\right\\Vert _{L^2_\\alpha (\\omega )}^2 := \\int _{\\omega } \\mathcal {Y}^{\\alpha }\\left|U(x,\\mathcal {Y}) \\right|^2 dx \\, d\\mathcal {Y}$ and denote by $L^2_\\alpha (\\omega )$ the space of square-integrable functions with respect to the weight $\\mathcal {Y}^\\alpha $ .", "The Caffarelli-Silvestre extension is conveniently described in terms of the Beppo-Levi space ${\\mathcal {B}}^1_{\\alpha }(\\mathbb {R}^d \\times \\mathbb {R}^+):= \\lbrace U \\in {\\mathcal {D}}^\\prime (\\mathbb {R}^{d} \\times \\mathbb {R}^+)\\,|\\,\\nabla U \\in L^2_\\alpha (\\mathbb {R}^d \\times \\mathbb {R}^+)\\rbrace $ .", "Elements of ${\\mathcal {B}}^1_\\alpha (\\mathbb {R}^d \\times \\mathbb {R}^+)$ are in fact in $L^2_{\\rm loc}(\\mathbb {R}^d \\times \\mathbb {R}^+)$ and one can give meaning to their trace at $\\mathcal {Y} = 0$ , which is denoted $\\operatorname{tr} U$ .", "Recalling $\\alpha = 1-2s$ , one has in fact $\\operatorname{tr} U \\in H^s_{\\rm loc}(\\mathbb {R}^d)$ (see, e.g., [25])." ], [ "Proof of Theorem ", "In order to make the proof of the main result more accessible, we sketch the main ingredients in the following, details are given in lemmas below.", "We also fix the notation for this section in the following listings.", "Throughout this section, we use the notation $\\lesssim $ to abbreviate $\\le $ up to a generic constant $C>0$ that does not depend on critical parameters in our analysis such as the mesh width $h$ .", "Localization with cut-off functions: Let $\\omega _0 \\subset \\omega _1 \\subset \\mathbb {R}^d$ be arbitrary open sets.", "We use cut-off functions with the properties $\\eta \\in C_0^\\infty (\\mathbb {R}^d), \\;\\; \\eta \\equiv 1 \\;\\; \\text{on} \\;\\omega _0, \\;\\; \\operatornamewithlimits{supp}\\eta \\subset \\omega _{1},\\text{ and }\\left\\Vert \\eta \\right\\Vert _{W^{t,\\infty }(\\omega _1)} \\lesssim \\frac{1}{\\operatornamewithlimits{dist}(\\omega _0,\\partial \\omega _{1})^t}\\text{ for } t\\in \\lbrace 1,2\\rbrace .$ In the following, whenever we use cut-off functions, they have the above properties, only the sets $\\omega _0$ and $\\omega _1$ will be specified.", "Mapping properties of commutators: Let the commutator of $\\mathcal {A}:\\widetilde{H}^s(\\Omega ) \\rightarrow H^{-s}(\\Omega )$ and an arbitrary cut-off function $\\eta $ be defined as the mapping $\\varphi \\mapsto \\eta (\\varphi ) := [\\mathcal {A},\\eta ](\\varphi ) := \\mathcal {A}(\\eta \\varphi ) - \\eta \\mathcal {A}(\\varphi ).$ The commutator $\\eta $ can be seen as a smoothed, localized version of $\\mathcal {A}$ : Lemma REF shows the improved mapping property $\\eta : \\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{1-s}(\\Omega ).$ We also use commutators of the full-space versions of the fractional Laplacian $(-\\Delta )^s$ with cut-off functions $\\eta $ defined by $\\varphi \\mapsto \\widetilde{_\\eta (\\varphi ) := [(-\\Delta )^s,\\eta ] := (-\\Delta )^s (\\eta \\varphi ) - \\eta (-\\Delta )^s \\varphi }$ Superapproximation: For $t \\in [0,1]$ , there is a linear operator $J_h : H^t(\\Omega )\\rightarrow S^{1,1}(h)$ such that we have for $v_h \\in S^{1,1}(h)$ and an arbitrary cut-off function $\\eta $ $\\left\\Vert \\eta v_h - J_h(\\eta v_h) \\right\\Vert _{H^t(\\Omega )} \\lesssim h^{2-t} \\left\\Vert v_h \\right\\Vert _{H^1(\\omega _\\eta )}.$ Here, we gain one power of $h$ , since $v_h$ is a discrete function.", "Remark REF gives an example of such an operator.", "Stability of the Galerkin projection: We define the Galerkin projection $\\Pi : \\widetilde{H}^s(\\Omega ) \\rightarrow S^{1,1}(h)$ by $a(\\Pi u,v_h) = a(u,v_h) \\quad \\forall v_h \\in S^{1,1}(h).$ For $t\\in [s,1]$ , we have by Lemma  REF $\\left|\\Pi v \\right|_{H^t(\\Omega )} \\lesssim \\left|v \\right|_{H^t(\\Omega )}.$ Remark 3.1 The Scott-Zhang projection $\\mathcal {J}_h: H^1(\\Omega ) \\rightarrow S^{1,1}(h)$ , introduced in [35], has the desired superapproximation properties: Let $v_h \\in S^{1,1}(\\mathcal {T}_h)$ , then $\\left|v_h \\right|_{H^2(T)} = 0$ for all elements $T \\in h$ , which for $t \\in [0,1]$ leads to $\\left\\Vert \\eta v_h - \\mathcal {J}_h(\\eta v_h) \\right\\Vert _{H^t(\\Omega )}^2 &\\lesssim h^{4-2t}\\sum _{T \\in \\eta )}\\left|\\eta v_h \\right|_{H^2(T)}^2 \\nonumber \\\\&\\lesssim h^{4-2t}\\sum _{T \\in \\eta )}\\Vert \\eta \\Vert _{W^{2,\\infty }(\\mathbb {R}^d)}^2\\left\\Vert v_h \\right\\Vert _{L^2(T)}^2 +\\Vert \\eta \\Vert _{W^{1,\\infty }(\\mathbb {R}^d)}^2\\left\\Vert \\nabla v_h \\right\\Vert _{L^2(T)}^2 \\nonumber \\\\&\\lesssim h^{4-2t}\\left\\Vert v_h \\right\\Vert ^2_{H^1(\\omega _\\eta )},$ where, in the last step, the bound on the derivatives of $\\eta $ from (REF ) was used.", "Together with Céa's Lemma and an inverse inequality, see, e.g., [20], this also implies $\\left\\Vert \\eta v_h - \\Pi (\\eta v_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim \\left\\Vert \\eta v_h - \\mathcal {J}_h(\\eta v_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\lesssim h^{2-s}\\left\\Vert v_h \\right\\Vert _{H^1(\\omega _\\eta )} \\lesssim h \\left\\Vert v_h \\right\\Vert _{H^s(\\omega _\\eta )},$ i.e., superapproximation properties of the Galerkin projection in the energy norm.", "The following lemma provides useful mapping properties of the commutator between the fractional Laplacian and a cut-off function as well as mapping properties for the commutator of second order.", "Lemma 3.2 Let $\\eta \\in C_0^\\infty (\\mathbb {R}^d)$ and let $\\eta $ be the commutator defined in (REF ) and $\\widetilde{_\\eta be the commutator defined in (\\ref {eq:commutatorFS}).\\begin{enumerate}[(i)]\\item The commutator \\widetilde{_\\eta :\\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{1-s}(\\mathbb {R}^d) is a bounded linear operator.\\item For the commutator \\mathcal {C}_\\eta , we have{\\begin{@align*}{1}{-1}\\eta : \\widetilde{H}^s(\\Omega ) \\rightarrow H^{1-s}(\\Omega ) \\quad \\text{ and by symmetry } \\quad \\eta : \\widetilde{H}^{s-1}(\\Omega ) \\rightarrow H^{-s}(\\Omega ).\\end{@align*}}An interpolation argument also gives{\\begin{@align*}{1}{-1}\\eta : \\widetilde{H}^{s-1+\\theta }(\\Omega ) \\rightarrow H^{-s+\\theta }(\\Omega ).", "\\qquad \\theta \\in [0,1].\\end{@align*}}\\item The commutator of second order \\widetilde{_{\\eta ,\\eta } is defined by\\widetilde{_{\\eta ,\\eta } \\varphi := \\widetilde{_\\eta (\\eta \\varphi ) - \\eta \\widetilde{_\\eta (\\varphi ) and a bounded linear operator{\\begin{@align*}{1}{-1}\\widetilde{_{\\eta ,\\eta } : \\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{2-s}(\\mathbb {R}^d).", "}\\end{@align*}}\\item For the commutator {\\eta ,\\eta } defined by{\\eta ,\\eta } \\varphi := \\eta (\\eta \\varphi ) - \\eta \\eta (\\varphi ), we have{\\begin{@align*}{1}{-1}{\\eta ,\\eta } : \\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{2-s}(\\Omega ) \\quad \\text{ and by symmetry and interpolation } \\quad {\\eta ,\\eta } : \\widetilde{H}^{s-1/2}(\\Omega ) \\rightarrow H^{3/2-s}(\\Omega ).\\end{@align*}}}}\\begin{proof}The commutator \\widetilde{_\\eta = [(-\\Delta )^s,\\eta ] and the commutator of second order \\widetilde{_{\\eta ,\\eta } have the representations(for sufficiently smooth e \\in \\widetilde{H}^{s}(\\Omega )){\\begin{@align*}{1}{-1}\\widetilde{_\\eta e (x) &= \\text{P.V.}", "\\int _{\\mathbb {R}^d} \\frac{\\eta (x)-\\eta (y)}{\\left|x-y \\right|^{d+2s}} e(y) dy, \\\\\\widetilde{_{\\eta ,\\eta } e(x) &= \\int _{\\mathbb {R}^d} \\frac{(\\eta (x)-\\eta (y))^2}{\\left|x-y \\right|^{d+2s}} e(y) dy.", "}}\\emph {Proof of (\\ref {item:lem:commutator-i}):}Using Taylor expansion, we may write for n \\in \\lbrace 1,2,3\\rbrace {\\begin{@align*}{1}{-1}\\eta (x)-\\eta (y) = \\sum _{\\alpha \\in \\mathbb {N}_0^d\\colon |\\alpha | \\le n} \\frac{1}{\\alpha !}", "D^\\alpha \\eta (x) (x-y)^\\alpha + R_n(x,y),\\end{@align*}}where the smooth remainder is O(|x - y|^{n+1}).Inserting this into the representation of \\widetilde{_\\eta shows that we have to analyze convolution type operators of the form\\kappa _\\alpha \\star e, where \\kappa _\\alpha (x) = \\frac{x^\\alpha }{\\left|x \\right|^{d+2s}} for some \\alpha \\in \\mathbb {N}_0^d with |\\alpha | > 0.Convolution-type operators of that kind are pseudodifferential operators, \\cite {taylor96}, and, in fact, the Fourier transform of\\kappa _\\alpha can be computed explicitly.By, e.g.,\\cite [Chap.~{II}, Sec.~3.3]{gelfand64}, we have for the Fourier transform{\\begin{@align}{1}{-1}({\\mathcal {F}} \\frac{1}{|z|^{d-t}})(\\zeta ) &= c_{t,d} \\left|\\zeta \\right|^{-t},\\qquad t \\ne -2m, \\quad m \\in \\mathbb {N}_0,\\\\({\\mathcal {F}} \\ln |z|)(\\zeta ) &= c_{0,d}^{\\prime } |\\zeta |^{-d} + c_{0,d}\\delta (\\zeta ) = c_{0,d}^{\\prime } |\\zeta |^{-d} + c_{0,d}\\mathcal {F}(1)(\\zeta ),\\end{@align}}where \\delta (\\cdot ) is the Dirac delta function.A special role is played by the case s = 1/2 for which the Riesz transform arises:{\\begin{@align}{1}{-1}{\\mathcal {F}} \\; \\text{P.V.}", "\\frac{z_i}{|z|^{d+1}} & = c^{\\prime } \\frac{\\zeta _i}{|\\zeta |}.\\end{@align}}}\\emph {1.~step:}Let e\\in \\widetilde{H}^s(\\Omega ), then \\operatornamewithlimits{supp}e \\subset \\overline{\\Omega }.For x with large \\left|x \\right|,the representation of the commutator \\widetilde{\\mathcal {C}}_\\eta e shows that it is a smooth function that decays like r^{-(d+2s)}.Consequently, in order to show that that \\widetilde{_\\eta e \\in H^{1-s}(\\Omega ), it suffices to assert themapping property \\widetilde{_\\eta :\\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{1-s}_{loc}(\\mathbb {R}^d).The same argument also applies to the commutator of second order, where it sufficesto show \\widetilde{_{\\eta ,\\eta } : \\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{2-s}_{loc}(\\mathbb {R}^d).", "}\\emph {2.~step:} Analysis of the remainder R_n:The remainder induces an operator with kernel r_n(x,y) = R_n(x,y)/|x - y|^{d+2s}.", "Its x-derivative satisfies{\\begin{@align*}{1}{-1}|\\partial _x r_n(x,y)|& \\le C |x - y|^{-(d-n+2s)}.\\end{@align*}}By the mapping properties of the Riesz potential (cf., e.g., \\cite [Lemma~{7.12}]{gilbarg-trudinger77a}), we therefore get the mapping propertyL^2(\\Omega ) \\rightarrow H^1_{loc}(\\mathbb {R}^d) provided 2s-n < 0, i.e., n \\ge 1 for s <1/2 and n \\ge 2 for s \\ge 1/2.", "}For the second derivative, we similarly have{\\begin{@align*}{1}{-1}|\\partial ^2_{x} r_n(x,y)| &\\le C |x - y|^{-(d-n+1+2s)},\\end{@align*}}and the mapping properties of the Riesz potential imply the mapping propertyL^2(\\Omega ) \\rightarrow H^2_{loc}(\\mathbb {R}^d) provided 1+2s-n < 0, i.e., n \\ge 2 for s <1/2 and n \\ge 3 for s \\ge 1/2.", "}\\emph {3.~step (estimating \\kappa _\\alpha for \\left|\\alpha \\right|=1):}For s \\ne 1/2 we note\\frac{z_i}{|z|^{d+2s}} = -\\frac{1}{d+2s-2} \\partial _{z_i} \\frac{1}{|z|^{d+2s-2}}.Let e \\in C^{\\infty }_0(\\Omega ).Using integration by parts in the first order term of the Taylor expansion gives for the principal value partupon setting c_{s,d}:= -(d+2s-2){\\begin{@align*}{1}{-1}c_{s,d} \\text{P.V.}", "\\int _{\\mathbb {R}^d} \\kappa _\\alpha (x-y) e(y) dy &=\\lim _{\\varepsilon \\rightarrow 0 } \\int _{\\mathbb {R}^d \\backslash B_\\varepsilon (x)}-\\nabla _{y} \\frac{1}{|x-y|^{d-(2-2s)}}\\cdot e_i e(y) dy \\\\&= \\lim _{\\varepsilon \\rightarrow 0 } \\int _{\\partial B_\\varepsilon (x)}\\frac{1}{|x-y|^{d-(2-2s)}} e_i\\cdot \\nu (y) e(y) ds_y + \\int _{\\mathbb {R}^d \\backslash B_\\varepsilon (x)}\\frac{1}{|x-y|^{d-(2-2s)}} \\partial _{y_i}e(y) dy,\\end{@align*}}where e_i is the i-th unit vector and \\nu (\\cdot ) denotes the outer normal vector to B_{\\varepsilon }(x).Using Taylor expansion, we write e(y)= e(x) + \\widetilde{r}_1(x,y), where the remainder \\widetilde{r}_1 = \\mathcal {O}(\\left|x-y \\right|).", "Then, the boundary integral converges to zero since{\\begin{@align*}{1}{-1}\\left|\\int _{\\partial B_\\varepsilon (x)}\\frac{1}{|x-y|^{d-(2-2s)}} e_i\\cdot \\nu (y) e(y) ds_y \\right| &\\le \\left| e(x)\\int _{\\partial B_\\varepsilon (x)}\\varepsilon ^{-d+2-2s} e_i\\cdot \\nu (y) ds_y \\right| \\\\&\\quad +\\left|\\int _{\\partial B_\\varepsilon (x)}\\varepsilon ^{-d+2-2s} \\nu (y)\\widetilde{r}_1(x,y) ds_y \\right| \\\\&= \\left|\\int _{\\partial B_\\varepsilon (x)}\\varepsilon ^{-d+2-2s} \\nu (y)\\widetilde{r}_1(x,y) ds_y \\right| \\\\&\\lesssim \\varepsilon ^{-d+3-2s} \\int _{\\partial B_\\varepsilon (x)} 1 dy \\lesssim \\varepsilon ^{-d+3-2s+d-1} =\\varepsilon ^{2-2s} \\rightarrow 0,\\end{@align*}}where the integral over e_i \\nu vanishes by symmetry.We conclude,{\\begin{@align*}{1}{-1}c_{s,d}\\text{P.V.}", "\\int _{\\mathbb {R}^d} \\kappa _\\alpha (x-y) e(y) dy &= \\int _{\\mathbb {R}^d}\\frac{1}{|x-y|^{d-(2-2s)}} \\partial _{y_i}e(y) dy,\\end{@align*}}so that by (\\ref {eq:landkof-1}) we get for |\\alpha | = 1\\begin{equation}c_{s,d}{\\mathcal {F}}(\\kappa _{\\alpha }\\star e)(\\zeta ) = {\\mathcal {F}}\\left(\\frac{1}{|z|^{d-(2-2s)}}\\star \\partial _{z_i} e\\right)(\\zeta ) = c_{\\alpha ,d} |\\zeta |^{-(2-2s)} \\left|\\zeta \\right| {\\mathcal {F}}(e) = c_{\\alpha ,d} |\\zeta |^{-1+2s}{\\mathcal {F}}(e),\\end{equation}which shows that \\kappa _{\\alpha }\\star e is an operator of order 2s-1.\\end{@align*}}\\emph {4.~step: The case 0 < s < 1:} Selecting n = 1, Steps 1 -- 3, show that [(-\\Delta )^s,\\eta ] has the mapping property\\widetilde{H}^s(\\Omega ) \\rightarrow H^{1-s}(\\mathbb {R}^d).", "}}\\emph {5.~step: The case 1/2 < s < 1:} We use n = 2.", "Again,the remainder R_2 maps L^2(\\Omega ) \\rightarrow H^1_{loc}(\\mathbb {R}^d) by Step~2 and for|\\alpha | = 1, the operator \\kappa _{\\alpha } is an operator of order 2s-1 by Step~3.The operator \\kappa _{\\alpha } with |\\alpha | = 2 is structurally similar to the case |\\alpha | = 1 sincewe can write\\begin{subequations}{\\begin{@align}{1}{-1}\\frac{z_i^2}{|z|^{d+2s}} & =- \\frac{1}{(d+2s-2)(d+2s-4)}\\left(\\partial ^2_{z_i}\\frac{1}{\\left|z \\right|^{d+2s-4}}+(d+2s-4)\\frac{1}{\\left|z \\right|^{d+2s-2}}\\right),\\\\\\frac{z_i z_j}{|z|^{d+2s}} & =- \\frac{1}{(d+2s-2)(d+2s-4)}\\left(\\partial _{z_i}\\partial _{z_j}\\frac{1}{\\left|z \\right|^{d+2s-4}}\\right).\\end{@align}}\\end{subequations}Using integration by parts, the first term on the right-hand side of both equations can be treated as in Step~3.", "For the second term on the left-hand side in the first formula, applying the Fourier transformation directly andusing (\\ref {eq:landkof-1}), provides that \\kappa _{\\alpha }\\star e is an operator of order 2s-2 for \\left|\\alpha \\right| =2.\\end{proof}}\\emph {6.~step: the case s = 1/2:}We use n = 2.", "For |\\alpha | = 1, the kernel \\kappa _\\alpha is the Riesz transform that is, bythe representation (\\ref {eq:riesz-transform}), an operator of order 0 = 1-2s.", "For |\\alpha | = 2, (\\ref {eq:derivative2})can be used for d \\notin \\lbrace 1,3\\rbrace showing that \\kappa _\\alpha induces an operator of order 2s-s.In the case d=1, the kernel \\kappa _\\alpha with |\\alpha | = 2 is bounded by 1.", "For d=3and |\\alpha | = 2, we use{\\begin{@align*}{1}{-1}\\frac{z_i^2}{|z|^{d+1}} = -\\partial _{z_i}^2 \\ln |z| + \\frac{1}{\\left|z \\right|^2}.\\end{@align*}}By (\\ref {eq:landkof-2}) and (\\ref {eq:landkof-1}) and integration by parts, we have as in (\\ref {eq:fourierorder}){\\begin{@align*}{1}{-1}{\\mathcal {F}}(\\kappa _{\\alpha }\\star e)(\\zeta ) &= -{\\mathcal {F}}\\left(\\ln |z|\\star \\partial ^2_{z_i} e\\right)(\\zeta ) + {\\mathcal {F}}\\left(\\left|z \\right|^{-2}\\star e\\right)(\\zeta ) \\\\ &= c_{\\alpha ,d}^{\\prime } |\\zeta |^{-3} {\\mathcal {F}}(\\partial ^2_{z_i} e) + c_{\\alpha ,d} {\\mathcal {F}(1)}{\\mathcal {F}}(\\partial ^2_{z_i} e) + \\tilde{c}_{\\alpha ,d} |\\zeta |^{-1} {\\mathcal {F}}(e) = \\widehat{c}_{\\alpha ,d} |\\zeta |^{-1}{\\mathcal {F}}(e),\\end{@align*}}which implies that \\kappa _{|\\alpha |} induces an operator of order 2s-2.", "}Altogether, this gives the boundedness of \\widetilde{_\\eta :\\widetilde{H}^{s}(\\Omega ) \\rightarrow H^{1-s}(\\mathbb {R}^d) for all s \\in (0,1).", "}\\emph {Proof of (\\ref {item:lem:commutator-iii}):} We use n=3.", "Taylor expansion andthe representation of \\widetilde{\\mathcal {C}}_{\\eta ,\\eta } shows that, due to (\\eta (x)-\\eta (y))^2 in the numerator, the leading order term produces \\kappa _\\alpha with \\left|\\alpha \\right| = 2 and leads to an operator of order 2s-2 as in Step~5 in the proof of (\\ref {item:lem:commutator-i}).The terms with \\left|\\alpha \\right| = 3 are structurally similar to those for \\left|\\alpha \\right| = 2.We have{\\begin{@align*}{1}{-1}\\frac{z_i^3}{|z|^{d+2s}} & =\\frac{1}{(d+2s-2)(d+2s-4)(d+2s-6)}\\left(\\partial ^3_{z_i}\\frac{1}{\\left|z \\right|^{d+2s-6}}-(d+2s-6)3\\partial _{z_i}\\frac{1}{\\left|z \\right|^{d+2s-4}}\\right),\\end{@align*}}and similar expressions hold for the mixed derivatives and the logarithm (for the case s=1/2).", "Therefore,we again can use integration by parts and (\\ref {eq:landkof-1}), (\\ref {eq:landkof-2}) to obtain that \\kappa _{\\alpha }\\star e is an operator of order 2s-3 for \\left|\\alpha \\right| =3.", "}The remainder R_3 maps L^2(\\Omega ) \\rightarrow H^2_{loc}(\\mathbb {R}^d) by Step~2 in the proof of (\\ref {item:lem:commutator-i}) and together this shows that \\widetilde{\\mathcal {C}}_{\\eta ,\\eta } is an operator of order 2s-2.\\end{enumerate}\\emph {Proof of (\\ref {item:lem:commutator-ii}) and (\\ref {item:lem:commutator-iv}):} As the operators \\widetilde{_\\eta and \\widetilde{\\mathcal {C}}_{\\eta ,\\eta } are extensionsof the operators \\eta and \\mathcal {C}_{\\eta ,\\eta } respectively, the boundedness \\eta : \\widetilde{H}^s(\\Omega ) \\rightarrow H^{1-s}(\\Omega ) follows from (\\ref {item:lem:commutator-i}) and the boundedness {\\eta ,\\eta } : \\widetilde{H}^s(\\Omega ) \\rightarrow H^{2-s}(\\Omega ) follows from (\\ref {item:lem:commutator-iii}).The symmetry of \\eta then immediately implies \\eta : \\widetilde{H}^{s-1}(\\Omega ) \\rightarrow H^{-s}(\\Omega ) as a bounded operator.", "Finally, both these mapping properties imply\\eta :L^2(\\Omega ) \\rightarrow H^{1-2s}(\\Omega ) by interpolation.", "The same argument gives the additional mapping property of the commutator of second order.", "}}$ We start with the proof of the first statement in Theorem REF , the local error estimate in the energy norm.", "[Proof of Theorem REF , (i)] We write using the Galerkin projection $\\Pi $ from (REF ), the symmetry of $\\mathcal {A}$ , and the definition of $\\eta $ from (REF ) $\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}^2 &\\lesssim \\left< \\mathcal {A}(\\eta e),\\eta e \\right>_{L^2(\\Omega )} =\\left< \\mathcal {A}(\\eta e-\\Pi (\\eta e)),\\eta e \\right>_{L^2(\\Omega )} + \\left< \\mathcal {A}(\\Pi (\\eta e)),\\eta e \\right>_{L^2(\\Omega )} \\\\&= \\left< \\mathcal {A}(\\eta e-\\Pi (\\eta e)),\\eta e \\right>_{L^2(\\Omega )} + \\left< \\Pi (\\eta e), \\eta e \\right>_{L^2(\\Omega )} +\\left< \\Pi (\\eta e), \\eta \\mathcal {A}e \\right>_{L^2(\\Omega )}=:{\\large {\\normalsize \\text{I}}} + {\\large {\\normalsize \\text{II}}}+ {\\large {\\normalsize \\text{III}}}.$ The mapping properties of $\\mathcal {A}$ , the stability of the Galerkin projection, the superapproximation property (REF ), $\\omega _\\eta \\subset \\Omega _1$ , and an inverse inequality give $\\left|{\\large {\\normalsize \\text{I}}} \\right| &= \\left|\\left< \\eta e-\\Pi (\\eta e),\\mathcal {A}(\\eta e) \\right>_{L^2(\\Omega )} \\right|\\\\ &\\le \\left\\Vert \\mathcal {A}(\\eta e) \\right\\Vert _{H^{-s}(\\Omega )}\\left(\\left\\Vert \\eta u - \\Pi (\\eta u) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert \\eta u_h - \\Pi (\\eta u_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\right) \\\\&\\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\left(\\left\\Vert \\eta u \\right\\Vert _{\\widetilde{H}^s(\\Omega )}+h^{1/2}\\left\\Vert u_h \\right\\Vert _{H^{s-1/2}(\\Omega _1)}\\right) \\\\&\\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\left((1+h^{1/2})\\left\\Vert \\eta u \\right\\Vert _{\\widetilde{H}^s(\\Omega )}+h^{1/2}\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right).$ The mapping properties of the commutator from Lemma REF as well as the stability of the Galerkin projection imply $\\left|{\\large {\\normalsize \\text{II}}} \\right| = \\left|\\left< \\Pi (\\eta e), \\eta e \\right>_{L^2(\\Omega )} \\right| \\lesssim \\left\\Vert \\Pi (\\eta e) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\left\\Vert \\eta e \\right\\Vert _{H^{-s}(\\Omega )}\\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ It remains to estimate $\\large {\\normalsize \\text{III}}$ .", "Here, we need an additional cut-off function $\\widehat{\\eta }$ satisfying $\\widehat{\\eta }\\equiv 1$ on $\\omega _\\eta $ and $\\omega _{\\widehat{\\eta }}\\subset \\Omega _{4/5}$ .", "Due to the assumption $h/R \\le 1/10$ a cut-off function with these properties exists.", "With the Galerkin orthogonality and the support properties of the Scott-Zhang projection $\\mathcal {J}_h$ , which imply $\\operatornamewithlimits{supp}\\mathcal {J}_h(\\eta \\Pi (\\eta e)) \\subset \\omega _\\eta $ , we obtain $\\left|\\large {\\normalsize \\text{III}} \\right|&=\\left|\\left< \\mathcal {A}e,\\eta \\Pi (\\eta e) \\right>_{L^2(\\Omega )} \\right|=\\left|\\left< \\widehat{\\eta }\\mathcal {A}e,\\eta \\Pi (\\eta e)-\\mathcal {J}_h(\\eta \\Pi (\\eta e)) \\right>_{L^2(\\Omega )} \\right|.$ Superapproximation of the Scott-Zhang projection (REF ), Lemma REF and the inclusions $\\widetilde{H}^{s-1}(\\Omega ) \\subset H^{s-1}(\\Omega ) \\subset H^{s-1/2}(\\Omega )$ , an inverse estimate, and the stability of the Galerkin projection lead to $\\left|\\large {\\normalsize \\text{III}} \\right|&\\lesssim \\left\\Vert \\widehat{\\eta }\\mathcal {A}e \\right\\Vert _{H^{-s}(\\Omega )}\\left\\Vert \\eta \\Pi (\\eta e)-\\mathcal {J}_h(\\eta \\Pi (\\eta e)) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\\\&\\lesssim \\left\\Vert \\mathcal {A}(\\widehat{\\eta }e) - \\mathcal {C}_{\\widehat{\\eta }}e \\right\\Vert _{H^{-s}(\\Omega )}h\\left\\Vert \\Pi (\\eta e) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} \\\\&\\stackrel{\\text{Lem.~\\ref {lem:commutator}}}{\\lesssim } \\left(\\left\\Vert \\widehat{\\eta }e \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right) h \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} \\\\&\\lesssim \\left(h\\left\\Vert \\widehat{\\eta }u \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+h^{1/2}\\left\\Vert u_h \\right\\Vert _{H^{s-1/2}(\\omega _{\\widehat{\\eta }})}+h\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right)\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\\\&\\lesssim \\left(h\\left\\Vert \\widehat{\\eta }u \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+h^{1/2}\\left\\Vert \\widetilde{\\eta } u_h \\right\\Vert _{H^{s-1/2}(\\Omega )}+h\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right)\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} \\\\&\\lesssim (h^{1/2}+h)\\left(\\left\\Vert \\widetilde{\\eta }u \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right)\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}$ Putting the estimates of the three terms together and using $h \\lesssim 1$ , we obtain $\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}^2 \\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\left(\\left\\Vert \\widetilde{\\eta }u \\right\\Vert _{\\widetilde{H}^s(\\Omega )}+\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right).$ Applying this estimate to $u - v_h$ for arbitrary $v_h \\in S^{1,1}_0(h)$ instead of $u$ and noting that the corresponding Galerkin error is $e = (u-v_h) +(v_h-u_h)$ leads to the desired estimate in the energy norm.", "Remark 3.3 The previous proof can easily be modified to account for non-uniform meshes $.Using the approximation properties of the Scott-Zhang projection in $ L2$ and $ H1$ from (\\ref {eq:superapproxSZ}) together with a cut-off function $$ with $ 1$ on $ 1/5$ and $ *supp2/5$, we can obtain$$\\left\\Vert \\eta v_h - \\Pi (\\eta v_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim h_{\\rm max}(\\Omega _1) \\left\\Vert \\chi v_h \\right\\Vert _{H^s(\\Omega )}$$by replacing the inverse inequality in (\\ref {eq:superapproxGalerkinProjection}) with an interpolation argument.", "Here, $ hmax(1) $ denotes the maximal mesh width in $ 1$, and the hidden constant additionally depends on $ 1/5$.Applying the inverse estimates for the terms {\\rm \\large {\\normalsize \\text{I}}} and {\\rm \\large {\\normalsize \\text{III}}} on non-uniform meshes produces factors $ hmin(1)-1/2$.$ Therefore, combining approximation properties with this inverse inequality in the same way as in the previous proof gives factors $\\frac{h_{\\rm max}(\\Omega _1) }{h_{\\rm min}(\\Omega _1)^{1/2} }$ instead of the factor $h^{1/2}$ for quasi-uniform meshes.", "Thus, if we have $\\frac{h_{\\rm max}(\\Omega _1) }{h_{\\rm min}(\\Omega _1)^{1/2} }\\le C$ with a constant independent of the local mesh sizes, the previous arguments give the sharp local error estimate $\\left\\Vert \\eta (u-u_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\le C \\left( \\operatornamewithlimits{inf}_{v_h \\in S^{1,1}_0(} \\left\\Vert \\widetilde{\\eta }(u- v_h) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} +\\left\\Vert u-u_h \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right).$ Here, $\\eta ,\\widetilde{\\eta }$ are cut-off functions as in Theorem REF and the assumption $6h\\le R$ for the quasi-uniform case is replaced by $6h_{\\rm max}(\\Omega _1) \\le R$ .", "In order to counteract the singular behavior of solutions of fractional PDEs near the boundary, graded meshes with $h_{\\rm min}(\\Omega ) \\sim h_{\\rm max}(\\Omega )^2$ are commonly employed, [1], [8].", "In the following, we focus on the case of local estimates in the stronger $H^1$ -seminorm stated in Theorem REF .", "We start with a stronger stability estimate for the Galerkin projection.", "Lemma 3.4 Let $\\Pi $ be the Galerkin projection defined in (REF ), and let $\\eta \\in C_0^{\\infty }(\\mathbb {R}^d)$ be an arbitrary cut-off function.", "For $v \\in H^1(\\Omega )$ , we have $\\left|\\eta v - \\Pi (\\eta v) \\right|_{H^{1}(\\Omega )} \\le C \\left|\\eta v \\right|_{H^{1}(\\Omega )}.$ For $v_h \\in S^{1,1}(\\mathcal {T}_h)$ , we have $\\left|\\eta v_h - \\Pi (\\eta v_h) \\right|_{H^1(\\Omega )} \\le C h\\left\\Vert v_h \\right\\Vert _{H^1(\\omega _\\eta )}.$ The constant $C>0$ depends only on $\\Omega $ , $d, s,$ the $\\gamma $ -shape regularity of $\\mathcal {T}_h$ , and $\\Vert \\eta \\Vert _{W^{2,\\infty }(\\mathbb {R}^d)}$ .", "Let $\\mathcal {J}_h$ be the Scott-Zhang projection from Remark REF .", "With an inverse inequality, see, e.g., [20], as well as Céa's Lemma, the superapproximation property (REF ) implies $\\left|\\eta v_h- \\Pi (\\eta v_h) \\right|_{H^{1}(\\Omega )}&\\le &\\left|\\eta v_h- \\mathcal {J}_h(\\eta v_h) \\right|_{H^{1}(\\Omega )} +\\left|\\mathcal {J}_h(\\eta v_h)- \\Pi (\\eta v_h) \\right|_{H^{1}(\\Omega )} \\\\&\\lesssim & h\\left\\Vert v_h \\right\\Vert _{H^{1}(\\omega _\\eta )} +h^{s-1}\\left\\Vert \\mathcal {J}_h(\\eta v_h)- \\Pi (\\eta v_h) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\nonumber \\\\&\\lesssim & h\\left\\Vert v_h \\right\\Vert _{H^{1}(\\omega _\\eta )} + h^{s-1}\\left(\\left\\Vert \\mathcal {J}_h(\\eta v_h)- \\eta v_h \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert \\eta v_h- \\Pi (\\eta v_h) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\right) \\\\&\\lesssim & h\\left\\Vert v_h \\right\\Vert _{H^{1}(\\omega _\\eta )} + h^{s-1}\\left\\Vert \\mathcal {J}_h(\\eta v_h)- \\eta v_h \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\lesssim h\\left\\Vert v_h \\right\\Vert _{H^{1}(\\omega _\\eta )}.$ For $v \\in H^1(\\Omega )$ , the same argument - replacing superapproximation with the classical approximation properties of the Scott-Zhang projection - gives (REF ).", "For the proof of the $H^1$ -error estimate, we exploit additional interior regularity provided by the following lemma.", "Lemma 3.5 Let $\\widehat{\\Omega }\\subset \\subset \\Omega $ be open and $\\eta $ be a cut-off function with $\\operatornamewithlimits{supp}\\eta \\subset \\widehat{\\Omega }$ .", "Assume $f \\in H^t(\\widehat{\\Omega })\\cap H^{-s}(\\Omega )$ for some $-s\\le t\\le 1-s$ and let $u$ solve (REF ).", "Then, $ \\eta u \\in H^{2s+t}(\\mathbb {R}^d)$ and $\\left\\Vert \\eta u \\right\\Vert _{H^{2s+t}(\\mathbb {R}^d)} \\lesssim \\left\\Vert u \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} + \\left\\Vert \\eta f \\right\\Vert _{H^{t}(\\mathbb {R}^d)}.$ By definition of the commutator $\\widetilde{\\mathcal {C}}_\\eta $ , the product $\\eta u$ solves the equation $(-\\Delta )^s(\\eta u) + \\eta u = \\eta (-\\Delta )^s u + \\widetilde{\\mathcal {C}}_{\\eta } u + \\eta u = \\eta f + \\widetilde{\\mathcal {C}}_{\\eta } u + \\eta u =: \\widetilde{f}$ Since $\\eta f \\in H^t(\\mathbb {R}^d)$ and $\\widetilde{\\mathcal {C}}_{\\eta } u \\in H^{1-s}(\\mathbb {R}^d)$ by Lemma REF and $\\eta u \\in H^s(\\mathbb {R}^d)$ , we have $\\widetilde{f} \\in H^{\\min (t,1-s,s)}(\\mathbb {R}^d)$ .", "Applying the Fourier transformation to (REF ) as in the proof of Lemma REF noting that all objects live in the full-space $\\mathbb {R}^d$ , gives $(1+\\left|\\zeta \\right|^{2s}) \\mathcal {F}(\\eta u) = \\mathcal {F}(\\widetilde{f})$ and the Fourier definition of Sobolev norms give $\\eta u \\in H^{2s+\\min (t,1-s,s)}(\\mathbb {R}^d)$ .", "Bootstrapping this argument until the minimum in the exponent is given by $t$ , then shows the claimed local regularity.", "The norm estimate follows directly from the above equation and the Fourier definition of Sobolev norms and the mapping properties of the commutator $\\widetilde{\\mathcal {C}}_\\eta $ from Lemma REF .", "We will repeatedly employ the $L^2$ -orthogonal projection $\\Pi ^{L^2}:L^2(\\Omega ) \\rightarrow S^{1,1}_0(h)$ defined by $\\left< \\phi -\\Pi ^{L^2}\\phi ,\\xi _h \\right>_{L^2(\\Omega )} = 0 \\qquad \\forall \\xi _h \\in S^{1,1}_0(h).$ There hold the following global stability and approximation estimates.", "Lemma 3.6 Let $s\\in [0,1]$ and $h$ a quasi-uniform mesh.", "Then, the $L^2$ -projection $\\Pi ^{L^2}$ defined in (REF ) is bounded in $\\widetilde{H}^s(\\Omega )$ for $s\\in [0,1]$ , i.e., there exists a constant $C_s>0$ depending only on $s$ such that $\\left\\Vert \\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\le C_s \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\quad \\text{ for all }\\phi \\in \\widetilde{H}^s(\\Omega ).$ Furthermore, there hold the approximation estimates in negative norms $\\left\\Vert \\phi -\\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} &\\le C h \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\quad \\text{ for } s\\in (1/2,1],\\\\\\left\\Vert \\phi -\\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} &\\le C h^{1+s-2\\varepsilon } \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\quad \\text{ for } s\\in [0,1/2],\\, \\varepsilon \\in [0,1/2].$ The stability estimate is a well-known property of the $L^2$ -orthogonal projection, cf.", "[13].", "The approximation estimate for $s\\in (1/2,1]$ follows by a simple duality argument and the fact that $H^{1-s}(\\Omega ) = \\widetilde{H}^{1-s}(\\Omega )$ with equivalent norms.", "In the case $s\\in [0,1/2]$ , we use $s-1<-1/2+\\varepsilon $ and again norm equivalence and duality, $\\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} & \\lesssim \\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{-1/2+\\varepsilon }(\\Omega )}\\lesssim \\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{H^{-1/2+\\varepsilon }(\\Omega )} \\lesssim h^{1+s-2 \\varepsilon } \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}.$ The $L^2$ -orthogonal projection has the advantage that it localizes very well, which is observed, e.g., in [38].", "The following lemma summarizes the local stability and approximation properties of the $L^2$ -projection used in the proof of our main result.", "Lemma 3.7 Let $\\Pi ^{L^2}$ be the $L^2$ -projection defined in (REF ) and $D_0\\subset D_1 \\subset \\Omega $ be nested open sets with $\\operatornamewithlimits{dist}(D_0,\\partial D_1)\\ge c_0h$ .", "Additionally, let $D_0$ be a union of elements of $h$ .", "Then, for $\\phi \\in H^1(D_1)\\cap L^2(\\Omega )$ , we have local stability $\\left\\Vert \\Pi ^{L^2}\\phi \\right\\Vert _{H^1(D_0)} \\le C\\left(\\left\\Vert \\phi \\right\\Vert _{H^1(D_1)} + e^{-c/h}\\left\\Vert \\phi \\right\\Vert _{L^2(\\Omega )}\\right)$ and approximation properties $\\left\\Vert \\phi -\\Pi ^{L^2}\\phi \\right\\Vert _{H^t(D_0)} \\le C\\left( h^{1-t}\\left\\Vert \\phi \\right\\Vert _{H^1(D_1)} + e^{-c/h}\\left\\Vert \\phi \\right\\Vert _{L^2(\\Omega )}\\right)$ for $t \\in [0,1]$ , where the constants $c,C>0$ depend only on $\\Omega ,d,D_0,D_1,c_0,t$ , and the $\\gamma $ -shape regularity of $h$ .", "Let $D_{1/2}$ be a set satisfying $D_0\\subset D_{1/2}\\subset D_1$ with $\\operatornamewithlimits{dist}(D_0,\\partial D_{1/2})\\ge c_0h/2$ and $\\operatornamewithlimits{dist}(D_{1/2},\\partial D_{1})\\ge c_0h/2$ .", "We use [38], which states that discrete functions $\\phi _h \\in S^{1,1}(h)$ satisfying the orthogonality $\\left< \\phi _h,\\xi _h \\right>_{L^2(\\Omega )} = 0$ for all $\\xi _h \\in S^{1,1}_0(h) $ , $\\operatornamewithlimits{supp}\\xi _h \\subset D_{1/2}$ are exponentially small locally, i.e., $\\left\\Vert \\phi _h \\right\\Vert _{L^2(D_0)} \\lesssim e^{-c_1/h} \\left\\Vert \\phi _h \\right\\Vert _{L^2(D_{1/2})}.$ We employ a cut-off function $\\eta $ with $\\eta \\equiv 1$ on $D_{1/2}$ and $\\operatornamewithlimits{supp}\\eta \\subset D_1$ .", "By definition of the $L^2$ -projection, we compute $\\left< \\Pi ^{L^2}\\phi ,\\xi _h \\right>_{L^2(\\Omega )} = \\left< \\phi ,\\xi _h \\right>_{L^2(\\Omega )} = \\left< \\eta \\phi ,\\xi _h \\right>_{L^2(\\Omega )} =\\left< \\Pi ^{L^2}(\\eta \\phi ),\\xi _h \\right>_{L^2(\\Omega )} \\forall \\xi _h \\in S^{1,1}_0(h), \\operatornamewithlimits{supp}\\xi _h \\subset D_{1/2}.$ Therefore, we may use $\\phi _h = \\Pi ^{L^2}\\phi -\\Pi ^{L^2}(\\eta \\phi )$ and conclude $\\left\\Vert \\Pi ^{L^2}\\phi -\\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{L^2(D_0)} \\lesssim e^{-c_1/h} \\left\\Vert \\Pi ^{L^2}\\phi -\\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{L^2(D_{1/2})}.$ Together with an inverse estimate this gives the local $H^1$ -stability $\\left\\Vert \\Pi ^{L^2}\\phi \\right\\Vert _{H^1(D_0)} &\\le \\left\\Vert \\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{H^1(D_0)} + \\left\\Vert \\Pi ^{L^2}\\phi - \\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{H^1(D_0)}\\\\& \\lesssim \\left\\Vert \\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{H^1(D_0)} + h^{-1}\\left\\Vert \\Pi ^{L^2}\\phi - \\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{L^2(D_0)} \\\\&\\lesssim \\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\Omega )} + h^{-1}e^{-c_1/h}\\left\\Vert \\Pi ^{L^2}\\phi - \\Pi ^{L^2}(\\eta \\phi ) \\right\\Vert _{L^2(D_{1/2})}\\lesssim \\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\Omega )} + e^{-c/h}\\left\\Vert \\phi \\right\\Vert _{L^2(\\Omega )},$ where we used the stability of the $L^2$ -projection both in $L^2$ and $H^1$ , cf.", "[13], and $h^{-1}e^{-c_1/h}\\lesssim e^{-c/h}$ .", "Replacing the $H^1$ -stability with approximation properties in the second to last estimate above, we obtain the local approximation property with the same argument.", "[Proof of Theorem REF (ii)] Throughout the proof we use four cut-off functions $\\chi ,\\zeta ,\\widehat{\\eta },\\eta $ satisfying $\\chi \\equiv 1$ on $\\Omega _0$ and $\\omega _\\chi \\subset \\Omega _{1/5}$ , $\\zeta \\equiv 1$ on $\\Omega _{1/5}$ and $\\omega _\\zeta \\subset \\Omega _{2/5}$ , $\\widehat{\\eta }\\equiv 1$ on $\\Omega _{2/5}$ and $\\omega _{\\widehat{\\eta }} \\subset \\Omega _{3/5}$ and $\\eta \\equiv 1$ on $\\Omega _{4/5}$ and $\\omega _\\eta \\subset \\Omega _{1}$ .", "The assumption $\\frac{h}{R} \\le \\frac{1}{10}$ allows for the construction of such cut-off functions that also satisfy $\\left\\Vert \\chi \\right\\Vert _{W^{1,\\infty }(\\mathbb {R}^d)},\\left\\Vert \\zeta \\right\\Vert _{W^{1,\\infty }(\\mathbb {R}^d)},\\left\\Vert \\widehat{\\eta } \\right\\Vert _{W^{1,\\infty }(\\mathbb {R}^d)}\\left\\Vert \\eta \\right\\Vert _{W^{1,\\infty }(\\mathbb {R}^d)} \\lesssim \\frac{1}{R}$ .", "With the triangle inequality and the $L^2$ -orthogonal projection $\\Pi ^{L^2}$ , we divide the error into three contributions $\\left|e \\right|_{H^1(\\Omega _0)} \\le \\left|\\chi e \\right|_{H^1(\\Omega )} \\le \\left|\\chi (\\eta e- \\Pi (\\eta e)) \\right|_{H^1(\\Omega )} + \\left|\\chi \\Pi ^{L^2}\\phi \\right|_{H^1(\\Omega )} +\\left|\\chi (\\Pi (\\eta e)-\\Pi ^{L^2}\\phi ) \\right|_{H^1(\\Omega )} =:{\\large {\\normalsize \\text{1}}}+{\\large {\\normalsize \\text{2}}}+{\\large {\\normalsize \\text{3}}} ,$ where $\\phi := \\mathcal {A}^{-1}(\\eta {\\eta } e)$ .", "These three terms are estimated as follows.", "Estimate of 1: We use Lemma REF and the triangle inequality to obtain $\\left|\\eta e- \\Pi (\\eta e) \\right|_{H^1(\\Omega )} &\\lesssim \\left|\\eta u \\right|_{H^1(\\Omega )} + h\\left\\Vert u_h \\right\\Vert _{H^1(\\Omega _1)} \\\\&\\lesssim (1+h)\\left\\Vert u \\right\\Vert _{H^1(\\Omega _1)} + h\\left\\Vert e \\right\\Vert _{H^1(\\Omega _1)}.$ Estimate of 2: By Lemma REF we have $\\eta \\mathcal {C}_\\eta e \\in H^{1-s}(\\Omega )$ , and Lemma REF then implies $\\eta \\phi = \\eta \\mathcal {A}^{-1}(\\eta \\mathcal {C}_\\eta e) \\in H^{1+s}(\\mathbb {R}^d)\\subset H^1(\\mathbb {R}^d)$ .", "The local $H^1$ -stability of the Galerkin projection from (REF ), applied with $D_0 = \\omega _\\chi $ and $D_1=\\Omega _{2/5}$ that satisfy $\\operatornamewithlimits{dist}(D_0,\\partial D_1) \\ge R/5 \\ge 2h$ , then implies $\\left|\\chi \\Pi ^{L^2}\\phi \\right|_{H^1(\\Omega )} \\lesssim \\left\\Vert \\phi \\right\\Vert _{H^1(\\Omega _{2/5})} + e^{-c/h} \\left\\Vert \\phi \\right\\Vert _{L^2(\\Omega )} \\lesssim \\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\mathbb {R}^d)} + e^{-c/h} \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )}.$ Lemma REF yields $\\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\mathbb {R}^d)}= \\left\\Vert \\eta \\mathcal {A}^{-1}(\\eta {\\eta } e) \\right\\Vert _{H^1(\\Omega )}&\\lesssim \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} + \\left\\Vert \\eta ^2 {\\eta } e \\right\\Vert _{\\widetilde{H}^{1-2s}(\\Omega )}\\lesssim \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+ \\left\\Vert \\eta {\\eta } e \\right\\Vert _{H^{1-2s}(\\Omega )},$ and as $1-2s<3/2-s$ , the mapping properties of $\\eta $ , ${\\eta ,\\eta }$ from Lemma REF lead to $\\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\mathbb {R}^d)} &\\lesssim \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert {\\eta }(\\eta e) \\right\\Vert _{H^{1-2s}(\\Omega )} +\\left\\Vert {\\eta ,\\eta } e \\right\\Vert _{H^{1-2s}(\\Omega )} \\nonumber \\\\&\\lesssim \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert {\\eta }(\\eta e) \\right\\Vert _{H^{1-2s}(\\Omega )} +\\left\\Vert {\\eta ,\\eta } e \\right\\Vert _{H^{3/2-s}(\\Omega )} \\nonumber \\\\&\\lesssim \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} + \\left\\Vert \\eta e \\right\\Vert _{L^2(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ It remains to bound $\\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}$ .", "The classical mapping property $\\mathcal {A}^{-1}:H^{-s}(\\Omega ) \\rightarrow \\widetilde{H}^s(\\Omega )$ and the mapping properties of $\\eta $ from Lemma REF imply $\\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )}&= \\left\\Vert \\mathcal {A}^{-1}(\\eta {\\eta } e) \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\lesssim \\left\\Vert \\eta {\\eta } e \\right\\Vert _{H^{-s}(\\Omega )} \\lesssim \\left\\Vert e \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} \\lesssim \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ Estimate of 3: Define the discrete function $\\psi _h := \\Pi (\\eta e)- \\Pi ^{L^2}\\phi $ and decompose $\\psi _h = v_h + w_h$ , where $v_h,$ $w_h \\in S^{1,1}_0(h)$ solve $\\left< \\mathcal {A}v_h,\\xi _h \\right>_{L^2(\\Omega )} &= \\left< \\zeta \\mathcal {A}\\psi _h,\\xi _h \\right>_{L^2(\\Omega )} \\quad \\forall \\xi _h \\in S^{1,1}_0(h), \\\\\\left< \\mathcal {A}w_h,\\xi _h \\right>_{L^2(\\Omega )} &= \\left< (1-\\zeta ) \\mathcal {A}\\psi _h,\\xi _h \\right>_{L^2(\\Omega )} \\quad \\forall \\xi _h \\in S^{1,1}_0(h).$ These definitions and the unique solvability of the Galerkin formulation indeed give $\\psi _h = v_h + w_h$ , and we call this the near-field $(v_h)$ /far-field $(w_h)$ splitting of $\\psi _h$.", "The ellipticity of $\\mathcal {A}$ gives the a priori bound $\\left\\Vert v_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim \\left\\Vert \\zeta \\mathcal {A}\\psi _h \\right\\Vert _{H^{-s}(\\Omega )} \\lesssim \\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^s(\\Omega )},$ and the triangle inequality then implies $\\left\\Vert w_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim \\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^s(\\Omega )}$ .", "Moreover, with the $\\widetilde{H}^s(\\Omega )$ -stability of the $L^2$ -projection and (REF ), we have $\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} &= \\left\\Vert \\Pi (\\eta e) - \\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\le \\left\\Vert \\Pi (\\eta e) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert \\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\nonumber \\\\&\\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ Now, with the triangle inequality, we write ${\\large {\\normalsize \\text{3}}} = \\left|\\chi \\psi _h \\right|_{H^1(\\Omega )} \\lesssim \\left\\Vert v_h \\right\\Vert _{H^1(\\Omega )} + \\left|\\chi w_h \\right|_{H^1(\\Omega )},$ and it remains to estimate the near-field and far-field parts.", "To that end, we note first that $\\psi _h$ satisfies a useful equation: For $\\xi _h \\in S^{1,1}_0(h)$ with $\\operatornamewithlimits{supp}\\xi _h \\subset \\overline{\\omega _\\zeta }\\subset \\Omega _{2/5}$ , the Galerkin orthogonality and $\\eta \\equiv 1$ on $\\omega _\\zeta $ imply $0 &= \\left< \\mathcal {A}e,\\xi _h \\right>_{L^2(\\Omega )} = \\left< \\mathcal {A}e,\\eta \\xi _h \\right>_{L^2(\\Omega )} = \\left< \\mathcal {A}(\\eta e)-{\\eta }e, \\xi _h \\right>_{L^2(\\Omega )} =\\left< \\mathcal {A}(\\Pi (\\eta e))-\\eta {\\eta }e, \\xi _h \\right>_{L^2(\\Omega )} \\nonumber \\\\ &=\\left< \\mathcal {A}(\\Pi (\\eta e)-\\mathcal {A}^{-1}(\\eta {\\eta }e)), \\xi _h \\right>_{L^2(\\Omega )} =\\left< \\mathcal {A}\\psi _h, \\xi _h \\right>_{L^2(\\Omega )} + \\left< \\mathcal {A}(\\Pi ^{L^2}\\phi -\\phi ), \\xi _h \\right>_{L^2(\\Omega )} .$ Estimate of the near-field: We exploit locality properties of the near-field to show the estimate $\\left\\Vert v_h \\right\\Vert _{H^1(\\Omega )} \\lesssim \\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ To see this, start with the inverse inequality $\\left\\Vert v_h \\right\\Vert _{H^1(\\Omega )} \\lesssim h^{s-1} \\left\\Vert v_h \\right\\Vert _{H^s(\\Omega )}.$ The mapping property and ellipticity of $\\mathcal {A}$ , the definition of $v_h$ , and (REF ) imply $\\left\\Vert v_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} &= \\sup _{w \\in H^{-s}(\\Omega )}\\frac{\\left< v_h,w \\right>_{L^2(\\Omega )}}{\\left\\Vert w \\right\\Vert _{H^{-s}(\\Omega )}} \\lesssim \\sup _{\\varphi \\in \\widetilde{H}^s(\\Omega )}\\frac{\\left< v_h,\\mathcal {A}\\varphi \\right>_{L^2(\\Omega )}}{\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}} =\\sup _{\\varphi \\in \\widetilde{H}^s(\\Omega )}\\frac{\\left< v_h,\\mathcal {A}\\Pi \\varphi \\right>_{L^2(\\Omega )}}{\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}} \\nonumber \\\\&= \\sup _{\\varphi \\in \\widetilde{H}^s(\\Omega )}\\frac{\\left< \\zeta \\mathcal {A}\\psi _h, \\Pi \\varphi \\right>_{L^2(\\Omega )}}{\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}} \\nonumber \\\\& = \\sup _{\\varphi \\in \\widetilde{H}^s(\\Omega )}\\frac{\\left< \\mathcal {A}\\psi _h, \\zeta \\Pi \\varphi -\\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )}-\\left< \\mathcal {A}(\\Pi ^{L^2}\\phi -\\phi ), \\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )}}{\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}}.$ For the first term in the numerator, we use superapproximation (REF ) to obtain $\\left|\\left< \\mathcal {A}\\psi _h, \\zeta \\Pi \\varphi -\\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )} \\right| \\lesssim \\left\\Vert \\mathcal {A}\\psi _h \\right\\Vert _{H^{-s}(\\Omega )}h\\left\\Vert \\Pi \\varphi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim h \\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^s(\\Omega )}.$ For the second term in the numerator on the right-hand side, we use the cut-off function $\\widehat{\\eta }$ satisfying $\\widehat{\\eta }\\equiv 1$ on $\\omega _\\zeta $ as well as the definition and mapping properties of the commutator $\\mathcal {C}_{\\widehat{\\eta }}$ to estimate $\\left|\\left< \\mathcal {A}(\\Pi ^{L^2}\\phi -\\phi ), \\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )} \\right| &=\\left|\\left< \\widehat{\\eta }\\mathcal {A}(\\Pi ^{L^2}\\phi -\\phi ), \\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )} \\right| \\nonumber \\\\&= \\left|\\left< \\mathcal {A}(\\widehat{\\eta }(\\Pi ^{L^2}\\phi -\\phi ))+\\mathcal {C}_{\\widehat{\\eta }}(\\Pi ^{L^2}\\phi -\\phi ), \\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right>_{L^2(\\Omega )} \\right|\\nonumber \\\\&\\lesssim \\left(\\left\\Vert \\mathcal {A}(\\widehat{\\eta }(\\Pi ^{L^2}\\phi -\\phi )) \\right\\Vert _{H^{-s}(\\Omega )} + \\left\\Vert \\mathcal {C}_{\\widehat{\\eta }}(\\Pi ^{L^2}\\phi -\\phi ) \\right\\Vert _{H^{-s}(\\Omega )}\\right)\\left\\Vert \\mathcal {J}_h(\\zeta \\Pi \\varphi ) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\nonumber \\\\&\\lesssim \\left(\\left\\Vert \\widehat{\\eta }(\\Pi ^{L^2}\\phi -\\phi ) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} + \\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )}\\right)\\left\\Vert \\varphi \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}.$ We conclude $\\left\\Vert v_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim h\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+ \\left\\Vert \\widehat{\\eta }(\\Pi ^{L^2}\\phi -\\phi ) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} + \\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )},$ and in view of (REF ), it remains to bound the last two terms on the right-hand side of (REF ).", "For the first term, we can exploit the localization of the $L^2$ -projection of Lemma REF .", "Applying (REF ) with $D_0 = \\omega _{\\widehat{\\eta }}$ and $D_1 = \\Omega _{4/5}$ that satisfy $\\operatornamewithlimits{dist}(D_0,\\partial D_1) \\ge R/5 \\ge 2h$ , we obtain $\\left\\Vert \\widehat{\\eta }(\\Pi ^{L^2}\\phi -\\phi ) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )} &\\lesssim h^{1-s}\\left\\Vert \\phi \\right\\Vert _{H^1(\\Omega _{4/5})} +e^{-c/h}\\left\\Vert \\phi \\right\\Vert _{L^2(\\Omega )}\\le h^{1-s}\\left\\Vert \\eta \\phi \\right\\Vert _{H^1(\\Omega )} +e^{-c/h}\\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\\\&\\lesssim h^{1-s} \\left( \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )} + \\left\\Vert \\eta e \\right\\Vert _{L^2(\\Omega )} \\right) + e^{-c/h}\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )} ,$ where we have used (REF ) and (REF ) in the last step.", "For the second term on the right-hand side of (REF ), we use the approximation properties of $\\Pi ^{L^2}$ given in Lemma REF .", "For $s>1/2$ , we obtain $\\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} \\lesssim h \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\stackrel{(\\ref {eq:estphihglob})}{\\lesssim } h\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ For the case $s\\le 1/2$ , we use the additional smoothness of $\\phi \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )$ from Assumption REF for $0< \\varepsilon < \\min \\lbrace s/2,1-s\\rbrace $ with right-hand side with $\\eta \\mathcal {C}_{\\eta } e \\in H^{1-s}(\\Omega )$ , and obtain $\\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} &=\\left\\Vert \\mathcal {A}^{-1}(\\eta \\mathcal {C}_{\\eta } e) \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}\\lesssim \\left\\Vert \\eta \\mathcal {C}_{\\eta } e \\right\\Vert _{\\widetilde{H}^{1/2-s-\\varepsilon }(\\Omega )}\\lesssim \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )},$ similarly as in (REF ).", "Lemma REF then gives $\\left\\Vert \\Pi ^{L^2}\\phi -\\phi \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )}\\lesssim h \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}\\lesssim h\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )},$ and combining all estimates, we get the required estimate of the near-field.", "Estimate of the far-field: For the far-field, we exploit additional smoothness properties of a suitable Caffarelli-Silvestre extension problem to show the estimate $\\left|\\chi w_h \\right|_{H^1(\\Omega )} \\lesssim h^{\\min \\lbrace s,1/2\\rbrace -2\\varepsilon }\\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )}+\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )}+\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}$ for $\\varepsilon > 0$ given by Assumption REF .", "Let $w\\in \\widetilde{H}^s(\\Omega )$ solve $\\mathcal {A}w = (1-\\zeta ) \\mathcal {A}\\psi _h$ and write, using the triangle inequality, $\\left|\\chi w_h \\right|_{H^1(\\Omega )} \\le \\left|\\chi w_h - \\Pi (\\chi w_h) \\right|_{H^1(\\Omega )} +\\left|\\Pi (\\chi (w- w_h )) \\right|_{H^1(\\Omega )} + \\left|\\Pi (\\chi w) \\right|_{H^1(\\Omega )} =:{\\large {\\normalsize \\text{a}}}+{\\large {\\normalsize \\text{b}}}+{\\large {\\normalsize \\text{c}}}.$ Superapproximation (REF ) together with an inverse estimate and the stability of the splitting of $\\psi _h$ first yields ${\\large {\\normalsize \\text{a}}} &= \\left|\\chi w_h - \\Pi (\\chi w_h) \\right|_{H^1(\\Omega )}\\lesssim h^{1-s} \\left\\Vert w_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim h^{1-s}\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\\\&\\stackrel{(\\ref {eq:estPsih})}{\\lesssim } h^{1-s}\\left(\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}\\right).$ To bound ${\\large {\\normalsize \\text{b}}}$ we write $e_w := w-w_h$ and apply an inverse estimate to obtain ${\\large {\\normalsize \\text{b}}} = \\left|\\Pi (\\chi (w- w_h )) \\right|_{H^1(\\Omega )} \\lesssim h^{s-1}\\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{H^s(\\Omega )}\\lesssim h^{s-1} \\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}.$ Note that, in fact, $w_h = \\Pi w$ , and hence Galerkin orthogonality, ellipticity of $\\mathcal {A}$ , the definition of the commutator ${\\chi }$ , and superapproximation of the Scott-Zhang projection $\\mathcal {J}_h$ (Remark REF ) lead to $\\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{\\widetilde{H}^s(\\Omega )}^2&\\lesssim \\left< \\mathcal {A}(\\Pi (\\chi e_w)),\\Pi (\\chi e_w) \\right>_{L^2(\\Omega )} = \\left< \\mathcal {A}(\\chi e_w),\\Pi (\\chi e_w) \\right>_{L^2(\\Omega )}=\\left< \\chi \\mathcal {A}e_w + {\\chi } e_w,\\Pi (\\chi e_w) \\right>_{L^2(\\Omega )} \\\\&=\\left< \\mathcal {A}e_w ,\\chi \\Pi (\\chi e_w)-\\mathcal {J}_h(\\chi \\Pi (\\chi e_w)) \\right>_{L^2(\\Omega )} +\\left< {\\chi } e_w,\\Pi (\\chi e_w) \\right>_{L^2(\\Omega )} \\\\&\\lesssim h\\left\\Vert \\mathcal {A}e_w \\right\\Vert _{H^{-s}(\\Omega )} \\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{\\widetilde{H}^s(\\Omega )} +\\left\\Vert {\\chi }e_w \\right\\Vert _{H^{-s}(\\Omega )}\\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\\\&\\stackrel{Lem.~\\ref {lem:commutator}}{\\lesssim } \\left\\Vert \\Pi (\\chi e_w) \\right\\Vert _{\\widetilde{H}^s(\\Omega )}\\left(h\\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}+\\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )}\\right).$ Since $\\psi _h \\in S^{1,1}_0(\\mathcal {T}_h)$ , we observe that $\\psi _h \\in \\widetilde{H}^{3/2-\\varepsilon }(\\Omega )$ for any $\\varepsilon > 0$ and consequently $\\mathcal {A}\\psi _h \\in H^{3/2-2s-\\varepsilon }(\\Omega )$ .", "In view of $s < 1$ , we conclude $(1-\\zeta )\\mathcal {A}\\psi _h \\in H^{1/2-s-\\varepsilon }(\\Omega )$ .", "Assumption REF then gives $w \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )$ .", "A duality argument, again using Assumption REF , the Galerkin orthogonality and approximation properties of the Scott-Zhang projection, provides $\\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )}& \\lesssim \\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{-1/2+s+\\varepsilon }(\\Omega )}= \\sup _{\\varphi \\in H^{1/2-s-\\varepsilon }(\\Omega )} \\frac{\\left< w-w_h,\\varphi \\right>_{L^2(\\Omega )}}{\\left\\Vert \\varphi \\right\\Vert _{H^{1/2-s-\\varepsilon }(\\Omega )}}\\\\&\\lesssim \\sup _{\\widehat{\\varphi }\\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\frac{\\left|\\left< w-w_h,\\mathcal {A}\\widehat{\\varphi } \\right>_{L^2(\\Omega )} \\right|}{\\left\\Vert \\widehat{\\varphi } \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}} =\\sup _{\\widehat{\\varphi }\\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\frac{\\left|\\left< \\mathcal {A}(w-w_h),\\widehat{\\varphi }-\\mathcal {J}_h \\widehat{\\varphi } \\right>_{L^2(\\Omega )} \\right|}{\\left\\Vert \\widehat{\\varphi } \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}}\\\\&\\lesssim h^{1/2-\\varepsilon } \\left\\Vert \\mathcal {A}(w-w_h) \\right\\Vert _{H^{-s}(\\Omega )} \\lesssim h^{1/2-\\varepsilon } \\left\\Vert w-w_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )}.$ With the additional regularity of $w$ , we can estimate $\\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} \\lesssim h^{1/2-\\varepsilon } \\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^s(\\Omega )} &\\lesssim h^{1-2\\varepsilon }\\left\\Vert w \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\nonumber \\\\ &\\lesssim h^{1-2\\varepsilon }\\left\\Vert (1-\\zeta )\\mathcal {A}\\psi _h \\right\\Vert _{H^{1/2-s-\\varepsilon }(\\Omega )}\\lesssim h^{1-2\\varepsilon }\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}.$ We note that $\\Vert e_w\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim \\Vert w\\Vert _{\\widetilde{H}^s(\\Omega )}\\lesssim \\Vert \\psi _h\\Vert _{\\widetilde{H}^{s}(\\Omega )}$ .", "We next distinguish the cases $1/2+s-\\varepsilon \\le 1$ and $1/2+s-\\varepsilon > 1$ .", "For $1/2+s-\\varepsilon \\le 1$ , we use (REF ), stability of the Galerkin projection and the $L^2$ -projection, and (REF ) to obtain ${\\large {\\normalsize \\text{b}}} &\\lesssim h^{s-2\\varepsilon }\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\lesssim h^{s-2\\varepsilon }\\left(\\left\\Vert \\Pi (\\eta e) \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} + \\left\\Vert \\Pi ^{L^2}\\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\right) \\\\&\\stackrel{1/2+s-\\varepsilon \\le 1}{\\lesssim } h^{s-2\\varepsilon } \\left( \\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )} + \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\right)\\stackrel{(\\ref {eq:NFtemp4})}{\\lesssim } h^{s-2\\varepsilon } \\left( \\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )} \\right),$ which is the required estimate for $1/2+s-\\varepsilon \\le 1$ .", "For $1/2+s-\\varepsilon > 1$ we employ an inverse estimate in (REF ) and obtain $\\left\\Vert e_w \\right\\Vert _{\\widetilde{H}^{s-1}(\\Omega )} \\lesssim h^{1-2\\varepsilon }\\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}\\lesssim h^{3/2-s-\\varepsilon }\\left\\Vert \\psi _h \\right\\Vert _{H^{1}(\\Omega )}.$ Using $\\phi \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega ) \\subset H^1(\\Omega )$ and the $H^1$ -stability of the $L^2$ -projection we can estimate $\\left\\Vert \\psi _h \\right\\Vert _{H^{1}(\\Omega )} \\lesssim \\left\\Vert \\Pi (\\eta e) \\right\\Vert _{H^1(\\Omega )} + \\left\\Vert \\phi \\right\\Vert _{H^{1}(\\Omega )} \\lesssim \\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )} + \\left\\Vert \\phi \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\stackrel{(\\ref {eq:NFtemp4})}{\\lesssim }\\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ Inserting everything into (REF ) leads to ${\\large {\\normalsize \\text{b}}} \\lesssim h^{1/2-\\varepsilon }\\left\\Vert \\psi _h \\right\\Vert _{H^1(\\Omega )}\\lesssim h^{1/2-2\\varepsilon }\\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )} + h^{1/2-\\varepsilon }\\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )},$ which is the required estimate for the case $s>1/2$ .", "To estimate c, we denote by $U_{\\rm far}$ the solution of the extension problem (REF ) with data $\\operatorname{tr}U_{\\rm far} = w$ .", "We apply a variation of [19] with open sets $B_x \\times B_y =: B \\subset B^{\\prime } := B_x^{\\prime } \\times B_y^{\\prime }$ and an extension $\\widehat{\\chi }$ of the cut-off function $\\chi $ satisfying $\\operatornamewithlimits{supp}\\widehat{\\chi } \\subset B^{\\prime }$ .", "Instead of prescribing support properties of $\\operatornamewithlimits{tr} U_{\\rm far}$ as in [19], an inspection of the proof therein shows that choosing the same test-function $V$ in the weak formulation of (REF ) for the difference quotient argument works, as long as $\\operatornamewithlimits{tr} V \\cdot (\\mathcal {Y}^\\alpha \\partial _{\\mathcal {Y}}U_{\\rm far})|_{y=0} \\equiv 0$ .", "The said test function $V$ is a second order difference quotient of $\\widehat{\\chi }U_{\\rm far}$ and therefore its trace is supported in $\\Omega _{1/5}$ .", "Since $-d_s(\\mathcal {Y}^\\alpha \\partial _{\\mathcal {Y}}U_{\\rm far})|_{y=0} = \\mathcal {A} w = (1-\\zeta ) \\mathcal {A}\\psi _h,$ the assumption $\\zeta \\equiv 1$ on $\\Omega _{1/5}$ indeed gives $\\operatornamewithlimits{tr} V \\cdot (\\mathcal {Y}^\\alpha \\partial _{\\mathcal {Y}}U_{\\rm far})|_{y=0} \\equiv 0$ and therefore the arguments of [19] imply $\\left\\Vert D_x(\\nabla U_{\\rm far}) \\right\\Vert _{L^2_\\alpha (B)} \\lesssim \\left\\Vert U_{\\rm far} \\right\\Vert _{H^1_\\alpha (B^{\\prime })}.$ The $H^1$ -stability of the Galerkin projection from Lemma REF , the multiplicative trace inequality from [25], and the Lax-Milgram Lemma then give $\\large {\\normalsize \\text{c}} = \\left|\\Pi (\\chi w) \\right|_{H^1(\\Omega )}&\\lesssim \\left|\\chi w \\right|_{H^1(\\Omega )} \\lesssim \\left\\Vert \\nabla _x U_{\\rm far} \\right\\Vert _{L^2_\\alpha (B)} + \\left\\Vert \\nabla _x U_{\\rm far} \\right\\Vert _{L^2_\\alpha (B)}^{(1-\\alpha )/2}\\left\\Vert \\partial _y\\nabla _x U_{\\rm far} \\right\\Vert _{L^2_\\alpha (B)}^{(1+\\alpha )/2} \\\\&\\lesssim \\left\\Vert U_{\\rm far} \\right\\Vert _{H^1_\\alpha (B^{\\prime })} \\lesssim \\left\\Vert w \\right\\Vert _{\\widetilde{H}^s(\\Omega )} \\lesssim \\left\\Vert (1-\\zeta )\\mathcal {A}\\psi _h \\right\\Vert _{H^{-s}(\\Omega )} \\lesssim \\left\\Vert \\psi _h \\right\\Vert _{\\widetilde{H}^{s}(\\Omega )}\\\\&\\stackrel{(\\ref {eq:estPsih})}{\\lesssim }\\left\\Vert \\eta e \\right\\Vert _{\\widetilde{H}^s(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ Putting the estimates for a, b and c and the near-field using $h \\lesssim 1$ together, we obtain $\\left|\\chi e \\right|_{H^1(\\Omega )} \\lesssim \\left\\Vert u \\right\\Vert _{H^1(\\Omega _1)} + h^{\\min \\lbrace s,1/2\\rbrace -2\\varepsilon }\\left\\Vert \\eta e \\right\\Vert _{H^1(\\Omega )}+\\left\\Vert \\eta e \\right\\Vert _{H^s(\\Omega )} + \\left\\Vert e \\right\\Vert _{H^{s-1/2}(\\Omega )}.$ Using the already shown statement (i) of Theorem REF for the third term, iterating the argument for the second term on the right-hand side (which is of higher order due to the additional positive powers of $h$ by choice of $\\varepsilon $ ) together with an inverse estimate, and replacing again $u$ with $u - v_h$ for arbitrary $v_h \\in S^{1,1}_0(h)$ and noting that $e = (u-v_h) +(v_h-u_h)$ leads to the desired estimate.", "With a classical duality argument and exploiting the (local and global) regularity of $u$ , Theorem REF immediately implies Corollary REF .", "[Proof of Corollary REF ] We only show the second statement of the corollary, the first statement follows with exactly the same arguments.", "The assumptions on the local and global regularity directly imply, see, e.g., [1], that $\\left\\Vert u-u_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )} &\\lesssim h^{\\alpha }\\left\\Vert u \\right\\Vert _{\\widetilde{H}^{s+\\alpha }(\\Omega )}, \\\\\\inf _{v_h \\in S^{1,1}_0(h)} \\left|u-v_h \\right|_{H^1(\\Omega _1)} &\\lesssim h^{\\beta }\\left\\Vert u \\right\\Vert _{H^{1+\\beta }(\\Omega _2)}.$ For the global term in Theorem REF and arbitrary $\\varepsilon >0$ , we use a duality argument together with Assumption REF , the Galerkin orthogonality and approximation properties of the Scott-Zhang projection $\\mathcal {J}_h$ to estimate $\\left\\Vert u-u_h \\right\\Vert _{H^{s-1/2}(\\Omega )} &\\lesssim \\left\\Vert u - u_h \\right\\Vert _{\\widetilde{H}^{s-1/2}(\\Omega )}\\lesssim \\left\\Vert u - u_h \\right\\Vert _{\\widetilde{H}^{-1/2+s+\\varepsilon }(\\Omega )}= \\sup _{w\\in H^{1/2-s-\\varepsilon }(\\Omega )} \\frac{\\left< u-u_h,w \\right>_{L^2(\\Omega )}}{\\left\\Vert w \\right\\Vert _{H^{1/2-s-\\varepsilon }(\\Omega )}}\\\\&\\lesssim \\sup _{v\\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\frac{\\left|\\left< u-u_h,\\mathcal {A}v \\right>_{L^2(\\Omega )} \\right|}{\\left\\Vert v \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}} =\\sup _{v\\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )} \\frac{\\left|\\left< \\mathcal {A}(u-u_h),v-\\mathcal {J}_h v \\right>_{L^2(\\Omega )} \\right|}{\\left\\Vert v \\right\\Vert _{\\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )}}\\\\&\\lesssim h^{1/2-\\varepsilon } \\left\\Vert \\mathcal {A}(u-u_h) \\right\\Vert _{H^{-s}(\\Omega )} \\lesssim h^{1/2-\\varepsilon } \\left\\Vert u-u_h \\right\\Vert _{\\widetilde{H}^s(\\Omega )},$ which finishes the proof of the corollary." ], [ "Numerical examples", "We illustrate our theoretical results of the previous sections with a numerical example in two dimensions.", "On the unit circle $\\Omega = B_1(0)$ with constant right-hand side $f = 2^{2s}\\Gamma (1+s)^2$ the exact solution is known to be $u(x) = (1-\\left|x \\right|^2)_+^s \\qquad \\text{where} \\;\\; g_+ = \\max \\lbrace g,0\\rbrace ,$ see, e.g., [5].", "We choose the subdomain $\\Omega _0 \\subset B_1(0)$ as a square centered at the origin with sidelength of $0.4$ as depicted in Figure REF .", "The exact solution satisfies $u \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )$ for any $\\varepsilon >0$ as well as $u \\in H^2(\\Omega _1)$ on every set $\\Omega _0 \\subset \\Omega _1 \\subset \\Omega $ satisfying $\\operatornamewithlimits{dist}(\\Omega _1,\\partial \\Omega ) > 0$ .", "Figures REF –REF show global and local errors in the $L^2$ -norm and the $H^1$ -seminorm.", "The discrete solutions are obtained from the MATLAB code [2], and the errors are computed elementwise using high precision Gaussian quadrature.", "Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm for s=0.1s=0.1.As predicted by the theory of [6] (for the global $H^1$ -error) and Corollary REF (for the local $H^1$ -error), we obtain – dropping the $\\varepsilon > 0$ , which we may expect to be arbitrary small in view of the shift theorem for smooth $\\Omega $ (cf.", "Remark REF ) – rates of $\\mathcal {O}(N^{1/4-s/2}) = \\mathcal {O}(h^{s-1/2})$ in the global $H^1$ -norm (provided the solutions are in $H^1$ ), $\\mathcal {O}(N^{1/4+s/2}) = \\mathcal {O}(h^{1/2+s})$ in the global $L^2$ -norm as well as $\\mathcal {O}(N^{1/2}) = \\mathcal {O}(h)$ both in the local $L^2$ -norm and $H^1$ -norm.", "Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm,left: s=0.3s=0.3;right: s=0.5s=0.5.Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm,left: s=0.7s=0.7;right: s=0.9s=0.9." ], [ "Acknowlegdgement", "The research of JMM is funded by the Austrian Science Fund (FWF) by the special research program Taming complexity in PDE systems (grant SFB F65).", "The research of MK was supported by Conicyt Chile through project FONDECYT 1170672." ], [ "Numerical examples", "We illustrate our theoretical results of the previous sections with a numerical example in two dimensions.", "On the unit circle $\\Omega = B_1(0)$ with constant right-hand side $f = 2^{2s}\\Gamma (1+s)^2$ the exact solution is known to be $u(x) = (1-\\left|x \\right|^2)_+^s \\qquad \\text{where} \\;\\; g_+ = \\max \\lbrace g,0\\rbrace ,$ see, e.g., [5].", "We choose the subdomain $\\Omega _0 \\subset B_1(0)$ as a square centered at the origin with sidelength of $0.4$ as depicted in Figure REF .", "The exact solution satisfies $u \\in \\widetilde{H}^{1/2+s-\\varepsilon }(\\Omega )$ for any $\\varepsilon >0$ as well as $u \\in H^2(\\Omega _1)$ on every set $\\Omega _0 \\subset \\Omega _1 \\subset \\Omega $ satisfying $\\operatornamewithlimits{dist}(\\Omega _1,\\partial \\Omega ) > 0$ .", "Figures REF –REF show global and local errors in the $L^2$ -norm and the $H^1$ -seminorm.", "The discrete solutions are obtained from the MATLAB code [2], and the errors are computed elementwise using high precision Gaussian quadrature.", "Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm for s=0.1s=0.1.As predicted by the theory of [6] (for the global $H^1$ -error) and Corollary REF (for the local $H^1$ -error), we obtain – dropping the $\\varepsilon > 0$ , which we may expect to be arbitrary small in view of the shift theorem for smooth $\\Omega $ (cf.", "Remark REF ) – rates of $\\mathcal {O}(N^{1/4-s/2}) = \\mathcal {O}(h^{s-1/2})$ in the global $H^1$ -norm (provided the solutions are in $H^1$ ), $\\mathcal {O}(N^{1/4+s/2}) = \\mathcal {O}(h^{1/2+s})$ in the global $L^2$ -norm as well as $\\mathcal {O}(N^{1/2}) = \\mathcal {O}(h)$ both in the local $L^2$ -norm and $H^1$ -norm.", "Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm,left: s=0.3s=0.3;right: s=0.5s=0.5.Figure: Local and global errors in the L 2 L^2- and H 1 H^1-norm,left: s=0.7s=0.7;right: s=0.9s=0.9." ], [ "Acknowlegdgement", "The research of JMM is funded by the Austrian Science Fund (FWF) by the special research program Taming complexity in PDE systems (grant SFB F65).", "The research of MK was supported by Conicyt Chile through project FONDECYT 1170672." ] ]

[ [ "Wave propagation in a strongly disordered 1D phononic lattice supporting\n rotational waves" ], [ "Abstract We investigate the dynamical properties of a strongly disordered micropolar lattice made up of cubic block units.", "This phononic lattice model supports both transverse and rotational degrees of freedom hence its disordered variant posses an interesting problem as it can be used to model physically important systems like beam-like microstructures.", "Different kinds of single site excitations (momentum or displacement) on the two degrees of freedom are found to lead to different energy transport both superdiffusive and subdiffusive.", "We show that the energy spreading is facilitated both by the low frequency extended waves and a set of high frequency modes located at the edge of the upper branch of the periodic case for any initial condition.", "However, the second moment of the energy distribution strongly depends on the initial condition and it is slower than the underlying one dimensional harmonic lattice (with one degree of freedom).", "Finally, a limiting case of the micropolar lattice is studied where Anderson localization is found to persist and no energy spreading takes place." ], [ "Introduction", "Wave propagation in heterogeneous media has attracted tremendous research interest in recent years.", "Families of one-dimensional (1D) continuous and discrete models have been extensively studied in this context [1], [2], [3], [4] focusing on the localization properties of both the normal modes of finite systems i.e., Anderson localization (AL) [5], and on the wave propagation in infinite media.", "Although the theory of AL was initially formulated for electronic systems, it has been successfully extended and applied to many other systems.", "Interestingly, recent experimental results on AL (see e.g., Refs.", "[7], [8], [6], [9], [10]) have opened new research frontiers and have revitalized the interest on studying AL both in quantum and classical systems.", "In the context of linear disordered 1D lattices, among different systems, special attention has been given to the tight binding electron model, the linear Klein-Gordon (KG) lattice [11], [3] and the harmonic lattice [12], [13].", "The interest in these models lies partly in the fact that they represent the linear limit of seminal nonlinear lattices such as the discrete nonlinear Schrödinger equation (DNLS), the quartic KG, and the Fermi-Pasta-Ulam-Tsingou (FPUT) lattices [14], [1], [15].", "Even more, these fundamental models have been adopted to describe a variety of physical systems and more recently, in the context of metamaterials, they have been extensively used as toy models for novel wave phenomena [16], [15].", "A typical route to study the wave properties of these heterogenous lattices is to monitor the time evolution of initially compact wave-packets.", "For the tight binding and the linear KG models, the dynamics after the excitation of such an initial condition is characterized by an initial phase of spreading, followed by a phase of total confinement to its localization length/volume.", "The width of the wave-packet is of the order of the maximum localization length [17].", "On the other hand, for the harmonic lattice, along with the localized portion of the energy, there is always a propagating part due to the existence of extended modes at low frequencies.", "A quantitative description of wave propagation in disordered 1D systems of one degree of freedom (DOF) per lattice site was formulated in Refs.", "[11], [12], [13] where wave-packet spreading was quantified using both analytical and numerical methods.", "Moreover, many variations of these 1D lattices have been studied extensively in several works including all the regimes from the periodic linear to the disordered nonlinear [21], [10], [18], [19], [22], [20], [23], [24], [25].", "A natural extension to the above studies is to investigate the corresponding behavior in disordered lattices with more than one DOF.", "Few such studies already exist in the literature especially as generalizations of the tight binding model by assuming a linear coupling between two (or more) 1D chains [26], [27] and illustrate how the coupling modifies the energy transport properties.", "Recent experiments also revealed the role of additional forces in disorder mechanical lattices. [28].", "On the other hand, the wave dynamics of disordered harmonic chains with two DOFs per site has merely been studied.", "Such models are relevant to macroscopic mechanical lattices (e.g., granular phononic crystals, lego and origami chains [29], [30], [31], [32], [33]), where the coupling between the DOFs stems from either the geometrical characteristics or from the material properties.", "Here, we present a thorough numerical study of a linear disordered system made up of square block elements that supports both translational and rotational waves [34], [35], [36].", "The model we investigate is used in bodies with beam-like microstructure to construct continuum models and in beam lattices [34], [37].", "The corresponding equations of motion bare close resemblance to other structures including 1D lattices of elastic cylinders [29] or spherical beads [30].", "Our goal is to unveil the role of the coupling between the DOFs regarding the energy transport in the presence of strong disorder and to identify the differences with the underlying 1D harmonic lattice.", "The rest of this paper is arranged as follows: In Section  we describe the model supporting both transverse and rotational motion.", "The static properties for the periodic and disorder cases are also discussed.", "In section  we investigate in detail the dynamical behavior of the system in the presence of strong disorder by initially exciting a single DOF at the center of the lattice.", "In section  we summarize and conclude the paper.", "Figure: (a) Schematic of the disorder phononic lattice with random shear stiffness indicated by the different spring thicknesses (colors).", "(b) Illustration of the transverse motion and the corresponding shear stiffness k (1) k^{(1)}.", "(c) Illustration of rotational motion and the corresponding bending stiffness k (2) k^{(2) }." ], [ " discrete model", "We consider a phononic structure composed of discrete block-spring elements such that the $n$ th element can be described by transverse and rotational DOFs as shown in Fig.", "REF .", "The transverse displacements are in the $y$ direction whilst the rotation is about an axis perpendicular to the $xy$ -plane.", "The blocks are coupled through a shear stiffness $k^{(1)}_n$ and a bending one $k^{(2)}_n$ [see Fig.REF (b,c)].", "In this work we consider $N$ identical cube blocks of mass $m$ with edges of length $2a$ and consequently a moment of inertia $I=2ma^2/3$ .", "Systems that could be potentially described by such a structure include models in micro- and nano-scale films [38], granular media [33], modeling of beam lattices [37] or the interaction of finite size particles with pre-designed connectors [39].", "The periodicity of the system is imposed by the distance $h$ between the center of each block as shown in Fig.", "REF , where $u_n$ and $\\phi _n$ respectively represent the transverse and rotational motion of the $n$ th block from equilibrium.", "The corresponding momenta are written as ${P_{n}^{(u)}}=m\\dot{u}_n$ and ${P_{n}^{(\\phi )}}=I\\dot{\\phi }_n$ for the former and latter motions, while $(~\\dot{}~)$ denotes derivative with respect to time.", "The total energy of the system $H$ , of the system is given by the following expression [35], [40] $H =\\sum _{n=1}^{N} \\frac{ 1 }{2} {P_{n}^{(u)}}^{2} +\\frac{ 1 }{2I} {P_{n}^{(\\phi )}}^{2}+ \\frac{ 1 }{2} K^{(1)}_{n+1} \\Big [ (u_{n+1} - u_{n} ) + \\frac{3}{2} (\\phi _{n+1} + \\phi _{n} ) \\Big ] ^2+ \\frac{ 1 }{2} K^{(2)}_{n+1} (\\phi _{n+1} - \\phi _{n} ) ^2.$ Here we have defined the constants $ K^{(1)}_n = 2 k^{(1)}_n (2a)^2/l^4_d $ , $ K_n^{(2)} = k^{(2)}_n2a^2/l^2 $ , the lengths $ l =h -2a$ and $ l_d = \\sqrt{l^2 + (2a)^2} $ and for simplicity we choose $ m =1 $ , $l=1$ and thus $h=3$ .", "The equations of motion for the two DOF are explicitly given by: $\\ddot{u}_{n} & = K^{(1)}_{n+1} \\big ( u_{n+1} - u_{n}\\big ) - K^{(1)}_{n} \\big (u_{n} - u_{n-1}\\big )\\nonumber \\\\&+ \\frac{ 3 K^{(1)}_{n+1}}{2} \\big ( \\phi _{n+1} +\\phi _{n} \\big ) - \\frac{3 K^{(1)}_{n} }{2} \\big ( \\phi _{n} +\\phi _{n-1} \\big ) ,\\\\I\\ddot{\\phi }_{n} & = \\frac{3 K^{(1)}_{n} }{2 } \\big ( u_{n-1} -u_{n} \\big ) + \\frac{3 K^{(1)}_{n+1}}{2} \\big ( u_{n} -u_{n+1} \\big )\\nonumber \\\\&- \\frac{9 K^{(1)}_{n+1}}{4} \\big ( \\phi _{n+1} + \\phi _{n}\\big ) - \\frac{9 K^{(1)}_{n} }{4}\\big (\\phi _{n} + \\phi _{n-1}\\big ) \\nonumber \\\\& + K^{(2)}_{n+1} \\big ( \\phi _{n+1} - \\phi _{n}\\big ) - K^{(2)}_{n} \\big (\\phi _{n} - \\phi _{n-1} \\big ).$ We first study the periodic phononic crystal [29] with $K^{(1)}_n\\equiv K^{(1)}=1$ and $K^{(2)}_n\\equiv K^{(2)}$ .", "In this case, we may look for Bloch like solutions of the form $ \\mathbf {X}_n = \\begin{pmatrix}u_n (t)\\\\\\phi _n (t)\\end{pmatrix} = \\mathbf {X}e ^{i\\Omega t - i Qn},$ where $\\mathbf {X}=[U,\\Phi ]$ is the amplitude vector, $\\Omega $ is the frequency and $ Q $ is the Bloch wave number.", "Inserting Eq.", "(REF ) into Eqs.", "(REF ) and  () we obtain the following eigenvalue problem for the allowed frequencies $\\mathbf {S} \\mathbf {X} = \\Omega ^2 \\mathbf {X}$ , where the resultant dynamical matrix is $\\mathbf {S} = \\begin{pmatrix}4 \\sin ^2 q & -6~i \\sin q \\cos q \\\\6~i \\sin q \\cos q & \\frac{2}{3} [ 9\\cos ^2 q + 4 K^{(2)} \\sin ^2 q ] \\\\\\end{pmatrix},$ with $q = Q / 2 $ .", "The corresponding expression for the eigenfrequencies is given by $\\Omega ^2_{\\pm } = \\frac{1}{2} \\Bigg \\lbrace 4 \\sin ^2 q +\\frac{2}{3} \\left( \\frac{9}{4} 4 \\cos ^2 q + 4 K^{(2)} \\sin ^2 q \\right) \\pm \\sqrt{ \\bigg [ 4 \\sin ^2 q + \\frac{2}{3}\\left( \\frac{9}{4} 4 \\cos ^2 q + 4 K^{(2)} \\sin ^2 q \\right) \\bigg ]^2 - 64 K^{(2)} p \\sin ^4 q } \\Bigg \\rbrace .", "$ The dispersion relation of Eq.", "(REF ) for $K^{(2)}=1$ is depicted by the solid curves plotted in Fig.", "REF (a).", "We directly observe the appearance of two branches separated by a band gap and terminated by a maximum allowed frequency.", "Since the two DOFs are coupled, the modes are composed by a mixture of transverse and rotational motion.", "Note that the constant $K^{(2)}$ , which depends on the bending stiffness, can be used as a tuning parameter to change the form of the dispersion relation and the dominant motion participating in each propagating mode [29].", "Figure: (a) Dispersion relations of the lattice for K (1) =1K^{(1)}=1 and K (2) =1K^{(2)} = 1 (K (2) =0K^{(2)} = 0) solid curves (dashed curves).", "(b) Corresponding eigenfrequencies for a single strongly disordered lattice (W=2)(W=2) with 〈K (1) 〉=1\\langle K^{(1)}\\rangle =1 and K (2) =1K^{(2)} = 1.", "The inset shows the mean value (200 realizations) of 〈P〉\\langle P \\rangle for each mode, and the standard deviation (shaded area).", "The vertical dashed line denotes the index where the quasi-extended modes appear.", "(c) Representative profiles of the eigenmodes of the disordered lattice with K (2) =1K^{(2)} = 1 for the three different cases indicated by the circle, square and triangle in (b).", "Here we show only profiles for u n u_n.In the rest of this work we introduce disorder to the system only through the shear spring stiffness's $K^{(1)}_n$ [see also Fig.", "REF (a)].", "We choose this particular disorder aiming to expose the role of each DOF and isolate its importance in the energy transfer.", "The values of $K^{(1)}_n$ are taken from a uniform probability distribution $f\\mathbf {\\big (} K^{(1)}_n \\mathbf { \\big ) } ={\\left\\lbrace \\begin{array}{ll}W^{-1}, &-W/2 < \\text{ $K^{(1)}_n$} - \\langle K^{(1)}\\rangle < W/2, \\\\0 & \\text{otherwise}.", "\\\\\\end{array}\\right.", "}$ The parameter $W$ determines the width of the distribution and thus the strength of the disorder.", "Fig.", "REF (b) illustrates the eigenfrequencies of a strongly disordered ($W=2$ ) finite chain of $N=10^3$ blocks.", "The eigenmodes have been sorted from lowest to highest frequency for increasing mode index $k$ .", "Due to the strength of the disorder, the middle band gap is filled with modes while the maximum frequency of the system is much bigger in comparison to the maximum frequency of the periodic chain.", "To further characterize the disordered finite lattice, for each mode we calculate the participation number [41] $ P = 1/ \\sum h_n^2 $ where $h_n = H_n / H$ is the normalization of the site energy $H_n$ .", "$P$ is an indicator of the localization of the mode and it becomes $P\\approx N$ for a mode with almost all sites excited, while $P = 1$ for a single site mode.", "The mean value of $P$ taken for 200 disorder realizations is shown in the inset of Fig.", "REF (b).", "It becomes clear that most of the modes are strongly localized throughout the spectrum except at very low frequencies where a rather small portion of the modes is extended.", "As such, we may loosely describe the modes as either localized or extended.", "Interestingly enough we obtain a set of what we coin as quasi-extended modes around the cut-off frequency of the upper branch of the periodic case.", "The appearance of these modes is due to the particular implemented disorder, which is only on the shear stiffness (see Appendix ).", "For illustration in Fig.", "REF (c) we show the normalized transverse profiles of an extended mode [$k= 50$ (circle)] and of two localized modes [$k=735$ (triangle), and $k=1700$ (square)].", "The normalized rotational profile follows the same patterns.", "As we will show below, both the low frequency extended modes and the quasi-extended modes contribute to the transfer of energy in the lattice." ], [ "Dynamics of the system", "To study the properties of energy transfer in the system we excite strongly disordered lattices using single site initial conditions.", "Our results are averaged over an ensemble of 200 disorder realizations and since we are interested in the effects of strong disorder we choose $W = 2$ such that $ K^{(1)}_n \\in (0, 2) $ .", "For all the time dependent simulations, each realization had $N=10^5$ lattice sites." ], [ "Momentum excitation", "We first study the dynamics of the lattice under two different initial momentum excitations $P_{N/2}^{(\\phi )}(0)=\\sqrt{I},\\;{\\rm or } \\; P_{N/2}^{(u)}(0)=1$ i.e., initially exciting either the transverse or the rotational momentum of the central site.", "Note that this choice of initial conditions corresponds to a total energy of $H=0.5$ for both cases.", "Some typical time evolutions of the energy densities are shown Figs.", "REF (a) and (b) for an initial (a) rotational and (b) transverse momentum excitation.", "We observe that for both cases, a large amount of energy remains localized in the region of the initial excitation at the lattice's center.", "This is expected due to the fact that most of the modes are localized and thus the implemented initial condition strongly excites localized modes around the central site.", "This is also quantified by the time evolution of the averaged participation number $\\langle P\\rangle $ shown in Figs.", "REF (c) and (d) for respectively the rotational and transverse initial momentum excitations.", "In fact we observe that in both cases $\\langle P\\rangle $ saturates to a small number compared to the total lattice size.", "However, comparing the two final values we observe a significant difference between the two cases as the transverse initial excitation [Fig.", "REF (d)] leads to a larger $\\langle P \\rangle $ .", "This behavior can be understood by studying the projection of the initial conditions onto the normal modes of a finite but large disordered lattice.", "For a given initial momentum excitation we define the vector $\\vec{V}(0)=[\\dot{u}_1(0),\\ldots , \\dot{u}_N(0),\\dot{\\phi }_1(0),\\ldots , \\dot{\\phi }_N(0)]^T$ whose projection on the system's normal modes is given by $\\vec{\\dot{R}}=\\mathbf {A}^{-1}\\vec{V}(0)$ with matrix $\\mathbf {A}$ having as columns the lattice eigenvectors.", "Using this projection, we can calculate the energy given to each normal mode as $E_k=\\dot{R}_k^2/2$ where $\\dot{ R}_k $ are the elements of projection vector $\\vec{\\dot{R}}$ .", "Obviously the system's total energy is $H= \\sum E_k$ .", "Figs.", "REF (e) and (f), although they appear to have a similar form, exhibit important differences regarding low index ($k$ ) modes (see also Appendix ).", "Since we sorted the modes with increasing frequency, low indices correspond to low frequency extended modes.", "In fact, for the initial rotational momentum [Fig.", "REF (e)], the low frequency extended modes (low index $k$ ) are the stronger excited ones with an energy up to the order of $10^{-6}$ .", "On the other hand, by initially exciting the transverse momentum, the low frequency modes (low index $k$ ) as shown in Fig.", "REF (f), are strongly excited acquiring energies up to $10^{-3}$ .", "These orders of magnitudes difference in energy of low frequency extended modes explains the differences in $\\langle P\\rangle $ shown in Figs.", "REF (c) and (d).", "Here we also observe major differences between the micropolar lattice and the well studied 1D harmonic lattice with disorder  [4], [12], [13].", "With an initial momentum excitation, instead of exciting all modes with the same energy as in the 1D harmonic lattice, here we observe a strong excitation of the low frequency modes and another set of modes around the cut-off frequency of the upper branch of the periodic case.", "As we will see below this has consequences to the energy transport.", "To quantify the energy spreading we compute the averaged second moment $\\langle m_2 \\rangle $  [12], [45], [46], [47], [48], [24] of the energy distributions, which for an initial excitation in the middle of the lattice is given by $m_{2} = \\sum _n (n-N/2)^2 h_n /H$ .", "Assuming a polynomial dependence of the spreading, for sufficiently long times, we may write $\\langle m_2 \\rangle \\propto t^{\\beta }$ and the parameter $\\beta $ is used to quantify the asymptotic behavior.", "The exponent $\\beta $ is calculated by first smoothing the $m_2 (t)$ values of each disorder realization through a locally weighted difference algorithm [49], [50].", "The estimate of the rate of change $\\beta = \\frac{ d \\log _{10}\\langle m_2 (t) \\rangle }{ d \\log _{10} t},$ is thus obtained numerically through a central finite difference scheme as the values of $m_2 (t)$ are analyzed in log-log scale.", "In Figs.", "REF (g) and (h) we observe that for both cases $\\beta $ reaches an asymptotic value.", "In fact $\\beta \\approx 0.75 ~(\\beta \\approx 1.25)$ for initial rotational (transverse) momentum excitations corresponding to subdiffusive (superdiffusive) transport.", "These values are quite different than the ones observed for the 1D harmonic lattice where momentum excitation is always found to be superdiffusive with $\\beta \\approx 1.5$  [12], [13], [51].", "To qualitatively explain this difference we first note that the exponent $\\beta $ has been shown to depend mainly on two factors: (i) the characteristics of extended modes (group velocity, localization length as function of frequency, total number) and (ii) the projection of the initial condition on the modes [12], [13], [51].", "Regarding point (i), for both models, there is a set of extended modes at $\\Omega \\ll 1$ .", "Major differences are thus expected since the dispersion relation of Eq.", "(REF ) for the micropolar lattice at low frequencies is quadratic with respect to the wavenumber i.e.,  $\\Omega \\approx 3\\sqrt{K^{(2)}}Q^2$ in contrast to the 1D harmonic lattice where $\\Omega \\approx Q$ .", "Furthermore, for the micropolar lattice, the quasi-extended modes at higher frequencies may influence the energy spreading, as it was shown for example in [12], [52] where additional extended modes were found either due to symmetries or resonances.", "As far as point (ii) is concerned, the results of Figs.", "REF (e) and (f) are substantially different from those of the 1D harmonic lattice indicating that differences between the two models are anticipated." ], [ "Displacement excitation", "To further compare the behavior of the micropolar model to that of the 1D harmonic lattice [11], [12], we now study the dynamics induced by the following initial conditions $\\phi _{\\frac{N}{2}} (0)=\\phi _{\\frac{N}{2}},\\;{\\rm or } \\; u_{\\frac{N}{2}}(0)=u_{\\frac{N}{2}},$ which correspond to initial rotation or transverse displacement of the central block.", "In this study, the values of $\\phi _{\\frac{N}{2}}$ and $u_{\\frac{N}{2}}$ are chosen such that the total energy for each realization is again $H=0.5$ .", "Similarly to Section REF , the evolution of the energy distribution [Figs.", "REF (a) and (b)] is characterized by a localized wave-packet at the region of the initial excitation in center of the lattice, and by a portion which is propagating.", "However, compared to the initial momenta excitations, here the energy carried away from the central site is substantially smaller.", "For both types of initial conditions, $\\langle P\\rangle $ attains an asymptotic value of less than 10 sites as shown in Figs.", "REF (c) and (d).", "This behavior can be also understood using the projection of the initial conditions to the normal modes of a large but finite lattice.", "This is now done by projecting the vector $\\vec{U}(0)=[u_1(0),\\ldots , u_N(0),\\phi _1(0),\\ldots ,\\phi _N(0)]^T$ onto the normal modes to yield $\\vec{R}=\\mathbf {A}^{-1}\\vec{U}(0)$ .", "In this case, the energy of the $k$ th normal mode is $ E_k = \\Omega _k^2R_k^2/2$ with $\\Omega _k$ being the $k$ th eigenfrequency.", "Again the system's total energy is $ H=\\sum E_k $ .", "The outcome of this projection is shown in Figs.", "REF (e) and (f) for the initial rotation and transverse displacement respectively.", "The results are similar to those of Figs.", "REF (e) and (f) with suppressed contributions of the low frequency modes leading to a small value of $\\langle P\\rangle $ during the evolution.", "Furthermore, we also calculated the exponent $\\beta $ for the energy propagation resulting from these two different initial excitations and the results are shown in Figs.", "REF (g) and (h).", "In a similar manner as in the 1D disordered harmonic lattice case, single site displacement excitation lead to subdiffusive behavior.", "However, our findings show that although the initial rotation excitations leads to the same value ($\\beta \\approx 0.5$ ) as in the 1D harmonic lattice, the initial transverse displacement features extremely slow energy transport with $\\beta \\approx 0.25$ .", "The discrepancy between the two models is anticipated as it was discussed at the end of Section REF .", "However our results for the displacement excitations of the micropolar lattice strongly suggest that the energy transport is indeed mediated by both the low frequency and the quasi-extended modes.", "To be more precise we compare the results of Fig.", "REF (e) with Fig.", "REF (e) and notice that in the latter lower frequency modes are less excited leading to a smaller value of $\\beta $ ($\\beta \\approx 0.75$ for the former and $\\beta \\approx 0.5$ for the latter).", "In the same spirit, by comparing Fig.", "REF (e) with Fig.", "REF (f) the main difference lies in the quasi-extended modes which are suppressed in the later case leading to a $\\beta \\approx 0.25$ instead of $0.75$ .", "Thus reducing the amount of energy allocated to either the low frequency extended modes or the quasi-extended modes results in a reduced $\\beta $ suggesting that both contribute to the energy spreading.", "Note that the result of initial transverse momentum, corresponding to Fig.", "REF (f) is not compared with the other three since in that case the low frequency modes are highly excited.", "We thus conclude that the complete picture is comprehended by casting one eye on the low frequency extended and the other onto the quasi-extended ones." ], [ "Energy contributions in the micropolar ", "The participation number measures the localization of the total energy and the exponent $\\beta $ is a measure of how fast the energy is spreading, however, none of them carries any information of what amount of this energy is attributed to the rotational or the transverse DOFs.", "We can have an indication of how much energy is attributed to each of the two DOFs by decomposing the total energy of the system into two parts i.e., $H = H_{R} + H_{T}$ as follows $&H_{R} = \\sum _{n=1}^{N} \\frac{1}{2I} {P^{(\\phi )}_{n}}^{2}+ \\frac{ 9 }{8} K^{(1)}_{n+1} ( \\phi _{n+1} + \\phi _{n} ) ^2 +\\frac{1 }{2} K^{(2)}_{n+1} ( \\phi _{n+1} - \\phi _{n} ) ^2 +\\frac{3 }{4} K^{(1)}_{n+1} ({ \\phi _{n+1} } { }+ { \\phi _{n} } ) ( { u_{n+1}} - { u_{n}} ), \\\\& H_{T} = \\sum _{n=1}^{N} \\frac{1}{2} {P^{(u)}_{n}}^{2}+ \\frac{1 }{2} K^{(1)}_{n+1} (u_{n+1} - u_{n})^2 +\\frac{3 }{4} K^{(1)}_{n+1} ({ \\phi _{n+1} } { }+ { \\phi _{n} } ) ( { u_{n+1}} - { u_{n}} ),$ Figure: (a) Averaged normalized energy contributions H R H_{R} and H T H_{T} of the normal modes for finite lattices of 1000 sites.", "(b) Time evolution of normalized averaged rotational (H R center H_R^{center}) and transverse (H T center H_T^{center}) energy contributions near the excitation region.", "Solid bolder (dashed lighter shaded) curves show rotational displacement (momentum) initial excitations.", "(c) Same as (b) but for transverse displacement (momentum) initial excitations.", "(d) Time evolution of averaged normalized energy contributions H R edge H_R^{edge} and H T edge H_T^{edge}, for transverse momentum excitations.", "Averaged values are over 200 disorder realizations and one standard deviation is indicated by the lightly shaded regions.separating the rotational $H_{R} $ and transverse $H_{T}$ energy contributions.", "Note that the coupling potential energy, which is described by the last terms in both Eqs.", "(REF ) and (), is equally shared between the two contributions.", "It is interesting to determine the nature of the lattice's energy in two different regions: i) around the initially excited central block and ii) sufficiently far away from the region of localization.", "For the central area we calculate the energy using Eqs.", "(REF ) and () but taking the sum for $n \\in [N/2 -100,N/2+100]$ to obtain $H_R^{center}$ and $H_T^{center}$ .", "Conversely, we also define the energies at the edges of the energy distribution $H_R^{edge}$ and $H_T^{edge}$ by summing Eqs.", "(REF ) and () for $n\\notin [N/2-5000,N/2+5000]$ .", "Before discussing the dynamical behavior of the system in these two distinct regions, it is relevant to show how the two different energy contributions are shared between the modes of a finite disordered lattice.", "The result is shown in Fig.", "REF (a) where the red (blue) curve depicts the transverse (rotational) energy contribution.", "As a general observation, we mention that the modes with lower $k$ values are dominated by transverse motion, while the high frequency ones are dominated by rotational motion.", "As it is shown in Fig.", "REF (b), for both initial conditions concerning the rotational DOFs [$P_{n}^{(\\phi )}(0)=\\sqrt{I}\\delta _{n,N/2}$ and $\\phi _{n} (0)=\\phi _{N/2} \\delta _{n,N/2}$ ], the central, localized part of the energy distribution is dominated by the rotational motion with almost the same ratios.", "This is so as the majority of the localized modes are dominated by rotation [see Fig.", "REF (a) and Fig.", "REF (b)].", "By initially exciting the transverse DOF we end up with the two energy contributions in the central part shown in Fig.", "REF (c).", "We find that the central part of the energy distribution for the initial displacement excitation [$u_{n} (0)=u_{N/2} \\delta _{n,N/2}$ ] is still dominated by rotation as indicated by the solid curve in Fig.", "REF (c).", "Interestingly, for the case of initial transverse momentum excitation [$P_{n}^{(u)}(0)=\\delta _{n,N/2}$ ], the energy contribution of each motion in the central part is inverted with respect to all other cases.", "This is due to the fact that this initial condition excites more strongly the low frequency modes [Fig.", "REF (f)] which according to Fig.", "REF (a) are dominated by transverse motion.", "Let us now turn our attention to the energy distribution far away from the central site following the propagating tails that are responsible for the energy transfer.", "The corresponding results for the initial transverse momentum excitation is shown in Fig.", "REF (d).", "After the arrival of the propagating front at the chosen sites $n=N/2\\pm 5000$ , it is readily seen that the energy at the edges is completely carried by the transverse motion.", "We have confirmed the same quantitative result for all types of initial conditions.", "This behavior can be comprehended since we have shown that energy is carried away mostly from the low frequency extended modes hence as shown in Fig.", "REF (a) these modes (corresponding to small $k$ ) are almost completely constituted by transverse motion and so is the energy at the edges." ], [ "The special case of $K^{(2)}=0$ ", "Now, we focus on a special case of the system i.e., in the limit of vanishing bending stiffness $K^{(2)}$ .", "Note that such a case is very relevant to situations where the bending stiffness is so small that it may be neglected, see for example Ref.", "[33].Then, the corresponding dispersion relation of the periodic system [dashed lines in Fig.", "REF (a)] is substantially altered.", "In particular instead of two propagating bands, it consists of a zero frequency non-propagative band and a dispersive band emerging after a cut-off frequency $\\Omega =2$ .", "The zero frequency branch is made possible due to a counterbalance between the shear and bending forces [29].", "To study the energy transfer for this special case of $K^{(2)}=0$ , we have performed simulations for the different single site initial conditions considered before and a characteristic example is shown in Fig.", "REF .", "There it is readily seen that the energy remains localized around the center, and in addition there is no energy transfer to the rest of the lattice.", "This is also confirmed by the time dependence of the exponent $\\beta $ in $\\langle m_2(t) \\rangle \\propto t^{\\beta }$ , which becomes zero (see inset of Fig.", "REF ) thus signaling no energy spreading.", "Trying to explain the absence of energy transport we found (by solving the corresponding eigenvalue problem numerically) that the lower branch of the system's frequency spectrum, in the limit case of $K^{(2)}=0$ , still remains at zero frequency even in the presence of strong disorder due to a counterbalance of the transverse and rotational motions.", "As such, in this limit, the micropolar lattice is similar to a 1D KG model i.e., featuring a single propagating band emerging after a lower cut-off frequency and thus the system is expected to exhibit AL.", "Figure: Time evolution of the energy density after an initial transverse momentum excitation P N/2 u (0)=1P_{N/2}^{u}(0)=1 with K (2) =0K^{(2)}=0 where n ˜=n-N/2\\tilde{n}=n-N/2.", "The inset depicts the evolution of the exponent, β\\beta in the relation 〈m 2 (t)〉∝t β \\langle m_2(t) \\rangle \\propto t^{\\beta } averaging over 200 disorder realizations, which is shown to be zero indicating no spreading.", "The horizontal dashed line indicates β=0\\beta = 0.We have demonstrated how energy is transported in a strongly disordered micropolar lattice subject to shear forces and bending moments when the shear stiffness are chosen randomly.", "The phononic crystal investigated was composed of connected blocks possessing two degrees of freedom corresponding to transverse and rotational motion.", "The dynamics of the energy density, under different single site initial excitations was characterized into two different regions: a localized energy distribution around the initially excited site and a propagating part at the edges of the lattice.", "The energy localization for each initial condition as quantified by the participation number $P$ was found to acquire a small (compared to the lattice length) asymptotic value.", "Depending on which motion or momentum we initially excited, energy spreading was found to be either superdiffusive or subdiffusive, as quantified by the energy's second moment $m_2$ .", "Compared to the underlying 1D harmonic case, energy transport is altered, and in general the micropolar lattice featured slower spreading.", "The modified energy transport characteristics are attributed to the differences of the dispersion relation between the two models in the low frequency limit, to the weight by which the modes of the system are excited depending on the initial condition and also to the existence of additional quasi-extended modes in the micropolar lattice.", "Furthermore, by measuring the parts of the total energy related to the rotational and transverse motions we showed that the propagating part is always carried by translation for any choice of initial condition.", "On the other hand, the localized part was found to be either dominated by rotation or translation depending on the initial conditions.", "Finally, the limiting case of vanishing bending force was found to be similar to a linear 1D KG lattice which exhibits AL and thus no energy spreading.", "Our results not only revealed interesting properties of 1D disordered micropolar lattices with bending forces, but also raised new questions for future investigations.", "A direct generalization of our results is to study the effect of other kinds of disorder i.e., disorder in the masses or in different combinations of the stiffnesses.", "Furthermore, the appearance of the extended modes at the edge of the upper band is worthy of its own investigation in relation to other known models where anomalous localization appears either due to correlations or symmetry." ], [ "acknowledgments", "Ch. S.", "thanks LAUM for its hospitality during his visits when part of this work was carried out.", "We also thank the center for High Performance Computing (https://www.chpc.ac.za) for providing computational resources for performing significant parts of this paper's computations." ], [ "Quasi-extended modes", "Here we focus our attention on the quasi-extended modes appearing close to the cut-off frequency of the upper band of the periodic case (see Fig.", "REF ).", "As depicted in Fig.", "REF (a), the participation number $\\langle P \\rangle $ features a peak around $k \\approx 1200$ which corresponds to the cut-off frequency of the upper branch of the periodic system [see Figs.", "REF (a) and (b)].", "Note that in many cases we found that these modes may be as extended and have $P$ values which are of the same order as the low index modes (small $k$ ) as indicated by the inset in Fig.", "REF (a) corresponding to a single realization.", "To further understand this phenomenon we now consider a characteristic profile of such a mode depicted in Fig.", "REF (b).", "We find that: (i) these modes consist almost solely of rotational motion (the contribution of the transverse DOFs is negligible i.e., $u_n \\approx 0$ ) and (ii) the profile of the modes consists of various regions with consecutive rotations of similar amplitude and opposite signs ($\\phi _{n+1}\\approx - \\phi _n$ ), as shown by the zoom in the inset of Fig.", "REF (b).", "Using this two observations and the functional form of the Hamiltonian (REF ), it is noticeable that for these modes, effectively only the bending potential term analogous to $K^{(2)}$ is present.", "But since there is no disorder in $K^{(2)}$ these modes are extended reminiscent of the periodic lattice." ], [ "Eigenmode Projections", "Here we take a closer look at the projections of the initial momentum excitations onto the normal modes especially in the low frequency regime.", "The results of Figs.", "REF (e) and (b) are combined together for comparison in Fig.", "REF (a) while a zoom in the low frequencies is shown in Fig.", "REF (b).", "From the latter we observe that for $k\\rightarrow 0$ the energy difference between the two types of excitation is as much as 10 orders of magnitude.", "This explains the larger values of participation number $\\langle P\\rangle $ shown in Figs.", "REF (c) and (d).", "Similarly for the initial displacement excitations the results are shown in Fig.", "REF (a) and (b).", "Although here the two curves are more similar again differences in the low frequency regime show that the initial transverse excitation (red curve) will acquire a larger $\\langle P\\rangle $ as it is found by comparing Figs.", "REF (c) and (d).", "Figure: (a) Figs.", "(e) and (b) combined together.", "(b) A zoom of some small kk values in (a).", "The lightly shaded red and blue regions indicate one standard deviation on either side of the mean value obtained over 200 disorder realizations." ] ]

[ [ "Time resolution and power consumption of a monolithic silicon pixel\n prototype in SiGe BiCMOS technology" ], [ "Abstract SiGe BiCMOS technology can be used to produce ultra-fast, low-power silicon pixel sensors that provide state-of-the-art time resolution even without an internal gain mechanism.", "The development of such sensors requires the identification of the main factors that may degrade the timing performance and the characterisation of the dependance of the sensor time resolution on the amplifier power consumption.", "Measurements with a $ \\mathrm{^{90}Sr} $ source of a prototype sensor produced in SG13G2 technology from IHP Microelectronics, shows a time resolution of 140 ps at an amplifier current of 7 $ \\mathrm{\\mu} $A and 45 ps at higher power consumption.", "A full simulation shows that the resolution on the measurement of the signal time-over-threshold, used to correct for time walk, is the main factor affecting the timing performance." ], [ "Time jitter comparison at low power operation of HBT and CMOS amplifiers", "The integration of pixelated sensors in a microelectronic process has opened to the possibility of producing sensors with small pixels for particle-physics experiments  [1] [2] without incurring in a complex hybridization process.", "An evolution of this approach is to combine a 100 or better time resolution to develop low-cost 4D sensors that can be produced in large volumes and assembled more rapidly and with simplified procedures.", "One of the main challenges for the development of such a detector is to reduce the power consumption of the front-end electronics while maintaining excellent timing performance, in a trade-off where the limits are set by the characteristics of the transistor.", "The selection of the technology for signal amplification and read-out is also primordial.", "The most direct approach to achieve excellent timing resolution with low power consumption is to use small-size transistors from down-scaled CMOS technologies.", "However the cost of an engineering run in these technologies is high and the improvement in transistor speed is limited by the vulnerability of the circuit to its parasitic capacitances.", "One alternative approach [3] to maximize the timing performance without scaling down the process node is to use SiGe BiCMOS technologies, which offer a standard silicon CMOS process with the addition of a high-performance SiGe Hetero-junction Bipolar Transistor (HBT).", "SiGe HBTs feature a transconductance and a transition frequency that can exceed those of most advanced CMOS nodes [4].", "Moreover, they exhibit low noise for fast signal integration typical of BJT technologies [5] and robustness to parasitic capacitance [4] that make them suitable for a pixel detector." ], [ "The SiGe BiCMOS monolithic pixel prototype", "To evaluate the analogue behavior of a pixel detector with 4D tracking capability, we designed the small-size monolithic silicon pixel prototype [3] of Figure REF in the IHP SG13G2 process [6] .", "The sensor was produced on a standard substrate resistivity of 50.", "It has matrices of large and small hexagonal pixels, with hexagon sides of 130 and 65.", "Each pixel is connected to the front-end circuit, which is placed in an independent deep-nwell to insulate it from the high-voltage substrate.", "A study of the timing performance at high power consumption of the prototype is described in [3].", "In this paper we focus on the contribution to the time resolution of the amplifier power consumption and of the time-walk correction method.", "Figure: The monolithic prototype in IHP SG13G2 technology used for the measurements.", "The pixels and front-end electronics are inside deep-nwells operated at positive low-voltage, while the p-doped substrate is operated at negative high-voltage.", "For the measurements presented here, the output of the discriminators was sent to a differential driver and directly readout by an oscilloscope.The prototype integrates a front-end electronic circuit consisting of a HBT-based amplifier and a CMOS-based discriminator originally developed to read large pixels of a silicon sensor for medical applications [7], which was not designed to operate at very low current.", "For this reason in this study the CMOS-based discriminator was kept at the original design current of 40.", "The HBT-based amplifier, on the other hand, has programmable bias components and it was operated with a supply current in the range between 7 and 150.", "The output of the discriminator was sent out from the chip using a differential CML driver, allowing the measurement of the signal Time of Arrival (ToA) and Time over Threshold (ToT).", "In this prototype, the measurement of the amplitude of the signal at the output of the amplifier was not possible." ], [ "Experimental setup with radioactive sources", "An experimental setup with radioactive sources (Figure REF ) was built to measure the gain, noise and time resolution of the prototype chip while operating the amplifier at a collector current of 7, 20 and 50 and the sensor at a bias voltage between 100 and 180.", "The expected depletion depth at a bias voltage of 140 for a bulk resistivity of 50 is 26, making the intrinsic sensor contribution to time resolution from the charge deposition profile and the sensor uniformity of response below 30 [8].", "The measurement of the gain of the amplifier at the different working points was performed using a 109Cd source, which generates 22 and 25keV photons.", "The time resolution was measured as the difference between the signal time of arrival for the pixel under test and a reference Low Gain Avalanche Diode (LGAD) using a 90Sr source.", "The LGAD was glued on a dedicated amplifier board, which had a 1mm-wide hole to allow for the passage of the 90Sr electrons and enable Time-of-Flight (ToF) measurement (see Figure REF ).", "The reference LGAD used for this test had a time resolution of 50 ps RMS [3], [9].", "The board with the monolithic SiGe prototype under test was precisely mounted so to have the hole in the LGAD board aligned with the small pixel called \"S0\" in Figure REF .", "Figure: Sketch of the experimental setup used for the ToF measurements showing the 90Sr source, the reference LGAD providing reference time and the monolithic prototype chip under test." ], [ "Amplifier noise and gain", "The measurement of the noise hit rate as a function of the discriminator threshold is reported in Figure REF , for thresholds above and below the baseline and for different amplifier currents.", "Although the noise hit rates show a compression for values larger than 500 due to a saturation of the counting-rate capability of the data-acquisition system, the measurements below 100kHz allow obtaining an indication of the RMS voltage noise as seen at the output of the discriminator.", "The CMOS-based discriminator acts as a filter for the amplifier noise.", "The ratio between the noise at the output of the amplifier ($ \\sigma _V $ ) and the one measured after the discriminator was estimated using a Cadence Spectrehttps://www.cadence.com/en_US/home.html simulation.", "The simulation shows that, in order to estimate the voltage noise at the output of the amplifier that we need to calculate the Equivalent Noise Charge (ENC), the standard deviation of the voltage noise obtained by the measurement of the noise hit rate at the output of the discriminator should be increased by 20% for an amplifier current of 7, 50% for a current of 20 and 60% for a current of 50.", "The resulting values after the scaling factor obtained from the simulation are reported in Table REF .", "Figure: Noise hit rate of the pixel under test as a function of the discriminator threshold.", "The acquisition setup saturated at a counting rate of 1.The charge gain $ A_Q $ of the amplifier was estimated by measuring the photon count rate as a function of the discriminator threshold using a 109Cd source.", "The results for the three amplifier currents considered are reported in Figure REF ; the data were fitted with the sum of two error functions, to account for the two main photon energies, and a constant, to account for the low-probability emission of 88 photons.", "A second degree polynomial was added to describe the low threshold region.", "The mean value of the error functions estimated by the fit was used to extract the amplifier gain values reported in Table REF .", "Then, the noise and gain measurements were used to estimate the Equivalent Noise Charge (ENC) values, also reported in Table REF .", "Figure: Photon count rate as a function of the discriminator threshold for the three values of the amplifier collector current with the sensor exposed to a 109Cd source.", "The lines represent the fit done with a composition of two error functions, a constant and a second degree polynomial function; the latter was introduced to describe the rise in the low-threshold region produced by low-energy photons.Table: Voltage noise at the output of the amplifier, charge gain and ENC for the different amplifier currents.", "The results for an amplifier current of 150 are those from .", "The uncertainties are statistical only." ], [ "Time resolution", "Electrons emitted by the 90Sr source were used to measure the jitter on the ToF between pixel \"S0\" and the reference LGAD detector.", "The output of the discriminator of the pixel under test was read by an oscilloscope with a sampling rate of 40GSa/s and an analogue bandwidth of 3GHz.", "The output signals of the LGAD were amplified using a discrete components amplifier based on SiGe HBTs [10] and sent to a channel of the oscilloscope.", "Pixels \"S1\" and \"S2\" were operated at the lowest possible threshold clean from noise.", "They were attenuated and read by one channel of the oscilloscope to generate a tag signal for events with charge shared between the pixel \"S0\" under test and the neighbouring ones.", "Pixel \"S3\" was not used and its discriminator threshold was set to the highest possible value to prevent it from firing.", "The oscilloscope trigger was configured to require a pulse from the pixel under test and a pulse from the reference LGAD within a 50 time window.", "At each trigger, the waveforms from all channels were recorded, allowing the calculation of the ToA of all pulses, the ToT of the pixel \"S0\" under test and the amplitude of the LGAD signal.", "During data analysis, each waveform was split in a signal region and a background region.", "The signal region was defined as the time interval in a window of $ \\pm $ 15 around the trigger time, containing all the pulses from the LGAD and the Device Under Test (DUT).", "The part of the waveform before and after the signal region –the background region– was used to estimate the baseline and remove noisy events: if one of the waveforms exceeded a given threshold in the background region, the event was considered to be noisy and removed from the analysis.", "The fraction on noisy events was within 0.1% of the total number of events.", "Coincidences between the reference LGAD and the DUT were acquired at several thresholds and bias voltages for different amplifier currents.", "To keep the data analysis as simple as possible, for each measurement a unique time-walk correction function was applied to the entire ToT range, although it was noticed that a slightly improved time resolution could be obtained by a special treatment of events at the edges of the TOT distribution.", "For the same reason, the only event selection applied to the prototype data was the removal of the very small fraction of noisy events mentioned above.", "To keep under control the reference time, only the events with LGAD pulse amplitudes in the interval in which the LGAD time resolution was measured to be 50ps [3] were considered.", "Figure REF shows the distribution of the ToT for the three amplifier-current working points after the simple data analysis described.", "The increase of the average value of the ToT at lower currents is an indication of the reduction of the amplifier bandwidth that is in part attributed to the increase of the output impedance and in part to a reduction of the gain-bandwidth product of the transistor.", "Figure: ToT distribution of the pixel under test for an amplifier current of 7, 20 and 50 at a threshold of 1043, 1093 and 1046 electrons, respectively, taken at a sensor bias voltage of 140.The charge-thresholds values were obtained dividing the discriminator threshold voltage by the amplifier gain.For each threshold, the ToF was measured with the oscilloscope as the difference between the ToA values of the signals from the prototype under test and the reference LGAD.", "The time-walk can be partly corrected by using the distribution of the correlation between the ToF and the prototype-signal ToT.", "As an example, Figure REF shows the TOF vs. TOT distribution obtained at a sensor bias voltage of 140V for an amplifier current of 50 and a threshold of 1046 electrons.", "The range of the time-walk correction depends on the discriminator threshold and on the amplifier current, as shown in Figure REF , with values ranging from 0.25 to 2.5.", "Figure: Difference of the pulse ToA at the oscilloscope between the pixel under test and the reference LGAD detector as a function of the ToT of the signal from the pixel under test.", "The data refer to a sensor bias voltage of 140 V, an amplifier current of 50 and a charge threshold of 1046 electrons.", "Blue dots represent events with cluster size one, orange dots events where at least one of the two adjacent pixels fired.Figure: Maximum time-walk correction at different charge threshold values for the amplifier working points.", "For each threshold the range of time-walk correction was estimated as the width at 10% of the amplitude of the distribution of ToF before correction - ToF after correction \\mathrm {ToF_{before~correction}-ToF_{after~correction}} .Figure REF shows the ToF distribution after time-walk correction for the same amplifier current and discriminator threshold of Figure REF .", "The mean value of the distribution was set close to zero by the time-walk correction.", "The time jitter of the distribution was estimated by a gaussian fit that excluded part of the tail at positive values of ToF that is caused by a systematic error in the time-walk correction [3].", "Indeed, due to its non-linear transfer function, the discriminator produces an asymmetrical jitter of the ToT, sometimes overestimating its value.", "When this happens, the signal ToA values are under-compensated for time walk, producing the non-gaussian tail observed in the ToF distribution.", "In all cases, the full width at half maximum (FWHM) of the distribution was found to be compatible with the $ \\sigma _{ToF} $ obtained from the Gaussian fit.", "The fraction of events in the tails exceeding the Gaussian distribution is typically below 5% of the total number of events.", "Figure: Distribution of the ToF between the pixel under test and the LGAD for an amplifier current of 50 and a discriminator threshold of 1046 electrons.", "The time jitter was estimated by a Gaussian fit - represented by the full red line - that excludes the tail of the distribution for ToF>125.", "The dashed line shows the continuation of the fitted Gaussian function outside the range of the fit.", "The fraction of events in the tail exceeding the Gaussian fit is f tail =4.8% f_{tail}=4.8\\% .The average value of the ToF was set close to zero by the time walk correction.The time resolution of the prototype under test with ToT-based time walk correction was finally estimated by subtracting in quadrature 50 (the time resolution of the LGAD) from the $ \\sigma _\\text{ToF} $ value obtained from the fit.", "Figure REF shows the time resolution for the different amplifier currents as a function of the discriminator threshold.", "In all cases the measured time resolution is below 180 and improves at larger thresholds.", "Cadence Spectre simulations suggest that the dependence on the discriminator threshold indicates that the poor ToT resolution, which limits the accuracy of the time-walk correction, is the dominating factor for the sensor time resolution.", "The reduction of the amplifier current enhances this effect, increasing both the range of the time walk and the signal fall time, with consequent deterioration of the ToT resolution at the lower thresholds.", "Therefore, we conclude that the intrinsic jitter of the amplifier is small in this prototype with respect to the resolution of the time-walk correction for all amplifier currents considered.", "Figure REF shows that the time resolution improves at higher sensor bias voltages.", "This improvement can be explained by two factors: An increase of the depletion depth, which is associated to a smaller pixel capacitance and a larger charge signal from the MIPs (up to 13% at the highest voltage).", "As a consequence it offers a better signal to noise ratio for the time walk correction.", "An increase of the average electric field in the active volume, with faster charge collection and improved intrinsic timing performance of the pixel sensor.", "The contribution from the second effect is small and it does not depend on the amplifier working point, therefore the large improvement at the lowest current can be attributed solely to the increase of the signal charge.", "Figure: Sensor time resolution vs. discrimination threshold for amplifier currents of 7, 20, 50 and 150 and sensor bias voltage of 140.", "The time resolution was obtained correcting for the time walk using the ToT at the output of the discriminator.", "The contribution of 50 from the reference LGAD to the time resolution was subtracted in quadrature from the ToF resolution.", "The results for an amplifier current of 150 are those from .Figure: Sensor time resolution vs. sensor bias for amplifier currents of 7, 20, 50 and 150 and discrimination thresholds of 1043, 710, 742 and 1080 electrons, respectively.", "The results for an amplifier current of 150 are those from ." ], [ "Conclusions and discussion on the impact of SiGe HBT in timing sensors", "The timing performance of a small-size monolithic silicon pixel prototype featuring an amplifier realised with SiGe HBT was measured with a 90Sr source setup.", "For a pixel capacitance of 70 fF, a time resolution of 140 was achieved for an amplifier current of 7, while 45 were measured at 150.", "These excellent results show that the low noise and high transition frequency of SiGe HBTs can be used to produce amplifiers for silicon pixel detectors with low time jitter at very low power consumption.", "An analysis of the results indicates that the measured time resolutions are limited by some design characteristics of this prototype and not by the performance of the SiGe HBT.", "In particular: A study of the sensor response with a CADENCE Spectre simulation suggests that the improvement of the timing performance for higher thresholds is indicative of a time resolution limited by the use of the measured ToT to compensate for the signal time walk.", "The improvement in timing performance at higher sensor bias indicates that an increase of the width of the depletion layer is beneficial, especially at very low power consumption.", "For the bias voltage and resistivity used here, the intrinsic sensor contribution to time resolution from the charge deposition profile and the sensor uniformity of response is expected to be below 30 [8].", "To quantify the ultimate performance of this technology, a single-transistor amplifier in common source (emitter) configuration was simulated using CADENCE Spectre.", "The simulation investigates the trade-off between the intrinsic timing jitter and the power consumption for a minimum-size NMOS transistor and a SiGe HBT in the same SG13G2 130nm BiCMOS process of IHP Microelectronics.", "Figure REF shows the results for two scenarios: a noiseless, ideal polarization of the transistors with a parasitic capacitive load of 2.5, and a realistic CMOS-based polarization circuit with the same capacitive load.", "In both cases a pixel capacitance of 80 was considered.", "Figure: CADENCE Spectre simulation of the time jitter as a function of the supply current for an amplifier based on a CMOS transistor in common source configuration (orange) and for a SiGe HBT in common emitter configuration (blue).", "The simulation was performed for an input capacitance of 80 and an input charge of 1600 electrons.", "The time jitter was defined as the ratio between the voltage noise at the output of the amplifier and the maximum slope of the rising edge of the pulse.", "Dashed lines: biasing circuit made of ideal, noiseless resistors.", "Full lines: realistic CMOS-based biasing circuit.The simulation shows that the HBT provides a lower intrinsic jitter than the CMOS transistor by more than a factor of two, despite the extra capacitance that was added to the standard HBT in the insulation process.", "This superior performance is further enhanced when the robustness to parasitic capacitance of the HBT is taken into account.", "The authors wish to thank the technical staff of the Department of Particle Physics of the University of Geneva for the assembly of the instrumentation and for the support in preparing the test, and our colleague Nicolò Cartiglia for providing the FBK reference LGAD." ] ]

[ [ "Demonstrating high-precision photometry with a CubeSat: ASTERIA\n observations of 55 Cancri e" ], [ "Abstract ASTERIA (Arcsecond Space Telescope Enabling Research In Astrophysics) is a 6U CubeSat space telescope (10 cm x 20 cm x 30 cm, 10 kg).", "ASTERIA's primary mission objective was demonstrating two key technologies for reducing systematic noise in photometric observations: high-precision pointing control and high-stabilty thermal control.", "ASTERIA demonstrated 0.5 arcsecond RMS pointing stability and $\\pm$10 milliKelvin thermal control of its camera payload during its primary mission, a significant improvement in pointing and thermal performance compared to other spacecraft in ASTERIA's size and mass class.", "ASTERIA launched in August 2017 and deployed from the International Space Station (ISS) November 2017.", "During the prime mission (November 2017 -- February 2018) and the first extended mission that followed (March 2018 - May 2018), ASTERIA conducted opportunistic science observations which included collection of photometric data on 55 Cancri, a nearby exoplanetary system with a super-Earth transiting planet.", "The 55 Cancri data were reduced using a custom pipeline to correct CMOS detector column-dependent gain variations.", "A Markov Chain Monte Carlo (MCMC) approach was used to simultaneously detrend the photometry using a simple baseline model and fit a transit model.", "ASTERIA made a marginal detection of the known transiting exoplanet 55 Cancri e ($\\sim2$~\\Rearth), measuring a transit depth of $374\\pm170$ ppm.", "This is the first detection of an exoplanet transit by a CubeSat.", "The successful detection of super-Earth 55 Cancri e demonstrates that small, inexpensive spacecraft can deliver high-precision photometric measurements." ], [ "Introduction", "ASTERIA, the Arcsecond Space Telescope Enabling Research In Astrophysics, is a small spacecraft designed to demonstrate enabling technologies for high-precision space-based photometry from small platforms.", "Space-based photometric measurements are a powerful tool for astrophysics, but time on existing large space telescopes is scarce.", "Small apertures in space can outperform ground-based telescopes in some metrics, such as temporal coverage and photometric precision.", "SmallSats and CubeSats have the potential to increase the availability of precision space-based photometric measurements, but their ability to perform measurements precise enough to be astrophysically useful must be demonstrated before that potential can be fully realized.", "The ASTERIA mission was launched to provide such a demonstration by measuring the transit of small exoplanets around nearby stars.", "In this paper we describe ASTERIA's photometric performance as demonstrated by observations of the transiting super-Earth 55 Cancri e. Section  presents an overview of the ASTERIA mission.", "Section  describes the 55 Cancri dataset.", "Section  describes image processing and transit fitting procedures; Section  presents the retrieved parameters.", "We summarize lessons learned from this mission in Section .", "Out of the thousands of exoplanets and thousands more planet candidates known to orbit main sequence stars, more than three quarters of them have been discovered by the transit technique.", "When a planet physically blocks light from its host star as it passes across the star's disk (transit), the star-to-planet area ratio can be measured photometrically as a small drop in the host star's brightness.", "The power of the transit technique implemented on a space platform is threefold.", "First, the planet-to-star size ratio is always more favorable than the planet-to-star mass ratio or the planet-to-star flux ratio.", "Second, transit discovery is not dependent on color or spectra such that a broad bandpass encompassing most of the visible light range can be used, increasing signal.", "Third, space-based missions above the blurring effects of Earth's atmosphere can reach much higher photometric precision than ground-based telescopes.", "Ideal satellite orbits do not suffer from the day/night cycle that breaks up transit observations for ground-based telescopes.", "For a review of space- and ground-based transit surveys see [13]." ], [ "ASTERIA in context: existing and future transit missions", "Several space-based missions have leveraged these advantages, each with distinct parameter space in terms of star type, star magnitude (and distance), and planet period (Table REF ).", "The pioneering Kepler Space Telescope [6] discovered thousands of exoplanets transiting Sun-like stars around relatively faint (V=10–15) and distant stars (typically over 1000 light years away).", "Nearly all of the Kepler planet host stars are too faint to permit follow-up measurements such as high-precision radial velocity to measure the planet mass.", "The MIT-led NASA Transiting Exoplanet Survery Satellite (TESS) mission (launched April 2018, [42]) is optimized for planets with orbital periods up to two weeks orbiting M dwarf stars; TESS will survey nearly the whole sky in a series of one month observation campaigns, with overlap and correspondingly longer temporal coverage at the ecliptic poles.", "The proposed ESA Planetary Transits and Oscillations of stars (PLATO) mission [10] aims to study bright (V of 4–11 mag) stars in a wide field of view (2256 square degrees), with 26 small (12 cm) telescopes mounted on the same platform.", "Both TESS and PLATO focus on bright stars amenable to spectroscopic follow-up observations.", "Accordingly, these missions use modest aperture sizes (10.5 cm for TESS, 12 cm for PLATO).", "ASTERIA, with a 6 cm aperture, is analogous to a single TESS or PLATO camera.", "ASTERIA is a 6U CubeSat technology demonstration mission [49].", "It was originally conceived as a prototype for one element of a constellation of stand-alone small space-based telescopes (Seager et al., in prep.).", "Note that the Bright Target Exoplorer (BRITE) constellation [55], while not having high-precision photometric capability, is an early example of a fleet of similar satellites for astronomy.", "The Fleet concept, described in Section REF , can be thought of as a distributed version of TESS or PLATO, where each camera is part of a free-flying spacecraft." ], [ "Space-based vs. ground-based transit photometry", "ASTERIA has reached an average photometric precision of 1000 ppm on a 60 second observation of a V$\\sim $ 6 star.", "For comparison we take HATNet [3] a collection of six ground-based telescopes with 11 cm apertures at three different geographic locations.", "One HATNet telescope can reach a photometric precision of $\\sim $ 3 mmag ($\\sim $ 3000 ppm) in a 3-minute observation for stars at the bright-star end (r$\\sim $ 9.5) [3].", "ASTERIA, despite a nearly 20 times smaller collecting area, performs slightly better than HATNet.", "With respect to larger ($>$ 1 m) ground-based telescopes, the only published 55 Cnc e transit detection from the ground is with the 2.5 m Nordic Optical Telescope (NOT) [12].", "The ALFOSC instrument on the NOT reaches a photometric precision of approximately 200 ppm in 7.5 minutes ($\\sim $ 800 ppm in 3 minutes); comparable to ASTERIA's best on the same star (1000 ppm in one minute).", "[12] determine a star-to-planet radius ratio of 0.0198$^{+0.0013}_{−0.0014}$ It may seem remarkable that a tiny space telescope such as ASTERIA has such a high photometric precision as compared to ground-based telescopes.", "This is because ASTERIA is free from scintillation and other effects of Earth's atmosphere and is therefore at an advantage over ground-based telescopes for observations of bright stars despite its small aperture size.", "See [35] for detailed discussion of the challenges of reaching sub-mmag photometric precision with ground-based telescopes.", "Scintillation is intensity fluctuation caused when starlight passes through regions of turbulence in Earth's upper atmosphere.", "Scintillation is seen by the naked eye as stars twinkling.", "Because scintillation is produced by high-altitude turbulence, the range of angles over which the scintillation is correlated is small, so correction using comparison stars is not usually helpful (but c.f.", "[31]).", "Estimated scintillation noise for a given star is described by [57] and [18] in units of relative flux, $ \\sigma _S = 0.09\\ D^{\\frac{-2}{3}} \\chi ^{1.75} (2\\ T_{int})^{\\frac{-1}{2}}e^{\\frac{-h}{h_0}},$ where $D$ is the diameter of the telescope in centimeters, $\\chi $ is the airmass of the observation, $T_{int}$ is the exposure time in seconds, $h$ is the altitude of the observatory in meters, and $h_0$ =8000 m is the atmospheric scale height.", "The constant 0.09 factor in front has a unit of cm$^{2/3}$ s$^{1/2}$ , such that the scintillation error in units of relative flux.", "Scintillation does not depend on stellar magnitude and therefore forms a noise floor that limits telescope performance, especially for bright stars.", "Using observatory site-specific atmospheric optical turbulence profiles, [37] showed that Equation (REF ) tends to underestimate the median scintillation noise by a mean factor of 1.5, and provides site-specific correction factors.", "Attempts to reduce scintillation noise by both scintillation noise correction concepts (e.g., [37] and observational strategies (e.g., [12] are ongoing.", "ASTERIA, free from atmospheric effects, focuses on mitigating other sources of non-Gaussian noise that limit photometric performance.", "We compare ground and space-based telescopes, assuming only photon and scintillation noise, and an air mass of 1.0 for an integration time of 100 seconds.", "ASTERIA is comparable to a $\\sim $ 2 m ground-based telescope, with these idealistic parameters.", "Here we have estimated fractional precision due to photon noise with $ \\sigma _P = 1/\\sqrt{At\\eta \\Delta \\lambda \\phi },$ where $A$ is collecting area in meters, $t$ is integration time in seconds, $\\eta $ is throughput (here taken to be 30%), $\\Delta \\lambda $ is wavelength range (V-band), and $\\phi $ is the incident photon flux for a star V=6.", "The total noise in this idealized comparison is $\\sigma = \\sqrt{\\sigma ^2_S + \\sigma ^2_P}$ .", "lcccccccc Space-based astronomy photometry-specific missions 0pt Mission Aperture Photometric Optimal mag.", "range FOV Bandpass Launch Orbit Ref Name [cm] precision [Vmag] [deg$^2$ ] [nm] MOST 15 1 350–750 Jun.", "2003 Polar LEO 1 CoRoT 27 5.4–16 6 400–1000 Dec. 2006 Polar LEO 2 Kepler/K2 95 8–16 116 420–900 Mar.", "2009 Earth trailing 3,4 BRITE 3 (x6) $\\le $ 4 24x24 390–460, 550–700 2013, 2014 Sun sync., LEO 5 ASTERIA 6.05 $<$ 6 11.2x9.6 500–900 Nov. 2017 LEO 6 TESS 10 (x4) 8–13 ($\\sim $ I band) 24x24 600–1000 Apr.", "2018 HEO 7 CHEOPS 32 6–12 0.16 400–1000 2019 Sun sync.", "LEO 8 PLATO 12 (x26) 4–11 1100 (x26) 500–1000 2026 Sun-Earth L2 9 References: (1) [54]; (2) [2]; (3) [6]; (4) [27]; (5) [55]; (6) [49]; (7) [42]; (8) [9]; (9) [41].", "Note that BRITE, ASTERIA, and CHEOPS focus observations on one specific target star at a time whereas the other missions are surveys for exoplanet discovery." ], [ "ASTERIA mission", "ASTERIA was designed to mitigate two key sources of systematic noise in space-based photometry: time-varying pointing errors and thermal variability.", "Errors in spacecraft pointing causes star centroids to drift across pixels on an array detector, inducing systematic variation in retrieved photometry due to intra- and inter-pixel gain variations [29].", "Much like scintillation for ground-based telescopes, pointing error results in a magnitude-independent noise floor for space-based observations.", "Thermal variations in the detector, optics, and electronics also induce systematic effects in photometry due to thermally dependent gains, mechanical expansion/contraction, and subtle changes in electronics performance.", "ASTERIA has its roots in the 3U CubeSat ExoplanetSat, developed by MIT in collaboration with Draper Lab [48].", "ASTERIA is a 6U CubeSat with dimensions 239 mm x 116 mm x 366 mm, and mass of 10.2 kg (Figure REF ).", "For a definition of the CubeSat form factor, see [24].", "See [49] for a detailed description of ASTERIA's system and subsystem design, [5] for a detailed description of the flight software framework, [17] for an overview of mission assurance and fault protection.", "ASTERIA was launched as cargo to the International Space Station (ISS) in August 2017 and deployed into space from ISS on November 20, 2017.", "The three month prime mission ended in February 2018 with the successful verification of all technology demonstration requirements.", "ASTERIA operations continued through several extended missions for a total of two years (eight times its nominal mission lifetime) until loss of contact in December 2019." ], [ "Technology demonstration goals", "ASTERIA was a technology demonstration mission rather than a science-driven mission, though the key technologies demonstrated were selected to enable future science missions.", "ASTERIA had two key technology demonstration goals: Line-of-sight pointing stability of 5 arcseconds RMS Thermal stability of $\\pm $ 10 milliKelvin RMS for a single point on the focal plane ASTERIA achieved 0.5 arcsecond RMS pointing stability on-orbit, approximately 10 times better pointing performance than achieved by other spacecraft in ASTERIA's mass and size category (e.g.", "BRITE).", "Fine pointing control was achieved with a two-stage control system.", "The coarse attitude of the spacecraft was controlled via reaction wheels and a star tracker.", "Residual pointing drift is mitigated by a closed-loop control system that measures centroids of guide stars on the payload detector and actuates a piezoelectric x-y translation stage holding the payload detector.", "See [38] for details of the pointing control system design and on-orbit performance.", "ASTERIA achieved $\\pm $ 5 milliKelvin thermal control on-orbit, measured over a 20-minute period at a single point on the back of the focal plane, approximately 100 times better than achieved by other spacecraft in ASTERIA's mass and size category.", "Like pointing control, thermal stability is also achieved using a two-stage system.", "The spacecraft payload (baffle, optics, focal plane, piezo stage, and readout electronics) is thermally isolated from the rest of the spacecraft using titanium bi-pod mounts and thermally insulating connectors.", "Thermal isolation reduces the orbital thermal variation to $\\sim $ 500 milliKelvin amplitude.", "A closed-loop control system raises the temperature of the focal plane to a set point slightly above the peak of orbital temperature variation using resistive heaters and holds the set temperature via feedback from high-precision thermal sensors mounted to the focal plane.", "See [49] for thermal control design and performance.", "Figure: ASTERIA interior layout (left) and fully assembled in the lab (right).", "In the interior view, payload components are listed in green; bus components in blue.", "Inset at top left shows exterior view, including deployed solar panels.", "The photograph on the right shows ASTERIA with solar panels deployed shortly before delivery to the launch provider.", "Photo credit: NASA/JPL." ], [ "Payload and camera design", "ASTERIA's payload is composed of the optical telescope assembly and the payload electronics.", "The optical telescope assembly includes a short baffle, a refractive optic (f/1.4, 85 mm), and the CMOS imager mounted to the piezoelectric stage.", "A refractive optic was chosen for ASTERIA both for its compactness and for its wide field of view.", "The pointing control algorithm needs several bright (V$<$ 6) guide stars to perform fine pointing adjustments, so a field of view several degrees across was required to ensure that sufficient guide stars were available on the detector for any given pointing.", "The telescope was deliberately defocused to oversample the PSF.", "The thermal control system is integrated with the optical telescope assembly, which is thermally isolated from the rest of the spacecraft.", "See Table REF for detailed specifications of the optical system and imager.", "ASTERIA used a Fairchild 5.5 megapixel CIS2521F CMOS imager as its science detector.", "A CMOS imager was chosen over a CCD for several reasons.", "The primary factor driving the detector choice is that the imager must be capable of fast readout both to accommodate the 20 Hz fine pointing control loop and to allow unsaturated observation of very bright science targets.", "At the time ASTERIA was in development, only CMOS detectors were capable of sufficiently fast readout.", "Additionally, the CMOS imager is designed to operate at room temperature, simplifying ASTERIA's thermal design.", "See [34] for a review of CCD vs CMOS technology.", "lc[h!]", "ASTERIA payload specifications 0pt Parameter Value Optics type Refractive Aperture diameter [mm] 60.7 Focal length [mm] 85 Pass band [nm] 500–900 Lens throughput 80% Detector dimensions [pixels] 2592 x 2192 Pixel size [$\\mu $ n] 6.5 x 6.5 Plate scale [arcsec/pixel] 15.8 Detector field of view [deg] 11.2 x 9.6 Quantum efficiency (mean across band) 42% Gain [e-/ADU] 6.44 ADC bit depth 11 The CMOS imager is divided into a top and bottom half; each half has a separate analog amplifier for each column.", "The gain of each column's amplifier is slightly different and must be corrected.", "There are eight optically dark (physically blocked from light) and eight electrically dark (electrically tied to ground so that photon-induced electrons cannot accumulate) rows at the top and bottom edges of the detector.", "The electrically dark pixels are used for bias and flat calibration (Section REF ).", "The detector also has 16 optically dark columns at the left and right edges of the detector for additional calibration; these are not used in the image calibration procedure.", "ASTERIA's CMOS-based camera was customized to read out subarrays of the full imager.", "ASTERIA uses this functionality for observations in fine pointing control mode.", "Eight 64 $\\times $  64 pixel windows, selected and defined on the ground, are centered on bright stars and read out at 20 Hz.", "The fine pointing control algorithm calculates centroid positions for each star and calculates the piezo stage motion needed to keep the designated target star motionless on the detector [39], [49], [38].", "See Figure REF for window layout used in 55 Cnc observations.", "Since the fine pointing control algorithm requires updates at 20 Hz, the maximum integration time in fine pointing mode for both science targets and guide stars is 50 msec.", "While some CMOS imagers are capable of reading out subarrays at different integration times, ASTERIA's detector and readout electronics do not allow that mode of operation, so all windows must have the same integration time and be read out at the same time.", "The integration time can be set to values less than 50 msec to avoid saturation for bright targets.", "Figure: Detector layout and window placement for 55 Cnc observations.", "Window 8 contains the target star; window 7 is a calibration window placed on the optically and electrically dark pixels at the bottom of the detector (indicated by double line border).", "The calibration window covers the same detector columns as the target star window.", "The remaining windows are used as guide stars by the fine pointing control system.", "The detector is 2592 ×\\times 2192 pixels, corresponding to 11.2 ∘ ^\\circ by 9.6 ∘ ^\\circ on the sky.The payload has the capability to sum, or co-add, many 50 msec frames together.", "Co-adding significantly reduces data volume while increasing signal-to-noise ratio (SNR) in each image.", "In typical operation, 1200 50 msec windowed frames, with eight 64 $\\times $  64 pixel windows, are summed into a one minute exposure for each 64 $\\times $  64 window.", "This one minute cadence data is downlinked to the ground for photometric analysis.", "No registration is performed prior to summing because the fine pointing system controls the payload pointing to 0.5 arcseconds RMS, approximately $\\frac{1}{30}$ th of pixel width (plate scale is 15”/pixel)." ], [ "Opportunistic science program", "The ASTERIA science team selected several target stars of scientific interest that met the needs of the pointing control and thermal control technology demonstrations in order to also demonstrate ASTERIA's photometric capability.", "Opportunistic science observations were carried out during ASTERIA's prime mission in parallel with verification of technology demonstration goals.", "The demonstration of the fine pointing control system required repeated observations of star fields since guide stars are used to control the piezo stage correction of small pointing errors.", "The thermal control system demonstration required the camera system (piezo stage, detector, and payload electronics) to be enabled since those components produce significant thermal loads." ], [ "Science targets", "ASTERIA's small aperture and maximum integration time of 50 msec limit useful science observations to stars brighter than V=7.", "Even brighter stars, V$<$ 6, were required as guide stars for the fine pointing control demonstration.", "Two star fields, 55 Cancri and HD 219134, were selected for the technology demonstration observations because they provided a favorable set of guide stars and were scientifically interesting targets.", "A third field, centered on $\\alpha $ Centauri, was observed after the technology demonstration observations had concluded.", "Several additional fields were observed during ASTERIA's extended mission, including the eclipsing binary Algol.", "Results from 55 Cancri observations are described in this paper; results from HD 219134 and $\\alpha $ Centauri will be presented in future publications.", "55 Cancri is a nearby Sun-like star (12.5 pc, spectral type G8 V [52]) that is host to five exoplanets, one of which, 55 Cancri e, is known to transit.", "55 Cancri e is a 2$R_{\\oplus }$ planet with an 18-hour orbital period; its transit was first detected by the Spitzer and MOST space telescopes [15], [56].", "ASTERIA observed 55 Cancri during the prime mission and first extended mission in an effort to detect the transit of 55 Cancri e and thereby demonstrate a high level of photometric precision.", "55 Cancri was the primary target star for the technology demonstration campaign.", "Table REF lists all observations used to create the final lightcurve described in this paper.", "All images consisted of 1200x50 ms (60 second) coadded exposures." ], [ "Observational planning and constraints", "ASTERIA observations must be planned within constraints imposed by both orbital geometry and spacecraft health and safety.", "ASTERIA's orbit is similar to that of the International Space Station (ISS), nearly circular with an altitude of $\\sim $ 400 km and orbital inclination of 51.6 degrees to the Earth's equator.", "The Earth occupies slightly less than half of the sky visible to ASTERIA.", "ASTERIA passes through the Earth's shadow (eclipse) for an average of 30 minutes out of its 92-minute orbit.", "Eclipse duration varies with solar $\\beta $ angle throughout the year, with brief periods of no eclipse and a maximum eclipse duration of 35 minutes.", "ASTERIA's baffle and thermal control system are designed for operation in eclipse only." ], [ "Observation planning", "ASTERIA observations are planned and sequenced for multiple spacecraft orbits.", "An observation is defined as the data collected during one eclipse period.", "There is one eclipse per spacecraft orbit, so there is a one-to-one correspondence between observations and orbits.", "When multiple observations/orbits are sequenced together, the thermal control system remains active across all observation orbits (both during eclipse and the sunlight portion of the orbit).", "Keeping the thermal control system running reduces thermal transients, though the first 1–2 orbits in a sequence exhibit a moderate thermal transient as the control system settles.", "Temperature data from sensors on the payload and throughout the spacecraft is available in time-tagged housekeeping data that is downlinked from the spacecraft separately from image data and later written into image FITS file headers.", "The camera system is power cycled and reinitialized at the beginning of each orbit just before image data is collected and then again at the end of an orbit after image data collection has finished.", "Power cycling and reinitialization of the camera puts the payload in a known good state and reduces the chance of sync loss between the camera FPGA and the flight computer.", "Power cycling the camera presents an impulse to the thermal control system since the camera dissipates $\\sim $ 2 W (see [49], Figure 27); the overshoot and settling from this impulse is observed in the photometric data (see Figure REF , top right, Section ).", "The camera is left on in a free running mode during the sunlit portion of the orbit to maintain a stable thermal environment because of the camera's large thermal dissipation.", "When all sequenced observations are complete, the camera and thermal control system are shut off and the spacecraft radio is turned back on.", "Image files resulting from the observation are losslessly compressed on board via the standard UNIX tar and gzip algorithms, and downlinked during subsequent communication passes." ], [ "Geometric and orbital constraints", "Four geometric conditions must be met for viable photometric observations: A clear line of sight exists between ASTERIA and the target star (no obstruction by the Earth, Sun, or Moon).", "ASTERIA is in Earth eclipse (full umbral shadow).", "The payload boresight must form an angle $>$ 90 degrees with the nadir vector (pointing from the spacecraft to the Earth center).", "The payload boresight must form an angle $>$ 20 degrees with the vector to the Moon.", "ASTERIA observes in eclipse only to avoid excess stray light from the SunTest observations have indicated that daylight observations may be feasible in the case where there is a large angle ($>$ 100 degrees) between the Sun and the target star, but such geometry is not common, so we do not perform daylight observations as part of normal operations..", "The third constraint on boresight to nadir angle prevents observations close to the Earth limb, where stray light from the illuminated limb degrades photometric data quality.", "This is a more restrictive requirement than simple line-of-sight (constraint 1).", "The fourth constraint seeks to avoid excess stray light from the Moon.", "The geometric constraints listed here are for data quality, not spacecraft safety.", "Pre-flight analysis showed that ASTERIA can safely point anywhere in the sky, including at the Sun, though doing so will temporarily reduce the effectiveness of the payload's thermal isolation and the active thermal control system." ], [ "Operational constraints", "Additional constraints are imposed on ASTERIA photometric observations in order to ensure spacecraft safety.", "Balancing the power needs of spacecraft subsystems during eclipse observations is critical since all electrical power for the spacecraft is drawn from the storage batteries during eclipse.", "The payload and the radio may not operate at the same time because their combined power draw quickly drains the battery.", "Furthermore, pre-flight testing revealed interference between the payload and the radio, so the radio must be turned off when the payload is on, and vice versa.", "Therefore ASTERIA cannot simultaneously perform observations and communicate with the ground.", "ASTERIA's fault protection system includes protection from battery undervoltage by means of a 43-minute `on-sun timeout' [17].", "Commanded slews to point ASTERIA at a target star must be carefully timed so that the total off-sun time, including slewing to to the target star at the beginning of an observation, eclipse time, and slewing back to a sun-pointed attitude at the end of an observation, does not exceed 43 minutes.", "Practically, this means that ASTERIA typically waits until it is already in eclipse to slew to the target star, slightly reducing the available time in eclipse for observations.", "The duration of observations is further limited by the size of the image memory buffer.", "Preallocated memory holds co-added images as they accumulate during an observation.", "During the prime mission, the hard-coded image memory buffer could hold up to 20 minutes of 1-minute co-added images.", "A flight software update at the beginning of ASTERIA's first extended mission increased the image memory buffer allocation to 24 minutes of 1-minute co-added data; a subsequent flight software update increased the capacity to 30 minutes.", "Images collected past the image buffer size are simply dropped and not written into memory.", "The final operational constraint on observations concerns the buildup of momentum in the spacecraft.", "If an attitude maneuver requires the wheels to spin faster than their maximum rate, the wheels are `saturated' and they cannot respond to the requested attitude change.", "Torque rods are used to desaturate the reaction wheels by exerting a torque against the Earth's magnetic field, allowing the reaction wheels to decrease their speed.", "Observations must be planned such that the momentum does not build up to the point where the reaction wheels become saturated and can no longer hold the commanded star-pointed attitude.", "ASTERIA has a residual dipole moment, measured at 0.17 Am$^2$ pre-flight [38], which interacts with the Earth's magnetic field and causes momentum to build up quickly in some attitudes.", "A MATLAB simulation tool is used to check the momentum build-up for each observation to ensure that the reaction wheels do not become saturated.", "Predicting reaction wheel speed during observations is important for data quality as well as spacecraft safety.", "Reaction wheels speed changes constantly to control spacecraft attitude and sometimes the speed of one or more wheels will pass through zero, meaning the wheel slows down and reverses its spin direction.", "Reaction wheel zero crossings induce a transient in spacecraft pointing (see [38], Figure 29), which causes an excursion in star centroid positions on the detector.", "We can bias the reaction wheel speeds before an observation in order to avoid zero crossings, though the bias must be carefully chosen to avoid wheel speed saturation.", "During the observation planning process, we adjust the wheel speed bias iteratively and examine the resulting wheel speed and momentum predictions in order to choose a bias that avoids wheel speed zero crossings while also keeping accumulated momentum within safe limits." ], [ "South Atlantic anomaly", "The South Atlantic Anomaly, while not a direct constraint on ASTERIA operations, can cause minor excursions in photometric data.", "The South Atlantic Anomaly (SAA) is a region over South America and the South Atlantic ocean where the Earth's radiation belts extend to low altitudes.", "Spacecraft in low Earth orbit passing through the SAA experience increased energetic proton flux, which can cause single event upsets in electronics and hot pixels in array detectors.", "ASTERIA does operate and observe through the SAA.", "Data collected at 20 Hz from the pointing control system indicate that transient hot pixels appear on the detector during SAA passages [38].", "These transient hot pixels last for one or two 50 msec frames before returning to background levels.", "The pointing disruptions caused by transient hot pixels are short compared to 1-minute co-added exposures and do not affect photometric precision [38].", "No build-up of persistent hot pixels has been observed.", "Data from ASTERIA observations are downlinked as binary files and then translated into FITS files using custom python software.", "Metadata, including spacecraft housekeeping telemetry (e.g.", "temperature measurements), are included in the FITS header for detrending.", "The primary data reduction challenge for ASTERIA is addressing column-dependent gain variation (CDGV).", "CDGV is specific to CMOS detectors because each column is tied to a specific amplifier and analog-to-digital (ADC) converter.", "Additionally, each pixel has its own amplifier.", "We performed extensive characterization of the detectors in the laboratory and tested various reduction recipes on the simulated data in order to find the optimal solution for ASTERIA data, which is presented below.", "For details on the laboratory characterization and reduction strategies assessed, see [33]." ], [ "Calibration data", "Traditional calibration frames (biases, darks, flats) were not collected during ground testing because they were not required to verify ASTERIA's technology demonstration requirements.", "Approximations for these calibration frames were collected in flight with mixed success.", "Bias (zero exposure time) and flat (uniform illumination) frames are typically used to correct for offsets and pixel-to-pixel gain variation, respectively.", "Dark frames are not needed for ASTERIA data reduction due to the short 50 msec integration time.", "Near-zero exposure time bias frames were collected on orbit in windowed mode at the minimum integration time the detector can support (22.6 $\\mu $ sec).", "Individual bias frames are coadded onboard the spacecraft in the same manner as light images.", "Bias frames collected in this manner (i.e.", "at a different time than the light images they are used to correct) were found to be less effective than calibration windows collected simultaneously with the target star images.", "The calibration window was placed at the upper or lower edge of the detector to capture the electrically/optically dark pixels.", "The calibration window covered the same detector columns as the light image.", "See Figure REF , where window 7 is the calibration window for window 8.", "Flat frames were approximated by using stray sunlight to illuminate the detector.", "The MOST space telescope used stray light illumination for on-orbit flat frames, although the stray light illumination was an unintended effect [43], while the Hubble Space Telescope has used the illuminated Earth for flat fielding [26].", "In order to capture stray light flats, ASTERIA was commanded to an attitude with an offset angle of 40 degrees from the sun vector during orbit day.", "The spacecraft then rotated about the camera boresight vector while collecting images to smear out the stray light illumination.", "Flat frames were also summed onboard the spacecraft to increase signal.", "This method produced images with 25-50% illumination as desired, but the illumination was not uniform across the detector." ], [ "Data reduction", "We begin the data reduction process by manually reviewing every coadded image and discarding frames that were obtained during the orbital “sunrise” or “sunset.” These frame appear either at the beginning or end of the orbit, and display significant background contamination from stray light.", "In a few cases, a bug in the pointing system caused the stars to be offset from the center of their windows; these data were discarded as well.", "We then proceed to bias and background correction.", "The median of the electrically dark bias column from the calibration window is subtracted from all pixels in the respective column in the science frame.", "We then subtract background/sky noise by selecting pixels in the target star window (rows 0-15) that are devoid of any stellar flux, taking the median of the pixels for each column, and subtracting that median value from every pixel in that column.", "Finally, we correct for column-dependent multiplicative gain.", "As noted in REF , the on-sky flatfield image was not evenly illuminated in region of the detector corresponding to the 55 Cnc window, and so could not be used for this purpose.", "Instead, for the 55 Cnc dataset, we divide each pixel in a given column by the normalized median of the corresponding bias column from the calibration window.", "The output of each processing step is shown in Figure REF .", "The target star is visually cleaner than the raw image, although there is still some residual column-dependent noise.", "Figure: 55 Cnc image calibration steps.", "The star closest to the center of the window is 55 Cancri; the star to the upper right is 53 Cancri.", "The frame is composed of 1200 50 msec exposures summed onboard ASTERIA.", "The window is 64x64 pixels square.", "See Figure for the locations of the light and calibration windows on the detector.Figure: (Left) Stellar image after performing removal of bias and background, and correction for flat field variations.", "The red circle indicates the photometric aperture.", "(Right) Reduced image in the log scale still showing column-dependent gain variations especially around the target.", "Note that 53 Cnc looks larger despite being slightly fainter than 55 Cnc in V band (V=6.23, ) due to its redder spectrum; chromatic aberration in the ASTERIA optics produces a larger PSF at longer wavelengths." ], [ "Centroiding and aperture photometry", "In order to measure the target star x/y position on the detector, we compute the flux-weighted 1st moments for each pixel in the subarray window along x and y.", "We use the mean centroid position to determine the target center, which is used as input to the aperture photometry procedure.", "We compute the fluxes using an array of apertures with radii ranging from 5 to 13 pixels.", "Observations of 55 Cnc are complicated by the spatial proximity of 53 Cnc (about 20 pixels), which is $\\delta $ V=0.28 fainter and partially blended with 55 Cnc.", "We compute for each aperture size the photometric precision given by the RMS of the light curve.", "As the flux contamination from 53 Cnc increases with aperture size, the contribution from background noise overtakes the improvement in photometric precision.", "An optimal aperture with a radius of 10 pixels is obtained based on the minimization of RMS.", "We use the CircularAperture routine from the photutils [8] python package to perform the photometric flux extraction from the subarray windows using a fixed hard-edged aperture with radius of 10 pixels for all 55 Cnc data.", "Visual inspection of the raw time-series shows that each visit starts with a telescope settling phase characterized by marked increased of the PSF FWHM for all stars of $\\sim $ 20%.", "This effect is likely due the thermal control system overcompensating for the drop in thermal load when the camera briefly turns off during reinitialization before the observation begins.", "We find however that the centroid positions and the overall background level are not affected.", "The proximity of 53 Cnc to 55 Cnc in ASTERIA's subarray aperture motivates us to attempt PSF fitting to extract the photometry.", "However, the ASTERIA PSF is field-dependent; guide stars from other windows cannot serve as PSF references.", "Thus, the only available PSF reference for 55 Cnc was 53 Cnc.", "Additionally, chromatic aberrations in ASTERIA's optics broaden the PSF of the redder 53 Cnc relative to 55 Cnc.", "We therefore find that aperture photometry outperforms PSF fitting.", "Figure: Initial (top left) and detrended (bottom left) lightcurves for one observation (so19).", "Data used for detrending, including sky background (center left), lens housing temperature (top right), and star centroid position (center, bottom right), are shown on the same time axis as the extracted photometry.", "A zoomed in inset (red) in the lens housing telemetry (top right) shows the orbit-to-orbit thermal behavior of the lens housing.", "The lens housing is not under active thermal control like the detector.", "The sharp rise in temperature seen in the second and fourth cluster of orbits happens when the camera first turns on and heats the lens housing; the sharp rise in temperature is not seen in the first and third sets of orbits because the camera failed to collect data during the first orbit of each set." ], [ "Photometric modeling", "For lightcurve detrending, we use the Markov Chain Monte Carlo (MCMC) algorithm implementation already presented in the literature [23].", "Inputs to the MCMC are the 55 Cnc photometric time-series and parameters described above.", "We explore different functional forms of the baseline model for each light curve (See Table REF for the results).", "These models can include a linear/quadratic trend with time, a second-order logarithmic ramp [30], [15] usually included for telescope settling with Spitzer and HST, a polynomial of the 1) centroid position and 2) onboard temperature measured at the lens housing, as well as linear combinations of the PSF FWHM along the x and y axes.", "We employ the Bayesian Information Criterion [44] to discriminate between the different baseline models.", "In the MCMC framework, the baseline model coefficients are determined at each step using a singular value decomposition method [40].", "During our tests, we find that the baseline model resulting in the lower Bayesian Information Criterion (Schwarz 1978) is a combination of a second order polynomial of the x/y centroid position, and a second order polynomial of the lens housing temperature.", "All results presented in this work have been obtained using this baseline model.", "We emphasise at this stage that these results are obtained over non-continuous visit durations of up to 5 hours on 55 Cnc, known to be a photometrically quiet star [21], [16].", "We expect the baseline model to become more complex for longer visits." ], [ "Transit fit", "We fit for the baseline model described above and the 55 Cnc e transit simultaneously in a same MCMC framework.", "We fit for the transit centre $T_0$ and the transit depth.", "We include the limb-darkening linear combinations $c_1=2u_1+u_2$ and $c_2=u_1-2u_2$ , where $u_1$ and $u_2$ are the quadratic coefficients drawn from the theoretical tables of [11] using published stellar parameters [53].", "We impose normal priors on the orbital period $P$ , impact parameter $b$ and the limb-darkening parameters $u_1$ and $u_2$ to the values shown in Table REF .", "We keep the eccentricity fixed to zero [14], [36].", "We execute two Markov chains of 50,000 steps each and assess their efficient mixing and convergence using the Gelman-Rubin statistic [22] by ensuring $r <1.001$ .", "Results for this MCMC fit are shown in Table REF ." ], [ "Noise properties", "The detector's average gain value of 6.44 e-/ADU, combined to the ADU measured counts translate to a photon-noise limit of about 400 ppm/min for 55 Cnc, depending on the visit.", "Our observations yield residual RMS between 620 ppm/min and 1450 ppm/min.", "The analysis of this noise excess shows that correlated noise contributes an average of 42% to the photometric noise budget over 15 to 100 min timescales.", "We attribute the remaining noise to astrophysical/instrumental noise that has not been characterized at this stage.", "We show in Figure REF the behavior of the photometric RMS with bin size, demonstrating the nominal contribution from correlated noise in the data.", "Figure: Photometric RMS versus bin size for all data sets.", "Black filled circles indicate the photometric residual RMS for different time bins.", "Each panel corresponds to an individual data set (see Table ).", "The expected decrease in Poisson noise normalized to an individual bin (1 min) precision is shown as a red dotted line." ], [ "55 Cancri e transit search", "The 55 Cancri e transit was chosen as a demonstration of ASTERIA's photometric capabilities since the transit ephemeris is known.", "We analyzed the data first using the literature values for the planet impact parameter ($b$ ), orbital period ($P$ ), and transit $T_0$ as priors, and then removed the priors to determine the transit detection efficiency in a blind search." ], [ "Transit search", "Our photometric analysis of the 55 Cnc ASTERIA data yields a transit detection at the $\\sim $ 2.2$\\sigma $ level.", "We conducted two MCMC fits: with and without priors on $T_0$ .", "Both fits used the priors on $b$ and $P$ listed in Table REF .", "When using the published $T_0$ as a prior [50], we find a transit depth median value of 374$\\pm $ 170 ppm (Figure REF , top).", "We show the probability distribution function of the retrieved transit depth in Figure REF .", "This is in agreement with the values published in the literature at similar wavelengths: [56] found 380$\\pm $ 52 ppm with MOST, [12] 361$\\pm $ 49 ppm between 457 and 671nm, 331$\\pm $ 36 ppm with HST STIS/G750L [7] and 346$\\pm $ 15 ppm in the global anaylsis of MOST data [50].", "Table REF lists all the retrieved parameters for the fit with a prior on $T_0$ .", "Without a prior on $T_0$ , we recover a transit depth median value of 637$\\pm $ 235 ppm and a transit center $T_0 = 8200.390^{+0.01}_{-0.01}$ (Figure REF (center)).", "The deviations in these two parameters are allowed, because the fit prefers parameters such that there is no coverage of the ingress and egress phases of the transit, leaving the transit parameters less constrained.", "The resulting best-fit value for the $T_0$ is off by 63 min compared to the most recent published ephemeris of 55 Cnc e [50]." ], [ "Blind transit search and injection-recovery test", "We then assessed our capability to detect 55 Cnc e transit in the ASTERIA data, without any prior knowledge on the planet (Figure REF , bottom).", "To this end we used our detrended data as input to the TLS code [25] to compute a periodogram (Figure REF .", "We identified the highest peak with an orbital period of 0.737 day, almost exactly the orbital period of 55 Cnc e. However this signal has a Signal Detection Efficiency [32] value of 4.9, which is not a firm detection (SDE$>$ 7).", "This finding is consistent with the MCMC fits conducted above.", "Figure: TLS Periodogram of the ASTERIA 55 Cnc e timeseries.", "The peak in the spectrum is marked in blue and matches 55 Cnc e's orbital period.", "The location of possible aliases are indicated in dash blue vertical lines.To assess the planet search completeness of our 55 Cnc dataset, we conducted an injection-recovery test using the allesfitter code (Günther & Tansu in prep).", "As the purpose of this step of the analysis was to determine the detection threshold at 55 Cnc e's ephemeris, we injected different planet sizes at the same $T_0$ and period.", "We find that we exceed a SDE of 7 [47] for planetary radii greater than 2.6 $R_{\\oplus }$ (vs. 55 Cnc e's $\\sim 1.9$  $R_{\\oplus }$ ).", "lcc[h!]", "55 Cnc e transit parameters 0pt Parameter Median & 1-$\\sigma $ credible interval Source Transit depth [ppm] $374 \\pm 170$ This work Impact parameter $b$ $0.40 \\pm 0.03$ prior [50] Transit T0 [BJD] $8200.4343^{0.0026}_{0.0024}$ prior [50] Period $P$ [days] $0.7365450 \\pm 0.0000001$ prior [50] Figure: Posterior distribution function of the transit depth of 55 Cnc e recovered from ASTERIA photometric observations, given priors on T 0 T_0, bb, and PP.Figure: Phase-folded lightcurves for 55 Cancri e. (a) Priors for PP, bb, and T 0 T_0 used; (b) Priors for PP, bb; (c) No transit model fit; phase-folded at the literature PP value." ], [ "Summary and discussion", "The ASTERIA CubeSat observed 55 Cancri e, a known transiting super-Earth orbiting a Sun-like star.", "The 55 Cnc data obtained with the ASTERIA CubeSat showed a marginal ($\\sim $ 2.5$\\sigma $ ) transit when fitted using priors from literature, but did not succeed in independently detecting the transit of 55 Cnc e, by a small margin.", "Our MCMC fits, completeness studies, and transit search demonstrate together that a signal is seen in ASTERIA data, but not at a level that is significant enough to claim independent detection without prior knowledge of the planet orbit and transit.", "Additional photometric data on other stars was obtained during ASTERIA's prime mission and extended missions; results will be presented in future publications using the photometric reduction framework described here.", "ASTERIA demonstrated subarcsecond pointing stability and $\\pm $ 10 milliKelvin thermal control using passive cooling and active heating [49], [38].", "ASTERIA has matured these key technologies to enable high-precision photometry in a small package for future astrophysics CubeSat/smallsat missions and demonstrated the utility of small spacecraft for cutting-edge photometric measurements above the blurring effects of the Earth's atmosphere." ], [ "Lessons learned", "ASTERIA's design was driven by technology demonstration requirements rather than science requirements.", "As such, there are several modifications and improvements that would enhance the capabilities of next generation astrophysics CubeSat missions.", "ASTERIA data showed excess systematic noise at timescales similar to the spacecraft orbit (90 minutes) and the duration of eclipse (20-30 minutes).", "The duration of the 55 Cnc e transit (96 minutes) is similar to the duration of ASTERIA's orbit, causing `clumping' of data around the transit phase over short timescales; multiple observations of the transit did not lead to uniform filling of the lightcurve in phase space.", "This effect led to a bias in the retrieved transit midpoint and depth (Section REF ).", "Current and future photometry missions with non-continuous coverage may be subject to similar bias if astrophysical periods (orbit, transit) are similar to instrumental periods (spacecraft orbit, observation duty cycle)." ], [ "Detector capability", "First, a detector architecture where subarrays could be read out at different rates would significantly expand the set of target stars available for observation.", "ASTERIA can only observe stars brighter than V=7 because there are not enough photons arriving from dimmer stars to reliably flip the first ADC bit in each 50 msec exposure.", "If no signal is registered by the ADC, no amount of co-adding will produce a usable signal.", "A CMOS imager and FPGA readout architecture capable of reading out guide star windows at 20 Hz for fine pointing control and science windows at a slower rate, allowing the collection of more photons per exposure, would allow a telescope with the same aperture size as ASTERIA to observe dimmer stars." ], [ "Absolute time tagging", "Future CubeSat photometry or spectroscopy missions would also benefit from improved time-tagging of photometric data.", "ASTERIA does not have a realtime clock or a functional GPS, so the flight computer clock resets to zero at every spacecraft reset.", "Absolute time-tagging was not necessary for the technology demonstration goals of the mission, so ASTERIA was launched with a non-functional GPS unit.", "The flight computer boot time in UTC must be calculated after each reset by comparing the UTC send time of a command with the logged receive time of the command in flight computer time, which is recorded as seconds from boot.", "Uncertainty in the UTC-spacecraft clock correlation is $\\ge $ 1 second because the ground data system records command send times with a precision of one second.", "A working GPS unit would solve this problem, as would an onboard realtime clock that does not reset when the flight computer reboots.", "One second time correlation precision is sufficient for the purposes of detecting transits with durations of many minutes to hours, but improved timing precision would open up additional applications, such as transit timing variation (TTV) measurement, for future missions." ], [ "Pre-flight camera characterization", "The test campaign for ASTERIA's payload was designed to ensure that it would be capable of demonstrating the key technologies of fine pointing control and thermal control.", "A future science-focused CubeSat photometry mission would benefit from pre-flight collection of bias, flat, and dark frames, and gain values, in multiple detector modes (full frame, windowed, co-added) over a range of flight-like temperatures.", "These calibration data could be used for reducing the photometric data rather than collecting calibration data on-orbit, where illumination conditions are not as easily controlled.", "The volume of available calibration data would also be much larger than what can be reasonably collected and downlinked from orbit.", "See [33]" ], [ "Orbit selection", "ASTERIA's orbit is not optimal for photometric timeseries observations because there is no continuous viewing zone and the observing duty cycle is capped at $\\sim $ 30% due to the requirement that observations take place in eclipse.", "A low Earth sun synchronous orbit would provide a stable thermal environment, near-continuous power generation, and longer continuous viewing.", "The MOST space telescope [54] used such an orbit and several upcoming CubeSat and smallsat missions will as well (SPARCS [46], [45], CHEOPS [9]).", "A low inclination orbit might be considered for future CubeSat space telescopes as well, as it provides continuous viewing of the celestial poles assuming sufficient baffling for stray light from the Sun and illuminated Earth.", "A low inclination orbit also avoids SAA passages, which would be important for sensitive detectors and/or electronics.", "See [49] and [17] for in-depth discussion of other lessons learned from ASTERIA development, integration and testing, and on-orbit operations." ], [ "Next steps", "Orbit decay projections show that ASTERIA will remain in orbit until at least spring 2020.", "Science operations ceased as of December 5, 2019 when contact with the spacecraft was lost.", "During its extended missions, ASTERIA also served as a testbed for new flight software and navigation techniques [20].", "The technologies and techniques ASTERIA has demonstrated are enabling for future astrophysics CubeSats and smallsats, including the ExoplanetSat Constellation (described below).", "A platform like ASTERIA with very stable pointing and thermal control could support a wide range of instruments, including photometers operating outside of the visible range (UV/IR) or low resolution spectrometers.", "It should be possible to scale the ASTERIA platform up in volume to a 12U+ form factor in order to increase aperture size without significantly changing the hardware and algorithms used to achieve subarcsecond pointing stability.", "ASTERIA was intended to be the first step toward a diverse fleet of small astrophysics missions, increasing access to space for the astrophysics community.", "ASTERIA was a successful technology demonstration of a future constellation of up to dozens of satellites, dubbed the ExoplanetSat Constellation.", "Each satellite would share ASTERIA's precision pointing and thermal control capabilities, operate independently from the others, but may have different aperture sizes in order to reach down to fainter stars than ASTERIA's current capability.", "The primary motivation is the fact that if there is a transiting Earth size planet in an Earth-like orbit about the nearest, brightest (V$<$ 7) Sun-like stars, we currently have no way to discover them; current missions saturate on these bright stars.", "The ultimate goal for the constellation is to monitor dozens of the brightest sun-like stars, searching for transiting Earth-size planets in Earth-like (i.e., up to one year) orbits.", "Because the brightest sun-like stars are spread all across the sky, a single telescope will not do.", "Instead, each satellite would monitor a single sun-like star target of interest for as long as possible, before switching to another star, with targets only limited by the Sun, Earth and Moon constraints.", "To narrow down the approximately 3,000 target stars brighter than V=7, one would have to find a way to constrain the stellar inclinations and assume the planets orbit within about 10 degrees of the star’s equatorial plane.", "This would reduce the number of target stars from about 3000 to about 300 [4], a much more tractable number of targets.", "The ExoplanetSat Constellation has a unique niche in context of existing and planned space transit surveys (Section ), but is still in concept phase.", "The authors thank the anonymous reviewer for their insightful and detailed comments which enhanced and improved the manuscript.", "The authors recognize the contributions of the extended team that supported ASTERIA development, integration and test, and operations, including Len Day, Maria de Soria Santacruz-Pich, Carl Felten, Janan Ferdosi, Kristine Fong, Harrison Herzog, Jim Hofman, David Kessler, Roger Klemm, Jules Lee, Jason Munger, Lori Moore, Esha Murty, Chris Shelton, David Sternberg, Rob Sweet, Kerry Wahl, Jacqueline Weiler, Thomas Werne, Shannon Zareh, and Ansel Rothstein-Dowden.", "We also recognize the JPL line organization and technical mentors for the expertise they provided throughout the project.", "We also wish to recognize JPL program management, especially Sarah Gavit and Pat Beauchamp, who oversaw ASTERIA within the Engineering and Science Directorate at JPL.", "We also thank Daniel Coulter and Leslie Livesay for their support.", "Finally, we would like to thank the DSS-17 ground station team at Morehead State University (MSU) in Kentucky.", "We acknowledge the outstanding efforts of the student operators, technical staff, and program management at Morehead State University, including Chloe Hart, Sarah Wilczewski, Alex Roberts, Maria Lemaster, Lacy Wallace, Rebecca Mikula, Bob Kroll, Michael Combs, and Benjamin Malphrus.", "We also thank Frank D. Lind and Mike Poirier of MIT Haystack observatory for their assistance in tracking ASTERIA during a communications anomaly.", "Funding was provided by the JPL Phaeton program and by the Heising-Simons Foundation.", "B.-O.", "D. acknowledges support from the Swiss National Science Foundation (PP00P2-163967).", "The research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).", "This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.", "astropy [51], astroquery [1], photutils [8], pyBJD (https://github.com/tboudreaux/pyBJD), matplotlib [28], IDL, STK, MATLAB clrchc 55 Cancri observations used in the light curve shown in Figure REF .", "0pt Observation First frame Last frame Number of 1-minute Phase range RMS name [UTC] [UTC] integrations [ppm/minute] Tech Demo 14 2018-01-07 11:33:36.7 2018-01-07 11:52:36.5 20 -0.063 – -0.045 1123 (td14) 2018-01-07 13:06:39.3 2018-01-07 13:25:39.1 20 0.025 – 0.042 2018-01-07 14:28:39.5 2018-01-07 14:47:39.3 19 0.102 – 0.120 Tech Demo 16 2018-01-10 10:26:35.5 2018-01-10 10:45:35.2 20 -0.053 – -0.035 986 (td16) 2018-01-10 12:01:39.9 2018-01-10 12:20:39.7 19 0.036 – 0.054 2018-01-10 13:25:57.3 2018-01-10 13:44:57.1 20 0.116 – 0.134 Tech Demo 18 2018-01-12 10:10:29.5 2018-01-12 10:29:29.3 19 -0.353 – -0.335 1088 (td18) 2018-01-12 11:43:03.9 2018-01-12 12:02:03.7 20 -0.266 – -0.248 2018-01-12 13:15:37.5 2018-01-12 13:34:37.3 20 -0.178 – -0.161 2018-01-12 14:54:42.8 2018-01-12 15:13:42.6 20 -0.085 – -0.067 2018-01-12 16:27:47.8 2018-01-12 16:46:47.6 20 0.003 – 0.021 2018-01-12 17:57:21.4 2018-01-12 18:16:21.2 20 0.087 – 0.105 2018-01-12 19:36:16.6 2018-01-12 19:55:16.4 20 0.180 – 0.198 2018-01-12 21:04:31.3 2018-01-12 21:23:31.1 19 0.264 – 0.282 2018-01-12 22:39:02.9 2018-01-12 22:57:02.7 18 0.353 – 0.370 2018-01-13 00:10:37.6 2018-01-13 00:29:37.4 20 0.439 – 0.457 Tech Demo 23 2018-01-22 06:24:41.2 2018-01-22 06:44:41.0 21 0.011 – 0.030 1016 (td23) 2018-01-22 07:57:17.0 2018-01-22 08:17:16.7 21 0.098 – 0.117 2018-01-22 09:29:55.1 2018-01-22 09:49:54.9 21 0.186 – 0.205 2018-01-22 10:59:28.3 2018-01-22 11:19:28.1 21 0.270 – 0.289 Tech Demo 24 2018-01-22 18:44:23.7 2018-01-22 19:04:23.5 21 -0.291 – -0.273 998 (td24) 2018-01-22 20:15:59.5 2018-01-22 20:35:59.3 21 -0.205 – -0.186 2018-01-22 21:47:35.2 2018-01-22 22:07:35.0 20 -0.119 – -0.100 2018-01-22 23:16:11.0 2018-01-22 23:36:10.7 20 -0.035 – -0.016 Science Observation 19 2018-04-13 20:44:02.5 2018-04-13 21:03:02.3 18 -0.205 – -0.187 798 (so19) 2018-04-13 22:16:36.3 2018-04-13 22:35:36.2 20 -0.118 – -0.100 2018-04-14 12:08:49.6 2018-04-14 12:27:49.4 20 -0.333 – -0.315 1193 2018-04-14 13:41:18.5 2018-04-14 14:00:18.3 20 -0.246 – -0.228 2018-04-14 15:13:46.3 2018-04-14 15:32:46.0 20 -0.159 – -0.141 2018-04-14 18:18:44.6 2018-04-14 18:37:44.4 20 0.0158 – 0.034 2018-04-15 09:42:29.4 2018-04-15 10:01:29.2 20 -0.113 – -0.095 1135 2018-04-15 12:47:29.5 2018-04-15 13:06:29.3 20 0.061 – 0.079 2018-04-16 01:07:25.4 2018-04-16 01:26:25.2 20 -0.241 – -0.223 1429 2018-04-16 02:39:50.4 2018-04-16 02:58:50.2 20 -0.154 – -0.136 2018-04-16 05:44:46.9 2018-04-16 06:03:46.7 20 0.020 – 0.038 2018-04-16 07:17:15.2 2018-04-16 07:36:15.0 20 0.108 – 0.125 Science Observation 20 2018-04-22 00:28:32.3 2018-04-22 00:43:32.1 16 -0.132 – -0.117 1956 (so20) 2018-04-22 02:01:01.9 2018-04-22 02:16:01.8 16 -0.044 – -0.030 2018-04-22 03:33:32.6 2018-04-22 03:48:32.5 15 0.043 – 0.057 2018-04-22 05:06:02.7 2018-04-22 05:21:02.5 16 0.130 – 0.144 Science Observation 24 2018-05-12 13:29:38.9 2018-05-12 13:48:38.6 19 -0.241 – -0.223 (not used) (so24) 2018-05-12 16:38:23.5 2018-05-12 16:57:23.3 20 -0.063 – -0.045 (not used) 2018-05-12 18:04:49.5 2018-05-12 18:23:49.3 18 0.018 – 0.036 639 Science Observation 25 2018-05-13 09:32:12.7 2018-05-13 09:51:12.5 19 -0.107 – -0.089 1031 (so25) 2018-05-13 11:08:36.4 2018-05-13 11:27:36.2 20 -0.016 – 0.002 2018-05-13 12:33:59.2 2018-05-13 12:52:59.0 19 0.064 – 0.082 2018-05-14 02:26:01.8 2018-05-14 02:45:01.6 19 -0.151 – -0.133 637 2018-05-14 19:26:05.3 2018-05-14 19:45:05.1 20 -0.190 – -0.172 1366 2018-05-14 20:58:34.1 2018-05-14 21:17:33.9 20 -0.102 – -0.085 2018-05-15 00:00:25.9 2018-05-15 00:19:25.7 20 0.069 – 0.087 Science Observation 26 2018-05-15 12:20:22.2 2018-05-15 12:41:22.0 22 -0.233 – -0.214 1487 (so26) 2018-05-15 15:25:19.8 2018-05-15 15:46:19.6 20 -0.059 – -0.039 2018-05-15 16:57:46.7 2018-05-15 17:18:46.4 22 0.028 – 0.048 Science Observation 28 2018-05-26 07:21:00.9 2018-05-26 07:35:00.7 15 (not used) (so28) 2018-05-26 08:53:20.2 2018-05-26 09:07:20.1 15 (not used) 2018-05-26 10:25:40.1 2018-05-26 09:07:20.1 15 (not used) 2018-05-27 09:29:45.0 2018-05-27 09:34:45.0 6 (not used) 2018-05-27 12:34:25.9 2018-05-27 12:39:25.8 6 (not used) 2018-05-28 20:53:39.3 2018-05-28 20:58:39.2 6 810 2018-05-28 22:26:00.8 2018-05-28 22:31:00.7 6 2018-05-28 23:58:21.4 2018-05-29 00:02:21.4 5 Science Observation 30 2018-06-08 18:36:29.9 2018-06-08 18:43:29.9 8 1535 (so30) 2018-06-08 20:08:52.2 2018-06-08 20:15:52.1 8 2018-06-08 21:42:13.2 2018-06-08 21:48:13.2 7 2018-06-08 23:13:35.9 2018-06-08T 3:20:35.8 8 2018-06-10 06:00:53.8 2018-06-10 06:07:53.7 8 1054 2018-06-10 07:33:17.4 2018-06-10 07:40:17.4 8 2018-06-10 09:05:37.9 2018-06-10 09:12:37.8 8 2018-06-10 10:38:01.3 2018-06-10 10:45:01.2 8 Observations designated `Tech Demo' or `td' took place during the initial phase of the mission when observations were focused on addressing ASTERIA's technology demonstration goals.", "After the technology demonstration objectives were completed, the naming scheme changed to 'Science Observation' or 'so' to indicate the change in focus from technology demonstration to collection of science data.", "Lightcurve and image data will be available at https://exoplanetarchive.ipac.caltech.edu/docs/ASTERIAMission.html" ] ]

[ [ "Promotion of Kreweras words" ], [ "Abstract Kreweras words are words consisting of n A's, n B's, and n C's in which every prefix has at least as many A's as B's and at least as many A's as C's.", "Equivalently, a Kreweras word is a linear extension of the poset ${\\sf V}\\times [n]$.", "Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration.", "Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane.", "We study Sch\\\"{u}tzenberger's promotion operator on the set of Kreweras words.", "In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the B's and C's.", "Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with `good' behavior under promotion, other than the four families of shapes classified by Haiman in 1992.", "We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg's $\\mathfrak{sl}_3$-webs, and Postnikov's trip permutation associated with any plabic graph.", "In this description, Sch\\\"{u}tzenberger's promotion corresponds to rotation of the web." ], [ "Introduction", "The famous ballot problem, whose history stretches back to the 19th century, asks in how many ways we can order the ballots of an election between two candidates Alice and Bob, who each receive $n$ votes, so that during the counting of ballots Alice never trails Bob.", "These ballot orderings correspond to words of length $2n$ in the letters $\\textrm {A}$ and $\\textrm {B}$ , with as many $\\textrm {A}$ 's as $\\textrm {B}$ 's, for which every prefix has at least as many $\\textrm {A}$ 's as $\\textrm {B}$ 's.", "Such words are called blueDyck words, and they are counted by the ubiquitous blueCatalan numbers $C_n \\frac{1}{n+1}\\binom{2n}{n}.$ In 1965, Kreweras  considered the following version of a 3-candidate ballot problem: in how many ways can we order the ballots of an election between three candidates Alice, Bob, and Charlie, who each receive $n$ votes, so that during the counting Alice never trails Bob and Alice never trails Charlie – although the relative position of Bob and Charlie may change during the counting?", "These ballot orderings correspond to words of length $3n$ in the letters $\\textrm {A}$ , $\\textrm {B}$ , and $, with equally many~$ A$^{\\prime }s, $ B$^{\\prime }s, and $ 's, for which every prefix has at least as many $\\textrm {A}$ 's as $\\textrm {B}$ 's and also at least as many $\\textrm {A}$ 's as $^{\\prime }s.We call such words {blue}{\\emph {Kreweras words}}.", "Kreweras proved that they are counted by the formula$$ K_n \\frac{4^n}{(n+1)(2n+1)}\\binom{3n}{n}.$$$ For many years Kreweras's formula seemed like an isolated enumerative curiosity, although simplified proofs were presented by Niederhausen , and Kreweras–Niederhausen  in the 1980s.", "Gessel  gave yet another proof which demonstrated that the generating function $\\sum _{n=0}^{\\infty }K_n \\, x^n$ for this sequence of numbers is algebraic.", "Interest in Kreweras's result was revived decades later in the context of lattice walk enumeration.", "Kreweras words evidently correspond to walks in $\\mathbb {Z}^2$ with steps of the form $A=(1,1)$ , $B=(-1,0)$ , and $C=(0,-1)$ from the origin to itself which always remain in the nonnegative orthant.", "Such walks are called blueKreweras walks.", "Bousquet-Mélou  gave another proof of Kreweras's product formula counting Kreweras walks using the kernel method from analytic combinatorics.", "Indeed, the Kreweras walks are nowadays a fundamental example in the study of “walks with small step sizes in the quarter plane,” a program successfully carried out over a number of years in the 2000s by Bousquet-Mélou and others (see, e.g., ).", "Finally, we note that Bernardi  gave a purely combinatorial proof of the product formula for the number of Kreweras walks via a bijection with (decorated) cubic maps.", "In this paper we study a cyclic group action on Kreweras words.", "Let $w=(w_1,w_2,\\ldots ,w_{3n})$ be a Kreweras word of length $3n$ .", "The bluepromotion of $w$ , denoted $\\operatorname{{Pro}}(w)$ , is obtained from $w$ as follows.", "Let $\\iota (w)$ be the smallest index $\\iota \\ge 1$ for which the prefix $(w_1,w_2,\\ldots ,w_{\\iota })$ has either the same number of $\\textrm {A}$ 's as $\\textrm {B}$ 's or the same number of $\\textrm {A}$ 's as $^{\\prime }s. Then$$ \\operatorname{{Pro}}(w) (w_2,w_3,\\ldots ,w_{\\iota (w)-1}, \\textrm {A}, w_{\\iota (w)+1}, w_{\\iota (w)+2}, \\ldots , w_{3n}, w_{\\iota (w)}).$$It is easy to verify that $ Pro(w)$ is also a Kreweras word, and that promotion is an invertible action on the set of Kreweras words.$ Example 1.1 Let $w = \\textrm {A}\\textrm {A}\\textrm {B}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g}\\textrm {B}};} \\textrm {B}$ .", "Here we circled the letter $w_{\\iota (w)}$ , and hence $\\operatorname{{Pro}}(w) = \\textrm {A}\\textrm {B}\\textrm {A}\\textrm {B}\\textrm {B}$ .", "We can further compute that the first several iterates of promotion applied to $w$ are $\\operatorname{{Pro}}(w) &= \\textrm {A}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g}\\textrm {B}};} \\textrm {A}\\textrm {B}\\textrm {B}\\\\\\operatorname{{Pro}}^2(w) &= \\textrm {A}\\textrm {A}{ \\textrm {B}\\textrm {B}\\textrm {B}\\\\\\operatorname{{Pro}}^3(w) &= \\textrm {A}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g};} \\textrm {A}\\textrm {B}\\textrm {B}\\textrm {B}\\operatorname{{Pro}}^4(w) &= \\textrm {A}\\textrm {A}\\textrm {B}\\textrm {B}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g}\\textrm {B}};} \\\\\\operatorname{{Pro}}^5(w) &= \\textrm {A}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g};} \\textrm {A}\\textrm {B}\\textrm {B}\\textrm {A}\\textrm {B}\\\\\\operatorname{{Pro}}^6(w) &= \\textrm {A}\\textrm {A}\\textrm {B}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g}\\textrm {B}};} \\textrm {A}\\textrm {B}\\operatorname{{Pro}}^7(w) &= \\textrm {A}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g}\\textrm {B}};} \\textrm {A}\\textrm {A}\\textrm {B}\\\\\\operatorname{{Pro}}^8(w) &= \\textrm {A}\\textrm {A}\\textrm {A}\\textrm {B}[baseline=(char.base)]{\\node [shape=circle, draw, inner sep=0pt,minimum height={*0.9},] (char) {\\vphantom{WAH1g};} \\textrm {B}\\textrm {B}\\\\\\operatorname{{Pro}}^9(w) &= \\textrm {A}\\textrm {A}\\textrm {B}\\textrm {A}\\textrm {B}\\textrm {B}{align*}Note that \\operatorname{{Pro}}^9(w) is obtained from w by swapping all \\textrm {B}^{\\prime }s for~^{\\prime }s and vice-versa.", "}}Our first result predicts the order of promotion on Kreweras words:}\\begin{thm} Let w be a Kreweras word of length 3n.", "Then \\operatorname{{Pro}}^{3n}(w) is obtained from~w by swapping all \\textrm {B}^{\\prime }s for ^{\\prime }s and vice-versa.", "In particular, \\operatorname{{Pro}}^{6n}(w)=w.\\end{thm}}Promotion of Kreweras words comes from the theory of partially ordered sets.", "In a series of papers from the 60s and 70s, Schützenberger~\\cite {schutzenberger1963quelques, schutzenberger1972promotion, schutzenberger1973evacuations} introduced and developed the theory of a cyclic action called {blue}{\\emph {promotion}}, as well as a closely related involutive action called {blue}{\\emph {evacuation}}, on the {blue}{\\emph {linear extensions}} of any poset.", "Let V(n) denote the Cartesian product of the 3-element ``V^{\\prime \\prime }-shaped poset~\\begin{tikzpicture}[scale=0.3] \\node [shape=circle,fill=black,inner sep=1] (B) at (-1,0) {}; \\node [shape=circle,fill=black,inner sep=1] (C) at (1,0) {}; \\node [shape=circle,fill=black,inner sep=1] (A) at (0,-1) {}; (B)--(A); (C)--(A); \\end{tikzpicture} and the n-element chain [n].", "Then, as observed by Kreweras--Niederhausen~\\cite {kreweras1981solution}, the linear extensions of V(n) are in obvious bijection with the Kreweras words of length~3n.", "And promotion of Kreweras words as described above is the same as Schützenberger^{\\prime }s promotion on the linear extensions of V(n).$ Previously there were only four known (non-trivial) families of posets for which the order of promotion can be predicted; see fig:haimanposets.", "These were classified by Haiman in the 1990s , .", "In a survey on promotion and evacuation, Stanley  asked whether there were any other families of posets for which the order of promotion is given by a simple formula.", "Our work shows that $V(n)$ is such an example.", "Figure: Promotion of Dyck words as rotation of noncrossing matchings.Figure: The local rule for growth diagrams of linear extensions." ] ]

[ [ "Scaling Participation -- What Does the Concept of Managed Communities\n Offer for Participatory Design?" ], [ "Abstract This paper investigates mechanisms for scaling participation in participatory design (PD).", "Specifically, the paper focuses on managed communities, one strategy of generification work.", "We first give a brief introduction on the issue of scaling in PD, followed by exploring the strategy of managed communities in PD.", "This exploration is underlined by an ongoing case study in the healthcare sector, and we propose solutions to observed challenges.", "The paper ends with a critical reflection on the possibilities managed communities offer for PD.", "Managed communities have much to offer beyond mere generification work for large-scale information systems, but we need to pay attention to core PD values that are in danger of being sidelined in the process." ], [ "Introduction", "In a recent article, Bødker and Kyng [2] propose an agenda for Participatory Design (PD) that Matters.", "One point they make is the need to scale PD to make it relevant for the design of large IT systems and infrastructures.", "The issue of scaling is not new in PD (see e.g.", "[14]), but the democratic implications of (not) being able to support PD on larger scales, especially in modern infrastructures, are rarely addressed: \"In our view, the centralization of (the major commercial platforms on) the Internet, big data, and large-scale infrastructuring challenge the core democratic ideals of PD\" [2].", "PD has historically focused on the democratisation of practices and equalisation of power relations [2], [9].", "These values need to be considered when discussing the scaling of PD.", "Our research interest is in investigating scaling mechanisms with these fundamental PD values in mind.", "In this paper we aim to explore a specific strategy, i.e.", "managed communities by Pollock et al.", "[11] as a scaling strategy for design work in PD.", "Software vendors actively engage managed communities – i.e.", "large groups of users from multiple current or future customer organisations – in continuous (re)design of their products.", "While managed communities in Pollock et al.", "'s original paper were described from a top-down managerial perspective, we believe the concept and its processes are valuable for discussing the scaling of PD as well.", "We investigate the managed communities strategy with insights from an ongoing case study of a software company that engages its customers and users in large yearly user conferences.", "In this early stage of our case study we look into how these conferences are organised, and what role they play as managed communities in PD.", "For this we combine Pollock et al.", "'s concept of managed communities with a framework proposed by Borum and Enderud [3].", "According to this framework, the suitability of user conferences as a PD tool depends on their agenda, participants, scope, and resources (such as time) available to users.", "Combining Pollock et al.", "'s concept of managed communities with the framework of Borum and Enderud, the contribution of our paper to PD research is a discussion of whether managed communities, originally described as a management tool, can be modified to become a scaling tool for PD.", "In the rest of this paper we first introduce some relevant related work on managed communities.", "We then present our case and our preliminary findings before we conclude with a discussion." ], [ "Managed communities and PD", "Achieving scale while at the same time not losing focus of core values has drawn recent attention in the PD community [2], [17], [13].", "Within this line of research, our inspiration for this paper is from Pollock et al.", "[11] exploring the following paradox: scholars often emphasize the importance of organisational contexts to the design of software systems, while at the same time there exist systems that work across these diverse contexts, even on a global scale.", "Pollock et al.", "use the term generification work to denote those strategies and processes that allow systems to be designed in such a way that they can travel across different organisational settings.", "User involvement plays a central role in generification work.", "A mechanism that companies use to keep users and customers in the loop is called managed communities.", "As part of the generification, \"the translation from a particular to a generic technology corresponds to a shift from a few isolated users to a larger extended ‘community’\" [11].", "During generification work, where users play an active role, mutual learning (witnessing) is observed as a positive by-product.", "Users become aware of the complexity of designing and implementing large-scale systems while the supplier of the system observes the differences and similarities across the settings.", "Vendors of systems for large markets often leave their systems open for customisation by the users, allowing particular requirements on a per-user basis.", "Further, users can potentially be empowered to participate, find new functionalities and help improve the product [16].", "Figure REF shows how generification shapes a system.", "In short, the generic blob in the figure denotes what is common across almost all user groups, the poly-generic is similar functionality varying little between user groups, the generic particular can be functionality specific for a user group but included in the core product due to strategic reasons, and the particular is not included in the system and left to the user groups to implement (or not).", "Figure: Generic solution as a 'black-blob'.", "Figure is from Instruments similar to managed communities are reported by other researchers as well.", "One such example are the academies in the Health Information System Programme (HISP) [4], an action research project that explicitly focuses on capacity building.", "HISP hosts academies at different expertise levels to include their users in the design process, facilitate requirement elicitation and provide a learning venue.", "Another example is reported by Eaton et al.", "[6], where they investigate online communities of third-party application developers who use Apple's AppStore platform and its APIs (Application Programming Interfaces).", "They show how Apple tries – and sometimes fails – to manage its communities of developers.", "Both cases also demonstrate PD in the context of the increasingly dominant platform model for software.", "The platform architecture supports particular requirements for specific users in its periphery, while the platform core holds generic requirements across different user groups.", "The concept of generification work is described by Pollock et al.", "from a managerial perspective.", "We want to investigate whether it can also be applied with the core values of PD in mind.", "Consequently, the users' role needs to shift from being informed to actively participating in decision-making.", "The term managed communities implies that they are managed.", "We need to ask: Managed, by whom and with what interest?", "If the power gap between the organisers and the participants is too big, managed communities will not support shared decision-making and rather strengthen the actor in power [5].", "Following the model of Borum and Enderud (1981) [3], organisers of managed communities need to be cautious regarding the four mechanisms that are influencing this power relation: (i) agenda control, (ii) participants, (iii) scope, and (iv) resources.", "A managed community comes with an agenda, commonly set by the company or organisation behind the development of the system.", "The agenda is designed with a certain outcome in mind, for example finding new requirements, smoothing requirements or gathering users' feedback on newly introduced functionalities.", "Large-scale communities have participants and users spread geographically.", "Selecting and inviting specific users shape the context in which the managed community and consequently the shaping of the system will take place.", "The scope of the community is also related to its agenda and participants, and how much freedom is given to them to define this scope.", "Connected to all three mechanisms used to exercise power is the mechanism of resources.", "Not all users who want to participate to the community have the resources required.", "For instance, time is a resource often scarcely available and time-intensive peak times vary across domains and regions." ], [ "The empirical case and methods", "The ongoing case study (started in March 2019) is part of an emerging multiple, embedded case study research design [18] investigating the design, implementation and use of assistive technologies in health care.", "The case takes place at a health IT company in one of Norway's largest municipalities.", "In the following, we will call this company HealthSoftWare.", "The municipality partners with HealthSoftWare to test, implement and deliver assistive technologies to the population, as specified by the Norwegian welfare technology program [7].", "In the following we will shortly describe the history of the company and the biography of the artefact that is of interest [10]." ], [ "The organisational context", "HealthSoftWare is specialised in providing self-reporting and self-management systems for healthcare organisations and clinical studies.", "The product idea was born in 2007 in the IT department of a private clinic.", "Initially, it focused on online patient-managed questionnaires for pain management and mental health.", "Professional development of the product started in 2009 and a business plan was put in place.", "While the previous version was driven by internal needs, although concepts were perceived as good and useful, the use of the system had remained low due to missing reimbursement possibilities.", "The system was redesigned to create volume and overcome reimbursement issues.", "From 2009 to 2014 the HealthSoftWare product changed from a solution to a platform for managing online healthcare-related questionnaires.", "The goal was to eventually target an international market, hence lock-in situations by exclusively supporting national standards were avoided.", "In addition, HealthSoftWare followed a modular, flexible architecture which allowed them to quickly react to market requirements.", "HealthSoftWare positioned themselves as an intermediary between their customers among healthcare service providers and researchers, and the providers of the standardised questionnaires.", "They did not pursue to become exclusive right holders to these questionnaires but aimed to provide a platform for the providers to reach a large audience – i.e.", "a platform model.", "The system was designed to be open, both partners and customers had access to the API for integration with their existing systems.", "By 2014, a studio module allowed customers to design and use their own questionnaires on the platform.", "User generated content was not any more isolated to a single instance but could travel across systems and organisations.", "In their major next version, HealthSoftWare focused on scalability and user-based customisation of the system.", "More advanced communication was introduced, such as secure chats with the ability to add attachments or the implementation of real time communication.", "Further, the user interface for patients was designed to be mobile first and responsive, in the light of increased use of tablets and smartphones.", "In 2018, one of Norway's largest municipalities was looking for a solution to implement self-reporting among patients with chronic conditions.", "By this time, HealthSoftWare fulfilled the major requirements and adding new devices and sensors to their system was supported through their API.", "They won the tender and extended their existing system to fulfil the project requirements." ], [ "Methods for data collection", "Data presented here was mainly collected by participant observation (during a user conference) and semi-structured interviews (1 interview with CEO and 3 interviews with the CTO).", "Collection of data took place between March and October 2019.", "The interviews were held in English and the information received at the user conference was in Norwegian.", "In addition to the observation, additional documents such as presentation slides and brochures from the user conference were collected and analysed.", "In our data collection we have focused on two instruments for managing communities, i.e.", "user conferences and training courses.", "We will describe these in the next section." ], [ "Findings", "HealthSoftWare organises regular yearly user conferences called HealthSoftWare Conference.", "These are spread over two days, each day covering one major part of the conference: (i) presentations mainly provided by user groups, i.e.", "the conference, and (ii) courses provided by HealthSoftWare, called course and inspiration.", "Table REF lists the theme of recent conferences.", "Table: Themes of HealthSoftWare conferencesParticipation at the HealthSoftWare conferences is open to all.", "Each part is subject to charge, and registration is mandatory.", "HealthSoftWare decides on the speakers to give a talk or presentation of their experience with the system.", "Presentations by user groups last 15 to 30 minutes and illustrate how the different sites implement and customise their HealthSoftWare.", "The presenters provide feedback to the company on what worked well and what to do differently.", "This also includes concrete requirements that are missing from their point of view such as integration to systems in their infrastructure, or criticism on current design, for example easier reporting functionality.", "The HealthSoftWare Conference exists not only of presentations.", "There are also other forms of events such as discussion rounds, courses and workshops.", "The user conference that we observed included a day of workshops.", "The scope of the workshops was defined by HealthSoftWare, and included two topics, i.e.", "integration of third-party systems with HealthSoftWare platform, and facilitating user customisation of the platform.", "In addition to the conference, HealthSoftWare provides courses under the name HealthSoftWare Academy.", "There are several thematic courses with fixed dates that target different user groups or provide in-depth knowledge about certain topics.", "The courses last one or two days and are subjected to charge.", "One example of such a course is the superuser course, which is for experienced users.", "The super-user course aims to impart knowledge and skills required to customise the platform by the user organisations for their local context.", "HealthSoftWare Academy is organised for a smaller number of participants (compared to the conference) and takes place close to the users' geographical location.", "While at user conferences the company expects to gain a greater insight into the users' requirements and their adapted solutions, the academy is designed for the users to learn certain skills for using or customising the system.", "However, as the courses are often \"hands-on\" and interactive, the organisers also learn from the participants how they use the system and where they might have problems.", "These insights allow HealthSoftWare designers to improve the platform.", "Table REF summarizes how the four dimensions agenda, participants, scope and resources play out for HealthSoftWare Conferences.", "Table: Organisation of user conferences described" ], [ "Discussion", "Genuine participation can be defined as \"the fundamental transcendence of the users' role from being merely informants to being legitimate and acknowledged participants in the design process\" [12].", "Managed communities as a strategy for generification work provide a potentially effective arena for discovering common needs across different user bases.", "At the same time, these communities legitimatise in a pseudo-democratic way the exclusion of the particular.", "Requirements relevant only for a few are down-prioritised, and their implementation is either delayed to unforeseen future or outsourced to the users themselves.", "This is probably the main departure of managed communities from traditional PD processes.", "Historically, the development of information systems has moved from in-house development to buying generic software [1].", "When IT is purchased, in form of a generic product or – in our case – a platform to be customised, the balance of power is shifted towards a new strong player, i.e.", "the owner of the product.", "This implies that managed communities carry multiple, sometimes contradictory roles with respect to PD.", "The shift in power is clearly visible with regard to the agenda and the scope of the observed user conference (see e.g.", "Table REF ).", "User conferences are normally hosted by a company developing a product with the aim to target a large market.", "This context influences strongly which information is presented during the conference, and what will become official knowledge, i.e.", "included in the roadmap for the product.", "Attention should be also drawn to what is not on the agenda [15], e.g.", "excess critique or request for odd functionality.", "The organisers are the actors in power.", "There are several ways to \"democratise\" managed communities.", "Transferring the concept of managed communities to PD would e.g.", "imply including the different user groups already at the stage of planning the agenda for the conferences and courses.", "As user conferences are normally taking place annually, following an iterative approach and reflecting on the previous conference during the planning process is a possible inclusion strategy.", "Topics for speakers can be decided together with the users – simple online voting tools can support this.", "Workshops and courses contribute to the goal of mutual learning, but again the selection process of workshop themes or course topics could be done together with the users and participants.", "Moreover, to overcome financial and geographic limitations, user conferences can be organised at alternating venues close to the users.", "For some inspiration for organising more democratic communities one can look at HISP, which has a record of successfully managing a globally distributed user community in a large-scale PD project [13], [4].", "However, we also need to ask for whose benefit such a democratisation process is, and who should pay for it.", "HealthSoftWare is not a large company.", "Managing a community is a relatively costly process, which needs to pay off in terms of new sales.", "It makes sense that the agenda and the scope of these conferences need to fit the product roadmap.", "Large customers – such as the municipality who just acquires their self-reporting platform – and their needs will be prioritised.", "At the same time, the survival of HealthSoftWare depends in finding new customers.", "So the conference is also used as a marketing tool.", "What is guiding a company such as HealthSoftWare is their product roadmap, which needs to be developed based on a sustainable business model for the company.", "This raises a fundamental challenge for PD.", "Traditionally, PD focused on individual user communities interacting with software designers in closed organisational contexts.", "The focus was on satisfying the needs of this community.", "Even if the system itself was large and complex, its ecosystem was relatively simple.", "Naturally, platform owners wish to sell their products to multiple user communities.", "They also want to maximise revenue by segmenting their user base [11], and prioritising those users who can pay more and are more aligned with the product roadmap.", "Products with ecosystems consisting of numerous communities need a more distributed community management.", "It is unrealistic to ask one actor in the ecosystem to guarantee value for all other actors through a fully participatory design process.", "This is valid not only for small companies but also large ones.", "In studying the ecosystem for Apple's AppStore, Eaton et al.", "[6] found that while Apple (similar to HealthSoftWare) played a central role in managing its own user communities, users created communities of their own and used these communities to put pressure on Apple.", "We have seen similar phenomena within the Norwegian public sector: small municipalities join together when acquiring large IT systems, in this way creating stronger communities managed by themselves and not the vendors.", "The concept of managed communities is useful for scaling PD and involving large groups of users and needs to be studied by PD researchers as a potentially new arena for future PD processes.", "Our research focused around the user conferences, which are part of a reflection of systems in use.", "The genuine use of the system in the social environment it is designed for is essential for uncovering the users' need.", "Further, regular user conferences can support a systematic and iterative design process which acknowledges the emergent use over time.", "[8] However, there are challenges related to (i) how to democratise these communities and the process of creating and maintaining them, and (ii) how to co-manage multiple communities with differing and sometimes conflicting agendas with the goal of creating better IT products.", "Implementing an iterative and continuous approach to the above two points can support the long-term scaling up of PD projects by establishing structures and routines that can go beyond episodes of participation." ], [ "Conclusions", "This paper gives a first glimpse into our investigation of how and if managed communities can support the scaling of participation in PD.", "The case study is ongoing, and data collection is limited, which can be seen as a limitation.", "However, we have seen similar patterns across other cases we are currently engaged with.", "Managed communities have the potential to support more than generifcation work and negotiating user requirements of large communities.", "In a next step, we will take the perspective of the users to explore this work further.", "This case is set in the healthcare sector, hence we will also take into account the voices of patients and indirect caregivers.", "Although they are not the direct users of the system, they are affected by the system in place and need to be heard." ] ]

[ [ "Dirac-type nodal spin liquid revealed by refined quantum many-body\n solver using neural-network wave function, correlation ratio, and level\n spectroscopy" ], [ "Abstract Pursuing fractionalized particles that do not bear properties of conventional measurable objects, exemplified by bare particles in the vacuum such as electrons and elementary excitations such as magnons, is a challenge in physics.", "Here we show that a machine-learning method for quantum many-body systems that has achieved state-of-the-art accuracy reveals the existence of a quantum spin liquid (QSL) phase in the region $0.49\\lesssim J_2/J_1\\lesssim0.54$ convincingly in spin-1/2 frustrated Heisenberg model with the nearest and next-nearest neighbor exchanges, $J_1$ and $J_2$, respectively, on the square lattice.", "This is achieved by combining with the cutting-edge computational schemes known as the correlation ratio and level spectroscopy methods to mitigate the finite-size effects.", "The quantitative one-to-one correspondence between the correlations in the ground state and the excitation spectra enables the reliable identification and estimation of the QSL and its nature.", "The spin excitation spectra containing both singlet and triplet gapless Dirac-like dispersions signal the emergence of gapless fractionalized spin-1/2 Dirac-type spinons in the distinctive QSL phase.", "Unexplored critical behavior with coexisting and dual power-law decays of N\\'{e}el antiferromagnetic and dimer correlations is revealed.", "The power-law decay exponents of the two correlations differently vary with $J_2/J_1$ in the QSL phase and thus have different values except for a single point satisfying the symmetry of the two correlations.", "The isomorph of excitations with the cuprate $d$-wave superconductors implies a tight connection between the present QSL and superconductivity.", "This achievement demonstrates that the quantum-state representation using machine learning techniques, which had mostly been limited to benchmarks, is a promising tool for investigating grand challenges in quantum many-body physics." ], [ "Introduction", "Collective excitations such as magnons and phonons consist of many elementary particles and provide us with fundamental understanding beyond the non-interacting picture, where the spontaneous symmetry breaking and associated Nambu-Goldstone bosons are required in many cases.", "Fractionalization, on the other hand, offers another route to realize emergent particles manifesting even in the absence of the symmetry breaking and serves as one of the central concepts in modern physics.", "The conventional elementary particles themselves can often be viewed as a bound state of more elementary objects, namely, the fractionalized particles, and such exotic particles emerge through the deconfinement.", "A prominent example of the deconfinement occurs in quantum chromodynamics: The proton and neutron that had been considered to be elementary particles before have turned out each to be a composite particle of three quarks with fractionalized charges, though quarks are hardly detected in experiments directly because of the confinement.", "In condensed matter, though the electron is an elementary particle in the vacuum, such deconfinement of electrons can be seen at low energies in specific circumstances followed by the ground-state structure of materials.", "Consequential emergent fractionalized particles were discovered in examples of polyacetylene soliton [1] and fractional quantum Hall states [2].", "The expectation would be that the emergent particles arising from the fractionalization still have particle character as low-energy excitations distinct from the elementary particles in the vacuum and the collective excitations in the symmetry broken states, and then would have novel functions in their many-body states, which may serve to future applications such as quantum computing.", "The QSL is a potential platform of such a fractionalization, where suppressed magnetic order by geometrical frustration of the spin interaction is expected to drive the fractionalization.", "The QSL phase was theoretically proposed both through numerical supports and mean-field theories [3], [4].", "Experimental efforts also supported the existence [4].", "However, theoretical and experimental efforts have not yet identified and established the nature of fractionalized particles in reality due to their hidden nature and various theoretical difficulties.", "So far, several different types of QSL have been proposed.", "One of the important properties to characterize the QSL is the excitation spectra: They are classified, first, by whether the excitations are gapped [as in the cases of gapped $Z_2$ spin liquids (short-ranged resonating valence bond (RVB) states) [5], [6] and chiral spin liquid [7]], or gapless [8], [9], [10], [11], [12], [13], [14].", "In the gapless case, one candidate is the gapless continuum of both of the singlet and triplet in an extended region of the Brillouin zone [11], [10], which may arise, for example, if spin-1/2 fermionic spinons emerging from the fractionalization constitute a large Fermi surface (or line) as in $U(1)$ spin liquid [12].", "Another proposal is the spinon nodal liquid, where a small number of spinon gapless points appear in the Brillouin zone, resulting in the discrete gapless points of the observable spin excitation as well [13], [14] (see Fig.", "REF shown later for illustration).", "At the gapless points, the dispersion may either be linear (Dirac dispersion) or quadratic.", "To establish the real existence of the QSL and then narrow down the nature of the QSL, we need to identify excitation spectra connected to experimental indications for a proper Hamiltonian that really accommodates the QSL state.", "However, it remains a challenge because of highly competing energies of various quantum states.", "We need a highly accurate framework for both ground and excited states in a momentum-resolved fashion.", "Such high accuracy is offered by a recently developed machine learning method for the ground state.", "Here, we extend this method to represent both the ground and excited states.", "To be more precise, we employ the restricted Boltzmann machine (RBM) combined with pair-product (PP) states [15].", "The RBM+PP method is further consolidated by efficient two independent state-of-the-art numerical procedures, namely the correlation ratio [16] and level spectroscopy [17] methods, to reach the thermodynamic limit quickly by reducing the finite-size effect.", "We then apply the RBM+PP to a candidate Hamiltonian of the spin-$1/2$ antiferromagnetic (AF) Heisenberg model on the square lattice with the nearest-neighbor and next-nearest-neighbor exchange interactions, $J_1$ and $J_2$ , respectively, called the $J_1$ -$J_2$ Heisenberg model.", "We employ two independent analyses to settle down the controversy and obtain firm evidence for the QSL phase: A finite range of the QSL phase in the region $0.49\\lesssim J_2/J_1\\lesssim 0.54$ is found.", "In the QSL phase, the singlet and triplet excitations are both gapless at four symmetric momenta in support of the nodal Dirac (or quadratic touching) dispersion of the fermionic spinon at $(\\pm \\pi /2,\\pm \\pi /2)$ in the Brillouin zone, which brings the coexisting power-law decay of spin-spin and dimer-dimer correlations.", "The isomorphic structure of the gapless excitations of spinons at $(\\pm \\pi /2,\\pm \\pi /2)$ with the $d$ -wave superconducting state in the cuprate superconductors are suggestive of a mutual profound connection." ], [ "The two-dimensional (2D) $J_1$ -$J_2$ Heisenberg Hamiltonian reads ${\\mathcal {H}} = J_1 \\sum _{ \\langle i, j \\rangle } {\\bf S}_i \\cdot {\\bf S}_j + J_2 \\sum _{ \\langle \\langle i, j \\rangle \\rangle } {\\bf S}_i \\cdot {\\bf S}_j,$ where ${\\bf S}_i$ is the spin-$1/2$ operator at site $i$ , whose $\\alpha $ ($\\alpha =x$ , $y$ , $z$ ) component is $S_i^\\alpha = \\frac{1}{2} {\\bf c}_i^\\dagger \\sigma _\\alpha {\\bf c}_i$ with the electron operator ${\\bf c}_i^\\dagger = (c_{i\\uparrow }^\\dagger ,c_{i\\downarrow }^\\dagger )$ and the Pauli matrix $ \\sigma _\\alpha $ .", "We set $J_1=1$ as the energy unit and we restrict the parameter range as $0 \\le J_2 \\le 1 $ .", "$ \\langle i, j \\rangle $ and $\\langle \\langle i, j \\rangle \\rangle $ denote nearest-neighbor and next-nearest-neighbor bonds, respectively.", "Despite many numerical efforts, there exists controversy on the ground state of this model in the literature.", "Although it is clear that Néel- and stripe-type AF phases exist for small and large $J_2$ regions, respectively, the estimate of phase boundary around $J_2=0.5$ , which is the classical boundary between the Néel and stripe phases, is still controversial.", "In fact, the issue is whether an unconventional quantum phase emerges around $J_2 = 0.5$ .", "The studies performing a systematic investigation of $J_2$ dependence with modern numerical techniques have proposed different scenarios: QSL (either gapless [18], [19] or gapped [20]), valence bond solid (VBS) (either columnar [21] or plaquette [22]), or both of them [23], [24].", "Deconfined quantum criticality was also proposed instead of the QSL phase [22], [25], which is interpreted as the QSL phase shrunk to a point and the fractionalization occurs only at this continuous phase transition point between the two symmetry-broken states.", "To settle the phase diagram after various highly controversial proposals, one needs to satisfy at least the following three requirements: Systematic investigation on finely-resolved $J_2$ dependence must be performed to establish whether the QSL exists as a phase in a finite $J_2$ interval because the QSL region is not expected to be wide.", "Calculation must be highly accurate because quite different states are highly competitive with near degeneracy of the energies.", "Reliable estimate of the thermodynamic limit to ensure it in the realistic bulk systems.", "When finite-size systems are studied, methods that are size-independent or have small finite-size effects are required to allow reliable extrapolation to the thermodynamic limit.", "Although recent rapid progress in variational numerical methods has contributed to better accuracy, previous studies satisfying all the points hardly exist.", "Most of the studies have argued whether the order parameter is finite or not in the thermodynamic limit; however, the order parameter is tiny around the continuous phase transitions, and it is hard to discuss whether the order parameter is really zero or not.", "Also, when finite-size systems are studied, the direct extrapolation of the order parameter has a large finite-size effect.", "Exceptionally, Ref.", "[24] employed the level spectroscopy analysis, which can mitigate the finite-size effect.", "However, it is an indirect method, which speculates phase transitions in the ground state indirectly from the excitation structure.", "Because there exists no rigorous proof for the one-to-one correspondence of the ground state and excitation structure, one needs to verify in ground-state quantities to settle the highly controversial issue.", "As is detailed below, we employ the RBM+PP method, which offers a unique way of calculating ground state and momentum-space excitation dispersion in a systematically improvable and tractable way, to satisfy the conditions 1 and 2.", "The high accuracy and tractable computational cost of the RBM+PP enable comprehensive correlation ratio (ground-state property) and level spectroscopy (excited-state property) analyses with small finite-size effects.", "To fulfill the condition 3, a crosscheck from the two independent analyses is essential." ], [ "Machine learning for quantum many-body systems", "Physical properties of many-body systems are governed by the eigenstates of the many-body Hamiltonian.", "Therefore, once the eigenstates of the Hamiltonian in Eq.", "(REF ) are known, we can predict the nature of the $J_1$ -$J_2$ model precisely.", "However, there is difficulty in obtaining eigenstates because the dimension of the eigenstates grows exponentially as the system size increases.", "In the present case where we consider the $J_1$ -$J_2$ Hamiltonian on the $L \\times L$ ($=N_{\\rm site}$ ) lattice with the periodic boundary condition, we cannot obtain the exact wave function when $N_{\\rm site} \\gtrsim 50$ .", "However, by using machine learning techniques, we can compress the data of eigenstates and approximate the wave functions accurately with a finite number of parameters.", "Here, we employ a newly developed machine learning method, RBM+PP [15], to obtain accurate representations for both the ground and excited states.", "The RBM is a type of artificial neural network having two (visible and hidden) layers [26].", "Using the machine learning technique, one can construct accurate many-body wave functions, which are systematically improvable toward the exact solution [27].", "Indeed, it has been shown both theoretically and numerically that the RBM variational state flexibly describes a variety of quantum states [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], including the states exhibiting the volume-law entanglement entropy [28], [30], which is advantageous to represent not only the ground state but also the excited states.", "Indeed, the RBM is shown to accurately describe excited states of quantum spin Hamiltonians [39], [37], for which existing numerical methods often encounter numerical difficulties.", "Meanwhile, the PP state (called “geminal\" in quantum chemistry) is represented by fermion wave functions, which can also accommodate volume-law entanglement.", "The PP state mapped onto bosonic spin space can represent RVB states [40], serving as a powerful starting point of the ground state approximation for the quantum spin systems [41].", "The combined wave function, RBM+PP, inherits advantages of both and acquires much better accuracy than those achieved by either of RBM or PP state separately [15].", "By the RBM+PP method with quantum number projections (see below), we can calculate momentum resolved excitations.", "The RBM+PP wave function $\\Psi (\\sigma ) = \\langle \\sigma | \\Psi \\rangle $ with $|\\sigma \\rangle =\\prod _{i}c_{i\\sigma }^{\\dagger }|0\\rangle $ is given by (we neglect normalization factor) [15] $\\Psi ( \\sigma ) = \\phi _{\\rm RBM} (\\sigma ) \\psi _{\\rm PP} (\\sigma )$ for each spin configuration $\\sigma = (\\sigma _1, \\sigma _2, \\ldots , \\sigma _{N_{\\rm site}} )$ with $\\sigma _i = 2 S_i^z = \\pm 1$ .", "The number of sites is given by $N_{\\rm site} =L \\times L$ and the periodic boundary condition is assumed.", "The RBM part is given by (we omit irrelevant bias term on the physical spins) $\\phi _{\\rm RBM} (\\sigma ) = \\sum _{ \\lbrace h_k\\rbrace } \\exp \\biggl ( \\sum _{i,k} W_{ik} \\sigma _i h_k +\\sum _{k} b_k h_k \\biggr )$ with the spin state of hidden units $h_k = \\pm 1$ , the interaction between physical and hidden variables $W_{ik}$ , and the bias on the hidden variables $b_k$ .", "The number of hidden units is taken to be 16.", "The sum over hidden variables can be evaluated analytically and Eq.", "(REF ) can be efficiently computed as $\\phi _{\\rm RBM} (\\sigma ) = \\prod _k 2 \\cosh \\bigl ( b_k + \\sum _i W_{ik} \\sigma _i \\bigr )$ .", "To make it possible to express the sign change of the wave function, we take the $b_k$ and $W_{ik}$ variational parameters to be complex.", "The PP state mapped onto spin systems reads $\\bigl | \\psi _{\\rm PP } \\bigl \\rangle = P_{\\rm G} \\biggl ( \\sum _{i,j} f_{ij}^{\\uparrow \\downarrow } c^\\dagger _{i \\uparrow } c^\\dagger _{j\\downarrow } \\biggr ) ^ { N_{\\rm site}/2 } \\bigl | 0 \\bigr \\rangle $ with real variational parameters $f_{ij}^{\\uparrow \\downarrow }$ .", "$\\psi _{\\rm PP}(\\sigma )$ in Eq.", "(REF ) is related as $\\psi _{\\rm PP}(\\sigma )\\equiv \\langle \\sigma | \\psi _{\\rm PP}\\rangle $ .", "Here, $P_{\\rm G} = \\prod _i ( 1 - n_{i\\uparrow } n_{i\\downarrow })$ with $n_{i \\uparrow } = c^\\dagger _{i \\uparrow } c_{i \\uparrow }$ and $n_{i \\downarrow } = c^\\dagger _{i \\downarrow } c_{i \\downarrow }$ is the Gutzwiller projection prohibiting double occupancy.", "We optimize the variational parameters $\\lbrace b_k, W_{ik}, f_{ij}^{\\uparrow \\downarrow } \\rbrace $ to minimize the energy $ E = \\frac{ \\langle \\Psi | { \\mathcal {H} } | \\Psi \\rangle }{ \\langle \\Psi | \\Psi \\rangle }$ .", "The energy is a highly nonlinear function with respect to the parameters $\\lbrace b_k, W_{ik}, f_{ij}^{\\uparrow \\downarrow } \\rbrace $ .", "Therefore, by interpreting the energy as a loss function, the task of obtaining the lowest-energy state can be recast as a machine-learning task, namely a high-dimensional optimization problem of the highly nonlinear function (RBM+PP) using the highly nonlinear loss function (energy) [42].", "The details of the optimization method and the calculation conditions can be found in Appendix ." ], [ "Strategy to overcome numerical challenges", "Various competing controversial scenarios have been proposed for the phase diagram and the nature of possible QSL as we mentioned above.", "The machine learning only is, despite its crucial importance, not enough to resolve these controversies.", "In fact, even when we obtain accurate representations of quantum states by the machine learning, (i) another challenge is how to reach quick convergence to the thermodynamic limit from available finite-size results (condition 3 listed in Sec. ).", "Furthermore, provided that the QSL phase exists, the next challenge is to elucidate its nature; (ii) it is essential to estimate the excitation gap structure and momentum resolved dispersion accurately.", "To overcome the challenge (i), the present paper employs an unprecedented combination of two methods and one supplementary analysis together and reaches quantitative agreements, which ensures the accuracy because the two methods are originally independent of each other.", "As a computational method to identify the quantum phases, this is the first attempt to use such combinations, and it successfully establishes a way to obtain the accurate phase diagram, which may serve as the standard method in the future.", "The first method is the correlation ratio method [16], which utilizes the ground state properties (see Sec.", "REF ).", "The second is the level spectroscopy [17], which detects the signature of the phase transition in the excitation spectra (see Sec.", "REF ).", "Both methods show small finite-size effects and quickly converge to the thermodynamic limit.", "These methods were developed independently and, in fact, measure the excited and ground-state properties, respectively, which are originally independent.", "However, the important point is that they have the one-to-one correspondence, conceptually similar to the fluctuation-dissipation theorem and Kubo formula, between the equilibrium and non-equilibrium excited states.", "Such correspondence and match in the calculated results help and ensure the reliability of the phase diagram.", "Further, the obtained phase boundary is supported by the standard finite-size scaling method thanks to the universal scaling relations (see Sec.", "REF ).", "For (ii), we use quantum number projection to reach the accuracy on the spectroscopy level [43] (see Sec.", "REF ).", "Here, we address the advantages of employing these methods." ], [ "Correlation ratio", "Correlation ratio $R$ quantifies how sharp the structure factor peak is.", "$R$ is given by $R=1-S({\\bf Q}+\\delta {\\bf q})/S({\\bf Q})$  [16], [44], where $S({\\bf q})$ is the structure factor, ${\\bf Q}$ is the peak momentum, and ${\\bf Q}+\\delta {\\bf q}$ is the neighboring momentum.", "In the case of the square lattice, $\\delta {\\bf q}= (2\\pi / L ,0)$ and $(0,2\\pi / L )$ .", "We see that the $R$ value approaches 1 (0) when the peak becomes sharp (broad).", "Therefore, with increasing system size, $R$ scales to 1 in the ordered phase with delta-function Bragg peak and 0 in the disordered phase.", "The crossing point of $R$ curves for different system sizes does not depend sensitively on system size.", "Thus it is suitable for an accurate estimate of the phase boundary between ordered and disordered phases in the thermodynamic limit [16], [44].", "We examine $R$ for both spin-spin and dimer-dimer correlations to detect Néel-AF and VBS transition points, respectively.", "The spin-spin correlation is given by $C_{\\rm s} ( {\\bf r}_i - {\\bf r}_j) = \\langle {\\bf S}_i \\cdot {\\bf S}_j \\rangle $ .", "The dimer-dimer correlation is defined as $C_{{ \\rm d}_\\alpha } ( {\\bf r}_i - {\\bf r}_j) = \\langle D^\\alpha _i D^\\alpha _j \\rangle - \\langle D^\\alpha _i \\rangle \\langle D^\\alpha _j \\rangle $ with the dimer operator $ D^\\alpha _i = {\\bf S}_i \\cdot {\\bf S}_{i + \\hat{\\alpha } }$ on the nearest-neighbor bonds for the $\\alpha $ -direction ($\\alpha =x$ , $y$ ).", "Hereafter, the subscripts “s\", “d$_x$ \", and “d$_y$ \" are used for spin-spin, dimer-dimer ($\\alpha =x$ and $\\alpha =y$ ) correlations, respectively.", "Then, the structure factor is calculated from $S_{{ \\rm \\gamma }} ( {\\bf q} ) = \\frac{1}{N_{\\rm site} } \\sum _{i,j} C_{{ \\rm \\gamma }} ( {\\bf r}_i - {\\bf r}_j) e ^{i {\\bf q} \\cdot ( {\\bf r}_i - {\\bf r}_j )}$ with $\\gamma = $ s, d$_x$ , and d$_y$ .", "The two correlation ratios $R_{\\rm N\\acute{e}el}$ and $R_{\\rm VBS}$ are defined from $S_{{ \\rm s}} ({\\bf q})$ and $S_{{\\rm d}_x} ({\\bf q})$ [or equivalently $S_{{\\rm d}_y} ({\\bf q})$ ] to determine the Néel-AF and VBS transition points, respectively.", "Close to the Néel-AF phase, the peak momentum is ${\\bf Q} = (\\pi ,\\pi )$ for $S_{\\rm s} ({\\bf q})$ .", "For VBS, ${\\bf Q} = (\\pi ,0)$ for $S_{{\\rm d}_x} ({\\bf q})$ and ${\\bf Q} = (0,\\pi )$ for $S_{{\\rm d}_y} ({\\bf q})$ ." ], [ "Level spectroscopy", "Quantum phases are characterized by their unique structure in excitation spectra.", "At finite sizes, if the phases are different, low-lying excitations will be characterized by different quantum numbers.", "Therefore, the transition point can be estimated by the size extrapolation of the crossing point of the low-lying excitation energies [17].", "This level spectroscopy method is known to have small system size dependence as well.", "Indeed, it has played an important role in precisely determining the Berezinskii-Kosterlitz-Thouless transition point for the sine-Gordon model [17].", "This method offers an analysis completely different but complementary to the correlation ratio method." ], [ "Quantum number projection", "The eigenstates of the Hamiltonian in finite size systems are labeled by quantum numbers.", "By optimizing the RBM+PP wave function for each quantum number sector, we can obtain both the ground state and low-lying excited states.", "We apply total-momentum and spin-parity projections to the RBM+PP wave functions to specify the quantum number [43]: $\\Psi _{\\bf K}^{S_\\pm } ( \\sigma ) = \\sum _{{\\bf R} } e^{ - i {\\bf K} \\cdot {\\bf R} } [ \\Psi ( T_{\\bf R} \\sigma ) \\pm \\Psi (- T_{\\bf R} \\sigma ) ]$ (double sign in the same order).", "$S_+$ ($S_-$ ) indicates even (odd) spin parity corresponding to even (odd) values of the total spin $S$ .", "${\\bf K}$ is the total momentum.", "$T_{\\bf R}$ is a translation operator shifting all the spins by ${\\bf R}$ .", "For each quantum number sector, we optimize the RBM+PP wave function to obtain the lowest-energy state.", "Although the spin-parity projection can only distinguish whether $S$ is even or odd, we always obtain a singlet state for the even $S$ sector and a triplet state for the odd $S$ sector (we confirm it by calculating $S$ expectation value for the obtained states).", "This is because the singlet (triplet) state is the lowest-energy state for each even (odd) $S$ sector.", "We note that the quantum number projection is helpful not only to distinguish quantum numbers but also to lower the variational energy [45].", "The ground state is given for ${\\bf K} = (0,0)$ and even $S$ sector.", "The energies for other quantum number sectors measured from the ground state energy determine the excitation spectra.", "Then, we can obtain singlet and triplet excitations separately with momentum resolution.", "Exceptionally, we need special treatments to obtain $S=0$ excited state at ${\\bf K} = (0,0)$ and $S=2$ excited states, which are described in detail in Appendix REF .", "As we mentioned above, the flexible representability of the RBM+PP gives accurate representations not only for ground states but also for excited states.", "The accurate estimate of momentum-resolved excitation gaps enables us to perform the above-mentioned level spectroscopy and also to elucidate the nature of the QSL phase.", "Near the quantum critical point, the susceptibility $\\chi $ at the ordering wave vector ${\\bf Q}$ in finite-sized systems follow the following finite-size scaling form [46] $\\frac{\\chi (t,{\\bf Q},L)}{L^{\\gamma /\\nu }}=f_{\\chi }(L^{1/\\nu }t),$ where the universal scaling function $f_{\\chi }$ appears with the correlation length exponent $\\nu $ and the susceptibility exponent $\\gamma $ .", "Here, $t$ assumed to satisfy $t\\ll 1$ is the dimensionless distance to the critical point.", "In the present case $t=(J_2-J_2^{\\rm N\\acute{e}el}) /J_1$ or $t=(J_2-J_2^{\\rm VBS} )/J_1$ .", "Through the relation between $\\chi $ and the structure factor $S(t,{\\bf Q},L)$ given by $\\chi (t,{\\bf Q},L) \\sim S(t,{\\bf Q},L)L^z$ with the dynamical exponent $z$ , we find that the squared order parameter $m^2=S(t,{\\bf Q},L)/L^d$ for the $d$ -dimensional system follows $m^2 L^{d+z-2+\\eta } =f_{\\chi }(L^{1/\\nu }t),$ if the Fisher's scaling relation $\\gamma /\\nu =2-\\eta $ holds for $\\eta $ associated with the anomalous dimension characterized by the power-law decay of the correlation, $C({\\bf r})\\sim 1/r^{d+z-2+\\eta }$ for distance $r = |{\\bf r}|$ at the critical point.", "Then the finite-size scaling plot should exhibit the universal scaling function $f_{\\chi }$ ." ], [ "Results", "First, we check the accuracy of the RBM+PP method in analyzing the $J_1$ -$J_2$ Heisenberg model (see Appendix ).", "We have confirmed that the RBM+PP achieves state-of-the-art accuracy not only among machine-learning-based methods but also among all available numerical methods.", "Indeed, the RBM+PP wave function marks the best precision for the ground state calculations among the compared methods for the $8 \\times 8$ and $10\\times 10$ lattices (see Fig.", "REF and Table REF ).", "We have also found that the RBM+PP represents excited states with unprecedented accuracy (Fig.", "REF ).", "Also, in our RBM+PP method, the computationally most demanding part is coming from the PP part, and the neural-network (RBM) part offers an efficient way of improving accuracy without increasing the scaling of computational cost, i.e., as compared to the PP only calculations with the computational cost of ${\\mathcal {O}}(N_{\\rm site}^3)$ , the computational cost increases only by ${\\mathcal {O}}(1)$ .", "This is in contrast to the Lanczos step combined with the variational Monte Carlo (VMC) method employed in Ref.", "hu13, which had marked the best variational ground-state energy for the $8 \\times 8$ and $10\\times 10$ lattices before the present study (see Appendix ): The Lanczos step increases the computational cost by $N_{\\rm site}^p$ times ($p$ : order of Lanczos step) compared to the VMC method with ${\\mathcal {O}}(N_{\\rm site}^3)$ computational cost, and hence the computational cost of VMC($p$ =1) and VMC($p$ =2) scales as ${\\mathcal {O}}(N_{\\rm site}^4)$ and ${\\mathcal {O}}(N_{\\rm site}^5)$ , respectively.", "Although we calculated the ground state and various excited states independently, a tractable computational-cost scaling of the RBM+PP method allowed us to perform numerous independent calculations for large system sizes within given computational resources.", "Thus obtained high-quality data contribute to a reliable determination of the phase diagram consistently from both ground-state and excitation analyses (see below).", "Figure: Ground-state phase diagram of square-lattice J 1 J_1-J 2 J_2 Heisenberg model (J 1 =1J_1=1) obtained by the RBM+PP method." ], [ "Ground-state phase diagram", "The RBM+PP method combined with the state-of-the-art numerical techniques convincingly uncovers the phase diagram of the $J_1$ -$J_2$ Heisenberg model as shown in Fig.", "REF .", "In the small (large) $J_2$ region, the Néel-type (stripe) AF long-range order appears as in the classical phase diagram.", "In between these two phases, nonmagnetic ground states, QSL and VBS, are found in the region $J_2^{\\rm N\\acute{e}el}\\approx 0.49\\le J_2 \\le J_2^{\\rm VBS} \\approx 0.54$ and $J_2^{\\rm VBS}\\approx 0.54\\le J_2 \\le J_2^{\\rm VS} \\approx 0.61$ , respectively.", "Whereas VBS breaks lattice symmetry, QSL does not break any.", "Clearly and notably, QSL is stabilized in a finite region of $J_2$ around $J_2 = 0.5$ .", "The phase transition between VBS and stripe-AF at $J_2^{\\rm VS}$ is of 1st order, which is characterized by the kink in the ground state energy, while the other two transitions are continuous (Fig.", "REF in Appendix ).", "Below, we describe the procedure to determine the continuous phase transition points.", "Results for the correlation ratios, $R_{\\rm N\\acute{e}el}$ and $R_{\\rm VBS}$ , are shown in Figs.", "REF (a) and REF (b), respectively (see Figs.", "REF and REF in Appendix  for the raw data of correlation functions).", "We see clear crossings of curves for three sizes at nearly the same points at $J_2 = J_2^{\\rm N\\acute{e}el} \\approx 0.49$ for $R_{\\rm N\\acute{e}el}$ and at $J_2 = J_2^{\\rm VBS} \\approx 0.54$ for $R_{\\rm VBS}$ .", "This standard procedure strongly supports that the two transitions associated with the Néel-AF and VBS ordering take place at the different points close to these system-size independent crossings.", "It supports the existence of an intermediate phase without any long-range ordering, i.e., QSL phase in the range $0.49 \\lesssim J_2 \\lesssim 0.54$ (see Appendix REF for the discussion of the system-size dependence of the crossing points).", "Figure: System-size dependence of correlation ratio for (a) spin-spin and (b) dimer-dimer correlations, which are used to detect the phase boundary of Néel-AF and VBS, respectively.In (a), the 18×1818\\times 18 data are added to reinforce the result.Figure: Low-lying excitation energies for J 1 J_1-J 2 J_2 Heisenberg model for (a) 12×1212\\times 12 and (b) 16×1616\\times 16 lattices.The red and black arrows indicate singlet-quintuplet and singlet-triplet level crossings, respectively.Figure: (a) System-size dependence of singlet-quintuplet (red dots) and singlet-triplet (black squares) level crossings indicated by red and black arrows in Fig.", ".The extrapolation to the thermodynamic limit is done by the polynomial fita+b/L 2 +c/L 4 a + b / L^2 + c/L^4 (solid curves).", "(b) System-size dependence of the excitation gap Δ\\Delta at the two level crossings.For the singlet-quintuplet level crossing in (a), the 18×1818\\times 18 data are added to corroborate the result." ], [ "Phase boundary determined by level spectroscopy", "The level spectroscopy method was applied to the 2D $J_1$ -$J_2$ Heisenberg model before [24].", "They interpreted the crossing between the lowest singlet and triplet excitations as the VBS-order boundary, following Ref.", "PhysRevB.94.144416.", "In addition, they found the singlet-quintuplet crossing and interpreted it as a signal of the disappearance of the AF long-range order, because the transition from the AF long-range order to quasi-long-range order in one-dimensional Heisenberg model with long-range interaction shows a similar behavior [24], [48].", "These two crossings extrapolated to $L \\rightarrow \\infty $ limit gave different $J_2$ values: $J_2= 0.463(2)$ and $J_2 = 0.519(2)$ for the singlet-quintuplet and singlet-triplet crossings, respectively.", "To critically crosscheck the consistency with the above correlation ratio result, we also reexamine the level spectroscopy analysis as a complementary check.", "We here enjoy the advantage of the momentum resolution in addition (contrary to Ref. Wang2018).", "Figure REF shows $J_2$ dependence of the excitation energies $\\Delta $ for sizes (a) $12\\times 12$ and (b) $16\\times 16$ at high-symmetry momenta.", "The singlet-quintuplet and singlet-triplet crossings signaling the AF-QSL and QSL-VBS transitions, respectively, are highlighted by arrows.", "The size extrapolation of the crossing points is shown in Fig.", "REF (a).", "We use $L^{-2}$ scaling as in Refs.", "Wang2018 and PhysRevB.94.144416.", "The extrapolated thermodynamic values are $J_2= 0.493(2)$ and $J_2 = 0.532(2)$ for the singlet-quintuplet and singlet-triplet crossings, respectively.", "The values are close to those of Ref.", "Wang2018 above.", "Quantitative differences may well be ascribed to the smaller system sizes calculated in Ref.", "Wang2018 than ours.", "As for the singlet-triplet crossing, our result is also consistent with a more recent estimate by the VMC method, which gives $J_2= 0.542(2)$  [49].", "More importantly, our phase boundary estimated by the level spectroscopy has a striking quantitative agreement with the correlation ratio result described above.", "It is of great significance to see the one-to-one correspondence between the ground-state phases and the excitation structures.", "We then safely conclude that a finite QSL region around $J_2 = 0.5$ emerges (see Supplementary Note 1 in Appendix  for additional noteworthy features found in the level spectroscopy).", "Figure REF (b) further shows the size dependence of the excitation gap $\\Delta $ at the crossing points.", "$\\Delta \\times L$ seems to converge at a finite value as $L \\rightarrow \\infty $ for both crossings.", "Therefore, the two critical points corresponding to AF-QSL and QSL-VBS transitions become gapless in the thermodynamic limit with the scaling $\\Delta \\propto 1/L$ .", "Figure: Low-lying excitation in the QSL phase.", "(a) Singlet and (b) triplet excitation gap along the symmetric line in the Brillouin zone at J 2 =0.5J_2 = 0.5.On top of the high-symmetry 𝐊{\\bf K} points (0,0)(0,0), (π,0)(\\pi ,0) and (π,π)(\\pi ,\\pi ), the excitations at intermediate points (π/2,0)(\\pi /2,0), (π,π/2)(\\pi ,\\pi /2) and (π/2,π/2)(\\pi /2,\\pi /2) are calculated.Black curves are expected dispersions in the thermodynamic limit (see text).Figure: (a) Weight of lowest branch in the dynamic spin structure factor for 𝐪=(π,0){\\bf q} = (\\pi ,0) and (π,π)(\\pi ,\\pi ) for J 2 /J 1 =0.5J_2/J_1=0.5.At each 𝐪{\\bf q} point, the weight is normalized by the total spectral weight ∫ 0 ∞ dωS s (𝐪,ω)\\int _0^\\infty d \\omega S_{\\rm s}({\\bf q}, \\omega ).", "(b) Schematic picture for plausible spinon dispersion around gapless points (±π/2,±π/2)(\\pm \\pi /2,\\pm \\pi /2), illustrated both for particle (pink) and hole (green) sides above and below the spinon Fermi energy.Two examples of two spinon excitations (two red and two black circles) are illustrated (see below).", "(c) The observable spin excitation is constructed from the two spinon excitations, which generates the gapless points at (π,0),(0,π),(0,0)(\\pi ,0), (0,\\pi ), (0,0) and (π,π)(\\pi ,\\pi ).", "For instance, the red circle with the momentum around (-π,0)(-\\pi ,0) is constructed from the two spinon excitations shown as the small red circles with the momenta around (-π/2,π/2)(-\\pi /2,\\pi /2) and (-π/2,-π/2)(-\\pi /2,-\\pi /2) in (b).The black circle is another example of spin excitation originated from the two spinon excitations shown as the small black circles in (b).Continuum incoherent spin excitations inside the cones are generated from the combinations of the two spinon excitations on the pink or green cone surfaces in (b).As we see in Fig.", "REF (b), the singlet excitation with ${\\bf K} = (\\pi ,0)$ and $(0,\\pi )$ becomes gapless at both AF-QSL and QSL-VBS critical points, implying that it is gapless through the QSL region sandwiched by these two critical points.", "In the QSL phase, the triplet excitation at ${\\bf K} = (\\pi ,\\pi )$ is the lowest excited state in finite-size systems [lower than the gapless singlet at $(\\pi ,0)$ ] lending support to the vanishing gap also for $(\\pi ,\\pi )$ triplet in the thermodynamic limit.", "By the excitation involving the triplet at $(\\pi ,\\pi )$ and the singlet at $(\\pi ,0)$ , one can construct the triplet $(0,\\pi )$ , which must be gapless if these two elementary excitations are excited far apart in the thermodynamic limit, even when they are repulsively interacting.", "In a similar way, one can construct a gapless singlet excitation at $(\\pi ,\\pi )$ and $(0,0)$ .", "Therefore, the singlet and triplet excitations are both gapless at $(0,0)$ , $(\\pi ,0)$ , $(0,\\pi )$ and $(\\pi ,\\pi )$ .", "To confirm this picture, we show in Fig.", "REF the results for (a) singlet and (b) triplet excitation energies for $8\\times 8$ , $12\\times 12$ , and $16\\times 16$ lattices at $J_2 =0.5$ in the QSL phase.", "We compute not only at high-symmetry ${\\bf K}$ points $(0,0)$ , $(\\pi ,0)$ and $(\\pi ,\\pi )$ but also at intermediate points $(\\pi /2,0)$ , $(\\pi ,\\pi /2)$ and $(\\pi /2,\\pi /2)$ [and symmetrically equivalent ${\\bf K}$ points such as $(-\\pi /2,0)$ , $(0,\\pi /2)$ , $(0,-\\pi /2)$ for $(\\pi /2,0)$ ].", "We find that the excitation gap decreases as $L$ increases at the high-symmetry ${\\bf K}$ points.", "The exceptional behavior at ${\\bf K} = (0,0)$ in the singlet sector is presumably an artifact, which arises from numerical difficulty in obtaining excited states in $S=0$ and ${\\bf K} = (0,0)$ sector (Supplementary Note 2 in Appendix ).", "On the other hand, the gap stays nearly constant at the intermediate ${\\bf K}$ points.", "By combining the gap analysis at the critical points (see above) and the size extrapolation of the gap by the scaling $a+b/L$ at the intermediate ${\\bf K}$ points, we draw dispersion expected in the thermodynamic limit.", "The excitation spectra in the thermodynamic limit exhibit unconventional behavior in which the gap vanishes at the four high-symmetry momenta.", "We find only these four points as the gapless excitations suggesting Dirac-type linear dispersion around these four points.", "To corroborate the conclusion about the four Dirac-type gapless points in the QSL phase, we have also calculated the excitation energies at $( m \\pi /3, n \\pi /3)$ with $m,n = 0$ , 1, 2, 3 for $12\\times 12$ lattice (Fig.", "REF in Appendix ).", "From the limited momenta we studied, although other possibilities such as the higher-order dispersion (e.g., quadratic band touching) or tiny but extended gapless regions rather than points are not excluded, the results in Fig.", "REF also support the Dirac-type nodal QSL." ], [ "Signature of fractionalization in quantum spin liquid", "In the present QSL phase, one can expect an exotic fractionalization of particles, where a charge-neutral spin-1/2 excitation, called spinon, emerges.", "Although the spinon excitation cannot be detected experimentally, the evidence of the fractionalization can be detected as an incoherent continuum in the dynamic spin structure factor $S_{\\rm s}({\\bf q}, \\omega )$ (spin-1 excitation) [50], which is interpreted by the two-particle (two-hole) or particle-hole excitation continuum of the spinons.", "We here compute the weight in $S_{\\rm s}({\\bf q}, \\omega )$ at ${\\bf q} = (\\pi ,0)$ and $(\\pi ,\\pi )$ for the lowest triplet excitation shown in Fig.", "REF .", "If the excitation were the conventional magnon branch of a magnetic phase, the weight would be the order 1.", "If the weight vanishes, most of the weight lies in incoherent continuum at higher energies, supporting the emergence of fractionalized spinons [50].", "Figure REF (a) shows the weight of the lowest branch in $S_{\\rm s}({\\bf q}, \\omega )$ for ${\\bf q} = (\\pi ,0)$ and $(\\pi ,\\pi )$ .", "We indeed see that the weight decreases as the system size increases.", "In particular, the weight at ${\\bf q} = (\\pi ,0)$ rapidly decreases to zero, which means that the spectral weight is dominated by the incoherent continuum.", "[We do not analyze the behavior at ${\\bf q} = (\\pi ,\\pi )$ in detail because of a numerical challenge due to the proximity to AF(Néel)-QSL phase boundary $J_2 = J_2^{\\rm N\\acute{e}el} \\approx 0.49$ (Supplementary Note 3 in Appendix )].", "This is a strong evidence that the fractionalization indeed occurs in the QSL phase of the $J_1$ -$J_2$ Heisenberg model.", "As we will discuss in Sec.", ", the dispersion of the emergent fractionalized spinon is expected to be gapless at the points $(\\pm \\pi /2,\\pm \\pi /2)$ [Fig.", "REF (b)].", "Figure: Real-space spin-spin (red dots) and dimer-dimer (green squares) correlation functions, |C s (𝐫)||C_{{ \\rm s}} ( {\\bf r})| and |C d x (𝐫)||C_{{ \\rm d}_x} ( {\\bf r})|, respectively, for the diagonal direction (r x =r y r_x \\!", "= \\!", "r_y) for 16×1616\\times 16 lattice at J 2 =0.5125J_2 = 0.5125 in the QSL phase.The solid and dashed lines are proportional to the power-law decayC(𝐫)∝1 |𝐫| z+η +∑ 𝐧/=(0,0) 1 |𝐫-L𝐧| z+η -1 |L𝐧| z+η C ({\\bf r}) \\propto \\frac{1}{| {\\bf r } |^{z+\\eta }} + \\sum _{{\\bf n} /= (0,0) } \\left( \\frac{1}{|{ \\bf r} - L {\\bf n} |^{z+\\eta }} - \\frac{1}{|L {\\bf n}|^{z+\\eta }} \\right) with z+η=1.52z+\\eta = 1.52 (solid) and 1.62 (dashed), in which we consider the effect of the periodic boundary condition .The values of z+ηz+\\eta are taken from the analysis in Fig.", "(c).The upturn at large |𝐫|| {\\bf r } | is due to the periodicity of the lattice." ], [ "Real-space correlation functions in the quantum spin liquid phase", "Figure.", "REF shows the real-space decay of spin-spin and dimer-dimer correlation functions, $|C_{{ \\rm s}} ( {\\bf r})|$ and $|C_{{ \\rm d}_x} ( {\\bf r})|$ , respectively, for the diagonal direction ($r_x \\!", "= \\!", "r_y$ ) for $16\\times 16$ lattice in the QSL phase ($J_2 = 0.5125$ ) (for the definition of the correlation function, see Methods).", "If the correlation function shows power-law decay, it is expressed by the exponent $z+\\eta $ , namely the spin-spin and dimer-dimer correlations should show $C({\\bf r}) \\sim r^{-(z + \\eta )}$ ($r = |{\\bf r}|$ ) in the real space as critical behavior.", "Both correlation functions indeed show consistent behaviors with the power-law decay.", "It evidences the dual critical nature of the VBS and Néel-AF correlations in the QSL phase, and this ground-state property is consistent with the gapless singlet and triplet excitations clarified independently (Sec.", "REF ).", "On top of the one-to-one correspondence between the ground-state phases and excitation structures revealed by the correlation ratio and level spectroscopy analyses (Secs.", "REF and REF ), we again demonstrate a nice correspondence between the ground-state and excitation properties.", "Figure: Finite-size scaling analysis.", "(a,c) Data collapse for Nèel-AF order parameter.We assume J 2 Ne ´ el /J 1 =0.49J_2^{\\rm N\\acute{e}el} /J_1 =0.49 and estimate the critical exponents z+ηz+\\eta and ν\\nu .The bayesian scaling analysis , gives z+η=1.410(4)z+\\eta = 1.410 (4) and ν=1.21(5)\\nu = 1.21 (5).", "(b,d) Data collapse for the VBS order parameter.The same analysis with assuming J 2 VBS /J 1 =0.54J_2^{\\rm VBS} /J_1 =0.54 gives z+η=1.436(6)z+\\eta = 1.436 (6) and ν=0.67(2)\\nu = 0.67(2).Solid curves are the inferred scaling functions.In (a) and (c), the 18×1818\\times 18 data are added to corroborate the result.The figure shows that the conventional finite-size scaling analysis consistently supports the results obtained by the correlation ratio and level spectroscopy.Figure: Size dependence of the squared order parameters for (a) Nèel-AF and (b) VBS.The solid black curves in (a, b) are the expected size dependence at the critical point m 2 ∝L -(z+η) m^2 \\propto L^{-(z+\\eta )}with z+ηz+\\eta estimatedby the finite-size scaling analysis shown in Fig.", ".The critical points are estimated from the analyses based on the correlation ratio and the level spectroscopy.", "(c) J 2 J_{2} dependence of the power-law exponent z+ηz+\\eta in the QSL phase obtained by fitting the size dependence of m 2 m^2 for L=8L=8, 12, 16 with a form m 2 =AL -(z+η) m^2 = A L^{-(z+\\eta )} (AA: constant)." ], [ "Finite-size scaling and size dependence of order parameter", "Figs.", "REF (a) and REF (b) show the data of finite-size scaling analysis of the Néel-AF and VBS order parameters, respectively.", "The squared order parameters for Néel-AF and VBS are given by $m_{\\rm N\\acute{e}el}^2 = S_{\\rm s} ({\\bf Q}) / N_{\\rm site} $ with ${\\bf Q} = (\\pi ,\\pi )$ and $m_{\\rm VBS}^2 = S_{{\\rm d}_x} ({\\bf Q}) / N_{\\rm site}$ with ${\\bf Q} = (\\pi ,0)$ [$= S_{{\\rm d}_y} ({\\bf Q}) / N_{\\rm site}$ with ${\\bf Q} = (0,\\pi )$ ], respectively [see Methods for the finite-scaling analysis method and the definition of the structure factor, $S_{\\rm s} ({\\bf Q})$ and $S_{{\\rm d}_x} ({\\bf Q})$ ].", "For the Néel-AF and VBS orderings, we assume that the critical points are at $J_2^{\\rm N\\acute{e}el} = 0.49$ and $J_2^{\\rm VBS} =0.54$ , respectively (see the phase diagram in Fig.", "REF ).", "The estimated critical exponents $z+\\eta $ and $\\nu $ deduced from the finite-size scaling are $z+\\eta = 1.410 (4)$ and $\\nu = 1.21 (5)$ for the Néel-AF order parameter, and $z+\\eta = 1.436 (6)$ and $\\nu = 0.67(2)$ for the VBS order parameter, respectively [The estimate does not depend significantly on the values of $J_2^{\\rm N\\acute{e}el}$ and $J_2^{\\rm VBS}$ (Supplementary Note 4 in Appendix )].", "These exponents do not belong to the known universality class and suggest unconventional criticality.", "Figs.", "REF (a) and REF (b) show the size dependence of the Néel-AF and VBS order parameters, respectively.", "Solid black curves are expected scaling curve $m^2 \\sim L^{-(z+\\eta )}$ at the critical points obtained by employing $z+\\eta =1.410$ and 1.436 for the Néel-AF and VBS critical points, respectively.", "As is discussed in Sec.", ", to settle the highly controversial situation on the phase diagram, the calculations need to fulfill three conditions: 1. systematic $J_2$ dependence survey, 2. high accuracy, and 3. reliable estimate of the thermodynamic limit.", "The RBM+PP data achieves the state-of-the-art accuracy level both for ground-state and excited states, satisfying the condition 2.", "With the high accuracy, we have performed a systematic investigation on the $J_2$ dependence both for ground-state and excited-state properties (condition 1).", "For the condition 3, both the correlation ratio and level spectroscopy have given consistent results, supporting the conclusion of the QSL phase in the region $0.49\\lesssim J_2/J_1\\lesssim 0.54$ .", "We do not find such quantitative consistency before, and we became convinced of the existence of the QSL phase only after finding their consistency.", "Nevertheless, we note that, at a qualitative level, an overall consensus on the existence of the QSL is being formed among the best accurate methods (Refs.", "hu13 and gong14 and ours) clarified in the benchmark shown in Appendix  (Note that Ref.", "[22] obtained essentially vanishing order consistent with our finite QSL region, though they considered alternative possibilities as well, which was not settled within their analyses of the size dependence of the order parameter correlation).", "The spin excitation dispersion has been rarely studied in the literature except for the studies obtained by assuming a priori a variational form of $Z_2$ nodal spin-liquid wave function [53], [54].", "In Ref.", "PhysRevB.98.134410, the spin cluster perturbation method is also employed to draw the dispersion.", "Our gapless structure lends support to these variational and the spin cluster perturbation studies in qualitative features, though our results have been obtained without such assumptions and approximations.", "Together with the consideration on the stability of the QSL phase [55] and the reason discussed below, our unbiased analysis evidences the QSL phase in the $J_1$ -$J_2$ Heisenberg model characterized by $Z_2$ nodal QSL (rather than $U(1)$ QSL) with gapless and fractionalized spin-$1/2$ spinon excitations at $(\\pm \\pi /2 ,\\pm \\pi /2)$ , proposed in an earlier study [18] (we did not exclude the possibility of $U(1)$ QSL just from the spin excitation spectra because the $Z_2$ and $U(1)$ QSL give very similar $S_{\\rm s}({\\bf q}, \\omega )$  [55]).", "The real spin excitations measurable in experiments must be made of two-spinon excitations, and thus the singlet and triplet gapless points are $(0,0)$ , $(\\pi ,0)$ , $(0,\\pi )$ and $(\\pi ,\\pi )$ [Fig.", "REF (c)].", "The gapless Dirac-type excitations in both singlet and triplet sectors show an excellent consistency with the dual critical nature of the VBS and Néel-AF correlations, decaying algebraically in the real space, in the QSL phase (Sec.", "REF ).", "The finite-size scaling analysis shown above suggests that the value for critical exponent $z + \\eta $ is about 1.4 for both of the AF-QSL and QSL-VBS critical points (Fig.", "REF ), which is suggestive of an emergent symmetry between the spin-spin and dimer-dimer correlations, associated with the Néel-AF and VBS orders, respectively.", "If the $U(1)$ QSL is realized as a phase, we will see the emergent symmetry within the whole QSL region as the critical phase [56], [57].", "However, the power-law exponent $z+\\eta $ seems to change in the QSL region: While it increases as $J_2$ increases for the spin-spin correlation, the dimer-dimer correlation shows the opposite trend [Fig.", "REF (c)].", "It supports that the QSL with the emergent $U(1)$ symmetry is absent for an extended $J_2$ region and implies the extended region of the $Z_2$ QSL instead.", "From Fig.", "REF (c), $U(1)$ symmetry is deduced to emerge at a single point $J_2^{U(1)} \\approx 0.52$ , where the values of $z+\\eta $ for the spin-spin and dimer-dimer correlations cross and coincide, and the $Z_2$ QSL may have different characters between $J_2>J_2^{U(1)}$ and $J_2<J_2^{U(1)}$ .", "It will be of great interest to investigate this issue further in future, especially by considering more detailed system size dependence to further establish the thermodynamic behavior.", "Since the excitation structure is isomorphic with the charge and spin excitations of the $d$ -wave superconducting state in the cuprate superconductors, it is suggestive of the connection of the two; the superconducting state could be borne out from the present QSL immediately when carriers are doped.", "The present accurate estimate of the spinon excitation, especially, incoherent nature of the spin excitations with continuum, will provide us with insights into the unsolved puzzles of the cuprate superconductors including the incoherent transport and charge dynamics." ], [ "Summary", "We have studied the 2D $J_1$ -$J_2$ Heisenberg model using a highly accurate machine-learning method, RBM+PP.", "Our achievements are summarized into the following points: the quantitative estimate of the phase diagram, useful insights into the QSL property to understand its nature, and the establishment of one-to-one correspondence between ground-state and excitation structure.", "First, by combining the RBM+PP with the correlation ratio and level spectroscopy methods, we have been able to extrapolate to the thermodynamic limit reliably by two independent analyses.", "The consistently reached between the two at an unprecedented level have given the firm evidence for a finite QSL region $0.49\\lesssim J_2/J_1\\lesssim 0.54$ .", "The phase diagram is summarized in Fig.", "REF .", "The QSL is characterized as the dual nature of the algebraic and coexisting correlations of the antiferromagnetic (associated with the Néel order) and dimer (associated with the VBS order) correlations, which had been thought incompatible before by the symmetry difference.", "The elucidated dual nature is also seen consistently in the excitation property: We have identified the Dirac-type dispersion with gapless points $(0,0)$ , $(\\pi ,0)$ , $(0,\\pi )$ and $(\\pi ,\\pi )$ in both the singlet and triplet excitation sectors (related each to the dimer-dimer and spin-spin correlations, respectively).", "The excitation structure is consistent with the emergence of the fractionalized spin-1/2 spinons with gapless Dirac dispersion.", "Interestingly, the power-law decay exponents of these two correlations change as a function of $J_2/J_1$ and do not coincide except for a single point around $J_2=0.52$ , which imposes a substantial constraint on the gauge structure of the QSL.", "Finally, our comprehensive calculations have revealed a fundamental “law of correspondence” between the ground-state and excitation structure in the $J_1$ -$J_2$ Heisenberg model.", "By establishing the phase diagram, we have demonstrated that the evolution of the ground state indeed maps to the change in the excitation structure induced by the level crossing in a fingerprint fashion with one-to-one correspondence.", "We have also shown that the coexisting power-law decay of the dimer-dimer and spin-spin correlation functions in the real space in the QSL phase (ground-state property) consistently corroborates the gapless structure of singlet and triplet excitations, respectively.", "Such one-to-one correspondence has a fundamental significance in physics, as the one-to-one correspondence between the equilibrium and non-equilibrium excited states addressed in the fluctuation-dissipation theorem and Kubo formula gives a foundation for the understanding of the linear response.", "Such accurate, systematic, and comprehensive elucidation of the QSL with insights into the duality of the gapless correlations and the law of correspondence has been enabled by the RBM wave function combined with the PP state and the quantum number projection that offers state-of-the-art accuracy within a tractable computational cost: The high accuracy and the tractable computational-cost scaling of the RBM+PP method [${\\mathcal {O}}(N_{\\rm site}^3)$ ] were necessary to prepare comprehensive high-quality data for large system sizes to accomplish our achievement.", "So far, the machine learning methods had been applied mostly to benchmark problems with known solutions.", "By combining the RBM+PP wave function and with cutting-edge methods to reduce finite-size corrections, we have succeeded in uncovering QSL in the long-standing challenging problem.", "This achievement opens a new avenue of numerical methods applicable to the grand challenges of quantum many-body systems.", "We acknowledge useful discussions with Satoshi Morita, Anders W. Sandvik, and Zi Yang Meng.", "We also thank Satoshi Morita for providing us with the raw data in Ref. Morita2015.", "Y.N.", "is grateful for fruitful discussions with Ribhu Kaul, Hidemaro Suwa, Yoshitomo Kamiya, Kenji Harada, Zheng-Cheng Gu, Giuseppe Carleo, and Ryui Kaneko.", "The implementation of the RBM+PP scheme is done based on the mVMC package [58].", "The computation was mainly done at Supercomputer Center, Institute for Solid State Physics, University of Tokyo, and RIKEN K computer.", "The authors are grateful to the financial support by a Grant-in-Aid for Scientific Research (Grant No.", "16H06345) from Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.", "Y.N.", "was supported by Grant-in-Aids for Scientific Research (JSPS KAKENHI) (Grants No.", "17K14336 and No.", "18H01158).", "This work is financially supported by the MEXT HPCI Strategic Programs, and the Creation of New Functional Devices and High-Performance Materials to Support Next Generation Industries (CDMSI) as well as by “Program for Promoting Researches on the Supercomputer Fugaku\" (Basic Science for Emergence and Functionality in Quantum Matter - Innovative Strongly-Correlated Electron Science by Integration of “Fugaku\" and Frontier Experiments -).", "We also acknowledge the support provided by the RIKEN Advanced Institute for Computational Science under the HPCI System Research project (Grants No.", "hp170263, hp180170, hp190145 and hp200132).", "1" ], [ "Optimization of RBM+PP wave function", "To search the lowest-energy quantum state for each quantum number sector, we optimize the variational parameters $\\lbrace b_k, W_{ik}, f_{ij}^{\\uparrow \\downarrow } \\rbrace $ to minimize the energy expectation value of the RBM+PP wave function.", "The energy expectation value $ E = \\frac{ \\langle \\Psi | { \\mathcal {H} } | \\Psi \\rangle }{ \\langle \\Psi | \\Psi \\rangle }$ can be calculated by the Monte Carlo sampling with weight $p(\\sigma ) = \\frac{ | \\Psi (\\sigma ) |^2 }{\\langle \\Psi |\\Psi \\rangle } $ $E =\\sum _{\\sigma } p (\\sigma ) E_{\\rm loc} (\\sigma ),$ where the local energy $E_{\\rm loc} (\\sigma )$ is given by $E_{\\rm loc} (\\sigma ) = \\sum _{\\sigma ^{\\prime }} \\langle \\sigma | { \\mathcal {H} } | \\sigma ^{\\prime } \\rangle \\frac{ \\langle \\sigma ^{\\prime } | \\Psi \\rangle }{ \\langle \\sigma | \\Psi \\rangle }$ .", "The $E$ value depends on the variational parameters.", "To optimize the variational parameters to minimize $E$ , we employ the stochastic reconfiguration (SR) method [59], which is equivalent to the imaginary-time Hamiltonian evolution $e^{-\\tau {\\mathcal {H}}} | \\Psi \\rangle $ within the Hilbert space spanned by the RBM+PP wave function.", "Because the imaginary-time Hamiltonian evolution $e^{-\\tau {\\mathcal {H}}} | \\Psi \\rangle $ always stably gives the lowest-energy state for each quantum number sector (as far as the initial RBM+PP state is not orthogonal to the lowest-energy state), the SR method enables stable optimizations.", "For further technical details of the SR optimization, we refer to Ref. PhysRevB.96.205152.", "The number of complex variational parameters in the RBM part is $N_{\\rm hidden} $ for $b_k$ and $N_{\\rm hidden} \\times N_{\\rm site}$ for $W_{ik}$ , respectively.", "As for the real variational parameters $f_{ij}^{\\uparrow \\downarrow }$ in the PP part, to reduce the number of parameters and the computational cost, we impose $4 \\times 4$ sublattice structure for the $8\\times 8$ , $12\\times 12$ , and $16\\times 16$ lattices, and $6 \\times 6$ sublattice structure for the $18\\times 18$ lattice, whereas we do not employ sublattice structure for the $6\\times 6$ lattice.", "In the case of the $4 \\times 4$ sublattice structure, the number of independent $f_{ij}^{\\uparrow \\downarrow } $ parameters is reduced from $N_{\\rm site}^2$ to $4 \\times 4 \\times N_{\\rm site} = 16N_{\\rm site}$ , and the other $f_{ij}^{\\uparrow \\downarrow } $ parameters are defined by spatial translation operations.", "In the presence of the Gutzwiller factor to map the PP state onto the spin system, the onsite $f_{ii}^{\\uparrow \\downarrow }$ parameters become completely redundant, i.e., the wave function does not depend on $f_{ii}^{\\uparrow \\downarrow }$ at all.", "Then, the number of relevant $f_{ij}^{\\uparrow \\downarrow }$ parameters are $16(N_{\\rm site} \\!", "- \\!", "1)$ .", "For the initial values for $\\lbrace b_k, W_{ik}, f_{ij}^{\\uparrow \\downarrow } \\rbrace $ , we put random numbers in order not to introduce bias in the initial variational state.", "More specifically, for each real and complex part of $b_k$ and $W_{ik}$ parameters, we put small random numbers from the interval $[-0.05,0.05]$ .", "In the case of the triplet state calculation, $b_k$ parameters are multiplied by 10 (note that if $b_k$ is zero, the RBM part is completely symmetric with respect to the global spin inversion).", "For the initial $f_{ij}^{\\uparrow \\downarrow }$ parameters, we put random numbers from $[-f_{\\rm max}, f_{\\rm max}]$ with $f_{\\rm max}$ depending on the distance $R_{ij}$ between $i$ th and $j$ th sites.", "We typically take $f_{\\rm max}$ to be proportional to $R_{ij}^{-a}$ with $a \\sim 2$ .", "Random $f_{ij}^{\\uparrow \\downarrow }$ parameters allow various spin ordering patterns whose period is within the sublattice size.", "A longer period structure than the system size is beyond the scope of this study, as are all the other earlier finite-system-size studies.", "For each $J_2$ point, we perform at least three independent optimizations of the RBM+PP wave functions from different initial variational parameters.", "We discuss the initial-parameter dependence in more detail in Appendix REF .", "The computational cost of the RBM+PP wave function employing sublattice structure in the PP part scales with ${\\mathcal {O}}( N_{\\rm site}^3)$ .", "In the RBM+PP method, a computationally demanding part is coming from the calculation of the PP wave function part and the neural network (RBM) part offers an efficient way of improving accuracy." ], [ "Special treatments to obtain some specific excited states", "As we describe in Sec.", "REF , we apply the spin-parity projection to distinguish whether the total spin $S$ is even or odd.", "Because the singlet (triplet) state is the lowest state for each even (odd) $S$ sector in the present study, we obtain a singlet (triplet) state for the even (odd) $S$ sector.", "Therefore, we can obtain the singlet ($S=0$ ) and triplet ($S=1$ ) excited states with momentum resolution.", "However, we need special treatment to obtain $S=0$ excited state at ${\\bf K} = (0,0)$ because the lowest-energy state in $S=0$ and ${\\bf K} = (0,0)$ quantum number sector is the ground state.", "We use additional simplified point-group projection on top of those in Eq.", "(REF ) to obtain excited states belonging to a different irreducible representation of the $C_{4v}$ point group of the square lattice than that of the ground state as follows: $\\Psi _{{\\bf K} = (0,0)}^{A, S_+} ( \\sigma ) &=&\\Psi _{{\\bf K} = (0,0)}^{S_+} ( \\sigma ) + \\Psi _{{\\bf K} = (0,0)}^{S_+} ( R_{\\pi /2} \\sigma ) \\\\\\Psi _{{\\bf K} = (0,0)}^{B, S_+} ( \\sigma ) &=&\\Psi _{{\\bf K} = (0,0)}^{S_+} ( \\sigma ) - \\Psi _{{\\bf K} = (0,0)}^{S_+} ( R_{\\pi /2} \\sigma ),$ where the $R_{\\pi /2}$ is an operator to rotate the spin configuration by 90 degrees.", "With this projection, we can distinguish whether the state belongs to $A$ (either $A_1$ or $A_2$ ) irreducible representation or $B$ (either $B_1$ or $B_2$ ) irreducible representation under the $C_{4v}$ point group (to distinguish between $A_1$ and $A_2$ or between $B_1$ and $B_2$ , we need full point group projection with 0, $\\pi /2$ , $\\pi $ , $3\\pi /2$ rotations).", "The ground state corresponds to the former, while the excited state corresponds to the latter.", "We also need special treatment to obtain $S=2$ excited states.", "To this end, we use the mVMC (many-variable variational Monte Carlo method) [58] based on the PP wave function.", "In the mVMC, the full spin projection to specify the total spin is available, and we apply it to get $S=2$ states.", "The full spin projection is time-consuming (at least about five times) compared to the spin-parity projection.", "At the cost of longer computational time for the full spin projection, the mVMC (only PP) gives comparable accuracy to the RBM+PP method.", "Figure: Comparison of the ground state energy for the J 1 J_1-J 2 J_2 Heisenberg model.The comparison is made among the variational energies under the periodic boundary condition.The system sizes are (a) 6×66\\times 6 and (b) 8×88\\times 8.Our RBM+PP results are compared with those obtained by the variational Monte Carlo (VMC) method combined with the pp-th order Lanczos steps ,the density-matrix renormalization group (DMRG) (with 8182 SU(2) states) , the convolutional neural network (CNN) , and the exact diagonalization (ED) .The CNN and ED results are available only for the 6×66\\times 6 lattice." ], [ "Calculation conditions", "In the present study, we fix the number of hidden units $N_{\\rm hidden}$ to be 16.", "We always apply the spin-parity and momentum projections during the optimization of the RBM+PP wave function.", "The special treatments to obtain $S=0$ excited state at ${\\bf K} = (0,0)$ and $S=2$ excited states are described above.", "To improve the quality of the data for the correlation function in Figs.", "REF , REF , REF , REF , REF , REF , REF , and REF quantitatively, we apply the simplified point-group projection in Eq.", "(REF ) to the optimized ground state RBM+PP wave function for the sector with $S=0$ and ${\\bf K} = (0,0)$ .", "The ground state energy in Fig.", "REF is also produced with the simplified point-group projection (see Appendix REF for the detail)." ], [ "Accuracy of the RBM+PP wave function", "By applying the RBM+PP method to the 2D $J_1$ -$J_2$ Heisenberg model on the square lattice, we confirm that the RBM+PP achieves state-of-the-art accuracy not only among machine-learning-based methods [62], [60], [63], [64] but also among all available numerical methods.", "Figure REF shows the comparison of the ground-state energy among various methods for the $6\\times 6$ and $8\\times 8$ lattices (see Table REF for the raw data).", "Here, the RBM+PP energy is obtained by optimizing the RBM+PP wave function with the momentum, spin-parity, and simplified-point-group projections.", "We do not employ the sublattice structure in the $f_{ij}^{\\uparrow \\downarrow }$ parameters (the sublattice structure used in the actual calculations is discussed in Appendix REF ), and the number of hidden units is 16 as commonly employed in the paper.", "Up to the $6\\times 6$ lattice, the exact diagonalization result is available.", "At $J_2 = 0.5$ , where the frustration is strong, the relative error of the RBM+PP energy is less than 0.01 %, demonstrating the high accuracy of the RBM+PP wave function.", "For the $8 \\times 8$ lattice, the RBM+PP wave function gives the best accurate energy among the compared variational methods for all $J_2$ values we studied.", "We have also performed the benchmark calculations for the $10 \\times 10$ lattice because the benchmarks of neural-network wave functions in the literature have mainly been performed using the $10 \\times 10$ lattice.", "We optimized the RBM+PP wave function with 16 hidden units (as used in the other system sizes) without introducing a sublattice structure in the PP part.", "We apply the momentum, spin-parity, and point-group projections.", "Table REF shows the comparison of the ground-state energy at $J_2=0.5$ among different wave functions.", "As in the $8 \\times 8$ lattice result, the RBM+PP gives the best accuracy among the various methods.", "From the systematic benchmarks on the $6\\times 6$ , $8\\times 8$ , and $10\\times 10$ lattices, we conclude that the RBM+PP achieves state-of-the-art accuracy.", "Table: Raw data of RBM+PP ground-state energy in Fig.", ".Table: Comparison of ground-state energy for the 10×1010\\times 10 lattice at J 2 =0.5J_2=0.5 among different wave functions.The wave functions in bold font use neural networks.In Ref.", ", pp-th order Lanczos steps are applied to the VMC wave function.In the actual calculations, we employ the sublattice structure in the $f_{ij}^{\\uparrow \\downarrow }$ parameters for the $8\\times 8$ , $12\\times 12$ , $16\\times 16$ , and $18\\times 18$ lattices, to reduce the computational cost from ${\\mathcal {O}}( N_{\\rm site}^4)$ to ${\\mathcal {O}}( N_{\\rm site}^3)$ (see Appendix REF ).", "The reduction of the computational time enables us to perform systematic calculations for various $J_2$ values and for different quantum-number sectors.", "By employing the sublattice structure, the accuracy becomes slightly worse compared to that without sublattice structure.", "For example, in the case of the $8\\times 8$ lattice at $J_2=0.5$ , the ground-state energy with the $4\\times 4$ sublattice structure is $-0.498460$ (6), which is compared to $-0.498886$ (1) obtained without a sublattice structure.", "The difference is less than 0.1 %; therefore, high accuracy is retained even with the sublattice structureFor large system sizes, we employ sublattice structures to make the calculations “practical” (making computational cost manageable).", "For example, in Ref.", "[18], the VMC($p$ =2) results are not available for large system sizes, and the “practical” calculation is VMC($p$ =1).", "We notice that, thanks to the retained accuracy, our calculations also show state-of-the-art accuracy at the “practical” level; for the ground-state energy for the $18 \\times 18$ lattice at $J_2 = 0.5$ , the VMC($p$ =1) wave function in [18] gives $E/N_{\\rm site} = - 0.49611(1)$ , whereas our RBM+PP wave function with $6\\times 6$ sublattice structure gives a better precision of $E/N_{\\rm site} = -0.496275(3)$.", "As for the spin-spin and dimer-dimer correlations, the obtained values of the order parameters are $m_{\\rm N\\acute{e}el}^2 = 0.06955(8)$ and $m_{\\rm VBS}^2 = 0.01720(3)$ in the case of the $4\\times 4$ sublattice structure, and $m_{\\rm N\\acute{e}el}^2 = 0.06724(8)$ and $m_{\\rm VBS}^2 = 0.01703(3)$ in the case of no sublattice structure.", "The actual calculations with sublattice structure tend to slightly overestimate the order parameters in the frustrated regime.", "We see a similar tendency in the case of the benchmark calculations of RBM only wave functions for the $6 \\times 6$ lattice at $J_2=0.5$ with changing the number of hidden units [67], where the Néel-AF order parameter tends to be overestimated for a small number of hidden units.", "With increasing the number of hidden units, the accuracy improves, and the order parameter shows an excellent agreement with the exact results [67].", "Considering the fact that improving accuracy tends to suppress the order parameter, our statement of the existence of the QSL phase with vanishing order parameters in the thermodynamic limit should be valid.", "Figure: Singlet, triplet and quintuplet excitation energies for 6×66\\times 6 lattice obtained by RBM+PP (filled symbols) and ED (open symbols).In ED, we calculate up to five excited states using ℋΦ{\\mathcal {H}}\\Phi  .At J 2 =0.40J_2 = 0.40, S=2S=2 excitation with the momentum (0,0)(0,0) is not included in the five lowest excited states.The same holds for S=1S=1 excitation with the momentum (π,π)(\\pi ,\\pi ) at J 2 =0.55J_2 = 0.55.The RBM+PP and exact results show a good agreement.Remarkably, we also find that the RBM+PP accurately represents excited states as well as the ground state.", "Figure REF shows the comparison of excitation energies for singlet, triplet, and quintuplet excitations between the exact and RBM+PP results for the $6\\times 6$ lattice.", "The agreement is excellent, where the difference in energy between the exact and the RBM+PP results is less than 0.01.", "Previously, there have been several attempts to obtain the excitation gap of the $J_1$ -$J_2$ model [18], [20], [22], [24].", "In Ref.", "hu13 using the combination of the VMC and Lanczos methods, the excited states are obtained by changing boundary condition, which limits the number of excited states that can be calculated [only $S=2$ with the momentum $(0,0)$ and $S=0$ with $(\\pi ,0)$ or $(0,\\pi )$ ].", "Also, the accuracy does not reach the level shown in Fig.", "REF even with the 2nd-order Lanczos being applied [VMC($p=2$ )].", "In Refs.", "jiang12,gong14,Wang2018 using the density-matrix renormalization group (DMRG), the open boundary condition is employed, and hence the dispersion is not available because the momentum is ill-defined.", "In the present study, we can obtain accurate excitation energies with momentum resolution.", "The accurate estimate of excitation gaps enables us to perform the level spectroscopy to estimate the phase boundary and elucidate the nature of the QSL phase.", "Figure: Initial-parameter dependence of the RBM+PP optimization curves for the ground state at (a) J 2 =0.5J_2=0.5 and (b) J 2 =0.6J_2=0.6 for the 8×88\\times 8 lattice.The results for the four independent optimizations are shown for each J 2 J_2.In (b), the dotted line indicates the total energy of a local-minimum solution." ], [ "Initial-parameter dependence of the RBM+PP optimization", "As we describe in Appendix REF , we perform several independent optimizations of the RBM+PP wave functions for each $J_2$ point.", "Here, using the $8\\times 8$ lattice, we show how the difference in initial variational parameters affects the optimization.", "Figure REF shows the initial-parameter dependence of the RBM+PP ground-state optimization curves for $J_2= 0.5$ and $J_2=0.6$ .", "For $J_2$ = 0.5, we see that four independent optimizations converge to the same energy stably.", "On the other hand, at $J_2 = 0.6$ , the RBM+PP wave function whose optimization curve is shown in blue color seems to be trapped in a local minimum.", "The green curve is also trapped at similar energy (dotted line), but it eventually gets out of the local minimum.", "The behavior seen at $J_2 = 0.6$ can be understood from the proximity to the 1st-order transition point around $J_2 = 0.61$ between the VBS and stripe-AF phases.", "At large system sizes, there exists an energy-level crossing between the states belonging to the same quantum-number sector (zero total momentum and singlet), which gives a kink in the $J_2$ dependence of the ground state energy (Fig.", "REF ).", "Therefore, different solutions are competing in small energy scale in the same quantum-number sector at $J_2 = 0.6$ , which makes the optimization more unstable as compared to that at $J_2 = 0.5$ .", "From this benchmark, we notice that it is important to perform several independent optimizations to avoid being trapped in local minima.", "In the present study, although the optimizations of the RBM+PP wave functions are done independently for different $J_2$ points, thanks to the several independent optimizations at each $J_2$ value, all the physical quantities change smoothly and continuously." ], [ "Ground state energy ", "The phase transition between the VBS and stripe-AF phases at $J_2^{\\rm VS}$ in Fig.", "REF is of 1st order.", "To see this, we show the ground state energy as a function of $J_2$ in Fig.", "REF .", "As the system size increases, we see a clear kink in the energy curve at $J_2^{\\rm VS} \\approx 0.61$ , giving evidence for the 1st-order phase transition.", "Figure: J 2 J_2 dependence of RBM+PP ground-state energy of square-lattice J 1 J_1-J 2 J_2 Heisenberg model.Figure: Structure factor for spin-spin correlation S s (𝐪)S_{\\rm s}({\\bf q}).Figure: Structure factor for dimer-dimer correlation S d x (𝐪)S_{{\\rm d}_x}({\\bf q}) and S d y (𝐪)S_{{\\rm d}_y}({\\bf q})." ], [ "Structure factors", "In Sec.", "REF , we discuss the crossing of the correlation ratio.", "The correlation ratio quantifies how sharp the structure factor peak is.", "In Figs.", "REF and REF , we show the raw data of the structure factors for spin-spin and dimer-dimer correlations, respectively, which are used in the correlation ratio analysis." ], [ "System-size dependence of the crossing $J_2$ points of the AF and VBS correlation ratios", "As described in Sec.", "REF , we determine the AF-QSL and QSL-VBS phase boundaries from the correlation ratio analysis.", "Figure REF shows the system-size dependence of the crossing points of the correlation-ratio curves.", "We see that the system-size dependence is small.", "The fits of the system-size dependence with $a+b/L$ dependence give the estimates of AF-QSL and QSL-VBS phase boundaries as $J_2^{\\rm N\\acute{e}el} = 0.492(8)$ and $J_2^{\\rm VBS} =0.548(1)$ .", "The fits using $a+b/L^2$ give $J_2^{\\rm N\\acute{e}el} = 0.490(4)$ and $J_2^{\\rm VBS} =0.542(1)$ .", "These results support our conclusions of $J_2^{\\rm N\\acute{e}el}\\approx 0.49$ and $J_2^{\\rm VBS}\\approx 0.54$ .", "Figure: System-size dependence of the crossing points of the correlation ratio for spin-spin (red dots) and dimer-dimer (black squares) correlations, which are used to determine the phase boundary of Néel-AF and VBS, respectively.We focus on the crossing of the curves between the L 1 ×L 1 L_1\\times L_1 and L 2 ×L 2 L_2 \\times L_2 lattices with (L 1 ,L 2 )=(8,12)(L_1, L_2) = (8,12), (8,16)(8,16), (12,16)(12,16) and (16,18)(16,18) for spin-spin correlations, and (L 1 ,L 2 )=(8,12)(L_1, L_2) = (8,12), (8,16)(8,16), and (12,16)(12,16) for the dimer-dimer correlations.L mid L_{\\rm mid} is defined as L mid =(L 1 +L 2 )/2L_{\\rm mid} = (L_1+L_2)/2." ], [ "Excitation gap at ", "As we mentioned in Appendix REF , we impose the $4 \\times 4$ sublattice structure in the $f_{ij}^{\\uparrow \\downarrow }$ parameters in the PP part.", "With this setting, we have momentum resolution of $4 \\times 4$ ${\\bf K}$ points: ${\\bf K} = ( m \\pi /2, n \\pi / 2)$ with $m,n= -1$ , 0, 1, 2.", "To investigate the sublattice-size dependence, for $12\\times 12$ lattice, we also calculate the excitation energies using $6 \\times 6$ sublattice structure.", "Then, we can calculate the excitation gaps at ${\\bf K} = ( m \\pi /3, n \\pi /3)$ with $m,n = -2$ , $-1$ , 0, 1, 2, 3.", "Figure REF shows the $f_{ij}^{\\uparrow \\downarrow }$ -sublattice-size dependence of the excitation energies.", "We see that the excitation gaps at high-symmetry ${\\bf K}$ points [$(0,0)$ , $(\\pi ,0)$ and $(\\pi ,\\pi )$ ] show good agreement between the $4 \\times 4$ and $6\\times 6$ sublattice structures.", "At the intermediate ${\\bf K}$ points, the excitation energies stay larger than those at high-symmetry ${\\bf K}$ points.", "This fact supports the scenario of Dirac-type nodal QSL.", "Figure: f ij ↑↓ f^{\\uparrow \\downarrow }_{ij}-sublattice-size dependence of excitation.", "(a) Singlet and (b) triplet excitation energy along the symmetric line in the Brillouin zone for 12×1212\\times 12 lattice.Red dots: 4×44\\times 4 sublattice structure.Black triangles: 6×66\\times 6 sublattice structure." ], [ "Supplementary Notes", " Around the AF-QSL and QSL-VBS phase boundaries, we see noteworthy features in singlet excitations at ${\\bf K}=(\\pi ,\\pi )$ and triplet ones at ${\\bf K}=(\\pi ,0), (0,\\pi )$ .", "First, around the AF-QSL boundary ($J_2 = J_2^{\\rm N\\acute{e}el} \\approx 0.49$ ), we see the kink in the excitation energy in the singlet ${\\bf K}=(\\pi ,\\pi )$ excitation [Figs.", "REF (a) and REF (b)].", "Actually, there is a level crossing in this quantum number sector, and the point-group irreducible representation of the lowest state changes at the kink.", "Also around the QSL-VBS boundary ($J_2 =J_2^{\\rm VBS}\\approx 0.54$ ), with increasing $J_2$ , there is an upturn of the excitation energy of triplet ${\\bf K}=(\\pi ,0)$ excitation for $16\\times 16$ lattice [Fig.", "REF (b)], which seems consistent with the fact that the triplet excitation has a gap in the VBS phase.", "These two supplementary features are suggestive of the connection to the phase transitions; it would be interesting to investigate them further.", "In Fig.", "REF , the excitation energy with $S=0$ and ${\\bf K} = (0,0)$ sector stays almost constant as the system size $L$ changes, in contrast with the behavior at the other high-symmetry ${\\bf K}$ points.", "The singlet excited state at ${\\bf K} = (0,0)$ must belong to a different irreducible representation than that of the ground state, because, in the present method, we cannot obtain the excited states with the same irreducible representation as that of the ground state.", "Such excited states with the same irreducible representation might show similar behavior to those at the other high-symmetry ${\\bf K}$ points.", "The weight of the triplet at $(\\pi ,\\pi )$ seems to be scaled naturally to a nonzero value, which might imply the remnant of the pole.", "This requires further clarification in larger system sizes in future.", "The reason could partly be that the calculation is done close to the AF(Néel)-QSL phase boundary $J_2 = J_2^{\\rm N\\acute{e}el} \\approx 0.49$ .", "Another origin might be a possible anisotropic (elliptic) Dirac dispersion of spinons with preserved $C_4$ symmetry, which makes the spinon particle-hole excitation denser for the momentum transfer $(\\pi ,\\pi )$ and makes the slow convergence to zero.", "The $J_2^{\\rm N\\acute{e}el}$ and $J_2^{\\rm VBS}$ dependence of the estimate of the critical exponents is as follows.", "For the Néel-AF order parameter, $z+\\eta = 1.384(3)$ , 1.410(4), 1.437(5) and $\\nu = 1.22(4)$ , 1.21(5), 1.18(5) for $J_2^{\\rm N\\acute{e}el} = 0.485$ , 0.490, 0.495, respectively.", "The $\\nu $ values for different $J_2^{\\rm N\\acute{e}el}$ values agree within the size of error bars.", "Although $z+\\eta $ increases as $J_2^{\\rm N\\acute{e}el}$ increases, the values lie around 1.4.", "For the VBS order parameter, $z+\\eta = 1.471 (8)$ , 1.436(6), 1.400(5) and $\\nu = 0.66(3)$ , 0.67(2), 0.65(2) for $J_2^{\\rm VBS} = 0.535$ , 0.540, 0.545, respectively.", "As in the case of the Néel-AF case, the $\\nu $ values for different $J_2^{\\rm VBS}$ values agree within the size of error bars.", "Though $z+\\eta $ decreases slightly as $J_2^{\\rm VBS}$ increases, it lies between 1.4 and 1.5, which are close to those at the Néel-AF critical point.", "Though $z+\\eta $ decreases slightly as $J_2^{\\rm VBS}$ increases, it lies between 1.4 and 1.5, which is close to those at the Néel-AF critical point." ] ]

[ [ "Inferring Signaling Pathways with Probabilistic Programming" ], [ "Abstract Cells regulate themselves via dizzyingly complex biochemical processes called signaling pathways.", "These are usually depicted as a network, where nodes represent proteins and edges indicate their influence on each other.", "In order to understand diseases and therapies at the cellular level, it is crucial to have an accurate understanding of the signaling pathways at work.", "Since signaling pathways can be modified by disease, the ability to infer signaling pathways from condition- or patient-specific data is highly valuable.", "A variety of techniques exist for inferring signaling pathways.", "We build on past works that formulate signaling pathway inference as a Dynamic Bayesian Network structure estimation problem on phosphoproteomic time course data.", "We take a Bayesian approach, using Markov Chain Monte Carlo to estimate a posterior distribution over possible Dynamic Bayesian Network structures.", "Our primary contributions are (i) a novel proposal distribution that efficiently samples sparse graphs and (ii) the relaxation of common restrictive modeling assumptions.", "We implement our method, named Sparse Signaling Pathway Sampling, in Julia using the Gen probabilistic programming language.", "Probabilistic programming is a powerful methodology for building statistical models.", "The resulting code is modular, extensible, and legible.", "The Gen language, in particular, allows us to customize our inference procedure for biological graphs and ensure efficient sampling.", "We evaluate our algorithm on simulated data and the HPN-DREAM pathway reconstruction challenge, comparing our performance against a variety of baseline methods.", "Our results demonstrate the vast potential for probabilistic programming, and Gen specifically, for biological network inference.", "Find the full codebase at https://github.com/gitter-lab/ssps" ], [ "Introduction", "Signaling pathways enable cells to process information rapidly in response to external environmental changes or intracellular cues.", "One of the core signaling mechanisms is protein phosphorylation.", "Kinases add phosphate groups to substrate proteins and phosphatases remove them.", "These changes in phosphorylation state can act as switches, controlling proteins' activity and function.", "A protein's phosphorylation status affects its localization, stability, and interaction partners [31].", "Ultimately, phosphorylation changes regulate important biological processes such as transcription and cell growth, death, and differentiation [24], [26].", "Pathway databases characterize the signaling relationships among groups of proteins but are not tailored to individual biological contexts.", "Even for well-studied pathways such as epidermal growth factor receptor-mediated signaling, the proteins significantly phosphorylated during a biological response can differ greatly from those in the curated pathway [27].", "The discrepancy can be due to context-specific signaling [23], cell type-specific protein abundances, or signaling rewiring in disease [35].", "Therefore, there is a need to learn context-specific signaling pathway representations from observed phosphorylation changes.", "In the clinical setting, patient-specific signaling pathway representations may eventually be able to guide therapeutic decisions [10], [19], [11].", "Diverse classes of techniques have been developed to model and infer signaling pathways [25].", "They take approaches including Granger causality [39], [4], information theory [6], [29], logic models [12], [18], [16], differential equations [38], [30], [20], non-parametric statistical tests [43], and probabilistic graphical models [36] among others.", "Some signaling pathway reconstruction algorithms take advantage of perturbations such as receptor stimulation or kinase inhibition.", "Although perturbing individual pathway members can causally link them to downstream phosphorylation changes, characterizing a complex pathway can require a large number of perturbation experiments.", "Inferring pathway structure from temporal phosphorylation data presents an attractive alternative.", "A single time series phosphorylation dataset can reveal important dynamics without perturbing individual pathway members.", "For instance, a kinase cannot phosphorylate substrates before it is activated.", "An alternative approach to pathway reconstruction selects a context-specific subnetwork from a general background network.", "These algorithms can use phosphorylation data to assign scores to protein nodes in a protein-protein interaction network.", "They then select edges that connect the high-scoring nodes, generating a subnetwork that may explain how the induced phosphorylation changes arise from the source of stimulation.", "Extensions accommodate temporal scores on the nodes [34], [2], [27], [32].", "Our present work builds on past techniques that formulate signaling pathway inference as a Dynamic Bayesian Network (DBN) structure estimation problem.", "This family of techniques relies on two core ideas: (i) we can use a DBN to model phosphorylation time series data; and (ii) the DBN's structure translates directly to a directed graph representing the signaling pathway.", "Rather than identifying a single DBN that best fits the data, these techniques take a Bayesian approach—they yield a posterior distribution over possible DBN structures.", "Some techniques use Markov Chain Monte Carlo (MCMC) to sample from the posterior [42], [17].", "Others use exact, enumerative inference to compute posterior probabilities [21], [33], [40].", "We present a new Bayesian DBN-based technique, Sparse Signaling Pathway Sampling (SSPS).", "It improves on past MCMC methods by using a novel proposal distribution specially tailored for the large, sparse graphs prevalent in biological applications.", "Furthermore, SSPS makes weaker modeling assumptions than other DBN approaches.", "As a result, SSPS scales to larger problem sizes and yields superior predictions in comparison to other DBN techniques.", "We implement SSPS using the Gen probabilistic programming language (PPL).", "Probabilistic programming is a powerful methodology for building statistical models.", "It enables the programmer to build models in a legible, modular, reusable fashion.", "This flexibility was important for prototyping and developing the current form of SSPS and readily supports future improvements or extensions.", "SSPS makes specific modeling assumptions.", "We start with the DBN model of [21], relax some assumptions, and modify it in other ways to be better-suited for MCMC inference.", "We first define some notation for clarity's sake.", "Let $G$ denote a directed graph with vertices $V$ and edges $E(G)$ .", "Graph $G$ will represent a signaling pathway, with vertices $V$ corresponding to proteins and edges $E(G)$ indicating their influence relationships.", "We use $\\text{pa}_G(i)$ to denote the parents of vertex $i$ in $G$ .", "Let $X$ denote our time series data, consisting of $|V|$ variables measured at $T$ timepoints.", "$X$ is a $T {\\times } |V|$ matrix where the $j$ th column corresponds to the $j$ th variable and the $j$ th graph vertex.", "As a convenient shorthand, let $X_+$ denote the latest $T{-}1$ timepoints in $X$ , and let $X_-$ denote the earliest $T{-}1$ timepoints in $X$ .", "Lastly, define $B_j \\equiv X_{-,\\text{pa}_G(j)}$ .", "In other words, $B_j$ contains the values of variable $j$ 's parents at the $T{-}1$ earliest timepoints.", "In general, $B_j$ may also include columns of nonlinear interactions between the parents.", "We will only include linear terms, unless stated otherwise." ], [ "Model derivation.", "In our setting, we aim to infer $G$ from $X$ .", "In particular, Bayesian approaches seek a posterior distribution $P(G|X)$ over possible graphs.", "From Bayes's rule we know $P(G|X) \\propto P(X|G) \\cdot P(G).$ That is, a Bayesian model is fully specified by its choice of prior distribution $P(G)$ and likelihood function $P(X|G)$ .", "We derive our model from the one used by [21].", "They choose a prior distribution of the form $P(G~|~G^\\prime , \\lambda ) \\propto \\exp \\left( -\\lambda | E(G) \\setminus E(G^\\prime )| \\right) $ parameterized by a reference graph $G^\\prime $ and inverse temperature $\\lambda $ .", "This prior gives uniform probability to all subgraphs of $G^\\prime $ and “penalizes” edges not contained in $E(G^\\prime )$ .", "$\\lambda $ controls the “importance” given to the reference graph.", "[21] choose a Gaussian DBN for their likelihood function.", "Intuitively, they assume linear relationships between variables and their parents: $X_{+,j} \\sim \\mathcal {N}(B_j \\beta _j, \\sigma _j^2) \\hspace{40.0pt} \\forall j \\in \\lbrace 1\\ldots |V|\\rbrace .$ A suitable prior over the regression coefficients $\\beta _j$ and noise parameters $\\sigma _j^2$ (Figure REF ) allows us to integrate them out, yielding this marginal likelihood function: $ P(X|G) \\propto \\prod \\limits _{j=1}^{|V|} T^{-\\frac{|\\text{pa}_G(j)|}{2}} \\left(X_{+,j}^{\\top } X_{+,j} - \\frac{T {-} 1}{T} X_{+,j}^\\top (B_j \\hat{\\beta }_{ols}) \\right)^{-\\frac{T-1}{2}} $ where $\\hat{\\beta }_{ols} = (B_j^\\top B_j)^{-1} B_j^\\top X_{+,j}$ is the ordinary least squares estimate of $\\beta _j$ .", "For notational simplicity, Equation REF assumes we have a single time course of length $T$ .", "In general, there may be multiple time course replicates with differing lengths.", "The marginal likelihood generalizes to that case in a straightforward way.", "In SSPS we use the same marginal likelihood function (Equation REF ), but a different prior distribution $P(G)$ .", "We obtain our prior distribution by decomposing Equation REF into a product of independent Bernoulli trials over graph edges.", "This decomposition in turn allows us to make some useful generalizations.", "Define edge existence variables $z_{ij} \\equiv \\mathbb {1}[(i,j) \\in E(G)]$ .", "Let $Z$ be the $|V|{\\times } |V|$ matrix of all $z_{ij}$ .", "Then we can rewrite Equation REF as follows: $P(G|G^\\prime , \\lambda ) ~\\equiv ~ P(Z|G^\\prime , \\lambda ) ~\\propto ~ \\prod _{(i,j) \\notin E(G^\\prime )}\\!\\!\\!", "e^{-z_{ij} \\lambda }$ $= \\prod _{(i,j) \\in E(G^\\prime )}\\!\\!\\!", "\\left(\\frac{1}{2}\\right)^{z_{ij}}\\!\\left(\\frac{1}{2}\\right)^{1 {-} z_{ij}}\\!\\!\\!\\!\\!", "\\prod _{(i,j) \\notin E(G^\\prime )} \\left( \\frac{e^{-\\lambda }}{1 {+} e^{-\\lambda }} \\right)^{z_{ij}}\\!", "\\left( \\frac{1}{1 {+} e^{-\\lambda }} \\right)^{1 {-} z_{ij}}$ where the last line is a true equality—it gives a normalized probability measure.", "We see that the original prior is simply a product of Bernoulli variables parameterized by a shared inverse temperature, $\\lambda $ .", "See Appendix REF for a more detailed derivation.", "Rewriting the prior in this form opens the door to generalizations.", "First, we address a shortcoming in the way reference graph $G^\\prime $ expresses prior knowledge.", "The original prior assigns equal probability to every edge of $G^\\prime $ .", "However, in practice we may have differing levels of prior confidence in the edges.", "We address this by allowing a real-valued prior confidence $c_{ij}$ for each edge: $P(Z|C,\\lambda ) = \\prod _{(i,j)} \\left( \\frac{e^{-\\lambda }}{e^{-c_{ij}\\lambda } {+} e^{-\\lambda }} \\right)^{z_{ij}}\\!", "\\left( \\frac{e^{-c_{ij}\\lambda }}{e^{-c_{ij}\\lambda } {+} e^{-\\lambda }} \\right)^{1 {-} z_{ij}} $ where $C$ is the matrix of all prior confidences $c_{ij}$ , replacing $G^\\prime $ .", "Notice that if every $c_{ij} {\\in } \\lbrace 0,1\\rbrace $ , then Equation REF is equivalent to the original prior.", "In effect, Equation REF interpolates the original prior, permitting a continuum of confidences on the interval $[0,1]$ .", "We make one additional change to the prior by replacing the shared $\\lambda $ inverse temperature variable with a collection of variables, $\\Lambda = \\lbrace \\lambda _j ~|~ j = 1, {\\ldots } ,|V| \\rbrace $ , one for each vertex of the graph.", "Recall that the original $\\lambda $ variable determined the importance of the reference graph.", "In the new formulation, each $\\lambda _j$ controls the importance of the prior knowledge for vertex $j$ and its parents: $P(Z|C,\\Lambda ) = \\prod _{(i,j)} \\left( \\frac{e^{-\\lambda _j}}{e^{-c_{ij}\\lambda _j} {+} e^{-\\lambda _j}} \\right)^{z_{ij}}\\!", "\\left( \\frac{e^{-c_{ij}\\lambda _j}}{e^{-c_{ij}\\lambda _j} {+} e^{-\\lambda _j}} \\right)^{1 {-} z_{ij}} $ We introduced $\\Lambda $ primarily to help MCMC converge more efficiently.", "Experiments with the shared $\\lambda $ revealed a multimodal posterior that tended to trap $\\lambda $ in high or low values.", "The introduction of vertex-specific $\\lambda _j$ variables yielded faster convergence with weaker modeling assumptions—an improvement in both respects.", "We implicitly relax the model assumptions further via our inference procedure.", "For sake of tractability, the original exact method of [21] imposes a hard constraint on the in-degree of each vertex.", "In contrast, we use a MCMC inference strategy with no in-degree constraints.", "In summary, our model departs from that of [21] in three important respects.", "It permits real-valued prior confidences $C$ , introduces vertex-specific inverse temperature variables $\\Lambda $ , and places no constraints on vertices' in-degrees.", "See the full model in Figure REF and Appendix REF for additional details.", "Figure: Our generative model.", "(top) Plate notation.", "DBN parameters β j \\beta _j and σ j 2 \\sigma _j^2 have been marginalized out.", "(bottom) Full probabilistic specification.We usually set λ min ≃3\\lambda _\\text{min} \\simeq 3 and λ max =15\\lambda _\\text{max} {=} 15.If λ min >0\\lambda _\\text{min}{>}0 is too small, Markov chains will occasionally be initialized with very large numbers of edges, causing computational issues.The method is insensitive to λ max \\lambda _\\text{max} as long as it's sufficiently large.Notice the improper prior 1/σ j 2 1/\\sigma _j^2.In this specification B j B_j denotes X -,pa Z (j) X_{-, \\text{pa}_Z(j)}; that is, the parents of vertex jj depend on edge existence variables ZZ.Our method uses MCMC to infer posterior edge existence probabilities.", "As described in Section REF , our model contains two classes of unobserved random variables: (i) the edge existence variables $Z$ and (ii) the inverse temperature variables $\\Lambda $ .", "For each step of MCMC, we loop through these variables and update them in a Metropolis-Hastings fashion." ], [ "Main loop.", "At a high level, our MCMC procedure consists of a loop over the graph vertices, $V$ .", "For each vertex $j$ , we update its inverse temperature variable $\\lambda _j$ and then update its parent set $\\text{pa}_G(j)$ .", "All of these updates are Metropolis-Hastings steps; the proposal distributions are described below.", "Each completion of this loop yields one iteration of the Markov chain." ], [ "Proposal distributions.", "For the inverse temperature variables we use a symmetric Gaussian proposal: $\\lambda _j^\\prime \\sim \\mathcal {N}(\\lambda _j, \\xi ^2)$ .", "In practice the method is insensitive to $\\xi $ ; we typically set $\\xi {=}3$ .", "The parent set proposal distribution is more complicated.", "There are two principles at work when we design a graph proposal distribution: (i) the proposal should efficiently traverse the space of directed graphs, and (ii) it should favor graphs with higher posterior probability.", "The most widely used graph proposal distribution selects a neighboring graph uniformly from the set of possible “add-edge,” “remove-edge,” and “reverse-edge” actions [42], [17].", "We'll refer to this traditional proposal distribution as the uniform graph proposal.", "In our setting, we expect sparse graphs to be much more probable than dense ones—notice how the marginal likelihood function (Equation REF ) strongly penalizes $|\\text{pa}_G(j)|$ .", "However, the uniform graph proposal exhibits a preference toward dense graphs.", "It proposes “add-edge” actions too often.", "This motivates us to design a new proposal distribution tailored for sparse graphs—one that operates on our sparse parent set graph representation.", "For a given graph vertex $j \\in V$ , the parent set proposal distribution updates $\\text{pa}_G(j)$ by choosing from the following actions: add-parent.", "Select one of vertex $j$ 's non-parents uniformly at random, and add it to $\\text{pa}_G(j)$ .", "remove-parent.", "Select one of vertex $j$ 's parents uniformly at random, and remove it from $\\text{pa}_G(j)$ .", "swap-parent.", "A simultaneous application of add-parent and remove-parent.", "Perhaps surprisingly, this action is not made redundant by the other two.", "It plays an important role by yielding updates that maintain the size of the parent set.", "Because the marginal likelihood (Equation REF ) changes steeply with $|\\text{pa}_G(j)|$ , Metropolis-Hastings acceptance probabilities will be higher for actions that keep $|\\text{pa}_G(j)|$ constant.", "These three actions are sufficient to explore the space of directed graphs, but we need another mechanism to bias the exploration toward sparse graphs.", "We introduce this preference via the probability assigned to each action.", "Intuitively, we craft the action probabilities so that when $|\\text{pa}_G(j)|$ is too small, add-parent moves are most probable.", "When $|\\text{pa}_G(j)|$ is too big, remove-parent moves are most probable.", "When $|\\text{pa}_G(j)|$ is about right, all moves are equally probable.", "We formulate the action probabilities for vertex $j$ as follows.", "As a shorthand, let $s_j = |\\text{pa}_G(j)|$ and define the reference size $\\hat{s}_j = \\sum _{i=1}^{|V|} c_{ij}$ .", "That is, $\\hat{s}_j$ uses the prior edge confidences $C$ to estimate an appropriate reference size for the parent set.", "Then, the action probabilities are $\\begin{aligned}p(\\texttt {add-parent}| s_j, \\hat{s}_j) & \\propto 1 - \\left(\\frac{s_j}{|V|}\\right)^{\\gamma (\\hat{s}_j)} \\\\p(\\texttt {remove-parent}| s_j, \\hat{s}_j) & \\propto \\left(\\frac{s_j}{|V|} \\right)^{\\gamma (\\hat{s}_j)}\\\\p(\\texttt {swap-parent}| s_j, \\hat{s}_j) & \\propto 2\\left(\\frac{s_j}{|V|} \\right)^{\\gamma (\\hat{s}_j)} \\cdot \\left(1 - \\left(\\frac{s_j}{|V|}\\right)^{\\gamma (\\hat{s}_j)}\\right)\\end{aligned}$ where $\\gamma (\\hat{s}_j) = 1 / \\log _2(|V|/ \\hat{s}_j)$ .", "We use these functional forms only because they have certain useful properties: (i) when $s_j{=}0$ , the probability of add-parent is 1; (ii) when $s_j{=}|V|$ , the probability of remove-parent is 1; and (iii) when $s_j{=}\\hat{s}_j$ , all actions have equal probability (Figure REF ).", "Beyond that, these probabilities have no particular justification.", "We provide additional information about the parent set proposal in Appendix REF .", "Figure: Action probabilities as a function of parent set size.The reference size s ^\\hat{s} is determined from prior knowledge.It approximates the size of a “typical” parent set.When s<s ^s {<} \\hat{s} , add-parent is most probable;when s>s ^s {>} \\hat{s} , remove-parent is most probable;and when s=s ^s {=} \\hat{s} , all actions have equal probability.Recall that Metropolis-Hastings requires us to compute the reverse transition probability for any proposal we make.", "This could pose a challenge given our relatively complicated parent set proposal distribution.", "However, Gen provides a helpful interface for computing reverse probabilities.", "The user can provide an involution function that returns the reverse of a given action.", "Gen then manages the reverse probabilities without further intervention.", "This makes it relatively easy to implement Metropolis-Hastings updates with unusual proposal distributions." ], [ "Termination, convergence, and inference.", "We follow the basic MCMC protocols described by [15].", "This entails running multiple (i.e., 4) Markov chains and discarding the first half of each chain as burnin.", "In all of our analyses, we terminate each Markov chain when it either (i) reaches a length of 100,000 iterations or (ii) the execution time exceeds 12 hours.", "These termination conditions are arbitrary but emulate a real-world setting where it may be acceptable to let the method run overnight.", "Upon termination, we assess convergence with two diagnostics: Potential Scale Reduction Factor (PSRF) and effective number of samples ($N_{\\text{eff}}$ ).", "PSRF identifies cases where the Markov chains fail to mix or achieve stationarity.", "$N_{\\text{eff}}$ provides a sense of “sample size” for our inferred quantities.", "It adjusts the number of MCMC samples by accounting for autocorrelation in each chain.", "For our purposes, we say a quantity has failed to converge if its PSRF $\\ge {1.01}$ or $N_{\\text{eff}}{<} 10$ .", "Note that satisfying these criteria does not guarantee convergence.", "However, failure to satisfy them is a reliable flag for non-convergence.", "Assuming a quantity hasn't failed to converge, we estimate it by simply taking its sample mean from all samples remaining after burnin.", "In our setting we are primarily interested in edge existence probabilities; i.e., we compute the fraction of samples containing each edge." ], [ "Probabilistic programming implementation", "We implemented SSPS using the Gen PPL.", "We briefly describe the probabilistic programming methodology and its advantages in our setting." ], [ "Probabilistic programming.", "Probabilistic programming is a methodology for building statistical models.", "It's based on the idea that statistical models are generative processes—sequences of operations on random variables.", "In probabilistic programming, we express the generative process as a program written in a PPL.", "This program is then compiled to produce a log-probability function, which can be used in inference tasks.", "Probabilistic programming systems typically provide a set of generic inference methods for performing those tasks—e.g., MCMC or Variational Bayes.", "Compare this with a more traditional approach, where the user must (i) derive and implement the log-probability function and (ii) implement an inference method that operates on the log-probability function.", "This process of manual derivation and implementation is error-prone and requires a high degree of expertise from the user.", "In contrast, probabilistic programming only requires the user to express their model in a PPL.", "The probabilistic programming system manages other details.", "Probabilistic programming also tends to promote good software engineering principles such as abstraction, modularity, and legibility.", "Most PPLs organize code into functions, which can be reused by multiple statistical models." ], [ "Probabilistic programming languages.", "Several PPLs have emerged in recent years.", "Examples include Stan [5], Edward2 [9], Pyro [1], PyMC3 [37], and Gen [7].", "PPLs differ in how they balance expressive power and ease of use.", "For example, Stan makes it easy to build hierarchical statistical models with continuous variables but caters poorly to other model classes.", "At the other extreme, Gen can readily express a large class of models—discrete and continuous variables with complex relationships—but requires the user to design a customized inference procedure." ], [ "Implementation in ", "We use the Gen PPL precisely for its expressive power and customizable inference.", "While implementing SSPS, the customizability of Gen allowed us to begin with simple prototypes and then make successive improvements.", "For example, our model initially used a dense adjacency matrix representation for $G$ , but subsequent optimizations led us to use a sparse parent set representation instead.", "Similarly, our MCMC method started with a naïve “add or remove edge” proposal distribution; we arrived at our sparse proposal distribution (Section REF ) after multiple refinements.", "Other PPLs do not allow this level of control (Table REF ).", "Table: A coarse comparison of some noteworthy PPLs.Gen provides expressiveness but requires the user to implement an inference program for their model.Cont's vars: continuous variables; HMC: Hamiltonian Monte Carlo." ], [ "Simulation study evaluation", "We use a simulation study to answer important questions about SSPS: How does its computational expense grow with problem size?", "Is it able to correctly identify true edges?", "What is its sensitivity to errors in the prior knowledge?", "Simulations allow us to answer these questions in a controlled setting where we have access to ground truth." ], [ "Data simulation process.", "We generate each simulated dataset as follows: Sample a random adjacency matrix $A \\in \\lbrace 0,1\\rbrace ^{|V|{\\times } |V|}$ , where each entry is the outcome of a $\\text{Bernoulli}(p)$ trial.", "$A$ specifies the structure of a DBN.", "We choose $p{=}5/|V|$ so that each vertex has an average of 5 parents.", "This approximates the sparsity we might see in signaling pathways.", "We denote the size of the original edge set as $|E_0|$ .", "Let the weights $\\beta $ for this DBN be drawn from a normal distribution $\\mathcal {N}(0,1/\\sqrt{|V|})$ .", "We noticed empirically that the $1/\\sqrt{|V|}$ scale prevented the simulated time series from diverging to infinity.", "Use the DBN defined by $A, \\beta $ to simulate $M$ time courses of length $T$ .", "We imitate the real datasets in Section REF by generating $M{=}4$ time courses, each of length $T{=}8$ .", "Corrupt the adjacency matrix $A$ in two steps: (i) remove $r \\cdot |E_0|$ of the edges from $A$ ; (ii) add $a \\cdot |E_0|$ spurious edges to the adjacency matrix.", "This corrupted graph simulates the imperfect prior knowledge encountered in reality.", "The parameters $r$ and $a$ control the “false negatives” and “false positives” in the prior knowledge, respectively.", "We use a range of values for parameters $|V|, r,$ and $a$ , yielding a grid of simulations summarized in Table REF .", "See Appendix REF and Figure REF for additional details.", "Table: These parameters define the grid of simulated datasets in our simulation study.There are 3×4×4=483 {\\times } 4 {\\times } 4 {=} 48 distinct grid points.For each one, we generate K=5K{=}5 replicatesfor a total of 240 simulated datasets.The graph corruption parameters, rr and aa, range from very little error (0.1) to total corruption (1.0)." ], [ "Performance metrics.", "We are primarily interested in SSPS's ability to correctly recover the structure of the underlying signaling pathway.", "The simulation study allows us to measure this in a setting where we have access to ground truth.", "We treat this as a probabilistic binary classification task, where the method assigns an existence confidence to each possible edge.", "We measure classification performance using area under the precision-recall curve (AUCPR).", "We use average precision to estimate AUCPR, as opposed to the trapezoidal rule (which tends to be overly-optimistic, see [8], [13]).", "Our decision to use AUCPR is motivated by the sparseness of the graphs.", "For sparse graphs the number of edges grows linearly with $|V|$ while the number of possible edges grows quadratically.", "Hence, as $|V|$ grows, the proportion of positive instances decreases and the classification task increasingly becomes a “needle-in-haystack” scenario.", "Performance measurements on simulated data come with many caveats.", "It's most instructive to think of simulated performance as a sanity check.", "Since our data simulator closely follows our modeling assumptions, poor performance would suggest serious shortcomings in our method." ], [ "HPN-DREAM network inference challenge evaluation", "We measure SSPS's performance on experimental data by following the evaluation outlined by the HPN-DREAM Breast Cancer Network Inference Challenge [22].", "Signaling pathways differ across contexts—e.g., cell type and environmental conditions.", "The challenge is to infer these context-specific signaling pathways from time course data." ], [ "Dataset.", "The HPN-DREAM challenge provides phosphorylation time course data from 32 biological contexts.", "These contexts arise from exposing 4 cell lines (BT20, BT549, MCF7, UACC812) to 8 stimuli.", "For each context, there are approximately $M{=}4$ time courses, each about $T{=}7$ time points in length.", "Cell lines have differing numbers of phosphosite measurements (i.e., differing $|V|$ ), ranging from 39 (MCF7) to 46 (BT20)." ], [ "Prior knowledge.", "Participants in the original challenge were free to extract prior knowledge from any existing data sources.", "As part of their analysis, the challenge organizers combined participants' prior graphs into a set of edge probabilities.", "These aggregate priors summarize the participants' collective knowledge.", "They were not available to participants in the original challenge, but we use them in our analyses of HPN-DREAM data.", "We provide them to each of the baseline methods (see Section REF ), so the resulting performance comparisons are fair.", "We do not compare any of our scores to those listed by [22] in the original challenge results." ], [ "Performance metrics.", "The HPN-DREAM challenge aims to score methods by their ability to capture causal relationships between pairs of variables.", "It estimates this by comparing predicted descendant sets against experimentally generated descendant sets.", "More specifically, the challenge organizers exposed cells to AZD8055, an mTOR inhibitor, and observed the effects on other phosphosites.", "From this they determined a set of phosphosites downstream of mTOR in the signaling pathway.", "These include direct substrates of the mTOR kinase as well as indirect targets.", "Comparing predicted descendants of mTOR against experimentally generated descendants of mTOR gives us a notion of false positives and false negatives.", "As we vary a threshold on edge probabilities, the predicted mTOR descendants change, which allows us to make a receiver operating characteristic (ROC) curve.", "We calculate the resulting area under the ROC curve (AUCROC) with the trapezoidal rule to score methods' performance on the HPN-DREAM challenge.", "[22] provide more details for this descendant set AUCROC scoring metric.", "AUCROC is sensible for this setting since each descendant set contains a large fraction of the variables.", "Sparsity is not an issue.", "Because SSPS is stochastic we run it $K{=}5$ times per context, yielding 5 AUCROC scores per context.", "Meanwhile the baseline methods are all deterministic, requiring only one execution per context.", "We use a simple terminology to compare SSPS's scores against those of other methods.", "In a given context, we say SSPS dominates another method if its minimum score exceeds that of the other method.", "Conversely, we say the other method dominates SSPS if its score exceeds SSPS's maximum score.", "This dominance comparison has flaws—e.g., its results depend on the sample size $K$ .", "However, it errs on the side of strictness and suffices as an aid for summarizing the HPN-DREAM evaluation results." ], [ "Baseline pathway inference algorithms", "Our evaluations compare SSPS against a diverse set of baseline methods." ], [ "Exact DBN {{cite:59d7564ee0c14ab86ac00c8cf1981b7e18681b5e}}.", "This method was an early inspiration for SSPS and is most similar to SSPS.", "However, the exact DBN method encounters unique practical issues when we run it on real or simulated data.", "The method's computational expense increases rapidly with problem size $|V|$ and becomes intractable unless the “max-indegree” parameter is set to a small value.", "For example, we found that the method used more than 32GB of RAM on problems of size $|V|{=}100$ , unless max-indegree was set ${\\le } 3$ .", "Furthermore, the exact DBN method only admits prior knowledge in the form of Boolean reference edges, rather than continuous-valued edge confidences.", "We overcame this by using a threshold to map edge confidences to 1 or 0.", "We chose a threshold of 0.25 for the HPN-DREAM challenge evaluation because it yielded a reasonable number of prior edges.", "We ran [21]'s implementation using MATLAB 2018a." ], [ "This method is based on the notion that two variables $X,Y$ have a causal relationship if there exists a functional dependence $Y{=}f(X)$ between them.", "It detects these dependencies using a chi-square test against the “no functional dependence” null hypothesis.", "FunChisq was a strong competitor in the HPN-DREAM challenge, despite the fact that it uses no prior knowledge.", "In order to use FunChisq, one must first discretize their time course data.", "We followed [43]'s recommendation to use 1D $k$ -means clustering for discretization.", "Detailed instructions are given in the HPN-DREAM challenge supplementary materials [22].", "We used the FunChisq (v2.4.9.1) and Ckmeans.1d.dp (v4.3.0) R packages." ], [ "LASSO.", "We included a variant of LASSO regression as a simple baseline.", "It incorporates prior knowledge into the typical primal formulation: $\\hat{\\beta _j} = \\text{argmin}_{\\beta } \\left\\lbrace \\Vert X_{+,j} - B_j \\beta \\Vert _2^2 ~+~ \\alpha \\sum _{i=1}^V e^{-c_{ij}} |\\beta _{i}| \\right\\rbrace $ where $c_{ij}$ is the prior confidence (either Boolean or real-valued) for edge $(i,j)$ .", "That is, the method uses penalty factors $e^{-c_{ij}}$ to discourage edges with low prior confidence.", "The method selects LASSO parameters, $\\alpha $ , using the Bayesian Information Criterion described by [44].", "We use GLMNet [14] via the GLMNet.jl Julia wrapper (v0.4.2)." ], [ "Prior knowledge baseline.", "Our most straightforward baseline simply reports the prior edge probabilities, performing no inference at all.", "Ideally, a Bayesian method should do no worse than the prior—new time course data should only improve our knowledge of the true graph.", "In reality, this improvement is subject to caveats about data quality and model fit." ], [ "SSPS software availability", "We provide the SSPS code, distributed under a MIT license, via GitHub (https://github.com/gitter-lab/ssps) and archive it on Zenodo (https://doi.org/10.5281/zenodo.3939287).", "It includes a Snakemake workflow [28] for our full evaluation pipeline, enabling the reader to reproduce our results.", "The code used in this manuscript corresponds to SSPS v0.1.1." ], [ "Results", "We describe evaluation results from the simulation study and HPN-DREAM network inference challenge.", "SSPS competes well against the baselines, with superior scalability to other DBN-based approaches." ], [ "Simulation study results", "We compare our method to the baselines listed in Section REF .", "We focus especially on the exact DBN method of [21], as SSPS shares many modeling assumptions with it.", "Table: Computational expense of SSPS as a function of problem size |V||V|.NN is the number of iterations completed by a Markov chain.N eff N_{\\text{eff}} accounts for burnin and autocorrelation in the Markov chains, giving a more accurate sense of the method's progress.The last column gives the approximate memory footprint of each chain.The non-monotonic memory usage is an artifact of the chain termination conditions (N>N{>}100,000 or time >12{>}12 hours).Table: Computational expense of the exact DBN method of measured in CPU-seconds, as a function of problem size |V||V| and various parameter settings.The method imposes an in-degree constraint on each vertex, shown in the “max indeg” column.The columns “linear” and “full” correspond to different regression modes, i.e., which interaction terms are included in the DBN's conditional probability distributions.“OOM” (Out Of Memory) indicates that the method exceeded a 32GB memory limit.“TIMEOUT” indicates that the method failed to complete within 12 hours." ], [ "Computational expense.", "Because SSPS uses MCMC, the user may allow it to run for an arbitrary amount of time.", "With this in mind, we summarize SSPS's timing with two numbers: (i) $N/\\text{cpu-hr}$ , the number of MCMC samples per CPU-hour; and (ii) $N_{\\text{eff}}/\\text{cpu-hr}$ , the effective number of samples per CPU-hour.", "We also measure the memory footprint per Markov chain, subject to our termination conditions.", "We measured these numbers for each simulation in our grid (see Table REF ).", "Table REF reports average values of $N/\\text{cpu-hr}$ , $N_{\\text{eff}}/\\text{cpu-hr}$ , and memory footprint for each problem size.", "As we expect, $N/\\text{cpu-hr}$ and $N_{\\text{eff}}/\\text{cpu-hr}$ both decrease approximately with the inverse of $|V|$ .", "In contrast, the non-monotonic memory usage requires more explanation.", "It results from two causes: (i) our termination condition and (ii) the sparse data structures we use to store samples.", "On small problems ($|V|{=}40$ ), the Markov chain terminates at a length of 100,000—well within the 12-hour limit.", "On larger problems ($|V|{=}100$ or 200) the Markov chain terminates at the 12-hour timeout.", "This accounts for the 500MB gap between small and large problems.", "The decrease in memory usage between $|V|{=}100$ and 200 results from our sparse representations for samples.", "Roughly speaking, the sparse format only stores changes in the variables.", "So the memory consumption of a Markov chain depends not only on $|V|$ , but also on the acceptance rate of the Metropolis-Hastings proposals.", "The acceptance rate is smaller for $|V|{=}200$ , yielding a net decrease in memory usage.", "Recall that SSPS differs from more traditional MCMC approaches by nature of its parent set proposal distribution, which is specially designed for sparse graphs (see Section REF ).", "When we modify SSPS to instead use a naïve uniform graph proposal, we see a striking difference in sampling efficiency.", "The uniform graph proposal distribution attains $N_{\\text{eff}}/\\text{cpu-hr}$ of 100, 10, and 0.2 for $|V|{=}40,100,$ and 200, respectively—drastically smaller than those listed in Table REF for the parent set proposal.", "It's possible that the traditional proposal could achieve higher $N_{\\text{eff}}/\\text{cpu-hr}$ by simply running faster.", "However, the more important consideration is how $N_{\\text{eff}}/\\text{cpu-hr}$ changes with $|V|$ .", "Our parent set proposal distribution's $N_{\\text{eff}}/\\text{cpu-hr}$ decays approximately like $O(1/|V|)$ .", "This is better than what we might expect from a simple analysis (Appendix REF ).", "Meanwhile, the traditional proposal distribution's $N_{\\text{eff}}/\\text{cpu-hr}$ decays faster than $O(1/|V|^4)$ .", "This gap between $O(1/|V|)$ and $O(1/|V|^4)$ sampling efficiencies makes an enormous difference on large problems.", "Table REF summarizes the computational expense of the exact DBN method [21].", "The method quickly becomes impractical as the problem size grows, unless we enforce increasingly strict in-degree restrictions.", "In particular, the exact DBN method's memory cost grows exponentially with its “max in-degree” parameter.", "The growth becomes increasingly sharp with problem size.", "When $|V|{=}200$ , increasing the maximum in-degree from 2 to 3 makes the difference between terminating in ${<}1$ minute and exceeding 32GB of memory.", "Such low bounds on in-degree are unrealistic, and will likely result in poor inference quality.", "In comparison, SSPS makes no constraints on in-degree, and its memory usage scales well with problem size.", "The other baseline methods—FunChisq and LASSO—are much less computationally expensive.", "Both finish in seconds and require less than 100MB of memory for each simulated task.", "This highlights the computationally intense nature of Bayesian approaches.", "Not every scenario calls for Bayesian inference.", "However, Bayesian inference is valuable in scientific settings where we're concerned with uncertainty quantification.", "Figure: Heatmap of AUCPR values from the simulation study.Both DBN-based techniques (SSPS and the exact method) score well on this, since the data is generated by a DBN.On large problems the exact DBN method needs strict in-degree constraints, leading to poor prediction quality.LASSO and FunChisq both perform relatively weakly.See Figure for representative ROC and PR curves." ], [ "Predictive performance.", "The simulation study provides a setting where we have access to “ground truth”—the true simulated graph.", "We use AUCPR to score each method's ability to recover the true graph's edges.", "Figure REF shows the AUCPR scores for our grid of simulations.", "Each heat map shows AUCPR as a function of graph corruption parameters, $r$ and $a$ .", "The heat maps are arranged by method and problem size $|V|$ .", "Each AUCPR value is an average over 5 replicates.", "SSPS maintains fairly consistent performance across problem sizes.", "In contrast, the other methods' scores decrease with problem size.", "For the exact DBN method, this is partially due to the small in-degree constraints imposed on the large problems.", "It is forced to trade model accuracy for tractability.", "Figure REF reveals further insights into these results.", "It plots differential performance with respect to the prior knowledge, in a layout analogous to Figure REF .", "Specifically, it plots the $t$ -statistic of each method's AUCPR, paired with the prior baseline's AUCPR.", "Whenever the prior graph has some informative edges, SSPS outperforms the prior.", "On the other hand, SSPS's performance deteriorates whenever the prior contains no true edges (i.e., $r{=}1$ ).", "Under those circumstances FunChisq may be a better choice.", "Since it doesn't rely on prior knowledge at all, it outperforms the other methods when the prior is totally corrupted.", "However, we expect that in most realistic settings there exists partially-accurate prior knowledge, in which case we expect FunChisq to perform worse than SSPS.", "These results confirm SSPS's ability to identify the true network, given partially-accurate prior knowledge and time series data consistent with the modeling assumptions.", "SSPS is fairly robust with respect to the prior's quality and has consistent performance across different problem sizes.", "Figure: Heatmap of differential performance against the prior knowledge, measured by AUCPR paired tt-statistics.SSPS consistently outperforms the prior knowledge across problem sizes and shows robustness to errors in the prior knowledge." ], [ "HPN-DREAM challenge results", "We evaluated SSPS on experimental data from the HPN-DREAM challenge.", "The challenge includes time series phosphorylation data from 32 biological contexts: 8 stimuli applied to 4 breast cancer cell lines.", "Methods are scored on their ability to correctly identify the experimentally derived descendants of mTOR.", "Figure REF shows bar charts comparing the methods' AUCROC scores in each context.", "Appendix REF provides additional details.", "Figure: Methods' performances across contexts in the HPN-DREAM Challenge.MCMC is stochastic, so we run SSPS 5 times; the error bars show the range of AUCROC scores.The other methods are all deterministic and require no error bars.See Figure for example predicted networks, Figure for AUCPR scores, and Figure for representative ROC and PR curves.SSPS performs satisfactorily on this task overall.", "Employing terminology from Section REF , SSPS dominates the exact DBN method in 18 of the 32 contexts, whereas the exact DBN method dominates SSPS in only 9 contexts.", "Meanwhile, SSPS dominates FunChisq in 11 contexts and is dominated by FunChisq in 15.", "This is not surprising because FunChisq was among the top competitors in the original challenge.", "LASSO, on the other hand, performs poorly.", "SSPS dominates LASSO in 18 contexts and is dominated in only 6.", "More puzzling is the strong performance of the prior knowledge baseline.", "SSPS dominates the aggregate prior in only 9 contexts and is dominated in 21.", "This is not isolated to our method.", "FunChisq outperforms and is outperformed by the prior knowledge in 11 and 21 contexts, respectively.", "The aggregate prior's strong performance is consistent with the results from the original HPN-DREAM challenge; this prior outperformed all individual challenge submissions [22].", "Even though the aggregate prior gives identical predictions for each context and totally ignores the time course data, it still attains better performance than the other methods.", "This suggests either (i) the data is relatively uninformative or (ii) the evaluation metric based on mTOR's descendants isn't sufficiently precise to measure context-specific performance.", "We suspect the latter, because FunChisq uses no prior knowledge but was the top performer in the HPN-DREAM challenge's in silico tasks.", "An evaluation based on one node's descendants is not as discriminative as an evaluation of the directed edges.", "Many different directed graphs can have equivalent or similar mTOR descendants.", "However, it is experimentally impractical to generate the context-specific gold standard networks that would be required for a more precise edge-based evaluation." ], [ "Discussion", "We presented SSPS, a signaling pathway reconstruction technique based on DBN structure estimation.", "It uses MCMC to estimate the posterior probabilities of directed edges, employing a parent set proposal distribution specially designed for sparse graphs.", "SSPS is a Bayesian approach.", "It takes advantage of prior knowledge with edge-specific confidence scores and can provide uncertainty estimates on the predicted pathway relationships, which are valuable for prioritizing experimental validation.", "SSPS scales to large problems more efficiently than past DBN-based techniques.", "On simulated data, SSPS yields superior edge predictions with robustness to flaws in the prior knowledge.", "Our HPN-DREAM evaluation shows SSPS performs comparably to established techniques on a community standard task.", "It is difficult to make stronger statements in the HPN-DREAM setting because the prior knowledge baseline performs so well and we can only evaluate the predicted mTOR descendants, not the entire pathway.", "However, SSPS's scalability among Bayesian methods, strong results in the simulation, and competitive performance in the HPN-DREAM challenge make it an attractive option for further investigation of real phosphorylation datasets.", "There are several potential limitations of SSPS relative to alternative pathway signaling models.", "Prior knowledge is not available in some organisms or biological conditions, reducing one advantage of our Bayesian approach.", "Although SSPS is more scalable than related DBN techniques, it would struggle to scale to proteome-wide phosphoproteomic data measuring thousands of phosphosites.", "For large datasets, we recommend running SSPS on a pruned version that includes only the highest intensity or most variable phosphosites.", "SSPS, like most DBN techniques, models only observed variables.", "It will erroneously exclude important pathway members, such as scaffold proteins, that are not phosphorylated.", "Latent variable models or background network-based algorithms are better suited for including unphosphorylated proteins in the pathway.", "Background network methods can also impose global constraints on the predicted pathway structure, such as controlling the number of connected components or proteins' reachability from relevant receptors [27].", "There are many possible ways to improve SSPS.", "For example, it could be extended to jointly model related pathways in a hierarchical fashion, similar to [33] and [23].", "Alternatively, SSPS could be modified to accommodate causal assumptions via Pearl's intervention operators; see the model of [40] for a relevant example.", "Combining temporal and interventional data [3] is another rich area for future work.", "On the algorithmic side, we could improve our MCMC procedure by adaptively tuning the parameters of its proposal distributions, as described by [15].", "Because SSPS is a probabilistic program, it is naturally extensible." ], [ "Acknowledgements", "We thank UW-Madison's Biomedical Computing Group for generously providing compute resources; the teams developing Snakemake and HTCondor [41] for empowering us to use those resources effectively; the Gen team [7] for designing a uniquely powerful probabilistic programming language; and the HPN-DREAM challenge organizers for providing experimental data for our evaluation.", "This work was funded by the National Institutes of Health (award T32LM012413) and National Science Foundation (award DBI 1553206)." ], [ "Model formulation details", "We provide additional information about our graph prior and marginal likelihood function.", "We also describe some implications of SSPS's model assumptions." ], [ "Derivation of graph prior (Equation ", "We step through a more detailed derivation of SSPS's new graph prior.", "We begin with the original graph prior (Equation REF ) and rewrite it in terms of the edge existence variables $Z$ : P(G|G, ) ( -| E(G) E(G)| ) = (-(i,j) E(G) zij ) = (i,j) E(G) e-zij = (i,j) E(G) (e-)zij ( 11 + e- )V2 - |E(G)| (i,j) E(G) (e-)zij = (i,j) E(G) ( 11 + e- ) (e-)zij = (i,j) E(G) ( 11 + e- )1 - zij ( e-1 + e- )zij Equation REF shows the original prior is in fact a product of independent Bernoulli variables—the edge existence variables $z_{ij}$ .", "Equation REF explicitly assigns probability to the edges not contained in $E(G^\\prime )$ .", "However, it also implicitly assigns uniform probability to every edge contained in $E(G^\\prime )$ .", "We deduce that they are $\\text{Bernoulli}(0.5)$ variables, allowing us to write the prior $P(Z~|~G^\\prime , \\lambda )$ in the following form: $\\prod _{(i,j) \\in E(G^\\prime )}\\!\\!\\!", "\\left(\\frac{1}{2}\\right)^{z_{ij}}\\!\\left(\\frac{1}{2}\\right)^{1 {-} z_{ij}}\\!\\!\\!\\!\\!", "\\prod _{(i,j) \\notin E(G^\\prime )} \\left( \\frac{e^{-\\lambda }}{1 {+} e^{-\\lambda }} \\right)^{z_{ij}}\\!", "\\left( \\frac{1}{1 {+} e^{-\\lambda }} \\right)^{1 {-} z_{ij}}$ just as shown in the main text.", "Now we modify the prior to use continuous-valued edge confidences $c_{ij}$ instead of Boolean reference edges $E(G^\\prime )$ .", "Intuitively, we want to restate Equation REF as a single product over all $Z$ variables, rather than two separate products.", "Our goal is to find a function $q(c_{ij})$ such that $ P(Z~|~C, \\lambda ) = \\prod _{(i,j)} q(c_{ij})^{z_{ij}} (1 - q(c_{ij}))^{1 - z_{ij}}.$ However, in order to remain consistent with the original prior $q(c_{ij})$ ought to be monotone-increasing and satisfy these criteria: $q(0) = e^{-\\lambda } / (1 + e^{-\\lambda }) \\hspace{50.0pt} \\text{and} \\hspace{50.0pt} q(1) = 1/2.$ It turns out that choosing $q(c_{ij}) = \\frac{e^{-\\lambda }}{e^{-c_{ij} \\lambda } + e^{-\\lambda }}$ satisfies these requirements.", "This brings us to Equation REF of the main text.", "From there, it is straightforward to replace the single shared $\\lambda $ variable with a set of vertex-specific $\\Lambda $ variables and arrive at Equation REF ." ], [ "Marginal likelihood function details.", "Equation REF is obtained by (i) using a Gaussian DBN as the likelihood function for $G$ , (ii) assuming certain prior distributions for the DBN parameters, and (iii) integrating the DBN parameters out.", "Specifically, let $\\beta _j$ and $\\sigma _j^2 ~ ~ \\forall j \\in \\lbrace 1\\ldots |V|\\rbrace $ be the DBN's weight and noise parameters, respectively.", "We assume an improper prior $\\sigma _j^2 \\propto 1 / \\sigma _j^2$ for the noise and a Gaussian prior for the weights: $ \\beta _j | \\sigma _j^2 \\sim \\mathcal {N}\\left(0, T\\sigma _j^2(B_j^\\top B_j)^{-1}\\right).$ In other words, SSPS uses an improper joint prior $P(\\beta _j, \\sigma _j^2) = P(\\beta _j | \\sigma _j^2) P(\\sigma _j^2)$ with $P(\\sigma _j^2) {\\propto } 1/\\sigma _j^2$ .", "This choice allows $\\beta _j$ and $\\sigma _j^2$ to be marginalized, yielding Equation REF .", "The power $-|\\text{pa}_G(j)|/2$ in Equation REF is correct when the DBN only uses linear terms.", "Recall that $B_j$ may in general contain columns of nonlinear interactions between parent variables.", "When that is true, the quantity $|\\text{pa}_G(j)|$ should be replaced by the width of $B_j$ .", "We elide this detail in the main text for brevity.", "Our implementation uses the correct exponent.", "Our implementation of the marginal likelihood function employs least recently used caching to reduce redundant computation.", "Code profiling shows that this yields a substantial improvement to efficiency.", "For additional in-depth discussion of Equation REF , we recommend the supplementary materials of [21]." ], [ "Additional insights about SSPS's model assumptions.", "SSPS's model has interesting properties that could lead to method improvements.", "For example, when we replace the shared $\\lambda $ variable with vertex-specific $\\Lambda $ variables, the model effectively becomes a set of $|V|$ independent models.", "The plate notation in Figure REF makes this clear; $X_-$ is the only shared variable, and it's fully observed.", "This has algorithmic implications.", "For example, future versions of SSPS could parallelize inference at the vertex level, allocating more resources to the parent sets that converge slowly.", "In the course of deriving Equation REF , we showed that our prior is a log-linear model over edge features.", "Equation REF shows this most clearly.", "Future versions of SSPS could use the expressiveness of log-linear densities over higher-order graph features to capture richer forms of prior knowledge." ], [ "Parent set proposal details", "A key component of SSPS is its novel parent set proposal distribution.", "We motivate its design and discuss its computational complexity in greater detail." ], [ "Parent sets instead of edges.", "The marginal likelihood (Equation REF ) is a function of the graph $G$ .", "However, it depends on $G$ only via its parent sets, which are encoded in the matrices $B_j$ .", "Accordingly, SSPS represents $G$ by storing a list of parents for each vertex.", "It makes sense to use a proposal distribution that operates directly on SSPS's internal parent set representation.", "This motivates our choice of the add-parent, remove-parent, and swap-parent proposals listed in Section REF .", "There is a natural correspondence between (i) likelihood function, (ii) data structure, and (iii) proposal distribution." ], [ "Sampling efficiency.", "We provide some intuition for the parent set proposal's superior sampling efficiency.", "Let $z_{ij}$ be a particular edge existence variable.", "The estimate for $z_{ij}$ converges quickly if MCMC updates $z_{ij}$ frequently.", "Hence, as a proxy for sampling efficiency, consider the number of times $z_{ij}$ gets updated per unit time.", "We decompose this quantity into three factors: $\\frac{z_{ij} \\text{ updates}}{\\text{unit time}} = \\epsilon \\cdot \\tau \\cdot \\alpha $ where $\\epsilon = \\frac{\\text{graph proposals}}{\\text{unit time}} \\hspace{40.0pt}\\tau = \\frac{z_{ij} \\text{ proposals}}{\\text{graph proposal}}$ $\\alpha = z_{ij} \\text{ acceptance probability}$ In other words, $\\epsilon $ is the time efficiency of the proposal distribution.", "The factor $\\tau $ is the probability that a given proposal touches $z_{ij}$ .", "Lastly, $\\alpha $ is the proposal's Metropolis-Hastings acceptance probability.", "For a given proposal distribution, we're interested in how these factors depend on $|V|$ .", "For simplicity of analysis, assume the Markov chain is in a typical state where the graph is sparse: $|E(G)| = O(|V|)$ .", "For the parent set proposal, execution time has no dependence on $|V|$ and hence $\\epsilon ~{=}~O(1)$ .", "Recall that the parent set proposal resides in an outer loop, which iterates through all $|V|$ vertices.", "It follows that for any particular proposal there is a $1/|V|$ chance that it acts on vertex $j$ .", "After choosing vertex $j$ , there is on average a $O(1/|V|)$ chance that the proposal affects $z_{ij}$ .", "This follows from the sparsity of the graph: vertex $i$ is typically a non-parent of $j$ and the probability of choosing it via an add-parent or swap-parent action is $O(1/|V|)$ .", "Hence, the parent set proposal has a probability $\\tau {=} O(1/|V|^2)$ of choosing $z_{ij}$ .", "Lastly, the acceptance probability $\\alpha $ has no dependence on $|V|$ and therefore $\\alpha = O(1)$ .", "The product of these factors gives an overall sampling efficiency of $O(1/|V|^2)$ for the parent set proposal.", "For the uniform graph proposal, $\\epsilon $ 's complexity depends on the particular implementation.", "For sake of generosity we assume an efficient implementation with $\\epsilon ~{=}~O(1)$ .", "The proposal chooses uniformly from $O(|V|^2)$ actions: add-, remove-, or reverse-edge.", "The probability of choosing one that affects $z_{ij}$ is $\\tau = O(1/|V|^2)$ .", "Recall that the marginal likelihood decreases steeply with parent set size.", "It follows that add-edge actions will typically have low acceptance probability.", "Since the graph is sparse, add-edge actions are overwhelmingly probable; the probability of not landing on one is $O(1/|V|^2)$ .", "If we assume the acceptance probability is high for remove-edge and reverse-edge actions, (i.e., they are accepted whenever they're proposed), then this suggests $\\alpha = O(1/|V|^2)$ , averaged over many proposals.", "The product of these factors suggests a sampling efficiency that decays like $O(1/|V|^4)$ .", "This gap between $O(1/|V|^2)$ and $O(1/|V|^4)$ sampling efficiencies explains most of the difference that we saw in Section REF .", "A more detailed analysis may reveal why the parent set proposal attains sampling efficiencies closer to $O(1/|V|)$ in practice." ], [ "Simulation study details", "We give additional details about the simulation study's methodology and results." ], [ "Simulation process.", "The simulation process described in Section REF differs from SSPS's modeling assumptions in several ways.", "Recall that the simulator constructs a DBN to generate time series data.", "This simulated DBN employs nonlinear interaction terms.", "The simulator assumes that the data at each timepoint is a cubic function of the data at the previous timestep.", "In contrast, all of our analyses ran SSPS with an assumption of linear dependencies.", "In other words, the data contained complexities that SSPS was unable to capture.", "SSPS's performance in the simulation study suggests that it has some robustness to modeling assumption mismatches.", "We provide an illustration of the simulation process in Figure REF .", "It is interesting to notice that the simulated networks do not resemble directed acyclic graphs (DAGs) in any way.", "They do not have any sense of directionality.", "Contrast this with the biological graphs shown in Figure REF .", "Strictly speaking these are not DAGs, but they do have an overall direction.", "Some vertices are source-like, and others are sink-like.", "Future simulations and models could be more biologically realistic if they incorporated this kind of structure." ], [ "Simulation study results.", "Figure REF gives some representative ROC and PR curves from the simulation study.", "On problem sizes up to $|V|=100$ , SSPS and the exact DBN method yield similar curves in both ROC and PR space—though SSPS's curves clearly dominate.", "On larger problems the exact DBN method's performance quickly deteriorates.", "Computational tractability requires the exact method to impose highly restrictive in-degree constraints.", "These observations are consistent with the heatmaps of Figures REF and REF in the main text." ], [ "HPN-DREAM challenge details", "We provide additional details for the methodology and results of the HPN-DREAM challenge evaluation." ], [ "Data preprocessing.", "The HPN-DREAM challenge data needed to be preprocessed before it could be used by the inference methods.", "The choices we made during preprocessing most likely affected the inference results.", "Many of the time series contain duplicate measurements.", "We managed this by simply averaging the duplicates.", "We log-transformed the time series since they were strictly positive and some methods (SSPS and exact DBN) assume normality.", "This probably made little difference for FunChisq, which discretizes the data as part of its own preprocessing." ], [ "Predicted networks.", "Figure REF visualizes networks from two biological contexts in the HPN-DREAM challenge evaluation.", "This gives a sense of how the different inference methods' predictions differ from each other.", "All of the predicted networks are fairly different, though the SSPS and exact DBN predictions are more similar to each other than they are to FunChisq.", "FunChisq predicts more self-edges than the other methods.", "In the BT549 cell line, the experimentally detected mTOR descendants include receptor proteins that would traditionally be considered upstream of mTOR in the pathway.", "The experimental results are reasonable due to the influence of feedback loops in signaling pathways.", "However, the number and positioning of the mTOR descendants highlights the differences between the coarse HPN-DREAM challenge evaluation, which is based on reachability in a directed graph, and the more precise evaluation in our simulation study, where we have the edges in the ground truth network.", "Figure: Prior and predicted pathways from the HPN-DREAM challenge.We show pathways from two contexts: cell lines BT549 (top row) and MCF7 (bottom row).The stimulus is EGF for both contexts.SSPS attained the best AUCROC of all methods in the (BT549, EGF) contextand the worst in the (MCF7, EGF) context.The yellow node is mTOR; red nodes are the experimentally generated (“ground truth”) descendants of mTOR." ], [ "HPN-DREAM AUCPR.", "For completeness, we complement the AUCROC results of Section REF with the corresponding AUCPR results.", "Figure REF shows AUCPR in bar charts, with an identical layout to Figure REF .", "AUCPR leads us to similar conclusions as those from AUCROC.", "SSPS dominates the exact DBN method in 19 contexts and is dominated in 10.", "Both SSPS and FunChisq dominate each other in 14 contexts.", "However, SSPS dominates the prior knowledge in only 9 contexts, and is dominated in 21.", "As before, we conclude that SSPS attains similar performance to established methods on this task.", "Figure: A bar chart similar to Figure except that it shows AUCPR rather than AUCROC.See Figure for details about the layout." ], [ "ROC and PR curves.", "Figure REF shows ROC and PR curves from our HPN-DREAM evaluation.", "We focus on two representative contexts: cell lines BT549 and MCF7, with EGF as the stimulus.", "The bar charts in Figure REF tell us that SSPS was the top performer in the (BT549, EGF) context.", "The ROC and PR curves are consistent with this.", "SSPS dominates the other methods in ROC and PR space.", "In contrast, SSPS was the worst performer in the (MCF7, EGF) context.", "The curves show SSPS performing worse than random.", "The LASSO ROC and PR curves are interesting.", "Its ROC curves show nearly random performance.", "Its PR curves are straight lines.", "Manually inspecting its predictions yields an explanation: (i) LASSO gives nonzero probability to a very small number of edges; (ii) that small set of edges results in a very small descendant set for mTOR; (iii) that small descendant set is incorrect.", "Figure: ROC curves (top) and PR curves (bottom) from the HPN-DREAM challenge.We show results for two contexts: cell line BT549 (left) and MCF7 (right).The stimulus is EGF for both contexts.Since SSPS is stochastic, we show all 5 of its curves in each plot.The other methods are all deterministic, and therefore only have one curve in each plot." ] ]

[ [ "Upper bounds on number fields of given degree and bounded discriminant" ], [ "Abstract Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$.", "In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (\\log n)^2})$ for an explicit constant $c$." ], [ "Introduction and statement of results", "Let $N_n(X) := \\#\\lbrace K/\\mathbb {Q} : [K:\\mathbb {Q}] = n, |\\mathrm {Disc}(K)| \\le X\\rbrace $ be the number of degree $n$ extensions of $\\mathbb {Q}$ with bounded absolute discriminant $\\mathrm {Disc}(K)$ .", "It follows from the Hermite–Minkowski theorem that $N_n(X)$ is finite, and in fact bounded by $O_n(X^n)$ .", "This was substantially improved by Schmidt [13], who shows that $N_n(X) \\ll X^{(n+2)/4}$ , by Ellenberg and Venkatesh [10], who obtain an exponent that is $O(\\exp (c\\sqrt{\\log n}))$ for some constant $c$ , and Couveignes [7], who shows that $N_n(X) \\ll X^{c (\\log n)^3}$ for some unspecified constant $c$ .", "Our main theorem improves on these results.", "Theorem 1.1 There is a constant $c>0$ such that $N_n(X) \\ll _n X^{c (\\log n)^2}$ for every $n \\ge 6$ .", "Explicitly, we may take $c=1.564$ , and for every $c^\\prime > 1/(4(\\log 2)^2) \\approx 0.52$ , there is some $N>0$ such that $N_n(X) \\ll _n X^{c^\\prime (\\log n)^2}$ for every $n \\ge N$ .", "The proof of Theorem REF follows the same general strategy employed by Ellenberg and Venkatesh and by Couveignes; see Section for a loose discussion of the differences.", "In fact, Theorem REF follows straightforwardly from a numerical computation and by combining the Schmidt bound with the following theorem that is somewhat more flexible than Theorem REF .", "Theorem 1.2 Let $n \\ge 2$ .", "1) Let $d$ be the least integer for which $\\left({d+2}\\atop {2}\\right) \\ge 2n+1$ .", "Then $N_n(X) \\ll _n X^{2d - \\frac{d(d-1)(d+4)}{6n}} \\ll X^{\\frac{8\\sqrt{n}}{3}}.$ 2) Let $3 \\le r \\le n$ and let $d$ be such that $\\left(d+r-1 \\atop r-1\\right) > rn$ .", "Then $N_n(X) \\ll _{n,r,d} X^{dr}$ .", "Optimizing the choice of $d$ , the second case of Theorem REF yields an exponent that is $O(r^2 n^{1/(r-1)})$ with an absolute implied constant.", "We note that, in applying Theorem REF to deduce Theorem REF , we will choose $r$ to be a suitable multiple of $\\log n$ .", "This improves upon the Schmidt bound $N_n(X) \\ll X^{\\frac{n + 2}{4}}$ for $n \\ge 95$ .", "For $n \\le 5$ , asymptotic formulas of the form $N_n(X) \\sim c_n X$ were proved by Davenport-Heilbronn [8], Cohen-Diaz y Diaz-Olivier [6], and Bhargava [2], [3]; although our method still applies in these cases, it yields a substantially weaker result.", "In general, for $6 \\le n \\le 94$ , the Schmidt bound remains the best known, but improvements are available for fields with restricted Galois structure due to work of Dummit [9]." ], [ "Setup for the proof", "We begin by recalling the central idea of Schmidt's proof in language that will be of use to us.", "Let $K$ be a number field of degree $n$ .", "The ring of integers $\\mathcal {O}_K$ is a lattice inside Minkowski space $K_\\infty := K \\otimes \\mathbb {R} \\simeq \\mathbb {R}^n$ with covolume $c_K \\sqrt{\\mathrm {Disc}(K)}$ , where $c_K$ is a constant depending only on the signature of $K$ .", "Similarly, the set $\\mathcal {O}_K^0$ of integers in $\\mathcal {O}_K$ with trace 0 forms a lattice inside the trace 0 subspace of $K_\\infty $ with covolume $c_K^\\prime \\sqrt{\\mathrm {Disc}(K)}$ for some $c_K^\\prime $ , and it follows that there is some $\\alpha \\in \\mathcal {O}_K^0$ all of whose embeddings are at most $O(\\mathrm {Disc}(K)^{\\frac{1}{2n-2}})$ .", "We assume for convenience of exposition that $\\mathbb {Q}(\\alpha ) = K$ ; if not, Schmidt proceeds by induction, counting both the possible extensions $F=\\mathbb {Q}(\\alpha )/\\mathbb {Q}$ and the possible $K/F$ .", "The minimal polynomial $p_\\alpha (x)$ of $\\alpha $ is given by $p_\\alpha (x) = \\prod _{\\sigma } (x-\\sigma (\\alpha )) = x^n + a_2(\\alpha ) x^{n-2} + \\dots + a_n(\\alpha ),$ where the coefficients $a_i$ are integers satisfying $|a_i(\\alpha )| \\ll \\mathrm {Disc}(K)^{ \\frac{i}{2n-2}}$ and the product runs over the $n$ embeddings $\\sigma \\colon K \\hookrightarrow \\mathbb {C}$ .", "It follows that, ignoring the issue of subfields, the number of degree $n$ fields with discriminant at most $X$ may be bounded by the number of integral polynomials $f(x) = x^n + a_2x^{n-2} + \\dots + a_n$ where each $a_i$ is bounded by $O(X^{\\frac{i}{2n-2}})$ .", "The number of such polynomials is $O(X^{\\frac{n+2}{4}})$ , and this is Schmidt's bound.", "The key idea of Ellenberg and Venkatesh's improvement, on which our work as well as Couveignes's is based, is that there are more invariants of small height attached to tuples of integers inside $\\mathcal {O}_K^0$ .", "For example, suppose $\\alpha $ and $\\beta $ are elements of $\\mathcal {O}_K^0$ whose maximum embedding is bounded by some $Y$ .", "Then $\\alpha $ and $\\beta $ (and hence $K$ ) are determined by their minimal polynomials, which in turn are determined by the traces $\\mathrm {Tr}(\\alpha ^i)$ and $\\mathrm {Tr}(\\beta ^i)$ for $1 \\le i \\le n$ .", "These traces are integers of size $O(Y^i)$ , and it follows that there are most $O(Y^{2\\sum _{i=2}^n i}) = O(Y^{n^2+n-2})$ possible pairs $(\\alpha ,\\beta )$ .", "It is possible to do much better by exploiting mixed traces $\\mathrm {Tr}(\\alpha ^i\\beta ^j)$ , however.", "By regarding the $n$ embeddings $\\alpha ^{(1)}, \\dots , \\alpha ^{(n)}$ and $\\beta ^{(1)},\\dots ,\\beta ^{(n)}$ of $\\alpha $ and $\\beta $ as variables, we might hope that once $2n$ of these mixed traces are specified, it is possible to recover the values $\\alpha ^{(1)}, \\dots , \\alpha ^{(n)}, \\beta ^{(1)},\\dots ,\\beta ^{(n)}$ , and hence the pair $(\\alpha ,\\beta )$ and the field $K$ .", "There are $\\left(d+2 \\atop 2\\right)-1$ mixed traces with $i + j \\le d$ , so in particular we might hope that the traces $\\mathrm {Tr}(\\alpha ^i\\beta ^j)$ with $i+j \\le d \\approx 2\\sqrt{n}$ suffice to determine $\\alpha $ and $\\beta $ .", "Since $\\mathrm {Tr}(\\alpha ^i\\beta ^j) \\ll Y^{i+j}$ , this would yield that there are at most $Y^{O(n^{3/2})}$ such pairs.", "In Lemma REF , we prove that the set of $2n$ traces $\\mathrm {Tr}(\\alpha ^i\\beta ^j)$ with smallest possible $i+j$ suffices to determine the pair for a “generic” choice of $\\alpha $ and $\\beta $ .", "We show that every field has such a choice of $\\alpha $ and $\\beta $ with small height, and this leads to the first case of Theorem REF .", "This also provides the first insight into our improvement over the work of Ellenberg and Venkatesh, who only prove in this context that a set of roughly $8n$ traces suffices.", "Consider now an $r$ -tuple $\\alpha _1,\\dots ,\\alpha _r \\in \\mathcal {O}_K$ , with $r \\ge 3$ .", "By a similar heuristic as above, we might hope that once $rn$ different traces $\\mathrm {Tr}(\\alpha _1^{i_1}\\dots \\alpha _r^{i_r})$ are specified, the tuple $\\alpha _1,\\dots ,\\alpha _r$ is determined.", "We show that this is the case in Lemma REF for a set of $rn$ traces with $i_1 + \\dots + i_r$ nearly as small as possible.", "By contrast, Ellenberg and Venkatesh only show that a set of roughly $2^{2r-1}n$ traces suffices.", "Couveignes's work is morally similar but structurally a little different; instead of working with mixed traces, he constructs a set of $r$ polynomials, each with about $rn$ coefficients, that determines each number field.", "Thus, his approach relies on taking roughly $r^2n$ invariants of a number field.", "By expressing the trace as a sum over embeddings, we may regard the mixed traces $\\mathrm {Tr}(\\alpha _1^{i_1}\\dots \\alpha _r^{i_r})$ as being governed by polynomial maps from $(\\mathbb {A}^n)^r$ to $\\mathbb {A}^1$ .", "The key lemma we use to determine the $\\alpha _i$ from these traces is then the following.", "Lemma 2.1 For $N \\ge 1$ , let $f_1, \\dots , f_N$ be polynomials from $\\mathbb {A}^N$ to $\\mathbb {A}^1$ .", "Suppose that the determinant of the matrix $(\\frac{\\partial f_i}{\\partial x_j})_{1 \\le i,j\\le N}$ is not identically 0.", "Then there is a nonzero polynomial $P\\colon \\mathbb {A}^N \\rightarrow \\mathbb {A}^1$ such that whenever $P(\\mathbf {x}_0) \\ne 0$ for some $\\mathbf {x}_0 \\in \\mathbb {A}^N$ , the variety $V:=V_{\\mathbf {x}_0}$ cut out by the equations $f_1(\\mathbf {x}) = f_1(\\mathbf {x}_0), \\dots , f_N(\\mathbf {x}) = f_N(\\mathbf {x}_0)$ consists of at most $\\prod _i (\\deg f_i)$ points.", "Let $F\\colon \\mathbb {A}^N \\rightarrow \\mathbb {A}^N$ be defined by $F(\\mathbf {x}) = (f_1(\\mathbf {x}),\\dots ,f_N(\\mathbf {x}))$ , so that $V_{\\mathbf {x}_0} = F^{-1}(F(\\mathbf {x}_0))$ .", "$F$ is dominant, since for example the image of a Euclidean neighborhood of any $\\mathbf {x}$ for which $\\det \\big ( \\frac{\\partial f_i}{\\partial x_j} \\big )(\\mathbf {x}) \\ne 0$ is a neighborhood of $F(\\mathbf {x})$ .", "By [12], there is a Zariski open set $U \\subseteq \\textnormal {im}(F)$ such that $F^{-1}(\\mathbf {y})$ is of dimension zero for $\\mathbf {y} \\in U$ .", "Choosing a polynomial $Q$ vanishing on the complement of $U$ , $P = Q \\circ F$ is our desired polynomial.", "Therefore, when $P(\\mathbf {x}_0) \\ne 0$ , the variety $V:=V_{\\mathbf {x}_0}$ has dimension 0 and consists of a finite number of points.", "The quantitative bound follows by using Bézout's theorem [11] to iteratively bound the number of affine components of the projectivization of $V\\big (f_1 - f_1(\\mathbf {x}_0), \\dots , f_m - f_m(\\mathbf {x}_0)\\big )$ for each $m \\le N$ ." ], [ "The dimension of mixed trace varieties", "In this section, we show that the varieties associated to fixed values of the “mixed traces” introduced in the previous section typically have dimension 0.", "We do so over $\\mathbb {C}$ .", "Thus, let $n \\ge 2$ be an integer, corresponding to the degree of the extensions we wish to count, and let $r \\ge 2$ be an integer corresponding to the number of elements of which we wish to take the mixed trace.", "To an $r$ -tuple $\\mathbf {a}=(a_1,\\dots ,a_r) \\in \\mathbb {Z}_{\\ge 0}^r$ , we associate the function $\\mathrm {Tr}_{n,\\mathbf {a}} \\colon (\\mathbb {A}^n)^r \\rightarrow \\mathbb {A}^1$ given by $\\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}_1,\\dots ,\\mathbf {x}_r):= \\sum _{i=1}^n x_{1,i}^{a_1} \\dots x_{r,i}^{a_r}.$ We let $|\\mathbf {a}| = a_1 + \\dots + a_r$ denote the total degree of $\\mathrm {Tr}_{n,\\mathbf {a}}$ .", "Motivated by Lemma REF , let $D \\mathrm {Tr}_{n,\\mathbf {a}}:= \\Big ( \\frac{\\partial }{\\partial x_{k,i}} \\mathrm {Tr}_{n,\\mathbf {a}} \\Big )_{\\begin{array}{c}1 \\le k \\le r \\\\ 1 \\le i \\le n\\end{array}}$ denote the (row) vector of partial derivatives of $\\mathrm {Tr}_{n,\\mathbf {a}}$ (i.e.", "its gradient).", "Our goal, then, is to find a set $A$ of $rn$ different vectors $\\mathbf {a}$ with small combined total degree for which the determinant of the matrix $(D \\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A}$ is not identically 0.", "We begin by considering the case $r=2$ , both to clarify ideas and because we obtain an essentially optimal result in this case.", "Lemma 3.1 Let $n \\ge 1$ , and let $A_n = (\\mathbf {a}_1, \\cdots , \\mathbf {a}_{2n})$ consist of the first $2n$ elements of the ordered set $\\lbrace (1,0),(0,1),(2,0),(1,1),(0,2),\\dots \\rbrace $ , that is, the set of ordered pairs $(i,j)$ ordered first by total degree $i+j$ , then by $j$ .", "Then with notation as above, $\\mathrm {det} (D\\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A_n} \\ne 0$ .", "Induction on $n$ , with $\\mathrm {det} (D\\mathrm {Tr}_{1,\\mathbf {a}})_{\\mathbf {a} \\in A_1} = 1$ .", "Every entry in the matrix $\\mathbf {D}=(D \\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A}$ is a monomial; the $(k, n)$ -entry $\\mathbf {D}_{k, n}$ equals $a_{k, 1} x_{1, n}^{a_{k, 1} - 1} x_{2, n}^{a_{k, 2}}$ , and $\\mathbf {D}_{k, 2n} = a_{k, 2} x_{1, n}^{a_{k, 1}} x_{2, n}^{a_{k, 2} - 1}.$ We write $\\det (\\mathbf {D})$ an alternating sum of products of these monomials.", "For each $k$ and $\\ell $ , the contribution of those terms involving either $\\mathbf {D}_{k, n}$ and $\\mathbf {D}_{\\ell , 2n}$ , or alternatively $\\mathbf {D}_{k, 2n}$ and $\\mathbf {D}_{\\ell , n}$ , is $& \\pm \\det \\begin{bmatrix} a_{k, 1} x_{1, n}^{a_{k, 1} - 1} x_{2, n}^{a_{k, 2}} & a_{k, 2} x_{1, n}^{a_{k, 1}} x_{2, n}^{a_{k, 2} - 1} \\\\a_{\\ell , 1} x_{1, n}^{a_{\\ell , 1} - 1} x_{2, n}^{a_{\\ell , 2}} & a_{\\ell , 2} x_{1, n}^{a_{\\ell , 1}} x_{2, n}^{a_{\\ell , 2} - 1} \\end{bmatrix} \\cdot \\delta _{k, \\ell } \\\\= & \\pm \\det \\begin{bmatrix} a_{k,1} & a_{k, 2} \\\\ a_{\\ell , 1} & a_{\\ell , 2} \\end{bmatrix} \\cdot x_{1, n}^{a_{k, 1} + a_{\\ell , 1} - 1} x_{2, n}^{a_{k, 2} + a_{\\ell , 2} - 1}\\cdot \\delta _{k, \\ell },$ where $\\delta _{k, \\ell }$ is the relevant $(2n - 2) \\times (2n - 2)$ matrix minor, which doesn't involve $x_{1,n}$ or $x_{2, n}$ .", "By construction, if $\\mathbf {a}_{k} + \\mathbf {a}_\\ell $ = $\\mathbf {a}_{2n - 1} + \\mathbf {a}_{2n}$ in $\\mathbb {Z}^2$ , then $\\lbrace k, \\ell \\rbrace = \\lbrace 2n - 1, 2n \\rbrace $ .", "Since the exponents of $x_{1, n}$ and $x_{2, n}$ in () are given by $\\mathbf {a}_{k} + \\mathbf {a}_{\\ell } - (1, 1)$ , this implies that the contribution () from $(l, \\ell ) = (2n - 1, 2n)$ is not cancelled by any other contribution.", "It therefore suffices to prove that this contribution is not zero.", "But this is immediate: the $2 \\times 2$ determinant in () is nonzero because consecutive elements of $A_n$ are never scalar multiples of one another, and $\\delta _{2n-1, 2n} = \\mathrm {det} (D\\mathrm {Tr}_{n-1,\\mathbf {a}})_{\\mathbf {a} \\in A_{n - 1}}$ .", "Example Let $n=3$ .", "Then $A = \\lbrace (1,0),(0,1),(2,0),(1,1),(0,2),(3,0)\\rbrace $ .", "The associated matrix is $\\mathbf {D} =\\begin{bmatrix}\\text{\\fbox{$1$}} & \\text{\\fbox{$1$}} & 1 & \\text{\\fbox{$0$}} & \\text{\\fbox{$0$}} & 0 \\\\\\text{\\fbox{$0$}} & \\text{\\fbox{$0$}} & 0 & \\text{\\fbox{$1$}} & \\text{\\fbox{$1$}} & 1 \\\\\\text{\\fbox{$2x_{1,1}$}} & \\text{\\fbox{$2x_{1,2}$}} & 2x_{1,3} & \\text{\\fbox{$0$}} & \\text{\\fbox{$0$}} & 0 \\\\\\text{\\fbox{$x_{2,1}$}} & \\text{\\fbox{$x_{2,2}$}} & x_{2,3} & \\text{\\fbox{$x_{1,1}$}} & \\text{\\fbox{$x_{1,2}$}} & x_{1,3} \\\\0 & 0 & \\text{\\fbox{$0$}} & 2x_{2,1} & 2x_{2,2} & \\text{\\fbox{$2x_{2,3}$}} \\\\3x_{1,1}^2 & 3x_{1,2}^2 & \\text{\\fbox{$3x_{1,3}^2$}} & 0 & 0 & \\text{\\fbox{$0$}} \\\\\\end{bmatrix}.$ The boxed entries comprise the $4 \\times 4$ and $2 \\times 2$ matrices in () for $(k, \\ell ) = (2n - 1, 2n)$ , whose determinants are nonzero and multiply to a summand of $\\det (\\mathbf {D})$ .", "For larger $r$ , we apply a theorem due to Alexander and Hirschowitz [1], that we state in the following manner to be consistent with our notation.", "(See also [4].)", "This theorem is also an important ingredient in Couveignes's work.", "Theorem 3.2 (Alexander–Hirschowitz) Let $V$ denote the complex vector space of homogeneous degree $d$ polynomials in $r$ variables.", "Given $n$ general points in $\\mathbb {P}^{r-1}$ , let $W \\subseteq V$ denote the subspace of polynomials whose first order partial derivatives all vanish at each of the $n$ points.", "Then $W$ has the “expected” codimension in $V$ , namely $\\mathrm {codim}\\,W= \\min \\lbrace \\mathrm {dim}\\,V, rn \\rbrace ,$ except for the following cases: $d=2$ , $2 \\le n \\le r-1$ ; $d=3$ , $r=5$ , $n=7$ ; $d=4$ , $(r,n) \\in \\lbrace (3,5),(4,9),(5,14) \\rbrace .$ With Theorem REF , we are able to find a good choice of the set $A$ in general.", "Lemma 3.3 Let $n \\ge 6$ , $3 \\le r \\le n$ , and suppose $d$ is such that $\\left( {d+r-1}\\atop {r-1}\\right) > rn$ .", "Then there is a set $A$ of $rn$ vectors $\\mathbf {a} \\in \\mathbb {Z}_{\\ge 0}^{r}$ of total degree $d$ for which the determinant $\\mathrm {det}(D\\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A} \\ne 0$ .", "As in Theorem REF , let $V$ denote the vector space of complex homogeneous polynomials in $r$ variables with degree $d$ .", "Choose an arbitrary set of $n$ general points in $\\mathbb {P}^{r-1}$ , and let $W$ denote the subspace with vanishing first order partials at each.", "Under the hypotheses of Lemma REF , $W$ has codimension $rn$ , since $\\mathrm {dim}\\,V = \\left( {d+r-1} \\atop {r-1}\\right)$ and none of the exceptional cases of Theorem REF apply.", "Let $\\mathfrak {M}_d$ denote the set of monomials of degree $d$ , which naturally forms a basis for $V$ .", "For each $m \\in \\mathfrak {M}_d$ , form an $rn$ -dimensional column vector $v_m$ by evaluating each of the first order partials of $m$ at the $n$ general points.", "Let $M$ denote the $rn \\times \\left( {d+r-1} \\atop {r-1}\\right)$ matrix whose columns are the vectors $v_m$ for $m \\in \\mathfrak {M}_d$ .", "The subspace $W$ may be identified with the kernel of $M$ , and since $W$ has the expected codimension, it follows that $M$ has full rank, namely $\\mathrm {rank}(M) = rn$ .", "Choose an $rn\\times rn$ minor $M^{\\prime }$ of $M$ of full rank.", "The columns of $M^{\\prime }$ are indexed by monomials of total degree $d$ that may be identified with elements of $\\mathbb {Z}_{\\ge 0}^r$ .", "Let $A$ consist of those associated elements in $\\mathbb {Z}_{\\ge 0}^r$ .", "Then $M^{\\prime }$ is the transpose of the matrix $(D \\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A}$ , evaluated at our set of $n$ points, and since $\\mathrm {det}(M^{\\prime }) \\ne 0$ we have $\\mathrm {det} (D \\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A} \\ne 0$ as well.", "Remark In Lemma REF (and hence also Theorem REF ) we may also allow $\\left( {d+r-1}\\atop {r-1}\\right) = rn$ , provided that $(d, r, n)$ is not $(3, 5, 7)$ or $(4, 5, 14)$ ." ], [ "Bounds on the number of number fields", "Let $K/\\mathbb {Q}$ be a number field of degree $n$ .", "Then $K$ may be embedded into $\\mathbb {C}^n$ .", "Lemmas REF and REF produce, for any $r \\ge 2$ , a set $A \\in \\mathbb {Z}_{\\ge 0}^r$ for which $\\mathrm {det}(D\\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A} \\ne 0$ .", "By Lemma REF , there is a hypersurface outside of which the variety cut out by specifying the mixed traces $\\mathrm {Tr}_{n,\\mathbf {a}}$ for $\\mathbf {a} \\in A$ consists of a bounded number of points.", "The following lemma, due to Ellenberg and Venkatesh [10], will be used to show that there is an $r$ -tuple of integers in $K$ , at least one of which cuts out $K$ , of small height that avoids this hypersurface.", "Lemma 4.1 (Ellenberg–Venkatesh) Let $P\\colon \\mathbb {A}^N \\rightarrow \\mathbb {A}^1$ be a polynomial of degree $d$ .", "Then there are integers $a_1,\\dots ,a_N$ with $|a_i| \\le (d+1)/2$ for which $P(a_1,\\dots ,a_N) \\ne 0$ .", "Lastly, we shall make use of the following upper bound on the height of the largest Minkowski minimum of a number field that follows from work of Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, Zhao [5].", "Lemma 4.2 Given a number field $K$ of degree $n$ , there is an integral basis $\\lbrace \\beta _1,\\dots ,\\beta _n\\rbrace $ of its ring of integers for which $|\\beta _i| \\ll _n \\mathrm {Disc}(K)^{1/n}$ in each archimedean embedding.", "Consider first the case $r \\ge 3$ and let $d$ be as in the statement of the theorem.", "Then by Lemma REF , there is a set $A$ of $\\mathbf {a} \\in \\mathbb {Z}_{\\ge 0}^r$ of size $rn$ and degree $d$ for which $\\mathrm {det}(D\\mathrm {Tr}_{n,\\mathbf {a}})_{\\mathbf {a} \\in A} \\ne 0$ .", "By Lemma REF , there is a polynomial $P\\colon (\\mathbb {A}^n)^r \\rightarrow \\mathbb {A}^1$ such that whenever $P(\\mathbf {x}_0) \\ne 0$ , the variety $\\lbrace \\mathbf {x} \\in (\\mathbb {A}^n)^r : \\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}) = \\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}_0) \\text{ for all $\\mathbf {a} \\in A$}\\rbrace $ consists of $O_{d,r,n}(1)$ points.", "Given $\\mathbf {x} \\in (\\mathbb {A}^n)^r$ , we also define $\\mathrm {Disc}^{(1)}(\\mathbf {x})$ to denote its discriminant in the first copy of $\\mathbb {A}^n$ , i.e.", "$\\mathrm {Disc}^{(1)}(\\mathbf {x}):= \\prod _{\\begin{array}{c}i,j \\le n \\\\ i \\ne j\\end{array}} (x_{1,i}-x_{1,j}).$ Now, let $K$ be a number field of degree $n$ and discriminant at most $X$ .", "By fixing an embedding $K \\hookrightarrow \\mathbb {C}^n$ , we may regard an $r$ -tuple of integers $\\alpha _1,\\dots ,\\alpha _r \\in \\mathcal {O}_K$ as giving rise to a point $\\mathbf {x}_\\alpha \\in (\\mathbb {A}^n)^r$ .", "Combining Lemmas REF and REF , we find there are $\\alpha _1,\\dots ,\\alpha _r \\in \\mathcal {O}_K$ with each $|\\alpha _i| \\ll _{n,d,r} X^{1/n}$ in each archimedean embedding for which the associated point $\\mathbf {x}_\\alpha $ satisfies both $P(\\mathbf {x}_\\alpha ) \\ne 0$ and $\\mathrm {Disc}^{(1)}(\\mathbf {x}_\\alpha ) \\ne 0$ .", "Since $\\mathrm {Disc}^{(1)}(\\mathbf {x}_\\alpha ) \\ne 0$ , $\\alpha _1$ cuts out a degree $n$ extension of $\\mathbb {Q}$ , which must therefore be equal to $K$ .", "It follows that any such $K$ is determined, up to $O_{n,r,d}(1)$ choices, by the $rn$ values $\\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}_\\alpha )$ for $\\mathbf {a} \\in A$ .", "Each of these quantities is an integer of size $O(X^{d/n})$ , whence there are $O(X^{rd})$ choices in total, and hence $O(X^{rd})$ number fields $K$ .", "The case $r=2$ is similar, except appealing to Lemma REF for the construction of the set $A$ , and noting that $\\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}_\\alpha )$ is an integer of size $O(X^{|\\mathbf {a}|/n})$ .", "For the set $A$ produced by Lemma REF , we find $\\sum _{\\mathbf {a} \\in A} |\\mathbf {a}|= 2nd - \\frac{d(d-1)(d+4)}{6},$ where $d$ is the least integer for which $\\left({d+2} \\atop {2}\\right) \\ge 2n+1$ .", "This yields the remaining case of the theorem; for the second inequality in (REF ), take $d = \\lfloor 2 \\sqrt{n} \\rfloor $ .", "We describe an asymptotically optimal choice of $r$ and $d$ as $n\\rightarrow \\infty $ in the second part of Theorem REF .", "This will show that the exponent in Theorem REF may be taken to be $(1/(4(\\log 2)^2) + o(1)) (\\log n)^2$ as $n\\rightarrow \\infty $ .", "Thus, let $n$ be large.", "We choose $d = \\alpha \\log n$ and $r-1 = \\beta \\log n$ for constants $\\alpha $ and $\\beta $ .", "So doing, a computation with Stirling's formula reveals $\\log \\left({d+r-1} \\atop {r-1}\\right)= ((\\alpha + \\beta ) \\log (\\alpha + \\beta ) - \\alpha \\log \\alpha - \\beta \\log \\beta ) \\log n + O(\\log \\log n).$ On the other hand, $\\log (rn) = \\log n + O(\\log \\log n)$ , so we find that asymptotically optimal choices of $\\alpha $ and $\\beta $ will satisfy $(\\alpha + \\beta ) \\log (\\alpha + \\beta ) - \\alpha \\log \\alpha - \\beta \\log \\beta = 1 + O\\left(\\frac{\\log \\log n}{\\log n}\\right).$ This expression is symmetric in $\\alpha $ and $\\beta $ , as is the exponent $dr = \\alpha \\beta (\\log n)^2 + O(\\log n)$ produced by Theorem REF .", "As a Lagrange multipliers computation shows, the exponent is minimized by choosing $\\alpha = \\beta = 1/2\\log 2$ .", "This yields the second part of the theorem.", "The first part of the theorem (with $c = 1.564$ ) follows for $n < e^{12}$ by a numerical computation, with $n = 805$ being the bottleneck.", "For $n > e^{12}$ we choose $d = r - 1 = \\lceil \\log (n) \\rceil $ , apply the bound $\\left(2d \\atop d\\right) \\ge \\frac{4^d}{2 \\sqrt{d}}$ , and verify that $\\left(2d \\atop d\\right) \\ge rn$ and $dr < 1.564(\\log n)^2$ with this choice." ], [ "Scope for improvement", "We list here three possible directions in which our results may be improved.", "First, for $r \\ge 3$ , it would be desirable to incorporate mixed traces of total degree at most $d$ , as opposed to restricting attention to mixed traces of total degree exactly $d$ as is done in Lemma REF .", "An optimistic version of this improvement would imply the bound $N_n(X) \\ll X^{dr}$ , say, whenever $\\left({d+r} \\atop {r}\\right) \\ge rn$ .", "This would substantially improve the efficiency of the method for fixed $r$ , as this exponent would be $O(r^2 n^{1/r})$ , as opposed to that obtained from Theorem REF , which is $O(r^2 n^{1/(r-1)})$ .", "It would not, however, yield an improved version of Theorem REF .", "Second, it would be worthwhile to incorporate greater input from the geometry-of-numbers.", "For $r < n$ , there are linearly independent $r$ -tuples of integers $\\alpha _1,\\dots ,\\alpha _r \\in \\mathcal {O}_K$ for which $|\\alpha _i| \\ll \\mathrm {Disc}(K)^{\\frac{1}{2(n-r+1)}}$ , which for $r \\asymp \\log (n)$ is significantly smaller than the bound $\\mathrm {Disc}(K)^{1/n}$ coming from Lemma REF .", "However, in applying Lemma REF , it is a priori necessary to work with the full ring of integers, and not a small rank sublattice, in which case Lemma REF is essentially optimal.", "It would be interesting to exclude the possibility that the hypersurface outside of which a mixed trace variety is a complete intersection contains these small rank sublattices for varying $K$ .", "This would lead to an improvement in the exponents in Theorems REF and REF essentially by a factor of 2.", "The authors hope to return to this question in future work.", "Lastly, unlike in the case of Schmidt's work, which considers minimal polynomials, it is not typically the case that algebraic points on a mixed trace variety cut out degree $n$ extensions, even when the variety is a complete intersection.", "That is, if integers $m_\\mathbf {a}$ are chosen for each $\\mathbf {a} \\in A$ , as is done in the proof of Theorem REF , the points $\\lbrace \\mathbf {x} \\in (\\mathbb {A}^n)^r : \\mathrm {Tr}_{n,\\mathbf {a}}(\\mathbf {x}) = m_\\mathbf {a} \\text{ for all $\\mathbf {a} \\in A$}\\rbrace $ need not define a degree $n$ field extension of the rationals.", "For example, let $n=5$ and $r=2$ , and consider the set $A$ produced by Lemma REF .", "Choosing integers $m_\\mathbf {a} \\in [-10^{|\\mathbf {a}|},10^{|\\mathbf {a}|}]$ randomly in Magma and performing a Groebner basis computation, the authors find that a “typical” choice of $\\lbrace m_\\mathbf {a} : \\mathbf {a} \\in A\\rbrace $ gives rise to solutions that define a degree 30 extension of the rationals that may be realized as a degree 5 extension of a sextic field $F/\\mathbb {Q}$ .", "Thus, integers corresponding to the mixed traces of elements in quintic fields should be such that the polynomial typically defining this sextic field admits a rational root.", "More generally, the authors speculate that mixed traces of $r$ -tuples of integers in degree $n$ extensions lie on thin subsets of $\\mathbb {Z}^{rn}$ in the sense of Serre, and that an understanding of these subsets could yield a substantial improvement to the resulting bounds on number fields." ], [ "Acknowledgments", "Although our work is mostly independent of Couveignes's, we learned of Theorem REF from his paper, allowing us to streamline our proof and to improve the constant in our exponents for $N_n(X)$ .", "We would like to thank Manjul Bhargava and Akshay Venkatesh for helpful feedback.", "FT was partially supported by grants from the Simons Foundation (Nos.", "563234 and 586594), and RJLO was partially supported by NSF grant DMS-1601398." ] ]

[ [ "Schwinger effect and false vacuum decay as quantum-mechanical tunneling\n of a relativistic particle" ], [ "Abstract We present a simple and intuitive description of both, the Schwinger effect and false vacuum decay through bubble nucleation, as tunneling problems in one-dimensional relativistic quantum mechanics.", "Both problems can be described by an effective potential that depends on a single variable of dimension length, which measures the separation of the particles in the Schwinger pair, or the radius of a bubble for the vacuum decay.", "We show that both problems can be described as tunneling in one-dimensional quantum mechanics if one interprets this variable as the position of a relativistic particle with a suitably defined effective mass.", "The same bounce solution can be used to obtain reliable order of magnitude estimates for the rates of the Schwinger pair production and false vacuum decay." ], [ "Introduction", "The Schwinger effect [1] and phase transitions through bubble nucleation [2], [3], [4], [5] are both nonperturbative effects in which a metastable state decays into an energetically favorable configuration.", "In the case of the Schwinger effect (see Ref.", "[6] for a review) the metastable state is a very strong static electric field that decays through spontaneous production of electron-positron pairs [1], [7], [8].", "Phase transitions can be modeled in scalar quantum field theory as the decay of a metastable state known as “false vacuum,\" where the scalar field represents an order parameter for the transition.", "In both problems, there is potential energy $U(x)$ that can be characterized by a single variable $x$ of dimension length, which measures the separation of the particles in the Schwinger pair or the radius of a bubble for the vacuum decay.", "The strong electric field in empty space and the false vacuum can both be identified with local minima in $U(x)$ at $x=0$ , and in both problems there is a critical distance $x_c > 0$ with $U(x_c) = U(0)$ beyond which the potential energy is smaller than $U(0)$ .", "For the Schwinger pair, this occurs because the rest masses of the electron-positron pair are overcompensated by their potential energy in the external field for $x>x_c$ .", "False vacuum decay occurs when the bubble radius is large enough that the volume-dependent energy difference between the two phases exceeds the surface-dependent energy needed to form the bubble wall.", "The decay of the metastable states can then be viewed as a quantum mechanical tunneling from $x=0$ to $x=x_c$ , and as usual for tunneling, the decay rate per unit volume is exponentially suppressed.", "It is common to express this rate as /V=A  e-B, where the coefficient $B$ is approximately a polynomial in the coupling constants in the theory, reflecting the nonperturbative nature of the decay.", "In the present work, we adopt a very simple picture and show that we can correctly reproduce the known expression for $B$ and obtain an order of magnitude $A$ in both problems by mapping them onto the tunneling of a relativistic particle in one-dimensional quantum mechanics.", "Here $x$ is interpreted as the position of the particle, and the potential energy $U(x)$ has to be complemented by a kinetic energy with a suitably defined effective mass.", "In this picture, the same bounce solution can be used to compute $B$ in both problems.", "This makes the analogy between these two phenomena very explicit.", "Before deriving our results, we briefly recollect some of the main results from previous computations that we compare to in Sec.", "REF .", "In Sec.", "REF , we show that Schwinger effect can be studied as quantum-mechanical tunneling for a relativistic particle.", "In Sec.", ", we show that this picture on Schwinger effect is analogous to false vacuum decay in the thin-wall regime.", "Section  is left for conclusions and discussions.", "To be self-contained, a brief review of the Callan-Coleman method on quantum tunneling is given in the Appendix.", "The original derivation of Schwinger effect was given in quantum electrodynamics (QED) and based on a calculation of the vacuum to vacuum transition (or vacuum persistence) amplitude in the presence of an external static electric field.", "This amounts to computing the QED effective action whose imaginary part can be related to the decay of the vacuum.Here the vacuum denotes the QED vacuum in the presence of an external classical electric field.", "If backreaction is taken into account this corresponds to the decay of the electric field, which could, e.g., be described by treating the field as a quantum object.", "We shall distinguish this phenomenon from false vacuum decay where we have multiple vacua in the absence of any external field.", "The widely used tool for this computation is the Schwinger proper time method [1].", "Inspired by string theory, a similar method to Schwinger proper time, called the worldline formalism, has been applied to the study of Schwinger effect in inhomogeneous external fields [9], [10].", "Schwinger effect can also be studied in the canonical way.", "In the time-dependent gauge for the electromagnetic potential, one studies the Bogolyubov transformation [11] between the in-modes and out-modes in the presence of a time-dependent external electromagnetic potential [12], [13], [14], [15], [16], [17], [18], [19].", "Setting the in-state to be the vacuum, the negative-frequency coefficient in the Bogolyubov transformation gives the number of the produced particles in the out-state.", "This method can also be used to study Schwinger effect in non-Abelian gauge theories [20], [21].", "The rate of pair production in all approaches is consistently determined to be proportional to $\\exp (-\\pi m^2/qE)$ .", "The nonanalytic dependence on the coupling $q$ indicates the nonperturbative nature of this effect.", "For comparison, the pair production rate was directly computed by Nikishov [22] and is given as =(qE)243e-m2/qE, where $m$ and $q$ are the positron mass and charge, respectively.", "$E$ is the magnitude of the static electric field.", "One may also consider the simpler $1+1$ -dimensional problem where the pair production rate is =qE2e-m2/qE.", "The tunneling interpretation on the basis of a potential energy and the analogy between both problems are of course well known.", "As was first pointed out by Brezin and Itzykson [12], the exponential dependence in the pair production rate is reminiscent of quantum tunneling suppression.", "This is usually illustrated with the qualitative picture of Dirac sea, where a negative-frequency state tunnels to a positive-frequency state, leading to pair production.", "The tunneling analogy has also been used to estimate thermal corrections to Schwinger effect [23] and in the context of the Dirac-Born-Infeld brane tunneling [24], [25].", "In Refs.", "[26], [27], [28], the authors use Schwinger effect to study conceptual subtleties appearing in bubble nucleation.", "A quantitative implementation of the tunneling picture perhaps comes from the canonical method in the space-dependent gauge.", "In this case, because the electromagnetic potential is time independent, one instead ends up with a static Schrödinger-like equation with a potential barrier from Klein-Gordon equation (in scalar QED) or Dirac equation.", "With certain assumptions for the interpretations of the incoming and outgoing waves, one can recover the pair production rate or the Schwinger formula for the vacuum decay rate [22], [29], [30], [31], [32], [33].", "Our analysis differs from this approach because it relies on neither the Klein-Gordon nor the Dirac equation." ], [ "Schwinger effect as quantum-mechanical tunneling", "From a quantum physics viewpoint, very strong electric fields represent an unstable state that must decay to the true ground state.", "Schwinger pair production is the dominant process that drives this decay.In many works in the literature, the vacuum decay rate and pair production rate are identified with each other, while in fact the latter is simply the leading contribution to the former.", "Processes with more particles in the final state give subdominant contributions [34].", "We now derive the pair production rate in the effective quantum-mechanical-tunneling picture as described in the Introduction.", "We start with the simpler $1+1$ -dimensional Schwinger effect.", "Let the positions of the positron and the electron be $x_1$ and $x_2$ respectively.", "Then the classical energy for a particle pair is E=m 1-x12+m 1-x22-qE(x1-x2), with $\\dot{x}_i=\\partial _t x_i$ , and we have neglected the Coulomb force between the two particles.", "Choosing the frame of mass center with $x_1=X+x$ and $x_2=X-x$ , we obtain E=2m 1-x2-2qEx.", "To study the tunneling problem, let us first look at the energy for a particle pair at rest which is $\\widetilde{U}_{\\rm eff}(x)=2m-2qEx$ .", "To take into account the fact that there are no particles initially, we modify the potential as $U_{\\rm eff}(x)=\\widetilde{U}_{\\rm eff}(x)-2m\\tilde{\\delta }(x)$ where $\\tilde{\\delta }(x)$ equals one for $x=0$ and zero otherwise so that $U_{\\rm eff}(x=0)=0$ .", "For $x\\ne 0$ , there is another point $x_c\\equiv m/(q E)$ at which the potential is zero, i.e., $U_{\\rm eff}(x_c)=0$ .", "For $0 < x < x_c$ , energy conservation cannot be satisfied.", "Schwinger effect can be described by the quantum tunneling of a relativistic particle from $x=x_0=0$ to $x=x_c$ .", "Equation (REF ) can be written as $\\dot{x}^2 = 1 - \\left(x/x_c + \\mathcal {E}/(2m)\\right)^{-2}.$ The Hamiltonian corresponding to Eq.", "(REF ) can be obtained from the following action: S=dt  [-2m1-x2+2qEx+2m(x)], where $2m\\tilde{\\delta }(x)$ has been included such that the energy vanishes for $x=0$ .", "This term does not affect the dynamics whenever $x\\ne 0$ .", "In principle, tunneling through arbitrarily shaped barriers in quantum mechanics can be described in terms of wave mechanics.", "However, the kinetic term in the action (REF ) is not canonical, so that known solutions of the Schrödinger cannot be used.", "We therefore resort to the principle of minimal action to compute the pair production rate $$ , taking advantage of the fact that the path integral for quantum mechanical amplitudes is dominated by trajectories near the classical solution.", "In the case of tunneling, this is somewhat problematic because there are no classical trajectories, which can directly be verified by noticing that (REF ) has no real solutions starting at $x=0$ for $\\mathcal {E}<2m$ ; for the case $\\mathcal {E}=0$ under consideration here, valid (real) classical solutions only exist for $x\\ge x_c$ .", "This problem can be circumvented by considering the analytic continuation to Euclidean space $t\\rightarrow -\\mathrm {i}\\tau $ and following the standard Callan-Coleman method [5].", "We give a brief summary of this method in the Appendix.", "In this approach, the strong field can be physically interpreted as an unstable bound state.", "Its decay rate $$ can be obtained from the imaginary part of the (approximate) eigenvalue $E_0$ of the Hamiltonian corresponding to the lowest bound state in the $\\tilde{\\delta }(x)$ -potential, which would be stable for $E=0$ or $q=0$ .", "This is in analogy to the “width\" of unstable particles in particle physics, where $q$ takes the role of a coupling constant that is responsible for the decay.", "A subtlety arises due to the fact that we have ignored the center of mass coordinate $X$ in the action (REF ).", "We account for this by first defining the transition rate $\\widetilde{}$ for $X=0$ , which we later relate to the full rate $$ that takes into account the fact that the transition can happen anywhere in space.", "The decay rate $\\widetilde{}$ is given as =-2Im E0=T2TIm(ZE[T]) with x0|e-HT|x0=x(=T)=x0Dx  e-SEZE[T], where $\\mathcal {T}$ is the amount of Euclidean time and $S_E$ is the Euclidean action, SE=d [2m1+(dxd)2-2qEx-2m(x)].", "$|x_0\\rangle $ is the state with the particle at $x=x_0=0$ .", "The path integral is to be taken over all trajectories that begin and end at $x=x_0$ at $\\tau =\\pm \\mathcal {T}$ .", "Formally, this corresponds to the Euclidean transition amplitude from $x=0$ to itself, but the expression is only needed as an auxiliary tool here.", "Figure: On the upper panel displays the instanton solution x B (τ)x_B(\\tau ) in ().The lower panel shows its analytic continuation to Minkowski space.If can be interpreted as the creation of a Schwinger pair that is created with separation x c x_c at t=0t=0.", "For t>0t>0, the particles are accelerated by the electric field to velocities that approach the speed of light, and the separation xx grows rapidly.One can estimate the path integral using the method of steepest descent.", "In contrast to the Minkowski space equation (REF ), its analytic continuation in Euclidean space for $\\mathcal {E}=0$ , (dxd)2=-1+xc2x2, has, besides the trivial solution $x_F(\\tau )\\equiv 0$ ,The trivial solution $x_F(\\tau )$ cannot be derived by Eq.", "(REF ) which is valid only for $x\\ne 0$ because we have omitted the $\\tilde{\\delta }(x)$ -term.", "Including the $\\tilde{\\delta }(x)$ -potential, we see that $x_F(\\tau )$ conserves energy.", "an instanton solution $x_B(\\tau )$ (named as bounce) for the boundary condition $x(\\tau \\rightarrow \\pm \\mathcal {T})=0$ ,We have assumed that, without loss of generality, the center of the bounce is at $\\tau =0$ .", "Further, we should note that only the real part of $x_B(\\tau )$ is physical; we have only done an analytic continuation in the time variable.", "xB()=xc2-2 for -xcxc, xB()=0 for others.", "The solution (REF ) is shown in Fig.", "REF .", "Its continuation to Minkowski space has a simple physical interpretation: the particle-antiparticle pair is spontaneously created at $t=0$ with vanishing velocity, but then accelerated due to the electric field, and their velocity asymptotically approaches the speed of light, cf. Fig.", "REF .", "In the Appendix, we briefly summarize how to estimate the path integral in the expression (REF ) by expanding the Euclidean path integral around the trivial solution $x_F(\\tau )$ and the bounce solution $x_B(\\tau )$ .", "After including the possibility of multiple subsequent bounces in the so-called dilute-gas approximation, one finally finds that the coefficient $B$ is simply given by the difference in the actions associated with the classical trajectories $x_F(\\tau )$ and $x_B(\\tau )$ , $B=S_E[x_B]-S_E[x_F]=S_E[x_B],$ where we have used $U_{\\rm eff}(x=0)=0$ .", "Substituting Eq.", "(REF ) into Eq.", "(REF ) with Eq.", "(REF ), we obtain B=m2qE, in agreement with the results (REF ) and (REF ) derived from the field theory.", "While the exponent $B$ in () can be computed from the action along the classical trajectories (REF ) alone, the prefactor $A$ depends on quantum fluctuations.", "For small (Gaussian) fluctuations, these can be obtained from a functional Taylor expansion around the classical path, cf.", "(), which yields the fluctuation operator G(1,2)=2 SEx(1)x(2).", "This amounts to computing the functional determinants of (REF ) evaluated for the bounce and false vacuum, cf. Eq.", "(REF ).", "This is rather involved in practice, and instead of a rigorous computation, we obtain an estimate based by exploiting the observation made in the original paper [5] that there is a zero mode in the eigensystem of ${G}$ for each spacetime symmetry.", "The action (REF ) is time translation invariant; the pair production can happen anytime.", "The zero mode corresponding to the time-translation symmetry gives a contribution $\\mathcal {T}\\sqrt{B/2\\pi }$ to $A$ .", "The fact that the pair creation can happen anywhere in space is not captured by the action (REF ) and has to be fixed by hand by introducing the analogous factor $V\\sqrt{B/2\\pi }$ , =(VB2).", "In total, we have a factor $V\\mathcal {T}(B/2\\pi )$ in $$ .", "The factor $V\\mathcal {T}$ will be canceled out by the $\\mathcal {T}$ and $V$ appearing in Eqs.", "() and (REF ).", "The contributions from all other modes (including the negative mode) are difficult to calculate; see, e.g., Ref. [35].", "By dimensional analysis, we know $A$ has dimension two in mass.", "Since the characteristic scale in the tunneling process is $x_c$ , we thus estimate $A$ as AB21xc2=qE2.", "Comparing with Eq.", "(REF ), we find that this gives a correct order of magnitude estimate.", "The somewhat surprising success of our very simple approach can be better understood by noting that the action (REF ) is similar to an action that can be derived by using the so-called Schwinger proper time representation [36].", "However, we emphasize that we do not need the field theoretical framework on which applications of this method to the Schwinger effect are usually based [9], [6], but we could simply guess (REF ) from (REF ) based on physical intuition.", "The analysis in $3+1$ -dimensional spacetime is similar since only the spatial direction with nonvanishing external electric field is most relevant.", "The main difference is that now we have four translation symmetries which contribute $V\\mathcal {T}(B/2\\pi )^2$ in $Z_B^E[\\mathcal {T}]$ , cf.", "(REF ), where $V$ is now the volume for three-dimensional space.", "Then, we estimate $A$ as A(B2)21xc4=(qE)242.", "A more careful comparison for $A$ in our method and other field-theoretical methods is left for future work." ], [ "Schwinger effect in spatially inhomogeneous electric fields", "Before ending this section, we note that our method can also be generalized to cases with inhomogeneous external fields.", "For example, in the $1+1$ -dimensional case, this means that the simple function $-2qEx$ may be replaced by a more complicated function $f(x)$ and the spatial-translation symmetry may be broken by the external field already.", "In that case, we shall study the total production rate for a specific event instead of the rate per unit volume.", "As an example for illustration, we consider a Sauter-type electric field [7] in $1+1$ -dimensional case, $E(x)=E{\\rm sech}^2(kx).$ We assume that the particle production is most efficient at the origin $x=0$ because that is where the field is the strongest.", "This electric field is symmetric with respect to the origin and as a result, one expects that the particle pair is likely nucleated with positions being at $x$ and $-x$ .", "The energy of the system for a given separation $x$ is obtained by simply integrating $E(x)$ over $x$ , E=2m1-x2-2qEk(kx).", "From the above equation, one immediately knows that the critical value of $x$ for the nucleated pair is (k xc)=mkqE = k xc .", "In order to have real finite $x_c$ , we must require $\\gamma <1$ , which amounts to $E > m k/q$ , i.e., there is a minimal field strength $E$ that is required for the effect to occur.", "Recalling that $1/k$ in (REF ) characterizes the spacial extension of the region where the electric field is present, we can also read this condition as $1/k > x_c$ , which simply means that there is no pair production if the region where the field is present is smaller than the critical distance $x_c$ .", "The corresponding Euclidean action now is SE=d[2m1+(dxd)2-2qEk(kx)-2m(x)], and the equation of motion is (dxd)2=-1+22(kx), which has the same form as (REF ) with the replacement $x\\rightarrow \\tanh (kx)$ .", "The above equation has an analytic bounce solution xB()=1karcsinh(2-2(1-2k||)1-2), for ||arcsin()1-2k, and $x_B(\\tau )=0$ for others.", "Substituting the bounce solution into the Euclidean action, one obtains the semiclassical tunneling rate B =m2qE 21+1-2 .", "The result (REF ) is physically intuitive if we interpret $1/\\gamma \\propto 1/k$ as a measure for the extension of the region where the electric field is present.", "For $\\gamma \\ll 1$ , this region is much larger than the critical distance $x_c$ , and we recover the result (REF ) for a homogeneous field.", "This makes sense because the function (REF ) is approximately constant over distance $x_c$ near the origin.", "For $\\gamma >1$ , the region where the field is present is smaller than the critical distance $x_c$ , and no tunneling happens.", "Note that in this regime the rate is strictly zero and not just exponentially suppressed.", "In other words, for any value of $k$ , there exists a critical field strength $E_{\\rm crit}=mk/q$ that is needed for pair creation to happen.", "For $E<E_{\\rm crit}$ , the pair creation rate vanishes, at least within the approximations made here.", "This is very different from the homogeneous field case where the pair creation rate is nonzero even for arbitrarily small $E$ , but simply becomes exponentially suppressed.", "This difference can be understood physically.", "In the homogeneous field case, the absolute value of the potential energy in (REF ) is unbound, it grows linearly with $x$ , and always exceeds the pairs' rest energy $2m$ for sufficiently large separation.", "In practice, pair creation does not happen for $E\\ll \\pi m^2/q$ because the distance $x_c$ that the system would have to tunnel becomes too large, but in principle it is possible.", "In contrast, for the localized field (REF ), the absolute value of the potential energy in Eq.", "(REF ) is bounded from above and cannot exceed $2qE/k$ .", "For $E<E_{\\rm crit}$ , this maximal value is smaller than $2m$ .", "Our result (REF ) is in agreement with the result given in Ref. [9].", "This shows that our simple method can be used to treat inhomogeneous electric fields.", "It would further be interesting to see whether our simple method can be further generalized to reproduce known results for the Schwinger effect in curved spacetime (see, e.g., [37], [38], [39], [28], [40], [41], [42], [43], [44], [45], [46], [48], [47]) or non-Abelian fields (see, e.g., [20]).", "This, however, clearly goes beyond the scope of the present paper.", "The above quantum-mechanical-tunneling picture on Schwinger effect can be used to draw a close analogy to false vacuum decay in quantum field theory, which can be mapped onto a simple tunneling problem in one-dimensional quantum mechanics in the same way.", "To see it, we recall the most important aspects of false vacuum decay in the thin-wall regime." ], [ "Brief review of false vacuum decay in field theory", "We consider a scalar field theory with the Euclidean action SE=d4x [12()2+U()].", "The quantum-mechanical picture that we develop in Sec.", "REF holds for a wide class of potentials.", "For the sake of definiteness we choose the same potential as in the work by Coleman to make connection to the known results in quantum field theory, $ U(\\Phi )=-\\frac{1}{2}\\mu ^2\\Phi ^2+\\frac{1}{3!", "}g\\Phi ^3+\\frac{1}{4!", "}\\lambda \\Phi ^4+U_0$ , with $\\mu ^2,g,\\lambda $ are positive real parameters.", "$U(\\Phi )$ exhibits a metastable minimum, as sketched in Fig.", "REF .", "The false and true vacua can be understood as local minima in the energy functional associated with different configurations of the background field $\\varphi \\equiv \\langle \\Phi \\rangle $ .", "For simplicity, we choose the constant $U_0$ such that $U(\\varphi _+)=0$ at the false vacuum configuration $\\varphi _+$ .", "The equation of motion for $\\varphi $ then reads -(t2 - 2)+U'()=0.", "The two configurations $\\varphi _+$ and $\\varphi _-$ must be spatially homogeneous and isotropic to avoid gradient energies.", "There are no classically allowed trajectories in field space that connect them, but tunneling through the barrier in Fig.", "REF is possible at the quantum level.", "Using the Coleman-Callan method, the transition can again be described by performing a continuation to Euclidean space.", "The decay rate per unit volume has the form as in Eq.", "(), and the semiclassical suppression factor is again given by B=SE[B], where $\\varphi _B$ is the bounce.", "The bounce satisfies the equation of motion (REF ) in Euclidean time $\\tau $ and the radial coordinate $r$ with the boundary conditions $\\varphi |_{\\tau \\rightarrow \\pm \\infty }=\\varphi _+$ and $\\dot{\\varphi }|_{\\tau =0}=0$ , where $^{\\prime }$ and the dot denote the derivatives with respect to the field $\\varphi $ and $\\tau $ , respectively.", "For the theory we consider, it can be shown that the bounce has $O(4)$ symmetry [4].", "The physical reason is that the false vacuum decay happens via bubble nucleation, and spherical bubbles are for energetic reasons the most likely configuration.", "Therefore, the equation of motion can be written as -d2d2-3dd+U'()=0, where $\\rho ^2=r^2+\\tau ^2$ , with the boundary conditions $\\varphi |_{\\rho \\rightarrow \\infty }=\\varphi _+$ and $\\mathrm {d}\\varphi /\\mathrm {d}\\rho |_{\\rho =0}=0$ .", "As did for Schwinger effect, we will only concern about $B$ .", "For the calculations of $A$ in false vacuum decay, see, e.g., Refs.", "[5], [49], [50], [51], [52], [53], [54], [55], [56] and especially Ref.", "[35] for a comparison of different methods.", "In the thin-wall approximation, the damping term in Eq.", "(REF ) can be neglected, and the solution is given by a “kink.\"", "For case $g^2/\\mu ^2\\ll \\lambda $ and $\\varphi _-\\simeq -\\varphi _+$ , one can find an analytic solution of the form $\\varphi = \\varphi _+\\tanh (\\gamma (\\rho -R_c)).$ The kink connects the false vacuum outside of the bubble with the true vacuum inside.", "$\\gamma $ and $R_c$ are parameters that depend on the details of the model and can be expressed in terms of the parameters in the potential, see e.g., Refs.", "[51], [55].", "In physical terms, $R_c$ is the radius of a critical bubble and $1/\\gamma $ is a measure for the thickness of the bubble wall.", "We denote the energy difference between the false vacuum and true vacuum as $\\epsilon $ , i.e., $\\epsilon =U(\\varphi _+)$ .", "Outside the wall, the Euclidean action $B_{\\rm outside}=0$ .", "Inside the wall, Binside=-22Rc4.", "The region near the wall contributes Bwall=22Rc3Rc-Rc+d[12(dd)2+U()] 22Rc3, where $\\delta $ is a large enough number compared with the characteristic scale of the bubble-wall width and we have defined the surface tension $\\sigma $ of the bubble wall.", "The radius of the critical bubble $R_c$ is given by the stationary point of $B$ , $\\mathrm {d}B/\\mathrm {d}R_c=0$ .", "We thus have $R_c=3\\sigma /\\epsilon $ and the decay suppression $B=27\\pi ^2\\sigma ^4/2\\epsilon ^3$ .", "Due to the $O(4)$ symmetry, the bounce has the profile shown in Fig.", "REF with the replacement $(x,x_c)\\rightarrow (\\rho ,R_c)$ , where the solid circle represents the bubble wall separating the false vacuum and the true vacuum defined by $\\rho =R_c$ .", "The analytic continuation $\\tau \\rightarrow \\mathrm {i}t$ of this condition into Minkowski space reads $r^2 - t^2 = R_c^2 $ .", "Solving for $r$ yields an expression for the radius $R(t)$ of an expanding bubble that nucleates at time $t=0$ with radius $R_c$ and then expands, $R(t) = (R_c^2 + t^2)^{1/2}$ .", "It can be seen in Fig.", "REF with the replacement $(x,x_c)\\rightarrow (R,R_c)$ ." ], [ "Quantum-mechanical model for false vacuum decay and analogy to Schwinger effect", "In the thin-wall regime, we can practically characterize the energy of the kink by the bubble radius $R$ and surface tension $\\sigma $ .", "We approximate $\\sigma $ as constant and use $R(t)$ as the dynamic variable.", "For a static thin-wall bubble, the energy is Ueff(R)=4R2-43R3, where the first term comes from the surface tension and the second term comes from the negative energy density inside the bubble.", "One can view ${U}_{\\rm eff}(R)$ as the effective potential for the bubble, shown in Fig.", "REF .", "If we interpret $R$ as the position of a particle, the transition from the false to the true vacuum can be viewed as a quantum mechanical tunneling problem in analogy to Schwinger effect in the previous section.", "Bubble nucleation occurs when the volume-dependent gain in energy due to $\\epsilon $ exceeds the surface-dependent energy due to the tension $\\sigma $ , i.e., for sufficiently large bubbles.", "The minimal radius $R_c$ for which this can happen is analogous to the critical distance $x_c$ for which pair creation becomes energetically favorable.", "Figure: The effective potential U eff (R)U_{\\rm eff}(R) for the bubble wall.We model the bubble as a particle with effective mass $m(R)=4\\pi R^2\\sigma $ in the potential (REF ) that initially stays at the origin $R=0$ with vanishing total energy.", "Classically, it is stable.", "However, quantum mechanically, it can tunnel to the exit point marked as $R_c$ in Fig.", "REF , followed by a motion away from the critical radius (bubble expansion).", "$R_c$ is determined by ${U}_{\\rm eff}(R)=0$ , giving us $R_c=3\\sigma /\\epsilon $ , in agreement with the result from the field theory.", "For a general moving bubble wall, energy conservation implies E=m(R)1-R2-43R3=0.", "Note that when $R(t)\\equiv 0$ , this equation is trivially satisfied.", "When $R$ varies from 0 to a number smaller than $R_c$ , the LHS of Eq.", "(REF ) is always positive and hence the conservation law cannot be satisfied.", "Thus, classically, the only solution is $R(t)\\equiv 0$ .", "For $R\\ne 0$ , Eq.", "(REF ) reduces to R2-1+Rc2/R2=0, taking the same form as Eq.", "(REF ).", "As did for Schwinger effect, we move to the Euclidean time $\\tau =\\mathrm {i}t$ and have (dRd)2=-1+Rc2R2 with the initial conditions $R(\\tau =0)=R_c$ and $\\dot{R}(\\tau =0)=0$ .", "We then obtain the same solution as (REF ) R()=Rc2-2 for -RcRc, R()=0 for others, which is in agreement with the result derived from the field theory.", "To obtain the decay rate, we note that Eq.", "(REF ) can be derived from the following Minkowskian action: S=dt[-m(R)1-R2+43R3].", "Taking $t\\rightarrow -\\mathrm {i}\\tau $ and $\\mathrm {i}S\\rightarrow -S_E$ , we obtain the Euclidean action SE=d[m(R)1+(dRd)2-43R3].", "Substituting the Euclidean motion (REF ) into the above action, one gets $B=27\\pi ^2\\sigma ^4/(2\\epsilon ^3)$ .", "One may also estimate the prefactor $A$ by including the contribution from the zero modes corresponding to spacetime-translation symmetries and dimensional analysis as we did for Schwinger effect, obtaining A(B2)21Rc4.", "Extending the model (REF ) beyond the thin-wall approximation may be carried out by using the functional Schrödinger equation and reducing the infinite field degrees of freedom to one or multiple degrees of freedom in a proper way [57], [58], [59], [60], [61], [62]." ], [ "Conclusions and Discussions", "In this work, we established an intuitive picture in which both, pair nucleation through Schwinger effect and false vacuum decay, can be mapped onto a quantum-mechanical tunneling problem for a relativistic particle in one dimension.", "This analogy is based on the well-known facts that the potential energy in both problems is related to a single distance variable (the separation $x$ between the members of the Schwinger pair or the radius $R$ of a nucleating bubble), and that there exists a critical distance $x_c, R_c$ beyond which the potential energy is lower than that of the zero distance configuration.", "For Schwinger pair, $x_c$ corresponds to the critical distance where the potential energy of the particles in the external field exceeds their rest mass.", "In the case of bubble nucleation, $R_c$ marks the critical radius for which the volume-dependent gain in energy exceeds the surface-dependent cost to make a bubble.", "The novelty of our approach lies in the observation that the transition rate in both cases can be described rather accurately by interpreting $x$ or $R$ as the position of a relativistic particle that tunnels from $x,R=0$ to $x,R = x_c,R_c$ if the kinetic energy is described with a suitably defined effective mass.", "For the Schwinger pair, the effective mass is simply $2m$ , i.e., the actual physical mass of the pair.", "For the vacuum decay, the mass of the effective particle is $4\\pi R^2\\sigma $ where $\\sigma $ is the surface tension of the bubble wall.", "This simple picture makes the analogy between Schwinger effect and false vacuum decay in the thin wall regime very clear.", "We expect that our approach can be generalized to more complicated situations, such as pair production in curved spacetime, or when considering non-Abelian fields." ], [ "ACKNOWLEDGMENTS", "We would like to thank Valerie Domcke and Björn Garbrecht for proofreading the paper and for their helpful comments.", "We also thank Oliver Gould for beneficial discussions." ], [ "Callan-Coleman method on quantum tunneling", "In this appendix, we briefly recall the Callan-Coleman formalism on quantum tunneling, closely following the original work [5].", "Suppose a particle initially occupies the ground state around the local minimum at $x_+$ shown in Fig.", "REF .", "While being stable classically, the particle can tunnel from local minimum $x_+$ through the barrier to the region around the global minimum $x_-$ .", "In the figure, we also indicate the escape point $\\mathbf {p}$ beyond which the motion of the particle can be described by a classically allowed trajectory.", "Figure: The classical potential U(x)U(x) in a theory with a metastable minimum.The tunneling rate can in principle be obtained by squaring the transition amplitude xp|e-iHT|x+=Dx  eiS.", "Here $H$ is the Hamiltonian, $T$ is the amount of time during the transition (typically taken to be infinity), and $S$ is the Minkowskian action.", "The path integral $\\mathcal {D}x$ is performed over all trajectories that start at $x_+$ and end at $x_{\\bf p}$ .", "A direct calculation of the tunneling transition amplitude is difficult because, for the boundary conditions of interest, there is no classical solution, i.e., no stationary point in the action that dominants the tunneling amplitude.", "So one cannot find a suitable way to perform perturbative expansion for the Minkowskian path integral.It has been shown recently that one can still apply the method of steepest descent to the Minkowskian path integral for the quantum-tunneling problem by generalizing the contour integral in complex analysis to path integral [35].", "See Ref.", "[63] for a related discussion for Schwinger effect.", "Fortunately, one can solve the problem in Euclidean space, using the Callan-Coleman method [5].", "Following Callan and Coleman, we instead consider the following Euclidean transition amplitude: x+|e-HT|x+=Dx  e-SE[x]ZE[T].", "Inserting in the Euclidean transition amplitude a complete set of energy eigenstates, i.e., x+|e-H T|x+= n e-En Tx+|nn| x+, and taking the large $\\mathcal {T}$ limit, we thus obtain E0=-T1T(ZE[T]|x+|0|2).", "Here $|0\\rangle $ denotes the quantum mechanical ground state in the potential minimum around $x_+$ , to be distinguished from the position eigenstate $|x_0\\rangle $ at $x=0$ in Sec.", "REF .A few comments on the intuitive prescription on extracting $$ from the imaginary part of the “energy\" for the metastable state are in place.", "First, the false vacuum state $|0\\rangle $ is not an eigenstate of the Hamiltonian and should not appear in the spectral resolution of the identity.", "Second, the path integral in Eq.", "() is apparently real and cannot give rise to a complex $E_0$ through Eq. ().", "The reason why the Callan-Coleman method works out is subtle and has been carefully explained recently in Ref. [35].", "$E_0$ has an imaginary part which gives the decay rate asAs noted in Ref.", "[35], this formula describes the tunneling from the false ground state around $x_+$ to all possible final states so that the exit point of the tunneling is not necessarily $x_{\\bf p}$ .", "However, the tunneling from $x_+$ to $x_{\\bf p}$ is the dominated process and this subtle difference is usually neglected.", "=-2Im E0=T2TIm(ZE[T]).", "Here we have used the fact that the amplitude squared does not contribute to the imaginary part.", "One can evaluate Eq.", "() through the method of steepest descent.", "We first need to identify all the stationary points in the path integral, i.e., the classical trajectories.", "In the Euclidean equation of motion, the potential is flipped upside down, cf. Fig.", "REF .", "This allows for, besides the trivial solution $x_F(\\tau )\\equiv x_+ = \\rm {const.", "}$ (which is called the false-vacuum solution in field theory), an instanton solution, named as bounce which starts at $x_+$ in the infinite past $\\tau \\rightarrow -\\infty $ , reaching the turning point $\\mathbf {p}$ at a time $\\tau =\\tau _c$ known as collective coordinate of the bounce and eventually bounces back to $x_+$ for $\\tau \\rightarrow \\infty $ , as shown in Fig.", "REF .", "We denote the bounce solution as $x_B(\\tau )$ .", "In addition, there can be multiple bounce solutions that also form stationary points.", "In the so-called dilute-gas approximation, their impact on the path integral is approximated by a combination of $n$ subsequent bounces that are separated by time intervals much larger than the duration of each single bounce.", "Figure: The potential is upside down in Euclidean space.Expanding the Euclidean path integral around all the stationary points gives ZE[T]=ZEF[T]+n=1 ZBnE[T], where the subscripts “$F$ ” and “$B_n$ ” indicate that the integral is evaluated by expanding $x(\\tau )$ around the “false-vacuum” and $n$ -bounce stationary points, i.e., $x(\\tau )\\equiv x_{F,B}(\\tau )+\\delta x(\\tau )$ .", "In the dilute-gas approximation, the partition function $Z_{B_n}$ factorizes as ZBnE[T]=ZFE[T]1n!", "(ZBE[T]ZFE[T])n, where the appearance of $Z_F^E[\\mathcal {T}]$ is due to the contribution from the trivial configurations between any two neighboring bounces.", "The factor $n!$ is due to the symmetry when exchanging the positions of the bounces in the $n$ -bounce configuration.", "All the terms can be recollected into an exponential function, which eliminates the $\\ln $ in ().", "One therefore finally finds the tunneling rate =T2T|Im  (ZEB[T]ZEF[T])|.", "$Z_B^E$ is imaginary because the bounce is not a stable stationary point but a saddle point such that there is a negative mode in the fluctuations about the bounce.", "Taking the absolute value is due to a sign ambiguity when extracting the imaginary part.", "Expansion to second order in $\\delta x(\\tau )$ gives the Gaussian approximation ZEF,B[T]Dx   e-SE[xF,B]-12d1d2 x(1) G[xF,B] x(2) AF,B e-SE[xF,B], where the quadratic fluctuation operator (REF ) has to be evaluated at the false vacuum and the bounce, ${G}[x_{F,B}]\\equiv {G}(\\tau _1,\\tau _2)|_{x_{F,B}}$ .", "From Eqs.", "() and (), one can read off the semiclassical suppression factor $B=S_E[x_B]-S_E[x_F]$ in the rate ().", "The prefactor $A$ in the decay rate $$ is determined by $A_{F,B}$ which can be expressed as the functional determinants of the operator (REF ) evaluated for fluctuations the bounce and false vacuum, i.e., ${G}[x_{F,B}]$ .", "For the “false vacuum,” the valuation is straightforward and yields AF=(G[xF])-1/2.", "For the bounce, however, there are several subtle points.", "First, the quadratic fluctuation operator evaluated at the bounce has a negative mode that originates from an unstable direction in ().", "The physical origin is the fact that the state $|0\\rangle $ in Eq.", "() is metastable, and this negative eigenvalue is the very reason why the imaginary part of $E_0$ in () is nonzero.", "Practically, this implies that the integral in () has to be solved by analytic continuation and the method of steepest descent to obtain a finite result.", "Second, except for the negative mode, the quadratic fluctuation operator evaluated at the bounce also has zero modes.", "These can be related to symmetries in the action that are spontaneously broken by the bounce solution.", "For example, the theories under consideration here are time translation invariant.", "This invariance is broken by the bounce solution that occurs at a specific time $\\tau _c$ , which we took to be zero before.", "An infinitesimal shift of $\\tau _c$ in $x_B(\\tau -\\tau _c)$ gives a different solution $x_B(\\tau -\\tau _c-\\delta \\tau _c)$ .", "On the other hand, $\\delta x(\\tau )\\equiv x_B(\\tau -\\tau _c-\\delta \\tau _c)-x_B(\\tau -\\tau _c)$ can be viewed as an infinitesimal fluctuation about the particular bounce $x_B(\\tau -\\tau _c)$ .", "Since the action has time-translation symmetry, both solutions give the same classical action; $\\delta x(\\tau )$ therefore generates a flat direction in the fluctuations about $x_B(\\tau -\\tau _c)$ and incurs a zero mode for the corresponding fluctuation operator.", "The integral over the zero modes can be traded for that over the collective coordinates with a Jacobian factor $\\sqrt{B/2\\pi }$ for each zero mode.", "For instance, in the example at hand, the zero mode corresponding to the time-translation symmetry gives a contribution $\\mathcal {T}\\sqrt{B/2\\pi }$ .", "Each spatial dimension yields a similar factor, where $\\mathcal {T}$ is replaced by the extension of spatial dimension.", "In a $d+1$ -dimensional spacetime this yields the overall factorNote that in quantum field theory, the $$ in Eqs.", "() and () should be replaced by $/V$ .", "The volume factor $V$ from the zero modes is then eliminated by the denominator in $/V$ and one ends up with Eq.", "(REF ) with $\\sqrt{B/2\\pi }$ replaced by $(\\sqrt{B/2\\pi })^{d+1}$ .", "$V\\mathcal {T}(\\sqrt{B/2\\pi })^{d+1},$ where $V$ represents the volume of $d$ -dimensional space.", "$$ is then given by $= \\sqrt{\\frac{B}{2\\pi }}\\left|\\frac{{\\rm det^{\\prime }}\\,{G}[x_B]}{{\\rm det}\\,{G}[x_F]}\\right|^{-1/2}\\mathrm {e}^{-B}, \\\\$ where a prime on $\\det $ indicates that the zero modes should be excluded when evaluating the functional determinant." ] ]

[ [ "Behavior of solutions to the 1D focusing stochastic $L^2$-critical and\n supercritical nonlinear Schr\\\"odinger equation with space-time white noise" ], [ "Abstract We study the focusing stochastic nonlinear Schr\\\"odinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise.", "Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (or the $L^2$-norm) conserved in the additive case.", "Therefore, we investigate the time evolution of these quantities.", "After that we study the influence of noise on the global behavior of solutions.", "In particular, we show that the noise may induce blow-up, thus, ceasing the global existence of the solution, which otherwise would be global in the deterministic setting.", "Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions.", "Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow-up happens, it has the same dynamics as in the deterministic setting, however, there is a (random) shift of the blow-up center, which can be described as a random variable normally distributed." ], [ "Introduction", "We consider the 1D focusing stochastic nonlinear Schrödinger (SNLS) equation, that is, the NLS equation subject to a random perturbation $f$ ${\\left\\lbrace \\begin{array}{ll}iu_t+u_{xx}+|u|^{2\\sigma }u= \\epsilon f(u),\\quad (x,t)\\in [0,\\infty ) \\times {\\mathbb {R}}, \\\\ u(0,x)=u_0(x).\\end{array}\\right.", "}$ Here, the term $f(u)$ stands for a stochastic perturbation driven by a space-time white noise $W(dt,dx)$ (described in Section REF ) and $u_0\\in H^1(\\mathbb {R})$ is the deterministic initial condition.", "In this paper we study the SNLS equation (REF ) with either an additive or a multiplicative perturbation driven by space-time white noise: its effect on the mass ($L^2$ norm) and energy (Hamiltonian), the influence of the noise on the global behavior of solutions and, in particular, its effect on the blow-up dynamics.", "In the deterministic setting the mass and the energy are typically conserved, however, these quantities may behave differently under stochastic perturbations, which might significantly change global behavior of solutions.", "The focusing stochastic NLS equation appears in physical models that involve random media, inhomogeneities or noisy sources.", "For example, the influence of the additive noise on the soliton propagation is studied in [14], multiplicative noise in the context of Scheibe aggregates is discussed in [34], the NLS studies in random media (via the inverse scattering transform) are discussed in [18], [1] (and references therein).", "Relevant analytical studies of the SNLS (REF ) have been done by de Bouard & Debussche in a series of papers [6], [7], [8], [9], and numerical investigations by Debussche & Di Menza and collaborators, can be found in [11], [10], [2].", "We consider two cases of the stochastic perturbation $f(u)$ in (REF ): $f(u)={\\left\\lbrace \\begin{array}{ll}u(x,t) \\circ {W}(dt,dx), \\quad \\mathrm {multiplicative \\,\\, case,} \\\\{W}(dt,dx), \\qquad \\mathrm {additive \\,\\, case}.\\end{array}\\right.", "}$ The notation $ u(x,t) \\circ W(dt,dx)$ stands for the Stratonovich integral, which makes sense when the noise is more regular (for example, when $W$ is replaced by its approximation $W_N$ ).", "This integral can be related to the Itô integral (using the Stratonovich-Itô correction term); for more details we refer the reader to [8].", "The reason for the Stratonovich integral is the mass conservation, which we discuss next while recalling the properties of the deterministic NLS equation.", "The deterministic case of (REF ), corresponding to $\\epsilon =0$ , has been intensively studied in the last several decades.", "The local wellposedness in $H^1$ goes back to the works of Ginibre and Velo [19], [20]; see also [25], [36], [5], and the book [4] for further details.", "During their lifespans, solutions to the deterministic equation (REF ) conserve several quantities, which include the mass $M(u)$ and the energy (or Hamiltonian) $H(u)$ defined as $M(u(t))=\\Vert u(t)\\Vert _{L^2}^2 \\equiv M(u_0) \\quad \\mbox{\\rm and } \\quad H(u(t))=\\frac{1}{2} \\Vert \\nabla u(t)\\Vert _{L^2}^2 - \\frac{1}{2\\sigma +2} \\Vert u(t)\\Vert _{L^{2\\sigma +2}}^{2\\sigma +2}\\equiv H(u_0).$ The deterministic equation is invariant under the scaling: if $u(t,x)$ is a solution to (REF ) with $\\epsilon =0$ , then so is $u_\\lambda (t,x) = \\lambda ^{1/\\sigma } \\, u(\\lambda ^2 t, \\lambda x)$ .", "This scaling makes the Sobolev $\\dot{H}^{s}$ norm of the solution invariant with the scaling index $s$ defined as $s= \\frac{1}{2}-\\frac{1}{\\sigma }.$ Thus, the 1D quintic ($\\sigma =2$ ) NLS is called the $L^2$ -critical equation ($s=0$ ).", "The nonlinearities higher than quintic (or $\\sigma >2$ ) make the NLS equation $L^2$ -supercritical ($s>0$ )The range of the critical index in 1D is $0<s<\\frac{1}{2}$ .", "; when $\\sigma <2$ , the equation is $L^2$ -subcritical.", "In this work we mostly study the $L^2$ -critical and supercritical SNLS equation (REF ) with quintic or higher powers of nonlinearity.", "In these cases, it is known that $H^1$ solutions may not exist globally in time (and thus, blow up in finite time), which can be shown by a well-known convexity argument on a finite variance ([37], [42], for a review see [35]).", "We next recall the notion of standing waves, that is, solutions to the deterministic NLS of the form $u(t,x) = e^{it} Q(x)$ .", "Here, $Q$ is a smooth positive decaying at infinity solution to $-Q+Q^{\\prime \\prime }+Q^{2\\sigma +1} = 0.$ This solution is unique and is called the ground state (see [38] and references therein).", "In 1D this solution is explicit: $Q(x) = (1+\\sigma )^{\\frac{1}{2\\sigma }} \\, \\operatorname{sech}^{\\frac{1}{\\sigma }}(\\sigma x)$ .", "In the $L^2$ -critical case ($\\sigma =2$ ) the threshold for globally existing vs. finite time existing solutions was first obtained by Weinstein [38], showing that if $M(u_0) < M(Q)$ , then the solution $u(t)$ exists globally in timeand scatters to a linear solution in $L^2$ , see [12] and references therein.", "; otherwise, if $M(u_0) \\ge M(Q)$ , the solution $u(t)$ may blow up in finite time.", "The minimal mass blow-up solutions (with mass equal to $M(Q)$ ) would be nothing else but the pseudoconformal transformations of the ground state solution $e^{it}Q$ by the result of Merle [30].", "While these blow-up solutions are explicit, they are unstable under perturbations.", "The known stable blow-up dynamics is available for solutions with the initial mass larger than that of the ground state $Q$ , and has a rich history, see [39], [41], [35], [16] (and references therein); the key features are recalled later.", "In the $L^2$ -supercritical case ($s>0$ ) the known thresholds for globally existing vs. blow-up in finite time solutions depend on the scale-invariant quantities such as $\\mathcal {ME}(u):=M(u)^{1-s} H(u)^{s}$ and $\\Vert u\\Vert _{L^2}^{1-s} \\Vert \\nabla u(t)\\Vert _{L^2}^s$ , where the former is conserved in time and the latter changes the $L^2$ -norm of the gradient.", "The original dichotomy was obtained in the fundamental work by Kenig and Merle [26] in the energy-critical case ($s=1$ in dimensions 3,4,5), where they introduced the concentration compactness and rigidity approach to show the scattering behavior (i.e., approaching a linear evolution) for the globally existing solutions under the energy threshold (i.e., $E(u_0)<E(Q)$ in the energy-critical setting).", "It was extended to the intercritical case $0<s<1$ in [22], [13], [21], followed by many other adaptations to various evolution equations and settings.", "A combined result for $0\\le s \\le 1$ is the following theorem (here, $X = \\lbrace H^1 ~ \\mbox{if} ~ 0<s<1;~ L^2 ~\\mbox{if}~ s=0; ~\\dot{H}^1 ~ \\mbox{if} ~ s=1 \\rbrace $ , for simplicity stated for zero momentum).", "Theorem 1 ([26], [22], [22], [13], [23],[21], [15], [12]) Let $u_0\\in X(\\mathbb {R}^N)$ and $u(t)$ be the corresponding solution to the deterministic NLS equation (REF ) ($\\epsilon =0$ ) with the maximal interval of existence $(T_*, T^*)$ .", "Suppose that $M(u_0)^{1-s} E(u_0)^s < M(Q)^{1-s} E(Q)^s $ .", "If $\\Vert u_0\\Vert _{L^2}^{1-s} \\Vert \\nabla u_0\\Vert _{L^2}^s < \\Vert Q\\Vert _{L^2}^{1-s} \\Vert \\nabla Q\\Vert _{L^2}^s$ , then $u(t)$ exists for all $t\\in \\mathbb {R}$ with $\\Vert u(t)\\Vert _{L^2}^{1-s} \\Vert \\nabla u(t)\\Vert _{L^2}^s < \\Vert Q\\Vert _{L^2}^{1-s} \\Vert \\nabla Q\\Vert _{L^2}^s$ and $u(t)$ scatters in $X$ : there exist $u_{\\pm }\\in X$ such that $\\lim \\limits _{t\\rightarrow \\pm \\infty }\\Vert u(t)-e^{it\\Delta }u_{\\pm }\\Vert _{X(\\mathbb {R}^N)}=0$ .", "If $\\Vert u_0\\Vert _{L^2}^{1-s} \\Vert \\nabla u_0\\Vert _{L^2}^s > \\Vert Q\\Vert _{L^2}^{1-s} \\Vert \\nabla Q\\Vert _{L^2}^s$ , then $\\Vert u(t)\\Vert _{L^2}^{1-s} \\Vert \\nabla u(t)\\Vert _{L^2}^s > \\Vert Q\\Vert _{L^2}^{1-s} \\Vert \\nabla Q\\Vert _{L^2}^s$ for $t\\in (T_*, T^*)$ .", "Moreover, if $|x|u_0\\in L^2(\\mathbb {R}^N)$ (finite variance) or $u_0$ is radial, then the solution blows up in finite time; if $u_0$ is of infinite variance and nonradial ($s>0$ ), then either the solution blows up in finite time or there exits a sequence of times $t_n\\rightarrow +\\infty $ (or $t_n\\rightarrow -\\infty $ ) such that $\\Vert \\nabla u(t_n)\\Vert _{L^2(\\mathbb {R}^N)}\\rightarrow \\infty $ .", "The focusing NLS equation subject to a stochastic perturbation has been studied in [8] in the $L^2$ -subcritical case, showing a global well-posedness for any $u_0 \\in H^1$ .", "Blow-up for $0 \\le s <1$ has been studied in [7] for an additive perturbation, and [9] for a multiplicative noise.", "The results in [9] state that in the multiplicative noise case for $s \\ge 0$ initial conditions with finite (analytic) variance and sufficiently negative energy blow up before some finite time $t>0$ with positive probability [7].", "For both additive and multiplicative noise, in the $L^2$ -supercritical case the authors prove that if noise is non-degenerate and regular enough as initial conditions, then blow-up happens with positive probability before a given fixed time $t>0$ (see further details in [7], which also discusses the $L^2$ -critical situation in the additive case, and [9]).", "This differs from the deterministic setting, where no blow-up occurs for initial data strictly smaller than the ground state (in terms of the mass).", "In [32] an adaptation of the above Theorem REF is obtained to understand the global behavior of solutions in the stochastic setting in the $L^2$ -critical and supercritical cases.", "One major difference is that mass and energy are not necessarily conserved in the stochastic setting.", "In the SNLS equation with multiplicative noise (defined via Stratonovich integral) the mass is conserved a.s. (see [8]), which allows to prove global existence of solutions in the $L^2$ -critical setting with $M(u_0) < M(Q)$ (see [32]).", "(A somewhat similar situation happens in the additive noise case, though mass is no longer conserved and actually grows linearly in time (see (REF ).)", "To understand global behavior in the $L^2$ -supercritical setting one needs to control energy (as can be seen from Theorem REF ).", "While it is possible to obtain some upper bounds on the energy on a (random) time interval (and in the additive noise it is also necessary to localize the mass on a random set, since it is not conserved), the exact behavior of energy is not clear.", "This is exactly what we investigate in this paper via discretization of both quantities (mass and energy) in various contexts, then obtaining estimates on the discrete analogs and tracking the dependence on several parameters.", "Once we track the growth (and leveling off in the multiplicative case) of energy (and mass in the additive setting), we study the global behavior of solutions.", "In particular, we investigate the blow-up dynamics of solutions in both $L^2$ -critical and supercritical settings and obtain the rates, profiles and other features such as a location of blow-up.", "Before we state these findings, we review the blow-up in the deterministic setting.", "A stable blow-up in deterministic setting exhibits a self-similar structure with a specific rate and profile.", "Thanks to the scaling invariance, the following rescaling of the (deterministic) equation is introduced via the new space and time coordinates $(\\tau , \\xi )$ and a scaling function $L(t)$ (for more details see [28], [35], [40]) $u(t,r)=\\frac{1}{L(t)^{\\frac{1}{\\sigma }}}\\,v(\\tau , \\xi ), \\quad \\mbox{where} \\quad \\xi =\\frac{r}{L(t)},~~r=|x|, \\quad \\tau =\\int _0^t\\frac{ds}{L(s)^2}.$ Then the equation (REF ) in the deterministic setting ($\\epsilon =0$ ) becomes $iv_{\\tau }+ia(\\tau )\\left(\\xi v_{\\xi }+\\frac{v}{\\sigma }\\right)+\\Delta v + |v|^{2\\sigma }v=0$ with $a(\\tau )=-L\\frac{dL}{dt} \\equiv -\\frac{d \\ln L}{d\\tau }.$ The limiting behavior of $a(\\tau )$ as $\\tau \\rightarrow \\infty $ (from the second term in (REF )) creates a significant difference in blowup behavior between the $L^2$ -critical and $L^2$ -supercritical cases.", "As $a(\\tau )$ is related to $L(t)$ via (REF ), the behavior of the rate, $L(t)$ , is typically studied to understand the blow-up behavior.", "Separating variables $v(\\tau , \\xi )=e^{i\\tau }Q(\\xi )$ in (REF ) and assuming that $a(\\tau )$ converges to a constant $a$ , the following problem is studied to gain information about the blow-up profile: ${\\left\\lbrace \\begin{array}{ll}\\Delta _{\\xi } Q -Q + ia\\left(\\dfrac{Q}{\\sigma } + \\xi Q_{\\xi } \\right) + |Q|^{2\\sigma }Q=0,\\\\Q_{\\xi }(0)=0,\\qquad Q(0)=\\mathrm {real}, \\qquad Q(\\infty )=0.\\end{array}\\right.", "}$ Besides the conditions above, it is also required to have $|Q(\\xi )|$ decrease monotonically with $\\xi $ , without any oscillations as $\\xi \\rightarrow \\infty $ (see more on that in [40], [35], [3]).", "In the $L^2$ -critical case the above equation is simplified (due to $a$ being zero) to the ground state equation (REF ).", "However, even in that critical context the equation (REF ) is still meticulously investigated (with nonzero $a$ but asymptotically approaching zero), since the correction in the blow-up rate $L(t)$ comes exactly from that.", "It should be emphasized that the decay of $a(\\tau )$ to zero in the critical case is extremely slow, which makes it very difficult to pin down the exact blow-up rate, or more precisely, the correction term in the blow-up rate, and it was quite some time until rigorous analytical proofs appeared (in 1D [33], followed by a systematic work in [31]-[17] and references therein; see review of this in [40] or [35]).", "In the $L^2$ -supercritical case, the convergence of $a(\\tau )$ to a non-zero constant is rather fast, and the rescaled solution converges to the blow-up profile fast as well.", "The more difficult question in this case is the profile itself, since it is no longer the ground state from (REF ), but exactly an admissible solution (without fast oscillating decay and with an asymptotic decay of $|\\xi |^{-\\frac{1}{\\sigma }}$ as $|\\xi | \\rightarrow \\infty $ ) of (REF ).", "Among all admissible solutions to (REF ) there is no uniqueness as it was shown in [3], [27], [40].", "These solutions generate branches of so-called multi-bump profiles, that are labeled $Q_{J,K}$ , indicating that the $J$ th branch converges to the $J$ th excited state, and $K$ is the enumeration of solutions in a branch.", "The solution $Q_{1,0}$ , the first solution in the branch $Q_{1,K}$ (this is the branch, which converges to the $L^2$ -critical ground state solution $Q$ in (REF ) as the critical index $s \\rightarrow 0$ ), is shown (numerically) to be the profile of stable supercritical blow-up.", "The second and third authors have been able to obtain the profile $Q_{1,0}$ in various NLS cases (see [40], also an adaptation for a nonlocal Hartree-type NLS [41]), and thus, we are able to use that in this work and compare it with the stochastic case.", "In the focusing SNLS case, in [10] and [11] numerical simulations are done when the driving noise is rough, namely, it is an approximation of space-time white noise.", "The effect of the additive and multiplicative noise is described for the propagation of solitary waves, in particular, it was noted that the blow-up mechanism transfers energy from the larger scales to smaller scales, thus, allowing the mesh size affect the formation of the blow-up in the multiplicative noise case (the coarse mesh allows formation of blow-up and the finer mesh prevents it or delays it).", "The probability of the blow-up time is also investigated and found that in the multiplicative case it is delayed on average.", "In the additive noise case (where noise is acting as the constant injection of energy) the blow-up seems to be amplified and happens sooner on average, for further details refer to [11].", "Other parameters' dependence (such as on the strength $\\epsilon $ of the noise) is also discussed.", "We note that the observed behavior of solutions as noted highly depends on the discretization and numerical scheme used.", "In this paper we design three numerical schemes to study the SNLS (REF ) driven by the space-time white noise.", "We then use these schemes to track the time dependence of mass and energy of the stochastic Schrödinger flow in each multiplicative and additive noise cases.", "After that we investigate the influence of the noise on the blow-up dynamics.", "In particular, we give positive confirmations to the following conjectures.", "Conjecture 1 ($L^2$ -critical case) Let $u_0 \\in H^1(\\mathbb {R})$ and $u(t)$ , $t>0$ , be an evolution of the SNLS equation (REF ) with $\\sigma =2$ and noise (REF ).", "In the multiplicative (Stratonovich) noise case, sufficiently localized initial data with $\\Vert u_0\\Vert _{L^2} > \\Vert Q\\Vert _{L^2}$ blows up in finite positive (random) time with positive probability.", "In the additive noise case, sufficiently localized initial data blows up in finite (random) time a.s.", "If a solution blows up at a random positive time $T(\\omega )>0$ for a given $\\omega \\in \\Omega $ , then the blow-up is characterized by a self-similar profile (same ground state profile $Q$ from (REF ) as in the deterministic NLS) and for $t$ close to $T(\\omega )$ $\\Vert \\nabla u(t,\\cdot ) \\Vert _{L^2_x} \\sim \\frac{1}{L(t)}, \\quad L(t) \\sim \\left( \\frac{2\\pi (T-t)}{\\ln |\\ln (T-t)|} \\right)^{\\frac{1}{2}} \\quad \\mbox{as} \\quad {t \\rightarrow T=T(\\omega )},$ known as the log-log rate due to the double logarithmic correction in $L(t)$ .", "Thus, the solution blows up in a self-similar regime with profile converging to a rescaled ground state profile $Q$ , and the core part of the solution $u_c(x,t)$ behaves as follows $u_c(t,x) \\sim \\dfrac{1}{L(t)^{\\frac{1}{2}}} Q\\left(\\frac{x-x(t)}{L(t)}\\right) e^{i\\gamma (t)}$ with parameters $L(t)$ converging as in (REF ), $\\gamma (t) \\rightarrow \\gamma _0$ , and $x(t) \\rightarrow x_c$ , the blow-up center $x_c$ .", "Furthermore, conditionally on the existence of blow-up in finite time $T(\\omega )> 0$ , $x_c$ is a Gaussian random variable; no conditioning is necessary in the additive case.", "Conjecture 2 ($L^2$ -supercritical case) Let $u_0 \\in H^1(\\mathbb {R})$ and $u(t)$ be an evolution of the SNLS equation (REF ) with $\\sigma > 2$ and noise (REF ).", "In the multiplicative (Stratonovich) noise case, sufficiently localized initial data blows up in finite positive (random) time with positive probability.", "In the additive noise case, any initial data leads to a blow up in finite (random) time a.s.", "If a solution blows up at a random positive time $T(\\omega )>0$ for a given $\\omega \\in \\Omega $ , then the blow-up core dynamics $u_c(x,t)$ for $t$ close to $T(\\omega )$ is characterized as $u_c(t,x) \\sim \\dfrac{1}{L(t)^{\\frac{1}{\\sigma }}} Q\\left(\\frac{x-x(t)}{L(t)}\\right) \\exp \\left({i \\theta (t) + \\frac{i}{2a(t)}\\log \\frac{T}{T-t}} \\right),$ where the blow-up profile $Q$ is the $Q_{1,0}$ solution of the equation (REF ), $a(t) \\rightarrow a$ , the specific constant corresponding to the $Q_{1,0}$ profile, $\\theta (t) \\rightarrow \\theta _0$ , $x(t) \\rightarrow x_c$ , the blow-up center, and $L(t)=(2a(T-t))^{\\frac{1}{2}}$ .", "Consequently, a direct computation yields that for $t$ close to $T(\\omega )$ $\\Vert \\nabla u( t, \\cdot )\\Vert _{L_x^2} \\sim \\frac{1}{L(t)^{1-s}} \\equiv {\\left(2a(T-t) \\right)^{-\\frac{1}{2}(\\frac{1}{2}+\\frac{1}{\\sigma })}}.$ Furthermore, conditionally on the existence of blow-up in finite time $T(\\omega )> 0$ , $x_c$ is a Gaussian random variable; no conditioning is necessary in the additive case.", "Thus, the blow-up happens with a polynomial rate (REF ) without correction, and with profile converging to the same blow-up profile as in the deterministic supercritical NLS case.", "As it was mentioned above, some parts of the above conjectures have been studied and partially confirmed in [11], [10], [9], [6] under various conditions.", "In this work we provide confirmation to both conjectures for various initial data (see also [11]).", "We note that this paper is the first work, where the dynamics of blow-up solutions such as profiles, rates, location, are investigated in the stochastic setting.", "The paper is organized as follows.", "In Section we give a description of the driving noise and recall analytical estimates for mass and energy in both multiplicative and additive settings.", "In Section we introduce three numerical schemes which are mass-conservative in deterministic and multiplicative noise settings, and one of them is energy-conservative in the deterministic setting.", "We discretize mass and energy and give theoretical upper bounds on those discrete analogs in §REF and REF ; this is followed by the corresponding numerical results, which track both mass and energy in various settings, and time dependence on the noise type and strength, and other discretization parameters (such as length of the interval, space and time step-sizes).", "In the following Section we create a mesh refinement strategy and make sure that it also conserves mass before and after the refinement, introducing a new mass-interpolation method.", "We then state our new full algorithm for the numerical study of solutions behavior for both deterministic and stochastic settings.", "We note that this algorithm is novel even in the deterministic case for studying the blow-up dynamics (typically the dynamic rescaling or moving mesh methods are used).", "The new algorithm is needed due to the stochastic setting, since noise creates rough solutions, compared with the deterministic case, and thus, the previous methods are simply not applicable.", "In Section we start considering global dynamics (for example, of solitons, and how noise affects the soliton solutions) and compare with the previously known results in the $L^2$ -subcritical case.", "Finally, in Section we study the blow-up dynamics in both the $L^2$ -critical ($\\sigma =2$ ) and $L^2$ -supercritical (e.g., $\\sigma =3$ ) cases.", "We observe that once a blow-up starts to form, the noise does not seem to affect either the blow-up profile or the blow-up rate.", "The only affect that we have observed is random shifting of the blow-up center of the rescaled ground state.", "With increasing number of runs, the variation in the center location appears to be distributed normally (we estimate the corresponding mean, which is very close to 0, and variance).", "Otherwise, there seems to be very little difference between the multiplicative/additive noise and deterministic settings.", "We give a summary of our findings in the last section.", "Acknowledgments.", "This work was partially written while the first author visited Florida International University.", "She would like to thank FIU for the hospitality and the financial support.", "A. M.'s research has been conducted within the FP2M federation (CNRS FR 2036).", "S.R.", "was partially supported by the NSF grant DMS-1815873/1927258 as well as part of the K.Y.", "'s research and travel support to work on this project came from the above grant." ], [ "Description of the driving noise", "The space-time white noise is defined in terms of a real-valued zero-mean Gaussian random field $ \\lbrace W(B)\\; : \\; B \\; \\mbox{\\rm bounded measurable subset of }\\; [0,+\\infty ) \\times \\mathbb {R}\\rbrace $ defined on a probability space $(\\Omega , {\\mathcal {F}},P)$ , with covariance given by $E\\big [ W(B) W(C)\\big ]= \\int _{B\\cap C} dt\\, dx$ for bounded measurable subsets $B,C$ of $[0,\\infty )\\times \\mathbb {R}$ .", "For $t\\ge 0$ let ${\\mathcal {F}}_t:= \\sigma (W(B): \\; B \\; \\mbox{\\rm bounded measurable subset of } \\; [0,t]\\times \\mathbb {R}).$ Given $0\\le t_1<t_2$ and a step function $h_N=\\sum _{l=1}^N a_l 1_{[y_l, y_{l+1})}$ , where $a_l\\in \\mathbb {R}$ and $y_1<y_2< \\cdots < y_{N+1}$ are real numbers, we let $\\int _{t_1}^{t_2}\\int _{\\mathbb {R}} h_N(x) W(ds,dx):=\\sum _{l=1}^N a_l W([t_1,t_2]\\times [y_l,y_{l+1}))$ .", "Given $0\\le t_1<t_2$ and a function $h \\in L^2(\\mathbb {R}; \\mathbb {R})$ , we can define the stochastic Wiener integral $ \\int _{t_1}^{t_2} \\int _{\\mathbb {R}} h(x) W(dt,dx) $ as the $L^2(\\Omega )$ limit of $\\int _{t_1}^{t_2} \\int _{\\mathbb {R}} h_N(x) W(dt,dx) $ for any sequence of step functions $h_N$ converging to $h$ in $L^2(\\mathbb {R};\\mathbb {R})$ .", "This stochastic integral is a centered Gaussian random variable with variance $[t_2-t_1] \\int _{\\mathbb {R}} |h(x)|^2 dx$ .", "Furthermore, if $h_1$ , $h_2$ are orthogonal functions in $L^2(\\mathbb {R};\\mathbb {R})$ with $\\Vert h_1\\Vert _{L^2}=\\Vert h_2\\Vert _{L^2}=1$ , the processes $\\lbrace \\int _0^t h_1(x) W(dt,dx)\\rbrace _{t\\ge 0}$ and $\\lbrace \\int _0^t h_2(x) W(dt,dx)\\rbrace _{t\\ge 0}$ are independent Brownian motions for the filtration $({\\mathcal {F}}_t , t\\ge 0)$ .", "Let $\\lbrace e_j\\rbrace _{j \\ge 0}$ be an orthonormal basis of $L^2(\\mathbb {R};\\mathbb {R})$ and let $\\beta _j(t)=\\int _0^t \\int _{\\mathbb {R}} e_j(x) W(ds,dx)$ , $j \\ge 0$ .", "The processes $\\lbrace \\beta _j\\rbrace $ are independent one-dimensional Brownian motions and we can formally write $W(t,x,\\omega ) =\\sum _{j\\ge 0} \\beta _j(t,\\omega ) e_j(x), \\quad t\\ge 0, \\; x\\in \\mathbb {R}, \\; \\omega \\in \\Omega .$ However, the above series does not converge in $L^2(\\mathbb {R})$ for a fixed $t > 0$ .", "To obtain an $L^2(\\mathbb {R};\\mathbb {R})$ -valued Brownian motion, we should replace the space-time white noise $W$ by a Brownian motion white in time and colored in space.", "More precisely, in the above series we should replace $e_j$ by $\\phi e_j$ for some operator $\\phi $ in $L^{0,0}_{2,\\mathbb {R}}$ , which is a Hilbert-Schmidt operator from $L^2(\\mathbb {R})$ to $L^2(\\mathbb {R})$ with the Hilbert-Schmidt norm $\\Vert \\phi \\Vert _{L_{2,\\mathbb {R}}^{0,0}}$ .", "This would yield $\\tilde{W}(t)=\\sum _{j\\ge 0} \\beta _j(t) \\phi e_j$ for some sequence $\\lbrace \\beta _j\\rbrace _{j\\ge 0} $ of independent one-dimensional Brownian motions and some orthonormal basis $\\lbrace e_j\\rbrace _{j\\ge 0}$ of $L^2(\\mathbb {R};\\mathbb {R})$ .", "The covariance operator ${\\mathcal {Q}}$ of $\\tilde{W}$ is of the trace-class with ${\\rm Trace}\\; {\\mathcal {Q}}=\\Vert \\phi \\Vert _{L_{2,\\mathbb {R}}^{0,0}}^2$ .", "For practical reasons, we will use approximations of the space-time white noise $W$ (thus, a more regular noise) using finite sums $W_N(t,x,\\omega ):= \\sum _{j=0}^N \\beta _j(t,\\omega ) e_j(x),$ with functions $\\lbrace e_j \\rbrace _j$ with disjoint supports, which are normalized in $L^2(\\mathbb {R}; \\mathbb {R})$ .", "This finite sum gives rise to an $L^2(\\mathbb {R};\\mathbb {R})$ -valued Brownian motion with the covariance operator ${\\mathcal {Q}}$ such that ${\\rm Trace}\\; {\\mathcal {Q}}=N+1$ .", "Unlike [11], [10], we will not suppose that the functions $\\lbrace e_j\\rbrace $ are indicator functions of disjoint intervals.", "Instead we will consider the following “hat\" functions, which belong to $H^1$ .", "For fixed $N\\ge 1$ we consider the hat functions $\\lbrace g_j\\rbrace _{0\\le j \\le N}$ defined on the space interval $[x_j, x_{j+1}]$ as follows.", "Let $x_{j+\\frac{1}{2}}:=\\frac{1}{2}\\big [ x_j+x_{j+1}]$ , $\\Delta x_j:= x_{j+1}-x_j$ , and for $j=0, \\cdots N-1$ , set $g_j(x):= {\\left\\lbrace \\begin{array}{ll}c_j (x-x_j) \\quad \\mbox{\\rm for} \\quad x \\in [x_j,x_{j+\\frac{1}{2}}], \\\\c_j (x_{j+1} -x) \\quad \\mbox{\\rm for} \\quad x \\in [x_{j+\\frac{1}{2}}, x_{j+1}],\\end{array}\\right.", "}$ where $c_j:= \\frac{2 \\sqrt{3}}{(\\Delta x_j)^{3/2}}$ is chosen to ensure $\\Vert g_j\\Vert _{L^2}=1$ .", "Given points $x_0< x_1< \\cdots <x_N$ , define the functions $e_j$ , $j=0, \\cdots , N$ , by $ {\\left\\lbrace \\begin{array}{ll} e_j = &g_{j-1} 1_{[x_{j-\\frac{1}{2}}, x_j]} + g_j 1_{[x_j, x_{j+\\frac{1}{2}}]}, \\; 1\\le j\\le N-1, \\\\e_0=& \\sqrt{2} g_0 1_{[x_0, x_{\\frac{1}{2}}]} , \\quad e_N= \\sqrt{2} g_{N-1} 1_{[x_{N-\\frac{1}{2}}, x_N]}.\\end{array}\\right.", "}$ Due to the symmetry of the functions $\\lbrace g_j\\rbrace $ , we have $\\Vert e_j\\Vert _{L^2}=1$ for $j=0, \\cdots , N$ .", "Since the functions $\\lbrace e_j\\rbrace $ 's have disjoint supports, they are orthogonal in $L^2(\\mathbb {R};\\mathbb {R})$ .", "We can now construct an orthonormal basis $\\lbrace e_k\\rbrace _{k\\ge 0}$ of $L^2(\\mathbb {R};\\mathbb {R})$ containing the above $\\lbrace e_j\\rbrace _{0\\le j\\le N}$ set as the first $N+1$ elements.", "Then the $L^{0,0}_{2,\\mathbb {R}}$ -Hilbert-Schmidt norm of the orthogonal projection $\\phi _N$ from $L^2(\\mathbb {R};\\mathbb {R})$ to the span of $\\lbrace e_j\\rbrace _{0\\le j\\le N}$ is equal to $N+1$ .", "Furthermore, unlike indicator functions, the above functions $\\lbrace e_j\\rbrace $ belong to $H^1$ .", "When the mesh size $\\Delta x_j$ is constant (equal to $\\Delta x$ ), an easy computation yields $\\Vert e_j\\Vert _{H^1}^2=1+ \\frac{12}{(\\Delta x)^2}$ and $\\Vert \\nabla e_j\\Vert ^2_{L^\\infty }= \\frac{12}{(\\Delta x)^3}$ .", "Therefore, if $\\Vert \\cdot \\Vert _{L^{0,1}_{2,\\mathbb {R}}}$ denotes the Hilbert-Schmidt norm from $L^2(\\mathbb {R};\\mathbb {R})$ to $H^1(\\mathbb {R};\\mathbb {R})$ , we have $\\Vert \\phi _N\\Vert _{L^{0,1}_{2,\\mathbb {R}}}^2 = (N+1) \\Big ( 1+ \\frac{12}{(\\Delta x)^{2}}\\Big ) \\sim \\frac{12\\, (N+1)}{(\\Delta x)^{2}} \\quad \\mbox{when} \\quad \\Delta x <<1,$ and $m_{\\phi _N} := \\sup _{x\\in \\mathbb {R}} \\sum _{k\\ge 0} \\big | \\nabla (\\phi _N e_k)(x)\\big |^2 = \\frac{12\\, (N+1)}{(\\Delta x)^3}.$" ], [ "Multiplicative noise", "We recall that this stochastic perturbation on the right-hand side of (REF ) is $f(u)= u(x,t) \\circ {W}(dt,dx)$ , where the multiplication understood via the Stratonovich integral, which makes sense for a more regular noise.", "When the noise $\\tilde{W}=\\sum _{j\\ge 0} \\beta _j \\phi e_j$ is regular in the space variable (that is, colored in space by means of the operator $\\phi \\in L^{0,0}_{2,\\mathbb {R}}$ ), the equation (REF ) conserves mass almost surely (see [8]), i.e., for any $t>0$ $M[u(t)]=M[u_0] \\quad \\mbox{\\rm a.s.}$ Using the time evolution of energy in the multiplicative case for a regular noise $\\tilde{W}$ (see [8], we have $H(u(t)) = & H(u_0) - \\mbox{\\rm Im } \\epsilon \\sum _{j\\ge 0} \\int _0^t \\int _{\\mathbb {R}} \\bar{u}(s,x)\\nabla u(s,x) \\cdot (\\nabla \\phi e_j)(x) \\, dx d\\beta _j(s) \\\\& +\\frac{\\epsilon ^2}{2} \\sum _{j\\ge 0} \\int _0^t \\int _{\\mathbb {R}} |u(t,x)|^2 \\; |\\nabla (\\phi e_j)|^2 \\, dx ds.$ Taking expected values and using the fact that $\\phi $ is Radonifying from $L^2(\\mathbb {R};\\mathbb {R})$ to $\\dot{W}^{1,\\infty }(\\mathbb {R};\\mathbb {R})$ , we deduce that $ {\\mathbb {E}}(H\\big ( u(t)\\big ) = H(u_0) + \\frac{\\epsilon ^2}{2} \\, {\\mathbb {E}} \\sum _{j\\ge 0} \\int _0^t \\int _{\\mathbb {R}} |u(s,x)|^2 \\big | \\nabla (\\phi e_j)(x)\\big |^2dx ds\\le H(u_0) + \\frac{\\epsilon ^2}{2} m_\\phi M(u_0)\\, t,$ where $m_\\phi := \\sup _{x\\in \\mathbb {R}} \\sum _{j\\ge 0} | \\nabla (\\phi e_j)(x)|^2<\\infty .$ We also consider the expected value of the supremum in time of the energy (Hamiltonian).", "However, the upper bound differs depending on the critical or supercritical cases.", "For exact statements and notation we refer the reader to [32], where it is shown that for any stopping time $\\tau <\\tau ^*(u_0)$ (here, $\\tau ^*(u_0)$ is the random existence time from the local theory), one has $\\mathbb {E} \\Big (\\sup _{s\\le \\tau } H\\big ( u(s)\\big ) \\Big )\\le \\mathbb {E}\\big (H(u_0)\\big )+\\frac{\\epsilon ^2}{2} m_\\phi \\, M(u_0) \\, \\mathbb {E}(\\tau ) \\nonumber \\; + 3 \\epsilon \\, \\sqrt{m_\\phi M(u_0)} \\,\\mathbb {E}\\Big ( \\sqrt{\\tau } \\, \\sup _{s\\le \\tau } \\Vert \\nabla u(s) \\Vert _{L^2(\\mathbb {R}^n)}\\Big ).", "$ Therefore, the bound on the energy depends on the growth of the last gradient term.", "In the $L^2$ -critical case, assuming that $\\Vert u_0\\Vert _{L^2} < \\Vert Q\\Vert _{L^2}$ , it is possible to control the kinetic energy $\\Vert \\nabla u(s)\\Vert _{L^2}^2$ in terms of the energy $H(u)$ (see e.g.", "[32]).", "Therefore, $\\tau ^*(u_0)=+\\infty $ a.s. (see [32]), and thus, for large times the upper estimate for the growth of the energy is at most linear: for any $t>0$ we have $\\mathbb {E} \\Big (\\sup _{s\\le t} H\\big ( u(s)\\big ) \\Big )\\le \\mathbb {E}\\big (H(u_0)\\big )+c_1 \\epsilon ^2 \\, m_\\phi M(u_0)\\, t + c_2 \\, \\epsilon \\, \\sqrt{m_\\phi M(u_0)} \\, \\sqrt{t}.$ In the $L^2$ -supercritical case it is more delicate to control the gradient; nevertheless, it is possible for some (random) time interval (for which we provide upper and lower bounds in [32]).", "The length of that time interval is inversely proportional to the strength of the noise $\\epsilon $ , the space correlation $m_{\\phi }$ , and the size of the initial mass $M(u_0)$ to some power depending on $\\sigma $ ." ], [ "Additive noise", "The additive perturbation in (REF ) is $f(u)= {W}(dt,dx)$ .", "In this case, mass is no longer conserved.", "It is easy to see that its expected value grows linearly in time.", "More precisely, the identity $M\\big ( u(t)\\big ) = M(u_0) + \\epsilon ^2 \\Vert \\phi \\Vert _{L_{2 \\mathbb {R}}^{0,0}}^2\\; t- 2 \\epsilon \\, \\mbox{\\rm Im } \\Big ( \\sum _{j= 0}^N \\int _0^t \\!\\!\\int _{\\mathbb {R}} u(s,x)\\overline{\\phi e_j(x)} dx d\\beta _j(s) \\Big )$ (see e.g.", "[8] or [32] ) implies $ {\\mathbb {E}}(M(u(t)))=M(u_0) + \\epsilon ^2 \\Vert \\phi \\Vert _{L^{0,0}_{2,\\mathbb {R}}}^2 t.$ For the energy bound, using [32] (see also [8]), we have $H(u(t)) \\le & H(u_0) + \\frac{\\epsilon ^2}{2} \\tau \\Vert \\phi \\Vert _{L_{2, \\mathbb {R}}^{0,1}}^2 + \\mbox{\\rm Im }\\epsilon \\Big (\\sum _{k\\ge 0} \\int _0^\\tau \\!\\!", "\\int _{\\mathbb {R}}\\nabla \\overline{u(s,x)} \\nabla (\\phi e_k)(x)\\, dx \\, d\\beta _k(s) \\Big )\\nonumber \\\\&\\; - \\mbox{\\rm Im } \\epsilon \\Big ( \\sum _{k\\ge 0} \\int _0^\\tau \\!\\!", "\\int _{\\mathbb {R}}| u(s,x)|^{2\\sigma } \\overline{u(s,x)} (\\phi e_k)(x) dx d\\beta _k(s) \\Big ).$ Taking expected values, we deduce the following linear upper bound for the time evolution of the expected (instantaneous) energy ${\\mathbb {E}}\\big ( H(u(t))\\big )\\le H(u_0) + \\frac{\\epsilon ^2}{2} \\Vert \\phi \\Vert _{L^{0,1}_{2,\\mathbb {R}}}^2 t .$ As in the multiplicative case, in order to have quantitative information on the expected time of the existence interval, we have to prove upper bounds on $\\mathbb {E} \\big (\\sup _{s<\\tau } H(u(s))\\big )$ .", "However, since in the additive noise case the mass is not conserved and grows linearly in time, we have to localize the energy estimate on a (random) set, where the mass can be controlled (for details see [32].", "With that localization and estimates on the time, the upper bound for the expected energy is linear in time; furthermore, the time existence of solutions is inversely proportional to $\\epsilon ^2$ and the correlation $m_{\\phi }$ .", "Next, we would like to investigate the mass and energy quantities numerically.", "For that we define discretized (typically referred to as discrete) analogs of mass and energy, we also introduce several numerical schemes, which we use to simulate solutions, and thus, track the above quantities.", "We first prove theoretical upper bounds on the discrete mass and energy in both multiplicative and additive noise cases, and then provide the results of our numerical simulations." ], [ "Numerical approach", "We start with introducing our numerical schemes for the SNLS (REF ).", "We present three numerical schemes that conserve the discrete mass in the deterministic and with a multiplicative stochastic perturbation.", "Furthermore, one of them also conserves the discrete energy (in the deterministic case).", "That mass-energy conservative (MEC) scheme is a highly nonlinear scheme, which involves additional steps of Newton iterations, slowing down the computations significantly and generating numerical errors.", "We simplify that scheme first to the Crank-Nicholson (CN) scheme, which is still nonlinear, though works slightly faster.", "Then after that we introduce a linearized extrapolation (LE) scheme, that is much faster (no Newton iterations involved) while producing tolerable errors.", "Before describing the schemes, we first define the finite difference operators on the non-uniform mesh." ], [ "Finite difference operator on the non-uniform mesh", "We start with the description of an efficient way to approximate the space derivatives $f_x$ and $f_{xx}$ .", "Let $\\left\\lbrace x_j \\right\\rbrace _{j=0}^N$ be the grid points on $[-L_c,L_c]$ (the points $x_j$ 's are not necessarily equi-distributed).", "From the Taylor expansion of $f(x_{j-1})$ and $f(x_{j+1})$ around $x_j$ , setting $f_j=f(x_j)$ and $\\Delta x_j=x_{j+1}-x_j$ , one has $f_x(x_j) \\approx \\frac{-\\Delta x_{j}}{\\Delta x_{j-1}(\\Delta x_{j-1}+\\Delta x_{j})} f_{j-1}+\\frac{\\Delta x_{j}-\\Delta x_{j-1}}{\\Delta x_{j-1}\\Delta x_{j}} f_j+\\frac{\\Delta x_{j-1}}{(\\Delta x_{j-1}+\\Delta x_{j})\\Delta x_{j}}f_{j+1},$ and $f_{xx}(x_j) \\approx \\frac{2}{\\Delta x_{j-1}(\\Delta x_{j-1}+\\Delta x_{j})} f_{j-1}-\\frac{2}{\\Delta x_{j-1}\\Delta x_{j}} f_j +\\frac{2}{(\\Delta x_{j-1}+\\Delta x_{j})\\Delta x_{j}}f_{j+1}.$ We define the second order finite difference operator $\\mathcal {D}_2 f_j := \\frac{2}{\\Delta x_{j-1}(\\Delta x_{j-1}+\\Delta x_{j})} f_{j-1}- \\frac{2}{\\Delta x_{j-1}\\Delta x_{j}} f_j +\\frac{2}{(\\Delta x_{j-1}+\\Delta x_{j})\\Delta x_{j}}f_{j+1}.$" ], [ "Discretization of space, time and noise", "We denote the full discretization in both space and time by $u_j^m:= u(t_m,x_j)$ at the $m^{\\rm th}$ time step and the $j^{\\rm th}$ grid point.", "We denote the size of a time step by $\\Delta t_{m-1}=t_m-t_{m-1}$ .", "To consider the Stratonovitch stochastic integral, we let $x_{j+\\frac{1}{2}}= \\frac{1}{2} \\big [x_j+x_{j+1}\\big ]$ , and we discretize the stochastic term in a way similar to that in [11], except that we use the basis $\\lbrace e_j\\rbrace _{0\\le j\\le N}$ defined in (REF ) instead of the indicator functions.", "Recall that $\\lbrace \\beta _j(t) \\rbrace _{0 \\le j \\le N}$ are the associated independent Brownian motions for the approximation $W_N$ of the noise $W$ (i.e., $\\beta _j(t)= \\int _0^t \\int _{\\mathbb {R}} e_j(x) W(dt,dx)$ ).", "Following a procedure similar to that in [11], we set $\\chi ^{m+\\frac{1}{2}}_j=\\frac{1}{\\sqrt{\\Delta t_m }}(\\beta _{j}(t_{m+1})-\\beta _{j}(t_{m})), \\quad 0\\le j \\le N.$ We note that the random variables $\\lbrace \\chi ^{m+\\frac{1}{2}}_j\\rbrace _{j,m}$ are independent Gaussian random variables ${\\mathcal {N}}(0,1)$ .", "In our simulation, the vector $(\\chi _0^{m+\\frac{1}{2}}, \\cdots , \\chi _N^{m+\\frac{1}{2}})$ is obtained by the Matlab random number generator normrnd.", "When computing a solution at the end points $x_0$ and $x_{N+1}$ , we set $u^m_0=u^m_{1}$ and $u^m_{N}=u^m_{N-1}$ for all $m$ .", "We also introduce the pseudo-point $x_{-1}$ satisfying $\\Delta x_{-1}=\\Delta x_0$ , and similarly, the pseudo-point $x_{N+1}$ satisfying $\\Delta x_{N-1}=\\Delta x_N$ .", "Let $ f^{m+\\frac{1}{2}}_j = \\frac{1}{2} \\big ( u^m_j+u^{m+1}_j\\big ) \\tilde{f}^{\\,m+\\frac{1}{2}}_j , \\quad \\mbox{\\rm where} \\quad \\tilde{f}^{\\,m+\\frac{1}{2}}_j := \\frac{\\sqrt{3}}{2}\\frac{\\big [ \\sqrt{\\Delta x_{j-1}}+\\sqrt{\\Delta x_j} \\big ] }{\\sqrt{\\Delta t_m}\\big [ \\Delta x_{j-1} + \\Delta x_j\\big ] } \\chi ^{m+\\frac{1}{2}}_j$ for $j=1, \\cdots N-1$ .", "Indeed, $\\tilde{f}^{\\, m+\\frac{1}{2}}_j =& \\frac{2}{\\Delta t_m (\\Delta x_{j-1} + \\Delta x_j)}\\; \\int _{t_m}^{t_{m+1}} d\\beta _j(s) \\int _{\\mathbb {R}} e_j(x) dx \\nonumber \\\\= & \\frac{2}{\\Delta t_m (\\Delta x_{j-1} + \\Delta x_j)}\\; \\big ( \\beta _j(t_{m+1}) -\\beta _j(t_{m} \\big )\\Big [ \\int _{x_{j-\\frac{1}{2}}}^{x_j} c_{j-1}(x_j-x) dx + \\int _{x_j}^{x_{j+\\frac{1}{2}}} c_j (x-x_j) dx \\Big ] \\nonumber \\\\=&\\frac{\\sqrt{3}}{2} \\frac{\\beta _j(t_{m+1})-\\beta _j(t_m)}{\\sqrt{\\Delta t_m}}\\frac{\\big [ \\sqrt{\\Delta x_{j-1}}+\\sqrt{\\Delta x_j} \\big ] }{ \\big [ \\Delta x_{j-1} + \\Delta x_j\\big ] } \\, \\chi ^{m+\\frac{1}{2}}_j.$ A similar computation gives $ f^{m+\\frac{1}{2}}_0 = \\frac{1}{2} \\big ( u^m_0+u^{m+1}_0\\big ) \\tilde{f}^{\\, m+\\frac{1}{2}}_0 , \\quad \\mbox{\\rm where} \\quad \\tilde{f}^{\\, m+\\frac{1}{2}}_0=\\frac{\\sqrt{3}}{2} \\frac{1}{\\sqrt{\\Delta t_m \\Delta x_0}} \\, \\chi ^{m+\\frac{1}{2}}_0,\\\\f^{m+\\frac{1}{2}}_N = \\frac{1}{2} \\big ( u^m_N+u^{m+1}_N\\big ) \\tilde{f}^{m+\\frac{1}{2}}_N , \\quad \\mbox{\\rm where} \\quad \\tilde{f}^{\\, m +\\frac{1}{2}}_N=\\frac{\\sqrt{3}}{2} \\frac{1}{\\sqrt{\\Delta t_m \\Delta x_{N}}}\\, \\chi ^{m+\\frac{1}{2}}_N.$ Note that in the definition of $f^{m+\\frac{1}{2}}_j$ , the factor $\\frac{1}{2} \\big ( u^m_j+u^{m+1}_j\\big )$ is related to the approximation of the Stratonovich integral, and that the expression of $\\tilde{f}^{\\, m +\\frac{1}{2}}_j $ differs from that in [11] and [10] for two reasons.", "On one hand, we have a non-constant space mesh, and on the other hand, even if the space mesh $\\Delta x_j$ is constant (equal to $\\Delta x$ ), the extra factor $\\frac{\\sqrt{3}}{2}$ comes from the fact that we have changed the basis $\\lbrace e_j\\rbrace _{0\\le j\\le N}$ .", "For a constant space mesh $h$ , we have $\\tilde{f}^{\\,m+\\frac{1}{2}}_j = \\frac{\\sqrt{3}}{2} \\frac{1}{\\sqrt{\\Delta t_m} \\;\\sqrt{ \\Delta x}} \\chi ^{m+\\frac{1}{2}}_j,\\quad j=0, \\cdots , N.$ Next, denote $V_j^m=|u^m_j|^{2\\sigma }$ and let $ {f}^{m+\\frac{1}{2}}_j$ be defined by (REF ), (REF ) and ().", "Note that $\\lbrace \\tilde{f}_j^{\\, m}\\rbrace $ then define additive noise.", "At the half-time step, we let $u_j^{m+\\frac{1}{2}}=\\frac{1}{2}(u_j^m+u_j^{m+1})\\quad \\mbox{\\rm and} \\quad V_j^{m+\\frac{1}{2}}= \\big | u_j^{m+\\frac{1}{2}} \\big |^{2\\sigma }.$ To summarize, the discrete version of noise that we consider in this work is defined as follows $g_j^{m+\\frac{1}{2}} ={\\left\\lbrace \\begin{array}{ll}\\epsilon \\, f_j^{m+\\frac{1}{2}},\\quad \\mathrm {multiplicative \\,\\, case,} \\\\\\epsilon \\, \\tilde{f}_j^{\\, m+\\frac{1}{2}},\\qquad \\mathrm {additive \\,\\, case},\\end{array}\\right.", "}$ where $\\lbrace f_j\\rbrace $ 's and $\\lbrace \\tilde{f}_j\\rbrace $ 's are defined in (REF ), (REF ) and ()." ], [ "Three schemes", "We now consider three schemes: the mass-energy conservative (MEC) scheme (also used in [11]) $ i \\, \\dfrac{u_j^{m+1}-u_j^m}{\\Delta t_m}+\\mathcal {D}_2 u_j^{m+\\frac{1}{2}} + \\frac{1}{\\sigma +1} \\frac{|u^{m+1}_j|^{2(\\sigma +1)}- |u^m_j|^{2(\\sigma +1)}}{|u^{m+1}_j|^2-|u^m_j|^2} \\; u^{m+\\frac{1}{2}}_j = g_j^{m+\\frac{1}{2}},$ the Crank-Nicholson (CN) scheme (which is a Taylor expansion of the previous one) $i \\, \\dfrac{u_j^{m+1}-u_j^m}{\\Delta t_m}+\\mathcal {D}_2 u_j^{m+\\frac{1}{2}}+V_j^{m+\\frac{1}{2}}u^{m+\\frac{1}{2}}_j= g_j^{m+\\frac{1}{2}},$ and the new linearized extrapolation (LE) scheme, which uses the extrapolation of $V_j^{m+\\frac{1}{2}}$ $i \\, \\dfrac{u_j^{m+1}-u_j^m}{\\Delta t_m}+\\mathcal {D}_2 u_j^{m+\\frac{1}{2}}+\\frac{1}{2}\\left(\\frac{2\\Delta t_{m-1}+\\Delta t_m}{\\Delta t_{m-1}} V^{m}_j- \\frac{\\Delta t_m}{\\Delta t_{m-1}} V^{m-1}_j \\right) u^{m+\\frac{1}{2}}_j= g_j^{m+\\frac{1}{2}},$ where $g_j^{m+\\frac{1}{2}}$ is defined in (REF ).", "To compare them, we note that the schemes (REF ) and (REF ) require to solve a nonlinear system at each time step, where the fixed point iteration or Newton iteration is involved (see [11] for details).", "To implement the scheme (REF ), only a linear system needs to be solved at each time step.", "Numerically, these three schemes generate similar results (for example, the discrete mass is conserved on the order of $10^{-10} - 10^{-12}$ ; see Figure REF .", "The Crank-Nicholson scheme (REF ) usually requires between 2 and 8 iterations at each time step, and thus, is about 3 times slower than the scheme (REF ), which requires no iteration.", "In its turn, the mass-energy conservative scheme (REF ) is about 2-3 times slower than the Crank-Nicholson (REF ).", "Thus, for the computational time, the last linearized extrapolation scheme (REF ) is the most convenient.", "We remark that the scheme (REF ) is a multi-step method.", "The first time step $u^1$ is obtained by applying either the scheme (REF ) or (REF ), and then we proceed with (REF )." ], [ "Discrete mass and energy", "We define the discrete mass by $M_{\\mathrm {dis}}[u^m]= \\frac{1}{2}\\sum _{j=0}^{N} |u^m_j|^2(\\Delta x_j+\\Delta x_{j-1}).$ For $m=0$ it is the first order approximation of the integral defining the mass $M(u_0)$ in (REF ).", "We also define the discrete energy (similar to [11]), which is adapted to non-uniform mesh as follows $H_{\\rm dis}[u^m]:= \\frac{1}{2} \\sum _{j=0}^N \\frac{\\big |u^m_{j+1}-u^m_j\\big |^2}{\\Delta x_j}- \\frac{1}{2(\\sigma +1)} \\sum _{j=0}^N\\frac{\\big ( \\Delta x_j + \\Delta x_{j-1}\\big )}{2} \\, \\, |u^m_j|^{2(\\sigma +1)}.", "$ In order to check our numerical efficiency, we define the discrepancy of discrete mass and energy as $\\mathcal {E}^m_1[M]:=\\max _m \\left\\lbrace M_{\\mathrm {dis}}[u^m] \\right\\rbrace -\\min _m \\left\\lbrace M_{\\mathrm {dis}}[u^m] \\right\\rbrace .$ And $\\mathcal {E}^m[H]:=\\max _m \\left\\lbrace H_{\\mathrm {dis}}[u^m] \\right\\rbrace -\\min _m \\left\\lbrace H_{\\mathrm {dis}}[u^m] \\right\\rbrace .$ In the deterministic case all three schemes conserve mass.", "In Figure REF we show that the linearized (LE) scheme has the smallest error in discrete mass, since unlike the other two schemes there is no nonlinear system to solve, and thus, only the floating error comes into play.", "In the MEC and CN schemes the error from solving the nonlinear systems accumulate at each time step.", "Consequently, the resulting error is accumulate slightly above ($10^{-10}$ ) (there, we take $|u^{m+1,k+1}-u^{m+1,k}|<10^{-10}$ as the terminal condition for solving the nonlinear system in these two schemes, where $k$ is the index of the fixed point iteration for computing $u^{m+1} =u^{m+1,\\infty }$ ).", "The MEC scheme (REF ) also conserves the discrete energy (REF ).", "While the other two schemes do not exactly conserve energy, the error of approximation is tolerable as shown on the right of Figure REF .", "(Again, as we set up the tolerance $|u^{m+1,k+1}-u^{m+1,k}|<10^{-10}$ in solving the resulting nonlinear system, the discrete energy error $\\mathcal {E}^m[H]$ stays around $10^{-10}$ .", "Note that $\\mathcal {E}^m[H]$ is a non-decreasing function in $m$ ; it increases slowly as time evolves.)", "Figure: Deterministic NLS (ϵ=0\\epsilon =0), L 2 L^2-critical case.", "Comparison of errors in the three schemes: mass-energy conservative (MEC) (), Crank-Nicholson (CN) () and linearized extrapolation (LE) ().", "Left: error in mass computation.", "Right: error in energy computation." ], [ "Discrete mass and energy for a multiplicative noise", "We now consider a multiplicative noise, or more precisely its discrete version as defined in (REF ).", "All three schemes conserve mass in this case.", "Lemma 3.1 The numerical schemes (REF ), (REF ) and (REF ) conserve the discrete mass, that is, $M_{\\mathrm {dis}}[u^{m+1}]=M_{\\mathrm {dis}}[u^m], \\quad m=0,1,\\cdots .$ Multiply the equations (REF ), (REF ) or (REF ) by $\\bar{u}^{m+\\frac{1}{2}}_j (\\Delta x_j + \\Delta x_{j-1})$ , sum among all indexes $j$ , and take the imaginary part.", "Note that we impose the Neumann BC on both sides by setting the pseudo-points $u_{-1}=u_{0}$ and $u_{N}=u_{N+1}$ .", "Then, with a straightforward computation, one obtains $M_{\\mathrm {dis}}[u^{m+1}]-M_{\\mathrm {dis}}[u^m]=0,$ which completes the proof.", "By Taylor's expansion, it is easy to see that the schemes (REF ) and (REF ) are of the second order accuracy $O((\\Delta t_m)^3)$ at each time step $\\Delta t_m$ .", "We say the scheme (REF ) is almost of the second order accuracy because the residue is on the order $O((\\Delta t_m)^2 \\Delta t_{m-1})$ .", "(Later, to make sure that blow-up solutions do not reach the blow-up time, we take the $m$ th time step $\\Delta t_m=\\min \\left\\lbrace \\Delta t_{m-1}, \\frac{\\Delta t_0}{\\Vert u^m\\Vert _{\\infty }^{2\\sigma }} \\right\\rbrace $ .", "Thus, $\\Delta t_m \\le \\Delta t_{m-1}$ .)", "Therefore, while the schemes (REF ) and (REF ) seem to be slightly more accurate than (REF ), all three give the same order accuracy in their application below." ], [ "Upper bounds on discrete energy", "We now study stability properties of the time evolution of the discrete energy (REF ) for the mass-energy conserving (MEC) scheme (REF ).", "Let $\\tau ^*_{\\rm dis}$ denote the existence time of the discrete scheme.", "For simplicity we take the uniform mesh in space and time, i.e., for each $j$ and $m$ , we set $\\Delta x= \\Delta x_j$ and $\\Delta t=\\Delta t_m$ .", "In that case the discrete energy is $ H_{\\rm dis}[u^m]:= \\Delta x \\Big ( \\frac{1}{2} \\sum _{j=0}^N \\Big | \\frac{u^m_{j+1}-u^m_j}{\\Delta x}\\Big |^2 - \\frac{1}{2(\\sigma +1)} \\sum _{j=0}^N|u^m_j|^{2(\\sigma +1)}\\Big ).$ Proposition 3.2 Let $u_0\\in H^1$ and $t_M<\\tau ^*_{\\rm dis} $ be a point of the time grid.", "Then for $\\Delta x \\in (0,1)$ $ {\\mathbb {E}}\\big ( H_{\\rm dis}[u^M] \\big ) &\\le \\, H_{\\rm dis}[u^0] + \\,\\frac{\\epsilon \\, \\sqrt{3}}{2\\, \\sqrt{2}} \\frac{\\sqrt{ \\ln (2\\, L_c) + |\\ln (\\Delta x)| }}{\\sqrt{\\Delta x}}\\, \\frac{1}{(\\Delta t)^{\\frac{3}{2}}} t_M,\\\\{\\mathbb {E}}\\Big ( \\max _{0\\le m\\le M} H_{\\rm dis}[u^m] \\Big ) &\\le \\, H_{\\rm dis}[u^0] + \\frac{\\epsilon \\, \\sqrt{3}}{\\sqrt{2}} \\,\\frac{\\sqrt{ \\ln (2\\, L_c) + |\\ln (\\Delta x)| }}{\\sqrt{\\Delta x}} \\,\\frac{1}{(\\Delta t)^{\\frac{3}{2}}} t_M.", "$ Multiplying equation (REF ) by $- \\Delta x\\, (\\bar{u}_j^{m+1}-\\bar{u}^m_j)$ , adding for $m=0, ..., M-1$ and $j=0, ..., N$ , and using the conservation of the discrete energy in the deterministic case, we deduce that for some real-valued random variable $R(M,N)$ , which changes from one line to the next, $H_{\\rm dis}[u^M]=&\\, H_{\\rm dis}[u^0] +i R(M,N) + \\epsilon \\, \\Delta x \\, \\sum _{m=0}^{M-1} \\sum _{j=0}^N (\\bar{u}^{m+1}_j-\\bar{u}^m_j)\\, \\frac{1}{2}\\big ( u^{m+1}_j+u^m_j\\big )\\tilde{f}^{\\,m+\\frac{1}{2}}_j \\nonumber \\\\=& H_{\\rm dis }[u^0] + i R(M,N) + \\frac{ \\epsilon \\, \\Delta x}{2} \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\, \\big ( |u^{m+1}_j|^2 - |u^{m}_j|^2\\big ) \\,\\tilde{f}^{\\,m+\\frac{1}{2}}_j, \\\\= &\\, H_{\\rm dis }[u^0] + i R(M,N) - \\frac{\\epsilon }{2} \\, \\frac{1}{\\Delta t} \\, \\int _0^{t_M}\\!", "\\int _{\\mathbb {R}}|{U}(s,x)|^2 W_N(ds,dx) + \\frac{ \\epsilon \\, \\Delta x}{2} \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\, |u^{m+1}_j|^2 \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j, $ where $U(s,x)$ is the step process defined by $U(s,x)=u^m_j$ on the rectangle $[t_m,t_{m+1})\\times [x_{j-\\frac{1}{2}} , x_{j+\\frac{1}{2}})$ .", "Since the discrete mass is preserved by the scheme (Lemma REF ), we have $\\frac{ \\epsilon \\, \\Delta x}{2} \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\, |u^{m+1}_j|^2 \\, \\tilde{f}^{\\, m+\\frac{1}{2}}_j \\le \\frac{\\epsilon }{2} \\sum _{m=0}^{M-1} \\max _{0\\le j\\le N} | \\tilde{f}^{\\, m+\\frac{1}{2}}_j| \\sum _{j=0}^N \\Delta x \\, |u^{m+1}_j|^2 =\\frac{\\epsilon \\, M_{\\rm dis}[u^0]}{2} \\sum _{m=0}^{M-1} \\max _{0\\le j\\le N} | \\tilde{f}^{\\, m+\\frac{1}{2}}_j|.$ Using the definition of $\\tilde{f}^{\\, m+\\frac{1}{2}}_j$ in (REF ), (REF ) and (), we deduce $E\\Big ( \\max _{0\\le j\\le N} | \\tilde{f}^{\\, m+\\frac{1}{2}}_j| \\Big ) = \\frac{\\sqrt{3}}{2\\, \\sqrt{\\Delta t} \\sqrt{\\Delta x}} E\\Big ( \\max _{0\\le j\\le N} | \\chi ^{m+\\frac{1}{2}}_j| \\Big ),$ where the random variables $\\chi ^{m+\\frac{1}{2}}_j$ are independent standard Gaussians.", "Using Pisier's lemma (see e.g.", "[29]), one observes that if $\\lbrace G_k\\rbrace _{ k=1, ...,n}$ are independent standard Gaussians and $M_n=\\max _{1\\le k\\le n} |G_k|$ , we have for $n\\ge 2$ $\\mathbb {E}(M_n) \\le \\sqrt{2 \\, \\ln (2\\, n)} .", "$ We enclose the proof below for the sake of completeness.", "For any $\\lambda >0$ , using the Jensen inequality and the fact that $x\\mapsto e^{\\lambda x}$ is increasing, we deduce $\\exp \\Big ( \\lambda \\, {\\mathbb {E}}\\Big [ \\max _{1\\le k\\le n} |G_k|\\Big ]\\Big ) \\le &\\; {\\mathbb {E}} \\Big ( \\exp \\Big [ \\lambda \\max _{1\\le k\\le n}\\big |G_k\\big | \\Big ] \\Big ) \\le {\\mathbb {E}} \\Big ( \\max _{1\\le k\\le n} \\exp \\big ( \\lambda |G_k|\\big ) \\Big ) \\\\\\le & \\; \\sum _{k=1}^n {\\mathbb {E}} \\Big (e^{\\lambda \\big | G_k\\big |}\\Big ) \\le n \\, 2\\, e^{\\frac{\\lambda ^2 }{2}}.$ Taking logarithms, we obtain $ {\\mathbb {E}}\\Big ( \\max _{1\\le k\\le n} |G_k | \\Big ) \\le \\frac{1}{\\lambda } \\ln \\Big (2\\, n\\, e^{\\frac{\\lambda ^2}{2}}\\Big )= \\frac{\\ln (2\\, n)}{\\lambda } + \\frac{\\lambda }{2},$ for every $\\lambda >0$ .", "Choosing $\\lambda = \\sqrt{2 \\ln (2n)}$ concludes the proof of (REF ).", "Keeping the real part of (), we obtain ${\\mathbb {E}} \\big ( H_{\\rm dis}[u^M]\\big ) \\le &\\; H_{\\rm dis}[u^0] + \\frac{\\epsilon }{2} M_{\\rm dis}[u^0] \\, M\\, \\frac{\\sqrt{3}}{2}\\,\\sqrt{2 \\ln \\big [ 2\\, (N+1)\\big ] } \\frac{1}{\\sqrt{\\Delta t} \\sqrt{\\Delta x}} \\\\\\le & \\;H_{\\rm dis}[u^0] + \\frac{\\epsilon \\, \\sqrt{3} }{2\\, \\sqrt{2}} \\frac{1}{\\sqrt{\\Delta x}} \\, \\sqrt{ \\ln \\Big ( \\frac{2\\, L_c}{\\Delta x}\\Big ) }\\, \\frac{1}{(\\Delta t)^{\\frac{3}{2}}} t_M.", "\\ $ This completes the proof of (REF ).", "To prove (), keeping the real part of (REF ) and estimating from above $| u^{m+1}_j|^2 - |u^m_j|^2$ by $| u^{m+1}_j|^2 + |u^m_j|^2$ , we get $\\max _{0\\le m\\le M} H_{\\rm dis}[u^M]= H_{\\rm dis}[u^0] + \\frac{ \\epsilon \\, \\Delta x}{2} \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\, \\big ( |u^{m+1}_j|^2 + |u^m_j|^2\\big )\\, |\\tilde{f}^{\\,m+\\frac{1}{2}}_j|,$ and the previous argument concludes the proof.", "Remark 3.3 Note that in (REF ) and (), the upper bound depends linearly on $\\epsilon $ , and for small $\\epsilon << 1$ so does the leading term of the theoretical estimate (REF ).", "There is also a very small dependence on $L_c$ , and a more important one on $\\Delta x$ and $\\Delta t$ .", "We remark that these are just the upper bounds, and to get a better idea about the growth and dependence of the energy on the various parameters, we investigate that numerically." ], [ "Numerical tracking of discrete mass and energy", "Our analytical results above provide mass conservation and upper bounds on the expected values of energy.", "We would like to check numerically behavior of these quantities.", "We start with testing the accuracy and efficiency of our schemes, for that we consider initial data $u_0 = A\\,Q$ , where $A>0$ and $Q$ is the ground state (REF ).", "Figure: Multiplicative noise.", "The error of the discrete mass computation ℰ 1 m \\mathcal {E}_1^m from (), ϵ=0.5\\epsilon =0.5, in both L 2 L^2-critical and supercritical cases for one trajectory.For the first test we take $u_0=0.95 Q$ and $\\epsilon =0.5$ in both $L^2$ -critical ($\\sigma =2$ ) and $L^2$ -supercritical ($\\sigma =3$ ) cases.", "The difference $\\mathcal {E}^n_1$ in both cases is shown in Figure REF .", "Observe that the error is on the order of $10^{-15}$ , which is almost at the machine precision ($10^{-16}$ ).", "Since not all of our three schemes conserve the discrete energy exactly (in the deterministic case), we study influence of the multiplicative noise onto the discrete energy (REF ).", "In Figure REF we show that in the $L^2$ -critical case and $\\epsilon =0.5$ , all three schemes produce the same result for the initial data $u_0=0.9Q$ , where the energy is growing and then starts leveling off around the time $t=15$ .", "On the right of the same figure we zoom on the time interval $[0,5]$ to see better the difference between the schemes, and we note that the linearized extrapolation (LE) scheme produces slightly lower values of the energy, even if the overall behavior is the same.", "In our further investigations we usually use the MEC scheme if we need to track the mass and energy, and when we investigate the more global features such as blow-up profiles or run a lot of simulations, then we utilize the LE scheme.", "Figure: Multiplicative noise, ϵ=0.5\\epsilon =0.5, L 2 L^2-critical case.", "Expected energy (averaged over 100 runs) using different schemes: mass-energy conservative(MEC) (), Crank-Nicholson (CN) () and linearized extrapolation (LE) ().Left: time 0<t<250<t<25.", "Right: zoom-in for time 0<t<50<t<5: note only a small difference with the LE scheme.We next study the growth of energy in time and the dependence on various parameters.", "In Figures REF , REF , REF we show the time dependence of solutions with initial data of type $u_0 = A\\, Q$ .", "Figure: Multiplicative noise in the L 2 L^2-critical case, σ=2\\sigma =2 (top)and L 2 L^2-supercritical case, σ=3\\sigma =3 (bottom); u 0 =0.8Qu_0=0.8Q, Δx=0.05\\Delta x=0.05, Δt=0.005\\Delta t=0.005, L c =20L_c=20.", "Time dependence of 𝔼(H(u(t)))\\mathbb {E} (H(u(t))) (left) vs.𝔼(sup s≤t H(u(s)) \\mathbb {E}( \\sup _{s\\le t} H(u(s)) (right) for various ϵ\\epsilon .In Figure REF we track the growth of the expected values of the instantaneous energy (on the left subplots) and of the supremum of energy (on the right subplots).", "To approximate the expected value, we average over 100 runs.", "Our simulations show that both start growing linearly at first (see zoom-in Figure REF ), then start slowing down until they peak and level off to some possibly maximum value.", "As expected the values of the maximal energy up to some specific time are larger.", "We observe that the stronger the noise is (i.e., the larger the coefficient $\\epsilon $ ), the shorter it takes for the expected energy to start leveling off.", "A similar behavior is seen in Figure REF for the gaussian initial data $u_0=A \\, e^{-x^2}$ and supergaussian data $u_0=A\\, e^{-x^4}$ in both critical and supercritical cases.", "From now on we only show expectations of instantaneous energy in our figures as plots for the maximal energy are very similar.", "Figure: Multiplicative noise in both L 2 L^2-critical and supercritical cases: gaussian (left two) u 0 =e -x 2 u_0=e^{-x^2} and supergaussian (right two) u 0 =e -x 4 u_0=e^{-x^4}; Δx=0.05\\Delta x=0.05, Δt=0.005\\Delta t=0.005, L c =20L_c=20.", "The time dependence of 𝔼(H(u(t)))\\mathbb {E} (H(u(t))) (left) vs. 𝔼(sup s≤t H(u(s)) \\mathbb {E}( \\sup _{s\\le t} H(u(s)) (right).We next investigate the dependence of the discrete energy (REF ) on computational parameters such as the length of the interval $L_c$ , the spatial step size $\\Delta x$ and the time step $\\Delta t$ .", "The results are shown in Figure REF for the expected energy values $\\mathbb {E}(H(u(t)))$ with varying sizes of $\\Delta x$ and $L_c$ ; in Figure REF the dependence on $\\Delta t$ is displayed.", "Figure: Multiplicative noise, u 0 =0.8Qu_0=0.8Q, ϵ=0.5\\epsilon =0.5.", "The growth of expected energy depends on Δx\\Delta xbut not on L c L_c in both L 2 L^2-critical (left) and supercritical (right) cases.Figure: Multiplicative noise, u 0 =0.9Qu_0=0.9Q, ϵ=0.5\\epsilon =0.5, L c =20L_c=20, Δx=0.05\\Delta x = 0.05, Δt=0.005\\Delta t=0.005.", "The growth of the expected energy for different Δt\\Delta t in both the L 2 L^2-critical and supercritical cases.Figure: Multiplicative noise.", "Zoom-in for small times (to track linear dependence): u 0 =0.9Qu_0=0.9Q, Δx=0.05\\Delta x=0.05, L c =20L_c=20, ϵ=0.5\\epsilon =0.5.", "The time dependence of 𝔼(H(u(t)))\\mathbb {E} (H(u(t)))for different values of Δt\\Delta t in both L 2 L^2-critical and supercritical cases.We remark that in both critical and supercritical cases, the computed values of expected energies (instantaneous and sup) are insensitive to the length of the computational domain $L_c$ .", "However, there is a dependence on the mesh size $\\Delta x$ : the smaller step size results in a larger value of energy; there is also a dependence on the time step $\\Delta t$ ." ], [ "Discrete mass and energy for an additive noise", "Our next endeavor is to study the additive stochastic perturbation $f(u)=W(dt,dx)$ , or its discretized version in (REF ).", "As in the multiplicative case, we replace the space-time white noise $W$ by its approximation $W_N$ defined in (REF ) in terms of the functions $\\lbrace e_j \\rbrace _{0 \\le j \\le {N}}$ described in (REF ).", "Then in our numerical schemes (REF ), (REF ) and (REF ) the right-hand side is $\\Big \\lbrace \\tilde{f}^{\\,m+\\frac{1}{2}}_j \\Big \\rbrace $ defined in (REF ) for $j=1, ..., N-1$ , in (REF ) for $j=0$ , and in () for $j=N$ .", "We show that for the schemes (REF ), (REF ) and (REF ) the time evolution of the expected value of the discrete mass on the time interval $[0,T]$ is estimated from above by an affine function $a+bt$ .", "We prove that the slope $b$ is a linear function of the length $L_c$ of the discretization interval $[-L_c,L_c]$ .", "Therefore, our upper bounds on the discrete mass and energy depend linearly on the total length $L_c$ ; they are inversely proportional to the constant time and space mesh sizes $\\Delta t$ and $\\Delta x$ .", "We do not claim that our upper bounds are sharp; this is the first attempt to upper estimate the discrete quantities." ], [ "Upper bounds on discrete mass and energy with additive noise", "Recall that the discrete mass of $u^m$ is defined by $M_{\\rm dis}[u^m]=\\Delta x \\sum _{j=0}^N |u^m_j|^2$ .", "Let $\\tau ^*_{\\rm dis}$ be the maximal existence time of a discrete scheme.", "Proposition 3.4 Let $u^m_j$ be the solution to the scheme (REF ), (REF ) or (REF ) with $\\tilde{f}^{m+\\frac{1}{2}}_j$ instead of $f^{m+\\frac{1}{2}}_j$ for a constant time mesh $\\Delta t$ and space mesh $\\Delta x$ .", "Then given $T\\in (0, \\tau ^*_{\\rm dis})$ and any element $t_M\\le T$ of the time grid, we have for any $\\alpha >0$ ${\\mathbb {E}} \\big (M_{\\mathrm {dis}}[u^M]\\big ) & \\le (1+\\alpha ) M_{\\rm dis}[u^0] + \\frac{3\\,T\\, (1+\\alpha )}{4\\, \\ln (1+\\alpha )} \\; \\epsilon ^2\\, \\frac{L_c}{\\Delta x} \\, \\frac{t_M}{\\Delta t},\\quad \\mbox{if} \\quad \\Delta t\\le \\frac{T}{1+\\alpha }, \\\\{\\mathbb {E}}\\Big ( \\max _{0\\le m\\le M} M_{\\mathrm {dis}}[u^m]\\Big ) &\\le \\Big (1+\\alpha + \\frac{\\alpha }{2T}\\Big ) M_{\\mathrm {dis}} [u^0]+ \\frac{3 \\, T\\, (1+\\alpha )^2}{2\\, \\alpha } \\; \\epsilon ^2 \\, \\frac{L_c}{\\Delta x} \\, \\frac{ t_M}{\\Delta t} .$ Recall that $u^{m+\\frac{1}{2}}_j= \\frac{1}{2} \\big ( u^m_j + u^{m+1}_j\\big )$ .", "Multiply the equation (REF ) by $- 2 i \\Delta t \\Delta x \\, \\bar{u}^{\\,m+\\frac{1}{2}}_j$ , sum on $j$ from $j=0$ to $N$ and then sum on $m$ from $m=0$ to $M-1$ .", "Then there exists a real-valued random variable $R(M,N)$ (changing from one line to the next) such that $\\Delta x \\sum _{j=0}^{N} |u^{M}_j|^2 & - \\Delta x \\sum _{j=0}^{N} |u^{0}_j|^2 = i R(M,N) \\; -2\\, \\, \\epsilon i \\sum _{m=0}^{M-1} \\sum _{j=0}^N\\Delta t \\, \\Delta x \\, \\frac{\\bar{u}^{\\, m}_j + \\bar{u}^{\\, m+1}_j}{2}\\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j \\nonumber \\\\&= i R(M,N) \\, - \\, \\epsilon \\int _0^{t_M} \\!\\!", "\\int _{\\mathbb {R}} \\mbox{\\rm Im}\\, \\big ( {U}(s,x) \\big )\\, W_N(ds,dx) - \\epsilon \\, \\sum _{m=0}^{M-1} \\sum _{j=0}^N\\Delta t \\, \\Delta x \\, \\mbox{\\rm Im}\\, ({u}^{m+1}_j) \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j, \\ $ where $U$ is the step process defined by ${U}(s,x)= u^m_j$ on the rectangle $[t_m,t_{m+1})\\times [x_{j-\\frac{1}{2}},x_{j+\\frac{1}{2}})$ .", "The Cauchy-Schwarz inequality applied to $\\sum _m\\, \\sum _j$ , the definition of $\\tilde{f}^{\\, m+\\frac{1}{2}}_j$ in (REF ), (REF ) and (), and Young's inequality imply that for $\\delta >0$ we have $\\Big |\\, \\epsilon \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\Delta t \\, \\Delta x \\, \\mbox{\\rm Im}\\, ({u}^{m+1}_j) \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j\\Big | \\le &\\, \\epsilon \\Big \\lbrace \\sum _{m=1}^M \\sum _{j=0}^N \\Delta t\\, \\Delta x\\, |u^m_j|^2 \\Big \\rbrace ^{\\frac{1}{2}}\\Big \\lbrace \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\Delta t\\, \\Delta x\\, |\\tilde{f}^{\\,m+\\frac{1}{2}}_j|^2\\Big \\rbrace ^{\\frac{1}{2}} \\nonumber \\\\\\le &\\, \\delta \\sum _{m=1}^M \\Delta t \\, M_{\\rm dis}[u^m] + \\frac{3\\, \\epsilon ^2}{16\\, \\delta } \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\frac{3}{4} | \\chi ^{\\,m+\\frac{1}{2}}_j|^2,$ where the random variables $\\chi ^{m+\\frac{1}{2}}_j$ are (as before) independent standard Gaussian random variables.", "Keeping the real part in (REF ), then plugging the above estimate into the (REF ) and taking expected values (note that the process $U$ is adapted), we deduce ${\\mathbb {E}}\\big ( M_{\\rm dis}[u^M]\\big ) \\le M_{\\rm dis}[u^0] + \\frac{3\\, \\epsilon ^2}{16\\, \\delta } M (N+1)+ \\delta \\sum _{m=1}^M \\Delta t {\\mathbb {E}}\\big ( M_{\\rm dis}[u^m]\\big ).$ Given $\\beta \\in (0,1)$ , we suppose that $ \\delta \\, \\Delta t \\le \\beta $ .", "Then the discrete version of the Gronwall lemma (see e.g.", "[24]) implies ${\\mathbb {E}}\\big ( M_{\\rm dis}[u^M]\\big ) \\le \\frac{1}{1-\\beta } \\Big [ M_{\\rm dis}[u^0] + \\frac{3\\, \\epsilon ^2}{16\\, \\delta } M (N+1) \\Big ] e^{\\delta (M-1) \\Delta t}.$ Fix $\\alpha \\in (0,1)$ and choose $\\beta \\in (0,1)$ such that $\\frac{1}{1-\\beta }= \\sqrt{1+\\alpha }$ , and choose $\\delta >0$ such that $e^{\\delta T} = \\sqrt{1+\\alpha }$ .", "Then $\\delta = \\frac{\\ln (1+\\alpha )}{2T}\\in \\big ( \\frac{\\alpha }{4T}, \\frac{\\alpha }{2T}\\big )$ , and $\\Delta t \\le \\frac{T}{1+\\alpha } \\le \\frac{2T }{(\\sqrt{1+\\alpha }+1)\\sqrt{1+\\alpha }} = \\frac{2T}{\\alpha } \\beta $ implies $\\delta \\, \\Delta t \\le \\beta $ .", "Furthermore, $M(N+1) \\le \\frac{t_M}{\\Delta t}\\, \\frac{2\\, L_c}{\\Delta x} $ , and we deduce (REF ).", "We next prove ().", "A similar computation, based on the first upper estimate in (REF ) and on (REF ), proves that for $\\delta >0$ we have $M_{\\rm dis}[u^M] = &\\, M_{\\rm dis}[u^0] + \\epsilon \\, \\Big | \\sum _{m=0}^{M-1} \\sum _{j=0}^N\\Delta t \\, \\Delta x \\, \\mbox{\\rm Im}\\, ({u}^{m}_j) \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j \\Big | + \\epsilon \\, \\sum _{m=0}^{M-1} \\sum _{j=0}^N\\Delta t \\, \\Delta x \\, \\mbox{\\rm Im}\\, ({u}^{m+1}_j) \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j, \\nonumber \\\\\\le &\\, M_{\\rm dis}[u^0] + \\delta \\Delta t \\, M_{\\rm dis}[u^0] + 2 \\delta \\sum _{m=1}^M \\Delta t \\, M_{\\rm dis}[u^m]+ 2 \\frac{\\epsilon ^2}{4\\, \\delta } \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\frac{3}{4}\\, |\\chi ^{\\,m+\\frac{1}{2}}_j|^2,$ where the random variables $\\chi ^{m+\\frac{1}{2}}_j$ (as before) are independent standard Gaussian random variables.", "Taking expected values, we deduce for any $\\delta >0$ ${\\mathbb {E}}\\Big (\\max _{1\\le m\\le M} M_{\\rm dis}[u^m]\\Big ) \\le (1+\\delta \\Delta t ) M_{\\rm dis}[u^0] +2\\, \\delta \\, t_M\\, {\\mathbb {E}}\\Big (\\max _{1\\le m\\le M} M_{\\rm dis}[u^m]\\Big )+ \\frac{3\\, \\epsilon ^2}{8\\, \\delta } \\, M\\, (N+1).$ Given $\\beta >0$ , choose $\\delta >0$ such that $2\\, \\delta T = \\beta $ ; this yields ${\\mathbb {E}}\\Big (\\max _{1\\le m\\le M} M_{\\rm dis}[u^m]\\Big ) \\le \\frac{1}{1-\\beta } \\Big [ \\big ( 1+\\delta \\Delta t\\big ) M_{\\rm dis}[u^0]+ \\frac{3\\, \\epsilon ^2}{8\\, \\delta } \\, M\\, (N+1)\\Big ].$ Given $\\alpha >0$ , choose $\\beta \\in (0,1)$ such that $\\frac{1}{1-\\beta }= 1+\\alpha $ ; then $\\delta = \\frac{\\alpha }{2T(1+\\alpha )}$ .", "This concludes the proof of () for the mass-energy conservative scheme.", "A similar argument is applied to the schemes (REF ) and (REF ) (with the additive right-hand side); the only difference is in the real-valued random variable $R(M,N)$ , which varies from one scheme to the next, but is not present in the final estimate.", "Remark 3.5 Note that the estimates (REF ) and () of the instantaneous and maximal mass are worse than the discrete analog of (REF ) by a factor of $\\frac{1}{\\Delta t}$ .", "One might try to solve this problem in the proof, changing $ 2 \\bar{u}^{\\, m+\\frac{1}{2}}_j \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j$ into $ \\big ( \\bar{u}^{m+1}_j- \\bar{u}^m_j\\big ) \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j + 2\\, \\bar{u}^m_j \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j $ , and using again the scheme to deal with the first term.", "This would introduce an extra $\\Delta t$ factor.", "However, if the product of the two stochastic Gaussian variables would give a discrere analog of the inequality (REF ), the deterministic part of the scheme would still create terms involving $\\bar{u}^{m+1}_j \\, \\tilde{f}^{\\,m+\\frac{1}{2}}_j $ .", "The corresponding non-linear “potential\" term would yield the mass to be raised to a large power to enable the use of the discrete Gronwall or Young lemma.", "We next study stability properties of the time evolution of the discrete energy defined by (REF ) for the mass-energy conserving (MEC) scheme (REF ) in the additive case.", "Proposition 3.6 Let $u^n_j$ be the solution to the scheme (REF ) with $\\tilde{f}^{\\, m+\\frac{1}{2}}_j$ in (REF ) for a constant time mesh $\\Delta t$ and space mesh $\\Delta x$ .", "Then given $T\\in (0, \\tau ^*_{\\rm dis})$ and any element $t_M\\le T$ of the time grid, we have for any $\\alpha >0$ ${\\mathbb {E}} \\big (H_{\\mathrm {dis}}[u^M]\\big ) & \\le (1+\\alpha ) H_{\\rm dis}[u^0] + \\frac{3\\,T\\, (1+\\alpha )}{4\\, \\ln (1+\\alpha )} \\; \\epsilon ^2\\, \\frac{L_c}{\\Delta x} \\, \\frac{t_M}{(\\Delta t)^2},\\quad \\mbox{if} \\quad \\Delta t\\le \\frac{T}{1+\\alpha }, \\\\{\\mathbb {E}}\\Big ( \\max _{0\\le m\\le M} H_{\\mathrm {dis}}[u^m]\\Big ) &\\le \\Big (1+\\alpha + \\frac{\\alpha }{2T}\\Big ) H_{\\mathrm {dis}} [u^0]+ \\frac{3 \\, T\\, (1+\\alpha )^2}{2\\, \\alpha } \\; \\epsilon ^2 \\, \\frac{L_c}{\\Delta x} \\, \\frac{ t_M}{(\\Delta t)^2} .$ Multiplying equation (REF ) by $- (\\bar{u}^{m+1}_j - \\bar{u}^m_j) \\Delta x $ , adding for $j=0, ..., N$ and $m=0, ..., M-1$ , and using the fact that in the deterministic case ($\\epsilon =0$ ) the scheme (REF ) preserves the discrete energy, we deduce the existence of a real-valued random variable $R(M,N)$ (changing from line to line) such that $H_{\\rm dis}[u^M] = & H_{\\rm dis }[u^0] + i R(M,N) - \\epsilon \\Delta x \\sum _{m=0}^{M-1} \\sum _{j=0}^N (\\bar{u}^{m+1}_j-\\bar{u}^m_j)\\tilde{f}^{\\, m+\\frac{1}{2}}_j \\\\= &H_{\\rm dis }[u^0] + i R(M,N) + \\, \\epsilon \\, \\frac{\\Delta x}{\\Delta t} \\int _0^{t_M} \\mbox{\\rm Re}\\, (u^m_j)\\, W_N(ds,dx)- \\epsilon \\, \\Delta x\\, \\sum _{m=0}^{M-1} \\sum _{j=0}^N \\mbox{\\rm Re}\\, u^{m+1}_j\\, \\tilde{f}^{\\, m+\\frac{1}{2}}_j .$ Notice that the last term in the above identity is similar to the last one in (REF ), except that the factor $\\Delta t$ is missing.", "Thus, the arguments used to prove Proposition REF conclude the proof." ], [ "Numerical tracking of discrete mass and energy, additive noise", "As in the multiplicative case, we start with testing the accuracy of our three numerical schemes (REF ), (REF ) and (REF ) with the additive forcing (REF ) on the right hand side and using the initial data $u_0=A\\,Q$ .", "In Figure REF we show the comparison of three schemes for the initial condition $u_0=0.9Q$ with the strength of the noise $\\epsilon = 0.05$ in the $L^2$ -critical case.", "We see that for both discrete mass and energy the schemes behave similarly with very little variation from one to another.", "Figure: Additive noise, ϵ=0.05\\epsilon =0.05, L 2 L^2-critical case.", "Time evolution of discrete mass (left) and energy (right) via different schemes:mass-energy conservative (MEC) (), Crank-Nicholson (CN) () and linearized extrapolation (LE) ().We first investigate dependence of mass and energy on the strength of the noise $\\epsilon $ .", "We take the initial condition $u_0=0.9 \\, Q$ and set $L_c=20$ , considering $x \\in [-L_c,L_c]$ ; we also set $\\Delta x=0.05$ and $\\Delta t=0.005$ .", "As before, we do 100 runs to approximate the expectation of either mass or energy.", "Recall that the identity (REF ) and the inequality (REF ) give linear dependence on time and square dependence on the noise strength $\\epsilon $ , similar to that in our upper estimates for the discrete quantities (REF ), () (for mass) and (REF ), () (energy).", "The results are shown in Figure REF , where we plot the expectation of the instantaneous quantities, $\\mathbb {E}(M(u(t)))$ and $\\mathbb {E}(H(u(t)))$ .", "We omit figures for $\\mathbb {E}(\\sup _{s \\le t}M(u(s)))$ and $\\mathbb {E}(\\sup _{s \\le t}H(u(s)))$ , since we get the same behavior as shown in Figure REF , and both discrete upper estimates (REF ) and () give similar dependence on all parameters.", "Figure: Additive noise.", "Dependence of the expected value of (instantaneous) mass (left) and energy (right) on the strength of the noise ϵ\\epsilon .", "Top: L 2 L^2-critical (σ=2\\sigma =2), bottom: L 2 L^2-supercritical (σ=3\\sigma = 3).Next, we show the dependence of the discrete mass and energy on the length of the computational interval $L_c$ and the step size $\\Delta x$ .", "We compare the growth of both expected mass and energy for two values of the length $L_c=20$ and $L_c=40$ , see Figure REF , which shows the linear dependence for both expected values of the mass and the energy: the $L^2$ -critical case ($\\sigma =2$ ) is shown in the top row, and the $L^2$ -supercritical case ($\\sigma =3$ ) is in the bottom row.", "Note that the slope doubles as we double the length of the computational interval $L_c$ .", "Figure: Additive noise, ϵ=0.05\\epsilon =0.05.", "Dependence of mass and energy on the time step-size Δt\\Delta t in the L 2 L^2-critical case.Figure: Additive noise, ϵ=0.05\\epsilon =0.05.", "Dependence of the expected value of the mass and energy on the length of the interval L c L_c and space step-size Δx\\Delta x.", "Top: L 2 L^2-critical (σ=2\\sigma =2), bottom: L 2 L^2-supercritical (σ=3\\sigma = 3).The dependence on the time step-size of both discrete mass and energy is shown in Figure REF .", "We show the dependence in the $L^2$ -critical case and omit the supercritical case it is similar.", "We also mention that we studied the growth of mass and energy for other initial data, for example, gaussian $u_0 = A \\, e^{-x^2}$ , and obtained similar results, see Figure REF .", "Figure: Additive noise, ϵ=0.05\\epsilon =0.05.", "Time dependence of the discrete mass (left) and energy (right) for the gaussian initial condition u 0 =0.5e -x 2 u_0=0.5 \\, e^{-x^2}.", "(Here, both mass and energy coincide regardless of the nonlinearity, σ=2\\sigma = 2 or 3, since the only dependence is in the potential part of energy,which creates a very small difference.", ")In this section, we investigated how used-to-be conserved quantities (mass and energy) in the deterministic setting behave in the stochastic case with both multiplicative and additive approximations of the space-time white noise.", "Our next goal is to look at a global picture and study how behavior of solutions is affected by the noise on a more global scale.", "We will see that in some cases the noise forces solutions to blow-up and in other instances, the noise will prevent blow-up formation (similar investigations were done in [11] and references therein).", "We confirm some of their findings, and then investigate the blow-up dynamics (rates, profiles, etc).", "Before we venture into that study, we need to refine our numerical method, which we do in the next section." ], [ "Numerical approach, refined", "To study solitons and their stability numerically, it is useful to have a non-uniform mesh to capture well certain spatial features.", "For that we use a finite difference method with non-uniform mesh.", "To study specific details of the evolution (such as formation of blow-up), we implement mesh refinement.", "However, to keep the algorithm efficient, the mesh refinement is applied only at certain time steps, when it is necessary.", "By a carefully chosen mesh-refinement strategy and a specific interpolation during the refinement (which we introduce below), we are able to keep the discrete mass at the same value before and after the mesh-refinement.", "Therefore, the discrete mass is exactly conserved at all times in our time evolution on $[0,T]$ (in the deterministic and multiplicative noise settings).", "We note that in the deterministic theory, solutions either exist globally in time or blow-up in finite time, and there are various results identifying thresholds for such a dichotomy.", "In the probabilistic setting blow up may hold in finite time with some (positive) probability even for small initial data.", "Indeed in [9] it is shown that for a multiplicative stochastic perturbation (driven by non-degenerate noise with a regular enough space correlation) given any non-null initial data there is blows up with positive probability.", "Therefore, when we study solutions of SNLS (REF ), we may refer to the types of solutions as globally existing (a.s.), long-time existing (perhaps with some estimates on the time of existence), and blow-up in finite time (with positive probability, or a.s.) solutions.", "We also mention that as an extra bonus for a multiplicative noise, our algorithm has very small fluctuation (on the order of $10^{-12}$ ) in the difference of the actual mass (REF ), which is approximated by the composite trapezoid rule; see (REF ).", "The tiny difference is observed in all scenarios of solutions: globally existing, long-time existing and blow-up in finite time (the difference is on the order of $10^{-12}$ ), which we demonstrate in Figure REF .", "This suggests that our algorithm is very accurate in all scenarios of solutions." ], [ "Mesh-refinement strategy", "When a solution starts concentrating or localizing spatially, in order to increase accuracy, it is necessary to put more points into that region.", "For example, as blow-up starts focusing towards a singular point as $t \\rightarrow T$ , the singular region will benefit from having more grid points.", "In this subsection, we discuss the mesh-refinement strategy.", "The idea comes from the scaling invariance of the NLS equation, or the dynamic rescaling method from [28], [35], [39] and [40].", "At time 0 the computational interval $[-L_c,L_c]$ is discretized into $N_0+1$ grid points $\\lbrace x^0_0,\\cdots ,x^0_{N_0} \\rbrace $ (which may be equi-distributed, since we typically begin with a uniform space mesh).", "When we proceed, we check at each time step if the scheme fulfills a tolerance criterion, described below.", "As we mentioned in the introduction, the stable blow-up dynamics for the deterministic NLS consists of the self-similar regime with the rescaled parameters $u(t,x)=\\frac{1}{L(t)^{1/ \\sigma }} v(\\tau ,\\xi ), \\quad \\xi =\\frac{x}{L(t)}, \\quad \\tau =\\int _0^t \\frac{1}{L^2(s)} ds,$ where $v(\\xi ,\\tau )$ is a globally (in $\\tau $ ) defined self-similar solution.", "We do not rescale the equation (REF ) into a new equation as we do not use the dynamic rescaling method due to regularity issues.", "However, we still adopt the rescaling idea for our mesh-refinement algorithm.", "Assume $\\xi $ is equi-distributed for all time steps $t_m$ and $\\Delta \\xi =\\xi _{1}-\\xi _0$ .", "Thus, we assume that there is a mapping $L(t_m)$ , which maps the point $x_j^m \\rightarrow \\xi _j$ .", "Using (REF ) or $L(t)^{1/\\sigma } u(t) = v(\\tau )$ with our discretization, we get $L(t_m)^{1/ \\sigma } \\left( u(x^m_j)- u(x^m_{j-1}) \\right) = v(\\xi _j)-v(\\xi _{j-1}),$ where both sides are well-behaved (since $v$ is now global), and thus, should have $O(1)$ value (referred to as the moderate value) for $j=0,1,\\cdots , N_m$ .", "(The rescaled solution $v(\\xi )$ is well-behaved as well).", "Using the second relation in (REF ), we define the discretization of the mapping of $L(t_m)$ at each interval $[\\xi _{j-1},\\xi _{j}]$ : $L^m_j= \\frac{x^m_j-x^m_{j-1}}{\\Delta \\xi }.$ Putting this into (REF ) and using the fact that $\\Delta \\xi $ is a constant, we obtain that $C_d:=\\lbrace {x^m_j-x^m_{j-1}} \\rbrace ^{1/ \\sigma } \\left( u(x^m_j)- u(x^m_{j-1}) \\right)$ remains moderate as time evolves for each $j=1,2,\\cdots , N_m$ .", "Therefore, we set the tolerance to be $M_{tol}^1=Tol_1 \\cdot \\max _j \\lbrace (\\Delta x^0_j)^{1/\\sigma } \\cdot |u^0_{j+1}-u^0_j| \\rbrace ,$ where $Tol_1$ is the constant we choose at $t=t_0$ (e.g., $Tol_1=2, 2.5$ or 5).", "This criterion is focused on the size of the quantity $u^m_{j+1}-u^m_{j}$ .", "As the solution reaches higher and higher amplitudes, we refine the grid and insert more points, in particular, to avoid the under-resolution issue.", "In a similar way, we set $M_{tol}^2=Tol_2 \\cdot \\max _j\\lbrace (\\Delta x^0_j)^{1/\\sigma } \\cdot |u^0_{j+1}+u^0_j| \\rbrace ,$ where $Tol_2$ is the constant we choose at the initial time $t=t_0$ (e.g., $Tol_2=0.5$ or 1).", "At each time step $t_m$ , we compute the quantities $\\gamma _j^m= (\\Delta x^m_j)^{1/\\sigma } \\cdot |u^{m}_{j+1}-u^{m}_j|$ and $\\eta _j^m= (\\Delta x^m_j)^{1/\\sigma } \\cdot |u^{m}_{j+1}+u^{m}_j|$ on each interval $[x_{j}^m,x_{j+1}^m]$ .", "If at time $t=t_m$ we have $\\gamma _j^m>M_{tol}^1$ , or $\\eta _j^m>M_{tol}^2$ for some $j$ 's, we divide the $j$ th interval $[x_{j}^m,x_{j+1}^m]$ into two sub-intervals $[x_{j}^m,x_{j+\\frac{1}{2}}^m]$ and $[x_{j+\\frac{1}{2}}^m,x_{j+1}^m]$ .", "Then, the new value $u^m_{j+\\frac{1}{2}}$ is needed.", "We discuss the strategy for obtaining $u^m_{j+\\frac{1}{2}}$ with the mass-preserving property in the next subsection.", "After using this midpoint refinement, we continue our time evolution to the next time step $t_{m+1}$ ." ], [ "Mass-conservative interpolation in the refinement", "Recall that when the tolerance is not satisfied at the $j^{th}$ interval, we refine the mesh by dividing that interval into two sub-intervals, and hence, we need an interpolation to find the new value of $u^m_{j+\\frac{1}{2}}$ at the point $x^m_{j+\\frac{1}{2}}= \\frac{1}{2}(x^m_j+x^m_{j+1})$ .", "A classical approach is to apply a linear interpolation (as, for example, in [11]): $u^m_{j+\\frac{1}{2}}=\\frac{1}{2}(u^m_j+u^m_{j+1}).$ When we add this middle point, the length of each interval $[x^m_j, x^m_{j+\\frac{1}{2}}]$ and $[x^m_{j+\\frac{1}{2}}, x^m_{j+1}]$ simply becomes $\\frac{1}{2} \\Delta x^m_j$ .", "Unfortunately, this widely used linear interpolation does not conserve the discrete mass.", "Indeed, let the discrete mass at the $j^{th}$ interval before the mesh refinement be $M_j= \\frac{1}{4}\\left[ |u^m_j|^2( \\Delta x^m_j+\\Delta x^m_{j-1} )+|u^m_{j+1}|^2( \\Delta x^m_{j+1}+\\Delta x^m_j ) \\right],$ and the mass after the mesh refinement be defined as $\\tilde{M}_j=\\frac{1}{4} \\left[ |u^m_j|^2\\Big (\\frac{1}{2} \\Delta x^m_j +\\Delta x^m_{j-1} \\Big ) + |u^m_{j+\\frac{1}{2}}|^2 \\Delta x^m_j+|u^m_{j+1} |^2\\Big (\\frac{1}{2} \\Delta x^m_j +\\Delta x^m_{j+1} \\Big ) \\right].$ Then a simple computation shows that $M_j - \\tilde{M}_j= \\frac{1}{4} |u^m_j-u^m_{j+1}|^2 \\Delta x^m_j.$ Hence, $\\tilde{M}_j<M_j$ on some subset of $\\Omega $ (where the random variables $u^m_j$ and $u^m_{j+1}$ differ), which is a non-empty set.", "In this linear interpolation, we suffer a loss of mass at each step of the mesh-refinement procedure.", "In another popular interpolation, via the cubic splines, a similar analysis shows that the scheme suffers the increase of mass at each step of the mesh-refinement procedure.", "To avoid these two problems, we proceed as follows.", "We set the two quantities (REF ) and (REF ) to be equal to each other, i.e., $M_j=\\tilde{M}_j$ , by solving this equation with the fact that $x^m_{j+1}-x^m_{j+\\frac{1}{2}}=x^m_{j+\\frac{1}{2}}-x^m_j=\\frac{1}{2} \\Delta x^m_j$ , we obtain $| u^m_{j+\\frac{1}{2}}|^2=\\frac{1}{2}\\left( |u^m_j|^2+|u^m_{j+1}|^2 \\right).$ Figure: Quadratic interpolation in () to obtain u j+1 2 m u^m_{j+\\frac{1}{2}} (index mm is omitted).To implement the condition (REF ), one choice is to set ${\\left\\lbrace \\begin{array}{ll}\\operatorname{Re}(u^m_{j+\\frac{1}{2}})=\\sqrt{\\frac{1}{2}\\left[ \\operatorname{Re}(u^m_j)^2+ \\operatorname{Re}(u^m_{j+1})^2 \\right] } \\, \\mathrm {sgn} \\big (\\operatorname{Re}(u^m_j)+\\operatorname{Re}(u^m_{j+1}) \\big ), \\\\\\operatorname{Im}(u^m_{j+\\frac{1}{2}})=\\sqrt{\\frac{1}{2}\\left[ \\operatorname{Im}(u^m_j)^2+ \\operatorname{Im}(u^m_{j+1})^2 \\right] } \\, \\mathrm {sgn} \\big (\\operatorname{Im}(u^m_j)+\\operatorname{Im}(u^m_{j+1}) \\big ).\\end{array}\\right.", "}$ This is what we use in our simulations.", "We next describe the steps of our full numerical algorithm." ], [ "The algorithm", "The full implementation of our algorithm proceeds as follows: 1.", "Discretize the space in the uniform mesh and set up the values of tolerance $Tol_1$ and $Tol_2$ .", "2.", "Apply the mass-conservative numerical schemes (REF ), (REF ) or (REF ) for the time evolution from $u^{m-1}$ to reach $u^m$ with the time step size $\\Delta t_{m-1}$ .", "3.", "At $t=t_m$ , change the time step size by $\\Delta t_{m}=\\frac{\\Delta t_0}{\\Vert u(\\cdot ,t_m) \\Vert _{\\infty }^{2 \\sigma }}$ for the next time evolution (thus, $t$ never reaches the blow-up time $T$ , in case there is a blow-up).", "4.", "If the solution meets the tolerance ($Tol_1$ or $Tol_2$ ) on some intervals $[x_j,x_{j+1}]$ , we divide those intervals into two sub-intervals.", "5.", "Apply the mass-conservative interpolation (REF ) to obtain the value of $u^m_{j+\\frac{1}{2}}$ .", "6.", "Continue with the time evolution to $t=t_{m+1}$ by applying (REF ), (REF ) or (REF ).", "A few remarks are due.", "First of all, this algorithm is applicable in the deterministic case.", "To our best knowledge, this is the first mesh-refinement numerical algorithm that conserves the discrete mass exactly before and after the refinement, which is especially important when simulating the finite time blow-up in the 1D focusing nonlinear Schrödinger equation with or without stochastic perturbation.", "Moreover, in the deterministic and multiplicative noise cases the discrete mass is conserved from the initial to terminal times.", "We note that in studying and simulating the blow-up solutions in the (deterministic) NLS equation, the dynamic rescaling or moving mesh methods are used (since solutions have some regularity); however, in the stochastic setting, those methods are simply not applicable because noise destroys regularity in the space variable.", "Secondly, its full implementation is needed for solutions that concentrate locally or blow up in finite time, where the refinement and mass-conservation are crucial features to ensure the reliability of the results.", "However, the algorithm is also applicable in the cases where the solution exists globally or long enough for numerical simulations.", "Indeed, if we start with the uniform mesh and remove the steps (1), (3), (4) and (5), it becomes a widely used second order numerical scheme for studying the NLS equation (in both deterministic and stochastic cases) without considering the singular solutions.", "When investigating solutions, which do not form singularities (exist globally in time or on sufficiently long time interval), the procedures (1), (3), (4) and (5) are not necessary and we omit them.", "When studying the blow-up solutions (in Section ), we incorporate fully all steps in order to obtain satisfactory results.", "When testing our simulations of blow-up solutions, not only the error of the discrete mass $\\mathcal {E}_1^m[M]$ from (REF ) a is checked, but also the discrepancy of the actual mass, approximated by the composite trapezoid rule at each time step, is checked, that is, $\\mathcal {E}_2^m[M]=\\max _m \\left\\lbrace M_{\\mathrm {app}}[u^m] \\right\\rbrace -\\min _m \\left\\lbrace M_{\\mathrm {app}}[u^m] \\right\\rbrace , ~~\\mbox{where}$ $M_{\\mathrm {app}}[u^m]=\\frac{1}{2}|u_0^m|^2 \\Delta x_0+\\sum _{j=1}^{N-2} |u_j^m| \\Delta x_j+\\frac{1}{2}|u_N^m|^2 \\Delta x_{N-1} .$ For this test, we choose $u_0=1.05Q$ and consider only the $L^2$ -critical case ($\\sigma =2$ ), comparing $\\epsilon =0$ (deterministic case) with $\\epsilon =0.1$ (multiplicative noise case).", "The initial spatial step-size is set to $\\Delta x=0.01$ , and the initial temporal step-size is set to $\\Delta t_0=\\Delta x/4$ .", "We take the computational domain to be $[-L_c,L_c]$ with $L_c=5$ .", "Figure REF shows the dependence of $\\mathcal {E}_1^m$ and $\\mathcal {E}_2^m$ on the focusing scaling parameter $L(t)=\\frac{1}{\\Vert u(t)\\Vert _{\\infty }^{\\sigma }}$ .", "Figure: The error of discrete and actual masses ℰ 1 m \\mathcal {E}_1^m and ℰ 2 m \\mathcal {E}_2^m for the L 2 L^2-critical case with or without the multiplicative noise.", "Left: ϵ=0\\epsilon =0.", "Right: ϵ=0.1\\epsilon =0.1.Observe that both the discrete mass and approximation of the actual mass are conserved well even when the focusing parameter reaches $\\sim 10^{-12}$ .", "Such high precision in mass conservation justifies well the efficiency of our schemes.", "We also tested other types of initial data (e.g., gaussian data $u_0=A \\, e^{-x^2}$ ), different noise strength ($\\epsilon =0.2,0.5$ ) and the supercritical power of nonlinearity ($\\sigma = 3$ ); the precision is similar to that shown in Figure REF .", "In the next two sections we discuss global behavior of solutions, showing how solitons behave for various nonlinearities (Section ), and then investigate the formation of blow-up (Section ) including our findings on profiles, rates and localization." ], [ "Numerical simulations of global behavior of solutions", "We again consider initial data of type $u_0 = A\\, Q$ , where $A>0$ and $Q$ is the ground state (REF ).", "In the deterministic setting one would consider two cases for numerical simulations, namely, $A<1$ (which guarantees the global existence and $A>1$ (which could be used to study blow-up solutions).", "In the stochastic setting we use similar data; however, as we will see (in Table REF ), we may not know a priori if the solution is global or blows up in finite time (a.s. or with some positive probability).", "For example, the condition $A<1$ does not necessarily guarantee global existence, or even sufficiently long (for numerical simulations) time existence as can be seen in Tables REF and REF .", "We consider additive noise first.", "Putting sufficiently large $\\epsilon $ and tracking for a sufficiently long time, we observe that small data leads to blow-up for the cases $\\sigma =2$ and $\\sigma =3$ .", "Figure: Additive noise, ϵ=0.1\\epsilon =0.1, u 0 =0.5Qu_0=0.5Q, L 2 L^2-critical case.", "The solution grows in time until the fixed point iteration fails.", "Bottom: time dependence of ∥u∥ ∞ \\Vert u\\Vert _{\\infty }, mass and energy.For example, in Figure REF , we take $u_0=0.5Q$ (far below the deterministic threshold) with sufficiently strong noise $\\epsilon =0.1$ and run for (computationally) long time: the fixed point iteration for solving the MEC scheme (REF ) fails to converge after 2000 iterations at time $t \\approx 19.485$ , which indicates that $u^{m+1}$ is far from $u^m$ at $t_m \\approx 19.485$ .", "The numerical scheme can not be run any further, and this is typically considered as the indication of the blow-up formation (see below comparison with the $L^2$ -subcritical case).", "Figure REF shows that the additive noise can create blow-up in finite time.", "In other words, the initial data, which in the deterministic case were to produce a globally existing scattering solution, in the additive forcing case could evolve towards the blow-up.", "This is partially due to the fact that the additive noise makes the mass and energy grow in time; see the bottom subplots in Figure REF , where both mass and energy grow linearly in time.", "Note that we start with a single soliton profile with a small amplitude ($0.5\\,Q$ ) and eventually the noise destroys the soliton profile with the growing $L^\\infty $ norm (left bottom subplot in Figure REF ).", "Figure: Additive noise, ϵ=0.1\\epsilon =0.1, u 0 =1.5Qu_0=1.5Q, L 2 L^2-subcritical case (σ=1\\sigma =1).Top: time evolution of |u(x,t)||u(x,t)|.", "Bottom: time dependence of ∥u(t)∥ ∞ \\Vert u(t)\\Vert _{\\infty }, mass and energy.It is also interesting to compare this behavior with the $L^2$ -subcritical case ($\\sigma =1$ ), where in the deterministic case all solutions are global (there is no blow-up for any data), see [8].", "Figure REF shows time evolution of the initial condition $u_0=1.5Q$ with the strength of the additive noise $\\epsilon =0.1$ (same as in Figure REF ).", "While the soliton profile is distinct for the time of the computation, it is obviously getting corrupted by noise: the $L^\\infty $ norm is slowly increasing with some wild oscillations.", "One can also observe that mass and energy grow linearly to infinity (as $t \\rightarrow \\infty $ ); see bottom plots of Figure REF .", "Note that while there is growth of mass and energy, as well as the $L^\\infty $ norm in this subcritical case, the fixed point iteration does not fail, indicating that there is no blow-up.", "For comparison we also show the influence of smaller noise $\\epsilon =0.05$ on a larger time scale ($0<t<50$ ) for the initial condition $u_0=Q$ ; see Figure REF .", "The smaller noise also seem to destroy the soliton with slow increase of the $L^\\infty $ norm and linearly growing mass and energy; however, the solution exists globally in time.", "Figure: Additive noise, ϵ=0.1\\epsilon =0.1, u 0 =Qu_0=Q, L 2 L^2-subcritical case (σ=1\\sigma =1).", "Top: time evolution of |u(x,t)||u(x,t)|.", "Bottom: time dependence of ∥u(t)∥ L ∞ \\Vert u(t)\\Vert _{L^\\infty }, mass and energy.Returning to the $L^2$ -critical and supercritical SNLS, we have seen that even small initial data can lead to blow-up.", "Therefore, we next compute the percentage of solutions that blow up until some finite time (e.g., $t=5$ ).", "We run $N_t = 1000$ trials to track solutions for various values of magnitude $A$ in the initial data $u_0 = A\\, Q$ , with $A$ close to 1.", "In Table REF we show the percentage of finite time blow-up solutions in the $L^2$ -critical case ($\\sigma =2$ ) with an additive noise ($\\epsilon = 0.01, 0.05, 0.1$ ): we take $A = 0.95, 1$ and $1.05$ (in the deterministic case these amplitudes would, respectively, lead to a scattering solution, a soliton, and a finite-time blow-up).", "Observe that blow-up occurs for $t<5$ even when $A=0.95 < 1$ with strong enough noise (e.g., when $\\epsilon =0.1$ , we get $98.4\\%$ of all solutions blow up in finite time; with $\\epsilon =0.05$ , we get $2.8\\%$ blow-up solutions, see Table REF ).", "This is in contrast with multiplicative noise as well as with the deterministic case in the $L^2$ -critical setting.", "Table: Additive noise.", "Percentage of blow-up solutions with initial data u 0 =AQu_0=A Q in the L 2 L^2-critical case (σ=2\\sigma =2) with N t =1000N_t=1000 trials and running time 0<t<50<t<5.Table REF shows the percentage of blow-up solutions in the $L^2$ -supercritical case ($\\sigma =3$ ) with additive noise.", "As in the $L^2$ -critical case, solutions with an amplitude below the threshold (e.g., $A=0.95$ ) can blow up in finite time (here, before $t=5$ ) with an additive noise of larger strength (for example, when $\\epsilon = 0.05$ , $3\\%$ of our runs blow up in finite time; for $\\epsilon = 0.1$ it is 98.6%).", "Table: Additive noise.", "Percentage of blow-up solutions with initial data u 0 =AQu_0=A Q in the L 2 L^2-supercritical case (σ=3\\sigma =3) with N t =1000N_t=1000 trials and running time 0<t<50<t<5.The effect of driving a time evolution into the blow-up regime (or in other words, generating a blow-up in the cases when a deterministic solution would exist globally and scatter) might be more obvious in the additive case, since the noise simply adds into the evolution and does not interfere with the solution.", "What happens in the multiplicative case, since the noise is being multiplied by the solution, is less obvious.", "Therefore, for completeness we mention the number of blow-up solutions we observe with $A<1$ in the multiplicative case.", "We tested the $L^2$ -supercritical case with $\\epsilon =0.1$ for a multiplicative perturbation, and observed the following: for $\\sigma =3$ , $u_0=0.99\\, Q$ , the number $N_t = 50\\, 000$ trial runs produced 2 blow-up trajectories.", "Thus, while the probability of (specific) finite time blow-up is extremely small (in this case it is 0.004%), it is nevertheless positive.", "The positive probability of blow-up in the $L^2$ -supercritical case is consistent with theoretical results of de Bouard and Debussche [9], which showed that in such a case any data will lead to blow-up in any given finite time with positive probability.", "In the $L^2$ -critical case it was shown in [32] that if $\\Vert u_0\\Vert _{L^2}<\\Vert Q\\Vert _{L^2}$ , then in the multiplicative (Stratonovich) noise case, the solution $u(t)$ is global, thus, no blow-up occurs.", "We tested the initial condition $u_0=0.99\\, Q$ , $\\epsilon =0.1$ (same as in the $L^2$ -supercritical case), and ran again $N_t = 50\\, 000$ trials.", "In all cases we obtained scattering behavior (or no blow-up trajectories), thus, confirming the theory.", "We next show how the blow-up solutions form and their dynamics in both cases of noise." ], [ "Blow-up dynamics", "In this section, we study the blow-up dynamics and how it is affected by the noise.", "We continue applying the numerical algorithms introduced in Section .", "We start with the $L^2$ -critical case and then continue with the $L^2$ -supercritical case.", "We first observe that, as the blow-up starts forming, there is less and less effect of the noise on the blow-up profile, and almost no effect on the the blow-up rate.", "However, we do notice that the noise disturbs the location of the blow-up center for different trial runs.", "In order to better understand the blow-up behavior (and track profile, rate, location), we run 1000 simulations and then average over all runs.", "For the location of the blow-up, we show the distribution of the location of the blow-up center shifts and its dependence on the number of runs.", "When using a very large number of trials, we obtain a normal distribution, see Figures REF and REF .", "For more details on the blow-up dynamics in the deterministic case we refer the reader to [39], [40], [35], [16]." ], [ "The $L^2$ -critical case", "We first consider the quintic NLS ($\\sigma =2$ ) and $\\epsilon =0$ (deterministic case), and then $\\epsilon =0.01, 0.05$ and $0.1$ with a multiplicative noise.", "We use generic Gaussian initial data ($u_0 =A e^{-x^2}$ ) as well as the ground state data ($u_0 = A Q$ ).", "Figure REF shows the blow-up dynamics of $u_0 = 3 e^{-x^2}$ with $\\epsilon =0.1$ .", "Observe that the solution slowly converges to the rescaled ground state profile $Q$ .", "Figure: Multiplicative noise, ϵ=0.1\\epsilon =0.1.", "Formation of blow-up in the L 2 L^2-critical case (σ=2\\sigma = 2): snapshots of a blow-up solution(given in pairs of actual and rescaled solution) at different times.", "Each pair of graphs shows in blue the actual solution |u||u| and its rescaled solutionL 1/σ |u|L^{1/\\sigma }|u|, which is compared to the absolute value of the normalized ground state solution e it Qe^{i t}Q in dashed red.Similar convergence of the profiles for other values of $\\epsilon $ is observed (we also tested $\\epsilon =0.01$ and $0.05$ , and compared with our deterministic work $\\epsilon =0$ in [39]).", "The last (right bottom) subplot on Figure REF shows that indeed the profile of blow-up approaches the rescaled $Q$ , however, one may notice that it converges slowly (compare this with the supercritical case in Figure REF ).", "This confirms the profile in Conjecture REF .", "We next study the rate of the blow-up by checking the dependence of $L(t)$ on $T-t$ .", "In Figure REF we show the rate of blow-up on the logarithmic scale.", "Note that the slope in the linear fitting in each case is $\\frac{1}{2}$ , thus, confirming the rate in Conjecture REF , $\\Vert \\nabla u(t)\\Vert _{L^2} \\sim \\left( T-t \\right)^{-\\frac{1}{2}}$ , possibly with some correction terms.", "This is similar to the deterministic $L^2$ -critical case; see more on that in [35] and [39].", "Figure: Multiplicative noise, L 2 L^2-critical case.", "The fitting of the rate L(t)L(t)v.s.", "(T-t)(T-t) on a log scale.", "The values of the noise strength ϵ\\epsilon are 0 (top left), 0.010.01 (top right), 0.050.05 (bottom left), 0.10.1 (bottom right).", "Observe that in all cases the linear fitting gives the slope 0.500.50.To provide a justification towards the claim that the correction in the stochastic perturbation case is also of a log-log type, see (REF ), we track similar quantities as we did in the dynamic rescaling method for the deterministic NLS-type equations; see [39], [40], [41].", "We track the quantity $a(t) =-LL_t$ , or equivalently, in the the rescaled time $\\tau =\\int _0^{t} \\frac{1}{L(s)^2} ds$ (or $\\frac{d\\tau }{dt}=\\frac{1}{L^2(t)}$ ), we have $a(\\tau )=-\\frac{L_{\\tau }}{L}$ .", "In the discrete version, by setting $\\Delta \\tau =\\Delta t_0$ , we get $\\tau _m=m \\cdot \\Delta t_0$ as a rescaled time.", "Consequently, at the $m$ th step we have $L(\\tau _m)$ , $u(\\tau _m)$ , and $a(\\tau _m)$ .", "As in [35], [16], [39], the parameter $a$ can be evaluated by setting $L(t)=\\left(1/\\Vert \\nabla u(t)\\Vert _{L^2}\\right)^{\\frac{2}{\\alpha }}$ with $\\alpha =1+\\frac{2}{\\sigma }=2-2s$ , since $s=\\frac{1}{2}-\\frac{1}{\\sigma }$ .", "Then, similar to [35] we get $a(t) =- \\frac{2}{\\alpha } \\frac{1}{( \\Vert \\nabla u(t)\\Vert _{L^2}^2)^{{\\frac{2}{\\alpha }+1}} }\\int |u|^{2\\sigma }\\operatorname{Im}(u_{xx}\\bar{u}) dx.$ Here, we specifically write a more general statement in terms of the dimension $d$ and nonlinearity power $\\sigma \\searrow 2$ , since the convergence of those parameters down to $d=1$ and $\\sigma =2$ is crucial in determining the correction in the blow-up rate (see more in [39]), as well as the value of $a(\\tau )$ for the profile identification in the supercritical case.", "The integral in (REF ) is evaluated by the composite trapezoid rule.", "Figure REF shows the dependence of the parameter $a$ with respect to $\\log L$ for a single trajectory (in dotted red) and for the averaged value over 2400 runs (in solid blue) on the left subplot (the strength of the multiplicative noise is $\\epsilon =0.1$ ).", "Observe that a single trajectory gives a dependence with severe oscillations due to noise in the beginning, but eventually smoothes out and converges to the average value as it approaches the blow-up time $T$ .", "This matches our findings in Figure REF , where eventually the blow-up profile becomes smooth.", "The right subplot shows the linear fitting for $a(\\tau )$ versus $1/\\ln (\\tau )$ .", "One may notice small oscillations in the blue curve: perhaps with the increase of the number of runs, the blue curve could have smaller and smaller oscillations, and would eventually approach a (yellow) line).", "We show one trajectory dependence in dotted red, the averaged value in solid blue and the linear fitting in solid yellow.", "This gives us first confirmation that the correction term is of logarithmic order.", "As in the deterministic case, we suspect that the correction is a double logarithm; however, this will require further investigations, which are highly nontrivial (even in the deterministic case).", "The above confirms Conjecture REF up to one logarithmic correction.", "Figure: Multiplicative noise, ϵ=0.1\\epsilon =0.1, L 2 L^2-critical case.", "Left: aa vs. log(L)\\log (L).", "Right: linear fitting for a(τ)a(\\tau ) vs. 1/ln(τ)1/\\ln (\\tau )." ], [ "Blow-up location", "So far we exhibited similarities in the blow-up dynamics between the multiplicative noise case and the deterministic case.", "A feature, which we find different, is the location of blow-up.", "We observe that the blow-up core, to be precise the spatial location $x_c$ of the blow-up center, shifts away from the zero (or rather wonders around it) for different runs.", "We record the values $x_c$ of shifts and plot their distribution in Figure REF for various values of $\\epsilon $ and for different number of trials $N_t$ to track the dependence.", "Our first observation is that the center shifts further away from zero when the strength of noise $\\epsilon $ increases.", "Secondly, we observe that the shifting has a normal distribution (see the right bottom subplot with the maximal number of trials in Figure REF ).", "The mean of this distribution approaches 0 when the number of runs $N_t$ increases.", "We record the variance of the shifts for different $\\epsilon $ 's and $\\sigma $ 's in Table REF .", "The variance seems to be an increasing function of the strength of the noise, which confirms our first observation above.", "In the same Table, we also record the $L^2$ -supercritical case that is discussed later.", "Figure: Multiplicative noise, ϵ=0.1\\epsilon =0.1, L 2 L^2-critical case.", "Left: shifts x c x_c of the blow-up center for different noise strength ϵ\\epsilon with the fixed N t =1000N_t=1000 number of runs.", "Right: dependence of shifts on the number of runs N t N_t for the same ϵ=0.1\\epsilon =0.1;observe that it approaches the normal distribution as the number of runs increases.Table: Multiplicative noise.", "The variance of the blow-up center shifts x c x_c in N t =1000N_t=1000 trials, see also Figure .In the case of an additive noise we obtain analogous results; for brevity we only include Figure REF to show convergence of the profiles, the other features remain similar and we omit them.", "Figure: Additive noise, ϵ=0.1\\epsilon =0.1.Formation of blow-up in the L 2 L^2-critical case (σ=2\\sigma =2): snapshots of a blow-up solution at different times.We conclude that in the $L^2$ -critical case, regardless of the type of stochastic perturbation (multiplicative or additive) and the strength (different values of $\\epsilon $ ) of the noise, the solution always blows up in a self-similar regime with the rescaled profile of the ground state $Q$ and the square root blow-up rate with the logarithmic correction, thus, confirming Conjecture REF ." ], [ "The $L^2$ -supercritical case", "In the $L^2$ -supercritical case we consider the septic NLS equation ($\\sigma =3$ ) as before with multiplicative or additive noise.", "We use either Gaussian-type initial data $u_0= A \\,e^{-x^2}$ or a multiple of the ground state solution $u_0=AQ$ , where $Q$ is the ground state solution with $\\sigma =3$ in (REF ).", "We consider the multiplicative noise of strength $\\epsilon = 0.01, 0.02$ and $0.1$ and investigate the blow-up profile.", "For the initial data $u_0=3\\,e^{-x^2}$ Figure REF shows the solution profiles at different times for $\\epsilon =0.1$ .", "The two main observations are: (i) the solution smoothes out faster compared to the $L^2$ -critical case (see Figure REF ); (ii) it converges to a self-similar profile very fast.", "To confirm this we compare the bottom right subplots in both Figure REF and Figure REF : in the supercritical case the profile of the rescaled solution (in solid blue) practically coincides with the absolute value of the re-normalized $Q \\equiv Q_{1,0}$ (in dashed red); this is similar to the deterministic case.", "Figure: Multiplicative noise, ϵ=0.1\\epsilon =0.1.", "Formation of blow-up in the L 2 L^2-supercritical case (σ=3\\sigma =3): snapshots at different times:the actual solution (blue) compared to the rescaled profile Q 1,0 Q_{1,0} (red).", "Note a visibly perfect match in the last right bottom subplot.Tests of other data and various values of $\\epsilon $ show that all observed blow-up solutions converge to the profile $Q_{1,0}$ .", "In Figure REF we show the linear fitting for the log dependence of $L(t)$ vs. $(T-t)$ , which gives the slope $\\frac{1}{2}$ .", "Note that even one trajectory fitting is very good.", "Further justification of the blow-up rate is done by checking the behavior of the quantity $a(\\tau )$ from (REF ).", "Figure REF shows that the quantity $a(\\tau )$ converges to a constant very fast (comparing with the decay to zero of $a(\\tau )$ in the $L^2$ -critical case in Figure REF ).", "Since $a(t) \\rightarrow a$ , a constant, we have $a=-LL_t$ and solving this ODE (with $L(T)=0$ ) yields $L(t)=\\sqrt{2a(T-t)}.$ Recall that $L(t)=\\left(1/\\Vert \\nabla u(t)\\Vert _{L^2}\\right)^{\\frac{2}{\\alpha }}$ , or equivalently, $L(t)=1/\\Vert u(t)\\Vert _{\\infty }^{\\sigma }$ , thus, we have the blow-up rate (REF ) for the super-critical case, or equivalently, $\\Vert u(t)\\Vert _{\\infty }=\\left(2a(T-t) \\right)^{-\\frac{1}{2\\sigma }} \\,\\, \\mbox{as} \\,\\, t\\rightarrow T, $ in the case when we evaluate the $L^{\\infty }$ norm.", "This indicates that solutions blows up with the pure power rate without any logarithmic correction, similar to the deterministic case (for details see [3], [40]).", "Figure: Multiplicative noise, L 2 L^2-supercritical case.", "A linear fitting of the rate L(t)L(t)v.s.", "(T-t)(T-t) on log scale.", "The values of the noise strength ϵ\\epsilon are 0 (top left), 0.010.01 (top right), 0.050.05 (bottom left), 0.10.1 (bottom right); the linear fitting gives 0.50 slope.Figure: Multiplicative noise, L 2 L^2-supercritical case, ϵ=0.1\\epsilon =0.1.", "Left: aa v.s.", "log(L)\\log (L), the focusing level.", "Right: numerical confirmation of the blowup rate ∥u(t)∥ ∞ =2a(T-t) -1 2σ \\Vert u(t)\\Vert _{\\infty }=\\left(2a(T-t) \\right)^{-\\frac{1}{2\\sigma }} (the limit has stabilized at 1).In the $L^2$ -supercritical case we also observe shifting of the blow-up center, show the distribution of shifts $x_c$ in the multiplicative noise; in particular, these random shifts have a normal distribution similar to the $L^2$ -critical case.", "The variance of shifts is shown in Table REF .", "Note that stronger noises (that is, larger values of $\\epsilon $ ) yield a larger shift away from the origin.", "Furthermore, comparing Figure REF with Figure REF , we find that the $L^2$ -supercritical case produces slightly larger variance of shifts.", "In other words, we observe that higher power of nonlinearity creates a larger variance, that is the blow-up location is more spread out.", "Figure: Multiplicative noise, L 2 L^2-supercritical case.", "Left: distribution of shifts x c x_c of the blow-up center for different ϵ\\epsilon 's with N t =1000N_t=1000 runs.", "Right: as N t N_t increases, it becomes more evident that the spread out of the blow-up location satisfies a normal distribution.We obtained similar results in the additive noise: the blow up occurs in a self-similar way at the rate $ L(t) =(2a(T-t))^{\\frac{1}{2}}$ , and the solution profile converges to the profile $Q_{1,0}$ relatively fast, see Figure REF for profile convergence.", "The quantities $a(\\tau )$ , $L(\\tau )$ also behave similar to the multiplicative noise parameters (and also to the deterministic cases).", "This confirms Conjecture REF .", "Figure: Additive noise, ϵ=0.1\\epsilon =0.1.", "Convergence of the blow-up in the L 2 L^2-supercritical case (σ=3\\sigma =3); actual solution and its rescaled version (blue), the rescaled profile solution Q 1,0 Q_{1,0} (red)." ], [ "Conclusion", "In this work we investigate the behavior of solutions to the 1d focusing SNLS subject to a stochastic perturbation which is either multiplicative or additive, and driven by space-time white noise.", "In particular, we study the time dependence of the mass ($L^2$ -norm) and the energy (Hamiltonian) in the $L^2$ -critical and supercritical cases.", "For that we consider a discretized version of both quantities and an approximation of the actual mass or energy.", "In the deterministic case these quantities are conserved in time, however, it is not necessarily the case in the stochastic setting.", "In the case of a multiplicative noise, which is defined in terms of the Stratonovich integral, the mass (both discrete and actual) is invariant.", "However, in the additive case the mass grows linearly.", "The energy grows in time in both stochastic settings.", "We give upper estimates on that time dependence and then track it numerically; we observe that energy levels off when the noise is multiplicative.", "We also investigate the dependence of the mass and energy on the strength of the noise, on the spatial and temporal mesh refinements and the length of the computational interval.", "For the above we use three different numerical schemes; all of them conserve discrete mass in the multiplicative noise setting, and one of them conserves the discrete energy in the deterministic setting, though that scheme involves fixed point iterations to handle the nonlinear system, thus, taking longer computational time.", "We introduce a new scheme, a linear extrapolation of the above and Crank-Nickolson discretization of the potential term, which speeds up significantly our computations, since the scheme is linear, and thus, avoiding extra fixed point iterations while having tolerable errors.", "We also introduce a new algorithm in order to investigate the blow-up dynamics.", "Typically in the deterministic setting to track the blow-up dynamics, the dynamic rescaling method is used.", "We use instead a finite difference method with non-uniform mesh and then mesh-refinement with mass-conservative interpolation.", "With this algorithm we are able to track the blow-up rate, profile and we find a new feature in the blow-up dynamics, the shift of the blow-up center, which follows normal distribution for large number of trials.", "We note that our algorithm is also applicable for the deterministic NLS equation, in particular, it can replace the dynamic rescaling or moving mesh methods used to track blow-up.", "We confirm previous results of Debussche et al.", "[11], [7], [9] showing that the additive noise can amplify or create blow-up (we suspect that this happens almost surely for any data) in the $L^2$ -critical and supercritical cases.", "In the multiplicative noise setting the blow-up seems to occur for any (sufficiently localized) data in the $L^2$ -supercritical case, and above the mass threshold in the $L^2$ -critical case.", "Finally, when the noise is present, a solution is likely to travel away from the initial `center', and, once the solution starts blowing up, the noise plays no role in the singularity structure, and the blow-up occurs with the rate and profile similar to the deterministic setting." ] ]

[ [ "Task-Based Information Compression for Multi-Agent Communication\n Problems with Channel Rate Constraints" ], [ "Abstract {We investigate the communications design in a multiagent system (MAS) in which agents cooperate to maximize the averaged sum of discounted one-stage rewards of a collaborative task.", "Due to the limited communication rate between the agents, each agent should efficiently represent its local observation and communicate an abstract version of the observations to improve the collaborative task performance.", "We first show that this problem is equivalent to a form of rate-distortion problem which we call task-based information compression (TBIC).", "We then introduce the state-aggregation for information compression algorithm (SAIC) to solve the formulated TBIC problem.", "It is shown that SAIC is able to achieve near-optimal performance in terms of the achieved sum of discounted rewards.", "The proposed algorithm is applied to a rendezvous problem and its performance is compared with several benchmarks.", "Numerical experiments confirm the superiority of the proposed algorithm." ], [ "Introduction", "This paper considers a collaborative task problem composed of multiple agents with local observations, while agents are allowed to communicate through a rate limited channel.", "The global state process of the environment, generated by a Markov decision process (MDP), is controlled by the joint actions of the agents.", "Moreover, the instantaneous reward signal to which all agents have access, is influenced by the global state and agents' joint actions.", "On one hand, maximizing the finite-horizon sum of discounted rewards, considered to be the unique goal of the network of agents, spurs them to act collaboratively.", "On the other hand, limited observability of the environment encourages the agents to effectively communicate to each other to acquire a better estimate of the global state of the environment.", "Due to the limited rate of the communication channel between the agents, it is necessary for agents to compactly represent their observations in communication messages.", "As we ultimately measure the performance of the multiagent system in terms of cumulative rewards, the loss of information caused by the compact representation of the agents' observations needs to be managed in such a way that we minimally compromise the cumulative rewards.", "As such, this form of information compression which we call task-based information compression is different from conventional compression algorithms whose ultimate aim is to reduce the distortion between the original and compressed data.", "We assume rate-limited but error-free communications - a noisy channel where traditional channel coding is used to ensure error free communications with the achievable rate that is, in fact, the rate constraint of the channel.", "The considered problem is formulated in a general form which can be directly applied to numerous problems in telecommunications (, control) and computer science.", "Suitable applications for this framework are those that include multiple cooperative decision makers that have to communicate through an ad-hoc network.", "While device to device communications between the agents enable collaboration and optimal joint action selection, in the current work, the achievable communication rate for these links is considered to be limited.", "As an illustration, consider a task in which multiple static cameras, see e.g., [1], [2], which are distantly positioned, are required to identify and track a particular target in a remote area.", "The cameras are subjected to local observations and they can select local actions, e.g.", "zoom in/out, changing the tilt of the camera in vertical/horizontal directions.", "They receive the largest reward if the target is located with high enough visibility and is tracked at all times.", "Communication between these decision-makers helps to easily track the target or adopt their action selection strategies when the target is not located yet.", "The considered framework can also be directly applied to object tracking by Unmanned Area Vehicle networks or by multi-agent systems, e.g.", "[3], [4], in which multiple UAVs/agents collaboratively track one/several moving object(s).", "Accordingly, framework is applicable to multiagent networked control scenarios where the realistic constraint of the rate-limited communications is in place.", "Another application for our problem is the rendezvous problem, drawn from computer science community [5], [6], where multiple agents, e.g.", "autonomous robots, want to get into a particular location at precisely the same time.", "The agents are unaware of the initial locations of each other but are allowed to communicate through a rate limited communication channel.", "The team of agents is rewarded if they achieve the task of arrival to the goal point at the same time, and will be punished if any of them arrives earlier.", "The given examples fall in the general category of multi-agent reinforcement learning, which is used in the literature as an effective framework to develop coordinated policies [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].", "The distributed decision-making of multi-agent systems has been addressed in [7], [8], [9], while many other works are focused on multiagent communications to enhance the joint action selection in partially observed environments [14], [10], [12], [15], [16], [17], [11], [18], [13], [19].", "Here we elaborate on some papers with focus on multi-agent communication.", "The work done in [10], [11], [12], [13], [19] has addressed the coordination of multiple agents through a noise-free communication channel, where the agents follow an engineered communication strategy.", "In [13] the impact of stochastic delays in multiagent communication is considered on the multiagent coordination, while [19] considers event-triggered local communications.", "Deep reinforcement learning with communication of the gradients of the agents' objective function is proposed in [14] to learn the communication among multiple agents.", "In contrast to the above mentioned works, the presence of noise in the inter-agent communication channel is studied by [16] and the absence of dedicated communication channels by [20].", "Authors of [16] report the emergence of unequal error protection by reinforcement learning agents.", "Papers [16] and [15] have contributed to the rapidly emerging literature on machine learning for communications [21], [22], [23].", "Some metrics are introduced in [17] to measure the positive signaling and positive listening amongst agents which learn how to communicate [16], [15], [14].", "In the current work we develop a state aggregation algorithm which enables each agent to reduce the entropy of its generated communication messages while maintaining their performance in the collaborative task.", "Classical state aggregation algorithms have been often used to reduce the complexity of the dynamic programming problems over MDPs [24], [25] as well as Partially Observable MDPs [26].", "To the best of our knowledge, they have never been used to design a task-based information compression algorithm over an MDP.", "One similar work is [27], which studies a task-based quantization problem.", "In contrast to our work, the assumption there is that the parameter to be quantized is only measurable and cannot be controlled.", "In our problem, agents' observations stem from a generative process with memory, an MDP.", "Similarly in [28], the authors have introduced a gated mechanism so that reinforcement learning-aided agents reduce the rate of their communication by removing messages which are not beneficial for the team.", "However, their proposed approach mostly relies on numerical experiments.", "In contrast, this paper relies on analytical studies to design a multiagent communication policy which efficiently coordinates agents over a rate limited channel.", "Conventionally, the communication system design is disjoint from the distributed decision making design [10], [11], [12], [13], [14], [29].", "This work can also be interpreted as a demonstration of the potential of considering the joint design of the source coding (compression) together with the multi-agent action policy design.", "Herein, we show a particular approach of joint design under which the joint problem decouples conditionally.", "Our particular approach is based on an indirect data-driven design exploiting multi-agent reinforcement learning." ], [ "Contributions", "The contributions of this paper are as follows: Firstly, we develop a general cooperative multiagent framework in which agents interact over an MDP environment.", "Unlike the existing works which assume perfect communication links [5], [29], [14], [15], we assume the practical rate-limited communications between the agents.", "We formulate a two-agents cooperative problem where agents interact over an MDP and can communicate over a channel with limited achievable rate.", "Our goal is to derive the optimal action selection and communication strategies to maximize to joint objective function.", "Secondly, we consider a learning based information compression (LBIC) policy that provides efficient communications over the rate limited channel.", "The principle of LBIC is to carry out the compact representation of an agent's local observation, based on reinforcement learning.", "The proposed LBIC can be trained and executed in a fully decentralized fashion.", "Thirdly, we show that the decentralized cooperative multi-agent problem can be conditionally decoupled to two decentralized problems of action policy selection and communication policy selection.", "Afterwards, we analytically transform one of these problems (the decentralized communications policy selection problem) into a so-called task-based rate distortion problem (a special form of the rate distortion problem) for which we propose State Aggregation for Information Compression (SAIC) as a solution.", "By leveraging the knowledge of the solution to the centralized problem we convert the task-based information compression to a k-median clustering problem.", "Solving this K-median problem numerically would then allow us to find a conditionally optimal communications (compression) policy; and after obtaining the communications policies, distributed training of the action policies follows separately.", "Finally, extensive numerical comparisons are made between the performance of the SAIC, LBIC and two benchmark schemes in terms of the optimality of the objective function, for the rendezvous problem.", "One of these benchmark schemes, titled conventional communication, is the result of a theorem developed by [18], adapted to our problem.", "The second benchmark is the control of the multiagent system by a centralized controller which is assumed to have perfect communications with the agents and executes the Q-learning.", "Due to the full observability of the environment, this reference scheme provides an upper-bound to the numerical problem.", "It is shown that both proposed schemes are of significant advantages over CIC and that the SAIC achieves the optimal performance.", "Organization: Section II describes the system model for a cooperative multi-agent task with rate constrained inter-agent communications.", "In Section III, LBIC is detailed.", "A conditionally optimal communication strategy is derived in Section IV by decomposing the original problem and proposing SAIC to solve the problem of decentralized communication policy optimization.", "The numerical results and discussions are provided in Section V. Finally, Section VI concludes the paper.", "For the reader's convenience, a summary of the notation that we follow in this paper is given in Table REF .", "Table: Table of notationsBold font is used for matrices or scalars which are random and their realizations follows simple font.", "We consider a two-agent system, where at any time step $t$ each agent $i \\in \\lbrace 1,2\\rbrace $ makes a local observation ${\\mathsf {o}}_i(t) \\in \\Omega $ on environment while the true state of environment is ${\\mathsf {s}}(t) \\in \\mathcal {S}$ .", "The alphabets $\\Omega $ and $\\mathcal {S}$ define observation space and state space, respectively.", "The true state of the environment ${\\mathsf {s}}(t)$ is controlled by the joint actions ${\\mathsf {m}}_i(t), {\\mathsf {m}}_j(t) \\in \\mathcal {M}$ of the agents, where each agent $i$ can only choose its local action ${\\mathsf {m}}_i(t)$ which is selected from the local action space $\\mathcal {M}$ .", "The environment runs on discrete time steps $t = 1, 2, ..., M$ , where at each time step, each agent $i$ selects its domain level action ${\\mathsf {m}}_i(t)$ upon having an observation ${\\mathsf {o}}_i(t)$ of environment.", "Dynamics of the environment are governed by a conditional probability mass function $ & T\\big ({\\mathsf {s}}(t+1),{\\mathsf {s}}(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big )=\\\\& p\\big ({\\bf s}(t+1)={\\mathsf {s}}(t+1)|{\\bf s}(t)={\\mathsf {s}}(t),{\\bf m}_i(t)={\\mathsf {m}}_i(t),{\\bf m}_j(t)={\\mathsf {m}}_j(t)\\big ) ,$ which is unknown to the agents.", "$T(\\cdot )$ defines the transition probability of the environment to state ${\\mathsf {s}}(t+1)$ given the current state of environment ${\\mathsf {s}}(t)$ and the joint actions ${\\mathsf {m}}_i(t) , {\\mathsf {m}}_j(t)$ .", "We recall that domain level actions ${\\mathsf {m}}_i(t)$ can, for instance, be in the form of a movement or acceleration in a particular direction or any other type of action depending on the domain of the cooperative task.", "We consider a particular structure for agents' observations, referred to as collective observations in the literature [18].", "Under collective observability, individual observation of an agent provides it with partial information about the current state of the environment, however, having knowledge of the collective observations acquired by all of the agents is sufficient to realize the true state of environment.", "To further elaborate, at all time steps $t$ agents' observation processes ${\\bf o}_i(t),{\\bf o}_j(t)$ follow eq.", "(REF ) and eq.", "().", "Note that even in the case of collective observability, for agent $i$ to be able to observe the true state of environment at all times, it needs to have access to the observations of the other agent $j \\ne i$ through communications at all times.", "$ H\\big ({\\bf o}_i(t)\\big ) \\le H\\big ({\\bf s}(t)\\big ), \\;\\;\\; i \\in \\lbrace 1,2\\rbrace , \\\\H\\big ({\\bf o}_i(t),{\\bf o}_j(t)\\big ) = H\\big ({\\bf s}(t)\\big ), \\;\\;\\; j\\ne i .$ A deterministic reward function $r(\\cdot ): \\mathcal {S} \\times \\mathcal {M}^2 \\rightarrow \\mathbb {R}$ indicates the reward of both agents at time step $t$ , where the arguments of the reward function are the global state of the environment ${\\mathsf {s}}(t)$ and the domain-level actions ${\\mathsf {m}}_i(t) , {\\mathsf {m}}_j(t)$ of both agents.", "We assume that the environment over which agents interact can be defined in terms of an MDP determined by the tuple $\\big {\\lbrace } \\mathcal {S}, \\mathcal {M}^2, r(\\cdot ), \\gamma , T(\\cdot ) \\big {\\rbrace }$ , where $\\mathcal {S}$ and $\\mathcal {M}$ are discrete alphabets, $r(\\cdot )$ is a function, $T(\\cdot )$ is defined in (REF ) and the scalar $\\gamma \\in [0,1]$ is the discount factor.", "The focus of this paper is on scenarios in which the agents are unaware of the state transition probability function $T(\\cdot )$ and of the closed form of the function $r(\\cdot )$ .", "However we assume that, further to the literature of reinforcement learning [30], a realization of the function $r\\big ({\\mathsf {s}}(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big )$ will be accessible for both agents $i$ and $j$ at each time step $t$ .", "Since the tuple $\\big {\\lbrace } \\mathcal {S}, \\mathcal {M}^2, r(\\cdot ), \\gamma , T(\\cdot ) \\big {\\rbrace }$ is an MDP and the state process ${\\bf s}(t)$ is jointly observable by agents, the system model of this cooperative multiagent setting is also referred to as a decentralized MDP in the literature of multiagent decision making [31].", "In what follows two problems regarding the above mentioned setup is detailed.", "The main intention of this paper is to address the decentralized control and inter-agent communications for a system of multiple agents.", "However, the problem of centralized control of the system is formalized in subsection REF as we keep comparing the analytical and numerical results obtained by decentralized algorithms with the centralized algorithm.", "Moreover, the simpler nature and mathematical notations used for the centralized problem, allow the reader to have a smoother transition to the decentralized problem which is of more complex nature and notation." ], [ "Centralized Control", "We consider a scenario in which a central controller has instant access to the observations ${\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t)$ of both agents through a free (with no cost on the objective function) and reliable communication channel.", "Following the Eq.", "() the joint observations ${\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t)$ are representative of the true state of environment ${\\mathsf {s}}(t)$ at time $t$ .", "From the central controller's point of view, the environment is an MDP introduced by the tuples $\\Big {\\lbrace } \\Omega ^2 , \\mathcal {M}^2, r(\\cdot ), \\gamma , T(\\cdot ) \\Big {\\rbrace }$ .", "The goal of the centralized controller is to maximize the expected sum of discounted rewards.", "The expectation is computed over the joint PMF of the whole system trajectory ${\\bf s}(1),{\\bf m}_i(1),{\\bf m}_j(1) , ...,{\\bf s}(M),{\\bf m}_i(M),{\\bf m}_j(M)$ from time $t=1$ to $t=M$ , where this joint PMF is generated if agents follow policy $\\pi (\\cdot )$ , eq.", "(REF ), for their action selections at all times and ${\\bf s}(1) \\in \\mathcal {S}$ is randomly selected by the environment.", "To have a more compact notation to refer to the system trajectory, hereafter, we represent the realization of a system trajectory at time $t$ by ${\\mathsf {tr}}(t)$ which corresponds to the tuple $\\langle {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t), {\\mathsf {m}}_i(t), {\\mathsf {m}}_j(t) \\rangle $ and the realization of the whole system trajectory by $\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}$ .", "Accordingly, the problem boils down to a single agent problem which can be denoted by $& \\underset{\\pi (\\cdot )}{\\text{max}}& & \\!\\!\\mathbb {E}_{p_\\pi \\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )} \\!\\Big {\\lbrace }\\!\\sum _{t=1}^M \\gamma ^{t-1} r\\big ({\\bf o}_i(t),{\\bf o}_j(t),{\\bf m}_i(t),{\\bf m}_j(t)\\big )\\!\\Big {\\rbrace } \\\\& \\text{s.t.}", "& & \\!\\!", "p_{\\pi }\\big ({\\mathsf {s}}(t+1) |{\\mathsf {s}}(t) \\big ) = \\sum _{m_i \\in \\mathcal {M}} \\sum _{m_j \\in \\mathcal {M}} \\pi \\big ({\\mathsf {m}}_i(t) , {\\mathsf {m}}_j(t) | {\\mathsf {s}}(t)\\big ) \\\\& & & \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; T\\big ( {\\mathsf {s}}(t+1), {\\mathsf {s}}(t) , {\\mathsf {m}}_i(t) , {\\mathsf {m}}_j(t) \\big ),$ where the policy $\\pi $ can be expressed by $ \\pi \\!\\Big (\\!", "{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t) \\Big {|} {\\mathsf {s}}(t) \\!", "\\Big )\\!\\!", "= \\!p \\Big (\\!", "{\\bf m}_i(t)\\!=\\!", "{\\mathsf {m}}_i(t),{\\bf m}_j(t)\\!={\\mathsf {m}}_j(t) \\Big {|} {\\bf s}(t)\\!={\\mathsf {s}}(t)\\!", "\\Big ),$ and $p_{\\pi }\\big ({\\mathsf {s}}(t+1)|{\\mathsf {s}}(t)\\big )$ is the probability of transitioning from ${\\mathsf {s}}(t)$ to ${\\mathsf {s}}(t+1)$ when the joint action policy $\\pi (\\cdot )$ is executed by the central controller.", "Similarly, $p_\\pi \\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )$ is the joint probability mass function (PMF) of ${\\mathsf {tr}}(1), {\\mathsf {tr}}(2),...,{\\mathsf {tr}}(M)$ when the joint action policy $\\pi (\\cdot )$ is followed by the central controller.", "The constraint of the problem (REF ) shows the limitations in place regarding the dynamics of environment, e.g.", "following the particular policy $\\pi (\\cdot )$ transition between some states might be unlikely.", "On one hand, problem (REF ) can be solved using single-agent Q-learning [30] and the solution $\\pi ^{*} (\\cdot )$ obtained by Q-learning is guaranteed to be the optimal control policy, given some non-restricting conditions [32].", "On the other hand, the use-cases of the centralized approach are limited to the applications in which there is a permanent communication link with unlimited rate between the agents and the controller.", "Whereas these conditions are not met in many remote applications, where there is no communication infrastructure to connect the agents to the central controller.", "While the centralized algorithm provides us with a performance upper bound in maximizing the objective function (REF ), the aim of this paper is to introduce decentralized approaches with comparable performance, where the agents are limited to local observations and decision making." ], [ "Problem Statement", " Here we consider a scenario in which the same objective function explained in Eq.", "(REF ) needs to be maximized by the two-agent system in a decentralized fashion, Fig.", "REF .", "Namely, agents with partial observability can only select their own actions.", "To prevail over the limitations imposed by the local observability, agents are allowed to have direct (explicit) communications, and not indirect (implicit) communications [20], [33].", "However, the communication is done through a channel with limited rate $C = I \\big ( {\\bf c}_i(t), \\tilde{{\\bf c}}_i(t) \\big ),$ where ${\\bf c}_i(t) \\in \\mathcal {C}=\\lbrace -1 , 1\\rbrace ^B$ stands for the communication message generated by agent $i$ before being encoded by error correction codes and $\\tilde{{\\bf c}}_i(t) \\in \\lbrace -1 , 1\\rbrace ^B$ corresponds to the same communication message after it is decoded by the channel coding block, of the same error correction code used at the transmitter, at the agent $j$ .", "Note that instead of maximal achievable rate of the channel, $C$ represents the channel maximal (non-)asymptotic achievable rate with any known channel coding scheme (of arbitrary code length).", "It should be noted that the design of the channel coding and modulation schemes are beyond the scope of this paper and the main focus is on the compression of agents' generated communication messages.", "In this problem, the limited rate of the channel is accepted as a constraint which is imposed by the given channel coding, length of code-words, modulation scheme and the available bandwidth.", "Here we assume, the achievable rate of information exchange for both inter-agent communication channels to be equal to $C$ , i.e., the communication resources are split evenly amongst the two agents.", "In particular we consider $C$ to be time-invariant and to follow: $\\text{}{\\left\\lbrace \\begin{array}{ll}C < H\\big ({\\bf o}_i(t)\\big ),\\\\C < H\\big ({\\bf o}_j(t)\\big ).\\end{array}\\right.", "}$ Figure: An illustration of the decentralized cooperative multiagent system with rate-limited inter-agent communications.Also for the convenience of our notation we define the function ${\\bf g}(t^{^{\\prime }})$ as follows: $ {\\bf g}(t^{^{\\prime }})= \\sum _{t=t^{^{\\prime }}}^{M}\\gamma ^{t-1} r\\big ({\\bf o}_i(t),{\\bf o}_j(t),{\\bf m}_i(t),{\\bf m}_j(t)\\big ).\\vspace{-2.84526pt}$ Note that ${\\bf g}(t^{^{\\prime }})$ is random variable and a function of $t^{^{\\prime }}$ as well as the trajectory $\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=t^{^{\\prime }}}^{t=M}$ .", "Due to the lack of space, here we drop a part of arguments of this function.", "Accordingly, the decentralized problem is formalized as $& \\underset{\\pi ^m_i, \\pi ^c_i}{\\text{max }}& & \\mathbb {E}_{p_{\\pi ^m_i, \\pi ^c_i}\\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )} \\Big {\\lbrace }{\\bf g}(1)\\Big {\\rbrace },\\; \\; \\; \\; i \\in \\lbrace 1,2\\rbrace \\; , i \\ne j \\\\& \\text{s.t.}", "& & p\\big ({\\mathsf {s}}(t+1) |{\\mathsf {s}}(t) \\big ) = \\sum _{m_i \\in \\mathcal {M}} \\sum _{m_j \\in \\mathcal {M}} T\\big ( {\\mathsf {s}}(t+1), {\\mathsf {s}}(t) , {\\mathsf {m}}_i , {\\mathsf {m}}_j \\big ) \\\\& & & \\pi ^m_i \\big ({\\mathsf {m}}_i | {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_j(t)\\big ) \\pi ^m_j \\big ({\\mathsf {m}}_j | {\\mathsf {o}}_j(t),\\tilde{{\\mathsf {c}}}_i(t)\\big ),\\\\& & &I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C,$ where in its general form, the domain level action policy $\\pi ^m_i: \\mathcal {M}\\times \\mathcal {C} \\times \\Omega \\rightarrow [0,1]$ of each agent $i$ is defined as $ & \\pi ^m_i\\Big ({\\mathsf {m}}_i(t) \\Big {|} {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\Big ) = \\\\& \\mathrm {Pr} \\Big ({\\bf m}_i(t)={\\mathsf {m}}_i(t) \\Big {|} {\\bf o}_i(t)={\\mathsf {o}}_i(t),\\tilde{{\\bf c}}_j(t)=\\tilde{{\\mathsf {c}}}_j(t) \\Big ),$ and the communication policy $\\pi ^c_i : \\mathcal {C}^2 \\times \\Omega \\rightarrow [0,1]$ of each agent $i$ follows: $& \\pi ^c_i\\Big ({\\mathsf {c}}_i(t) \\Big {|} {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\Big ) = \\\\& \\mathrm {Pr} \\Big ({\\bf c}_i(t)={\\mathsf {c}}_i(t) \\Big {|} {\\bf o}_i(t)={\\mathsf {o}}_i(t),\\tilde{{\\bf c}}_j(t)=\\tilde{{\\mathsf {c}}}_j(t) \\Big ).$ Similar to the centralized problem, ${p_{\\pi ^m_i, \\pi ^c_i}\\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )}$ is the joint probability mass function of ${\\mathsf {tr}}(1), {\\mathsf {tr}}(2),...,{\\mathsf {tr}}(M)$ when each agent $i \\in \\lbrace 1,2\\rbrace $ follows the action policy $\\pi ^m_i(\\cdot )$ and the communication policy $\\pi ^c_i(\\cdot )$ and ${\\bf s}(1) \\in \\mathcal {S}$ is randomly selected by the environment.", "To make the problem more concrete, further to (REF ) and (REF ), here we assume the presence of an instantaneous communication between agents, which is in contrast to delayed communication models [16], [34].", "Fig.", "REF demonstrates this communication model.", "As such, each agent $i$ at any time step $t$ prior to the selection of its action ${\\mathsf {m}}_i(t)$ receives a communication message $\\tilde{{\\mathsf {c}}}_j(t)$ that includes some information about the observations of agent $j$ at time $t$ .", "Under the instantaneous communication scenario, the generation of the communication message ${\\mathsf {c}}_i(t)$ by an agent $i$ cannot be conditioned on the received communication message $\\tilde{{\\mathsf {c}}}_j(t)$ from agent $j$ , as it causes an infinite regress.", "Moreover, for agent $j$ there will be no new information in $\\tilde{{\\mathsf {c}}}_i(t) \\sim \\pi ^c_i\\Big ({\\mathsf {c}}_i(t) \\Big {|} {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\Big )$ given $\\tilde{{\\mathsf {c}}}_i(t) \\sim \\pi ^c_i\\Big ({\\mathsf {c}}_i(t) \\Big {|} {\\mathsf {o}}_i(t) \\Big )$ , as agent $j$ has already full access to its own observation ${\\mathsf {o}}_j(t)$ .", "Figure: Ordering of observation, communication and action selection for instantaneous communication concerning a UAV object tracking example, with 0<t ' <t '' <t ''' <10<t^{\\prime }<t^{\\prime \\prime }<t^{\\prime \\prime \\prime }<1.", "At time t=t 0 t=t_0 both agents (UAVs) make local observations on the environment.", "At time t=t 0 +t ' t=t_0+t^{\\prime } both agents select a communication signal to be generated.", "At time t=t 0 +t '' t=t_0+t^{\\prime \\prime } agents receive a communication signal from the other agent.", "At time t=t 0 +t ''' t=t_0+t^{\\prime \\prime \\prime } agents select a domain level action, here it can be the movement of UAVs or rotation of their cameras etc." ], [ "Learning Based Information Compression in Multiagent Coordination Tasks", "In this section we consider a strategy that jointly learns the communication and the environment action policies of both agents, with the aim of addressing problem (REF ).", "To this end we adapt the distributed Q-learning algorithm to the setup at hand [35].", "Accordingly, given the local observation ${\\mathsf {o}}_i(t)$ , each agent $i$ follows the communication policy $\\pi ^c_i(\\cdot )$ to select its communication message ${\\mathsf {c}}_i(t)$ , whereas in the same time step $t$ it receives a communication message $\\tilde{{\\mathsf {c}}}_j(t)$ generated by agent $j$ .", "After the receipt of the communication message $\\tilde{{\\mathsf {c}}}_j(t)$ each agent $i$ follows its policy $\\pi ^m_i(\\cdot )$ to select the domain level action ${\\mathsf {m}}_i(t)$ .", "The aim is to optimize the expected sum of rewards along two block coordinates, where these block coordinates are policies $\\pi ^m_i(\\cdot )$ and $\\pi ^c_i(\\cdot )$ described in subsection REF .", "Note that we can view the policy functions also as tensors, e.g.", "$\\pi ^m_i\\Big ({\\mathsf {m}}_i(t) \\Big {|} {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\Big )$ can be viewed as a tensor with three dimensions.", "As demonstrated in Algorithm 1, the idea is to successively optimize the objective function over coordinate blocks similar to coordinate descent algorithms [36], [37], with the difference that at each iteration the exact/inexact optimized coordinate is obtained using an approximation of Bellman optimality equations rather than gradient descent.", "Learning-Based Information Compression [1] Input parameters: $\\gamma $ , $\\alpha $ Initialize all-zero Q-tables $Q^{m,(0)}_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ , $Q^{c,(0)}_{i}\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big )$ , for $i=1,2$ each iteration $k=1:K$ find $\\pi ^{m,(k)}_{i}(\\cdot )$ by solving (REF ), for $i=1,2$ find $\\pi ^{c,(k)}_{i}(\\cdot )$ by solving (REF ), for $i=1,2$ $\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!", "\\!\\!\\!\\!$ $\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\text{\\textbf {Output: }} \\pi _{i}^{m,(K)}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )\\text{ for } i=1,2$ $\\quad \\,\\,\\,\\pi _{i}^{c,(K)}\\big ({\\mathsf {c}}_i(t)|{\\mathsf {o}}_i(t)\\big )\\text{ for } i=1,2$ Following Algorithm 1, during the iteration $k$ , each agent $i$ initially optimizes the objective function along the block coordinate $\\pi _{i}^{m}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ by solving the problem of decentralized action policy optimization i.e, (REF ), for $i \\!\\in \\!\\lbrace 1,2\\rbrace , i \\!\\ne j \\!$ .", "$ & \\pi ^{m,(k)}_i\\big ( \\cdot \\big ) = \\underset{\\pi ^m_i(\\cdot )}{\\text{argmax}}& & \\!\\!\\mathbb {E}_{p_{\\pi ^m_i, \\pi ^{c,(k-1)}_i}\\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )} \\Big {\\lbrace }{\\bf g}(1)\\Big {\\rbrace }, \\\\& \\text{s.t.", "}& & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p\\big ({\\mathsf {s}}(t+1) |{\\mathsf {s}}(t) \\big ) = \\!\\!\\!\\!\\sum _{m_i \\in \\mathcal {M}} \\sum _{m_j \\in \\mathcal {M}}\\!\\!\\!\\!", "T\\big ( {\\mathsf {s}}(t+1), {\\mathsf {s}}(t) , {\\mathsf {m}}_i , {\\mathsf {m}}_j \\big ) \\\\& & & \\;\\;\\;\\;\\;\\;\\pi ^m_i \\big ({\\mathsf {m}}_i | {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_j(t)\\big ) \\pi ^m_j \\big ({\\mathsf {m}}_j | {\\mathsf {o}}_j(t),\\tilde{{\\mathsf {c}}}_i(t)\\big ) \\\\& & & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C$ Afterwards, by following the action policy $\\pi ^{m,(k)}_i\\big ( \\cdot \\big )$ obtained earlier within the same iteration $k$ , each agent $i$ optimizes the objective function along the block coordinate $\\pi ^{c}_i\\big ( \\cdot \\big )$ by solving the problem of decentralized action policy optimization (REF ), for $i \\!\\in \\!\\lbrace 1,2\\rbrace , i \\!\\ne j \\!$ .", "$& \\pi ^{c,(k)}_i\\big ( \\cdot \\big ) = \\underset{\\pi ^c_i(\\cdot )}{\\text{argmax}}& & \\!\\!\\mathbb {E}_{p_{\\pi ^{m,(k)}_i(\\cdot ), \\pi ^{c}_i(\\cdot )}\\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )} \\Big {\\lbrace }{\\bf g}{1}\\Big {\\rbrace } \\\\& \\text{s.t.", "}& & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!p\\big ({\\mathsf {s}}(t+1) |{\\mathsf {s}}(t) \\big ) = \\sum _{m_i \\in \\mathcal {M}} \\sum _{m_j \\in \\mathcal {M}} T\\big ( {\\mathsf {s}}(t+1), {\\mathsf {s}}(t) , {\\mathsf {m}}_i , {\\mathsf {m}}_j \\big ) \\\\& & & \\!\\!\\!\\!", "\\pi ^{m,(k)}_i \\big ({\\mathsf {m}}_i | {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_j(t)\\big ) \\pi ^{m,(k)}_j \\big ({\\mathsf {m}}_j | {\\mathsf {o}}_j(t),\\tilde{{\\mathsf {c}}}_i(t)\\big ) \\\\& & & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C\\vspace{-8.53581pt}$ In both problems (REF ) and (REF ), both agents need to act cooperatively such that they can maximize the sum of discounted rewards.", "Agents, however, are unable to acquire any information about the actions of the other agent.", "Accordingly, as proposed by [35], these problems can be solved by distributed Q-learning and the exact (or inexact) solutions can be reached after training the Q-tables of both agents through enough a sufficient number of episodes of training.", "Algorithms 2 and 4 detail the way we apply distributed Q-learning to solve problems (REF ) and (REF ) respectively.", "Distributed Learning of Action Policies [1] Input: $\\gamma $ , $\\alpha $ , $c$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ and $Q^{m,(k-1)}_i(\\cdot )$ , $Q^{c,(k-1)}_i(\\cdot )$ , for $i=1,2$ Initialize all-zero table $N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ , for $i=1,2$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and Q-table $Q^{m,(k)}_{i}(\\cdot ) \\leftarrow Q^{m,(k-1)}_{i}(\\cdot )$ , for $i=1,2$ each episode $l=1:L$ Randomly initialize local observation ${\\mathsf {o}}_i(t=1)$ , for $i=1,2$ $t_l = 1:M$ Select ${\\mathsf {c}}_i(t)$ following $\\pi ^{c,(k-1)}_i(\\cdot )$ , for $i=1,2$ Obtain message $\\tilde{{\\mathsf {c}}}_j(t)$ , for $i=1,2 \\;\\; j \\ne i$ Update $Q^{m,(k)}_{i}\\big ({\\mathsf {o}}_i(t-1),\\tilde{{\\mathsf {c}}}_j(t-1),{\\mathsf {m}}_i(t-1)\\big )$ , for $i=1,2$ Select ${\\mathsf {m}}_i(t) \\in \\mathcal {M}$ by solving (REF ), for $i=1,2$ Increment $N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big ) $ , for $i=1,2$ Obtain reward $r\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t) \\big )$ , for $i=1,2$ Make a local observation ${\\mathsf {o}}_i(t)$ , for $i=1,2$ $t_l=t_l+1$ end Compute $\\sum ^{M}_{t=1} \\gamma ^t r_t$ for the $l$ th episode end Output: $Q^{m,(k)}_{i}(\\cdot )$ , $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and $\\pi _{i}^{m,(k)}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ for $i=1,2$ In order to obtain each agent $i$ 's policies $\\pi ^m_i(\\cdot ),\\pi ^c_i(\\cdot )$ we leverage state-action value functions $Q^m_i\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ and $Q^c_i\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big )$ .", "As expressed in (REF ) a state-action function $Q^m_i \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ estimates the conditional expectation of sum of discounted rewards, given observation ${\\mathsf {o}}_i(t)$ , received communication $,\\tilde{{\\mathsf {c}}}_j(t)$ and selected action ${\\mathsf {m}}_i(t)$ .", "The $Q$ -table $Q^c_i\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big )$ can also be explained the same way.", "$Q^m_i \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big ) =\\mathbb {E}_{p_{\\pi ^m_i, \\pi ^c_i}\\big (\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}\\big )} \\Big {\\lbrace } \\\\\\sum _{t=1}^M \\gamma ^{t-1} r\\big ({\\bf o}_i(t), {\\bf o}_j(t),{\\bf m}_i(t), {\\bf m}_j(t)\\big )| {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) , {\\mathsf {m}}_i(t)\\Big {\\rbrace },$ As detailed, the action-value function is trained based on its interaction with environment.", "For a given action-value function, to control the trade-off between exploitation and exploration, we adopt the Upper Confidence Bound (UCB) method [30].", "UCB, when applied to the distributed learning of action policies, selects the actions ${\\bf m}_i(t)$ as $ {\\bf m}_i(t) = \\underset{{\\mathsf {m}}}{\\text{argmax}} \\,\\, Q^m_i\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}\\big ) +k \\sqrt{\\frac{\\mathrm {ln}(T_t)}{N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}\\big )}},$ where $k>0$ is a constant; $T_t$ is the total number of time steps in the episodes considered up to the current time $t$ in a given training epoch; and table $N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ counts the total number of times that the observation ${\\mathsf {o}}_i(t)$ has been made together with the receipt of communication signal $\\tilde{{\\mathsf {c}}}_j(t)$ and the selection of ${\\mathsf {m}}_i(t)$ among the previous $T_t$ steps.", "When $k$ is large enough, UCB encourages the exploration of the state-action tuples that have been experienced fewer times.", "The update of the action value function based on the available observations at time $t$ follows the off-policy Q-learning algorithm, i.e., [30] $\\!", "\\!", "\\,Q^m_i \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big ) \\leftarrow (1-\\alpha ) Q^m_i \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big ) +$ $\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\!\\!", "\\, \\alpha \\gamma \\Big (\\!", "{r_t}+ \\underset{{\\mathsf {m}}\\in \\mathcal {M}}{\\text{max}}\\, Q^m_i \\big ({\\mathsf {o}}_i(t+1),\\tilde{{\\mathsf {c}}}_j(t+1),{\\mathsf {m}}\\big )\\!\\Big ),$ where $\\alpha > 0$ is a learning rate parameter.", "The full algorithm of distributed Q-learning of action policies is detailed in Algorithm 2.", "At the end of the training process, the policy $\\pi _{i}^{m,(k)}(\\cdot )$ can be obtained by $\\pi _{i}^{m,(k)}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )=\\delta \\Big (\\!", "{\\mathsf {m}}_i(t) -\\underset{{\\mathsf {m}}\\in \\mathcal {M}}{\\text{argmax}}\\, Q^m_i \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}\\big )\\!\\Big )\\!,$ where $\\delta (\\cdot )$ is the Dirac delta function.", "Algorithm 4, will bear a close resemblance to Algorithm 2, with the difference that in it, for each agent $i$ the $Q^c_i(\\cdot )$ is trained rather than action selection Q-table.", "For the sake of improved readability, we have included the Algorithm 4 in Appendix A.", "Remark 1: Distributed Q-learning algorithms have been shown to converge to the optimal joint solution, if applied on deterministic MDPs [35].", "The rate constraint of the communication channel can be seen as a form of aggregation of state of the MDP.", "As an MDP after being aggregated can still be an MDP, problem (REF ) can be optimally solved by distributed Q-learning conditioned on the tuple $\\Big {\\lbrace } \\Omega \\times \\mathcal {C} , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ being an MDP.", "The problem is escalated when we face similar issues in problem (REF ) too.", "Despite the good performance of LBIC scheme and its capability to be implemented fully distributedly, no guarantee for its convergence to optimal solution is provided in this paper." ], [ "State Aggregation for Information Compression in Multiagent Coordination Tasks", "This section tackles the constraint on the rate of inter-agent information exchange in the problem (REF ) by introducing state aggregation for compression of agents observations.", "State aggregation in this paper is applied as a method to carry out a task-based information compression.", "We design the state aggregation algorithm such that it can suppress part of observation information that results in the smallest possible loss in the performance of the multi-agent system, where this loss is measured in terms of regret from maximum achievable expected cumulative rewards.", "As mentioned before, we intend to perform compression in a manner that the entropy of compressed communication messages fits the rate of the inter-agent communication channel $C$ .", "Similar to some other recent papers with focus on multi-agent coordination [8], [14], here a centralized training phase for the two-agent system is required, however, the execution can still be done in a decentralized fashion.", "Although in a more general approach the selection of communication actions ${\\mathsf {c}}_i(t)$ could be conditioned on both ${\\mathsf {o}}_i(t)$ and $\\tilde{{\\mathsf {c}}}_j(t)$ , here we focus on just communication policies of type $\\pi ^c_i\\big ({\\mathsf {c}}_i(t) | {\\mathsf {o}}_i(t) \\big )$ , where communication actions of each agent at each time are selected only based on its observation at that time, further to the explanations given in subsection REF .", "Here we assume that the communication resources are split evenly amongst the two agents, by considering the achievable rate of information exchange of both communication channels to be equal to $C$ .", "As such, both agents compress their observations to acquire communication messages of equal entropy.", "We also assume observations of both agents to have equal entropy $H\\big ( {\\mathsf {o}}_i(t)\\big ) = H\\big ( {\\mathsf {o}}_j(t)\\big )$ .", "To solve problem (REF ), in this section, we first solve the problem (REF ) for a general policy $\\pi ^c_i(\\cdot )$ being followed by each agent $i$ .", "Therefore, the solution of this problem is not drawn based on a specific $\\pi ^c_i(\\cdot )$ being agent $i$ 's communication policy, but based on a parametric $\\pi ^c_i(\\cdot )$ .", "Accordingly, the obtained solution to problem (REF ) can be plugged into (REF ) which leaves us with only one policy function $\\pi ^c_i(\\cdot )$ to be optimized.", "The advantage of this approach is that if both of the mentioned problems can be solved optimally, then the problem (REF ) has been separable and the obtained solutions are the optimal solutions of it.", "According to REF , the objective function of the decentralized problem (REF ), which is a function of system trajectory $\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M}$ can also be written as $ & \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M})}\\big {\\lbrace } {\\bf g}(1) \\big {\\rbrace } = \\\\& \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}({\\mathsf {o}}_i(1),\\tilde{{\\mathsf {c}}}_j(1))} \\Big {\\lbrace }\\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=2}^{t=M}) | {\\bf o}_i(1),\\tilde{{\\bf c}}_j(1))} \\big {\\lbrace } \\\\& \\;\\;\\;\\;\\;\\;\\; \\;\\;\\;\\;\\;\\;\\;\\; \\;\\;\\;\\;\\;\\;\\; \\;\\;\\;\\;\\;\\;\\;\\; {\\bf g}(1) | {\\bf o}_i(1) , \\tilde{{\\bf c}}_j(1)\\big {\\rbrace }\\Big {\\rbrace } =\\\\& \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}({\\mathsf {o}}_i(1),\\tilde{{\\mathsf {c}}}_j(1))} \\Big {\\lbrace }V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace },$ where $V_{\\pi ^m_i,\\pi ^c_i}\\big ( {\\mathsf {o}}_i(t) ,\\tilde{{\\mathsf {c}}}_j(t) \\big )$ is the unique solution to the Bellman equation corresponding to the joint action and communication policies $\\pi ^m_i,\\pi ^c_i$ of both agents $\\begin{aligned}& V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big ) = \\underset{{\\mathsf {m}}_j(t)}{\\sum _{{\\mathsf {m}}_i(t),}} r\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big ) \\\\& \\pi _i^m\\big ({\\mathsf {m}}_i(t)\\big {|} {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\big ) \\pi _j^m\\big ({\\mathsf {m}}_j(t)\\big {|} {\\mathsf {o}}_j(t),\\tilde{{\\mathsf {c}}}_i(t) \\big )+ \\\\& \\gamma \\underset{{\\mathsf {o}}_j(t+1)\\in \\Omega }{\\sum _{{\\mathsf {o}}_i(t+1) \\in \\Omega ,}} V\\big ({\\mathsf {o}}_i(t+1),\\tilde{{\\mathsf {c}}}_j(t+1)\\big ) \\pi ^c_{i}\\big (\\tilde{{\\mathsf {c}}}_j(t+1) | {\\mathsf {o}}_j(t+1) \\big ) \\\\& T\\big ( \\underbrace{{\\mathsf {o}}_i(t+1), {\\mathsf {o}}_j(t+1)}_{{\\mathsf {s}}(t+1)}, \\underbrace{{\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t) }_{{\\mathsf {s}}(t)}, {\\mathsf {m}}_i(t) , {\\mathsf {m}}_j(t) \\big ), \\;\\;\\;\\\\&\\forall {\\mathsf {o}}_i(t) \\in \\Omega ,\\tilde{{\\mathsf {c}}}_j(t) \\in \\mathcal {C}.\\end{aligned}$ In light of eq.", "(REF ) the objective function of the problem (REF ) can be expressed as $ \\begin{aligned}& \\underset{\\pi ^m_i}{\\text{max }}& \\!\\!\\!\\!\\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace } \\!V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\!", "\\Big {\\rbrace }, \\; i \\in \\lbrace 1,2\\rbrace , i \\ne j.", "\\\\\\end{aligned}$ Lemma 1, lets us to obtain the solution of (REF ) by finding the optimal value function $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ .", "This function can be found either by applying Bellman optimality equations for a sufficient number of times on $V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ or by Q-learning.", "It is important, however, to note that the value function $ V_{\\pi ^{m^{*}}_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ obtained by Q-learning will be optimal only if, the environment, explained by the tuple $\\Big {\\lbrace } \\Omega \\times \\mathcal {C} , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ , can be proven to be an MDP.", "Accordingly, we assume that the aggregated MDP denoted by $\\Big {\\lbrace } \\Omega \\times \\mathcal {C} , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ which is obtained by doing state aggregation on the original MDP denoted by $\\Big {\\lbrace } \\Omega ^2 , \\mathcal {M}^2, r(\\cdot ), \\gamma , T(\\cdot ) \\Big {\\rbrace }$ is an MDP itself.", "Lemma 1 The maximum of expectation of value function $V_{\\pi ^m_i,\\pi ^c_i} \\big ({\\bf o}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\big )$ , over the joint distribution of ${\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)$ is equal to the expectation of value function of optimal policy $\\underset{\\pi ^m_i}{\\text{max }}\\;\\;\\;& \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace } V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace } = \\\\& \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace }V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace }$ Remember that numerical methods such as value iteration or Q-learning, cannot normally provide parametric solutions which is in contrast to our requirements in SAIC, as explained earlier in this section.", "Lemma 2, allows us to acquire a parametric approximation of $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ by leveraging the value function $V^*\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t)\\big )$ corresponding to the optimal solution of the centralized problem (REF ).", "Note that policy $\\pi ^*(\\cdot )$ as described in Section , is the optimal solution to the centralized problem (REF ) which can be obtained using Q-learning.", "Accordingly, following lemma 2, we propose to derive an off-policy approximation of $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )$ having the knowledge of policy $\\pi ^*(\\cdot )$ .", "Lemma 2 The optimal value of $V^{*}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ can be obtained using the solution $\\pi ^*(\\cdot )$ and its corresponding value function $V^{*}\\big ( {\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t) \\big )$ following $V^{*}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )= \\sum _{{\\mathsf {o}}_j(t) \\in Q_k} V^{*}\\big ( {\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t) \\big ) p\\big ({\\mathsf {o}}_j(t)| \\tilde{{\\mathsf {c}}}_j(t)\\big ).$ Based on the results of lemma 1 and lemma 2, theorem 3 is constructed such that it allows us to compute the communication policies of agents independent from their action policies.", "The proposed communication policy by theorem 3, is conditionally the optimal communication policy.", "Theorem 3 The communication policy that can maximize the achievable expected cumulative rewards in the decentralized coordination problem (REF ) can be obtained by solving the k-median clustering problem $\\begin{aligned}& \\underset{\\mathcal {P}_i}{\\text{min}}& & \\sum _{k=1}^{2^B} \\;\\; \\sum _{{\\mathsf {o}}_i(t) \\in \\Omega } \\Big {|} V^{*}\\big ({\\mathsf {o}}_i(t)\\big ) - \\mu ^{^{\\prime }}_k \\Big {|},\\end{aligned}$ where $\\mathcal {P}_i$ corresponds to a unique $\\pi ^c_i(\\cdot )$ .", "Further to eq.", "(REF ), problem (REF ) can be expressed by $ \\begin{aligned}& \\underset{\\pi ^m_i(\\cdot ),\\pi ^c_i(\\cdot )}{\\text{max }}& & \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace }V_{\\pi ^m_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace }, \\\\& \\text{s.t.}", "& &I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C,\\end{aligned}$ for $i \\in \\lbrace 1,2\\rbrace ,\\, i \\ne j$ .", "We can now plug $\\pi ^{m^{*}}_i(\\cdot )$ , which is the result of solving problem (REF ), into problem (REF ) to obtain the following problem $\\begin{aligned}& \\underset{\\pi ^c_i(\\cdot )}{\\text{max }}& &\\mathbb {E}_{p_{\\pi ^{m^{*}}_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace }V_{\\pi ^{m^{*}}_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace }, \\\\& \\text{s.t.}", "& &I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C,\\end{aligned}$ for $i \\in \\lbrace 1,2\\rbrace ,\\, i \\ne j$ .", "By substituting $V_{\\pi ^{m^{*}}_i,\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ with its approximator $V^{*}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )$ , we will have $\\begin{aligned}& \\underset{\\pi ^c_i(\\cdot )}{\\text{max }}& &\\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )} \\Big {\\lbrace }V^{*}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )\\Big {\\rbrace }\\; \\; \\; \\; i \\in \\lbrace 1,2\\rbrace \\; , i \\ne j \\\\& \\text{s.t.}", "& &I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C.\\end{aligned}$ Note that the optimizers of the problem (REF ) and (REF ) are identical due to the fact that the additional term $\\mathbb {E} \\big {\\lbrace }V^{*}\\big ({\\bf o}_i(t),{\\bf o}_j(t)\\big ) \\big {\\rbrace }$ is independent from the communication policy $\\pi ^c_i(\\cdot )$ .", "Furthermore, the problem (REF ) is now expressed as a form of rate distortion problem with mean absolute difference of the value functions $V^{*}\\big ({\\bf o}_i(t), {\\bf o}_j(t)\\big )$ and $V^{*}\\big ({\\bf o}_i(t), \\tilde{{\\bf c}}_j(t)\\big )$ as the measure of distortion.", "This interpretation of problem (REF ) can be understood later by seeing the eq.", "(REF ).", "$\\begin{aligned}& \\underset{\\pi ^c_i(\\cdot )}{\\text{min}}& & \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}\\big ({\\mathsf {o}}_i (1),\\tilde{{\\mathsf {c}}}_j (1)\\big )}\\Big {\\lbrace }V^{*}\\big ({\\bf o}_i(1), {\\bf o}_j(1)\\big ) - V^{*}\\big ({\\bf o}_i(1), \\tilde{{\\bf c}}_j(1)\\big ) \\Big {\\rbrace }\\\\& \\text{s.t.", "}& & I\\big ( {\\bf c}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C ,\\\\\\end{aligned}$ The expectation $\\mathbb {E}_{p_{\\pi ^*}({\\mathsf {o}}_i, {\\mathsf {o}}_j) } \\Big {\\lbrace } V^{*}\\big ({\\bf o}_i(1), {\\bf o}_j(1)\\big ) - V^{*}\\big ({\\bf o}_i(1), \\tilde{{\\bf c}}_j(1)\\big ) \\Big {\\rbrace }$ can be estimated by computing it over the empirical distribution of ${\\bf o}_1(1)$ , ${\\bf o}_2(1)$ .", "Note that the empirical joint distribution of ${\\bf o}_1(1)$ , $\\tilde{{\\bf c}}_2(1)$ can be obtained by following the communication policy $\\pi ^c_i(\\cdot )$ on the empirical distribution of ${\\bf o}_1(1)$ , ${\\bf o}_2(1)$ .", "Therefore, the problem (REF ) can be rewritten as $\\begin{aligned}& \\underset{\\pi ^c_i(\\cdot )}{\\text{min}}& & \\sum _{{\\mathsf {o}}_i(1) \\in \\Omega } \\;\\; \\sum _{{\\mathsf {o}}_j(1) \\in \\Omega } \\Big {|} V^{*}\\big ({\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t)\\big ) - V^{*}\\big ({\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_j(t)\\big )\\Big {|}\\\\& \\text{s.t.}", "& & I\\big ( {\\bf o}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C.\\\\\\end{aligned}$ Selection of the optimal $\\pi ^c_i(\\cdot )$ here, means to extract all the information of ${\\mathsf {o}}_i(t)$ which is useful in the task and to include them in $\\tilde{{\\mathsf {c}}}_i(t)$ .", "This task-based information compression problem can be interpreted as a quantization problem in which distortion is defined in terms of value functions as seen in (REF ), where a certain number of quantization levels, $2^C$ , is allowed.", "Note that the term $V^{*}\\big ({\\bf o}_i(t), {\\bf o}_j(t)\\big ) - V^{*}\\big ({\\bf o}_i(t), \\tilde{{\\bf c}}_j(t)\\big )$ , being always non-negative, is equal to its absolute value $\\Big {|} V^{*}\\big ({\\bf o}_i(t), {\\bf o}_j(t)\\big ) - V^{*}\\big ({\\bf o}_i(t), \\tilde{{\\bf c}}_j(t)\\big ) \\Big {|}$ .", "To gain more insight about the meaning of this task-based information compression, we have also detailed a conventional quantization problem which is adapted to our problem setting in eq.", "(REF ), where ${{\\bf c}}_j \\sim \\pi ^c_j\\big ({\\bf c}_j(1) | {\\bf o}_j(1)\\big )$ .", "In fact, the compression scheme applied in the CIC, explained later on in subsection (REF ), is obtained by solving a similar problem, with the caveat that distance in case of CIC is computed in terms of squared euclidean distance.", "$\\begin{aligned}& \\underset{\\pi ^c_i(\\cdot )}{\\text{min}}& & \\sum _{{\\mathsf {o}}_j(1) \\in \\Omega } \\Big {|} {\\mathsf {o}}_j(t) - \\tilde{{\\mathsf {c}}}_j(t)\\Big {|}\\\\& \\text{s.t.}", "& & I\\big ( {\\bf o}_j(t) ; \\tilde{{\\bf c}}_j(t) \\big ) \\le C.\\\\\\end{aligned}$ Quantization levels are disjoint sets $\\mathcal {P}_i \\subset \\Omega $ , where their union $\\cup _{k=1}^{2^C}\\mathcal {P}_{i,k}$ will cover the entire $\\Omega $ .", "Each quantization level is represented by only one communication message ${\\mathsf {c}}_j(t)={\\mathsf {c}}_k \\in \\mathcal {C}$ .", "Further to lemma 2, the value of $V^{*}\\big ({\\bf o}_i(t), \\tilde{{\\bf c}}_j(t)\\big )$ can be computed by empirical mean $\\mu _k = \\frac{1}{|\\mathcal {P}_{i,k}|} \\sum _{{\\mathsf {o}}_j \\in \\mathcal {P}_{i,k}} V^{*}\\big ({\\bf o}_i, {{\\bf o}}_j \\big )$ .", "The quantization problem (REF ) becomes a k-median clustering problem as illustrated by (REF ) $\\begin{aligned}& \\underset{\\mathcal {P}_i}{\\text{min}}& & \\sum _{{\\mathsf {o}}_i(t) \\in \\Omega } \\;\\;\\sum _{k=1}^{2^C} \\;\\; \\sum _{{\\mathsf {o}}_j(t) \\in \\mathcal {P}_{i,k}} \\Big {|} V^{*}\\big ({\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t)\\big ) - \\mu _k \\Big {|}, \\\\\\end{aligned}$ where $\\mathcal {P}_i = \\lbrace \\mathcal {P}_{i,1}, ..., \\mathcal {P}_{i,2^C}\\rbrace $ is a partition of $\\Omega $ .", "By taking the mean of $V^{*}\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t) \\big )$ over the empirical distribution of ${\\mathsf {o}}_j(t)$ we can also marginalize out ${\\mathsf {o}}_j(t)$ .", "Again, it does not change the solution of the problem and we will have $\\begin{aligned}& \\underset{\\mathcal {P}_i}{\\text{min}}& & \\sum _{k=1}^{2^C} \\;\\; \\sum _{{\\mathsf {o}}_i(t) \\in \\Omega } \\Big {|} V^{*}\\big ({\\mathsf {o}}_i(t)\\big ) - \\mu ^{^{\\prime }}_k \\Big {|},\\end{aligned}$ in which $\\mu ^{^{\\prime }}_k = \\sum _{{\\mathsf {o}}_j(t) \\in \\Omega } \\mu _k$ will approximate $V^{*}\\big ( {\\mathsf {c}}_j(t) \\big )$ .", "Theorem REF allows us to compute a communication policy $\\pi ^{c^{}}_i(\\cdot )$ by clustering values of $V^{*}\\big ( {\\mathsf {o}}_i(t)\\big )$ , where this policy can be the optimal communication policy under some conditions which are further discussed later on in this section.", "As mentioned in Theorem REF , one way to obtain $V^{*}\\big ( {\\mathsf {o}}_i(t)\\big )$ is to solve the centralized problem (REF ) by Q-learning.", "By solving this problem $Q^{*}\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big )$ can be obtained.", "Accordingly, following Bellman optimality equation, we can compute $V^{*}\\big ( {\\mathsf {o}}_i(1)\\big )$ by $\\begin{aligned}\\underset{{\\mathsf {m}}}{\\text{max }} \\; Q^{e^*}\\big ({\\mathsf {o}}_i(1),{\\mathsf {m}}_i(1)\\big ) = V^{*}\\big ({\\mathsf {o}}_i(1)\\big ),\\end{aligned}$ where $V^{*}\\big ({\\mathsf {o}}_i(1)\\big )$ can be expressed as $ V^{*}\\big ({\\mathsf {o}}_i(1)\\big ) = \\mathbb {E}_{p_{\\pi ^{*}}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=2}^{t=M} | {\\mathsf {o}}_i(1))} \\Bigg {\\lbrace }{\\bf g}(1)| {\\bf o}_i(1) = {\\mathsf {o}}_i(1)\\Bigg {\\rbrace }$ and further to the law of iterated expectations, it can also be written as $ \\begin{aligned}& V^{*}\\big ({\\mathsf {o}}_i(1)\\big ) = \\mathbb {E}_{p({\\bf o}_j(1))} \\Bigg {\\lbrace } \\mathbb {E}_{p_{\\pi ^{*}}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=2}^{t=M} | {\\mathsf {o}}_i(1), {\\bf o}_j(1))} \\bigg {\\lbrace } \\\\& {\\bf g}(1)| {\\bf o}_i(1) = {\\mathsf {o}}_i(1) , {\\bf o}_j(1)\\bigg {\\rbrace } \\Bigg {\\rbrace } =\\\\& \\sum _{{\\mathsf {o}}_j(1) \\in \\Omega } p({\\bf o}_j(t) = {\\mathsf {o}}_j(1))\\underbrace{ \\mathbb {E}_{\\pi ^*} \\bigg {\\lbrace }{\\bf g}(1)| {\\bf o}_i(1) = {\\mathsf {o}}_i(1) , {\\bf o}_j(t) = {\\mathsf {o}}_j(1) \\bigg {\\rbrace }}_{V^{*}({\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1))}\\end{aligned}$ and $V^{*}\\big ({\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1)\\big )$ can be approximated by centralized $Q$ -learning following $ \\begin{aligned}& V^{*}\\!\\big ({\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1)\\big ) \\!=\\!", "V^{*}\\big ({\\mathsf {s}}(1)\\big )\\!", "= \\!", "\\underset{{\\mathsf {m}}_1,{\\mathsf {m}}_2}{\\text{max }} \\; Q^{*}\\!\\big ({\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1),{\\mathsf {m}}_i(1),{\\mathsf {m}}_j(1)\\big ) \\\\& = \\mathbb {E}_{p_{\\pi ^{*}}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=2}^{t=M} | {\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1))} \\Bigg {\\lbrace }{\\bf g}(1)| {\\bf o}_i(t) = {\\mathsf {o}}_i(1), {\\bf o}_j(t) = {\\mathsf {o}}_j(1)\\Bigg {\\rbrace }.\\end{aligned}$ Using (REF ) and (REF ) we can simply compute $V^{*}\\big ({\\mathsf {o}}_i(1)\\big )$ by $ \\begin{aligned}& V^{*}\\big ({\\mathsf {o}}_i(1)\\big ) = \\\\ &\\sum _{{\\mathsf {o}}_j(1) \\in \\Omega } \\underset{{\\mathsf {m}}_1,{\\mathsf {m}}_2}{\\text{max }} \\; Q^{*}\\big ({\\mathsf {o}}_i(1),{\\mathsf {o}}_j(1),{\\mathsf {m}}_i(1),{\\mathsf {m}}_j(1)\\big ) p\\big ({\\bf o}_j(t) = {\\mathsf {o}}_j(1)\\big ).", "\\\\\\end{aligned}$ Based on (REF ), $V^{*}\\big ({\\mathsf {o}}_i(1)\\big )$ can be computed both analytically (if transition probabilities of environment are available ) and numerically.", "A remarkable feature of computing $V^{*}\\big ({\\mathsf {o}}_i(1)\\big )$ using this method is its independence from the used inter-agent communication algorithm.", "By following this scheme, detailed in Algorithm 3, we first compute the value $V^{*}({\\mathsf {o}})$ for all ${\\mathsf {o}}\\in \\Omega $ .", "Afterwards, by solving the k-median clustering problem (REF ), an observation aggregation scheme indicated by $\\mathcal {P}_i$ is computed.", "By following this aggregation scheme, the observations ${\\mathsf {o}}_i(t) \\in \\Omega $ will be aggregate such that the performance of the multi-agent system in terms of the the objective function it attains is optimized.", "Upon the availability of the exact $\\pi ^{c^{}}_i(\\cdot )$ , which is also the aggregation scheme executed by the agent $i$ , we need to find an exact action policy for both agents corresponding to it.", "That is, we now solve the problem (REF ) not for a parametric communication policy, but for the exact communication policy $\\pi ^c_i(\\cdot )$ which was obtained by solving problem (REF ).", "As such, the second training phase in which the action $Q$ -tables $Q^m_i(\\cdot )$ for $i=\\lbrace 1,2\\rbrace $ are obtained as well as the execution phase of the algorithm can be done distributively.", "Further to the assumption mentioned for lemma 1, remark 2 is stated bellow and remark 3 and 4 will follow with no proof given.", "State Aggregation for Information Compression (SAIC) [1] Input: $\\gamma $ , $\\alpha $ , $c$ Initialize all-zero table $N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big )$ , for $i=1,2$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and Q-table $Q^{m}_{i}(\\cdot ) \\leftarrow Q^{m,(k-1)}_{i}(\\cdot )$ , for $i=1,2$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and all-zero Q-table $Q\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big )$ .", "Obtain $\\pi ^{*}(\\cdot ) \\text{ and } Q^{*}(\\cdot )$ by solving (REF ) using Q-learning [30].", "Compute $V^{*}\\big ( {\\mathsf {o}}_i(t) \\big )$ following eq.", "(REF ), for $\\forall {\\mathsf {o}}_i(t) \\in \\Omega $ .", "Solve problem (REF ) by applying k-median clustering to obtain $\\pi ^c_i(\\cdot )$ , for $i=1,2$ .", "each episode $k=1:K$ Randomly initialize local observation ${\\mathsf {o}}_i(t=1)$ , for $i=1,2$ $t_k = 1:M$ Select ${\\mathsf {c}}_i(t)$ following $\\pi ^{c}_i(\\cdot )$ , for $i=1,2$ Obtain message $\\tilde{{\\mathsf {c}}}_j(t)$ , for $i=1,2 \\;\\; j \\ne i$ Update $Q^{m}_{i}\\big ({\\mathsf {o}}_i(t-1),\\tilde{{\\mathsf {c}}}_j(t-1),{\\mathsf {m}}_i(t-1)\\big )$ , for $i=1,2$ Select ${\\mathsf {m}}_i(t) \\in \\mathcal {M}$ by solving (REF ), for $i=1,2$ Increment $N^m_{i}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t),{\\mathsf {m}}_i(t)\\big ) $ , for $i=1,2$ Obtain reward $r\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t) \\big )$ , for $i=1,2$ Make a local observation ${\\mathsf {o}}_i(t)$ , for $i=1,2$ $t_k=t_k+1$ end Compute $\\sum ^{M}_{t=1} \\gamma ^t r_t$ for the $l$ th episode end Output: $Q^{m}_{i}(\\cdot )$ , $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and $\\pi _{i}^{m}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ by following (REF ) for $i=1,2$ Remark 2: The the action $Q$ -tables $Q^m_i(\\cdot )$ for $i=\\lbrace 1,2\\rbrace $ which are obtained by Q-learning are optimal only if the original MDP denoted by $\\Big {\\lbrace } \\Omega ^2 , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ which is aggregated following the equivalence relation defined by $\\mathcal {P}_i$ will remain an MDP even in its aggregated form.", "Equivalently, if $\\Big {\\lbrace } \\Omega \\times \\mathcal {C} , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ is an MDP, using Q-learning, we can find the optimal action policy $\\pi ^{m^{*}}_i(\\cdot )$ corresponding to $\\pi ^{c}_i(\\cdot )$ .", "This also equivalent to say that if $\\Big {\\lbrace } \\Omega \\times \\mathcal {C} , \\mathcal {M}^2, r(\\cdot ), \\gamma , T^{\\prime }(\\cdot ) \\Big {\\rbrace }$ is an MDP, using Q-learning, one can find the optimal solution for (REF ) when each agent $i$ follows $\\pi ^{c}_i(\\cdot )$ corresponding to $\\mathcal {P}_i$ .", "Remark 3: If the communication policies $\\pi ^c_i(\\cdot )$ obtained by solving (REF ) are optimal, and the condition of remark 2 is met, the action policies obtained by Q-learning in the decentralized training phase will also be globally optimal.", "Remark 4: The optimality of the communication policies $\\pi ^c_i(\\cdot )$ and $\\pi ^c_j(\\cdot )$ obtained by solving (REF ) is conditioned on the optimal performance of the numerical algorithm used to solve (REF ) and the accuracy of the approximation of $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ by value function $V^*\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ ." ], [ "Numerical results", "In this section, we evaluate our proposed schemes via numerical results for the popular rendezvous problem, in which the inter-agent communication channel is set to have a limited rate.", "Rendezvous problem is of particular interest as it allows us to consider a cooperative multiagent system comprising of two agents that are required to communicate for their coordination task.", "In particular, as detailed in subsection REF , if the communication between agents is not efficient, at any time step $t$ each agent $i$ will only have access to its local observation ${\\mathsf {o}}_i(t)$ , which is its own location in the case of rendezvous problem.", "This mere information is insufficient for an agent to attain the larger reward $R_2$ , but is sufficient to attain the smaller reward $R_1$ .", "Accordingly, compared with cases in which no communication between agents is present, in the set up of the rendezvous problem, efficient communication policies can increase the attained objective function of the multiagent system up to six-folds, as will be seen in Fig.", "REF .", "The system operates in discrete time, with agents taking actions and communicating in each time step $t=1,2,...$ .", "We consider a variety of grid-worlds with different size values $N$ and different locations for the goal-point $\\omega ^T$ .", "We compare the proposed SAIC and LBIC with (i) Centralized Q-learning scheme and (ii) the Conventional Information Compression (CIC) scheme which is explained in subsection REF ." ], [ "Rendezvous Problem", "As illustrated in Fig.", "REF , two agents operate on an $N \\times N$ grid world and aim at arriving at the same time at the goal point on the grid.", "Each agent $i \\in \\lbrace 1,2\\rbrace $ at any time step $t$ can only observe its own location ${\\mathsf {o}}_i(t) \\in \\Omega $ on the grid, where the observation space is $\\Omega = \\lbrace 0,2,...,n^2-1\\rbrace $ .", "Each episode terminates as soon as an agent or both visit the goal point which is denoted as $\\omega ^T \\in \\Omega $ .", "That is, at any time step $t$ that the observations tuple $\\langle {\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t)\\rangle $ is a member of $\\Omega ^T= \\lbrace \\omega ^T\\rbrace \\times \\Omega \\cup \\Omega \\times \\lbrace \\omega ^T\\rbrace $ , the episode will be terminated.", "At time $t=1$ , the initial position of both agents is randomly and uniformly selected amongst the non-goal states, i.e.", "for each agent $i \\in \\lbrace 1,2\\rbrace $ the initial position of the agent is ${\\mathsf {o}}_i(1) \\in \\Omega - \\lbrace \\omega ^T \\rbrace $ .", "At any time step $t=1,2,...$ each agent $i$ observes its position, or environment state, and acquires information about the position of the other agent by receiving a communication message $\\tilde{{\\mathsf {c}}}_j(t)$ sent by the other agent $j \\ne {i}$ at the time step $t$ .", "Based on this information, agent $i$ selects its environment action ${\\mathsf {m}}_i(t)$ from the set $\\mathcal {M} = \\lbrace \\text{Right},\\text{Left},\\text{Up},\\text{Down},\\text{Stop}\\rbrace $ , where an action ${\\mathsf {m}}_i(t) \\in \\mathcal {M}$ represent the horizontal/vertical move of agent $i$ on the grid at time step $t$ .", "For instance, if an agent $i$ is on a grid-world as depicted on Fig.", "REF (a), and observes ${\\mathsf {o}}_i(t)=4$ and selects \"Up\" as its action, the agent's observation at the next time step will be ${\\mathsf {o}}_i(t+1)=8$ .", "If the position to which the agent should be moved is outside the grid, the environment is assumed to keep the agent in its current position.", "We assume that all these deterministic state transitions are captured by $T\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big )$ , which can determine the observations of agents in the next time step $t+1$ following $\\langle {\\mathsf {o}}_i(t+1),{\\mathsf {o}}_j(t+1) \\rangle = T\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big ).$ Accordingly, given observations ${\\mathsf {o}}_i(t), {\\mathsf {o}}_j(t)$ and actions $ {\\mathsf {m}}_i(t), {\\mathsf {m}}_j(t) $ , both agents receive a single team reward $r\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t) \\big )={\\left\\lbrace \\begin{array}{ll}R_1, & \\text{if $P_1$}\\\\R_2, & \\text{if $P_2$},\\\\0, & \\text{otherwise},\\\\\\end{array}\\right.", "}$ where $R_1 < R_2$ and the propositions $P_1$ and $P_2$ are defined as $P_1: T\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big ) \\in \\Omega ^T - \\lbrace \\omega ^T \\rbrace ^2$ and $P_2: T\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t)\\big ) \\in \\lbrace \\omega ^T \\rbrace ^2$ .", "When only one agent arrives at the target point $\\omega ^T$ , the episode will be terminated with the smaller reward $R_1$ being obtained, while the larger reward $R_2$ is attained when both agents visit the goal point at the same time.", "Note that this reward signal encourages coordination between agents which in turn can benefit from inter-agent communications.", "Furthermore, at each time step $t$ agents choose a communication message to send to the other agent by selecting a communication action ${\\mathsf {c}}_i(t) \\in \\mathcal {C} = \\lbrace 0,1\\rbrace ^C$ of $C$ bits, where $C$ is the maximum achievable rate of the inter-agent communication channel following specific channel conditions and error correction methods which are used.", "That is, $C$ is imposed to the problem as a constraint.", "The goal of the multiagent system is, thus, to maximize the average discounted cumulative rewards by solving the problem (REF )." ], [ "Conventional Information Compression In Multiagent Coordination Tasks", "As a baseline, we consider a conventional scheme that selects communications and actions separately.", "For communication, each agent $i$ sends its observation ${\\mathsf {o}}_i(t)$ to the other agent by following policy $\\pi ^c_i(\\cdot )$ .", "According to this policy the agent's observation ${\\mathsf {o}}_i(t)$ will be mapped to a binary bit sequence ${\\mathsf {c}}_i(t)$ , using an injective (and not necessarily surjective) mapping $f_1: \\Omega \\rightarrow \\lbrace -1,1\\rbrace ^C$ .", "Consequently, the communication policy $\\pi ^c_i$ becomes deterministic and follows $\\pi ^c_i\\big ({\\mathsf {c}}_i(t+1)| {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\big ) = \\delta \\Big (\\!", "{\\mathsf {c}}_i(t+1) - f_1\\big ( {\\mathsf {o}}_i(t) \\big )\\!\\Big )\\!.$ Agent $i$ obtains an estimate $\\tilde{{\\mathsf {c}}}_j(t)$ of the observation of $j$ by having access to a quantized version of ${\\mathsf {o}}_j(t)$ .", "This estimate is used to define the environment state-action value function $Q^o_j \\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_i(t),{\\mathsf {m}}_i(t)\\big )$ .", "This function is updated using Q-learning and the UCB policy in a manner similar to Algorithm 2, with no communication policy to be learned.", "This communication strategy is proven to be optimal [18], if the inter-agent communication does not impose any cost on the cooperative objective function and the communication channel is noise-free.", "Under these conditions, and when the dynamics of the environment are deterministic, each agent $i$ can distributively learn the optimal policy $\\pi ^m_i(\\cdot )$ , using value iteration or its model-free variants e.g.", "Q-learning [35].", "This communication policy requires a channel rate $C \\ge H({\\bf o}_j)$ , whereas in this paper, we are focused on the scenarios with $C \\le H({\\bf o}_j)$ .", "Therefore, due to the limited rate of the communication channel, a form of information compression is required to be carried out.", "Note that compression before a converged action policy is not possible, since all observations are a priori equally likely.", "Thus, we first train the CIC on a communication channel with unlimited capacity.", "Afterwards, when a probability distribution for observations is obtained, by applying Lloyd's algorithm, we define an equivalence relation on the observation space $\\Omega $ with $2^{C}$ numbers of equivalence classes $\\mathcal {Q}_1,..., \\mathcal {Q}_{2^{C}}$ .", "According to the defined equivalence relation by Lloyd's algorithm, we can uniquely indicate the mapping $f_2: \\Omega \\rightarrow \\lbrace -1,1\\rbrace ^C$ that maps agent $j$ 's observation ${\\mathsf {o}}_j(t)$ .", "The mapping $f_2(\\cdot )$ that maps agent $j$ 's observation ${\\mathsf {o}}_j(t)$ into a quantized communication ${\\mathsf {c}}_j(t)$ is not an injective mapping anymore.", "That is, by receiving the communication message $\\tilde{{\\mathsf {c}}}_j(t) \\in \\mathcal {Q}_k \\subset \\mathcal {C}$ agent $i$ can not retrieve ${\\mathsf {o}}_j(t)$ but understands the observation of agent $j$ has been a member of $\\mathcal {Q}_k$ .", "Upon, an optimal performance of Lloyd's algorithm in defining the equivalence relation, we expect this algorithm to perform optimally, as long as $H\\big ({\\bf o}_i(t)\\big ) \\le C$ .", "Note that this algorithm has a limitation, as it requires the first round of training to be done over communication channels with unlimited capacity." ], [ "Results", "To perform our numerical experiments, rewards of the rendezvous problem are selected as $R_1=1$ and $R_2=10$ , while the discount factor is $\\gamma = 0.9$ .", "A constant learning rate $\\alpha =0.07$ is applied, and the UCB exploration rate $c=12.5$ .", "In any figure that the performance of each scheme is reported in terms of the averaged discounted cumulative rewards, the attained rewards throughout training iterations are smoothed using a moving average filter of memory equal to 20,000 iterations.", "Regardless of the grid-world's size and goal location, the grids are numbered row-wise starting from the left-bottom as shown in Fig.", "REF -a.", "Fig.", "REF illustrates the performance of the two proposed schemes LBIC and SAIC as well as the two benchmark schemes centralized Q-learning and CIC.", "The performance is measured in terms of the achievable sum of discounted rewards in a rendezvous problem.", "The grid-world is considered to be of size $N=8$ and its goal location to be $\\omega ^T=22$ .", "The rate budget of the channel between the two agents is $C=2$ bits per time step.", "Since centralized Q-learning is not affected by the limitation on channel's achievable bit rate, it achieves optimal performance after enough training, 160k iterations.", "The CIC, due to insufficient rate of the communication channel never achieves the optimal solution.", "The LBIC, however, is seen to outperform the CIC, although it is trained and executed fully distributedly.", "It is observed that the SAIC by less than 1% gap achieves optimal performance and does that remarkably fast, where the performance gap for the LBIC and CIC are roughly 20% and 30% respectively.", "The yellow curve showing the performance of the CIC with no communication between agents, would show us the best performance that can be achieved if no communication between agents is in place.", "In fact, the better performance of any scheme compared with the yellow curve, is the sign that the scheme benefits from some effective communication between agents.", "Note that, when inter-agent communication is unavailable, i.e., $C=0$ bit per time step, there would be no difference in the performance of the CIC, SAIC or LBIC as all of them use the same algorithm to find out the action policy $\\pi ^m_i(\\cdot )$ .", "We also recall the fact that both the CIC and SAIC require a separate training phase which is not captured by Fig.", "REF .", "SAIC requires a centralized training phase and CIC a distributed training phase with unlimited capacity of inter-agent communication channels.", "The performance of these two algorithms in Fig.", "REF is plotted after the first phase of training.", "Figure: A comparison between all four schemes in terms of the achievable objective function with channel rate constraint C=2C=2 bits per time steps and number of training iterations/episodes K=200kK=200k.To understand the underlying reasons for the remarkable performance of the SAIC, Fig.", "REF is provided so that equivalence classes computed by the SAIC can be seen, all the locations of the grid shaded with the same colour belonging to the same equivalence class.", "The SAIC is extremely efficient, in performing state aggregation such that the loss of observation information does not incur any loss of achievable sum of discounted rewards.", "Fig.", "REF -(a), illustrates the state aggregation obtained by the SAIC, for which the achievable sum of discounted rewards is illustrated in Fig.", "REF .", "It is illustrated in Fig.", "REF -(a) that how the SAIC performs observation compression with ratio $R= 3:1$ , while it leads to nearly no performance loss for the collaborative task of the multiagent system.", "Here the definition of compression ratio follows $R = \\frac{*{H\\big ( {\\bf o}_i(t) \\big )}}{*{H\\big ( {\\bf c}_i(t) \\big )}}.$ Figure: State aggregation for multi-agent communication in a two-agent rendezvous problem with grid-worlds of varied sizes and goal locations.", "The observation space is aggregated to four equivalence classes, C=2C=2 bits, and number of training episodes has been K=1500kK=1500k, K=1000kK=1000k and K=500kK=500k for figure (a) and (b) and (c) respectively.", "Locations with similar color represent all the agents' observations which are grouped into the same equivalence class.", "The information compression ratio RR has seen to be 6:2, 5:2 and 4:2 in subplots a), b) and c) respectively.", "Fig.", "REF , allows us to see how precise the approximation of $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ by the value function $V^{*}\\big ({\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ is.", "The figure illustrates the values for both $V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(1),\\tilde{{\\bf c}}_j(1)\\big )$ and $V^{*}\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t)\\big )$ , where ${\\mathsf {o}}_i(t)=21$ and for all ${\\mathsf {o}}_j(t)$ takes all possible values in $ \\Omega $ .", "For instance the values $7.2$ mentioned on the right down corner of the grid demonstrates the value of $V^{*}\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t)\\big )$ when ${\\mathsf {o}}_i(t)=21$ and ${\\mathsf {o}}_j(t)=8$ .", "Figure: The approximation of the value function V π i m * ,π i c 𝐨 i (t) , 𝐜 ˜ j (t)V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big ) by V * 𝗈 i (t) , 𝗈 j (t)V^{*}\\big ({\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t) \\big ) following eq.", "() is evaluated here.", "Left grid-world shows the observation space of Ω\\Omega , amongst which one particular observation is chosen 𝗈 i (t)=21{\\mathsf {o}}_i(t)=21.", "While agent ii makes this observation, agent jj can potentially be at any other 64 locations of the greed.", "The value function V * 𝗈 i (t) = 21 , 𝗈 j (t)V^{*}\\big ({\\mathsf {o}}_i(t)=21,{\\mathsf {o}}_j(t) \\big ) for all 𝗈 j (t)∈Ω{\\mathsf {o}}_j(t) \\in \\Omega is depicted in the right grid-world, e.g.", "a number at location 23, shows the value function V * 𝗈 i (t) = 21 , 𝗈 j (t) = 23=13V^{*}\\big ({\\mathsf {o}}_i(t)=21,{\\mathsf {o}}_j(t)=23 \\big ) = 13.", "You can also see the values of V π i m * ,π i c 𝐨 i (t) , 𝐜 ˜ j (t)V_{{\\pi ^m_i}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big ) for 𝗈 i (t)=21{\\mathsf {o}}_i(t)=21 and all possible 𝖼 ˜ j (t)∈𝒞\\tilde{{\\mathsf {c}}}_j(t) \\in \\mathcal {C} with C=2C=2 bits.We also investigate the impact of achievable bit rate $C$ on the achievable value of objective function for the LBIC, SAIC and CIC, in Fig.", "REF .", "In this figure, the normalized value of achieved objective function for any scheme at any given $C$ is shown.", "As per (REF ), the average of the attained objective function for the scheme of interest is computed by $\\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M})}\\big {\\lbrace } {\\bf g}(1) \\big {\\rbrace }$ , where $\\pi ^m_i(\\cdot )$ and $\\pi ^c_i(\\cdot )$ are obtained by the scheme of interest after solving (REF ) with a given value of $C$ .", "The attained objective function for the scheme of interest is then normalized by dividing it to the average of objective function $ \\mathbb {E}_{p_{\\pi ^{*}}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M})}\\big {\\lbrace } {\\bf g}(1) \\big {\\rbrace } $ that is attained by the optimal centralized policy $\\pi ^{*}(\\cdot )$ .The policy policy $\\pi ^{*}(\\cdot )$ is the optimal solution to (REF ) under no communications constraint.", "$ \\frac{ \\mathbb {E}_{p_{\\pi ^m_i,\\pi ^c_i}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M})}\\big {\\lbrace } {\\bf g}(1) \\big {\\rbrace } }{ \\mathbb {E}_{p_{\\pi ^{*}}(\\lbrace {\\mathsf {tr}}(t)\\rbrace _{t=1}^{t=M})}\\big {\\lbrace } {\\bf g}(1) \\big {\\rbrace } }$ Accordingly, when the normalized objective function of a particular scheme is seen to be close to the value 1, it implies that the scheme has been able to compress the observation information with almost zero loss in the achieved objective function.", "On one hand, it is demonstrated that the SAIC soon achieves the optimal performance, while it takes the CIC at least $C=4$ bits to get close to achieve a sub-optimal value of the objective function.", "The LBIC, on the other hand, provides more than 10% performance gain in very low rates of communication $C \\in \\lbrace 1,2,3\\rbrace $ bits per time step, compared with CIC and 20% performance gain compared with SAIC at $C=1$ bits per time step.", "Figure: A comparison between the performance of several multi-agent communication and control schemes in terms of the achieved value of the objective function under different achievable bit rates.", "All experiments are performed on a grid-world of size N=8N=8, where the goal point is located on the grid no.", "22, similar to the one depicted on Fig.", "-a.", "The number of training episodes/iterations for any scheme at any given value of CC has been K=200KK=200K.Fig.", "REF , being closely related to Fig.", "REF , studies the normalized objective functions attained by the LBIC, SAIC and CIC under different compression ratios $R$ .", "A whopping 40% performance gain is acquired by the SAIC, in comparison to the CIC, at high compression ratio $R=3:1$ .", "This means 66% of data rate saving with no performance drop in attaining the collaborative objective function.", "The SAIC, however, underperforms the LBIC and CIC at very high compression ratio of $R=6:1$ .", "This is due to the fact that the condition mentioned in remark 2 is not met at this high rate of compression.", "Moreover, the CIC scheme is seen not to achieve the optimal performance even at compression rate of $R=6:5$ which is due to the fact that by exceeding the compression ratio $R=1:1$ each agent $i$ may loose some information about the observation ${\\mathsf {o}}_j(t)$ of the other agent which can helpful in taking the optimal action decision.", "Figure: A comparison between the performance of several multi-agent communication and control schemes in terms of the achieved value of the objective function under different rates of information compression.", "All experiments are performed on a grid-world of size N=8N=8, where the goal point is located on the grid no.", "22, similar to the one depicted on Fig.", "-a.", "The number of training episodes/iterations for any scheme at any given value of CC has been K=200KK=200K.Fig.", "REF , illustrates the advantage of high resolution grid for a rendezvous problem.", "Although the channel rate $C$ and information compression scheme has been similar in both curves, a large performance gap is visible between the two.", "In fact, the only difference is that the centralized training of the SAIC LR has been carried out on a grid world with reduced resolution.", "This experiment reveals that by naively reducing the resolution of the grids, an acceptable performance in the rendezvous problem cannot be achieved.", "All curves which were obtained by applying the SAIC in the previous experiments, shown in the earlier figures, were the result of SAIC HR.", "Figure: Effect of grid/spatial resolution in the rendezvous problem.", "In SAIC HR, the centralized training phased is performed on a grid-world of size N=8N=8 with the goal point ω T =21\\omega ^T=21, similar to the one depicted on Fig.", "-a.", "In SAIC LR, however, the centralized training phased is performed on a grid-world of size N=4N=4 with the goal point being ω T =7\\omega ^T=7 and K=200 ' 000K=200^{\\prime }000.As demonstrated through a range of numerical experiments, the weakness of conventional schemes for compression of agents' observations is that they may lose/keep information regardless of how useful they can be towards achieving the optimal objective function.", "In contrast, the task-based compression schemes SAIC and LBIC, for communication rates (very) lower than the entropy of the observation process, manage to compress the observation information not to minimize the distortion but to maximize the achievable value of the objective function." ], [ "Conclusion", "This paper has investigated a distributed multiagent reinforcement learning problem in which agents share a unique task, maximizing the average discounted cumulative one-stage rewards.", "Since we consider a limited rate for the multiagent communication channels, task-based compression of agents observations has been of the essence.", "The so-called task-based compression schemes introduced in this paper, are different from the conventional source coding algorithms in the sense that they do not aim at achieving minimum possible distortion given a communication rate, but rather, they maximize the objective function of the multiagent system given a communication rate between agents.", "The proposed schemes are seen to outperform conventional source coding algorithms, by up to a remarkable 40% difference in the achieved objective function, when being imposed with (very) tight constraint on the communication rate.", "The introduced information compression schemes can have a substantial impact in many communication applications, e.g.", "device to device communications, where the ultimate goal of communication is not a reliable transfer of information between two ends but is to acquire information which is useful to improve an achievable team objective.", "Our scheme is of more relevance to scenarios where observation process which is to be compressed is generated by an MDP and is not and i.i.d process.", "The studies in this paper have been limited to a team of two agents with symmetric constraints on the communication rates.", "Accordingly, considering a system composed of larger number of agents as well as non-symmetric rate constraints for the same problem can be useful avenues to extend the applicability of the introduced schemes." ], [ " In Algorithm 4, each agent $i$ selects its communication action ${\\mathsf {c}}_i(t)$ by solving $ {\\mathsf {c}}_i(t) = \\underset{{\\mathsf {c}}\\in \\mathcal {C}}{\\text{argmax}} \\,\\, Q^{c,(k)}_i\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}\\big ) +k \\sqrt{\\frac{\\mathrm {ln}(T_t)}{N^c_{i}\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}\\big )}}.$ The communication Q-table $Q^{c,(k)}_i(\\cdot )$ is updated following off-policy Q-learning $\\begin{aligned}&Q^{c,(k)}_i \\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big ) \\leftarrow (1-\\alpha ) Q^{c,(k)}_i \\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big ) +\\\\&\\alpha \\gamma \\Big (\\!", "{r_t}+ \\underset{{\\mathsf {m}}\\in \\mathcal {M}}{\\text{max}}\\, Q^{c,(k)}_i \\big ({\\mathsf {o}}_i(t+1),\\tilde{{\\mathsf {c}}}_j(t+1),{\\mathsf {m}}\\big )\\!\\Big ).\\end{aligned}$ The output of the algorithm includes communication policy $\\pi ^{c,(k)}_i(\\cdot )$ for each agent $i$ , where policy can be computed by $\\pi _{i}^{c,(k)}\\big ({\\mathsf {c}}_i(t)|{\\mathsf {o}}_i(t)\\big )=\\delta \\Big (\\!", "{\\mathsf {c}}_i(t) -\\underset{{\\mathsf {c}}\\in \\mathcal {C}}{\\text{argmax}}\\, Q^{c,(k)}_i \\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}\\big )\\!\\Big )\\!.$ Distributed Learning of Communication Policies [1] Input: $\\gamma $ , $\\alpha $ , $c$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ and $Q^{m,(k)}_i(\\cdot )$ , $Q^{c,(k-1)}_i(\\cdot )$ , for $i=1,2$ Initialize all-zero table $N^c_{i}\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big )$ , for $i=1,2$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and Q-table $Q^{c,(k)}_{i}(\\cdot ) \\leftarrow Q^{c,(k-1)}_{i}(\\cdot )$ , for $i=1,2$ each episode $l=1:L$ Randomly initialize local observation ${\\mathsf {o}}_i(t)$ , for $i=1,2$ $t_l = 1:M$ Select ${\\mathsf {c}}_i(t)$ by solving (*), for $i=1,2$ Increment $N^c_{i}\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t)\\big ) $ , for $i=1,2$ Obtain message $\\tilde{{\\mathsf {c}}}_j(t)$ , for $i=1,2 \\;\\; j \\ne i$ Select ${\\mathsf {m}}_i(t) $ by following $\\pi _{i}^{m,(k)}\\big ({\\mathsf {m}}_i(t)|{\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\big )$ , $\\;\\;$ for $i=1,2$ Obtain reward $r\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t),{\\mathsf {m}}_i(t),{\\mathsf {m}}_j(t) \\big )$ , for $i=1,2$ Make a local observation ${\\mathsf {o}}_i(t)$ , for $i=1,2$ Update $Q^{c,(k)}_{i}\\big ({\\mathsf {o}}_i(t),{\\mathsf {c}}_i(t-1)\\big )$ by following (REF ), $\\;\\;\\;\\;\\;\\;$ for $i=1,2$ $t_l=t_l+1$ end Compute $\\sum ^{M}_{t=1} \\gamma ^t r_t$ for the $l$ th episode end Output: $Q^{c,(k)}_{i}(\\cdot )$ , for $i=1,2$ $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\!$ and $\\pi _{i}^{c,(k)}\\big (\\cdot )$ by following (REF ), for $i=1,2$" ], [ "Proof of Lemma 1", "We know that by following the optimal action policy ${\\pi _i^m}^{*}$ , the following holds $ \\begin{aligned}& V_{{\\pi _i^m}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big ) \\ge V_{{\\pi _i^m},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big ),\\\\& \\forall \\pi ^m_i(\\cdot ), \\,\\, \\forall \\tilde{{\\bf c}}_i(t)\\in \\lbrace -1,1\\rbrace ^B , \\,\\, \\forall {\\bf o}_i(t) \\in \\Omega .\\end{aligned}$ According to (REF ), a weighted sum of $ V_{{\\pi _i^m}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )$ will always remain larger than or equal to the sum of $ V_{{\\pi _i^m},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )$ weighted with the same coefficients: $ \\begin{aligned}& \\mathrm {E}_{p\\big ( {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t) \\big )} \\Big {\\lbrace }V_{{\\pi _i^m}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )\\Big {\\rbrace } \\ge \\\\& \\mathrm {E}_{p\\big ( {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t) \\big )} \\Big {\\lbrace }V_{{\\pi _i^m},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )\\Big {\\rbrace },\\end{aligned}$ where $p\\big ( {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t) \\big )$ can be any joint probability mass function of $ {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t)$ .", "That is to say $\\begin{aligned}& \\mathrm {E}_{p\\big ( {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t) \\big )} \\Big {\\lbrace }V_{{\\pi _i^m}^{*},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )\\Big {\\rbrace } = \\\\& \\underset{\\pi ^m_i}{\\text{max }}\\;\\;\\;\\mathrm {E}_{p\\big ( {\\mathsf {o}}_i(t), \\tilde{{\\mathsf {c}}}_i(t) \\big )} \\Big {\\lbrace }V_{{\\pi _i^m},\\pi ^c_i}\\big ({\\bf o}_i(t),\\tilde{{\\bf c}}_j(t)\\big )\\Big {\\rbrace }.\\end{aligned}$" ], [ "Proof of Lemma 2", " $& V^{*}\\big ({\\mathsf {o}}_i(t^{^{\\prime }}),\\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }})\\big ) = \\\\& \\mathbb {E}_{p\\big ( \\lbrace tr\\rbrace ^{M}_{t^{^{\\prime }}} | {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\big )} \\Big {\\lbrace } \\\\& \\sum _{t=t^{^{\\prime }}}^M \\gamma ^{t-1} r\\big ({\\bf o}_i(t), {\\bf o}_j(t),{\\bf m}_i(t), {\\bf m}_j(t)\\big )| {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\Big {\\rbrace } = \\\\& \\mathbb {E}_{p\\big ( \\lbrace {\\mathsf {tr}}\\rbrace ^{M}_{t^{^{\\prime }}} | {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t) \\big )}\\Big {\\lbrace }{\\bf g}(t^{^{\\prime }})| {\\mathsf {o}}_i(t),\\tilde{{\\mathsf {c}}}_j(t)\\Big {\\rbrace } = \\\\& \\sum _{\\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M}} {\\bf g}(t^{^{\\prime }})p\\Big ( \\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M} | {\\mathsf {o}}_i(t^{^{\\prime }}), \\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }}) \\Big ),$ where the conditional probability $p\\Big ( \\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M} | {\\mathsf {o}}_i(t^{^{\\prime }}), \\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }}) \\Big )$ can be extended following the law of total probabilities $& V^{*}\\big ({\\mathsf {o}}_i(t^{^{\\prime }}),\\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }})\\big ) =\\sum _{\\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M}} {\\bf g}(t^{^{\\prime }})\\Bigg [ \\\\& \\sum _{{\\mathsf {o}}_j(t) \\in \\Omega } p\\Big ( \\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M} | {\\mathsf {o}}_i(t^{^{\\prime }}), {\\mathsf {o}}_j(t^{^{\\prime }}),\\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }}) \\Big ) p\\big ( {\\mathsf {o}}_j(t) | \\tilde{{\\mathsf {c}}}_j(t) \\big )\\Bigg ]$ in which ${\\mathsf {o}}_i(t^{^{\\prime }}),{\\mathsf {o}}_j(t^{^{\\prime }})$ are sufficient statistics and the second summation can be shifted to have $& V^{*}\\big ({\\mathsf {o}}_i(t^{^{\\prime }}),\\tilde{{\\mathsf {c}}}_j(t^{^{\\prime }})\\big ) = \\\\& \\sum _{{\\mathsf {o}}_j(t) \\in \\Omega }\\underbrace{\\sum _{\\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M}} {\\bf g}(t^{^{\\prime }})p\\Big ( \\lbrace {\\mathsf {tr}}\\rbrace _{t^{^{\\prime }}}^{M} | {\\mathsf {o}}_i(t^{^{\\prime }}), {\\mathsf {o}}_j(t^{^{\\prime }}) \\Big )}_{V^{*}\\big ( {\\mathsf {o}}_i(t),{\\mathsf {o}}_j(t) \\big )} p\\big ( {\\mathsf {o}}_j(t) | \\tilde{{\\mathsf {c}}}_j(t) \\big ).$" ] ]

[ [ "Intelligent Residential Energy Management System using Deep\n Reinforcement Learning" ], [ "Abstract The rising demand for electricity and its essential nature in today's world calls for intelligent home energy management (HEM) systems that can reduce energy usage.", "This involves scheduling of loads from peak hours of the day when energy consumption is at its highest to leaner off-peak periods of the day when energy consumption is relatively lower thereby reducing the system's peak load demand, which would consequently result in lesser energy bills, and improved load demand profile.", "This work introduces a novel way to develop a learning system that can learn from experience to shift loads from one time instance to another and achieve the goal of minimizing the aggregate peak load.", "This paper proposes a Deep Reinforcement Learning (DRL) model for demand response where the virtual agent learns the task like humans do.", "The agent gets feedback for every action it takes in the environment; these feedbacks will drive the agent to learn about the environment and take much smarter steps later in its learning stages.", "Our method outperformed the state of the art mixed integer linear programming (MILP) for load peak reduction.", "The authors have also designed an agent to learn to minimize both consumers' electricity bills and utilities' system peak load demand simultaneously.", "The proposed model was analyzed with loads from five different residential consumers; the proposed method increases the monthly savings of each consumer by reducing their electricity bill drastically along with minimizing the peak load on the system when time shiftable loads are handled by the proposed method." ], [ "Introduction", "Energy generated from power grids fuels the modern lifestyle.", "Per-user consumption of energy is ever increasing.", "People, nowadays, use a lot of modern appliances for their day to day chores.", "With technological advances, the invention of new appliances, and the ever-increasing interest of the new generations in the gadget market, investment in the household appliance market has increased manifold.", "With most of these appliances running on electricity, the rising electricity consumption also increases the load on power grids.", "People nowadays are willing to pay more for electricity rather than live without it.", "Household appliances in the US are responsible for approximately 42% of the total energy consumption [1].", "Given the high demand for electricity, efforts are being made continuously to improve smart grids with advanced research in power system and computer science.", "Rising energy requirement increases the load on power grids.", "Also, energy consumption follow specific trends that lead to disparity in demands from grids based on the time of the day, i.e, energy demand during particular periods can be higher than usual, whereas during other periods of the day energy requirements can be pretty low.", "During peak hours, the load on power grids increases drastically.", "To avert this problem, Demand Side Management (DSM) strategies are used.", "DSM strategies involve Demand Response (DR), energy efficiency, and conservation schemes.", "Figure: Proposed architecture of smart gird with RL-DSM at consumer end.DR [2], [3] focuses on modifying consumers' demand on the benefit of reducing peak load on the grid and in turn, giving some incentives back to customers who participate in it.", "DR encourages consumers to use less power during peak hours of the day or shift their time of energy usage to off-peak hours of the day.", "Examples of DR may include storing energy during the off-peak periods from the grid and using the stored energy during the peak period.", "One could also make use of renewable sources of energy by storing these energies such as solar, wind, geothermal and biogas, which could be used during the peak periods of the day.", "The benefits of DSM are discussed in [4].", "Home Energy Management (HEM) [5] is a part of DSM.", "HEM systems manage the usage of electricity in smart homes.", "DR is a crucial component of HEM systems.", "DR [6] is one of the techniques used in DSM.", "It involves scheduling loads on the timescale by moving high wattage loads to a different time to reduce the maximum load on the grid without changing the net energy consumed.", "It focuses on changing the “when consumed\" rather than changing the “how much consumed\".", "Electricity generation involves several generating stations employing different generation technologies working in conjunction, thus making the process dynamic in its behavior.", "This translates to varying costs of electricity generation at any given point in time.", "This is where load shifting comes into play.", "Scheduling the load on the timescale that would reduce the overall peak demand on the grid and also saves electricity bills for the consumers.", "Most of the DR based optimization models are based on two broad categories: price-based and incentive-based.", "Price based optimization is discussed in the study conducted in this paper.", "The price-based models consider Time of Use (ToU) pricing, peak load pricing, critical peak pricing, and real-time pricing [7], [8], which take into account the peak and off-peak tariffs.", "The varying tariffs of energy consumption during the entire duration of a day based on the aggregate load on the electrical systems act as a motivation for consumers to adjust their appliance usage to take advantage of the lower prices during specific periods.", "Incentive-Based DR programs are two types: Classical programs and Market-based programs.", "Classical programs include Direct Load Control and Interruptible programs, Market-based are Emergency DR, Demand Bidding, Capacity Market, and Ancillary services market [9].", "In [10], the authors proposed a pricing scheme for consumers with incentives to achieve a lower aggregate load profile.", "They also studied the load demand minimization possible with the amount of information that consumers share.", "In [11] linear and nonlinear modeling for incentive-based DR for real power markets was proposed.", "System-level dispatch of demand response resources with a novel incentive-based demand response model was proposed by [12].", "In [13], the authors proposes an Interruptible program, including penalties for customers in case they do not respond to load reduction.", "A real-time implementation of incentive-based DR programs with hardware for residential buildings is shown in [10].", "In [14] a novel DR program targeting small to medium size commercial, industrial, and residential customers is proposed.", "Reinforcement Learning (RL) [15] is an area of machine learning in computer science where a learning agent interacts with an environment and receives rewards as feedback of the interaction with the ultimate goal of maximizing the cumulative reward.", "To achieve this, RL agents attempt to come up with a policy mapping states to the best action at any given state, which would result in maximum expected future rewards.", "Well designed RL agents have displayed impeccable decision-making capabilities, such as Google's AlphaGo and OpenAI Five, in complex environments without the requirement of any prior domain knowledge.", "Deep Reinforcement Learning (DRL) [16] is a merger of deep learning and reinforcement learning where deep learning architectures such as neural networks are used with reinforcement learning algorithms like Q-learning, actor-critic, etc.", "[17] discusses building DRL agents for playing Atari games and how DQN (Deep Q-Network) shows exceptional performance in playing the games.", "The proposed work aim at applying deep reinforcement learning techniques to the scenario of load shifting and comparing the results obtained with that of the MILP [18], [19], [20] based methods.", "We also propose a smart grid architecture, as shown in Figure REF where RL based DSM controller can be placed at the consumer end.", "The grid end DSM with RL is an extension of this work, where raw loads come from residential microgrids instead of individual homes.", "An automated RL agent performing the task of load shifting would go a long way into optimizing load consumption by distributing loads from peak to off-peak periods, thus reducing the total load on the power grid during peak periods and reducing the energy bills of consumers.", "The main contributions of the proposed work are: Introduced Deep reinforcement learning in Demand Side Management (RL-DSM) for DR.", "Analyzed that the impact of a well-calculated reward system is crucial for Demand Side Management Reinforcement Learning models.", "The proposed reinforcement learning model surpassed traditional methods with a single objective by saving 6.04% of the monthly utility bill.", "The proposed reinforcement learning model with multi-objective saved 11.66% of the monthly utility bill, which shows the superior ability of the learning model over traditional methods for Demand Side Management." ], [ "Related Work", "Demand response optimization is extensively explored in literature.", "[21] gives an overview of DSM with various types and the latest demonstration projects in DSM.", "[22] discusses a greedy iterative algorithm that enables users to schedule appliances.", "[23] presents linear programming based load scheduling algorithms.", "Mixed-integer linear programming model for optimal scheduling of appliances has been discussed in [18], [19], [20].", "Heuristic-based scheduling algorithms that aim at cost minimization, user comfort maximization, and peak to average ratio minimization have been discussed in detail in [24].", "A constrained multi-objective scheduling model for the purpose of optimizing utility and minimizing cost is proposed in [25].", "A dynamic pricing model for energy consumption cost minimization and comfort maximization in the context of smart homes has been explored in [26].", "A study of various control algorithms and architectures applied to DR was carried out in [27].", "[28] proposes a demand response cost minimization strategy with an air source heat pump along with a water thermal storage system in a building.", "[29] studies various energy demand prediction machine learning models like feed-forward neural network, support vector machine, and multiple linear regression.", "[30] proposes a systematic method to quantify building electricity flexibility, which can be an assuring demand response resource, especially for large commercial buildings.", "[31] studies the effect of demand response on energy consumption minimization with the heat generation system in small residential builds in a more frigid climate.", "Increasing demand for energy has shifted the focus of the power industry to alternative renewable sources of energy for power generation.", "Integrating renewable sources of energy into HEM systems can be beneficial for both customers and power generation companies.", "Renewable sources of energy can fill in the excess demand for electricity by customers, thus moderating the peak loads on the grids.", "Aghaei et al.", "carried out studies on DR using renewable sources of energy [32].", "[33] introduces a two-stage power-sharing framework.", "[34] talks about a cloud-based DSM method where consumer also has local power generation with batteries.", "[35] discusses the design for smart homes with the integration of renewable energy sources for peak load reduction and energy bill minimization.", "[36] introduces a real-time decentralized DSM algorithm that takes advantage of energy storage systems (ESS), renewable energy, and regularizing charging/discharging.", "[37] proposes a method to improve DSM by optimizing power and spectrum allocation.", "[38] proposes a hierarchical day-ahead DSM model with renewable energy.", "[39] discusses an Intelligent Residential Energy Management System (IREMS), which offers a model with in-house power generation using renewable sources of energy.", "[40] proposes an algorithm to minimize consumer cost by facilitating energy buying and selling between utility and residential community.", "[41] offers a multi-objective to demand response program by reducing the energy cost of residential consumers and peak demand of the grid.", "[42] introduces collaborative energy trading and load scheduling using a game-theoretic approach and genetic algorithm.", "[43] propose a game-theoretic approach to minimize consumer energy cost and discomfort in a heterogeneous residential neighborhood.", "[44] reduces the overall cost of the system by optimizing the load scheduling and energy storage control simultaneously with Lyapunov optimization.", "[45] proposes an intelligent residential energy management system for residential buildings to reduce peak power demand and reducing prosumers' electrics bills.", "[46] introduces an automated smart home energy management system using L-BFGS-B (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) algorithm with time-of-use pricing to optimize the load schedule.", "[47] introduces an RL model to meet the overall demand with the current load, and the next 24 hrs load predicted information by shifting loads.", "[48] formed a fully automated energy management system by decomposing rescheduling loads over device clusters.", "[49] proposes the Consumer Automated Energy Management System (CAES), an online learning model that estimates the influence of future energy prices and schedules device loads.", "[50] proposes a decentralized learning-based multi-agent residential DR for efficient usage of renewable sources in the grid.", "Lu [51] dynamic pricing DR algorithm formulated as an MDP to promote service providers profit, reduce costs for consumers, and attain efficiency in energy demand and supply.", "[52] introduced RL based scheduling of controllable load under a day-ahead pricing scheme.", "This work encroaches into the territory of applying DRL to DR.", "Existing deep reinforcement learning-based model for DR, which came out recently are [53], [54].", "[53] discusses an RL and DNN based algorithm modeled on top of real-time incentives.", "A complete review of a collection of all control algorithms and their assessment in DR strategies in the residential sector is given in [54].", "It discusses machine learning-based predictive approaches and rule-based approaches.", "Applying deep reinforcement learning models to the setting of DR can be a lucrative field of research.", "The proposed model draws inspiration from DRL agents used to play the game of Tetris.", "We have explored models of DRL automated agents playing the game of Tetris and have tried to look for areas of similarity in the formulation of the reward function.", "[55] discusses the design of a DQN model for playing Tetris and presents the reward system of an automated Tetris agent.", "Figure: Demonstrates the difference in block settling in game environment (left) and proposed DR simulation (right).", "Red, Green and Blue blocks aren't supported by the grid base or another load block, thus they are allow to be broken down and slid down to lower level in the simulation.", "Figure best viewed in color." ], [ "Methods", "Reinforcement learning has shown tremendous strides in learning to take action in game environments surpassing humans.", "Bringing RL capability to DR created the need to model DR into a game environment.", "An Atari game called Tetris falls in very close with the simulation environment needed for DR.", "In Tetris, the player is asked to move blocks of different sizes and shapes in a 2D grid.", "Actions the player can take on the block are rotation, moving left, right, or down.", "The player is rewarded with points when a full line in the grid is filled, and the game terminates when the block placed touches the maximum height of the grid.", "We adapted this game environment to build a simulation to perform DR.", "The blocks in the game will be device loads in the DR simulation.", "The player will be replaced by an RL agent who will take action like moving load blocks down, left, and right.", "Unlike solid blocks in the game environment, the load blocks in the DR simulation are flexible solids, i.e., if part of the load in the grid is not supported by the grid base or another load block, it is allowed to slide down to the lower level as shown in Figure REF .", "The agent reward is determined by the load peak generated when a block is placed in the simulation grid.", "A positive reward is given if it doesn't increase the current maximum load peak and a negative reward when the current maximum load peak increases.", "The simulation ends when the load placed crosses the maximum height, which motivates the RL agent to place more load in the simulation grid without generating peaks." ], [ "Simulation Design", "The simulation is modeled to represent loads on a timescale of 24 hours.", "The simulation environment consists of 24 columns, each column representing an hour on the 24-hour clock.", "The height of each column depicts the aggregate load on the electrical system at that particular hour.", "Ideally the height of the simulation grid is decided by the maximum aggregate load from the historical load profiles of the consumers in a specific grid.", "As a proof of concept, a maximum of 25kW of the aggregate load is set while training and testing, this can be scaled according to the size of the power grid.", "If the aggregate load at any hour exceeds the constraints on maximum load, the simulation terminates." ], [ "Delineating States and Actions", "The learning algorithm stores data in the form of (state, action, next state, reward) tuples.", "The state is an image of the current simulation screen.", "The action space is defined in terms of three actions: left, right and drop which imply left move, right move and dropping the block onto the simulation respectively.", "All actions are single actions; i.e., for any state transition, only a single action can be performed, there are no cases where a combination of the individual actions can be performed for a state transition.", "Taking action at any state causes the game to transition to a new state that generates a reward.", "At any point of time if the agent decides on one of the actions the simulation moves to a new state.", "A right action shifts the current load one cell (representing timestamp) to the right and the new state is the load block shifted one block to the right.", "A left action shifts the block of load one cell to the left and the simulation transitions to a new state with the load block shifted one cell to the left.", "A drop action results in the load block being placed on the load profile and a subsequent state transition with the load block now placed on the load profile immediately below the load block.", "Actions in this sense are discrete and not continuous.", "At any point of time the state of the simulation is the load profile and the load block.", "This is captured on every action.", "The state transitions are finite as there can be finite shapes of blocks and they can be placed in on 24 timestamps.", "State transitions are deterministic in the the sense that given a state, block, action we can always predict the next state." ], [ "Deep-Q-Learning (DQN)", "DQN [16] agent based on the idea of using a neural network to approximate the below Q as shown in Equation REF , and the pipeline of this method can be found in Algorithm REF .", "$\\begin{aligned}Q^{\\pi }(s, a) = \\max _{\\pi } \\mathbb {E}_{\\pi }[ r_{t} + \\gamma r_{t+1} +\\gamma ^ {2} r_{t+2} + \\\\ ..... | s_{t}=s, a_{t}=a]\\end{aligned}$ where $a_{t}$ is the action and $s_{t}$ is the state at time t, $r_{t}$ is the reward obtained by taking action at given state $s_{t}$ , $\\pi $ denote policy function defines the learning agent's way of behaving at a given time and $\\gamma $ is the discount factor.", "The Q-function can be simplified as: $Q^{*}(s, a) = \\mathbb {E}_{s^{\\prime }} [ r + \\gamma \\max _{a^{\\prime }} Q^{*}(s^{\\prime }, a^{\\prime }) | s, a]$ where $s^{\\prime }$ is the state generated by performing action $a$ in state $s$ and $a^{\\prime }$ denoted the action taken in state $s^{\\prime }$ .", "The DQN model used in the proposed method is a Double DQN [56].", "Double DQNs handles the problem of the overestimation of Q-value.", "This consists of two networks, namely a policy network and a target network where all changes to the gradients are made on the policy network, which is synced with the target network at regular intervals of episodes.", "Episode is the length of the simulation at the end of which the system ends in a terminal state.", "DQN network is used to select the best action to take for the next state (the action with the highest Q-value).", "The target network is used to calculate the Q-value of taking that action at the next state.", "Figure: Network architecture used for agent.", "The policy network and target network have the same architecture.", "ConvlayerConvlayer consist of convolution block, batch normalization layer and ReLU activation.", "FCFC stands for fully connected layer.The network consists of three convolution layers and one fully connected layer as shown in Figure REF .", "All convolution layers (CNN) use a $5\\times 5$ kernel and a stride of 2 with batch normalization and ReLU (Rectified Linear Unit) as the activation function.", "The last hidden layer is a fully connected (FC) layer with 192 units, and the output layer consists of three units for three actions.", "CNN is used over other network architectures because CNN is memory efficient and best for feature extraction.", "The number of parameters in FC drastically increases with the increase in hidden layers when compared to CNN, thus increases computation and memory used.", "Designing a faster model is crucial for applications like load shifting, thus memory efficient and fast neural networks like CNNs are used.", "According to the design of the proposed RL state, CNN is more apt as it is efficient in feature extraction from raw data with spatial structures like simulation screens.", "We have also done an extensive analysis of how the selected CNN network over other affects the load shifting problem in the proposed model, as shown in Table REF and Figure REF .", "Even though the shallow network was able to minimize cost, it creates high load peaks.", "With batch normalization, the model normalizes the input layer by adjusting and scaling the activation.", "Batch normalization reduces the amount by what the hidden unit values of the network shift around (covariance shift), and this is proven in [57] to speed up the learning process.", "RMSProp [58] is used for optimization in the network.", "For training, the network approximates the Q-values for each action and take action with the maximum Q-value.", "The target network estimate the Q-values of the optimal Q-function Q* by using the Bellman Equation REF , since Q-function for any policy obeys the Bellman equation.", "Temporal difference error $\\delta $ is computed by taking the difference of predicted Q-value and the optimal Q-value computed from the Bellman equation using the target network.", "$\\delta = Q(s, a) - ( r + \\gamma \\max _{a^{\\prime }} Q^{*}(s^{\\prime }, a^{\\prime }))$ By minimizing the loss between the Q-value and the optimal Q-value, the agent arrives at the optimal policy.", "Huber loss is used to minimize the error defined as below: $\\mathcal {L} = \\frac{1}{|B|} \\sum _{b \\epsilon B} \\mathcal {L}(\\delta )$ where batches $B$ of experiences (knowledge acquired in the past) where sampled from the agent's replay buffer and $\\mathcal {L}(\\delta )$ is defined as $\\mathcal {L}(\\delta )={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\delta ^{2} & |\\delta |\\le 1 \\\\|\\delta |-\\frac{1}{2} & Otherwise\\end{array}\\right.", "}$ The Huber loss is robust to outliers.", "When the error is small, it takes the definition of mean squared error but acts like the mean absolute error when the error is significant.", "Figure: DQN algorithm" ], [ "Epsilon Greedy Policy and Experience Replay", "Epsilon greedy policy and Experience replay are two crucial techniques that help the learning process of the RL agent drastically.", "Given the fact that the state space is vast, the agent is enabled to explore the available state space initially without taking too many predicted actions from the network.", "When the agent takes a random step at any given state its called exploration and when it uses already accumulated experience to make a predicted decision from the network is called exploitation.", "Exploration and exploitation should not be run exclusively.", "The agent explores many states at first by taking random actions, but with each successive epoch, it increases the number of informed decisions exploiting the known information to maximize the reward.", "The decay function $\\gamma $ used for this purpose is the exponential decay function which is defined below: $\\gamma = \\epsilon _{e} + \\frac{\\epsilon _{s} + \\epsilon _{e}}{e^{\\frac{sd}{\\epsilon _{d}}}}$ where $sd$ is the total number of iterations till now and $\\epsilon _{d}$ is a hyperparameter controlling the rate of decay from $\\epsilon _{s}$ to $\\epsilon _{e}$ .", "One iteration translates to taking one action.", "Note that episodes and iterations have different meanings in this context.", "Hyperparameters are parameters that are manually set by the practitioner and tuned according to a specific goal.", "Table: Daily load demand of five different consumers.", "Rated power 1.0, 0.5 indicate the power rating at first and second hour of the device.Table: Hyper parameters used in experiment.Table: Monthly energy billing scheme.The Experience Replay [59] is used to avoid the agent from forgetting previous experiences as it trains on newer states and to reduce correlations between experiences.", "In Experience Replay, a buffer of the old experiences is maintained and the agent is trained on it.", "By sampling batches of experience from the buffer at random, the correlation between the experiences can be reduced.", "The reason for this is that if you were to sample transitions from the online RL agent as they are being experienced, there would be a strong temporal/chronological relationship between them.", "During our experiments, we fixed buffer size at $3 \\times 10^{4}$ .", "We empirically found the optimal buffer size that works best for our model without allotting huge amount of reserved memory space.", "We have done extensive analysis on picking the right buffer size, which can be found in Table REF and Figure REF ." ], [ "Reward Systems", "The proposed reward system consists of three important aspects, maximum height $h^{max}$ , variance of the height distribution $var(h)$ and number of complete lines $l$ formed from the distribution.", "The reward system $R$ is summarized as: $R = \\alpha _1 * var(h) + \\alpha _2 * l - \\alpha _3 * h^{max}$ The variance of the height distribution $var(h)$ is given by: $var(h) = \\frac{1}{1 + var(h_a)}$ where $var(h_a)$ the variance of the height distribution after taking the action.", "The variance of load distribution reward $var(h)$ would encourage the agent to shift load such that more uniform distribution of height of the load profile can be attained.", "Complete lines $l$ reward complements the agent to increase the spread to have more complete lines (rows) in the simulation.", "This reward component enables the agent to decide to shift a load to either side edges of the simulation.", "Maximum high reward component $h^{max}$ helps the agent to avoid making a peak in the load profile as this reward contributes negative reward if the current maximum high increased.", "Other hyperparameters used for the experiments are shown in Table REF .", "The variance of load distribution encountered some cases where the agent realised that placing the blocks on the same position also lead to increase in variance for a sufficiently well spread out distribution.", "Due to this, initially the agent would perform as expected by placing blocks in a manner which caused the distribution to spread out, but after the certain point it would start placing all the blocks at the same position, resulting in the game terminating with poor results.", "To counter this, standard deviation was utilized to properly scale the impact of distribution on the overall reward vis-à-vis the other reward parameters.", "For simultaneous cost and peak minimization, an additional reward term $c$ is introduced to the peak minimization reward system.", "The cost is calculated with the help of the piece-wise pricing adapted from [60] as shown in Table REF .", "As the price of the load schedule increases, the reward for the agent decreases; the term helps the agent to shift load that results in lower billing price, as shown in the Equation REF .", "$R = \\alpha _1 * var(h) + \\alpha _2 * l - \\alpha _3 * h^{max} - \\alpha _4 * c$ [1]Continuous curves have been used for better visualization instead of the discretized plot even though the load demand is only captured at 24 timestamps.", "Figure: RL-DSM results[1] of residential loads for five different consumers (a), (b), (c), (d) and (e) respectively with Case 1 objective of peak minimization which are compared with MILP.", "(f) is the RL-DSM results of aggregated residential loads for five different consumers to minimize daily peak load with Standard deviation(std RL) and Variance(var RL).", "The figure best viewed in color.Table: Energy bill minimization with peak reduction (Case 1).Figure: RL-DSM results[1] of residential loads for five different consumers (a), (b), (c), (d) and (e) respectively with Case 2 objective of peak and cost minimization.", "(f) is the RL-DSM results of aggregated residential loads for five different consumers to minimize daily peak load and cost.", "The figure best viewed in color.Table: Energy bill minimization with peak and cost reduction (Case 2).Figure: RL-DSM results[1] of aggregated residential loads of five different consumer to minimize daily peak load and cost (a) with Shallow (After with Shallow) and Deep network (After with Deep) (b) using different memory buffer size like 3000 (3k), 10000 (10k), 30000 (20k) and 50000 (50k).", "The figure best viewed in color.Table: Demonstration of deep CNN network's efficiency compared to shallow network.", "Shallow Net: Shallow network and Deep Net: Deep network.Table: Demonstration of the effect of different memory buffer size like 3000 (3k), 10000 (10k), 30000 (20k) and 50000 (50k) in RL-DSM for peak and cost minimization." ], [ "Simulation Results and Discussion", "The proposed RL model is tested in two different case studies with five customers load data adapted from [18], as shown in Table REF .", "In case 1, the utility energy bill is reduced with peak minimization objective.", "Case 2 reduces the utility energy bill by lowering peak and cost simultaneously.", "The models based on the proposed peak minimization technique showcase better results than MILP." ], [ "Case 1", "The objective of this case study is to minimize peak load demand with RL and thereby reducing the energy bill of the consumer.", "The model is tested with one objective on five different consumers, as discussed in the above sections.", "DR with RL (using DQN agent) scheduled load is compared with MILP scheduling for all consumers in Figure REF .", "Figure REF shows how the system reacts to the aggregated load of all consumers.", "The effect of selecting the right reward function of the RL model for scheduling each consumer load is shown in Table REF by comparing variance and standard deviation as a reward function.", "The variance reward system outperformed the standard deviation for all test consumer data.", "As shown in Table REF , the aggregated load reduced the overall bill cost from $432 to $391.95, saving around 9.27%.", "From the results, it can be inferred that the proposed method could reduce the monthly bill dramatically when compared to other traditional methods on single peak objective minimization." ], [ "Case 2", "In this case study, two different objectives are minimized simultaneously with RL.", "Here, the model reduces the peak load and cost of the load schedule.", "This hybrid multi-objective function guarantee that the load profile does not have high peaks, and at the same time, the aggregated price of the load per day is minimum, as seen in Figure REF .", "It is observed that the loads from peak hours have been moved to min peak or low peak time to minimize high power demand.", "Taking load cost into account trains the model to understand that moving most loads from peak to non-peak hours is not entirely ideal for the consumer as it may not shift the load to the time duration at lower prices.", "Adding the cost factor helps the agent to move the blocks to the lower-priced time duration, which currently has fewer loads.", "However, the experiments with single cost objective, which only considers cost and disregards peak minimization entirely, shows that non-peak hours started to get high peak loads as expected because the reward system for the agent focused solely on cost minimization.", "This multi-objective function solves this issue.", "Adding peak minimization with cost minimization assures that the agent is not fixated on cost minimization, causing peaks to be formed during a time when the cost of electricity consumption is less.", "Thus, both the parameters of the multi-objective function, peak and cost minimization are mutually beneficial for one another.", "Figure REF shows how the system reacts to the aggregated load of all consumers.", "As shown in Table REF , the aggregate load reduced bill cost from $432 to $381.6, saving around 11.66%.", "The current limitation of the proposed system is that it does not support dynamic preferred time for each device, and load demand is on the scale of 0.5kW.", "This is the reason that the daily test load demand shown in Table REF has a 1hour/unit time scale and 0.5kW/unit power demand scale to fit the proposed RL simulator.", "Figure: RL-DSM results[1] on loads of 14 different devices to demonstrate the scalability of the proposed method.As traditional methods like MILP does not have a holistic knowledge of the system, it can be only used for a fixed number of devices.", "If the traditional system needs to accommodate more devices at some point in time after deployment, the method needs to be rerun, and scheduling factors should be computed again.", "The proposed methods are more flexible as it has the learning capability to handle any load given to it.", "Even after the proposed RL model is deployed, there is no need for any adjustment to add or remove devices from the system at any time instant.", "The time complexity of the proposed method is better than other traditional methods and its linearly proportional to number of devices, unlike traditional methods that have exponential growth in time concerning the number of devices to be scheduled.", "A trained RL agent always operates with a fixed number of floating-point arithmetic (FLOPs) and multiply-adds (MAdd) for any number of loads to be scheduled.", "The number of FLOPs and MAdd only depend on the architecture of the network.", "This makes the proposed model schedule hundreds of devices time and space-efficient than traditional methods.", "The time complexity of RL can be defined as $\\mathcal {O}(kp)$ and MILP would be $\\mathcal {O}(k^2p)$ where $p$ is the number of parameters associated with the model and $k$ is the number of devices to be scheduled.", "The space complexity of RL can be formulated as $\\mathcal {O}(p)$ , whereas for MILP would be $\\mathcal {O}(k^2)$ .", "The scalability of the proposed methodology is demonstrated by scheduling 46 loads of 14 different devices but with power-hungry devices with RL agent as shown in Figure REF .", "Deploying trained RL agents in an embedded device is as easy as deploying traditional models." ], [ "Conclusion", "The exponential growth in demand for power in the household has increased the stress on the power grid to meet its demand.", "DR can help a smart grid to improve its efficiency to meet the power need of the customer.", "This paper proposes a residential DR using a deep reinforcement learning method where both load profile deviation and utility energy bills are minimized simultaneously.", "An extensive case study with a single objective and multi-objective cases is conducted.", "In both cases, the proposed method outperformed the traditional MILP method.", "This paper exhibited the potential of reinforcement learning for better smart grid operations and tested all these cases on five different consumers and showcased the ability of the proposed method.", "In the future, this work can be extended to introduce variable preferred time for each device and improve the RL simulation grid to accommodate devices with lower power demand with a scale of 0.1kW.", "This work can also be extended to schedule more granular timed devices with less than 1hr to complete its task.", "In this work, renewable energy sources and energy storage are not considered.", "Designing an RL agent to manage renewable energy and energy storage can bring out the full potential of AI models in energy management.", "Supplementary Material: Intelligent Residential Energy Management System using Deep Reinforcement Learning Figure: RL-DSM results[1] of residential loads for five different consumers (a), (b), (c), (d) and (e) respectively with Shallow (After with Shallow) and Deep network (After with Deep).", "(f) is the RL-DSM results of aggregated residential loads for five different consumers to minimize daily peak load and cost.", "The figure best viewed in color.Figure: RL-DSM results[1] of residential loads for five different consumer (a), (b), (c), (d) and (e) respectively using different memory buffer size like 3000(3k), 10000(10k), 30000(20k) and 50000(50k).", "(f) is the RL-DSM results of aggregated residential loads for five different consumers to minimize daily peak load and cost.", "The figure best viewed in color." ] ]

[ [ "Depth-aware Blending of Smoothed Images for Bokeh Effect Generation" ], [ "Abstract Bokeh effect is used in photography to capture images where the closer objects look sharp and every-thing else stays out-of-focus.", "Bokeh photos are generally captured using Single Lens Reflex cameras using shallow depth-of-field.", "Most of the modern smartphones can take bokeh images by leveraging dual rear cameras or a good auto-focus hardware.", "However, for smartphones with single-rear camera without a good auto-focus hardware, we have to rely on software to generate bokeh images.", "This kind of system is also useful to generate bokeh effect in already captured images.", "In this paper, an end-to-end deep learning framework is proposed to generate high-quality bokeh effect from images.", "The original image and different versions of smoothed images are blended to generate Bokeh effect with the help of a monocular depth estimation network.", "The proposed approach is compared against a saliency detection based baseline and a number of approaches proposed in AIM 2019 Challenge on Bokeh Effect Synthesis.", "Extensive experiments are shown in order to understand different parts of the proposed algorithm.", "The network is lightweight and can process an HD image in 0.03 seconds.", "This approach ranked second in AIM 2019 Bokeh effect challenge-Perceptual Track." ], [ "Introduction", "Depth-of-field effect or Bokeh effect is often used in photography to generate aesthetic pictures.", "Bokeh images basically focus on a certain subject and out-of-focus regions are blurred.", "Bokeh images can be captured in Single Lens Reflex cameras using high aperture.", "In contrast, most smartphone cameras have small fixed-sized apertures that can not capture bokeh images.", "Many smartphone cameras with dual rear cameras can synthesize bokeh effect.", "Two images are captured from the cameras and stereo matching algorithms are used to compute depth maps and using this depth map, depth-of-field effect is generated.", "Smartphones with good auto-focus hardware e.g.", "iPhone7+ can generate depth maps which helps in rendering Bokeh images.", "However, smartphones with single camera that don't have a good auto-focus sensor have to rely on software to synthesize bokeh effect.", "Also, already captured images can be post-processed to have Bokeh effect by using this kind of software.", "That is why generation of synthetic depth-of-field or Bokeh effect is an important problem in Computer Vision and has gained attention recently.", "Most of the existing approaches[16], [17], [20] work on human portraits by leveraging image segmentation and depth estimation.", "However, not many approaches have been proposed for bokeh effect generation for images in the wild.", "Recently, [14] proposed an end-to-end network to generate bokeh effect on random images by leveraging monocular depth estimation and saliency detection.", "In this paper, one such algorithm is proposed that can generate Bokeh effect from diverse images.", "The proposed approach relies on a depth-estimation network to generate weight maps that blend the input image and different smoothed versions of the input image.", "The generated bokeh images by this algorithm are visually pleasing.", "The proposed approach ranked 2nd in AIM 2019 challenge on Bokeh effect Synthesis- Perceptual Track[6].", "Depth estimation from a single RGB image is a significant problem in Computer Vision with a long range of applications including robotics, augmented reality and autonomous driving.", "Recent advances in deep learning have helped in the progress of monocular depth estimation algorithms.", "Supervised algorithms rely on ground truth Depth data captured from depth sensors.", "[12] formulated monocular depth estimation as a combination of two sub-problems: view synthesis and stereo matching.", "View synthesis network creates another view of the given RGB image which is utilized by the stereo matching network to estimate the depth map.", "[2] train a multi-scale deep network that predicts pixelwise metric depth with relative-depth annotations from images in the wild.", "[1] trains a depth estimation network on large amount of synthetic data and use image style transfer to map real world images to synthetic images.", "[4] poses Monocular Depth Estimation as an image reconstruction problem.", "They train a Convolutional Neural Network on easy-to-obtain binocular stereo data rather than training on depth ground truth and introduce a novel loss function that checks left-right consistency in generated disparity maps.", "[5] introduces a minimum reprojection loss, an auto masking loss and a full-resolution multi-scale sampling method to train a deep network in self-supervised way.", "Their minimum reprojection loss handles occlusion in a monocular video and auto-masking loss helps network to ignore confusing and stationary pixels.", "[8] created a training dataset using Structure from Motion and Multi-view Stereo methods.", "They trained three different networks using scale-invariant reconstruction and gradient matching loss and ordinal depth loss which achieves good generalization performance.", "Figure: Pipeline of the proposed method.Figure: MegaDepth Architecture.", "Different colors denote different convolutional blocks.", "The last convolutional block is modified in the proposed system." ], [ "Depth Effect in Images", "Early work in Depth-of-field generation in images uses automatic portrait segmentation.", "[16] used a FCN[11] network to segment the foreground in the images and use the foreground for different kind of background editing such as background replacement, BW one color and depth-of-field effect.", "[17] uses a neural network to segment person and its accessories in an image to generate a foreground mask in the case of person images.", "Then they generate dense depth maps using a sensor with dual-pixel auto-focus hardware and use this depth map along with foreground mask (if available) for depth-dependent rendering of shallow depth-of-field images.", "[20] too uses both learning based and traditional techniques to synthesize depth-of-field effect.", "They use off-the-shelf networks for portrait segmentation and single image depth estimation and use SPN[9] to refine the estimated segmentation and depth maps.", "They split the background into different layers of depth using CRF[22] and render the bokeh image.", "For fast rendering, they also learn a spatially varying RNN filter[10]) with CRF-rendered bokeh result as ground truth.", "[14] proposed an end-to-end network for Bokeh Image Synthesis.", "First, they compute depth map and saliency map using pretrained monocular depth estimation and saliency detection networks respectively.", "These maps and the input image are fed to a densely-connected hourglass architecture with pyramid pooling module.", "This network generates spatially-aware dynamic filters to synthesize corresponding bokeh image.", "In this paper, an end-to-end deep learning approach is proposed for Bokeh effect rendering.", "In this approach, the corresponding bokeh image is thought to be a weighted sum of different smoothed versions of the input image.", "The spatially varying weights are learned with the help of a pretrained monocular depth estimation network by modifying the last convolutional layer.", "Compared to some other methods in the literature, this approach doesn't rely on foreground or salient region segmentation, which makes the proposed algorithm lightweight.", "The proposed method generates good bokeh effect rendering both qualitatively and quantitatively.", "This approach came 2nd in Perceptual track of AIM 2019 challenge on Bokeh effect synthesis.", "Figure: Monocular depth estimation result comparison among different state-of-the-art models." ], [ "Depth Estimation Network", "Megadepth[8] is used as Monocular Depth estimation network in this work.", "The authors use an hourglass architecture which was originally proposed in [2].", "The architecture is shown in Figure-REF .", "The encoder part of this network consists of a series of convolutional modules (which is a variant of inception module) and downsampling.", "In the decoder part, there is a series of convolutional modules and upsampling with skip connections that add back features from higher resolution in between.", "Megadepth was trained using a large dataset collected from internet using structure-from-motion and multi-view stereo methods.", "Megadepth works quite well in a wide variety of datasets such as KITTI[3], Make3D[15] and DIW[2].", "This generalization ability of Megadepth makes it suitable for bokeh effect rendering for diverse scenes.", "Figure-REF shows predicted depth maps by Megadepth and other state-of-the-art models.", "We can see that Megadepth produces better quality depth estimation compared to other models." ], [ "Bokeh effect rendering", "The depth-of-field image can be formulated as a weighted sum of the original image and differently smoothed versions of original image.", "These different smoothed images can be generated using different size of blur kernels.", "Concretely, intensity value of each pixel position in the bokeh image can be thought of as a weighted sum of intensity values from original image and smoothed images in that pixel position.", "Hence, the generated bokeh image is given by, $\\hat{I}_{bokeh} = W_0\\odot I_{org} + \\sum _{i=1}^{n} W_i \\odot B(I_{org},k_i)$ where $I_{org}$ is the original image, $B(I_{org},k_i)$ is image $I_{org}$ smoothed by blur kernel of size $k_i\\times k_i$ and $\\odot $ stands for elementwise multiplication, such that for each pixel position $(x,y)$ , $\\sum _{i=0}^{n} W_{i}[x,y] = 1$ In the proposed approach, the weights $W_i$ are predicted with the help of a neural network.", "This can be achieved by modifying the last layer of the pretrained depth estimation network.", "Specifically, in the last layer a convolutional layer with $3\\times 3$ kernel is used along with spatial softmax activation to learn the weights.", "The proposed approach is summarized in Figure-REF .", "In the experiments, $n$ is chosen to be 3 i.e.", "three different smoothed images are used.", "The smoothed images were obtained by applying Gaussian Blur with Blur kernels of size $25\\times 25$ , $45\\times 45$ and $75\\times 75$ .", "The effect of using less number of kernels and kernels of different size are discussed in the following section." ], [ "System Configuration", "The codes were written in Python and Pytorch([13]) is used as the deep learning framework.", "The models were trained on a machine with Intel Xeon 2.40 GHz processor, 64 GB RAM and NVIDIA GeForce TITAN X GPU card with approximately 12GB of GPU memory." ], [ "Dataset Description", "We use ETH Zurich Bokeh dataset[6], which was used in AIM 2029 Bokeh Effect Synthesis Challenge.", "This dataset contains 4893 pairs of bokeh and bokeh-free images.", "Training set contains 4493 pairs whereas Validation and Testing datasets contain 200 pairs each.", "Validation and Testing Bokeh images are not publicly available at this moment.", "This dataset contains a variety of outdoor scenes.", "The normal images were captured using narrow aperture ($f/16$ ) and a high aperture ($f/1.8$ ) was used to generate the Bokeh effect in images.", "The images in this dataset have resolution of around $1024 \\times 1536$ pixels.", "A snapshot of this dataset is given in Figure-REF .", "The training set of ETH Zurich Bokeh Dataset is divided to a train set containing 4400 pairs and a validation set containing 294 pairs for experiments in this paper.", "This validation set is denoted as Val294." ], [ "Loss functions", "Reconstruction loss: $l_1$ loss is used as reconstruction loss to guide the network to generate images with pixel values close to ground truth.", "Reconstruction loss is given by, $l_r = \\left\\Vert \\hat{I}_{bokeh}-I_{bokeh}\\right\\Vert _{1}$ Perceptual loss: Negative SSIM[18] loss is used to improve perceptual quality of the generated images.", "Perceptual loss is given by, $l_p = -SSIM(\\hat{I}_{bokeh}, I_{bokeh})$" ], [ "Training Strategy", "The model is trained in 3 phases.", "In phase-1, the model is trained on smaller resolution images to save training time.", "Each image from training set is resized to a resolution of $384 \\times 512 $ and the network is trained using reconstruction loss.", "In phase-2, the network is trained using images of resolution $768 \\times 1024 $ with reconstruction loss.", "In phase-3, the model is further fine-tuned with perceptual loss $l_p$ ." ], [ "Other training details", "Adam optimizer([7]) is used to train the network.", "The values of $\\beta _1$ and $\\beta _2$ are chosen to be $0.9$ and $0.999$ respectively.", "The initial learning rate is set to be 1e-3 and gradually decreased to 1e-5.", "Each image in the training set is horizontally and vertically flipped to augment the dataset." ], [ "Testing Strategy", "During testing, the input image is first resized to the original dimension in which the network was trained($384 \\times 512 $ for Phase-1 and $768 \\times 1024 $ for Phase-2 and Phase-3) and then passed to the network.", "The synthesized image is then scaled back to the input image resolution using bilinear interpolation." ], [ "Evaluation metrics", "Both fidelity and perceptual metrics are used to evaluate the model's performance.", "Fidelity measures include Peak Signal-to-Noise Ratio(PSNR) and Structural Similarity Index(SSIM)[18].", "LPIPS[21] is used to measure perceptual quality of the generated results.", "Figure: Baseline results from Saliency detection and comparison against the proposed model." ], [ "Quantitative and Qualitative Results", "Baseline.", "Bokeh effect can be generated by simply segmenting the foreground in the image and blurring the rest of the image.", "One such system is used as baseline in this paper.", "A state-of-the-art saliency detection method, Stacked Cross Refinement Network(SCRN)[19] is used for segmenting the foreground in the input image.", "The background is blurred using $75 \\times 75 $ Gaussian kernel.", "As SCRN generates soft saliency maps, bokeh images are generated using Equ-REF .", "Quantitative comparison between Saliency detection based baseline and the proposed method is shown in Table-REF .", "We can see that the proposed approach performs significantly better than the baseline on all three metrics.", "Figure-REF shows saliency maps generated by SCRN and corresponding bokeh effect rendering by baseline and the proposed model.", "We can see that the performance of the baseline is limited to accuracy of saliency detection which leads to artifacts near edges and incorrect blurring of foreground in rendered images.", "Table: Quantitative comparison between Baseline and the proposed approach on Val294 set." ], [ "Visualization", "Examples of generated bokeh images and the corresponding predicted weight maps by our network are shown in Figure-REF .", "We can see that, image regions that are closer have higher weights in $W_0$ and $W_1$ compared to $W_2$ and $W_3$ .", "This allows the model to generate blur effect that is varying with respect to depth i.e.", "closer objects look sharp and distant regions get more blurred with increasing depth in the rendered bokeh image.", "Figure: Visualization of weight maps generated by proposed model.", "Closer objects have larger weights in W 0 W_0 and W 1 W_1 compared to W 2 W_2 and W 3 W_3." ], [ "Ablation Study", "Less number of kernels vs. more number of kernels.", "The main intuition behind using a number of kernels is to try to synthesize bokeh images in a way that regions that are closer to the camera have no or less blur effect whereas distant objects have more blur effect.", "Using less number of kernels will guide network to learn lesser number of such blur levels in the image.", "Using more number of blur kernels helps the network two generate smoother background in synthesized bokeh images.", "Table-REF shows quantitative comparison between using just one blur kernel of size 75 and the proposed approach.", "We can observe in Figure-REF using more number of kernels produces more visually pleasing background.", "Figure: Qualitative Comparison between using different number of blur kernels.Table: Quantitative comparison between using different number of kernels on Val294 set.Smaller kernels vs bigger kernels.", "To demonstrate the effect of using bigger kernels, two different sets of blur kernels were used.", "In one experiment, blur kernel sizes are respectively 5, 25 and 45 and in another experiment blur kernels of size 25, 45 and 75 are used.", "Table-REF shows that using bigger kernels yields better performance.", "Qualitative comparison between these two settings are shown in Figure-REF .", "By analysing weight maps $W_1$ and $W_0$ in both settings, we observe that the value of $W_0$ is more in the in-focus region in case of bigger kernels than of smaller kernels.", "This helps network that uses bigger blur kernels to generate bokeh images that preserve more details in closer objects.", "Figure: Qualitative comparison between Bokeh Images generated using smaller and bigger kernels.", "(e), (f), (g), (h) show weight maps using smaller kernels and (i), (j), (k), (l) show weight maps using bigger kernels.Table: Quantitative comparison between bokeh effect generated by smaller kernels and bigger kernels on Val294 set.Effect of three-phase training.", "The model is trained in three phases.", "As we can see in Table-REF , results from Phase-1 scores the highest PSNR among all the phases, whereas Phase-3 produces best results with respect to SSIM and LPIPS.", "We observe in Figure-REF that Phase-2 and Phase-3 results preserve more details in the closer object than Phase-1 as Phase-2 and 3 models are trained on higher resolution images.", "Also, through fine-tuning with SSIM loss in Phase-3, the model generates a smoother blur effect in the background, as shown in Figure-REF .", "Table: Quantitative Comparison among results from different phases on Val294 set.Figure: Qualitative Comparison between results at the end of three phases of training." ], [ "AIM 2019 Challenge on Bokeh Effect Synthesis", "The proposed solution participated in AIM 2019 Challenge on Bokeh Effect Synthesis which is a competition on example-based Bokeh effect generation.", "This competition had two tracks- namely Fidelity track and Perceptual track.", "In Fidelity track, submissions where judged based on PSNR and SSIM scores, whereas Mean Opinion Score(MOS) based on a user study was used to rank submissions in Perceptual track.", "80 participants registered in this competition among which 9 teams entered the final phase.", "Two submissions were made to the separate tracks of the competition.", "Phase-1 results are submitted to Fidelity Track and Phase-3 results are submitted to Perceptual Track.", "The test results are shown in Table-REF .", "The submission to Fidelity track (CVL-IITM-FDL) ranked 6th and 3rd among all the submissions with respect to PSNR and SSIM respectively.", "The Perceptual Track submission (CVL-IITM-PERC) ranked 2nd among all the submissions based on MOS score.", "Table: AIM 2019 Challenge on Bokeh Effect Synthesis Leaderboard." ], [ "Efficiency", "The proposed model is lightweight.", "It takes 28 MB to store the model on hard drive.", "The model can process an HD image of resolution $1024 \\times 1575$ in $0.03$ seconds (I/O time is excluded)." ], [ "Conclusion", "In this paper, an end-to-end deep learning approach for Bokeh effect synthesis is proposed.", "The synthesized bokeh image is rendered as a weighted sum of the input image and a number of differently smoothed images, where the corresponding weight maps are predicted by a depth-estimation network.", "The proposed system is trained in three phases to synthesize realistic bokeh images.", "It is shown through experiments that using more number of blur kernels and bigger blur kernels produce better quality bokeh images.", "The proposed algorithm is lightweight and can post-process an HD image in 0.03 seconds.", "The proposed approach has been ranked 2nd among the solutions proposed in Perceptual Track of AIM 2019 challenge on Bokeh effect synthesis.", "In future, the effect of using different kinds of blur kernel (e.g.", "average blur, disk blur) can be explored.", "It is also interesting to see how one can incorporate both monocular depth estimation and saliency detection to produce a lightweight system that generates high-quality bokeh effect." ] ]

[ [ "Network Partitioning and Avoidable Contention" ], [ "Abstract Network contention frequently dominates the run time of parallel algorithms and limits scaling performance.", "Most previous studies mitigate or eliminate contention by utilizing one of several approaches: communication-minimizing algorithms; hotspot-avoiding routing schemes; topology-aware task mapping; or improving global network properties, such as bisection bandwidth, edge-expansion, partitioning, and network diameter.", "In practice, parallel jobs often use only a fraction of a host system.", "How do processor allocation policies affect contention within a partition?", "We utilize edge-isoperimetric analysis of network graphs to determine whether a network partition has optimal internal bisection.", "Increasing the bisection allows a more efficient use of the network resources, decreasing or completely eliminating the link contention.", "We first study torus networks and characterize partition geometries that maximize internal bisection bandwidth.", "We examine the allocation policies of Mira and JUQUEEN, the two largest publicly-accessible Blue Gene/Q torus-based supercomputers.", "Our analysis demonstrates that the bisection bandwidth of their current partitions can often be improved by changing the partitions' geometries.", "These can yield up to a X2 speedup for contention-bound workloads.", "Benchmarking experiments validate the predictions.", "Our analysis applies to allocation policies of other networks." ], [ "Introduction", "Network contention frequently dominates the run time of parallel algorithms and limits scaling performance [6].", "Most previous studies mitigate or eliminate contention by utilizing one of several approaches: communication-minimizing algorithms (cf.", "[29], [8], [14]); hotspot-avoiding routing schemes (cf.", "[28]); topology-aware task mapping (cf.", "[10]); or improving global network properties such as bisection bandwidth, edge-expansion (cf.", "[9], [30]), partitioning (cf.", "[18]), and network diameter (cf.", "[20]).", "Parallel jobs running on a supercomputer or a cloud platform often do not utilize the entire machine at once.", "Rather, the job is assigned a subset of the system's compute nodes and associated resources for its exclusive useSome cloud platforms allow `multi-tenancy', in which case exclusivity is not guaranteed.", "This adds further challenge which we do not address in this paper.. Optimizing the internal bisection bandwidth of allocated partitions can decrease or completely eliminate the link contention of a parallel computation, improving overall performance for contention-bound workloads." ], [ "Our contribution", "Using isoperimetric analysis, we study torus networks and characterize partition geometries that maximize internal bisection bandwidth.", "Our analysis utilizes a novel generalization of Bollobás and Leader's bounds on the edge-isoperimetric problem on torus graphs [11].", "A solution was known for tori with dimensions of equal size, whereas our new bound applies to torus graphs with arbitrary dimension sizes.", "This is useful, as the vast majority of torus networks with 3 dimensions or more have unequal dimensions.", "We apply isoperimetric analysis to compute node partition allocations allowed by the allocation of Mira and JUQUEEN, the two largest publicly-accessible Blue Gene/Q torus-based supercomputers.", "Our analysis demonstrates that the bisection bandwidth of their current partitions can often be improved by changing the partitions' geometries, yielding up to a $\\times 2$  speedup for contention-bound workloads.", "Benchmarking experiments on both systems validate the predictions.", "We show an impact of 10% speedup for fast matrix multiplication.", "We also applied the analysis to the Blue Gene/Q machine Sequoia, but no experiments were performed as it is no longer available for scientific research.", "Lastly, we discuss network configurations for hypothetical Blue Gene/Q systems which, despite having fewer compute and network resources, may perform better.", "They are predicted to improve upon the network performance of JUQUEEN by increasing the bisection bandwidth of partitions.", "This work focuses first on the Blue Gene/Q supercomputer series, but the application of our method to other networks topologies such as hypercubes, Dragonfly, Slim Fly, and HyperX, is also described in detail.", "Bisection bandwidth is a standard metric for network performance.", "An in-depth description of the Blue Gene/Q topology appears in [12], which also includes an analysis of its bisection bandwidth, and outlines traffic patterns that are challenging for the network to efficiently route.", "However, they do not discuss the bisection bandwidth of network partitions.", "Finding worst-case traffic patterns for an arbitrary network topology can be non-trivial in the general case.", "A method for generating “near-worst-case\" traffic patterns is shown in [19].", "The edge-isoperimetric problem (see definition in Section ) is a well-known problem in combinatorics, and general solutions have been shown for several graphs that either directly correspond or are very similar to network topologies used in practice.", "These include: hypercubes [16], cubic tori [11], Cartesian products of cliques [24], and 2-dimensional mesh grids [1].", "The edge-isoperimetric problem provides a tight bound on the bandwidth between two arbitrary sides of the network.", "Since both sides may have the same size, the edge-isoperimetric problem generalizes the problem of determining the bisection bandwidth of a graph.", "Closely related to the edge-isoperimetric problem is the notion of small-set expansion in graphs (see Section ).", "Indeed, if a graph $G$ is $d$ -regular then the two problems are essentially equivalent.", "Spectral methods that can be used to approximate the small-set expansion of arbitrary graphs are described in [23].", "The small set expansion of a network graph is used in [7] to derive lower bounds on the contention costs, and potentially determine when a given parallel algorithm on a given system is inevitably asymptotically contention-bound.", "In Section  we provide preliminaries on the edge-isoperimetric problem on torus graphs, and on the IBM Blue Gene/Q architecture.", "In Section  we present isoperimetric analysis of general tori and apply it to the partitions of Mira and JUQUEEN, concluding with alternative, improved partitions.", "In Section  we perform experiments on Mira and JUQUEEN, and discuss the results.", "In Section  we present implications of our analysis on networks design, discuss the applicability of our methods to other network topologies including ToFu, Dragonfly, Fat-Tree, and HyperX, and outline future work." ], [ "Preliminaries", "A main application of our method is improving allocation policies of the torus-based Blue Gene/Q systems.", "We begin by defining torus graphs and our primary analysis tool of the edge-isoperimetric problem." ], [ "Torus graphs", "Let $D$ and $a_1, \\ldots , a_D$ be integers, and let $G = \\left(V,E\\right)$ be a graph.", "If $V = \\left[a_1\\right] \\times \\ldots \\times \\left[a_D\\right]$ , and every two vertices $u = \\left(u_1, \\ldots , u_D\\right), v = \\left(v_1, \\ldots , v_D\\right)$ are adjacent if and only if $\\exists k$ such that $u_k = v_k \\pm 1 \\bmod a_k$ and $\\forall j \\ne k, u_j = v_j$ , then $G$ is said to be a $D$ -torus (also, $D$ -dimensional torus).", "If $a_1 = \\ldots = a_D$ then $G$ is said to be a cubic torus.", "Let $G=\\left(V,E\\right)$ be a graph, and let $A,B\\subset V$ .", "Then, the perimeter of $A$ is $E\\left(A, \\overline{A}\\right) = \\left\\lbrace u,v \\mid u \\in A, v \\notin A\\right\\rbrace $ and the interior of $A$ is $E\\left(A, A\\right) = \\left\\lbrace u,v \\mid u \\in A, v \\in A\\right\\rbrace $ .", "For any $k$ -regular graph, the following equation holds: $\\forall A \\subseteq V, k \\left|A\\right| = 2 \\left|E\\left(A,A\\right)\\right| + \\left|E\\left(A,\\overline{A}\\right)\\right|$ The edge-isoperimetric problem is defined as follows: given a graph $G=\\left(V, E\\right)$ and some integer $t \\le \\frac{\\left|V\\right|}{2}$ , find $S \\subset V$ with $\\left|S\\right| = t$ of minimal perimeter size.", "That is, find $S$ such that: $\\left|E\\left(S, \\bar{S}\\right)\\right| = \\min _{\\begin{array}{c}A \\subset V \\\\ \\left|A\\right| = t\\end{array}} {\\left\\lbrace \\left|E\\left(A, \\bar{A}\\right)\\right|\\right\\rbrace }$ Such a set $S$ is said to be isoperimetric.", "Note that by Equation REF , for $k$ -regular graphs, minimizing the perimeter is equivalent to maximizing the interior.", "If $G$ is a cubic torus, then the following bound of Bollobás and Leader [11] applies: [Edge-isoperimetric ineq.", "for cubic tori] Let $G=\\left(V,E\\right)$ be a cubic $D$ -dimensional torus such that $V = \\left[n\\right]^D$ , and let $t \\le \\frac{n^D}{2}$ .", "Then $\\forall S\\subset V$ with $\\left|S \\right|= t$ : $\\left|E\\left(S,\\overline{S}\\right)\\right| \\ge \\min _{r\\in \\left\\lbrace 0,\\ldots ,D-1\\right\\rbrace }2\\left(D-r\\right)\\cdot n^{\\frac{r}{D-r}}\\cdot t^{\\frac{D-r-1}{D-r}}$ For $r$ such that $\\left(\\frac{t}{n^r}\\right)^{\\frac{1}{D-r}}$ is an integer, we define $S^{\\prime }\\subset V$ such that: $S^{\\prime }= \\left[n\\right]^r \\times \\left[\\left(\\frac{t}{n^r}\\right)^{\\frac{1}{D-r}}\\right]^{D-r}$ In this case, $S^{\\prime }$ is a $D$ -dimensional cuboid with $r$ dimensions of length $n$ , and $D-r$ dimensions of length $\\left(\\frac{t}{n^r}\\right)^{\\frac{1}{D-r}}$ .", "Each vertex contributes $2\\left(D-r\\right)$ edges to the cut.", "Since $\\left|S^{\\prime } \\right|= t$ , a simple counting argument leads to: $\\left|E(S^{\\prime }, \\bar{S^{\\prime }})\\right|= 2r \\cdot t^{\\frac{D-r-1}{D-r}} \\cdot n^{\\frac{r}{D-r}}$ Therefore the bound presented in Theorem  is tight for certain values of $t$ .", "The small-set expansion of a graph $G = \\left(V,E \\right)$ , denoted $h_t\\left(G\\right)$ , is defined: $h_t \\left(G\\right) = \\min _{\\begin{array}{c}A \\subset V \\\\ \\left|A\\right| \\le t\\end{array}}\\frac{\\left|E\\left(A, \\overline{A}\\right)\\right|}{\\left|E\\left(A, A\\right)\\right| + \\left|E\\left(A, \\overline{A}\\right)\\right|}$ Small-set expansion can be used to test whether a given network will be inevitably asymptotically contention-bound when executing a parallel algorithm with known per-processor communication costs [7].", "Since the small-set expansion is attained by the bisection for all networks and partitions considered in this work, it will suffice for us to consider only the bisection bandwidth.", "IBM Blue Gene/Q systems [13] have 5D torus network topologies where the size of at least one dimension is exactly 2.", "The bisection bandwidth of a Blue Gene/Q system is $2 \\cdot \\frac{N}{L} \\cdot B$ , where $N$ is the number of nodes, $L$ is size of the longest dimension, and $B$ is the capacity of a single bidirectional link [12].", "A midplane in the Blue Gene/Q topology is a physical arrangement of 512 compute nodes, internally connected by a 5D torus network with dimensions $4 \\times 4 \\times 4 \\times 4 \\times 2$ .", "The last dimension, of length 2, is internal to the midplane.", "A physical rack in a Blue Gene/Q system consists of two midplanes.", "13 Blue Gene/Q systems appear in the November 2017 list of top 500 supercomputers [15].", "The network is physically constructed in such a way that partitions may have wrap-around links in a given dimension even when they do not fully cover that dimension in the entire network.", "All partitions discussed in this work are 4-dimensional sub-tori where some dimensions may have size 1.", "There are no published limits to the maximal size of a Blue Gene/Q system or to the lengths of anyExcept the 5th dimension, which has size 2 and is internal to each midplane.", "dimension [12].", "To simplify notation, we always present the dimensions of a torus network and its partitions in sorted order by length.", "This canonical representation treats partitions whose geometries are identical up to rotations as one.", "Therefore, a machine with network size $2 \\times 2 \\times 2 \\times 1 \\times 1$ fits 4 partitions with geometry $2 \\times 1 \\times 1 \\times 1 \\times 1$ .", "Outside of jobs which require an exceptionally small amount of compute nodes, all partitions in Blue Gene/Q systems are defined by cuboids (Cartesian products of chains and cycles) consisting of whole midplanes.", "We therefore represent the Blue Gene/Q network and its partitions as 4-dimensional tori of midplanes.", "For example, consider a 6-midplane system of dimensions $3 \\times 2 \\times 1 \\times 1$ .", "In terms of compute nodes, this system has 3072 compute nodes and network size $12 \\times 8 \\times 4 \\times 4 \\times 2$ .", "The best possible 1536-compute node partition of this system has dimensions $12 \\times 4 \\times 4 \\times 4 \\times 2$ and 256 links in its bisection.", "An alternate partition with dimensions $8 \\times 6 \\times 4 \\times 4 \\times 2$ would have the same node count, but a greater bisection of 384.", "However, since its largest dimension consists of $1.5$ midplanes it is not supported by the Blue Gene/Q topology.", "Such a partition could be constructed by over-provisioning an additional midplane and defining a partition with dimensions $8 \\times 8 \\times 4 \\times 4 \\times 2$ .", "Our benchmarks and applications all use message-passing communication with MPI, which allows the individual processes (often referred to as ranks) in the computations to communicate directly with each other.", "Unless explicitly stated otherwise, each compute node is assigned only one MPI rank.", "This allows an improved bisection bandwidth of 512 links, but at the cost of requiring additional compute nodes.", "We next introduce the Blue Gene/Q systems Mira and JUQUEEN." ], [ "Mira", "Installed at Argonne National Laboratory [5], it is the largest Blue Gene/Q system accessible for scientific research.", "Mira is ranked 24th in the July 2019 Top 500 supercomputers [15].", "It has 49152 compute nodes, with network size $16 \\times 16 \\times 12 \\times 8 \\times 2$ , or $4 \\times 4 \\times 3 \\times 2$ midplanes." ], [ "JUQUEEN", "Installed at Jülich Supercomputing Centre, JUQUEEN is the second-largest Blue Gene/Q system accessible for scientific research.", "It wasJUQUEEN was since dismantled and does not appear in later lists.", "ranked 22nd in the November 2017 Top 500 supercomputers [15].", "JUQUEEN has 28672 compute nodes, and network size $28 \\times 8 \\times 8 \\times 8 \\times 2$ , or $7 \\times 2 \\times 2 \\times 2$ midplanes." ], [ "Theoretical Analysis", "Using isoperimetric analysis, we identify allocation policies that are not optimal; namely, we point to partitions with sub-optimal internal bisection bandwidth.", "Whenever such partitions exist, we find partition geometries with optimal bisection bandwidth that are likely to reduce link contention." ], [ "The Edge-Isoperimetric Problem", "We obtain a novel generalization of Theorem  to arbitrary torus graphs.", "We show that the bound is optimal for cuboid subsets, and conjecture that it is optimal for arbitrary subsets as well.", "[Edge-isoperimetric ineq.", "for tori] Let $G=\\left(V,E\\right)$ be a $D$ -dimensional torus with $V = \\left[a_1\\right] \\times \\left[a_2\\right] \\times \\ldots \\times \\left[a_D\\right]$ , and $t \\le \\frac{\\left|V\\right|}{2}$ .", "Suppose, without loss of generality, that $a_1 \\ge a_2 \\ge \\ldots \\ge a_D$ .", "Then, for any cuboid $S\\subset V, \\left|S \\right|= t$ : $\\left|E\\left(S,\\overline{S}\\right)\\right|\\ge \\min _{r\\in \\left\\lbrace 0,\\ldots ,D-1\\right\\rbrace }2\\left(D-r\\right)\\left(\\prod _{i=0}^{r-1} a_{D-i}\\right)^{\\frac{1}{D-r}}t^{\\frac{D-r-1}{D-r}}$ Like Theorem , our bound can be attained in some cases.", "Let $k = \\prod _{i=0}^{r-1} a_{D-i}$ .", "If $\\exists r$ such that $\\left(\\frac{t}{k}\\right)^{\\frac{1}{D-r}}$ is an integer, define the cuboid $S_r = \\left[\\left(\\frac{t}{k}\\right)^\\frac{1}{D-r}\\right]^{D-r} \\times \\left[a_{D-r+1} \\right] \\times \\ldots \\times \\left[a_D \\right]$ .", "Our proof strategy for Theorem REF is as follows: in Lemma REF we show an explicit construction of a class of cuboid sets in general torus graphs whose cut size matches Equation REF .", "In Lemma REF we show these sets are isoperimetric, thereby completing the proof.", "Let $G = \\left(V,E\\right)$ be a $D$ -dimensional torus with $V = \\left[a_1\\right] \\times \\ldots \\times \\left[a_D\\right]$ , and let $t,k,r^{\\prime }$ be integers such that $t \\le \\frac{\\left|V\\right|}{2}$ and $\\left(\\frac{t}{k}\\right)^\\frac{1}{D-r^{\\prime }}$ is an integer.", "Define $S_{r^{\\prime }}$ as in Theorem REF , with $\\arg \\min = r^{\\prime }$ .", "Then, $S_{r^{\\prime }}$ maintains: $\\left|E\\left(S_{r^{\\prime }},\\overline{S_{r^{\\prime }}}\\right)\\right| = 2\\left(D-r^{\\prime }\\right) \\left(\\prod _{i=0}^{r^{\\prime }-1} a_{D-i}\\right)^{\\frac{1}{D-r^{\\prime }}} t^{\\frac{D-r^{\\prime }-1}{D-r^{\\prime }}}$ If $a_1 = \\ldots = a_D = 2$ , then by Harper [16], $S_{r^{\\prime }}$ is an isoperimetric set maintaining the desired cut size, and the lemma follows.", "If all dimension lengths are strictly greater than 2, then we show the cut size directly: We count the edges in $E\\left(S_{r^{\\prime }}, \\overline{S_{r^{\\prime }}}\\right)$ by considering the size of each $\\left(D-1\\right)$ -dimensional face of the cuboid $S_{r^{\\prime }}$ .", "Faces in dimensions where $S_{r^{\\prime }_i} = a_i$ contribute no edges to the cut.", "There are $\\frac{t}{\\left(\\frac{t}{k}\\right)^\\frac{1}{D-r}} = \\left(\\prod _{i=0}^{r-1} a_{D-i}\\right)^{\\frac{1}{D-r}} t^{\\frac{D-r-1}{D-r}}$ vertices on each remaining face.", "Then: $\\left|E\\left(S_{r^{\\prime }},\\overline{S_{r^{\\prime }}}\\right)\\right| = 2\\left(D-r^{\\prime }\\right) \\left(\\prod _{i=0}^{r^{\\prime }-1} a_{D-i}\\right)^{\\frac{1}{D-r^{\\prime }}} t^{\\frac{D-r^{\\prime }-1}{D-r^{\\prime }}}$ $S_{r^{\\prime }}$ is similar to $S^{\\prime }$ as defined in Equation REF , but instead of having $r^{\\prime }$ dimensions of length $n$ , the dimension lengths are $a_D, \\ldots , a_{D-r^{\\prime }+1}$ .", "If only some dimensions $a_{D-k+1}, \\ldots , a_{D}$ have lengths 2, then we choose $S_{r^{\\prime }}$ such that they are all covered, and then proceed as before with $t^{\\prime } = \\frac{t}{2^k}$ , and the same cut is attained.", "Let $G, t, D, k, r, S_r$ be defined as in Lemma REF .", "Let $A \\subset V$ be some cuboid $\\left[A_1 \\right] \\times \\ldots \\times \\left[A_D\\right]$ with $\\left|A\\right| = t$ .", "Suppose there exist exactly $r$ indices $i_1, \\ldots , i_r$ that maintain $A_{i_k} = a_{i_k}$ .", "Then, $\\left|E\\left(S_r,\\overline{S_r}\\right)\\right| \\le \\left|E\\left(A,\\overline{A}\\right)\\right|$ .", "If $A$ can be transformed into $S_r$ by changing the order of equal-sized dimensions (i.e., $A$ is a rotation of $S_r$ ) then the equality is trivial.", "By assumption, $A$ and $S_r$ both fully cover exactly $r$ dimensions.", "Then, since $A$ is not a rotation of $S_r$ , there must exist dimensions $i,j$ such that $A_i < t^\\frac{1}{D-r} < A_j$ .", "That is, the projection of $A$ on the dimensions $i, j$ is an oblong rectangle and not a square.", "In this case, the lemma follows directly by applying to $A$ the same counting argument used in the proof of Lemma REF .", "Suppose $\\exists j$ such that $A_j = a_j$ and $S_{r_j} \\ne a_j$ .", "Since $A$ is not a rotation of $S_r$ , and by definition, $S_{r_i} = a_i$ for $a_{D-r}, \\ldots , a_D$ , then a face of $A$ contributes at least a factor of $\\frac{a_j}{a_{D-r}}$ more edges to the perimeter than a face of $S_r$ .", "Thus, $\\left|E\\left(S_r,\\overline{S_r}\\right)\\right| \\le \\left|E\\left(A,\\overline{A}\\right)\\right|$ with equality if and only if $A$ is a rotation of $S_r$ .", "A central implication of Theorem REF is that for large values of $t$ with $r_{\\text{opt}} = D-1$ , the size of the perimeter is bounded below by $2\\cdot \\prod _{i=2}^{D} a_i$ .", "In particular, the bisection bandwidth is improved the closer $\\frac{a_1}{t}$ is to $t^\\frac{D-1}{D}$ .", "This is consistent with the result of [12] regarding the bisection bandwidth of the Blue Gene/Q network, and leads us to an easy corollary.", "Let $G = \\left(V, E\\right)$ be the network graph of a Blue Gene/Q machine, and let $A$ be a cuboid of midplanes with dimensions $A_1 \\times \\ldots \\times A_4$ .", "If $\\exists B \\subset V$ with dimensions $B_1 \\times \\ldots \\times B_4$ such that $\\left|A\\right| = \\left|B\\right|$ and $\\frac{B_1}{\\left|A\\right|} < \\frac{A_1}{\\left|A\\right|}$ , then $B$ has strictly greater internal bisection bandwidth than $A$ ." ], [ "Analysis of Blue Gene/Q Systems", "We apply Lemma REF to the partitions in Mira and JUQUEEN, and obtain their bisection bandwidths.", "Using Lemma REF , partition geometries with improved bisection bandwidth are found." ], [ "Mira", "Not all cuboids of midplanes are permitted by Mira's scheduler.", "There is a predefined list of partitions that may be used (see Table REF in Appendix B).", "Where possible, we propose partitions of identical size and greater internal bisection bandwidth (see Table REF and Figure REF ).", "With the assistance of the operators of Mira, we were able to allocate the new partitions for the duration of our experiments.", "This allowed us to conduct benchmark comparisons (see Section ).", "Partitions of JUQUEEN's network can be any cuboids of midplanes that fit inside the full network.", "Users may request partitions by specifying either their exact geometry in midplanes, or by specifying only the overall size.", "For some sizes, partitions both optimal and sub-optimal in terms of internal bisection bandwidth are permissible by the job scheduler.", "When only a partition size is specified, inconsistent performance may occur if one part of a user's executions is allocated optimal partitions, and another part is not.", "Figure: Mira: Normalized bisection bandwidth of currently-defined and proposed partition geometries.", "Each link contributes 1 unit of capacity.Table: Mira: partial list of normalized bisection bandwidths of current and proposed partitions, showing only rows where the bisection is increased.", "Full list in Table , Appendix B.Figure: JUQUEEN: Normalized bisection bandwidth of best and worst-case partition geometries.", "Each link contributes 1 unit of capacity.", "The `spiking' drops correspond to partitions whose size requires them to be ring-shaped, and hence have small bisection bandwidth.Table: JUQUEEN: partial list of normalized bisection bandwidths of optimal and worst-case partitions, showing only rows where best and worst cases differ.", "Full list in Table , Appendix B." ], [ "Experiments", "We support our theoretical predictions with the following experiments: [(A)] bisection pairing; matrix multiplication; and simulation of strong scaling test.", "The proposed partitions on Mira were made available for our experiments by the generous assistance of the system operators of Mira, who let us use a temporarily modified processor allocation policy.", "This did not require modifying the network's physical structure, but rather only the software-defined policy.", "Bisection pairing experiment Experimental settings We performed a ping-pong benchmark using the furthest-node scheme outlined in [12], which pairs nodes that are located at a maximal number of hops from each other.", "The benchmark was performed as follows: each pair of compute nodes simultaneously sends to and receives from its counterpart a message of fixed size.", "This was repeated for 30 rounds without synchronization across distinct pairs.", "The first 4 rounds were treated as warm-up, and were not counted in the total time.", "To prevent unexpected behaviors due to caching effects, the messages were randomly generated between each round.", "A single link in the Blue Gene/Q network topology has a bandwidth of 2 Gigabyte per second per direction [12], and so the total communication volume between each pair of ranks was set to 2 Gigabytes, broken into 16 chunks sized $0.1342$ Gigabytes each, to maximize the induced contention and its visibility.", "We compared the performance of currently-used partitions against the proposed partitions on Mira, and measured the average time required for a pair of nodes to complete all rounds.", "This was replicated on JUQUEEN, where we compared best-case and worst-case partitions.", "Figure: Mira: Bisection pairing experiment, using 4 warm-up rounds and 26 communication rounds, with messages of size 0.13420.1342 Gigabyte.Figure: JUQUEEN: Bisection pairing experiment, using 4 warm-up rounds and 26 communication rounds, with messages of size 0.13420.1342 Gigabyte.", "As shown in Table 2, the average bisection bandwidth per node is identical for the 4 and 8 midplane partitions, but is 50%50\\% smaller for the 6 midplane partition.", "This is consistent with the observed results.", "Results The results for Mira and JUQUEEN are described in Figure REF and Figure REF , respectively.", "On both Mira and JUQUEEN, the difference in average execution time is at least a factor of $1.92$ where the predicted factor is $2.00$ (except for 24 midplanes on Mira, where it is $1.44$ and $1.50$ , respectively).", "This confirms the impact of partition geometry on the network contention and execution time, and shows our predicted speedup is attainable for contention-bound workloads on both systems.", "An unexpected difference can be seen between the currently-defined and proposed partitions on Mira when running on 16 and 24 midplanes.", "The $9.7\\%$ increase between the currently-defined 16 and 24 partitions may be attributed to a combination of noise and low path diversity relative to the other partition geometries worsening the contention.", "In addition, the fact that some of the network links of the size 3 dimension in the 24 midplane partition are only utilized in one direction may have also caused a mild increase in effective resource contention.", "For the proposed partitions, the increase between 16 and 24 partitions is expected, since the node count was increased by a factor of $1.5$ while the bisection bandwidth remained constant.", "In summary, the bisection pairing experiment results agree almost perfectly with our predictions.", "Matrix multiplication experiment Experimental settings In order to measure the impact of our findings on real-life applications, we benchmark the performance of the Strassen-Winograd matrix multiplication algorithm.", "We used the same set of parameters for equal-sized partitions.", "However, parameters were adjusted based on the partition sizes.", "Table REF details the parameters of each execution.", "Table: Parameters of the matrix multiplication experiment on Mira.We used a parallel implementation by [8], [25] on random inputs between the different partition geometries.", "The experimental constraints of that [8], [25] hold here.", "Namely, there must be exactly $f\\cdot 7^k$ MPI ranks, where $f$ and $k$ are integers and $1 \\le f \\le 6$ , and the matrix dimension must be a multiple of $f \\cdot 2^r \\cdot 7^{\\left\\lceil \\frac{k}{2}\\right\\rceil }$ .", "We could not disable some of the compute nodes in a partition, as the additional network resources belonging to those `disabled' nodes would still be utilized by the system.", "However, Mira has no partitions that contain exactly $7^k$ midplanes for any $k > 1$ .", "We therefore used multiple cores in each processor in order to create the required rank count, and tried to minimize the imbalance in compute and communication costs between the physical processors.", "Parameter selection is described in Table REF .", "For example, the execution on 8 midplanes had a total of $31,213$ MPI ranks, and each compute node was allowed to use up to 8 cores (where each core may only be associated with a single rank).", "Results Time spent performing computation does not significantly differ between partition geometries of the same size.", "These computation costs are $0.554, 0.5115, 0.4965$ and $0.0604$ seconds for $4, 8, 16$ and 24 midplanes, respectively.", "Communication costs of runs using proposed partitions were smaller by factors of $\\times 1.37$ up to $\\times 1.52$ than all executions which utilized currently-defined partitions (see Figure REF ).", "The total wall-clock time was smaller by factors of $\\times 1.08$ up to $\\times 1.22$ , due to the common computation costs.", "Figure: Mira: Matrix multiplication experiment.", "Three executions were performed for each midplane count and each partition type.Costs offset by communication-hiding are not presented.", "For both partition types, they are 0.059,0.067,0.0990.059, 0.067, 0.099 and 0 seconds for 4,8,164,8,16 and 24 midplanes, respectively.", "Results are described using communication time instead of wallclock as the additional computation time for identical workloads is not relevant to the contention costs.", "Simulation of strong scaling test Motivation When optimal and sub-optimal partitions are randomly selected by the job scheduler, unnoticed variations in the bisection bandwidth can potentially result in false conclusions regarding the scaling behavior of an algorithm.", "The purpose of this experiment is to test the possibility of contention costs to cause a parallel algorithm that has good strong scaling properties to appear as though it does not.", "For example, consider the results of the bisection pairing experiment on JUQUEEN (see Figure REF ) and suppose the runs on up to 6 midplanes are assigned only proposed partitions, but the runs on 8 and 12 midplanes are assigned only worst-case partitions.", "Without knowledge of the bisection bandwidths available to each execution, the runtime may seem to increase linearly with midplane count, which is clearly incorrect.", "Experimental settings We used the same code as in Experiment REF , and were subject to the same constraints in parameter selection.", "We could not use more than 3 distinct midplane counts in the experiment without altering other parameters such as the matrix dimension.", "Table: Strong scaling experiment parameters on Mira, performing matrix multiplication with dimension 9408.In order to show two scalability plots with a common point, we chose midplane sizes $2, 4$ and 8, and a matrix dimension of 9408 storing double-precision values.", "There is only one way to define a cuboid of 2 midplanes, and so the smallest execution used a partition common to both the current and proposed geometries.", "Figure: Mira: Strong-scaling experiment.", "The run on 2 midplanes allows only one partition geometry.", "Floating values indicate communication times.", "Communication costs hidden by computations were not counted.", "Results This experiment was partly successful, as the results indicate a linear decrease of communication costs when strong scaling from 2 midplanes to 8 using the proposed partition geometry, but only a sub-linear decrease when using the current partitions.", "Therefore, a user should be advised that a test of the computation's strong scaling behavior using the currently-defined partitions may incorrectly indicate it cannot linearly scale beyond 4 midplanes.", "Runs on all partition types exhibited super-linear scaling of the communication costs between 2 and 4 midplanes, regardless of partition geometry used.", "As the difference in bisection bandwidth is exactly $\\times 2$ , link contention alone cannot account for the effect.", "The exact source of the super-linear speedup is not fully understood.", "It may be related to the fact that the data fits entirely within the shared L2 cache when using 4 and 8 midplanes, but not when using 2 midplanes.", "Perhaps this allowed the dedicated communications core to more efficiently move the data.", "More specifically, each of Mira's processors has 32 Megabyte of L2 cache storage shared by all its cores, and an additional core used for communications.", "This means $32, 64,$ and 128 Gigabyte of combined L2 storage for $2,4,$ and 8 midplanes, respectively.", "The execution pattern the BFS-DFS matrix multiplication algorithm used was 4 BFS steps, requiring a combined minimum of $3 \\cdot \\left(\\frac{7}{4}\\right)^4 \\cdot 8 \\cdot 9408^2$ bytes, or $18.55$ Gigabyte in order to store all matrices across all processors, added with a similar amount of space for the communications library buffers.", "When the overall space requirement exceeds the L2 memory of 2 midplanes, this results in cache misses and use of the slower RAM, hence a slowdown for the executions on 2 midplanes.", "This somewhat muddles the visibility of scalability properties of the fast matrix multiplication computation.", "It remains evident that the scaling is better when the proposed geometries are used.", "Specifically, the computation on 2 midplanes exhibits a $\\times 4.4$ decrease in communication costs on 8 midplanes in a proposed geometry, and a $\\times 3.3$ decrease when using the current geometry.", "Therefore, the fast matrix multiplication algorithm may be incorrectly surmised to have a smaller strong scaling range on Mira if evaluated only using the current partition geometries.", "More extreme disparities are possible: given the bisection bandwidth of the 2 and 4 midplane partitions in Table REF , a computation's wallclock time may remain identical on both 2 and 4 midplanes even if the computation can linearly scale to 16 midplanes and beyond.", "Discussion In this work, we focus on potential performance boosts due to improved internal bisection bandwidth of partitions.", "Determining the importance of such speedups relative to other possible machine design optimization goals is beyond the scope of this work.", "Particularly, there are many motivations to the design and installation of specific supercomputer systems, as well as to setting a processor allocation policy.", "Such motivations may include computational kernels or specific job sizes deemed particularly important for the system, or even specific software that is intended to be executed often.", "Further reasons may include packing of jobs affecting overall system utilization, cabling complexity, cooling, and ease of access and maintenance.", "Application to other topologies We discussed the IBM Blue Gene/Q network topology in great detail, but our method applies to arbitrary network topologies if edge-isoperimetric problems can be efficiently solved on their network graphs.", "When the network graph is regular and has uniform link capacity – which is the case in almost all networks of supercomputers (except Dragonfly; see below) – isoperimetric analysis is sufficient to determine the small-set expansion of the graph.", "This provides additional information to merely the bisection bandwidth, and can predict contention bottlenecks at locations other than the network bisection [7].", "The ToFu interconnect used by the K Computer [3] is a high-dimensional torus with certain similarities to Blue Gene/Q.", "Torus networks of lower dimension, such as the Cray XK7 $3D$ -torus machine Titan [27], may require a formulation of the edge-isoperimetric problem that considers weighted edges.", "For hypercube-based supercomputers such as Pleiades [26], the edge-isoperimetric problem is long solved in [16], and so our method is directly usable.", "For Fat-Tree topologies, the application of our method is more challenging.", "If the processor allocation policy permits distinct jobs to share network resources, then the available link capacity may be smaller than isoperimetric analysis alone would indicate.", "If sharing of network resources is forbidden, then the policy is expected to be so constrained that our method will not be able to obtain improvements.", "HyperX networks are Cartesian products of cliques $K_{a_1},\\ldots ,K_{a_D}$ .", "The number of cliques in the product and their exact sizes are both variable.", "Each clique may have a different link capacity; when all links have the same capacity, the HyperX network is said to be regular.", "Finding an optimal HyperX structure for a fixed vertex count is performed by exhaustive search [2].", "The network bisection bandwidth is attained by selecting half of the vertices in $K_i$ for some $1 \\le i \\le D$ and all vertices in $K_{j}$ for $i \\ne j$  [2].", "The edge-isoperimetric problem for regular HyperX network graphs is solved in [24], by choosing vertices of the product cliques in order of descending size.", "Dragonfly networks [20] as implemented in the Cray XC series [4] are a collection of `groups' each containing up to 96 Aries routers, where each group is an instance of $K_{16} \\times K_6$ .", "Links belonging to the $K_6$ clique have a normalized capacity of 3 relative to the $K_{16}$ links, requiring a weighted version of the edge-isoperimetric problem to be used.", "Unlike HyperX, the Dragonfly network also contains inter-group links with a normalized capacity of 4.", "To apply our method to a Dragonfly-based system, it is necessary to model the inter-group links to create the network graph.", "We are unaware of any public description of the inter-group link arrangement, but [17] discusses three possible schemes for such systems.", "A further minor challenge is the pairing of Aries routers: unlike an edge in a simple graph, each endpoint of an inter-group link is a pair of adjacent Aries routers.", "It may thus be necessary to introduce the constraint of $t$ even when considering edge-isoperimetric problems on these networks.", "The Slim Fly network topology is more difficult to analyze in the general case, since the cabling layout varies greatly based on the global network size, necessitating exhaustive search [9].", "Given the complexity of finding such constructions, the existence of a general solution to edge-isoperimetric problems that fits all possible constructions seems unlikely.", "Sequoia Installed at Lawrence Livermore National Laboratory, it is the largest Blue Gene/Q system in production.", "Sequoia is ranked 6th in the November 2017 list of top 500 supercomputers [15].", "It has 98304 compute nodes, with network size $16 \\times 16 \\times 16 \\times 12 \\times 2$ , or $4 \\times 4 \\times 4 \\times 3$ midplanes [22].", "Sequoia's scheduler seems to support all partition geometries supportable that the Blue Gene/Q network allows (similarly to JUQUEEN).", "Hence, both optimal and sub-optimal permissible partitions may be defined for certain midplane counts.", "Sequoia transitioned into classified work in 2013 [21].", "We thus could not perform experiments on that system, but depending on its allocation policy it may be possible to improve its network performance using our analysis.", "Machine design The ratio of maximal dimension and total machine sizes influences the global bisection bandwidth, we can reason about the design of an entire Blue Gene/Q network.", "Recall that JUQUEEN has a network size of $7 \\times 2 \\times 2 \\times 2$ (56 midplanes in total).", "We consider similar machines with 48 and 54 midplanes (denoted here JUQUEEN-48 and JUQUEEN-54, respectively), with more balanced dimension sizes.", "JUQUEEN-54 has dimensions $3 \\times 3 \\times 3 \\times 2$ , and JUQUEEN-48 has dimensions $ 4 \\times 3 \\times 2 \\times 2$ .", "Mira has network size $4 \\times 4 \\times 3 \\times 2$ .", "As the networks of JUQUEEN-54 and JUQUEEN-48 are both subgraphs of Mira's, their physical construction is clearly feasible.", "Both these machines have fewer midplanes than JUQUEEN, but have better (greater) bisection bandwidth due to their network sizes.", "Figure: Normalized bisection bandwidth comparison between JUQUEEN and hypothetical machines JUQUEEN-48 and JUQUEEN-54.", "We assume that JUQUEEN always uses best-case partitions.A comparison of bisection bandwidths of partitions of those theoretical machines against the optimal allocations of JUQUEEN is presented in Figure REF , and a full listing of optimal partitions for all three machines appears in Table REF .", "The network bandwidths of partitions of both theoretical machines are identical to those of JUQUEEN when utilizing smaller partitions, and strictly greater on the largest partition sizes.", "Since JUQUEEN is a larger system, applications that are able to perfectly strong scale to its full size will still exhibit superior performance when executed on the entire machine.", "However, in nearly all other cases, JUQUEEN cannot outperform the smaller theoretical machines.", "On contention-bound workloads, the suggested machines are predicted to perform at least as well as JUQUEEN, and attain speedup factors up to $\\times 2$ and $\\times 1.5$ for JUQUEEN-54 and JUQUEEN-48, respectively.", "The partition geometries of proposed machines are described in Table REF .", "Table: Full list of best-case partitions in JUQUEEN and the two proposed machines JUQUEEN-54 and JUQUEEN-48.", "Dimensions are listed in sorted order, BW is normalized bisection bandwidth.Future Work Our conjecture about the optimality of Equation REF for arbitrary subsets remains open.", "We believe further speedups on Blue Gene/Q can be demonstrated for several kernels of interest.", "Direct $N$ -body simulation have greater asymptotic contention cost lower bounds than fast matrix multiplication [7], increasing the impact of the internal bisection bandwidth.", "High-performance implementations of FFT, classical matrix multiplication, and other common kernels may better utilize the available hardware resources, decreasing the ratio of time spent performing computation.", "For both those cases, the impact of internal bisection bandwidth on wallclock time is predicted to be greater than in Experiment REF .", "Similar isoperimetric analysis can be conducted on other networks to potentially improve processor allocation policies, and to ensure contention-related effects do not unnecessarily inhibit scaling.", "Testing bisection sensitivity of machine benchmarks can be done by comparing the score of equal-sized partitions with different bisection bandwidths.", "Designing new network topologies, and evaluating existing ones, should be done with their partitioning constraints and internal bisection bandwidths in mind.", "Such considerations can reveal specific partition sizes for which the network performs poorly, and makes it easier to solve such issues.", "Processor allocation policy decisions of job schedulers can be improved if they are informed whether a given computation is expected to be network-bound or not.", "For example, if a partition with sub-optimal bisection bandwidth is currently available for use, a scheduler may decide whether to allocate it to a pending job, or to wait for a partition with better bisection bandwidth.", "This decision can be contingent on a user-provided hint which indicates whether the job is expected to be contention-bound or not.", "Conclusions We presented a method for analyzing processor allocation policies using an isoperimetric analysis of the network graph, and determining whether any partition geometries induce sub-optimal internal bisection bandwidth.", "We applied our method to two leading Blue Gene/Q supercomputers; demonstrated performance improvements for various workloads; and have shown how to apply our method to other networks.", "Acknowledgment We thank Ivo Kabadshow and Dorian Krause of Jülich Supercomputing Centre for their help in arranging the JUQUEEN experiments.", "We thank Adam Scovel of Argonne National Laboratory for his help and support in setting up custom partitions on Mira.", "Our experiments could not have been done without their help.", "The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.", "(www.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC).", "This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.", "Research is supported by grants 1878/14, and 1901/14 from the Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) and grant 3-10891 from the Ministry of Science and Technology, Israel.", "Research is also supported by the Einstein Foundation and the Minerva Foundation.", "This work was supported by the PetaCloud industry-academia consortium.", "This research was supported by a grant from the United States-Israel Bi-national Science Foundation (BSF), Jerusalem, Israel.", "This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 818252).", "This work was supported by The Federmann Cyber Security Center in conjunction with the Israel national cyber directorate.", "Machine Partitions Table: Mira: normalized bisection bandwidths of all current and proposed partitions.Table: Full list of JUQUEEN allocation best and worst cases by compute node count PP.", "Dimensions are listed in sorted order, BW is bisection bandwidths normalized by link capacity." ], [ "Discussion", "In this work, we focus on potential performance boosts due to improved internal bisection bandwidth of partitions.", "Determining the importance of such speedups relative to other possible machine design optimization goals is beyond the scope of this work.", "Particularly, there are many motivations to the design and installation of specific supercomputer systems, as well as to setting a processor allocation policy.", "Such motivations may include computational kernels or specific job sizes deemed particularly important for the system, or even specific software that is intended to be executed often.", "Further reasons may include packing of jobs affecting overall system utilization, cabling complexity, cooling, and ease of access and maintenance." ], [ "Application to other topologies", "We discussed the IBM Blue Gene/Q network topology in great detail, but our method applies to arbitrary network topologies if edge-isoperimetric problems can be efficiently solved on their network graphs.", "When the network graph is regular and has uniform link capacity – which is the case in almost all networks of supercomputers (except Dragonfly; see below) – isoperimetric analysis is sufficient to determine the small-set expansion of the graph.", "This provides additional information to merely the bisection bandwidth, and can predict contention bottlenecks at locations other than the network bisection [7].", "The ToFu interconnect used by the K Computer [3] is a high-dimensional torus with certain similarities to Blue Gene/Q.", "Torus networks of lower dimension, such as the Cray XK7 $3D$ -torus machine Titan [27], may require a formulation of the edge-isoperimetric problem that considers weighted edges.", "For hypercube-based supercomputers such as Pleiades [26], the edge-isoperimetric problem is long solved in [16], and so our method is directly usable.", "For Fat-Tree topologies, the application of our method is more challenging.", "If the processor allocation policy permits distinct jobs to share network resources, then the available link capacity may be smaller than isoperimetric analysis alone would indicate.", "If sharing of network resources is forbidden, then the policy is expected to be so constrained that our method will not be able to obtain improvements.", "HyperX networks are Cartesian products of cliques $K_{a_1},\\ldots ,K_{a_D}$ .", "The number of cliques in the product and their exact sizes are both variable.", "Each clique may have a different link capacity; when all links have the same capacity, the HyperX network is said to be regular.", "Finding an optimal HyperX structure for a fixed vertex count is performed by exhaustive search [2].", "The network bisection bandwidth is attained by selecting half of the vertices in $K_i$ for some $1 \\le i \\le D$ and all vertices in $K_{j}$ for $i \\ne j$  [2].", "The edge-isoperimetric problem for regular HyperX network graphs is solved in [24], by choosing vertices of the product cliques in order of descending size.", "Dragonfly networks [20] as implemented in the Cray XC series [4] are a collection of `groups' each containing up to 96 Aries routers, where each group is an instance of $K_{16} \\times K_6$ .", "Links belonging to the $K_6$ clique have a normalized capacity of 3 relative to the $K_{16}$ links, requiring a weighted version of the edge-isoperimetric problem to be used.", "Unlike HyperX, the Dragonfly network also contains inter-group links with a normalized capacity of 4.", "To apply our method to a Dragonfly-based system, it is necessary to model the inter-group links to create the network graph.", "We are unaware of any public description of the inter-group link arrangement, but [17] discusses three possible schemes for such systems.", "A further minor challenge is the pairing of Aries routers: unlike an edge in a simple graph, each endpoint of an inter-group link is a pair of adjacent Aries routers.", "It may thus be necessary to introduce the constraint of $t$ even when considering edge-isoperimetric problems on these networks.", "The Slim Fly network topology is more difficult to analyze in the general case, since the cabling layout varies greatly based on the global network size, necessitating exhaustive search [9].", "Given the complexity of finding such constructions, the existence of a general solution to edge-isoperimetric problems that fits all possible constructions seems unlikely.", "Installed at Lawrence Livermore National Laboratory, it is the largest Blue Gene/Q system in production.", "Sequoia is ranked 6th in the November 2017 list of top 500 supercomputers [15].", "It has 98304 compute nodes, with network size $16 \\times 16 \\times 16 \\times 12 \\times 2$ , or $4 \\times 4 \\times 4 \\times 3$ midplanes [22].", "Sequoia's scheduler seems to support all partition geometries supportable that the Blue Gene/Q network allows (similarly to JUQUEEN).", "Hence, both optimal and sub-optimal permissible partitions may be defined for certain midplane counts.", "Sequoia transitioned into classified work in 2013 [21].", "We thus could not perform experiments on that system, but depending on its allocation policy it may be possible to improve its network performance using our analysis.", "The ratio of maximal dimension and total machine sizes influences the global bisection bandwidth, we can reason about the design of an entire Blue Gene/Q network.", "Recall that JUQUEEN has a network size of $7 \\times 2 \\times 2 \\times 2$ (56 midplanes in total).", "We consider similar machines with 48 and 54 midplanes (denoted here JUQUEEN-48 and JUQUEEN-54, respectively), with more balanced dimension sizes.", "JUQUEEN-54 has dimensions $3 \\times 3 \\times 3 \\times 2$ , and JUQUEEN-48 has dimensions $ 4 \\times 3 \\times 2 \\times 2$ .", "Mira has network size $4 \\times 4 \\times 3 \\times 2$ .", "As the networks of JUQUEEN-54 and JUQUEEN-48 are both subgraphs of Mira's, their physical construction is clearly feasible.", "Both these machines have fewer midplanes than JUQUEEN, but have better (greater) bisection bandwidth due to their network sizes.", "Figure: Normalized bisection bandwidth comparison between JUQUEEN and hypothetical machines JUQUEEN-48 and JUQUEEN-54.", "We assume that JUQUEEN always uses best-case partitions.A comparison of bisection bandwidths of partitions of those theoretical machines against the optimal allocations of JUQUEEN is presented in Figure REF , and a full listing of optimal partitions for all three machines appears in Table REF .", "The network bandwidths of partitions of both theoretical machines are identical to those of JUQUEEN when utilizing smaller partitions, and strictly greater on the largest partition sizes.", "Since JUQUEEN is a larger system, applications that are able to perfectly strong scale to its full size will still exhibit superior performance when executed on the entire machine.", "However, in nearly all other cases, JUQUEEN cannot outperform the smaller theoretical machines.", "On contention-bound workloads, the suggested machines are predicted to perform at least as well as JUQUEEN, and attain speedup factors up to $\\times 2$ and $\\times 1.5$ for JUQUEEN-54 and JUQUEEN-48, respectively.", "The partition geometries of proposed machines are described in Table REF .", "Table: Full list of best-case partitions in JUQUEEN and the two proposed machines JUQUEEN-54 and JUQUEEN-48.", "Dimensions are listed in sorted order, BW is normalized bisection bandwidth.Our conjecture about the optimality of Equation REF for arbitrary subsets remains open.", "We believe further speedups on Blue Gene/Q can be demonstrated for several kernels of interest.", "Direct $N$ -body simulation have greater asymptotic contention cost lower bounds than fast matrix multiplication [7], increasing the impact of the internal bisection bandwidth.", "High-performance implementations of FFT, classical matrix multiplication, and other common kernels may better utilize the available hardware resources, decreasing the ratio of time spent performing computation.", "For both those cases, the impact of internal bisection bandwidth on wallclock time is predicted to be greater than in Experiment REF .", "Similar isoperimetric analysis can be conducted on other networks to potentially improve processor allocation policies, and to ensure contention-related effects do not unnecessarily inhibit scaling.", "Testing bisection sensitivity of machine benchmarks can be done by comparing the score of equal-sized partitions with different bisection bandwidths.", "Designing new network topologies, and evaluating existing ones, should be done with their partitioning constraints and internal bisection bandwidths in mind.", "Such considerations can reveal specific partition sizes for which the network performs poorly, and makes it easier to solve such issues.", "Processor allocation policy decisions of job schedulers can be improved if they are informed whether a given computation is expected to be network-bound or not.", "For example, if a partition with sub-optimal bisection bandwidth is currently available for use, a scheduler may decide whether to allocate it to a pending job, or to wait for a partition with better bisection bandwidth.", "This decision can be contingent on a user-provided hint which indicates whether the job is expected to be contention-bound or not.", "We presented a method for analyzing processor allocation policies using an isoperimetric analysis of the network graph, and determining whether any partition geometries induce sub-optimal internal bisection bandwidth.", "We applied our method to two leading Blue Gene/Q supercomputers; demonstrated performance improvements for various workloads; and have shown how to apply our method to other networks.", "We thank Ivo Kabadshow and Dorian Krause of Jülich Supercomputing Centre for their help in arranging the JUQUEEN experiments.", "We thank Adam Scovel of Argonne National Laboratory for his help and support in setting up custom partitions on Mira.", "Our experiments could not have been done without their help.", "The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.", "(www.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC).", "This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.", "Research is supported by grants 1878/14, and 1901/14 from the Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) and grant 3-10891 from the Ministry of Science and Technology, Israel.", "Research is also supported by the Einstein Foundation and the Minerva Foundation.", "This work was supported by the PetaCloud industry-academia consortium.", "This research was supported by a grant from the United States-Israel Bi-national Science Foundation (BSF), Jerusalem, Israel.", "This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 818252).", "This work was supported by The Federmann Cyber Security Center in conjunction with the Israel national cyber directorate.", "Machine Partitions Table: Mira: normalized bisection bandwidths of all current and proposed partitions.Table: Full list of JUQUEEN allocation best and worst cases by compute node count PP.", "Dimensions are listed in sorted order, BW is bisection bandwidths normalized by link capacity." ] ]

[ [ "Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT\n correspondence with a simple surface operator" ], [ "Abstract The quantum toroidal algebra of $gl_1$ provides many deformed W-algebras associated with (super) Lie algebras of type A.", "The recent work by Gaiotto and Rapcak suggests that a wider class of deformed W-algebras including non-principal cases are obtained by gluing the quantum toroidal algebras of $gl_1$.", "These algebras are expected to be related with 5d AGT correspondence.", "In this paper, we discuss quantum deformation of the W-algebras obtained from $\\widehat{su}(N)$ by the quantum Drinfeld-Sokolov reduction with su(2) embedding [N-1,1].", "They were studied by Feigin and Semikhatov and we refer to them as Feigin-Semikhatov's W-algebras.", "We construct free field realization and find several quadratic relations.", "We also compare the norm of the Whittaker states with the instanton partition function under the presence of a simple surface operator in the N=3 case." ], [ "Introduction", "The study of deformed chiral algebras has begun in 1990s.", "It was pioneered in [1], where quantum deformation of Virasoro algebra was constructed.", "It has been generalized to the principal W-algebras for type A in [2] and for simple Lie algebras in [3].", "After the discovery of the AGT correspondence [4], [5], [6], a new way to deal with chiral algebras was developed [7], [8].", "In this correspondence, the action of the chiral algebra on the instanton moduli space is described well by $W_{1+\\infty }$ or the affine Yangian.", "When the correspondence is extended to 5d supersymmetric gauge theories [9], the dual algebras lift to deformed W-algebras and they can be understood as truncations of the quantum toroidal algebras.", "In particular, the quantum toroidal algebra of $\\mathfrak {gl}_1$ (also called Ding-Iohara-Miki algebra [10], [11]) provides the unified way to deal with all of the deformed W-algebras of type A.", "It was also found that it contained the deformed W-algebras q-$W(\\widehat{\\mathfrak {gl}}_{n|m})$ associated with $\\mathfrak {gl}_{n|m}$ [12].", "Further, it has been recently clarified that the quantum toroidal algebra of $\\mathfrak {gl}_1$ reproduces the deformed W-algebras for the other simple Lie algebras [13].", "The CFT limit of q-$W(\\widehat{\\mathfrak {gl}}_{n|m})$ has been studied also from the different viewpoints [14], [15], [16].", "In [14], Gaiotto and Rapčák constructed these algebras in the context of supersymmetric gauge theoreis.", "They considered the system consisting of a 5-brane junction and several D3 branes.", "The chiral algebra referred to as Y-algebra appears at the 2d corner in this configuration and its explicit form can be described by the quantum Drinfeld-Sokolov reduction and the coset construction.", "Although the definition is quite different from $W(\\widehat{\\mathfrak {gl}}_{n|m})$ , the character analysis suggests that they are the same algebras.", "One of the interesting features in Gaiotto-Rapčák's construction is that one can extend the algebras by considering more general brane-webs [17], [18], [19].", "There are two ways to define these algebras.", "In the same manner as the case of Y-algebra, one can define them by the quantum Drinfeld-Sokolov reduction and the coset construction.", "On the other hand, one can also see the system as the combination of the Y-algebras and the operators connecting two junctions.", "The relation between these two descriptions are known as so-called module extension.", "All of the non-principal W-algebras of type A can be realized in the above framework.", "In the AGT correspondence, these algebras appear when we consider the supersymmetric gauge theories with surface operators [20], [21], [22].", "When the gauge group is SU(N), the surface operator is labelled by the partition of $N$ .", "In 2d side, it corresponds to $\\mathfrak {su}(2)$ embedding of Drinfeld-Sokolov reduction.", "For the partition $[N-1,1]$ , the surface operator is called simple and the corresponding W-algebra was studied by Feigin and Semikhatov in [23].", "This algebra has the two currents $e_N(z),f_N(z)$ with spin $\\frac{N}{2}$ and all of the other currents with spin $1,2\\cdots N-1$ can be obtained by taking the OPE of $e_N(z)$ and $f_N(w)$ .", "In this paper, we construct quantum deformation of Feigin-Semikhatov's W-algebras by lifting the Gaiotto-Rapčák's construction to the q-deformed case.", "We implement it by gluing two quantum toroidal algebras of $\\mathfrak {gl}_1$ .", "In the Gaiotto-Rapčák's framework, the currents $e_N(z),f_N(z)$ are given by the product of the primary operators for the Y-algebras.", "For the purpose of q-deformation, we need to deform them into the appropriate vertex operators.", "However, that is difficult to deal with directly because the definition for q-analogue of a primary field is still vague.", "Then we first analyze the case of $N=2$ which corresponds to the known algebra $U_q(\\widehat{\\mathfrak {s}l}_2)$ .", "It is known that $U_q(\\widehat{\\mathfrak {sl}}_2)$ can be embedded in the quantum toroidal algebra of $\\mathfrak {gl}_2$ [24] and how it contains two quantum toroidal algebras of $\\mathfrak {gl}_1$ as subalgebras has been already studied in [25].", "By using these results, we obtain the deformed vertex operators.", "By extending them to the cases for general $N$ , we construct quantum deformation of Feigin-Semikhatov's W-algebras.", "We also compare the norm of the Whittaker states with the 5d instanton partition function under the existence of a simple surface operator.", "This paper is organized as follows.", "In section 2, we provide a review of the quantum toroidal algebra of $\\mathfrak {gl}_1$ and Gaiotto-Rapčák's VOA.", "In section 3, we first review the property of the quantum toroidal algebra of $\\mathfrak {gl}_2$ .", "Then we decompose $U_q(\\widehat{\\mathfrak {gl}}_2)$ into the two quantum toroidal algebras of $\\mathfrak {gl}_1$ , from which we obtain their vertex operators.", "As an application of the vertex operator for q-$W(\\widehat{\\mathfrak {gl}}_1)$ , we demonstrate that the gluing construction reproduces the known representation for the quantum toroidal algebra of $\\mathfrak {gl}_2$ .", "In section 4, we extend them to the cases for general $N$ by requiring the commutativity with the screening charges.", "For the part which cannot be determined from the screening charges, we fix it by demanding that the quadratic relations has the desired property.", "For the case of $N=3$ which corresponds to deformed Bershadsky-Polyakov algebra [26], [27], we write down the quadratic relation in detail.", "In section 5, we compare the norm of the Whittaker states for the deformed Bershadsky-Polyakov algebra with the 5d SU(3) instanton partition function under the existence of a simple surface operator." ], [ "W-algebras associated with $\\mathfrak {gl}_{n|m}$ and quantum toroidal algebra of {{formula:a937621b-915b-44f4-bd9e-828f0bd8ed02}}", "In [12], a new kind of quantum W-algebras was defined as the commutant of the screening charges associated with $\\mathfrak {gl}_{n|m}$ root system.", "These screening charges appeared in the study of Fock representation of quantum toroidal algebra of $\\mathfrak {gl}_1$ .", "On the other hand, the same (but undeformed) W-algebras were constructed in a totally different way in [14].", "They arise in the system of the brane junction in type IIB string theory.", "In this section, we review their definitions and some properties." ], [ "Quantum toroidal algebra of $\\mathfrak {gl}_1$", "In this subsection, we follow the convention in [12].", "The quantum toroidal algebra $\\mathcal {E}_1(q_1,q_2,q_3)$ of $\\mathfrak {gl}_1$ is generated by the Drinfeld currents, $E(z)=\\sum _{m \\in \\mathbb {Z}} E_mz^{-m},\\;\\; F(z)=\\sum _{m \\in \\mathbb {Z}} F_mz^{-m},\\;\\; K^\\pm (z)=(C^\\perp )^{\\pm 1} \\exp \\left(\\sum _{r>0} \\mp \\frac{\\kappa _r}{ r} H_{\\pm r} z^{\\mp r} \\right),$ and the centers $C,C^{\\perp }$ .", "Here, we set $\\kappa _r=\\prod _{i=1}^3(q_i^{r/2}-q_i^{-r/2})=\\sum _{i=1}^3(q_i^{r}-q_i^{-r}).$ The parameters $q_i=\\mathrm {e}^{\\epsilon _i}\\ (i=1,2,3)$ are not independent due to the condition $\\epsilon _1+\\epsilon _2+\\epsilon _3=0$ .", "The defining relation is given as follows: $\\begin{split}&g(z,w)E(z)E(w)+g(w,z)E(w)E(z)=0, \\qquad \\quad g(w,z)F(z)F(w)+g(z,w)F(w)F(z)=0,\\\\&K^\\pm (z)K^\\pm (w) = K^\\pm (w)K^\\pm (z),\\quad \\qquad \\frac{g(C^{-1}z,w)}{g(C z,w)}K^-(z)K^+ (w)=\\frac{g(w,C^{-1}z)}{g(w,C z)}K^+(w)K^-(z),\\\\&g(z,w)K^\\pm (C^{(-1\\mp 1)/2}z)E(w)+g(w,z)E(w)K^\\pm (C^{(-1\\mp 1) /2}z)=0,\\\\&g(w,z)K^\\pm (C^{(-1\\pm 1)/2}z)F(w)+g(z,w)F(w)K^\\pm (C^{(-1\\pm 1)/2}z)=0\\,,\\\\&[E(z),F(w)]=\\frac{1}{\\kappa _1}(\\delta \\bigl (\\frac{Cw}{z}\\bigr )K^+(w)-\\delta \\bigl (\\frac{Cz}{w}\\bigr )K^-(z)),\\end{split}$ with some Serre relations.", "Here, we set $g(z,w)=\\prod _{i=1}^3 (z-q_iw),\\quad \\delta (z)=\\sum _{m \\in \\mathbb {Z}}z^m.$ Some of the above relations can be also written as follows: $\\begin{split}&[H_r,H_s]=\\delta _{r+s,0}r\\frac{C^r-C^{-r}}{\\kappa _r},\\\\&[H_r,E(z)]=-C^{(-r-|r|)/2}E(z)z^r,\\\\&[H_r,F(z)]=C^{(-r+|r|)/2}F(z)z^r.\\end{split}$ For later convenience, we introduce the current $t(z)$ [28] $\\begin{split}&t(z)=\\alpha (z)E(z)\\beta (z),\\\\&\\alpha (z)=\\mathrm {exp}\\bigl (\\sum _{r=1}^{\\infty }\\frac{-\\kappa _r}{r(1-C^{2r})}H_{-r}z^r\\bigr ),\\\\&\\beta (z)=\\mathrm {exp}\\bigl (\\sum _{r=1}^{\\infty }\\frac{-C^{-r}\\kappa _r}{r(1-C^{-2r})}H_rz^{-r}\\bigr ),\\end{split}$ which commutes with Heisenberg subalgebra $H_r$ .", "The algebra $\\mathcal {E}_1$ is equipped with the coproduct, $\\begin{aligned}&\\Delta (H_r)=H_r\\otimes 1+C^{-r}\\otimes H_r,\\quad \\Delta (H_{-r})=H_{-r}\\otimes C^{r}+1\\otimes H_{-r}, \\quad r>0\\\\&\\Delta (E(z))=E\\left(C_2^{-1}z\\right)\\otimes K^+\\left(C_2^{-1}z\\right)+ 1\\otimes E\\left(z\\right),\\\\&\\Delta (F(z))=F\\left(z\\right)\\otimes 1 + K^-\\left(C_1^{-1}z\\right)\\otimes F(C_1^{-1}z),\\\\&\\Delta (X)=X\\otimes X,\\;\\; \\text{for $X= C, C^\\perp $},\\end{aligned} $ where $C_1 =C\\otimes 1$ , $C_2 =1\\otimes C$ .", "The first line implies $\\Delta K^+(z)=K^+(z)\\otimes K^+(C_1z),\\quad \\Delta K^-(C_2z)=K^-(z)\\otimes K^-(z).$ The algebra $\\mathcal {E}_1$ has the MacMahone module [29] where the bases are labelled by a plane partition.", "In this representation, the centers are set to $C=1$ and $C^{\\perp }=c$ .", "The currents $K^{\\pm }(z)$ are diagonalized and the eigenvalue for the highest weight state $|{\\rm hw}\\rangle $ is given by $K^{\\pm }(z)|{\\rm hw}\\rangle =c\\frac{1-c^{-2}v/z}{1-v/z}|{\\rm hw}\\rangle .$ The MacMahone module is irreducible for generic parameters, but, if the condition $c=q_i^{\\frac{m}{2}}q_j^{\\frac{n}{2}}$ $(i\\ne j, n,m\\in \\mathbb {Z}_{\\ge 0})$ is satisfied, the states containing the box at $x_i=m+1,x_j=n+1,x_k=1$ , ($k\\ne i,j$ ) become singularWe set the coordinate of the box at the origin to $(x_1,x_2,x_3)=(1,1,1)$ .. That implies the irreducible module is described by a plane partition with a \"pit\" [12].", "One can also introduce an asymptotic Young diagram to each axis, but it must be compatible with the pit condition.", "Following [12], we denote by $\\mathcal {M}_{\\mu ,\\nu ,\\lambda }(v,c)$ the MacMahon module with the asymptotic Young diagrams $(\\mu ,\\nu ,\\lambda )$ .", "In the case of $c=q_i^{\\frac{1}{2}}$ , the MacMahone representation reduces to the Young diagram representation which is known as the vertical representation $V^{(i)}_v$ in the literature.", "One can reconstruct the MacMahone module with $c=q_i^{\\frac{m}{2}}q_j^{\\frac{n}{2}}$ from the tensor product $V^{(i)}_{v_1}\\otimes \\cdots \\otimes V^{(i)}_{v_m}\\otimes V^{(j)}_{v_{m+1}}\\otimes \\cdots \\otimes V^{(j)}_{v_{n+m}}$ , where $v_l={\\left\\lbrace \\begin{array}{ll}vq_i^{1-l}\\quad (l\\le m)\\\\vq_i^{1-m}q_j^{m-l}\\quad (m<l\\le n).\\end{array}\\right.", "}$ The representation parameters are determined so that the condition (REF ) will hold.", "We note that the order of the tensor product can be freely changed, which is assured by the existence of the universal R-matrix.", "In the similar way, one can determine the parameters when the asymptotic Young diagram is inserted.", "For generic parameters, there are no constraints on the shape of the Young diagrams.", "In that sense, the MacMahon module corresponds to the degenerate module.", "There is another important representation realized by free bosons.", "We introduce three types of free boson oscillators $a_n$ , $n\\in \\mathbb {Z}$ with the relationsIn this notation, we do not put the subscript $i$ on $a_r$ because one can read it off from the representation space.", "$[a_r,a_s]=r\\frac{(q_i^{r/2}-q_i^{-r/2})^3}{-\\kappa _{r}}\\delta _{r+s,0}\\qquad i=1,2,3.$ For later use, we also introduce an operator $Q$ and the parameter $u$ with the relation $[a_n,Q]=\\epsilon _i\\delta _{n,0},\\quad u=e^{a_0}.$ For each $i$ , we have the following free boson representation: $\\begin{split}\\rho _u^{(i)}(E(z))&=\\frac{u(1-q_i)}{\\kappa _1}\\exp \\left(\\sum _{r=1}^\\infty \\frac{q_i^{-r/2}\\kappa _r}{r(q_i^{r/2}-q_i^{-r/2})^2}a_{-r}z^r\\right)\\exp \\left(\\sum _{r=1}^\\infty \\frac{\\kappa _r}{r(q_i^{r/2}-q_i^{-r/2})^2}a_{r}z^{-r}\\right), \\\\\\rho _u^{(i)}(F(z))&=\\frac{u^{-1}(1-q_i^{-1})}{\\kappa _1}\\exp \\left(\\sum _{r=1}^\\infty \\frac{-\\kappa _r}{r(q_i^{r/2}-q_i^{-r/2})^2}a_{-r}z^r\\right)\\exp \\left(\\sum _{r=1}^\\infty \\frac{-q_i^{r/2}\\kappa _r}{r(q_i^{r/2}-q_i^{-r/2})^2}a_{r}z^{-r}\\right), \\\\\\rho _u^{(i)}(H_r)&=\\frac{a_r}{q_i^{r/2}-q_i^{-r/2}}, \\qquad \\rho _u^{(i)}(C^\\perp )=1,\\qquad \\rho _u^{(i)}(C)=q_i^{1/2}.\\end{split}$ We denote the representation space by $\\mathcal {F}^{(i)}_u$ , or simply by $\\mathcal {F}^{(i)}$ if it does not cause confusion.", "Finally, we mention the automorphism which changes the centers as $C^{\\perp }\\rightarrow C$ , $C\\rightarrow {(C^{\\perp })}^{-1}$ .", "Under the automorphism, the vertical representation $V_u^{(i)}$ transforms into the free boson representation $\\mathcal {F}_u^{(i)}$ .", "That implies the MacMahon representation with $c=q_i^{\\frac{m}{2}}q_j^{\\frac{n}{2}}$ is isomorphic to the tensor product of free boson representations $\\underbrace{\\mathcal {F}^{(1)}_{u_1}\\otimes \\cdots \\otimes \\mathcal {F}^{(1)}_{u_m}}_{m}\\otimes \\underbrace{\\mathcal {F}^{(2)}_{u_{m+1}}\\otimes \\cdots \\otimes \\mathcal {F}^{(2)}_{u_{m+n}}}_{n}$ , where the order of the Fock spaces can be changed again.", "In this representation, the Drinfeld currents act on the Fock space as vertex operators.", "They can be characterized by the screening charges introduced in [12].", "Let us first see the simple case $\\mathcal {F}^{(i)}_{u_1}\\otimes \\mathcal {F}^{(i)}_{u_2}$ which is known to realize q-Virasoro and Heisenberg algebra.", "It is widely known that there are two screening charges, $\\begin{aligned}S_{\\pm }^{ii}&=\\oint S_{\\pm }^{ii}(z)\\mathrm {d}z,\\\\S_+^{ii}(z)&=e^{\\frac{\\epsilon _{i+1}}{\\epsilon _i}Q_{12}}z^{\\frac{\\epsilon _{i+1}}{\\epsilon _i}a_{12}+\\frac{\\epsilon _i}{\\epsilon _{i-1}}}\\exp \\left(\\sum _{r=1}^\\infty \\frac{-(q_{i+1}^{r/2}{-}q_{i+1}^{-r/2})}{r(q_i^{r/2}{-}q_i^{-r/2})}b_{-r}z^r\\right)\\exp \\left(\\sum _{r=1}^\\infty \\frac{(q_{i+1}^{r/2}{-}q_{i+1}^{-r/2})}{r(q_i^{r/2}{-}q_{i}^{-r/2})} b_{r}z^{-r}\\right),\\\\S_-^{ii}(z)&=e^{\\frac{\\epsilon _{i-1}}{\\epsilon _i}Q_{12}}z^{\\frac{\\epsilon _{i-1}}{\\epsilon _i}a_{12}+\\frac{\\epsilon _i}{\\epsilon _{i+1}}}\\exp \\left(\\sum _{r=1}^\\infty \\frac{-(q_{i-1}^{r/2}{-}q_{i-1}^{-r/2})}{r(q_i^{r/2}{-}q_i^{-r/2})}b_{-r}z^r\\right)\\exp \\left(\\sum _{r=1}^\\infty \\frac{(q_{i-1}^{r/2}{-}q_{i-1}^{-r/2})}{r(q_i^{r/2}{-}q_{i}^{-r/2})} b_{r}z^{-r}\\right),\\end{aligned}$ where $\\begin{split}&b_{-r}=q_i^{-r} (a_{-r}\\otimes 1)-q_i^{-r/2} (1\\otimes a_{-r}), \\quad b_r=q_i^{r/2} (a_r\\otimes 1)-q_i^{r} (1\\otimes a_r), \\quad r>0,\\\\&Q_{12}=Q\\otimes 1-1\\otimes Q,\\quad a_{12}=a_0\\otimes 1-1\\otimes a_0.\\end{split}$ The next simple case is $\\mathcal {F}^{(i)}_{u_1}\\otimes \\mathcal {F}^{(j)}_{u_2}$ ($i\\ne j$ ).", "The screening charge is given by $\\begin{split}&S^{ij}=\\oint S^{ij}(z)\\mathrm {d}z,\\\\&S^{ij}(z)=e^{Q^{\\prime }_{12}}z^{a^{\\prime }_{12}+\\frac{\\epsilon _j}{-\\epsilon _i-\\epsilon _j}}\\exp \\left(\\sum _{r=1}^\\infty \\frac{1}{-r} c_{-r}z^r\\right)\\exp \\left(\\sum _{r=1}^\\infty \\frac{1}{r} c_{r}z^{-r}\\right),\\end{split}$ where $\\begin{split}& c_{-r}=\\frac{q_1^{-r}(q_2^{r/2}-q_2^{-r/2})}{q_1^{r/2}-q_1^{-r/2}}(a_{-r}\\otimes 1)-\\frac{q_1^{-r/2}(q_1^{r/2}-q_1^{-r/2})}{q_2^{r/2}-q_2^{-r/2}}(1\\otimes a_{-r}),\\\\& c_r=\\frac{q_1^{r/2}(q_2^{r/2}-q_2^{-r/2})}{q_1^{r/2}-q_1^{-r/2}}(a_r\\otimes 1)-\\frac{q_3^{-r/2}(q_1^{r/2}-q_1^{-r/2})}{q_2^{r/2}-q_2^{-r/2}}(1\\otimes a_r),\\quad r>0,\\\\&Q^{\\prime }_{12}=\\frac{\\epsilon _j}{\\epsilon _i} Q\\otimes 1-\\frac{\\epsilon _i}{\\epsilon _j} 1\\otimes Q,\\quad a^{\\prime }_{12}=\\frac{\\epsilon _j}{\\epsilon _i} a_0\\otimes 1-\\frac{\\epsilon _i}{\\epsilon _j} 1\\otimes a_0.\\end{split}$ One can check it satisfies a fermionic relation $S^{ij}(z)S^{ij}(w)=-S^{ij}(w)S^{ij}(z)$ .", "For the generic case $\\mathcal {F}^{(i_1)}\\otimes \\mathcal {F}^{(i_2)}\\otimes \\cdots \\otimes \\mathcal {F}^{(i_n)}$ , the screening currents are given by the union of those for the neighboring Fock spaces $\\mathcal {F}^{(i_l)}\\otimes \\mathcal {F}^{(i_{l+1})}$ $(l=1,\\cdots n-1)$ .", "For $i_1=i_2=\\cdots =i_n$ , it is known that the screening charges have the structure of $\\mathfrak {sl}_n$ root system and give the quantum $W_n$ algebra.", "For $(\\mathcal {F}^{(i)})^{\\otimes n}\\otimes (\\mathcal {F}^{(j)})^{\\otimes m}$ ($i\\ne j$ ), the screening charges have the structure similar with the root system of $\\mathfrak {sl}_{n|m}$ .", "The corresponding W-algebras have not been well studied so far and the authors of [12] referred to them as q-$W(\\widehat{\\mathfrak {gl}}_{n|m})$ .", "The reason why it is not q-$W(\\widehat{\\mathfrak {sl}}_{n|m})$ but q-$W(\\widehat{\\mathfrak {gl}}_{n|m})$ is that there is an additional Heisenberg algebra." ], [ "Gaiotto-Rapčák's VOA and its extension", "In [14], a family of W-algebras $Y_{L,M,N}[\\Psi ]$ was constructed from the system of the 5-brane junction with several D3-branes inserted: Figure: The nonnegative integers L,M,NL,M,N indicate the number of the D3-branes.", "The 5-branes and the D3-branes meet at the three-dimensional intersections.", "These three intersections have the common two-dimensional boundary at the junction, where Y L,M,N [Ψ]Y_{L,M,N}[\\Psi ] appears.From the viewpoint of D3-branes, we can see the system as $\\mathcal {N}=4$ U(L), U(M) and U(N) SYM with the 3d interfaces.", "By considering twisted $\\mathcal {N}=4$ SYM with a coupling constant $\\Psi \\in \\mathbb {C}$ , the authors of [14] obtained the W-algebras $Y_{L,M,N}[\\Psi ]$ to which they referred as Y-algebra.", "The explicit form is given as follows: $Y_{L,M,N}[\\Psi ]={\\left\\lbrace \\begin{array}{ll}\\frac{W_{N-M,1,\\cdots ,1}[\\widehat{U}(N|L)_{\\Psi }]}{\\widehat{U}(M|L)_{\\Psi -1}}\\quad (N>M)\\\\\\quad \\frac{\\widehat{U}(N|L)_{\\Psi }\\otimes {\\rm Sb}^{U(N|L)}}{\\widehat{U}(N|L)_{\\Psi -1}}\\qquad (N=M)\\\\\\frac{W_{M-N,1,\\cdots ,1}[\\widehat{U}(M|L)_{-\\Psi +1}]}{\\widehat{U}(N|L)_{-\\Psi }}\\quad (N<M).\\end{array}\\right.", "}$ Here, we need to explain the notation.", "The symbol $W_{N_1,N_2\\cdots N_n}$ denotes Drinfeld-Sokolov reduction with $\\mathfrak {su}(2)$ embedding labelled by the partition $[N_1,N_2,\\cdots ,N_n]$ of $N=N_1+\\cdots +N_n$ .", "The level $\\Psi $ of $\\widehat{U}(N|L)$ is defined so that the subalgebra $\\widehat{SU}(N|L)$ will have the level $k=\\Psi +L-N$ .", "The matter fields ${\\rm Sb}^{U(N|L)}$ consist of $N$ symplectic bosons and $L$ symplectic fermions.", "They form $\\widehat{U}(N|L)$ and the level of its subalgebra $\\widehat{SU}(N|L)$ is $-1$The level of $\\widehat{SU}(0|L)$ in this notation is different from the standard one by minus sign.. One can also see Y-algebra as a truncation of $W_{1+\\infty }$ or affine Yangian of $\\mathfrak {gl}_1$[17]The claim is subtle when $L,M,N>0$ .", "In this paper, we only consider the cases where one of $L,M$ and $N$ is zero..", "This claim is based on the observation that the vacuum character of $Y_{L,M,N}[\\Psi ]$ is the same as that of a plane partition with a pit at ($L+1, M+1, N+1$ ).", "The relation between the parameters is given by $\\Psi =-\\frac{\\epsilon _2}{\\epsilon _1}.$ The module of Y-algebra is realized by some operators at the junction.", "The characteristic one comes from the line operator introduced in the three-dimensional intersections with one of its end points attached to the junction.", "The interfaces between, for example, $L$ D3-branes and $M$ D3-branes can be described by U(M$|$ L) CS theory and the line operator is labelled by the weight of the gauge group.", "In terms of a plane partition, we can interpret it as an asymptotic Young diagram.", "When we insert the line operators as in Figure REF , they give the MacMahon module $\\mathcal {M}_{\\lambda ,\\mu ,\\nu }$ .", "We note that the shape of the Young diagram is restricted due to the pit and it indeed gives the representation of super Lie group.", "Figure: The line operators are denoted by their weights in this figure.", "When one of L,ML,M or NN is zero, it corresponds to the MacMahon module ℳ λ,μ,ν \\mathcal {M}_{\\lambda ,\\mu ,\\nu } with a pit at (L+1,M+1,N+1)(L+1,M+1,N+1).We explain the relation between $Y_{0,M,N}$ and $W(\\widehat{\\mathfrak {gl}}_{N|M})$ in more detail.", "The simplest case is $Y_{0,0,N}$ , which gives $W_N$ algebra and Heisenberg algebra according to (REF ).", "On the other hand, the MacMahon module with a pit at $(1,1,N+1)$ is equivalent to $(\\mathcal {F}^{(3)})^{\\otimes n}$ , which gives the same algebra.", "The asymptotic Young diagrams give the completely degenerate module (see [30]).", "The next example is $Y_{0,1,2}$ , which gives $\\frac{\\widehat{U}(2)}{\\widehat{U}(1)}$ known as SU(2) parafermion with an extra Heisenberg algebra.", "In terms of the quantum toroidal algebra, it should correspond to the vertex operator acting on the Fock space $\\mathcal {F}^{(2)}\\otimes (\\mathcal {F}^{(3)})^{\\otimes 2}$ .", "When we fix the order of the Fock spaces to $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}$ , there are two fermionic screening charges.", "As we will explain, they are indeed those of SU(2) parafermion.", "One can extend Y-algebra to a wider family of W-algebras by gluing trivalent vertices [17].", "In this paper, we mainly focus on the following diagram: Figure: The diagram for Feigin-Semikhatov's W-algebra W N (2) W^{(2)} _NWe first need to comment on the trivalent vertex with different slope.", "It can be obtained from the standard one through the IIB S-duality.", "When the transformation matrix is given by $M=\\left(\\begin{array}{c}p\\ q\\\\r\\ s\\end{array}\\right)\\in SL(2,\\mathbb {Z})$ , the D5-brane and NS5-brane are transformed into $\\left(\\begin{array}{c}p\\\\r\\end{array}\\right)$ -brane and $\\left(\\begin{array}{c}q\\\\s\\end{array}\\right)$ -brane, respectively.", "At the same time, the coupling constant changes as $\\Psi \\rightarrow \\frac{p\\Psi +q}{r\\Psi +s}$ .", "In terms of the affine Yangian's parameters, the transformation law is expressed as $(\\epsilon _1,\\epsilon _2)\\rightarrow (\\epsilon _1,\\epsilon _2)M^{-1}= (s\\epsilon _1-r\\epsilon _2,-q\\epsilon _1+p\\epsilon _2).$ In the q-deformed case, it is rewritten as $(q_1,q_2)\\rightarrow (q_1^sq_2^{-r},q_1^{-q}q_2^p).$ According to the BRST procedure, the algebra associated with figure REF is $W_{N-1,1}[\\widehat{U}(N)_\\Psi ]$ .", "It gives affine Kac-Moody algebra $\\widehat{U}(2)_\\Psi $ for $N=2$ and Bershadsky-Polyakov algebra [26], [27] for $N=3$ .", "In [23], Feigin and Semikhatov studied the algebra from the two perspectives: screening charges and coset construction.", "Our concern is on the first one and we will discuss its q-deformation We mention that the Y-algebra provides also the second description.", "In figure REF , the upper corner algebra is $Y_{0,1,N}[\\Psi ]$ , which is equivalent under the IIB S-duality to $Y_{N,0,1}[\\Psi ]=\\frac{\\widehat{U}(N|1)_{-1+\\frac{1}{\\Psi }}}{\\widehat{U}(N)_{\\Psi }}$ .", "This is exactly the coset construction proposed in [23].", "This duality has been proved in [31], [32].", "We also mention that the level-rank duality between the minimal models of $W_M$ and $W_{N-1,1}[\\widehat{U}(N)]$ discussed in [23] can be described in terms of a plane partition [30].. One may also consider that the system consists of $Y_{0,1,N}[\\Psi ]$ , $Y_{0,0,1}[\\Psi -1]$ and the line operators connecting the two junctions.", "As we have seen, their module is described by two plane partitions.", "The line operators can be interpreted as the asymptotic Young diagrams connecting the two plane partitions.", "Then the entire representation space is given by $\\oplus _{\\lambda \\in \\mathbb {Z}}\\mathcal {M}_{\\varnothing ,\\varnothing ,\\lambda }\\otimes \\mathcal {M}_{\\varnothing ,\\lambda ,\\varnothing }$ .", "We note that Young diagrams are usually labelled by positive integers, but it was claimed in [17] that negative weight diagrams should be also summed up.", "While each Y-algebra acts on a single plane partition, we have the other elements which change the asymptotic Young diagram.", "They correspond to the currents $e_N(z),f_N(z)$ of $W_{N-1,1}[\\widehat{U}(N)_\\Psi ]$ with spin $\\frac{N}{2}$ .", "In the following, we sometimes refer to them as the gluing fields.", "In terms of free boson realization, they behave as the vertex operators which change the zero mode of the Fock space.", "The above description is familiar for $N=2$ ; it is known that $\\widehat{SU}(2)$ can be realized by SU(2) parafermion $\\psi (z),\\psi ^{\\dagger }(z)$ and U(1) boson $\\phi (z)$ as follows, $J^+(z)=\\psi (z)e^{\\phi (z)},\\quad J^-(z)=\\psi ^{\\dagger }(z)e^{-\\phi (z)},\\quad J^3(z)=\\partial \\phi (z).$ In terms of the gluing construction, the currents $J^{\\pm }(z)$ correspond to the line operators with $U(1)$ weight $\\pm 1$ and are expressed as the product of the primary fields for $Y_{0,1,2}[\\Psi ]$ and $Y_{0,0,1}[\\Psi -1]$Precisely speaking, the U(1) boson in (REF ) is different from that of $Y_{0,0,1}$ .", "Because the Y-algebra contains Heisenberg algebra, the primary field of $Y_{0,1,2}$ is given as the product of $\\psi (z)$ (or $\\bar{\\psi }(z)$ ) and an extra vertex operator $e^{\\tilde{\\phi }(z)}$ .", "For $Y_{0,0,1}$ , the primary field is just $e^{\\overline{\\phi }(z)}$ .", "Then we find $\\phi (z)=\\tilde{\\phi }(z)+\\overline{\\phi }(z)$ .", "Another combination of $\\tilde{\\phi }(z)$ and $\\overline{\\phi }(z)$ commutes with all of the algebra and gives U(1) factor..", "In this construction, the non-localness of the parafermion are resolved thanks to the additional U(1) boson.", "The same is true for the other $N$ and the conformal dimension of the gluing fields are (half-)integer $\\frac{N}{2}$ .", "It is natural to expect that the algebra associated with the brane-web in figure REF is also a truncation of some $W_{\\infty }$ -like algebra or affine Yangian.", "We explain this point along with [17], [33].", "In general, we can insert D3-branes in the four places as follows: Figure: The diagram where D3-branes are inserted in the four places.The conformal dimension of the gluing field with fundamental or anti-fundamental weight is $h=1+\\frac{L+N-2M}{2}$ .", "That implies the algebras with different values of $\\rho =\\frac{L+N-2M}{2}$ should be truncations of different $W_{\\infty }$ -like algebras.", "Let us consider the $\\rho =0$ case.", "For $L=M=N=0$ , the corresponding algebra is associated with AGT on $\\mathbb {C}^2/\\mathbb {Z}_2$ [34].", "The relation to AGT becomes more explicit through the string duality, which leads to the system in which D4-branes stack on the divisor inside the toric Calabi-Yau manifold under the existence of D0-branes [16], [33].", "For $L=2,M=1,K=N=0$ , it corresponds to $\\widehat{U}(2)$ .", "These facts imply that they are truncations of affine Yangian of $\\mathfrak {gl}_2$In this paper, we use the terms $\\mathfrak {gl}_n$ and U(n) interchangeably..", "This observation will play an important role in the next section.", "For the $\\rho \\ne 0$ case, it includes the non-principal W-algebras $W_{M+2\\rho ,M}[\\widehat{U}(L)]$ .", "The relation between the deformed W-algebras for general $\\mathfrak {su}(2)$ embedding and the quantum toroidal algebras has been discussed in [35].", "We also mention that the deformation of the W-algebra $W_{N,N,\\cdots ,N}[\\widehat{U}(Nn)]$ has been recently proposed in [36] together with the relation to the quantum toroidal algebra of $\\mathfrak {gl}_n$ ." ], [ "Decomposition of $U_q(\\widehat{\\mathfrak {gl}}_2)$ by quantum toroidal algebras", "In this section, we explore whether the gluing construction works even after q-deformation.", "We study how $U_q(\\widehat{\\mathfrak {gl}}_2)$ is decomposed into q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ and q-$W(\\widehat{\\mathfrak {gl}}_{1})$ .", "For that purpose, it is better to embed $U_q(\\widehat{\\mathfrak {gl}}_2)$ into the quantum toroidal algebra of $\\mathfrak {gl}_2$ .", "We first review the definition of the quantum toroidal algebra of $\\mathfrak {gl}_2$ and its important properties following [25], [37]." ], [ "Quantum toroidal algebra of $\\mathfrak {gl}_2$", "The quantum toroidal algebra $\\mathcal {E}_2(q_1,q_2,q_3)$ of $\\mathfrak {gl}_2$ has two parameters $d,q$ .", "The following parameters are also used, $& q_1=d q^{-1},\\ q_2=q^2,\\ q_3=d^{-1}q^{-1}\\ .$ It has the following generators, $E_{i,k},\\ F_{i,k},\\ H_{i,r},\\ K_i^{\\pm 1},\\ C^{\\prime \\pm 1} \\quad (i\\in \\lbrace 0,1\\rbrace ,\\ k\\in {\\mathbb {Z}},\\ r\\in {\\mathbb {Z}}/\\lbrace 0\\rbrace ).$ By introducing the Drinfeld currents $E_i(z) =\\sum _{k\\in {\\mathbb {Z}}}E_{i,k}z^{-k}, \\quad F_i(z) =\\sum _{k\\in {\\mathbb {Z}}}F_{i,k}z^{-k}, \\quad K_i^{\\pm }(z) = K_i^{\\pm 1} \\exp (\\pm (q-q^{-1})\\sum _{r=1}^\\infty H_{i,\\pm r}z^{\\mp r}),$ the defining relations are written as follows: $\\begin{split}&\\hspace{113.81102pt}\\text{$C^{\\prime \\pm 1}$ are central},\\\\&\\hspace{28.45274pt}K_iE_j(z)K_i^{-1}=q^{a_{i,j}}E_j(z),\\quad K_iF_j(z)K_i^{-1}=q^{-a_{i,j}}F_j(z),\\\\&\\hspace{85.35826pt}K^\\pm _i(z)K^\\pm _j (w) = K^\\pm _j(w)K^\\pm _i (z), \\\\&\\hspace{28.45274pt}\\frac{g_{i,j}(C^{\\prime -1}z,w)}{g_{i,j}(C^{\\prime }z,w)}K^-_i(z)K^+_j (w) =\\frac{g_{j,i}(w,C^{\\prime -1}z)}{g_{j,i}(w,C^{\\prime }z)}K^+_j(w)K^-_i (z),\\\\&(-1)^{i+j}g_{i,j}(z,w)K_i^\\pm (C^{\\prime (-1\\mp 1)/2}z)E_j(w)+g_{j,i}(w,z)E_j(w)K_i^\\pm (C^{\\prime (-1\\mp 1)/2}z)=0,\\\\&(-1)^{i+j}g_{j,i}(w,z)K_i^\\pm (C^{\\prime (-1\\mp 1)/2}z)F_j(w)+g_{i,j}(z,w)F_j(w)K_i^\\pm (C^{\\prime (-1\\mp 1)/2}z)=0\\,,\\\\&\\hspace{28.45274pt}[E_i(z),F_j(w)]=\\frac{\\delta _{i,j}}{q-q^{-1}}(\\delta \\bigl (C^{\\prime }\\frac{w}{z}\\bigr )K_i^+(w)-\\delta \\bigl (C^{\\prime }\\frac{z}{w}\\bigr )K_i^-(z)),\\\\&\\hspace{28.45274pt}(-1)^{i+j}g_{i,j}(z,w)E_i(z)E_j(w)+g_{j,i}(w,z)E_j(w)E_i(z)=0, \\\\&\\hspace{28.45274pt}(-1)^{i+j}g_{j,i}(w,z)F_i(z)F_j(w)+g_{i,j}(z,w)F_j(w)F_i(z)=0,\\\\\\end{split}$ where $a_{i,j}=\\begin{pmatrix}2 & -2\\\\-2 & 2\\\\\\end{pmatrix}$ is the Cartan matrix of $\\widehat{\\mathfrak {sl}}_2$ and $g_{i,j}(z,w)={\\left\\lbrace \\begin{array}{ll}z-q_2w & (i= j),\\\\(z-q_1w)(z-q_3w)& (i\\ne j).\\end{array}\\right.", "}$ It is possible to rewrite the above relations using $H_{i,r}$ instead of $K^{\\pm }_i(z)$ as follows: $&[H_{i,r},E_j(z)]=a_{i,j}(r) z^r E_j(z)C^{\\prime -(r+|r|)/2}\\,,\\\\&[H_{i,r},F_j(z)]=-a_{i,j}(r)z^r F_j(z)C^{\\prime -(r-|r|)/2}\\,,\\\\&[H_{i,r},H_{j,s}]=\\delta _{r+s,0}\\,a_{i,j}(r)\\frac{C^{\\prime r}-C^{\\prime -r}}{q-q^{-1}},$ where $a_{i,i}(r)=[r]_q(q^r+q^{-r})/r$ , $a_{i,j}(r)=-[r]_q(d^r+d^{-r})/r$ ($i\\ne j$ ) and $[r]_q=\\frac{q^r-q^{-r}}{q-q^{-1}}$ .", "The algebra $\\mathcal {E}_2(q_1,q_2,q_3)$ has the subalgebras $\\mathcal {E}_1^{(1)}(\\tilde{q}_1,\\tilde{q}_2,\\tilde{q}_3)$ and $\\mathcal {E}_1^{(3)}(\\hat{q}_1,\\hat{q}_2,\\hat{q}_3)$ commuting with each other.", "Their generators and parameters are given, respectively, as follows [25]: $\\text{For $\\mathcal {E}_1^{(1)}$},\\quad &E(z)=E_{1|0}^{(1)},\\ F(z)=F_{1|0}^{(1)}(z),\\ K(z)=K_{1|0}^{\\pm \\ (1)}(z), C=C^{\\prime }, C^{\\perp }=K_1K_2,\\\\&\\tilde{q}_1=q_2,\\ \\ \\tilde{q}_2=q_1^2,\\ \\ \\tilde{q}_3=q_3q_1^{-1}.", "\\\\\\ \\nonumber \\\\\\text{For $\\mathcal {E}_1^{(3)}$},\\quad &E(z)=E_{1|0}^{(3)},\\ F(z)=F_{1|0}^{(3)}(z),\\ K(z)=K_{1|0}^{\\pm \\ (3)}(z), C=C^{\\prime }, C^{\\perp }=K_1K_2,\\\\&\\hat{q}_1=q_2,\\ \\ \\hat{q}_2=q_1q_3^{-1},\\ \\ \\hat{q}_3=q_3^2.$ Here, we set $\\begin{split}&E_{1|0}^{(1)}(z)=\\lim _{z^{\\prime }\\rightarrow z}(1-\\frac{z}{z^{\\prime }})(1-\\frac{q_3z}{q_1z^{\\prime }})E_1 (q_1z^{\\prime })E_0(z), \\\\&F_{1|0}^{(1)}(z)=\\lim _{z^{\\prime }\\rightarrow z}(1-\\frac{z^{\\prime }}{z})(1-\\frac{q_1z^{\\prime }}{q_3z})F_0(z) F_1 (q_1z^{\\prime }), \\\\&K_{1|0}^{\\pm ,(1)}(z)=K_1^{\\pm }(q_1z)K_0^{\\pm }(z),\\\\&E_{1|0}^{(3)}(z)=\\lim _{z^{\\prime }\\rightarrow z}(1-\\frac{z}{z^{\\prime }})(1-\\frac{q_1z}{q_3z^{\\prime }})E_1 (q_3z^{\\prime })E_0(z),\\\\&F_{1|0}^{(3)}(z)=\\lim _{z^{\\prime }\\rightarrow z}(1-\\frac{z^{\\prime }}{z})(1-\\frac{q_3z^{\\prime }}{q_1z})F_0(z) F_1 (q_3z^{\\prime }), \\\\&K_{1|0}^{\\pm ,(3)}(z)=K_1^{\\pm }(q_3z)K_0^{\\pm }(z).\\end{split}$ Another important subalgebra is the quantum affine algebra $U_q(\\widehat{\\mathfrak {sl}}_2)$ .", "If we set $\\begin{split}&x^+(z)=E_1(\\alpha z),\\ \\ x^-(z)=F_1(\\alpha z),\\ \\ h_r=(C^{\\prime 1/2}\\alpha ^{-1})^rH_{1,r}\\ (r\\in \\mathbb {Z}/0),\\ \\ q^{h_0}=K_1,\\\\& \\sum _{m=0}^\\infty \\psi _m z^{-m}=q^{h_0}\\mathrm {exp}\\bigl ((q-q^{-1})\\sum _{m=1}^\\infty h_m z^{-m}\\bigr ),\\\\&\\sum _{m=0}^\\infty \\varphi _{-m}z^{m}=q^{-h_0}\\mathrm {exp}\\bigl (-(q-q^{-1})\\sum _{m=1}^\\infty h_{-m} z^{m}\\bigr ),\\end{split}$ these elements satisfy the following defining relation of $U_q(\\widehat{\\mathfrak {sl}}_2)$ : $\\begin{split}&[h_m,h_n]=\\delta _{m+n,0}{{[2m]_q[mc]_q}\\over {m}},\\ [h_m,q^{h_0}]=0,\\\\&q^{h_0}x_m^\\pm q^{-h_0}=q^{\\pm 2}x_m^\\pm ,\\\\& [h_m,x_n^\\pm ]=\\pm \\frac{[2m]_q}{m}q^{\\mp \\frac{|m|c}{2}}x_{m+n}^\\pm ,\\\\&x_{m+1}^\\pm x_{n}^\\pm -q^{\\pm 2}x_n^\\pm x_{m+1}^\\pm =q^{\\pm 2} x_{m}^\\pm x_{n+1}^\\pm -x_{n+1}^\\pm x_{m}^\\pm ,\\\\&[x_m^+,x_n^-]={1\\over {q-q^{-1}}}(q^{{{(m-n)}\\over 2}c}\\psi _{m+n}-q^{{{(n-m)}\\over 2}c}\\varphi _{m+n}),\\end{split}$ where $C^{\\prime }=q^c$ .", "The parameter $\\alpha $ can take an arbitrary value, but we fix it to $\\alpha =q_3^{1/2}q^{-1}$ in the following.", "In [38], [39], the free field realization of $U_q(\\widehat{\\mathfrak {sl}}_2)$ was constructed.", "These results were promoted to the representation $\\rho $ of $\\mathcal {E}_2$ in [24], [37], which is so-called evaluation homomorphism.", "In this representation, the center takes the value $C^{\\prime }=q_3$ and the generators behave as the following vertex operators: $\\begin{split}&-(q-q^{-1})\\rho (E_1(z))=:e^{u_1^+(z)}:-:e^{u_1^-(z)}:,\\quad (q-q^{-1})\\rho (F_1(z))=:e^{v_1^+(z)}:-:e^{v_1^-(z)}:,\\\\&-(q-q^{-1})\\rho (E_0(z))=:e^{\\widehat{u}_1^+(z)}:-:e^{\\widehat{u}_1^-(z)}:,\\quad (q-q^{-1})\\rho (F_0(z))=:e^{\\widehat{v}_1^+(z)}:-:e^{\\widehat{v}_1^-(z)}:,\\\\&\\rho (H_{1,n})=h_{1,n},\\quad \\rho (H_{0,n})=\\widehat{h}_{1,n}, \\quad \\rho (K_1)=q^{h_{1,0}},\\quad \\rho (K_0)=q^{-h_{1,0}},\\end{split}$ where the following free boson fields are introduced $&u^\\pm _1(z)=Q_{u_1}+u_{1,0}\\log z\\mp a_{3,0}\\log q-\\sum _{n\\ne 0}\\frac{u^\\pm _{1,n}}{n}z^{-n}\\,,\\\\&v^\\pm _1(z)=-\\bigl (Q_{u_1}+u_{1,0}\\log z\\bigr )\\pm a_{2,0}\\log q -\\sum _{n\\ne 0}\\frac{v^\\pm _{1,n}}{n}z^{-n},\\\\&\\hat{u}^\\pm _1(z)=-\\bigl (Q_{u_1}+u_{1,0}\\log z\\bigr )\\mp a_{2,0}\\log q-\\sum _{n\\ne 0}\\frac{\\hat{u}^\\pm _{1,n}}{n}z^{-n}\\,,\\\\&\\hat{v}^\\pm _1(z)=Q_{u_1}+u_{1,0}\\log z\\pm a_{3,0}\\log q-\\sum _{n\\ne 0}\\frac{\\hat{v}^\\pm _{1,n}}{n}z^{-n}.$ We note that the above bosons are not independent and are expressed by four independent bosons.", "One of them corresponds to $\\widehat{\\mathfrak {gl}}_1$ and the other three correspond to the bosons used in Wakimoto representation.", "The concrete expression for these bosons are not necessary in what follows and we omit it.", "See [37] for details." ], [ "Decomposition of $U_q(\\widehat{\\mathfrak {gl}}_2)$", "Let us explore the decomposition of $\\mathcal {E}_2$ into the two $\\mathcal {E}_1$ s in the case where they satisfy the truncation conditions.", "Because the centers of the subalgebras $\\mathcal {E}_1^{(1)}$ and $\\mathcal {E}_1^{(3)}$ are the same, their truncation conditions are not independent: the nonnegative integer parameters $K,M,L,K^{\\prime },N,M^{\\prime }$ labelling each truncation condition must satisfy the relation, $C=\\tilde{q}_1^{K}\\tilde{q}_2^{M}\\tilde{q}_3^{L}=\\hat{q}_1^{K^{\\prime }}\\hat{q}_2^{N}\\hat{q}_3^{M^{\\prime }}.$ One can see from () and () that it implies $K=K^{\\prime },\\quad M=M^{\\prime }, \\quad L+N-2M=0$ for generic $q_1,q_2,q_3$ .", "Let us compare these results with Gaiotto-Rapčák's construction.", "In figure REF , there are two Y-algebras, $Y_{K,M,L}$ and $Y_{K,N,M}$ .", "This combination matches the first two relations in (REF ).", "The last condition also matches the discussion in the last part of section ; as we have explained, the $W_{\\infty }$ -like algebra associated with figure REF depends on the parameter $\\rho =\\frac{L+N-2M}{2}$ and it is expected that the affine Yangian of $\\mathfrak {gl}_2$ is realized in the $\\rho =0$ case.", "Further, the relation between the parameter of $\\mathcal {E}_1^{(1)}$ and that of $\\mathcal {E}_1^{(3)}$ is also consistent with the relation (REF ) in the gluing construction.", "In the remaining part of this section, we focus on the case of $C=q_3$ where $\\mathcal {E}_2$ truncates to $U_q(\\widehat{\\mathfrak {gl}}_2)$ .", "The subalgebras $\\mathcal {E}_1^{(1)}$ and $\\mathcal {E}_1^{(3)}$ truncate to q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$The algebra q-$W({\\mathfrak {sl}_{2|1}})$ was originally proposed in [40] and further studied in [41].", "and q-$W(\\widehat{\\mathfrak {gl}}_1)$ , respectively.", "As we have seen, q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ (resp.", "q-$W(\\widehat{\\mathfrak {gl}}_1)$ ) can be realized by three bosons (resp.", "one boson).", "In our notation, we express the boson for q-$W(\\widehat{\\mathfrak {gl}}_1)$ with a tilde and the boson for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ acting on $i$ -th Fock space with a subscript $(i)$One should be careful not to confuse the subscript $(i)$ with that in (REF ).", "Recall that we do not put the subscript which distinguish the type of the boson..", "In order to understand how the gluing fields of $U_q(\\widehat{\\mathfrak {gl}}_2)$ decompose into the vertex operators for the two q-W algebras, let us express the bosons used in (REF ) by these bosons.", "We can implement that by comparing the two free boson representations for $\\mathcal {E}_1$ : the first one is obtained from (REF ) and (REF ) while the other one is directly obtained from (REF ) and (REF ).", "The result is as follows (see Appendix for the detail of the computation): $\\begin{split}&u_{1,n}^+={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})q_3^n}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})q_3^n}{d^n-d^{-n}}a_n^{(3)}-\\frac{(q^n-q^{-n})q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n>0) \\\\-\\frac{(q^n-q^{-n})q^{-2n}}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})q_3^nq^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\&u_{1,n}^-={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})q_1^n}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n>0) \\\\-\\frac{(q^n-q^{-n})q_1^{2n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})q^{-n}}{d^n-d^{-n}}a_n^{(3)}-\\frac{(q^n-q^{-n})q_3^nq^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\\\end{split}$ $\\begin{split}&v_{1,n}^+={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})q^{-n}}{d^n-d^{-n}}a_n^{(1)}+\\frac{(q^n-q^{-n})q_1^{2n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})q^{-n}q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n>0) \\\\\\frac{(q^n-q^{-n})q_1^n}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\&v_{1,n}^-={\\left\\lbrace \\begin{array}{ll}\\frac{(q^n-q^{-n})q^{-2n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})q^{-n}q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n>0) \\\\\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})q_3^n}{d^n-d^{-n}}a_n^{(1)}+\\frac{(q^n-q^{-n})q_3^n}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\\\end{split}$ $\\begin{split}h_{1,n}={\\left\\lbrace \\begin{array}{ll}&\\hspace{-8.53581pt}\\frac{[n]_q}{n}\\left(\\frac{q_1^n-q_1^{-n}}{d^n-d^{-n}}a_n^{(1)}-d^na_n^{(2)}+\\frac{(q_1^n-q_1^{-n})q^n}{d^n-d^{-n}}a_n^{(3)}-\\tilde{a}_n\\right)\\ (n>0)\\hspace{85.35826pt}\\\\&\\hspace{-8.53581pt}\\frac{[n]_q}{n}\\left(\\frac{(q_1^n-q_1^{-n})q^n}{d^n-d^{-n}}a_n^{(1)}-d^na_n^{(2)}+\\frac{q_1^n-q_1^{-n}}{d^n-d^{-n}}a_n^{(3)}-\\tilde{a}_n\\right)\\ (n<0)\\end{array}\\right.", "}\\end{split}$ $\\begin{split}&Q_{u_1}=Q^{(2)}+\\tilde{Q},\\quad u_{1,0}=\\frac{1}{\\epsilon _1-\\epsilon _3}(a_0^{(2)}-\\tilde{a}_0),\\quad h_{1,0}=\\frac{1}{\\epsilon _1-\\epsilon _3}(-\\epsilon _3a_0^{(2)}+\\epsilon _1\\tilde{a}_0)\\\\&a_{2,0}=\\frac{1}{\\epsilon _2}(-a_0^{(1)}+a_0^{(2)}),\\quad a_{3,0}=\\frac{1}{\\epsilon _2}(-a_0^{(2)}+a_0^{(3)}),\\end{split}$ where the commutation relations are given as follows: $\\begin{split}&[a_n^{(1)},a_m^{(1)}]=[a_n^{(3)},a_m^{(3)}]=-n\\frac{(d^n-d^{-n})^2}{(q_1^n-q_1^{-n})(q^n-q^{-n})}\\delta _{n+m,0},\\\\&[a_n^{(2)},a_m^{(2)}]=n\\frac{(q_1^n-q_1^{-n})^2}{(d^n-d^{-n})(q^n-q^{-n})}\\delta _{n+m,0},\\quad [\\tilde{a}_n,\\tilde{a}_m]=-n\\frac{(q_3^n-q_3^{-n})^2}{(d^n-d^{-n})(q^n-q^{-n})}\\delta _{n+m,0},\\\\&[a_n^{(2)},Q^{(2)}]=[\\tilde{a}_n,\\tilde{Q}]=\\epsilon _2\\delta _{n,0}.\\end{split}$ The other commutators vanish.", "The zero modes satisfy the condition, $\\epsilon _1u_{1,0}=\\frac{1}{2}(a_0^{(1)}+a_0^{(3)}).$ We note that this condition is compatible in the entire Fock space.", "The vertex operator for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ is realized on the Fock space $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}$ while that for q-$W(\\widehat{\\mathfrak {gl}}_1)$ is realized on $\\tilde{\\mathcal {F}}^{(3)}$ .", "The reason why the order of Fock spaces for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ is fixed to $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}$ is that the Wakimoto representation of $\\widehat{\\mathfrak {sl}}_2$ can be characterized by two fermionic screening chargesWe note that there is an additional screening charge which is not an exponential type but dressed by a polynomial.", "; for another ordering of the Fock spaces, we only have one fermionic screening charge.", "It is convenient to introduce the boson $a_n^{\\mathfrak {u}(1)}$ associated with Heisenberg subalgebra of q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ as follows: $\\frac{a_n^{\\mathfrak {u}(1)}}{q_3^n-q_3^{-n}}={\\left\\lbrace \\begin{array}{ll}-\\frac{a_n^{(1)}}{d^n-d^{-n}}+\\frac{d^na_n^{(2)}}{q_1^n-q_1^{-n}}-\\frac{q^na_n^{(3)}}{d^n-d^{-n}}\\quad (n>0)\\\\-\\frac{q^na_n^{(1)}}{d^n-d^{-n}}+\\frac{d^na_n^{(2)}}{q_1^n-q_1^{-n}}-\\frac{a_n^{(3)}}{d^n-d^{-n}}\\quad (n<0).\\end{array}\\right.", "}$ Then, the relation (REF ) can be rewritten into the simpler form: $h_{1,n}=-\\frac{[n]_q}{n}J_n,\\qquad J_n=\\frac{q_1^n-q_1^{-n}}{q_3^n-q_3^{-n}}a_n^{\\mathfrak {u}(1)}+\\tilde{a}_n.$ The other linear combination of $a_n^{\\mathfrak {u}(1)}$ and $\\tilde{a}_n$ which commutes with $J_n$ is $j_n=a_n^{\\mathfrak {u}(1)}+\\tilde{a}_n.$ It commutes with all the elements of $U_q(\\widehat{\\mathfrak {sl}}_2)$ and forms $U_q(\\widehat{\\mathfrak {gl}}_2)$ together with $U_q(\\widehat{\\mathfrak {sl}}_2)$ .", "We can extract the Heisenberg part from the currents $x^{\\pm }(z)$ as follows: $&x^+(z)=\\psi _q(z) :\\exp \\left(\\sum _{n\\ne 0}\\frac{q_3^{-\\frac{|n|}{2}}(q^n-q^{-n})}{n(q_3^n-q_3^{-n})}J_nz^{-n}\\right):,\\\\&x^-(z)=\\overline{\\psi }_q(z) :\\exp \\left(-\\sum _{n\\ne 0}\\frac{q_3^{\\frac{|n|}{2}}(q^n-q^{-n})}{n(q_3^n-q_3^{-n})}J_nz^{-n}\\right):.$ The vertex operators $\\psi _q(z), \\overline{\\psi }_q(z)$ can be interpreted as q-deformed SU(2) parafemions.", "Finally, we mention the screening currents of the deformed Wakimoto representation.", "As we have explained, there are two fermionic screening currents in $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}$ .", "We checked that they are indeed the same as those for $U_q(\\widehat{\\mathfrak {sl}}_2)$ obtained in [42]." ], [ "The vertex representation of $\\mathcal {E}_2$ from the gluing construction", "As an application of the vertex operator for q-$W(\\widehat{\\mathfrak {gl}}_1)$ , we consider the gluing construction for $K=1,L=M=N=0$ : (2,0)–(0,0); (0,1.5)–(0,0)–(-1.3,-1.3)–(-3.3,-2.4); (-1.3,-1.3)–(2,-1.3); 2) at (1,0.8) 0; 1) at (0.8,-0.8) 0; 01) at (-1.8,-0.2) 1; 02) at (0,-2) 0; The branching rule of the module has been already studied in [25], where the vertical representation ($C=1$ ) was considered.", "In this section, we consider the free boson representation.", "From the truncation condition, we find that the center should be $C^{\\prime }=q$ .", "In the previous case, the Drinfeld currents $E_1(z)$ , $F_1(z)$ of $\\mathcal {E}_2$ are realized as the product of the vertex operators for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ and q-$W(\\widehat{\\mathfrak {gl}}_1)$ .", "In the following, we examine whether the gluing construction works also for the present case.", "We refer to q-$W(\\widehat{\\mathfrak {gl}}_1)$ associated with the upper (resp.", "the lower) junction as the upper (resp.", "the lower) q-W.", "The vertex operators $V^{\\pm }(z)$ for q-$W(\\widehat{\\mathfrak {gl}}_1)$ which we found in the previous subsection are rewritten as $V^{\\pm }(z)=\\exp \\left(\\pm (\\tilde{Q}^{\\prime }+\\tilde{a}_0^{\\prime }\\log z-\\sum _{n\\ne 0}q_3^{\\mp \\frac{|n|}{2}}\\frac{\\tilde{a}^{\\prime }_n}{n}z^{-n})\\right),$ where $[\\tilde{a}_0^{\\prime },\\tilde{Q}^{\\prime }]=\\frac{\\epsilon _2}{\\epsilon _3-\\epsilon _1},\\quad [\\tilde{a}_n^{\\prime },\\tilde{a}_m^{\\prime }]=-n\\frac{q^n-q^{-n}}{d^n-d^{-n}}\\delta _{n+m,0}.$ To apply these vertex operators to the present case, we need slight modification because the parameters of q-$W(\\widehat{\\mathfrak {gl}}_1)$ are not the same.", "In the MacMahon language, the vertex operators (REF ) correspond to $\\mathcal {M}_{\\emptyset , \\mu ,\\emptyset }$ with a pit at $(1,1,2)$ , where $\\mu $ is either the fundamental weight or the anti-fundamental weight.", "On the other hand, the vertex operators for the lower q-W correspond to $\\mathcal {M}_{\\emptyset ,\\mu ,\\emptyset }$ with a pit at $(2,1,1)$ .", "Then they are obtained from (REF ) by exchanging the parameters $\\hat{q}_1$ and $\\hat{q}_3$ defined in (), which leads to $&&V^{\\pm }_l(z)=\\exp \\left(\\pm (Q_l+l_0\\log z-\\sum _{n\\ne 0}q^{\\mp \\frac{|n|}{2}}\\frac{l_n}{n}z^{-n})\\right),\\\\&&[l_0,Q_l]=\\frac{2\\epsilon _3}{\\epsilon _3-\\epsilon _1},\\quad [l_n,l_m]=-n\\frac{q_3^n-q_3^{-n}}{d^n-d^{-n}}\\delta _{n+m,0}.$ We can obtain the vertex operators $V^{\\pm }_u(z)$ for the upper q-W in the similar way.", "They correspond to $\\mathcal {M}_{\\emptyset ,\\emptyset ,\\mu }$ with a pit at $(2,1,1)$ and we need to exchange $\\hat{q}_2$ and $\\hat{q}_3$ .", "We also need to change $\\hat{q}_i$ to $\\tilde{q}_i$ .", "As a result, we have $&&V^{\\pm }_u(z):=\\exp \\left(\\pm (Q_u+u_0\\log z-\\sum _{n\\ne 0}q^{\\mp \\frac{|n|}{2}}\\frac{u_n}{n}z^{-n})\\right),\\\\&&[u_0,Q_u]=\\frac{2\\epsilon _1}{\\epsilon _1-\\epsilon _3},\\quad [u_n,u_m]=n\\frac{q_1^n-q_1^{-n}}{d^n-d^{-n}}\\delta _{n+m,0}.$ If the gluing construction works, the following vertex operators should give the representation of $\\mathcal {E}_2$ : $E_1(z)\\rightarrow V_l^+(z)V_u^+(z),\\quad F_1(z)\\rightarrow V_l^-(z)V_u^-(z).$ We can simplify them as follows: $V_d^{\\pm }(z)\\equiv V_l^{\\pm }(z)V_u^{\\pm }(z)=\\exp \\left(\\pm (Q_d+d_0\\log z-\\sum _{n\\ne 0}q^{\\mp \\frac{|n|}{2}}\\frac{d_n}{n}z^{-n})\\right),$ where $[d_0,Q_d]=2,\\quad [d_n,d_m]=n(q^n+q^{-n})\\delta _{n+m,0}.$ One can check that they obey the quadratic relations of $\\mathcal {E}_2$ if we set $H_{1,n}\\rightarrow \\frac{q^{n/2}-q^{-3n/2}}{q-q^{-1}}\\frac{d_n}{n}.$ The other currents $E_0(z),F_0(z),H_{0,n}$ are given in the same way by introducing the boson whose commutators with $d_n$ and itself are proportional to (REF ).", "One can check that they indeed give the $C^{\\prime }=q$ representation of $\\mathcal {E}_2$ .", "This type of the vertex representation was studied in [43] (see also [44]).", "The above result suggests that the gluing construction works well at least for q-$W(\\widehat{\\mathfrak {gl}}_1)$ ." ], [ "Quantum deformation of Feigin-Semikhatov's W-algebra", "In this section, we discuss quantum deformation of Feigin-Semikhatov's W-algebra $\\mathcal {W}_N^{(2)}$ .", "The generating currents $e_N(z), f_N(z)$ of $\\mathcal {W}_N^{(2)}$ were constructed as the commutant of a set of screening charges in [23].", "In this construction, the additional Heisenberg algebra which trivially commutes with the screening charges is also introduced to make the currents local.", "As we have seen, one can construct $\\mathcal {W}_N^{(2)}$ by gluing $Y_{0,1,N}$ and $Y_{0,0,1}$ .", "In the following, we construct its q-deformation by gluing q-$W(\\widehat{\\mathfrak {gl}}_{N|1})$ and q-$W(\\widehat{\\mathfrak {gl}}_1)$ .", "In the following, the deformation of $e_N(z)$ and $f_N(z)$ is denoted by ${E}_N(z)$ and ${F}_N(z)$ ." ], [ "Quantum deformation of Bershadsky-Polyakov algebra", "Let us first consider the case of $N=3$ which corresponds to Bershadsky-Polyakov algebra.", "There are several possibilities for the order of Fock spaces of q-$W(\\widehat{\\mathfrak {gl}}_{3|1})$ .", "In this section, we consider $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(3)}$ .", "For each pair of neighboring Fock spaces, we have a screening charge.", "The screening charges for the first three Fock spaces are the same as those of q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ and we have already studied them in the previous section.", "The new one is $S^{33}_{\\pm }$ which acts on the last two Fock spaces.", "Because we already know that the vertex operators for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ commute with the screening charges except $S^{33}_{\\pm }$ , it is natural to consider that the vertex operators of q-$W(\\widehat{\\mathfrak {gl}}_{3|1})$ can be obtained by some modification.", "Actually, we do not need any modification for ${F}_3(z)$ .", "That is because it acts only on the first two Fock spaces as we can see from (REF ) and it trivially commutes with $S^{33}_{\\pm }$ .", "Here, there is a subtle problem.", "The screening charges do not completely determine ${F}_3(z)$ because they do not impose any constraint on the Heisenberg part.", "At least, it is true in the CFT limit that the free field expression is the same for different values of $N$ .", "We assume that is also true after q-deformation.", "One of the reasonings for this assumption is that the relevant quadratic relation in the shifted toroidal algebra defined in [35] does not depend on the \"shift\".", "Then we haveWe need to explain the notation.", "In (REF ), we interpret $\\rho (F_1)(z)$ as an operator acting on $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(3)}$ by considering it acts on the last Fock space as an identity operator.", "${F}_3(z)=\\rho (F_1)(z).$ Next, we consider the Heisenberg parts.", "We have two Heisenberg algebras, one of which is the subalgebra of q-$W(\\widehat{\\mathfrak {gl}}_{3|1})$ while the other is q-$W(\\widehat{\\mathfrak {gl}}_1)$ itself.", "As in the $U_q(\\widehat{\\mathfrak {gl}}_2)$ case, one of the linear combinations $J_n$ serves as the element of the deformed $\\mathcal {W}_3^{(2)}$ while the other one $j_n$ gives the $\\mathfrak {gl}_1$ factor which commute with all the elements of the deformed $\\mathcal {W}_3^{(2)}$ .", "We can determine their explicit forms from the commutation relation with ${F}_3(z)$ : the linear combination which commutes with $\\mathcal {F}_3(z)$ is $j_n$ .", "To express them, we introduce the following boson, $\\begin{split}\\frac{a_n^{\\mathfrak {u}(1)}}{q^{-n}d^{-2n}-q^nd^{2n}}={\\left\\lbrace \\begin{array}{ll}&-\\frac{a_n^{(1)}}{d^n-d^{-n}}+\\frac{d^na_n^{(2)}}{q_1^n-q_1^{-n}}-\\frac{q^na_n^{(3)}}{d^n-d^{-n}}-\\frac{q_3^{-n}a_n^{(4)}}{d^n-d^{-n}}\\quad (n>0)\\\\&-\\frac{q_3^{-n}a_n^{(1)}}{d^n-d^{-n}}+\\frac{d^{2n}a_n^{(2)}}{q_1^n-q_1^{-n}}-\\frac{d^na_n^{(3)}}{d^n-d^{-n}}-\\frac{a_n^{(4)}}{d^n-d^{-n}}\\quad (n<0).\\end{array}\\right.", "}\\end{split}$ We note that we have already used the same symbol in (REF ).", "Here and below, we extend the definition of $a_n^{\\mathfrak {u}(1)}$ to all $N$ by letting it be the representation of $H_n$ as in (REF ).", "Then we have $&&J_n={\\left\\lbrace \\begin{array}{ll}&\\frac{(q_1^n-q_1^{-n})(q_3^n-q_3^{-n})d^{-n}}{(q^{-n}d^{-2n}-q^nd^{2n})^2}a_n^{\\mathfrak {u}(1)}+\\tilde{a}_n\\quad (n>0)\\\\&\\frac{(q_1^n-q_1^{-n})(q_3^n-q_3^{-n})}{(q^{-n}d^{-2n}-q^nd^{2n})^2}a_n^{\\mathfrak {u}(1)}+\\tilde{a}_n\\quad (n<0),\\end{array}\\right.", "}\\\\\\ \\nonumber \\\\&&j_n={\\left\\lbrace \\begin{array}{ll}&\\frac{a_n^{\\mathfrak {u}(1)}}{q^{-n}d^{-2n}-q^nd^{2n}}+\\frac{\\tilde{a}_n}{q_3^n-q_3^{-n}}\\quad (n>0)\\\\&\\frac{a_n^{\\mathfrak {u}(1)}}{q^{-n}d^{-2n}-q^nd^{2n}}+\\frac{d^n\\tilde{a}_n}{q_3^n-q_3^{-n}}\\quad (n<0).\\end{array}\\right.", "}$ Finally, we consider ${E}_3(z)$ .", "We will see that the following operator commutes with the screening charges: ${E}_3(z)=\\frac{1}{(q-q^{-1})^2}\\left(e^{A(z)}-e^{B(z)}-e^{C(z)}+e^{D(z)}\\right),$ where $\\begin{split}&A(z)=Q_{u_1}+u_{1,0}\\log z-\\frac{\\epsilon _2}{2}u_{1,0}+\\frac{1}{2}(3a_0^{(2)}-a_0^{(3)}-a_0^{(4)})-\\sum _{n\\ne 0}\\frac{A_n}{n}z^{-n},\\\\&B(z)=Q_{u_1}+u_{1,0}\\log z-\\frac{\\epsilon _2}{2}u_{1,0}+\\frac{1}{2}(a_0^{(2)}+a_0^{(3)}-a_0^{(4)})-\\sum _{n\\ne 0}\\frac{B_n}{n}z^{-n},\\\\&C(z)=Q_{u_1}+u_{1,0}\\log z-\\frac{\\epsilon _2}{2}u_{1,0}+\\frac{1}{2}(a_0^{(2)}-a_0^{(3)}+a_0^{(4)})-\\sum _{n\\ne 0}\\frac{C_n}{n}z^{-n},\\\\&D(z)=Q_{u_1}+u_{1,0}\\log z-\\frac{\\epsilon _2}{2}u_{1,0}-\\frac{1}{2}(a_0^{(2)}-a_0^{(3)}-a_0^{(4)})-\\sum _{n\\ne 0}\\frac{D_n}{n}z^{-n},\\\\\\end{split}$ and $\\begin{split}&A_n={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})d^{-3n}}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-3n}}{d^n-d^{-n}}a_n^{(3)}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{d^n-d^{-n}}a_n^{(4)}-\\frac{q^n-q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n>0) \\\\-\\frac{(q^n-q^{-n})q^{-n}d^{-2n}}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\&B_n={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})d^{-n}}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{d^n-d^{-n}}a_n^{(4)}-\\frac{q^n-q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n>0) \\\\-\\frac{(q^n-q^{-n})q^{-n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{d^n-d^{-n}}a_n^{(3)}-\\frac{(q^n-q^{-n})q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\&C_n={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})d^{-n}}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{d^n-d^{-n}}a_n^{(3)}-\\frac{q^n-q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n>0) \\\\-\\frac{(q^n-q^{-n})q^{-n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{d^n-d^{-n}}a_n^{(4)}-\\frac{(q^n-q^{-n})q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\&D_n={\\left\\lbrace \\begin{array}{ll}-\\frac{(q^n-q^{-n})d^n}{q_1^n-q_1^{-n}}a_n^{(2)}-\\frac{q^n-q^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n>0) \\\\-\\frac{(q^n-q^{-n})q^{-n}d^{2n}}{q_1^n-q_1^{-n}}a_n^{(2)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})}{d^n-d^{-n}}a_n^{(3)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{d^n-d^{-n}}a_n^{(4)}-\\frac{(q^n-q^{-n})q_3^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0)\\end{array}\\right.", "}\\\\\\end{split}$ Here, the zero modes satisfy $\\epsilon _1u_{1,0}=\\frac{1}{2}(a_0^{(1)}+a_0^{(3)}+a_0^{(4)}).$ The definition of $a_n^{(4)}$ is the same as that of $a_n^{(1)}$ and $a_n^{(3)}$ given in (REF ).", "We note that these vertex operators do not commute with $j_n$ for $n>0$ .", "One can remove the $\\mathfrak {gl}_1$ factor as follows: ${E}_3(z)=\\exp {\\left(-\\frac{1}{n}\\sum _{n=1}^{\\infty }\\frac{q^nd^{2n}(q^n-q^{-n})(q_1^n-q_1^{-n})}{q_3^n+q_3^{-n}+q_1^{-n}}j_{-n}z^n\\right)}\\tilde{{E}}_3(z),$ where the new current $\\tilde{{E}}_3(z)$ does not contain $j_n$ .", "However, the coefficients in (REF ) becomes messy and that is why we include the extra factor.", "We note that it does not change the commutation relations because we add only the negative mode.", "One can see the commutativity with the screening charges for the first three Fock spaces by rewriting (REF ) as follows: $e^{A(z)}-e^{B(z)}={E}_2(dz)\\lambda _1(z),\\quad e^{C(z)}-e^{D(z)}={E}_2(d^{-1}z)\\lambda _2(z),$ where we set ${E}_2(z)=\\rho (E_1(q_1z)).$ Here, the vertex operators $\\lambda _1(z),\\lambda _2(z)$ consist of the oscillators $a_n^{(4)}, \\tilde{a}_n$ which trivially commute with the screening charges.", "For the commutativity with $S^{33}_{\\pm }$ , we see that $e^{A(z)}$ and $e^{D(z)}$ trivially commute with it because the boson oscillators in $A(z)$ and $D(z)$ are orthogonal to those in $S^{33}_{\\pm }(z)$ .", "The remaining term $e^{B(z)}+e^{C(z)}$ also commutes with $S^{33}_{\\pm }$ .", "To see that, we rewrite it as follows: $\\begin{split}&e^{B(z)}+e^{C(z)}\\\\=\\ \\ &\\lambda _3(z)\\Biggl (e^{\\frac{1}{2}(a_0^{(3)}-a_0^{(4)})}\\exp \\Bigl (-\\sum _{n<0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{n(d^n-d^{-n})}a_n^{(3)}z^{-n}\\Bigr )\\exp \\Bigl (\\sum _{n>0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{n(d^n-d^{-n})}a_n^{(4)}z^{-n}\\Bigr )\\Biggr .\\\\&\\Biggl .+e^{-\\frac{1}{2}(a_0^{(3)}-a_0^{(4)})}\\exp \\Bigl (-\\sum _{n<0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{n(d^n-d^{-n})}a_n^{(4)}z^{-n}\\Bigr )\\exp \\Bigl (\\sum _{n>0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{n(d^n-d^{-n})}a_n^{(3)}z^{-n}\\Bigr )\\Biggr )\\\\=\\ \\ &e^{-\\frac{1}{2}(a_0^{(3)}+a_0^{(4)})}\\lambda _3(z)\\rho (\\Delta (E(z)))\\exp \\left(\\sum _{n>0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{n(d^n-d^{-n})}(a_n^{(3)}+d^na_n^{(4)})z^{-n}\\right),\\end{split}$ where $\\lambda _3(z)$ is the vertex operator which trivially commutes with $S_{\\pm }^{33}$ .", "Here, we need to give some explanation for $\\rho (\\Delta (E(z)))$ .", "The symbol $\\Delta (E(z))$ means the coproduct for the current $E(z)$ of $\\mathcal {E}_1$ defined in (REF ).", "The symbol $\\rho $ denotes the representation on the third and fourth Fock spaces defined in (REF ).", "By construction, $\\rho (\\Delta (E(z)))$ commutes with $S_{\\pm }^{33}$ .", "The other factors trivially commute with $S_ {\\pm }^{33}$ .", "As in the case of ${F}_3(z)$ , there is ambiguity in the Heisenberg part.", "We fixed it by demanding the Weyl symmetry as follows.", "From the formulae in Appendix , we have $\\begin{split}&(z-q^2w){E}_3(z){E}_3(w)+(w-q^2z){E}_3(w){E}_3(z)\\\\=\\ &q(1-q^2)(1-q_1^2)z\\left(\\delta \\left(\\frac{w}{d^2z}\\right)(:e^{B(z)+C(w)}:-:e^{A(z)+D(w)}:)+d^{-2}\\delta \\left(\\frac{d^2w}{z}\\right)(:e^{C(z)+B(w)}:-:e^{D(z)+A(w)}:)\\right)\\end{split}$ Because of the relation $A_n+d^{-2n}D_n=B_n+d^{-2n}C_n,$ the RHS of (REF ) vanishes and the relation (REF ) is simplified as follows: $(z-q^2w){E}_3(z){E}_3(w)+(w-q^2z){E}_3(w){E}_3(z)=0.$ This relation becomes the same as that of ${F}_3(z)$ if we change $q$ to $q^{-1}$ .", "That is interpreted as the Weyl transformation which flips the Dynkin diagram of $\\mathfrak {gl}_{3|1}$ .", "The quadratic relation between ${E}_3(z)$ and ${F}_3(w)$ is given as follows: $\\begin{split}&[{E}_3(z),{F}_3(w)]\\\\=\\ &-\\delta \\left(\\frac{w}{d^3z}\\right)\\frac{1-q_3^2}{(1-d^{-2})(q-q^{-1})^2}{K}_3(w)+\\delta \\left(\\frac{w}{dz}\\right)\\frac{1}{(q-q^{-1})^2}{X}_3(w)-\\delta \\left(\\frac{q^2dw}{z}\\right)\\frac{1}{(q-q^{-1})^2}{\\tilde{K}}^-_3(w),\\end{split}$ where we define $\\begin{split}&{K}_3(w)=:e^{A(d^{-3}w)+v_1^+(w)}:,\\\\&{X}_3(w)=q^{-2}:e^{A(d^{-1}w)+v_1^-(w)}:-\\frac{1-q_1^2}{1-d^2}:e^{A(d^{-1}w)+v_1^+(w)}:+:e^{B(d^{-1}w)+v_1^+(w)}:+:e^{C(d^{-1}w)+v_1^+(w)}:,\\\\&{\\tilde{K}}^-_3(w)=:e^{D(q^2dw)+v_1^-(w)}:.\\end{split}$ The currents ${K}_3(w)$ and $\\tilde{K}^-_3(w)$ are the vertex operators of the Heisenberg subalgebra as follows: $&&{K}_3(w)=e^{k_0}:\\exp {\\left(-\\sum _{n\\ne 0}\\frac{1}{n}k_nw^{-n}\\right)}:,\\quad {\\tilde{K}}^-_3(w)=e^{\\tilde{k}_0}\\exp {\\left(-\\sum _{n\\ne 0}\\frac{1}{n}\\tilde{k}_nw^{-n}\\right)},\\\\&&k_n=d^{3n}A_n+v_{1,n}^+={\\left\\lbrace \\begin{array}{ll}&\\frac{(q^n-q^{-n})(q^{-n}d^{-2n}-q^nd^{2n})}{q_3^n-q_3^{-n}}J_n\\quad (n>0)\\\\&-\\frac{(q^n-q^{-n})(d^n-d^{-n}q_1^n)q_1^n}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n<0),\\end{array}\\right.", "}\\\\&&\\tilde{k}_n=q^{-2n}d^{-n}D_n+v_{1,n}^-={\\left\\lbrace \\begin{array}{ll}\\qquad 0\\qquad \\qquad (n>0)\\\\-\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})q_3^{2n}}{q^{-n}d^{-2n}-q^nd^{2n}}a_n^{u(1)}+\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\quad (n<0),\\end{array}\\right.", "}\\\\&&k_0=-2\\tilde{k}_0=-\\frac{\\epsilon _1-\\epsilon _3}{\\epsilon _1}(a_0^{(1)}+a_0^{(3)}+a_0^{(4)})+2a_0^{(2)}.$ For later use, we split the current ${K}_3(w)$ into the positive and negative parts as follows: $&&{K}_3(w)={K}^-_3(w){K}^+_3(w),\\\\&&{K}^-_3(w)=e^{k_0}:\\exp {\\left(\\sum _{n>0}\\frac{1}{n}k_{-n}w^n\\right)},\\qquad {K}^+_3(w)=\\exp {\\left(-\\sum _{n>0}\\frac{1}{n}k_nw^{-n}\\right)}.$ The current ${X}_3(w)$ can be written as follows: $\\begin{split}{X}_3(w)=&\\exp {\\left(\\sum _{n=1}^{\\infty }\\frac{q_3^{-n}(q^n-q^{-n})(q_1^n-q_1^{-n})(d^n-d^{-n})}{n(q^{-n}d^{-2n}-q^nd^{2n})^2}a_{-n}^{u(1)}w^n\\right)}\\rho \\bigl (\\Delta ^{(4)}(t(d^{-1}w))\\bigr )\\\\&\\hspace{200.0pt}e^{\\tilde{a}_0}\\exp {\\left(-\\sum _{n=1}^{\\infty }\\frac{(q^n-q^{-n})q^{-n}}{n}J_nw^{-n}\\right)},\\end{split}$ where the current $t(z)$ is defined in (REF ).", "Here, we introduce the notation for the higher coproduct $\\Delta ^{(n+1)}=(1\\otimes \\Delta )\\Delta ^{(n)}$ , $\\Delta ^{(2)}=\\Delta $ .", "The symbol $\\rho $ denotes the representation on $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes \\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(3)}$ .", "We can interpret it as q-analogue of Virasoro generator.", "It is convenient to change the normalization as follows to make the the Weyl symmetry clear, $\\begin{split}&G^+(z)={E}_3(d^{-3}z),\\quad G^-(z)={F}_3(z),\\quad \\tilde{\\psi }^-(z)=\\tilde{{K}}^-_3(q^{-1}d^{-2}z),\\quad T(z)={X}_3(d^{-1}z)\\\\&\\psi (z)=\\psi ^-(z)\\psi ^+(z),\\quad \\psi ^-(z)=q^{-1}{K}^-_3(z),\\quad \\psi ^+(z)={K}^+_3(z).\\end{split}$ Then the equation (REF ) is rewritten into $\\begin{split}&[G^+(z),G^-(w)]\\\\=\\ &-\\delta \\left(\\frac{w}{z}\\right)\\frac{q(1-q_3^2)}{(1-d^{-2})(q-q^{-1})^2}\\psi (w)+\\delta \\left(\\frac{d^2w}{z}\\right)\\frac{1}{(q-q^{-1})^2}T(dw)-\\delta \\left(\\frac{q^2d^4w}{z}\\right)\\frac{1}{(q-q^{-1})^2}\\tilde{\\psi }^-(qd^2w).\\end{split}$ The quadratic relations between two of these currents are given as follows: $&&\\psi (z)G^-(w)-q^2\\frac{1-\\frac{z}{q^2w}}{1-\\frac{q^2z}{w}}G^-(w)\\psi (z)\\\\&=&-\\frac{(1-q^{-4})(1-q^4d^2)}{1-q_3^{-2}}\\delta \\left(\\frac{w}{q^2z}\\right)\\psi ^-(z)G^-(w)\\psi ^+(z)+\\frac{(1-q^2)(1-q_1^2)}{1-q_3^{-2}}\\delta \\left(\\frac{d^2w}{z}\\right)\\psi ^-(z)G^-(w)\\psi ^+(z),\\nonumber \\\\&& \\nonumber \\\\&&\\tilde{\\psi }^-(z)G^-(w)=q^2\\frac{1-\\frac{z}{q^3d^2w}}{1-\\frac{z}{q^{-1}d^2w}}G^-(w)\\tilde{\\psi }^-(z),\\\\&& \\nonumber \\\\&&q^2\\frac{1-\\frac{w}{q^2d^{-1}z}}{1-\\frac{q^2dw}{z}}T(z)G^-(w)-G^-(w)T(z)=\\frac{(1-q_1^2)(q-q^{-1})}{1-d^2}\\delta \\left(\\frac{w}{dz}\\right)\\psi ^-(w)G^-(d^{-2}w)\\psi ^+(w),\\\\&& \\nonumber \\\\&&\\psi (z)G^+(w)-q^{-2}\\frac{1-\\frac{q^2z}{w}}{1-\\frac{z}{q^2w}}G^+(w)\\psi (z)\\\\&=&-\\frac{(1-q^4)(1-q^{-4}d^{-2})}{1-q_3^2}\\delta \\left(\\frac{q^2w}{z}\\right)\\psi ^-(z)G^+(w)\\psi ^+(z)+\\frac{(1-q^{-2})(1-q_1^{-2})}{1-q_3^2}\\delta \\left(\\frac{w}{d^2z}\\right)\\psi ^-(z)G^+(w)\\psi ^+(z),\\nonumber \\\\&& \\nonumber \\\\&&\\tilde{\\psi }^-(z)G^+(w)=q^{-2}\\frac{1-\\frac{q^3d^2z}{w}}{1-\\frac{q^{-1}d^2z}{w}}G^+(w)\\tilde{\\psi }^-(z),\\\\&& \\nonumber \\\\&&q^{-2}\\frac{1-\\frac{q^2d^{-1}w}{z}}{1-\\frac{q^{-2}d^{-1}w}{z}}T(z)G^+(w)-G^+(w)T(z)=-\\frac{(1-q_1^{-2})(q-q^{-1})}{1-d^{-2}}\\delta \\left(\\frac{dw}{z}\\right)\\psi ^-(w)G^+(d^2w)\\psi ^+(w),$ One can see again that these relations are invariant under the permutation of the currents, $G^+(z)$ and $G^-(z)$ , and the inversion of the parameters $q_i\\rightarrow q_i^{-1}$ $(i=1,2,3)$ ." ], [ "General $N$", "We consider the cases for general $N$ .", "We consider $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes (\\mathcal {F}^{(3)})^{\\otimes N-1}$ for the Fock space of q-$W(\\widehat{\\mathfrak {gl}}_{N|1})$ .", "As in the previous case, the current ${F}_N(z)$ is given by ${F}_N(z)=\\rho (F_1(z)).$ For the current ${E}_N(z)$ , we claim that it is possible to decouple from ${E}_N(z)$ the vertex operator $\\Lambda (z)$ acting on $\\tilde{\\mathcal {F}}^{(3)}$ as follows: $&&{E}_N(z)={E^{\\prime }}_N(z)\\Lambda (d^{3-N}z),\\\\&&\\Lambda (z)=\\exp \\left(-\\sum _{n>0}\\frac{(q^n-q^{-n})q_3^{-n}}{n(q_3^n-q_3^{-n})}\\tilde{a}_{-n}z^n\\right)\\exp \\left(\\sum _{n>0}\\frac{q^n-q^{-n}}{n(q_3^n-q_3^{-n})}\\tilde{a}_nz^{-n}\\right).$ The vertex operator ${E^{\\prime }}_N(z)$ is determined by the following recursion relation: $&&{E^{\\prime }}_{N+1}(z)={E^{\\prime }}_N(dz)e^{\\phi ^+_{(N+2)}(z)}-e^{\\phi ^-_{(N+2)}(z)}{E^{\\prime }}_N(d^{-1}z),$ where $&&\\phi ^+_{(m)}(z)=-\\frac{1}{2}a_0^{(m)}+\\tilde{a}_0+\\sum _{n>0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^{-2n}}{n(d^n-d^{-n})}a_n^{(m)}z^{-n}\\hspace{30.0pt}\\\\&&\\phi ^-_{(m)}(z)=-\\sum _{n>0}\\frac{(q^n-q^{-n})(q_1^n-q_1^{-n})d^n}{n(d^n-d^{-n})}a_{-n}^{(m)}z^n+\\frac{1}{2}a_0^{(m)} $ for $m\\ge 3$ .", "We note that we have changed the normalization of ${E}_N(z)$ for simplicity.", "The zero modes satisfy the condition, $\\epsilon _1u_{1,0}=\\frac{1}{2}\\sum _{\\begin{array}{c}j=1\\\\j\\ne 2\\end{array}}^{N+1}a_0^{(j)}.$ The initial condition is given by $&&{E^{\\prime }}_1(z)=:e^{\\phi _{(2)}(z)}:,\\\\&&\\phi _{(2)}(q_3^{1/2}z)=Q_{u_1}+u_{1,0}\\log z+\\frac{3\\epsilon _1}{2}u_{1,0}-\\frac{1}{2}a_0^{(2)}+\\sum _{n\\ne 0}\\frac{(q^n-q^{-n})q_1^{-\\frac{|n|}{2}}}{n(q_1^n-q_1^{-n})}a_n^{(2)}z^{-n}.$ We note that the $N=1$ case corresponds to q-analogue of $\\beta \\gamma $ CFT.", "By induction, we can show that ${E}_{N+1}(z)$ commutes with these screening charges in the following way.", "We assume the claim is true for ${E}_N(z)$ .", "From the recursion relation (REF ), it is obvious that the current ${E}_{N+1}(z)$ commutes with the screening charges except the last one $S_{\\pm }^{33}$ .", "To show the commutativity with the remaining one, we use the explicit form of the current, $&&{E^{\\prime }}_N(z)=\\sum _{s_3=\\pm 1}\\cdots \\sum _{s_{N+1}=\\pm 1}(-1)^{P_s}:e^{\\phi ^{s_3,s_4\\cdots ,s_{N+1}}(z)}:,\\\\&&\\phi ^{s_3,s_4\\cdots ,s_{N+1}}(z)=\\phi _{(2)}(d^{\\sum _{i=3}^{N+1}s_i}z)+\\sum _{j=3}^{N+1}\\phi _{(j)}^{s_j}(d^{\\sum _{k=j+1}^{N+1}s_k}z),$ where we use the notation, $P_s=&\\sum _{i=3}^{N+1}\\frac{1-s_i}{2},\\qquad \\phi _{(i)}^p={\\left\\lbrace \\begin{array}{ll}\\phi _{(i)}^+\\qquad p=1\\\\\\phi _{(i)}^-\\qquad p=-1\\end{array}\\right.", "}\\quad (i\\ge 3).$ By rewriting this into $\\begin{split}{E^{\\prime }}_N(z)=&\\sum _{s_3=\\pm 1}\\cdots \\sum _{s_{N-1}=\\pm 1}(-1)^{P_s}\\left(:e^{\\phi ^{s_3,s_4\\cdots ,+,+}(z)}:+:e^{\\phi ^{s_3,s_4\\cdots ,-,-}(z)}:\\right.\\\\&\\left.\\hspace{70.0pt}-:e^{\\phi ^{s_3,s_4\\cdots ,s_{N-1}}(z)}:(:e^{\\phi ^-_{(N)}(dz)+\\phi ^+_{(N+1)}(z)}:+:e^{\\phi ^+_{(N)}(d^{-1}z)+\\phi ^-_{(N+1)}(z)}:)\\right),\\end{split}$ we can see the commutativity in the same manner as in the $N=3$ case.", "We note that the zero modes $\\tilde{a}_0=a_0^{(2)}+\\frac{\\epsilon _3-\\epsilon _1}{2\\epsilon _1}\\sum _{\\begin{array}{c} j=1\\\\ j\\ne 2\\end{array}}^{m}a_0^{(j)}$ included in (REF ) do not play any role in the relation to the screening charges, but they are necessary to produce the desired quadratic relations as we will see in the following.", "Let us derive the quadratic relations.", "We first consider the one between ${E}_N(z)$ and ${E}_N(w)$ .", "One can show by induction that it is given by $(z-q_2w){E}_N(z){E}_N(w)=(w-q_2z){E}_N(w){E}_N(z).$ For the detail of the computation, see Appendix .", "Next, we consider the relation between ${E}_N(z)$ and ${F}_N(z)$ .", "We set ${E}_N^{s_3,s_4\\cdots s_{N+1}}(z)=:e^{\\phi ^{s_3,s_4\\cdots ,s_{N+1}}(z)}:\\Lambda (d^{3-N}z).$ Then we have $\\begin{split}[{E}_N(z),{F}_N(w)]=&(-1)^N\\delta \\left(\\frac{q^2d^{N-2}w}{z}\\right)\\tilde{K}_N^-(w)-q^{4-2N}\\delta \\left(\\frac{w}{d^Nz}\\right)\\Biggl (\\prod _{\\begin{array}{c}j=0\\\\ j\\ne i\\end{array}}^{N-2}\\frac{1-q^2d^{2(i-j)}}{1-d^{2(i-j)}}\\Biggr ){K}_N(w)\\\\&\\hspace{120.0pt}-\\displaystyle {\\sum _{i=0}^{N-3}\\delta \\left(\\frac{d^{N-4-2i}w}{z}\\right)}{X}_N^{(N-1-i)}(w),\\end{split}$ where $&&\\tilde{K}_N^-(w)=:{E}_N^{s_3=s_4=\\cdots =s_{N+1}=-1}(q^2d^{N-2}w)e^{v_1^-(w)}:\\\\&&{K}_N(w)=:{E}_N^{s_3=s_4=\\cdots =s_{N+1}=1}(d^{-N}w)e^{v_1^+(w)}:\\\\&&{X}_N^{(N-1-i)}(w)=\\hspace{-11.38109pt}\\displaystyle {\\sum _{\\begin{array}{c}s_3,\\cdots ,s_{N+1}\\\\P_s\\le N-2-i\\end{array}}}\\hspace{-8.53581pt}(-1)^{P_s}q^{2P_s+4-2N}\\Biggl (\\prod _{\\begin{array}{c}j=0\\\\ j\\ne i\\end{array}}^{N-2-P_s}\\frac{1-q^2d^{2(i-j)}}{1-d^{2(i-j)}}\\Biggr ):{E}_N^{s_3,s_4\\cdots s_{N+1}}(d^{N-4-2i}w)e^{v_1^+(w)}: \\nonumber \\\\&&\\hspace{50.0pt}-\\hspace{-14.22636pt}\\displaystyle {\\sum _{\\begin{array}{c}s_3,\\cdots ,s_{N+1}\\\\P_s\\le N-3-i\\end{array}}\\hspace{-8.53581pt}(-1)^{P_s}q^{2P_s+4-2N}\\Biggl (\\prod _{\\begin{array}{c}j=0\\\\ j\\ne i\\end{array}}^{N-3-P_s}\\frac{1-q^2d^{2(i-j)}}{1-d^{2(i-j)}}\\Biggr )}:{E}_N^{s_3,s_4\\cdots s_{N+1}}(d^{N-4-2i}w)e^{v_1^-(w)}:.$ This relation can be derived from (REF ) and (REF ).", "The currents ${K}_N(w)=e^{k_0}\\exp \\left(-\\sum _{n\\ne 0}\\frac{1}{n}k_nw^{-n}\\right),\\qquad \\tilde{K}_N^-(w)=e^{\\tilde{k}_0}\\exp \\left(-\\sum _{n\\ne 0}\\frac{1}{n}\\tilde{k}_nw^{-n}\\right)$ are the vertex operators for the Heisenberg subalgebras as before.", "To see it, we introduce the $\\mathfrak {u}(1)$ boson, $\\begin{split}\\frac{a_n^{\\mathfrak {u}(1)}}{q^{-n}d^{-(N-1)n}-q^nd^{(N-1)n}}={\\left\\lbrace \\begin{array}{ll}-\\frac{a_n^{(1)}}{d^n-d^{-n}}+\\frac{d^na_n^{(2)}}{q_1^n-q_1^{-n}}-\\sum _{m=3}^{N+1}\\frac{q^nd^{(m-3)n}a_n^{(m)}}{d^n-d^{-n}}\\quad (n>0)\\\\-\\frac{q^nd^{(N-2)n}a_n^{(1)}}{d^n-d^{-n}}+\\frac{d^{(N-1)n}a_n^{(2)}}{q_1^n-q_1^{-n}}-\\sum _{m=3}^{N+1}\\frac{d^{(N+1-m)n}a_n^{(m)}}{d^n-d^{-n}}\\quad (n<0).\\end{array}\\right.", "}\\end{split}$ Then we have $&&k_n={\\left\\lbrace \\begin{array}{ll}(1-q^{-2n})\\left(\\frac{(q_1^n-q_1^{-n})}{q^{-n}d^{-(N-1)n}-q^nd^{(n-1)n}}a_n^{\\mathfrak {u}(1)}+\\frac{q_3^n-q_3^{-n}d^{(2N-4)n}}{q_3^n-q_3^{-n}}\\tilde{a}_n\\right)\\qquad (n>0)\\\\\\frac{(1-q^{-2n})(1-d^{(2N-4)n})}{q_3^n-q_3^{-n}}\\tilde{a}_n\\qquad (n<0),\\end{array}\\right.", "}\\\\&&\\tilde{k}_n={\\left\\lbrace \\begin{array}{ll}\\qquad 0\\qquad \\quad (n>0)\\\\-(q^n-q^{-n})q^{-2n}\\left(\\frac{(q_1^n-q_1^{-n})d^{(1-N)n}}{q^{-n}d^{-(N-1)n}-q^nd^{(N-1)n}}a_n^{\\mathfrak {u}(1)}+d^{-n}\\tilde{a}_n\\right)\\qquad (n<0),\\end{array}\\right.", "}\\\\&&\\tilde{k}_0=-\\frac{1}{N-1}k_0=-a_0^{(2)}+\\frac{\\epsilon _1-\\epsilon _3}{2\\epsilon _1}\\sum _{\\begin{array}{c}i=1\\\\ i\\ne 2\\end{array}}^{N+1}a_i.$ Apart from the Heisenberg part, we have $N-2$ currents in the RHS of (REF ).", "That is the same as the undeformed case [23].", "The currents ${X}_N^{(n)}$ can be seen as q-analogue of the $W_n$ current.", "Finally, we mention the representation associated with another Dynkin diagram of $\\mathfrak {gl}_{N|1}$ .", "So far, we have fixed the order of the Fock space to $\\mathcal {F}^{(3)}\\otimes \\mathcal {F}^{(2)}\\otimes (\\mathcal {F}^{(3)})^{\\otimes N-1}$ , but it can be easily generalized to another ordering.", "As one can see from (REF ), the current ${E}_N(z)$ does not act on the first Fock space, which implies that it commutes with the screening charges of $(\\mathcal {F}^{(3)})^{\\otimes M}\\otimes \\mathcal {F}^{(2)}\\otimes (\\mathcal {F}^{(3)})^{\\otimes N-1}$ without any modification.", "For the current ${F}_N(z)$ , we need modification, but that can be similarly done as previously discussed.", "We expect that the quadratic relations do not depend on the ordering of the Fock space." ], [ "The norm of Whittaker states and Nekrasov's instanton partition function", "In the AGT correspondence, non-principal W-algebras appear as the dual of 4d $\\mathcal {N}=2$ supersymmetric gauge theories with surface operators.", "The half-BPS surface operators in SU($N$ ) gauge theories are labelled by the partition of $N$ , which represents the preserved gauge symmetry.", "For the partition $[n_1,n_2\\cdots n_M]$ , the preserved gauge group is $S[{\\rm U}(n_1)\\times {\\rm U}(n_2)\\times \\cdots \\times {\\rm U}(n_M)].$ In 2d side, the partition is related with $\\mathfrak {su}(2)$ embedding in the Drinfeld-Sokolov reduction of $\\widehat{\\mathfrak {su}}(N)$ [20], [21], [22].", "Feigin-Semikhatov's W-algebra is associated with the partition $N=(N-1)+1$ and the corresponding surface operator is called simple.", "It is considered that the above correspondence lifts to 5d $\\mathcal {N}=1$ supersymmetric gauge theory on $\\mathbb {C}_{\\epsilon _1}\\times \\mathbb {C}_{\\epsilon _2}\\times S^1$ and the corresponding chiral algebra should lift to its deformation.", "In this section, we focus on q-deformed Bershadsky-Polyakov algebra and examine whether it matches with 5d AGT by comparing the norm of Whittaker states (Gaiotto states) with the instanton partition function." ], [ "Nekrasov's partition function under the existence of a surface operator", "In this section, we review the instanton partition function with a surface operator following [22].", "In the following, we consider ${\\rm U}(N)$ instead of ${\\rm SU}(N)$ .", "Let us first see the case without a surface operator.", "The moduli space is given by the ADHM data.", "To describe it, we introduce the $k$ -dimensional vector space $V$ and $N$ -dimensional vector space $W$ for $k$ -instantons in ${\\rm U}(N)$ gauge theory.", "Then, the moduli space is described by the data $B_1,B_2\\in {\\rm Hom}(V,V),I\\in {\\rm Hom}(W,V),J\\in {\\rm Hom}(V,W)$ with ADHM constraint imposed on.", "The theory has the rotational symmetry SO(4) and the gauge symmetry ${\\rm U}(N)$ .", "The action of their Cartan torus ${\\rm U}(1)^{N+2}$ on the moduli space is the key ingredient for the computation of the instanton partition function.", "Concretely, the contribution to the instanton partition function comes only from the fixed points of the torus action labelled by $N$ Young diagrams and can be evaluated from the character of the tangent spaces there.", "The character of the vector spaces $V,W$ at the fixed point $\\vec{Y}=(Y_1,\\cdots Y_N)$ are given as follows: $W=\\sum _{\\alpha =1}^Nu_\\alpha ,\\quad V=\\sum _{\\alpha =1}^Nu_{\\alpha }\\sum _{(i,j)\\in Y_{\\alpha }}q_1^{1-i}q_2^{1-j},$ where $u_{\\alpha }=e^{a_\\alpha }$ and $q_{1,2}=e^{\\epsilon _{1,2}}$ are the equivalent parametersIn this subsection, we use the parameters $q_{1,2}$ independently of the quantum toroidal algebra..", "Here, we abuse the notation and identify the vector spaces with their characters.", "The character of the tangent space is given by $\\chi _{\\vec{Y}}=-(1-q_1)(1-q_2)V^*V+W^*V+q_1q_2V^*W,$ where the parameters $u_\\alpha , q_i$ are replaced with their inverse in $V^*$ and $W^*$ .", "Substituting (REF ) into (REF ), one finds that the character can be expressed as $\\chi _{\\vec{Y}}=\\sum _{i=1}^{2Nk}e^{w_i(\\vec{Y})}.$ Then the Nekrasov partition function [45] for 5d SYM is given by $Z=\\sum _{\\vec{Y}}x^{|\\vec{Y}|}\\frac{1}{\\prod _{i=1}^{2Nk}(1-e^{w_i(\\vec{Y})})}.$ The 4d case is obtained by rescaling the parameters $a_\\alpha \\rightarrow Ra_\\alpha $ , $\\epsilon _{1,2}\\rightarrow R\\epsilon _{1,2}$ and taking the limit $R\\rightarrow 0$ .", "Next, we review the instanton partition function with a surface operator.", "The crucial point in [22] is that the instanton moduli space under the existence of a surface operator labelled by $[n_1,n_2,\\cdots n_M]$ can be described as that on the orbifold $\\mathbb {C}_{\\epsilon _1}\\times \\mathbb {C}_{\\epsilon _2}/\\mathbb {Z}_M$ .", "Due to the orbifolding, it is natural to rescale $q_2$ to $q_2^{1/M}$ and redefine $u_{1,\\cdots N}$ as $u_{s,I}$ , where $s=1,\\cdots n_I$ and $I=1,\\cdots M$ .", "The vector spaces $V,W$ are decomposed into the subspaces by the $\\mathbb {Z}_M$ charges, $W=\\oplus _{I=1}^MW_I, \\quad V= \\oplus _{I=1}^MV_I.$ The vector space $W_I$ is $n_I$ -dimensional and its character is given as follows: $W_I=\\sum _{s=1}^{n_I}q_2^{-I/M}u_{s,I}.$ The vector space $V_I$ consists of the vector corresponding to the box $(i,j^{\\prime })\\in Y_{s,I-j^{\\prime }+1}$ and its character is given by $V_I=\\sum _{J=1}^M\\sum _{s=1}^{n_{I-J+1}}q_2^{\\lfloor \\frac{I-J}{M}\\rfloor -I/M}u_{s,I-J+1}\\sum _{(i,jM+J)\\in Y_{s,I-J+1}}q_1^{1-i}q_2^{-j},$ where $\\lfloor \\frac{I-J}{M}\\rfloor ={\\left\\lbrace \\begin{array}{ll}\\ \\ 0\\quad (I\\ge J)\\\\-1\\quad (I<J)\\end{array}\\right.", "}$ .", "We note that the index should be understood modulo $M$ .", "The fixed point is still labelled by $N$ Young diagrams, but the dimension of the tangent space becomes smaller because the ADHM data is restricted to $B_1\\in {\\rm Hom}(V_I,V_I),\\quad B_2\\in {\\rm Hom}(V_I,V_{I+1}),\\quad I\\in {\\rm Hom}(W_I,V_I),\\quad J\\in {\\rm Hom}(V_I,W_{I+1}).$ Then the character of the tangent space is given by $\\chi _{\\vec{Y}}=\\sum _{I=1}^M\\left(-(1-q_1)V_I^*V_I+(1-q_1)q_2^{1/M}V_{I-1}^*V_I+W_I^*V_I+q_1q_2^{1/M}V_{I-1}^*W_I\\right).$ The instanton partition function is obtained in the similar way with the previous case, but we should change the fugacity parameter $x$ to $\\prod _{I=1}^Mx_I^{d_I(\\vec{Y})}$ , where $d_I(\\vec{Y})$ denotes the dimension of $V_I$ .", "These fugacity parameters measure the magnetic charges associated with the ${\\rm U}(1)$ subgroups in (REF ) as well as the instanton number.", "As an example, we consider the ${\\rm U}(N)$ instanton partition function under the presence of the surface operator of the type $[N-1,1]$ .", "We focus on the coefficients of $x_1^{d_1}$ and $x_2^{d_2}$ .", "For $x_2^{d_2}$ , the Young diagrams except the last one must be empty because of $d_1=0$ .", "The last one consists of only one column.", "For such Young diagram, the character of each vector space is given as follows: $W_1=\\sum _{s=1}^{N-1}u_{s,1}q_2^{-1/2},\\quad W_2=u_{1,2}q_2^{-1},\\quad V_1=0,\\quad V_2=\\sum _{i=1}^{d_2}q_1^{1-i}q_2^{-1}u_{1,2},$ which leads to $\\chi _{\\vec{Y}}=\\sum _{i=1}^{d_2}q_1^i\\left(1+\\sum _{s=1}^{N-1}\\frac{q_2u_{s,1}}{u_{1,2}}\\right).$ Then the instanton partition function is given by $Z^{(S)}(\\vec{u},(\\emptyset ,\\cdots \\emptyset ,1^{d_2}))=\\Biggl (\\prod _{i=1}^{d_2}\\Bigl ((1-q_1^i)\\prod _{s=1}^{N-1}(1-\\frac{u_{s,1}}{u_{1,2}}q_1^iq_2)\\Bigr )\\Biggr )^{-1}.$ For $x_1^{d_1}$ , the computation becomes more complicated.", "In this case, the last Young diagram is empty while all of the others are in the shape of one-column Young diagrams.", "For simplicity, we consider the case of $N=3$ .", "After some computations, one can see that the final expression can be written in a simple form.", "From the result for $d_1=1,2,3,4$ , we conjecture the following form: $\\sum _{n=0}^{d_1}Z^{(S)}(\\vec{u},(1^n,1^{d_1-n},\\emptyset ))=\\Biggl (\\prod _{i=1}^{d_1}\\Bigl (1-q_1^i\\Bigr )\\Bigl (1-\\frac{u_{1,2}}{u_{1,1}}q_1^i\\Bigr )\\Bigl (1-\\frac{u_{1,2}}{u_{2,1}}q_1^i\\Bigr )\\Biggr )^{-1}.$ The similarity between (REF ) and (REF ) can be understood in terms of the chiral algebra.", "It corresponds to the Weyl transformation which we discussed in the previous section." ], [ "Comparison", "In this subsection, we compare the norm of the Whittaker states for the deformed Bershadsky-Polyakov algebra with the instanton partition function of 5d $\\mathcal {N}=1$ U(3) SYM with the surface operator of the type $3=2+1$ .", "We can define the Whittaker state in the same way as the 4d case [20], [22].", "We set the mode expansion as follows: $\\begin{split}&G^+(z)=e^{u_{1,0}\\log z}\\sum _{n\\in \\mathbb {Z}}G_n^+z^{-n},\\quad G^-(z)=e^{-u_{1,0}\\log z}\\sum _{n\\in \\mathbb {Z}}G_n^-z^{-n},\\quad T(z)=\\sum _{n\\in \\mathbb {Z}}T_nz^{-n},\\\\&\\psi (z)=\\sum _{n\\in \\mathbb {Z}}\\psi _nz^{-n},\\quad \\tilde{\\psi }(z)=\\sum _{n\\ge 0}\\tilde{\\psi }_{-n}z^n.\\end{split}$ The highest weight state is defined by the conditions, $\\begin{split}&G_{n-1}^+\\mathinner {|{\\rm hw}\\rangle }=G_n^-\\mathinner {|{\\rm hw}\\rangle }=T_n\\mathinner {|{\\rm hw}\\rangle }=k_n\\mathinner {|{\\rm hw}\\rangle }=0,\\quad {\\rm for\\ }n\\ge 1,\\\\&\\mathinner {\\langle {\\rm hw}|}G_{-n+1}^-= \\mathinner {\\langle {\\rm hw}|}G_{-n}^+= \\mathinner {\\langle {\\rm hw}|}T_{-n}= \\mathinner {\\langle {\\rm hw}|}k_{-n}=0,\\quad {\\rm for\\ } n\\ge 1,\\\\&T_0\\mathinner {|{\\rm hw}\\rangle }=h\\mathinner {|{\\rm hw}\\rangle },\\quad e^{k_0}\\mathinner {|{\\rm hw}\\rangle }=J\\mathinner {|{\\rm hw}\\rangle }.\\end{split}$ The Whittaker state is then defined by the conditions, $\\begin{split}&G^+_n\\mathinner {|{W}\\rangle }=t_1^{1/2}\\delta _{n,0}\\mathinner {|{W}\\rangle },\\quad \\mathinner {\\langle {W}|}G^-_{-n}=\\mathinner {\\langle {W}|}t_1^{1/2}\\delta _{n,0},\\quad {\\rm for\\ } n\\ge 0,\\\\&G^-_n\\mathinner {|{W}\\rangle }=t_2^{1/2}\\delta _{n,1}\\mathinner {|{W}\\rangle },\\quad \\mathinner {\\langle {W}|}G^+_{-n}=\\mathinner {\\langle {W}|}t_2^{1/2}\\delta _{n,1},\\quad {\\rm for\\ } n\\ge 1,\\\\&T_n\\mathinner {|{W}\\rangle }=k_n\\mathinner {|{W}\\rangle }=0,\\qquad \\mathinner {\\langle {W}|}T_{-n}=\\mathinner {\\langle {W}|}k_{-n}=0\\quad {\\rm for\\ } n\\ge 1,\\end{split}$ where it is assumed that the Whittaker state can be expanded by the descendant states of a single highest weight $\\mathinner {|{\\alpha ,\\beta }\\rangle }$ state as follows: $&&\\mathinner {|{W}\\rangle }=\\sum _{n,m\\ge 0}t_1^{n/2}t_2^{m/2}\\mathinner {|{n,m,\\alpha ,\\beta }\\rangle },\\\\&&\\mathinner {\\langle {W}|}=\\sum _{n,m\\ge 0}t_1^{n/2}t_2^{m/2}\\mathinner {\\langle {n,m,\\alpha ,\\beta }|}.$ Here, the normalization is fixed by $&&\\mathinner {|{0,0,\\alpha ,\\beta }\\rangle }=\\mathinner {|{\\alpha ,\\beta }\\rangle },\\\\&&\\mathinner {\\langle {\\alpha ,\\beta |\\alpha ,\\beta }\\rangle }=1,$ and the parameters $\\alpha ,\\beta $ are associated with $h,J$ as we will see below.", "Each term in the above expansion is expected to be fixed recursively.", "In general, it is a tedious work to compute the states at higher order, but we can easily obtain the states for $n=0$ or $m=0$ because the states with such charges are uniquely fixed to $&&\\mathinner {|{n,0,\\alpha ,\\beta }\\rangle }\\propto (G_{0}^-)^n\\mathinner {|{\\alpha ,\\beta }\\rangle },\\qquad \\mathinner {\\langle {n,0,\\alpha ,\\beta }|}\\propto \\mathinner {\\langle {\\alpha ,\\beta }|}(G_0^+)^n\\\\&&\\mathinner {|{0,m,\\alpha ,\\beta }\\rangle }\\propto (G_{-1}^+)^m\\mathinner {|{\\alpha ,\\beta }\\rangle },\\qquad \\mathinner {\\langle {n,0,\\alpha ,\\beta }|}\\propto \\mathinner {\\langle {\\alpha ,\\beta }|}(G_1^-)^n.$ The proportional factors are immediately read off from the conditions such as $(G_0^+)^n\\mathinner {|{n,0,\\alpha ,\\beta }\\rangle }=\\mathinner {|{\\alpha ,\\beta }\\rangle }$ .", "Then the norm of the Whittaker states is written as $\\mathinner {\\langle {W|W}\\rangle }=1+\\sum _{n=1}^{\\infty }\\frac{t_1^n}{\\mathinner {\\langle {\\alpha ,\\beta |(G_0^+ )^n(G_0^-)^n|\\alpha ,\\beta }\\rangle }}+\\sum _{m=1}^{\\infty }\\frac{t_2^m}{\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^m(G_{-1}^+)^m|\\alpha ,\\beta }\\rangle }}+\\cdots .$ In the following, we compare (REF ) with the instanton partition function (REF ) and (REF ).", "We first compute the coefficient of $t_1^n$ .", "The easiest way is to use the free field representation.", "For that, we need to describe the highest weight state in terms of the free bosons.", "The free boson module is defined as follows: $&&a_n^{(i)}\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=0,\\quad \\tilde{a}_{n}\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=0,\\quad (n>0,\\ i=1,2,3,4)\\\\&&\\frac{1}{2}(3a_0^{(2)}-a_0^{(3)}-a_0^{(4)})\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=\\alpha \\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle },\\\\&&\\frac{1}{2}(a_0^{(2)}+a_0^{(3)}-a_0^{(4)})\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=\\gamma \\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle },\\\\&&\\frac{1}{2}(a_0^{(2)}-a_0^{(3)}+a_0^{(4)})\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=\\delta \\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle },\\\\&&\\frac{1}{2}(a_0^{(2)}-a_0^{(1)})\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=\\beta \\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }.$ We also impose $u_{1,0}\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }=0.$ We note that the condition $u_{1,0}=0$ holds in the whole Fock space because of $[Q_{u_1},u_{1,0}]=0$ .", "This process is so-called picture fixing.", "Combining it with (REF ), we have $a_0^{(1)}+a_0^{(3)}+a_0^{(4)}=0,$ which is rewritten into $\\alpha +\\beta =2(\\gamma +\\delta ).$ The state $\\mathinner {|{\\alpha ,\\beta ,\\gamma ,\\delta }\\rangle }$ satisfies all the anihilation conditions except the one for $G_0^+$ .", "It requires the relation $(e^{\\alpha }-e^{\\gamma })(1-e^{\\delta -\\alpha })=0.$ We have the two solutions, $\\alpha =\\gamma $ and $\\alpha =\\delta $ , but the difference does not cause any effect on the representation theory.", "From the above constraint, all the parameters can be expressed by $\\alpha $ and $\\beta $ .", "In the following, we denote the highest weight state just by $\\mathinner {|{\\alpha ,\\beta }\\rangle }$ .", "The eigenvalues $h,J$ are given by $h=q^{-2}e^{\\alpha -\\beta }-\\frac{1-q_1^2}{1-d^2}e^{\\alpha +\\beta }+e^{\\alpha +\\beta }+e^{\\frac{3\\beta -\\alpha }{2}},\\qquad J=e^{\\alpha +\\beta }.$ We note that the module defined above is analogue of the Wakimoto module for $\\widehat{\\mathfrak {sl}}_2$ .", "Using the free boson representation, we can easily compute the coefficient of $t_1^n$ as follows: $\\begin{split}&\\quad \\mathinner {\\langle {\\alpha ,\\beta |(G_0^+ )^n(G_0^-)^n|\\alpha ,\\beta }\\rangle }\\\\=\\ \\ &\\prod _{i=0}^{n-1}\\frac{q^{-i}e^{\\beta }-q^ie^{-\\beta }}{q-q^{-1}}\\prod _{j=1}^n\\frac{q^{-3i}e^{\\alpha }-q^{-i}e^{\\alpha }-q^{-i}e^{\\frac{\\beta -\\alpha }{2}}+q^ie^{\\frac{\\beta -\\alpha }{2}}}{(q-q^{-1})^2}\\\\=\\ \\ &\\prod _{i=1}^{n}\\frac{(q^{-i+1}e^{\\beta }-q^{i-1}e^{-\\beta })(e^{\\frac{\\beta -\\alpha }{2}}-q^{-2i}e^\\alpha )(q^i-q^{-i})}{(q-q^{-1})^3}\\end{split}$ Next, we consider the coefficient of $t_2^m$ in (REF ).", "One can see that the way using free boson representation requires a large amount of computation.", "In the following, we mainly use the algebraic relation.", "From (REF ), we have $\\begin{split}&\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^n(G_{-1}^+)^n|\\alpha ,\\beta }\\rangle }\\\\=\\ &\\sum _{m=0}^{n-1}\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^{n-1}(G_{-1}^+)^m[G_1^-,G_{-1}^+](G_{-1}^+)^{n-m-1}|\\alpha ,\\beta }\\rangle }\\\\=\\ &\\sum _{m=0}^{n-1}\\mathinner {\\langle {\\alpha ,\\beta |(G_1^-)^{n-1}(G_{-1}^+)^m\\left(\\frac{q(1-q_3^2)}{(1-d^{-2})(q-q^{-1})^2}\\psi _0-\\frac{1}{d^2(q-q^{-1})^2}T_0+\\frac{1}{q^2d^4(q-q^{-1})^2}\\tilde{\\psi }^-_0\\right)(G_{-1}^+)^{n-m-1}|\\alpha ,\\beta }\\rangle }.\\end{split}$ We see from this equation that we can implement the computation if we obtain the eigenvalues of $\\psi _0$ , $T_0$ and $\\tilde{\\psi }_0$ .", "We note that arbitrary descendant states are not necessarily eigenvectors of the zero modes as opposed to CFT case because the degeneracy is resolved after q-deformation.", "For the state $(G_{-1}^+)^n\\mathinner {|{\\alpha ,\\beta }\\rangle }$ , we do not need to care about that because there is no other state with the same charge.", "We begin with $\\psi _0$ .", "From (), we have $\\begin{split}&\\psi _0G^+_{-1}-q^{-2}G^+_{-1}\\psi _0+(q^2-q^{-2})\\sum _{n\\ge 1}^{\\infty }q^{-2n}G^+_{-n-1}\\psi _n\\\\=\\ &\\sum _{n\\in \\mathbb {Z}}\\left(-\\frac{(1-q^4)(1-q^{-4}d^{-2})q_2^n}{1-q_3^2}+\\frac{(1-q^{-2})(1-q_1^{-2})d^{-2n}}{1-q_3^2}\\right)\\sum _{m\\ge {\\rm max}(0,-n)}^{\\infty }\\psi ^-_{-m-n}G^+_{n-1}\\psi ^+_{m}.\\end{split}$ When we consider the action of (REF ) on $(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }$ , most of the terms vanish and are simplified as follows: $\\begin{split}&(\\psi _0G^+_{-1}-q^{-2}G^+_{-1}\\psi _0)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }\\\\=\\ &\\left(-\\frac{(1-q^4)(1-q^{-4}d^{-2})}{1-q_3^2}+\\frac{(1-q^{-2})(1-q_1^{-2})}{1-q_3^2}\\right)\\psi ^-_{0}G^+_{-1}\\psi ^+_{0}(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }.\\end{split}$ We can simplify the RHS as $(\\psi _0G^+_{-1}-q^{-2}G^+_{-1}\\psi _0)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }=q^{4l}(q^3-q^{-3})e^{\\alpha +\\beta }(G_{-1}^+)^{l+1}\\mathinner {|{\\alpha ,\\beta }\\rangle }.$ Then we can obtain the eigenvalue of $\\psi _0$ recursively from (REF ), $\\begin{split}\\psi _0(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }=&\\left(q^{-2l-1}e^{\\alpha +\\beta }+(q^3-q^{-3})\\sum _{i=0}^{l-1}q^{-2i}q^{4(l-i-1)}e^{\\alpha +\\beta }\\right)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }\\\\=&\\ q^{4l-1}e^{\\alpha +\\beta }(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }.\\end{split}$ We can apply the similar discussion to $T_0$ , which lead to $(q^{-2}T_0G^+_{-1}-G^+_{-1}T_0)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }=-\\frac{(1-q_1^{-2})(1-q^{-2})d^2}{1-d^{-2}}q^{4l}e^{\\alpha +\\beta }(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }.$ The we have $\\begin{split}&\\quad T_0(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }\\\\=\\ &\\left(q^{2l}\\bigl (q^{-2}e^{\\alpha -\\beta }-\\frac{1-q_1^2}{1-d^2}e^{\\alpha +\\beta }+e^{\\alpha +\\beta }+e^{\\frac{3\\beta -\\alpha }{2}}\\bigr )-\\frac{(1-q_1^{-2})(1-q^{-2})q_3^{-2}}{1-d^{-2}}\\sum _{i=0}^{l-1}q^{4l-2i-4}e^{\\alpha +\\beta }\\right)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }\\\\=\\ &\\left(q^{2l-2}e^{\\alpha -\\beta }+\\biggl (q^{2l-2}d^2-\\frac{(1-q_1^2)d^2}{1-d^2}q^{4l}\\biggr )e^{\\alpha +\\beta }+q^{2l}e^{\\frac{3\\beta -\\alpha }{2}}\\right)(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }.\\end{split}$ For $\\tilde{\\psi }^-_0$ , we have $\\tilde{\\psi }^-_0G^+_{-1}=q^{-2}G^+_{-1}\\tilde{\\psi }^-_0,$ which leads to $\\tilde{\\psi }^-_0(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }=q^{-2l}e^{-\\frac{\\alpha +\\beta }{2}}(G_{-1}^+)^l\\mathinner {|{\\alpha ,\\beta }\\rangle }.$ By substituting (REF ), (REF ) and (REF ) into (REF ), we have the following recursion relation, $\\begin{split}\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^n(G_{-1}^+)^n|\\alpha ,\\beta }\\rangle }=f_n\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^{n-1}(G_{-1}^+)^{n-1}|\\alpha ,\\beta }\\rangle } ,\\\\\\end{split}$ where $\\begin{split}f_n&=\\frac{1}{(q-q^{-1})^2}\\sum _{l=0}^{n-1}\\left((q^{4l}+q^{4l-2}-q^{2l-2})e^{\\alpha +\\beta }-q^{2l-2}d^{-2}e^{\\alpha -\\beta }+q^{-2l-2}d^{-4}e^{-\\frac{\\alpha +\\beta }{2}}-q^{2l}d^{-2}e^{\\frac{3\\beta -\\alpha }{2}}\\right)\\\\&=\\frac{-q^{-3}}{(q-q^{-1})^3}\\left((q^{2n}-q^{4n})e^{\\alpha +\\beta }-d^{-2}(1-q^{2n})e^{\\alpha -\\beta }-q^2d^{-4}(1-q^{-2n})e^{-\\frac{\\alpha +\\beta }{2}}-q^2d^{-2}(1-q^{2n})e^{\\frac{3\\beta -\\alpha }{2}}\\right)\\\\&=\\frac{-q^{-3}(1-q^{2n})(q^{2n}e^{\\alpha }-q_1^{-2}e^{\\frac{\\beta -\\alpha }{2}})(e^{\\beta }-q^{-2n}d^{-2}e^{-\\beta })}{(q-q^{-1})^3}.\\end{split}$ Then we have $\\mathinner {\\langle {\\alpha ,\\beta |(G_1^- )^n(G_{-1}^+)^n|\\alpha ,\\beta }\\rangle }=(-1)^n\\prod _{i=1}^n\\frac{q^{-3}(1-q^{2i})(q^{2i}e^{\\alpha }-q_1^{-2}e^{\\frac{\\beta -\\alpha }{2}})(e^{\\beta }-q^{-2i}d^{-2}e^{-\\beta })}{(q-q^{-1})^3}.$ Let us compare (REF ) and (REF ) with the partition function (REF ) and (REF ).", "For convenience, we denote the Omega background parameters by ($q_1^{\\prime },q_2^{\\prime }$ ) here.", "If we set $&&q_1^{\\prime }=q^2,\\qquad q_2^{\\prime }=q_1^2,\\\\&&\\frac{u_{1,1}}{u_{1,2}}=e^{-2\\beta }d^{-2},\\qquad \\frac{u_{2,1}}{u_{1,2}}=e^{\\frac{\\beta -3\\alpha }{2}}q_1^{-2}$ and tune the fugacity parameters appropriately, the instanton partition function matches with the norm of the Whittaker states up to the difference between \"$1-e^X$ \" and \"$2\\sinh \\frac{X}{2}$ \".", "We expect it can be resolved by introducing the Chern-Simons termWe note that 5d supersymmetric gauge theories with surface operators and Chern-Simons terms were discussed in [46], [47].. We note that the parameters in (REF ) are exactly the same as those of the subalgebra $\\mathcal {E}_1^{(1)}$ given in ()." ], [ "Conclusion and Discussion", "In this paper, we proposed the quantum deformation of Feigin-Semikhatov's W-algebras.", "We first studied the simplest case $U_q(\\widehat{\\mathfrak {sl}}_2)$ in terms of the gluing construction proposed in [14], [17].", "We found that the currents ${E}_2(z),{F}_2(z)$ can be expressed as the product of the vertex operators for q-$W(\\widehat{\\mathfrak {gl}}_{2|1})$ and q-$W(\\widehat{\\mathfrak {gl}}_1)$ .", "Based on this result, we constructed the vertex operators for q-$W(\\widehat{\\mathfrak {gl}}_{N|1})$ by demanding the commutativity with the screening charges proposed in [12].", "We fixed the ambiguity for the Heisenberg part by requiring the Weyl transformation which flips the Dynkin diagram of $\\mathfrak {gl}_{n|1}$ .", "We also checked the relation to 5d AGT correspondence with a simple surface operator for the deformed Bershadsky-Polyakov algebra.", "There are several directions for the future work.", "It is interesting to extend the discussion of this paper to another (super) W-algebra.", "In the case of Feigin-Semikhatov's W-algebras, one of $\\mathcal {E}_1$ used in the gluing construction is merely Heisenberg algebra, which allows us to assume the gluing currents ${E}_N(z),{F}_N(z)$ to be the product of the two vertex operators.", "It would be not true for the another W-algebra and we may need alternative method.", "In this paper, we use free field representation in the gluing construction, but it may be possible to do the same process at the level of the quantum toroidal algebra.", "In fact, it is expected that Gaiotto-Rapčák's VOA at least for a certain brane-web is a truncation of a quantum toroidal algebra [33].", "We expect that the deformed Feigin-Semikhatov's W-algebras can be realized from the shifted toroidal algebra of $gl_2$ defined in [35].", "There is a close relation between 2d CFT and integrable systems [48], [49], [50].", "It was also studied in the q-deformed case [51], [52] and has been recently studied in terms of the quantum toroidal algebra [53], [54], [55], [56], [13].", "It would be interesting to study the integrals of motion for the deformed Feigin-Semikhatov's W-algebras using the quantum toroidal algebras.", "It would be also interesting to explore the relation to symmetric functions.", "The insertion of a simple surface operator can be realized in two ways in terms of 6d theory.", "If we consider a codimension 4 operator instead of condimension 2, it does not change the dual algebra but serves as the insertion of degenerate primary fields of $W_N$ [57], [58].", "The relation between the correlation functions for different W-algebras has been recently studied well in [59].", "We hope that the relation is helpful to define q-analogue of the degenerate primary fields of $W_N$ ." ], [ "Acknowledgement", "The author would like to thank H. Awata, M. Fukuda, H. Kanno, Y. Matsuo and J. Shiraishi for helpful discussions and comments.", "He is partially supported by JSPS fellowship." ], [ "The free boson representation of $\\mathcal {E}_1$ from the deformed Wakimoto representation", "In this section, we use the following relations for the Wakimoto bosons: $\\begin{split}{\\rm for\\ } n>0\\qquad &[u_{1,n}^+,\\hat{u}_{1,m}^+]=[u_{1,n}^-,\\hat{u}_{1,m}^-]=nd^n(q^n-q^{-n})\\delta _{n+m,0},\\qquad [u_{1,n}^-,\\hat{u}_{1,m}^+]=0,\\\\&[u_{1,n}^+,\\hat{u}_{1,m}^-]=n(q^n-q^{-n})(d^n+d^{-n})\\delta _{n+m,0},\\\\\\end{split}$ $[a_{2,0},Q_{u_1}]=1,\\qquad [a_{3,0},Q_{u_1}]=-1.\\hspace{50.0pt}$ The free boson representation of the current (REF ) is given as follows: $\\begin{split}&\\rho (E_{1|0}^{(1)}(z))\\\\=\\ &\\lim _{z^{\\prime }\\rightarrow z}\\frac{(1-\\frac{z}{z^{\\prime }})(1-\\frac{q_3z}{q_1z^{\\prime }})}{(q-q^{-1})^2}\\left(\\frac{1-\\frac{q^2z}{z^{\\prime }}}{q(1-\\frac{z}{z^{\\prime }})}:e^{u_1^+(q_1z^{\\prime })+\\hat{u}_1^+(z)}:+\\frac{(1-\\frac{z}{q_1^2z^{\\prime }})(1-\\frac{d^2z}{z^{\\prime }})}{q(1-\\frac{z}{z^{\\prime }})(1-\\frac{z}{d^2z^{\\prime }})}:e^{u_1^+(q_1z^{\\prime })+\\hat{u}_1^-(z)}:\\right.\\\\&\\hspace{280.0pt}\\left.+\\frac{q(1-\\frac{q^2z}{z^{\\prime }})}{1-\\frac{z}{z^{\\prime }}}:e^{u_1^-(q_1z^{\\prime })+\\hat{u}_1^-(z)}:\\right), \\\\=\\ &-\\frac{1-d^{-2}}{q-q^{-1}}:e^{u_1^+(q_1z^{\\prime })+\\hat{u}_1^+(z)}:+\\frac{(1-q_1^{-2})(1-d^2)}{q(q-q^{-1})^2}:e^{u_1^+(q_1z)+\\hat{u}_1^-(z)}:-\\frac{q^2(1-d^{-2})}{q-q^{-1}}:e^{u_1^-(q_1z^{\\prime })+\\hat{u}_1^-(z)}:\\end{split}$ $\\begin{split}\\rho (E_{1|0}^{(3)}(z))&=\\lim _{z^{\\prime }\\rightarrow z}\\frac{(1-\\frac{z}{z^{\\prime }})(1-\\frac{q_1z}{q_3z^{\\prime }})}{(q-q^{-1})^2}\\frac{(1-\\frac{d^4z}{z^{\\prime }})(1-\\frac{q^2z}{z^{\\prime }})}{q(1-\\frac{z}{z^{\\prime }})(1-\\frac{d^2z}{z^{\\prime }})}:e^{u_1^+(q_3z^{\\prime })+\\hat{u}_1^+(z)}:\\hspace{30.0pt}\\\\&=-\\frac{1-d^4}{q-q^{-1}}:e^{u_1^+(q_3z)+\\hat{u}_1^+(z)}:.\\end{split}$ The coefficients do not matter because we can absorb them into the zero mode." ], [ "Useful formulae", "The commutation relations among the oscillators $A_n,B_n,C_n,D_n$ are given as follows: $\\begin{split}{\\rm for\\ }n\\ge 0,\\ \\ &[A_n,A_m]=[B_n,B_m]=[C_n,C_m]=[D_n,D_m]=-n(q^n-q^{-n})q^n\\delta _{n+m,0},\\\\&[A_n,B_m]=[A_n,C_m]=[B_n,D_m]=[C_n,D_m]=-n(q^{2n}-q^{-2n})\\delta _{n+m,0},\\\\&[C_n,B_m]=-n(q^n-q^{-n})q_1^nd^n\\delta _{n+m,0},\\\\&[D_n,A_m]=n(q^n-q^{-n})q^nd^{2n}\\delta _{n+m,0},\\\\&[A_n,D_m]=-n(q^n-q^{-n}))(q^n+q^{-n}+q^{-n}d^{-2n})\\delta _{n+m,0},\\\\&[B_n,C_m]=-n(q^n-q^{-n}))(q^n+q^{-n}-q^nd^{-2n})\\delta _{n+m,0},\\\\&[B_n,A_m]=[C_n,A_m]=[D_n,B_m]=[D_n,C_m]=0,\\end{split}$ $\\begin{split}{\\rm for\\ }n\\in \\mathbb {Z},\\ \\ &[A_n,v_{1,m}^+]=n(q^n-q^{-n})q^n(d^{-n}+d^{-3n})\\delta _{n+m,0},\\hspace{120.0pt}\\\\&[B_n,v_{1,m}^+]=[C_n,v_{1,m}^+]=n(q^n-q^{-n})q^nd^{-n}\\delta _{n+m,0},\\\\&[A_n,v_{1,m}^-]=n(q^n-q^{-n})q^nd^{-n}\\delta _{n+m,0},\\\\&[D_n,v_{1,m}^-]=-n(q^n-q^{-n})q^nd^n\\delta _{n+m,0},\\\\&[D_n,v_{1,m}^+]=[B_n,v_{1,m}^-]=[C_n,v_{1,m}^-]=0.\\end{split}$ We can extend all of the relations (REF ) to $n\\in \\mathbb {Z}$ as follows: $[Y_n,Z_m]=({\\rm RHS\\ of\\ (\\ref {eq:ABCDcom})})+n\\theta (n<0)(q^{2n}-q^{-2n})\\delta _{n+m,0}\\quad (Y,Z\\in {\\lbrace A,B,C,D\\rbrace }),$ where $\\theta (n<0)={\\left\\lbrace \\begin{array}{ll}1\\ (n <0)\\\\0\\ (n\\ge 0)\\end{array}\\right.", "}$ .", "Due to the additional term in (REF ), we need a rational factor in the quadratic relations in order to produce the delta function $\\delta (\\frac{w}{z})$ .", "The contractions between the two vertex operators are given as follows: $\\begin{split}&{K}(z)e^{v_1^+(w)}=q^{-4}\\frac{(1-\\frac{w}{q_3^2z})(1-\\frac{q^2w}{z})}{(1-\\frac{d^2w}{z})(1-\\frac{w}{q^2z})}:{K}(z)e^{v_1^+(w)}:,\\quad {K}(z)e^{v_1^-(w)}=q^{-4}\\frac{(1-\\frac{w}{q_3^2z})(1-\\frac{q^2w}{z})}{(1-\\frac{d^2w}{z})(1-\\frac{w}{q^2z})}:{K}(z)e^{v_1^-(w)}:,\\\\&e^{v_1^+(w)}{K}(z)=\\frac{1-\\frac{q_3^2z}{w}}{1-\\frac{z}{d^2w}}:e^{v_1^+(w)}{K}(z):,\\quad e^{v_1^-(w)}{K}(z)=\\frac{1-\\frac{q_3^2z}{w}}{1-\\frac{z}{d^2w}}:e^{v_1^-(w)}{K}(z):,\\\\&\\tilde{K}^-(z)e^{v_1^+(w)}=q^2:\\tilde{K}^-(z)e^{v_1^+(w)}:,\\quad \\tilde{K}^-(z)e^{v_1^-(w)}=q^2:\\tilde{K}^-(z)e^{v_1^-(w)}:,\\\\&e^{v_1^+(w)}\\tilde{K}^-(z)=\\frac{1-\\frac{q^2z}{w}}{1-\\frac{z}{q^2w}}:e^{v_1^+(w)}\\tilde{K}^-(z):,\\quad e^{v_1^-(w)}\\tilde{K}^-(z)=\\frac{1-\\frac{q^2z}{w}}{1-\\frac{z}{q^2w}}:e^{v_1^-(w)}\\tilde{K}^-(z):,\\\\&{K}(z)e^{X(w)}=q^4\\frac{(1-\\frac{q^{-2}dw}{z})(1-\\frac{q^{-2}d^3w}{z})}{(1-\\frac{dw}{z})(1-\\frac{q^2d^3w}{z})}:{K}(z)e^{X(w)}:,\\quad (X=A,B,C,D)\\\\&e^{X(w)}{K}(z)=\\frac{1-\\frac{z}{q^{-2}dw}}{1-\\frac{z}{dw}}:e^{X(w)}{K}(z):,\\quad (X=A,B,C,D)\\\\&\\tilde{K}^-(z)e^{X(w)}=q^{-2}:\\tilde{K}^-(z)e^{X(w)}:\\quad (X=A,B,C,D)\\\\&e^{X(w)}\\tilde{K}^-(z)=\\frac{1-\\frac{dz}{w}}{1-\\frac{q^4dz}{w}}:e^{X(w)}\\tilde{K}^-(z):,\\quad (X=A,B,C,D)\\\\&{X}(z)e^{v^+_1(w)}=-\\frac{q^{-4}(1-\\frac{w}{q_1^2z})(1-\\frac{q^2w}{z})}{(1-\\frac{w}{d^2z})(1-\\frac{w}{q^2z})}\\frac{ 1-q_1^2}{1-d^2}:e^{A(d^{-1}z)+v^+_1(z)+v^+(w)}:+\\frac{q^{-2}(1-\\frac{q^2w}{z})}{1-\\frac{w}{q^2z}}:e^{B(d^{-1}z)+v^+(z)+v^-(w)}:\\\\&\\hspace{65.0pt}+q^{-2}\\frac{1-\\frac{q^2w}{z}}{1-\\frac{w}{q^2z}}:e^{C(d^{-1}z)+v^+_1(z)+v^+_1(w)}:+q^{-4}\\frac{(1-\\frac{q^2w}{z})(1-\\frac{w}{q_1^2z})}{(1-\\frac{w}{z})(1-\\frac{w}{d^2z})}:e^{A(d^{-1}z)+v^-_1(z)+v^+(w)}:\\\\&{X}(z)e^{v^-_1(w)}=q^{-4}\\frac{(1-\\frac{q^2w}{z})^2}{(1-\\frac{w}{z})(1-\\frac{w}{q^2z})}:e^{A(d^{-1}z)+v^+_1(z)+v^-(w)}:\\\\&\\hspace{65.0pt}-q^{-2}\\frac{1-\\frac{q^2w}{z}}{1-\\frac{w}{q^2z}}(:e^{B(d^{-1}z)+v_1^+(z)+v_1^-(w)}:+:e^{C(d^{-1}z)+v_1^+(z)+v_1^-(w)}:+q^{-2}:e^{A(d^{-1}z)+v_1^+(z)+v_1^-(w)}:)\\end{split}$ For generic $\\mathcal {W}_N^{(2)}$ , the contractions are given as follows: $\\begin{split}&{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^+(w)}\\\\=\\ &q^{2-N-\\sum _{i=3}^{N+1}s_i}\\exp \\left(\\sum _{n>0}\\frac{(1-q^{-2n})(d^{(N-3)n}-d^{-(2+\\sum _{i=3}^{N+1}s_i )n})}{n(d^n-d^{-n})}\\biggl (\\frac{w}{z}\\biggr )^n\\right):{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^+(w)}:,\\end{split}$ $\\begin{split}&e^{v_1^+(w)}{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)\\\\=\\ &q\\exp \\left(-\\sum _{n>0}\\frac{(1-q^{2n})(d^{(3-N)n}-d^{(2+\\sum _{i=3}^{N+1}s_i) n})}{n(d^n-d^{-n})}\\biggl (\\frac{z}{w}\\biggr )^n\\right):{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^+(w)}:,\\hspace{60.0pt}\\end{split}$ $\\begin{split}&{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^-(w)}\\\\=\\ &q^{2-N-\\sum _{i=3}^{N+1}s_i}\\exp \\left(\\sum _{n>0}\\frac{(1-q^{-2n})(d^{(N-3)n}-d^{-\\sum _{i=3}^{N+1}s_i n})}{n(d^n-d^{-n})}\\biggl (\\frac{w}{z}\\biggr )^n\\right):{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^-(w)}:,\\end{split}$ $\\begin{split}&e^{v_1^-(w)}{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)\\\\=\\ &q^{-1}\\exp \\left(-\\sum _{n>0}\\frac{(1-q^{2n})(d^{(3-N)n}-d^{\\sum _{i=3}^{N+1}s_i n})}{n(d^n-d^{-n})}\\biggl (\\frac{z}{w}\\biggr )^n\\right):{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^-(w)}:,\\hspace{60.0pt}\\end{split}$ From (REF ) and (REF ), we have $\\begin{split}&[{E}_N^{s_3,s_4\\cdots s_{N+1}}(z),e^{v_1^+(w)}]\\\\=\\ &{\\left\\lbrace \\begin{array}{ll}\\quad 0\\qquad \\qquad (s_3=s_4=\\cdots =s_{N+1}=-1)\\\\q^{2-N-\\sum _{i=3}^{N+1}s_i}(1-q^2)\\displaystyle {\\sum _{i=0}^{\\frac{N-3+\\sum _{k=3}^{N+1}s_k}{2}}}\\delta \\left(\\frac{d^{N-4-2i}}{z}\\right)\\Biggl (\\prod _{\\begin{array}{c}j=0\\\\ j\\ne i\\end{array}}^{\\frac{N-3+\\sum _{k=3}^{N+1}s_k}{2}}\\frac{1-q^2d^{2(i-j)}}{1-d^{2(i-j)}}\\Biggr ):{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^+(w)}:.\\end{array}\\right.", "}\\\\&\\hspace{400.0pt}({\\rm otherwise})\\end{split}$ From (REF ) and (REF ), we have $\\begin{split}&[{E}_N^{s_3,s_4\\cdots s_{N+1}}(z),e^{v_1^-(w)}]\\\\=\\ &{\\left\\lbrace \\begin{array}{ll}(q-q^{-1})\\delta \\left(\\frac{q^2d^{N-2}w}{z}\\right):{E}_N^{s_3,s_4,\\cdots ,s_{N+1}}(z)e^{v_1^-(w)}:\\qquad (s_3=s_4=\\cdots =s_{N+1}=-1)\\\\\\quad 0\\qquad (\\#\\lbrace i\\in \\lbrace 3,4\\cdots N+1\\rbrace |s_i=1\\rbrace =1)\\\\q^{2-N-\\sum _{i=3}^{N+1}s_i}(1-q^2)\\displaystyle {\\sum _{i=0}^{\\frac{N-5+\\sum _{k=3}^{N+1}s_k}{2}}\\delta \\left(\\frac{d^{N-4-2i}}{z}\\right)\\Biggl (\\prod _{\\begin{array}{c}j=0\\\\ j\\ne i\\end{array}}^{\\frac{N-5+\\sum _{k=3}^{N+1}s_k}{2}}\\frac{1-q^2d^{2(i-j)}}{1-d^{2(i-j)}}\\Biggr )}:{E}_N^{s_3,s_4\\cdots s_{N+1}}(z)e^{v_1^-(w)}:.\\end{array}\\right.", "}\\\\&\\hspace{400.0pt}({\\rm otherwise})\\end{split}$" ], [ "The proof of (", "We introduce the following notation: $\\Lambda (z)\\Lambda (w)=f\\left(\\frac{w}{z}\\right):\\Lambda (z)\\Lambda (w):,\\quad f\\left(\\frac{w}{z}\\right)=\\exp \\left(\\sum _{n>0}\\frac{(q^n-q^{-n})q_3^{-n}}{n(d^n-d^{-n})}\\biggl (\\frac{w}{z}\\biggr )^n\\right).$ Then (REF ) is equivalent to $(z-q_2w)f\\left(\\frac{w}{z}\\right){E^{\\prime }}_N(z){E^{\\prime }}_N(w)-(w-q_2z)f\\left(\\frac{z}{w}\\right){E^{\\prime }}_N(w){E^{\\prime }}_N(z)=0.$ Let us show (REF ) by induction.", "We assume (REF ) is true for $N$ .", "When we express the quadratic relation for ${E^{\\prime }}_{N+1}(z)$ by ${E^{\\prime }}_N(z)$ using (REF ), it is expanded into the four parts.", "The first one is $\\begin{split}&(z-q_2w){E^{\\prime }}_N(dz)e^{\\phi _{(N+1)}^+(z)}\\Lambda (d^{2-N}z){E^{\\prime }}_N(dw)e^{\\phi ^+_{(N+1)}(w)}\\Lambda (d^{2-N}w)-(z\\leftrightarrow w)\\\\=\\ &q_2(z-q_2w)f\\left(\\frac{w}{z}\\right){E^{\\prime }}_N(dz){E^{\\prime }}_N(dw)e^{\\phi _{(N+1)}^+(z)}e^{\\phi ^+_{(N+1)}(w)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):-(z\\leftrightarrow w)\\\\=\\ &0,\\end{split}$ The second one is $\\begin{split}&(z-q_2w){E^{\\prime }}_N(dz)e^{\\phi _{(N+1)}^+(z)}\\Lambda (d^{2-N}z)e^{\\phi _{(N+1)}^-(w)}{E^{\\prime }}_N(d^{-1}w)\\Lambda (d^{2-N}w)\\\\&\\hspace{80.0pt}-(w-q_2z)e^{\\phi _{(N+1)}^-(w)}{E^{\\prime }}_N(d^{-1}w)\\Lambda (d^{2-N}w){E^{\\prime }}_N(dz)e^{\\phi _{(N+1)}^+(z)}\\Lambda (d^{2-N}z)\\\\=\\ &q_2(z-q_2w){E^{\\prime }}_N(dz){E^{\\prime }}_N(d^{-1}w)\\frac{(1-\\frac{w}{q_2z})(1-\\frac{w}{q_1^2z})}{(1-\\frac{w}{z})(1-\\frac{w}{d^2z})}e^{\\phi _{(N+1)}^-(w)}e^{\\phi _{(N+1)}^+(z)}f\\left(\\frac{w}{z}\\right):\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):\\\\&\\hspace{80.0pt}-(w-q_2z){E^{\\prime }}_N(d^{-1}w){E^{\\prime }}_N(dz)e^{\\phi _{(N+1)}^-(w)}e^{\\phi _{(N+1)}^+(z)}f\\left(\\frac{z}{w}\\right):\\Lambda (d^{2-N}w)\\Lambda (d^{2-N}z):\\\\=\\ & q_2\\frac{1-\\frac{w}{q_2z}}{1-\\frac{w}{d^2z}}(z-q_1^{-2}w)f\\left(\\frac{w}{d^2z}\\right){E^{\\prime }}_N(dz){E^{\\prime }}_N(d^{-1}w)e^{\\phi _{(N+1)}^-(w)}e^{\\phi _{(N+1)}^+(z)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):\\\\&\\hspace{40.0pt}-\\frac{1-\\frac{q_2z}{w}}{1-\\frac{d^2z}{w}}(w-q_3^{-2}z)f\\left(\\frac{d^2z}{w}\\right){E^{\\prime }}_N(d^{-1}w){E^{\\prime }}_N(dz)e^{\\phi _{(N+1)}^-(w)}e^{\\phi _{(N+1)}^+(z)}:\\Lambda (d^{2-N}w)\\Lambda (d^{2-N}z):\\\\=\\ &q_2(1-q_1^2)\\delta \\left(\\frac{w}{d^2z}\\right)\\Biggl ((z-q_1^{-2}w)f\\left(\\frac{w}{d^2z}\\right){E^{\\prime }}_N(dz){E^{\\prime }}_N(d^{-1}w)\\Biggr )e^{\\phi _{(N+1)}^-(w)}e^{\\phi _{(N+1)}^+(z)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):\\\\=\\ &0.\\end{split}$ To obtain the last line, we use the relation $(z-q_2w)f\\left(\\frac{w}{z}\\right){E^{\\prime }}_N(z){E^{\\prime }}_N(w)\\propto z-w$ which is derived from (REF ).", "The third one is $\\begin{split}&(z-q_2w)e^{\\phi _{(N+1)}^-(z)}{E^{\\prime }}_N(d^{-1}z)\\Lambda (d^{2-N}z){E^{\\prime }}_N(dw)e^{\\phi _{(N+1)}^+(w)}\\Lambda (d^{2-N}w)\\\\&\\hspace{80.0pt}-(w-q_2z){E^{\\prime }}_N(dw)e^{\\phi _{(N+1)}^+(w)}\\Lambda (d^{2-N}w)e^{\\phi _{(N+1)}^-(z)}{E^{\\prime }}_N(d^{-1}z)\\Lambda (d^{2-N}z)\\\\=\\ &\\frac{1-\\frac{q_2w}{z}}{1-\\frac{d^2w}{z}}(z-q_3^{-2}w){E^{\\prime }}_N(d^{-1}z){E^{\\prime }}_N(dw)e^{\\phi _{(N+1)}^-(z)}e^{\\phi _{(N+1)}^+(w)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):\\\\&\\hspace{80.0pt}-q_2\\frac{1-\\frac{z}{q_2w}}{1-\\frac{z}{d^2w}}(w-q_1^{-2}z){E^{\\prime }}_N(dw){E^{\\prime }}_N(d^{-1}z)e^{\\phi _{(N+1)}^+(w)}e^{\\phi _{(N+1)}^-(z)}:\\Lambda (d^{2-N}w)\\Lambda (d^{2-N}z):\\\\=\\ &(1-q_1^{-2})\\delta \\left(\\frac{d^2w}{z}\\right)\\Biggl ((z-q_3^{-2}w)f\\left(\\frac{d^2w}{z}\\right){E^{\\prime }}_N(d^{-1}z){E^{\\prime }}_N(dw)\\Biggr )e^{\\phi _{(N+1)}^-(z)}e^{\\phi _{(N+1)}^+(w)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):\\\\=\\ &0.\\end{split}$ The last one is $\\begin{split}&(z-q_2w)e^{\\phi _{(N+1)}^-(z)}{E^{\\prime }}_N(d^{-1}z)\\Lambda (d^{2-N}z)e^{\\phi _{(N+1)}^-(w)}{E^{\\prime }}_N(d^{-1}w)\\Lambda (d^{2-N}w)-(z\\leftrightarrow w)\\\\=\\ &(z-q_2w)f\\left(\\frac{w}{z}\\right){E^{\\prime }}_N(d^{-1}z){E^{\\prime }}_N(d^{-1}w)e^{\\phi _{(N+1)}^-(z)}e^{\\phi ^-_{(N+1)}(w)}:\\Lambda (d^{2-N}z)\\Lambda (d^{2-N}w):-(z\\leftrightarrow w)\\\\=\\ &0.\\end{split}$ From the above, the relation (REF ) holds also for $N+1$ .", "$\\Box $" ] ]

[ [ "The R-matrix of the quantum toroidal algebra" ], [ "Abstract We consider the R-matrix of the quantum toroidal algebra of type gl_1, both abstractly and in Fock space representations.", "We provide a survey of a certain point of view on this object which involves the elliptic Hall and shuffle algebras, and show how to obtain certain explicit formulas." ], [ "Introduction", "The quantum toroidal algebra ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ is quite a fascinating object: ubiquitous, but not completely belonging to a single area of mathematics and physics.", "It can be interpreted as the quantum affinization of the deformed Heisenberg algebra: ${U_q(\\dot{{\\mathfrak {gl}}}_1)}= U_q(\\widehat{{\\mathfrak {gl}}}_1)$ although since the latter is not of Drinfeld-Jimbo type, this interpretation is a bit ad-hoc.", "The quantum toroidal algebra was studied by Ding-Iohara ([14]) and Miki ([35]), and appeared in numerous places in both the mathematical and physical literature, where it is sometimes known as the deformed $W_{1+\\infty }$ algebra (see [4], [5], [6], [7], [10], [11], [12], [18], [23], [24], [46], [47] and many other works).", "It is connected with geometric representation theory ([21], [37], [43]) and from there with the $q$ -deformed Alday-Gaiotto-Tachikawa relations ([1], [2], [8], [34], [40], [45]).", "Last but not least, the quantum toroidal algebra is related to double affine Hecke algebras in type A ([44]).", "The main purpose of this note is to exploit two other incarnations of ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ : the elliptic Hall algebra ([13], [42]) and the double shuffle algebra ([16], [20], [36]), in order to study the universal$^*$ $R$ -matrixThe terminology “universal$^*$ $R$ -matrix\" means that ${\\ddot{R}}$ differs from the actual universal $R$ -matrix by certain powers of $q$ , see (REF ) and (REF ), as shown in Section 2.2 of [18].", "This is a well-known feature of quantum groups, where the analogous notion is the quasi $R$ -matrix of [33].", ": ${\\ddot{R}}\\in {U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\widehat{\\otimes }{U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ Using the tools developed in [13], one obtains the following formula: Theorem 1 The universal$^*$ $R$ -matrix can be factored as: ${\\ddot{R}}= \\prod _{\\text{coprime } (a,b) \\in {\\mathbb {N}}\\times {\\mathbb {Z}}\\sqcup (0,1)} \\exp \\left[ \\sum _{d=1}^\\infty \\frac{P_{da,db} \\otimes P_{-da,-db}}{d} \\frac{\\left( q^{\\frac{d}{2}} - q^{-\\frac{d}{2}} \\right)}{\\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right)} \\right] \\qquad $ in terms of the generators $\\lbrace P_{n,m} \\in {U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\rbrace _{(n,m) \\in {\\mathbb {Z}}^2 \\backslash (0,0)}$ constructed by [13] and [42] (see Subsection REF for a review).", "The product is taken in increasing order of $\\frac{b}{a}$ .", "Formula (REF ) arises from the fact that products of the $P_{n,m}$ 's in increasing order of slope form an orthogonal basis of the quantum toroidal algebra, which itself stems from the fact that coherent sheaves on an elliptic curve have Harder-Narasimhan filtrations.", "Moreover, (REF ) may be interpreted as a quantum toroidal version of the celebrated product formulas for universal $R$ -matrices of quantum groups from [15], [29], [30], [31], [32], [41].", "The $\\ddot{{\\mathfrak {gl}}}_n$ analogue of the (REF ) was studied in [39].", "Combining (REF ) with the shuffle algebra computations developed in [36], one can obtain explicit formulas for the image of ${\\ddot{R}}$ in two types of Fock spacesThe representations $F^\\uparrow _u$ and $F^\\rightarrow _u$ were denoted by $F^{(0,1)}_u$ and $F^{(1,0)}_u$ , respectively, in [3].", ": ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\curvearrowright F^\\uparrow _u \\text{ and } F^\\rightarrow _u$ These formulas can be found in Theorems REF and REF , respectively, and can be used to understand tensor products of Fock spaces as representations of the quantum toroidal algebra.", "As explained in [2], such tensor products govern the five-dimensional AGT relations ([1], [8]).", "I would like to thank Mikhail Bershtein, Jean-Emile Bourgine, Boris Feigin, Alexandr Garbali, Roman Gonin, Andrei Okounkov, Francesco Sala, Olivier Schiffmann, Junichi Shiarishi, Yan Soibelman and Alexander Tsymbaliuk for numerous wonderful discussions on the subject of quantum toroidal algebras over the years.", "I gratefully acknowledge NSF grants DMS-1760264 and DMS-1845034, as well as support from the Alfred P. Sloan Foundation.", "Let ${\\mathbb {F}}$ be a field, implicitly the ground field of all our constructions.", "Recall that a bialgebra is an algebra $A$ with unit 1 which is endowed with homomorphisms: $\\Delta : A \\rightarrow A \\otimes A \\qquad \\text{and} \\qquad \\varepsilon : A \\rightarrow {\\mathbb {F}}$ called coproduct and counit, respectively, which satisfy certain compatibility properties.", "We will often employ Sweedler notation for the coproduct: $\\Delta (a) = a_1 \\otimes a_2$ the meaning of which is that there is an implied summation of tensors in the right-hand side.", "Given two bialgebras $A^+$ and $A^-$ , a pairing between them: $\\langle \\cdot , \\cdot \\rangle : A^+ \\otimes A^- \\rightarrow {\\mathbb {F}}$ is called a bialgebra pairing if it intertwines the product and coproduct as below: a a', b = a a', op(b) a, b b' = (a), b b' for all $a,a^{\\prime } \\in A^+$ , and $b,b^{\\prime } \\in A^-$ .", "Given such a bialgebra pairing, we can form the Drinfeld double ([15]) of the bialgebras $A^+$ and $A^-$ , namely the vector space: $A = A^+ \\otimes A^-$ One can make $A$ into a bialgebra by requiring that $A^+ \\cong A^+ \\otimes 1$ and $A^- \\cong 1 \\otimes A^-$ be sub-bialgebras, and that the multiplication of elements coming from different tensor factors be constrained by the relation: $a_1 \\cdot b_1 \\langle a_2, b_2 \\rangle = \\langle a_1, b_1 \\rangle b_2 \\cdot a_2$ for all $a_1,b_1 \\in A^+$ and $a_2,b_2 \\in A^-$ .", "Remark 2 To define the bialgebra structure on (REF ) using relation (REF ), one needs all bialgebras involved to be Hopf algebras.", "In other words, there must exist antipode maps $S : A^\\pm \\rightarrow A^\\pm $ which satisfy certain compatibility conditions with the product, coproduct and pairing, and then relation (REF ) will be equivalent to: $\\langle S^{-1}(a_1), b_1 \\rangle a_2 b_2 \\langle a_3, b_3 \\rangle = b \\cdot a$ for all $a \\in A^+$ and $b \\in A^-$ .", "Formula (REF ) is the one which allows to unambiguously define the product on the vector space (REF ).", "The reason why we do not write down the antipode explicitly is that in all cases studied in the present paper, it exists and is uniquely determined by the bialgebra structure, and it is a straightforward exercise to write it down and to check that it satisfies all the required compatibility properties." ], [ "If the pairing (REF ) is non-degenerate, then we may define: $R = \\sum _{i} a_i \\otimes b_i \\in A^+ \\otimes A^- \\hookrightarrow A \\otimes A$ as $\\lbrace a_i, b_i\\rbrace _i$ go over any set of dual bases with respect to the pairing.", "The canonical tensor (REF ) is called the universal $R$ -matrix of $A$ , and satisfies the properties: $R \\cdot \\Delta (a) = \\Delta ^{\\text{op}}(a) \\cdot R$ for any $a \\in A$ (here $\\Delta ^{\\text{op}}$ denotes the opposite coproduct, which is obtained from $\\Delta $ by switching the two tensor factors), as well as: (1)R = R13 R23 (1 )R = R13 R12 where $R_{12} = R \\otimes 1$ , $R_{23} = 1 \\otimes R$ , and $R_{13}$ is defined analogously.", "The importance of this construction is the following: property (REF ) implies that for any representations $V,W\\in \\text{Rep}(A)$ , the operator $R^V_W$ given by: $A \\otimes A \\rightarrow \\text{End}(V \\otimes W), \\quad R \\mapsto R^V_{W}$ intertwines the $A$ -module structures $V \\otimes W$ and $W \\otimes V$ (up to a swap of the factors).", "We may also perform this construction for a single representation $V \\in \\text{Rep}(A)$ : A A A End(V),       R RV A A End(V) A,       R RV Explicitly, the way one defines $R_V$ (respectively $R^V$ ) is to write $R$ as a sum of tensors $a\\otimes b \\in A \\otimes A$ , and replace each $b$ (respectively $a$ ) that appears in such tensors by the corresponding endomorphism of $V$ prescribed by the $A$ -module structure of $V$ .", "Given a vector and covector $v \\in V$ , $\\lambda \\in V^\\vee $ , we may therefore consider: $_\\lambda R_v \\in A \\qquad \\text{and} \\qquad ^\\lambda R^v \\in A$ obtained by taking the $\\langle \\lambda | v \\rangle $ matrix coefficient of $R_V$ (respectively $R^V$ ) in the second (respectively first) tensor factor." ], [ "Consider two formal parameters $q_1$ and $q_2$ , and set: $q = q_1 q_2$ We will slightly abuse notation by writing ${\\mathbb {Q}}(q_1,q_2)$ instead of ${\\mathbb {Q}}(q_1^{\\frac{1}{2}}, q_2^{\\frac{1}{2}})$ , which the reader should interpret as the fact that we fix square roots of $q_1$ and $q_2$ .", "As these square roots are simply cosmetic, and not essential, features of the theory, this abuse seems acceptable.", "Let us consider the deformed Heisenberg algebra: ${U_q(\\dot{{\\mathfrak {gl}}}_1)}= {\\mathbb {Q}}(q_1,q_2) \\Big \\langle P_n, c^{\\pm 1} \\Big \\rangle _{n \\in {\\mathbb {Z}}\\backslash 0} \\Big /^{c \\text{ central}}_{\\text{relation (\\ref {eqn:heis})}}$ where: $\\Big [ P_n, P_{n^{\\prime }} \\Big ] = \\delta _{n+n^{\\prime }}^0 \\frac{n \\left(q_1^{\\frac{n}{2}} - q_1^{-\\frac{n}{2}} \\right) \\left(q_2^{\\frac{n}{2}} - q_2^{-\\frac{n}{2}} \\right)}{\\left(q^{-\\frac{n}{2}} - q^{\\frac{n}{2}} \\right)} \\left(c^n - c^{-n} \\right)$ It is a bialgebra with respect to the coproduct determined by: $\\Delta (c) = c \\otimes c$ $\\Delta (P_n) = {\\left\\lbrace \\begin{array}{ll} P_n \\otimes 1 + c^n \\otimes P_{n} &\\text{if } n > 0 \\\\ P_n \\otimes c^n + 1 \\otimes P_n &\\text{if } n < 0 \\end{array}\\right.", "}$ and the counit determined by $\\varepsilon (c) = 1$ , $\\varepsilon (P_n) = 0$ for all $n$ .", "Moreover: Uq(gl1)= Q(q1,q2)[Pn,c1]n N Uq(gl1)= Q(q1,q2)[P-n,c1]n N (here ${\\mathbb {N}}= {\\mathbb {Z}}_{>0}$ ) are sub-bialgebras of ${U_q(\\dot{{\\mathfrak {gl}}}_1)}$ , and there is a bialgebra pairing: $\\langle \\cdot , \\cdot \\rangle : {U_q^\\ge (\\dot{{\\mathfrak {gl}}}_1)}\\otimes {U_q^\\le (\\dot{{\\mathfrak {gl}}}_1)}\\rightarrow {\\mathbb {Q}}(q_1,q_2)$ determined by: $\\langle c, - \\rangle = \\langle -, c\\rangle = \\varepsilon (-)$ $\\Big \\langle P_n, P_{-n^{\\prime }} \\Big \\rangle = \\delta _{n^{\\prime }}^n \\frac{n \\left(q_1^{\\frac{n}{2}} - q_1^{-\\frac{n}{2}} \\right) \\left(q_2^{\\frac{n}{2}} - q_2^{-\\frac{n}{2}} \\right)}{\\left( q^{\\frac{n}{2}} - q^{-\\frac{n}{2}} \\right)}$ and properties (REF )-(REF ).", "It is easy to show that: ${U_q(\\dot{{\\mathfrak {gl}}}_1)}= {U_q^\\ge (\\dot{{\\mathfrak {gl}}}_1)}\\otimes {U_q^\\le (\\dot{{\\mathfrak {gl}}}_1)}\\Big / (c \\otimes 1 - 1 \\otimes c)$ is the Drinfeld double constructed with respect to the pairing (REF ).", "Remark 5 Note that one can change the right-hand side of (REF ) to any scalars that depend on $n$ , and alternatively, this can be achieved by changing the right-hand side of (REF ).", "The reason why we prefer the scalars above is that they naturally appear in Macdonald polynomial theory (see [38] for a brief survey of the connection) and in the study of the quantum toroidal algebra (see Section )." ], [ "It is easy to see that the restriction of the pairing (REF ) to the subalgebras: Uq+(gl1)= Q(q1,q2)[Pn]n N Uq-(gl1)= Q(q1,q2)[P-n]n N is non-degenerate.", "Indeed, as $\\bar{n}= (n_1 \\ge \\dots \\ge n_t)$ goes over partitions, the products: $P_{\\pm \\bar{n}} = P_{\\pm n_1} \\dots P_{\\pm n_t}$ give rise to orthogonal bases with respect to (REF ): $\\Big \\langle P_{\\bar{n}}, P_{-\\bar{n}^{\\prime }} \\Big \\rangle = \\delta _{\\bar{n}^{\\prime }}^{\\bar{n}} z_{\\bar{n}}$ where: $z_{\\bar{n}} = \\bar{n}!", "\\prod _{i=1}^t \\frac{n_i \\left(q_1^{\\frac{n_i}{2}} - q_1^{-\\frac{n_i}{2}} \\right) \\left(q_2^{\\frac{n_i}{2}} - q_2^{-\\frac{n_i}{2}} \\right)}{\\left( q^{\\frac{n_i}{2}} - q^{-\\frac{n_i}{2}} \\right)}$ and $\\bar{n}!$ is the product of factorials of the number of times each positive integer $n$ appears in the partition $\\bar{n}$ .", "One would like to invoke formula (REF ) to conclude that the universal $R$ -matrix of the deformed Heisenberg algebra is given by: $\\dot{R}:= \\sum _{\\bar{n}\\text{ partition}} \\frac{P_{\\bar{n}} \\otimes P_{-\\bar{n}}}{z_{\\bar{n}}} = \\exp \\left[ \\sum _{n=1}^\\infty \\frac{P_n \\otimes P_{-n}}{n} \\frac{\\left( q^{\\frac{n}{2}} - q^{-\\frac{n}{2}} \\right)}{\\left(q_1^{\\frac{n}{2}} - q_1^{-\\frac{n}{2}} \\right) \\left(q_2^{\\frac{n}{2}} - q_2^{-\\frac{n}{2}} \\right)} \\right] \\qquad $ Because the exponential is an infinite sum, (REF ) lies in a certain completion: $\\dot{R}\\in {U_q(\\dot{{\\mathfrak {gl}}}_1)}\\widehat{\\otimes }{U_q(\\dot{{\\mathfrak {gl}}}_1)}$ We call (REF ) the universal$^*$ $R$ -matrix of ${U_q(\\dot{{\\mathfrak {gl}}}_1)}$ , and note that it slightly differs from the actual universal $R$ -matrix because it is not equal to the canonical tensor of the pairing (REF ).", "The reason for this is that it does not account for the powers of the central element $c$ .", "To make matters worse, the expressions $c^n - c^{n-1}$ lie in the kernel of the bialgebra pairing, making it degenerate, and thus not even allowing us to construct the canonical tensor.", "We will now show how to fix the issue." ], [ "To construct the actual universal $R$ -matrix, one needs to introduce an “almost central\" element $d$ into the deformed Heisenberg algebra, and consider: $\\tilde{U}_q(\\dot{{\\mathfrak {gl}}}_1) = {U_q(\\dot{{\\mathfrak {gl}}}_1)}\\otimes _{\\mathbb {Q}}{\\mathbb {Q}}[d^{\\pm 1}] \\Big / \\Big ( [c,d] = 0, \\ d P_n = q^{-n} P_n d \\Big )$ with coproduct $\\Delta (d) = d \\otimes d$ .", "Moreover, we must replace (REF ) by: $\\langle c,d \\rangle = \\langle d,c \\rangle = q, \\qquad \\langle c \\text{ or } d ,P_n \\rangle = \\langle P_n, c \\text{ or } d \\rangle = 0$ for all $n \\ne 0$ .", "It is straightforward to check that this gives rise to a bialgebra pairing between the two halves of (REF ).", "However, now we have a second problem, in that it is not clear to construct the canonical tensor on the infinite-dimensional vector space ${\\mathbb {Q}}(q_1,q_2)[c^{\\pm 1}, d^{\\pm 1}]$ .", "To remedy this issue, we set: $q = e^{\\hbar }$ and work over ${\\mathbb {Q}}((\\hbar ))$ instead of over ${\\mathbb {Q}}(q)$ .", "Then we replace the elements $c$ and $d$ by their logarithms $\\gamma $ and $\\delta $ , explicitly defined by: $c = e^{\\hbar \\gamma }, \\quad d = e^{\\hbar \\delta }$ Set $\\Delta (\\gamma ) = \\gamma \\otimes 1 + 1 \\otimes \\gamma $ and $\\Delta (\\delta ) = \\delta \\otimes 1 + 1 \\otimes \\delta $ , and define the pairing by: $\\langle \\gamma , \\delta \\rangle = \\langle \\delta , \\gamma \\rangle = \\frac{1}{\\hbar }$ and all other pairings involving $\\gamma $ and $\\delta $ are set equal to 0.", "With this in mind, the canonical tensor restricted to the subalgebra generated by $\\gamma , \\delta $ takes the form: $\\sum _{n,n^{\\prime }=0}^\\infty \\gamma ^n \\delta ^{n^{\\prime }} \\otimes \\delta ^n \\gamma ^{n^{\\prime }} \\cdot \\frac{\\hbar ^{n+n^{\\prime }}}{n!n^{\\prime }!}", "= q^{\\gamma \\otimes \\delta + \\delta \\otimes \\gamma }$ Therefore, the correct formula for the universal $R$ -matrix of ${U_q(\\dot{{\\mathfrak {gl}}}_1)}$ is: $R_{{U_q(\\dot{{\\mathfrak {gl}}}_1)}} = \\dot{R}\\cdot q^{\\log _q c \\otimes \\log _q d + \\log _q d \\otimes \\log _q c}$ where $\\dot{R}\\in {U_q(\\dot{{\\mathfrak {gl}}}_1)}\\widehat{\\otimes }{U_q(\\dot{{\\mathfrak {gl}}}_1)}$ is defined in (REF ).", "We stress once again the fact that in order to properly define the expression (REF ), one needs to make all the modifications explained in the present Subsection: introduce the “almost central\" element $d$ , work over power series in $\\log q$ and replace the elements $c$ and $d$ by their logarithms in base $q$ .", "Since the power of $q$ in (REF ) will always act by a simple operator in all representations we are concerned with, we will henceforth focus on providing formulas for $\\dot{R}$ (which will be the interesting part of the $R$ -matrix for us)." ], [ "The basic representation of ${U_q(\\dot{{\\mathfrak {gl}}}_1)}$ is the Fock space: $F = {\\mathbb {Q}}(q_1,q_2)[p_1,p_2,\\dots ]$ with the action given by: $c \\mapsto q^{\\frac{1}{2}}, \\qquad d \\mapsto q^{\\text{deg}}$ (here, $\\deg $ denotes the grading on the polynomial ring which sets $\\deg p_n = n$ ) and: P-n multiplication by pn Pn - n (q1n2 - q1-n2 ) (q2n2 - q2-n2 ) pn The universal$^*$ $R$ -matrix (REF ) in a tensor product of Fock modules is therefore: $\\dot{R}^{F}_{F} = \\sum _{\\bar{n}= (n_1 \\ge \\dots \\ge n_t)} \\frac{\\partial }{\\partial p_{\\bar{n}}} \\otimes p_{\\bar{n}} \\cdot \\frac{1}{\\bar{n}!}", "\\prod _{i=1}^t \\left( q^{-\\frac{n_i}{2}} - q^{\\frac{n_i}{2}} \\right)$ where $p_{\\bar{n}} = p_{n_1} \\dots p_{n_t}$ and $\\frac{\\partial }{\\partial p_{\\bar{n}}} = \\frac{\\partial }{\\partial p_{n_1}} \\dots \\frac{\\partial }{\\partial p_{n_t}}$ .", "We have: $\\dot{R}^{F}_{F} \\in \\text{End}(F \\otimes F)$ because all but finitely many of the summands in (REF ) act trivially on any vector of $F \\otimes F$ , thus making formula (REF ) a well-defined endomorphism.", "This will be the case with all infinite sums that we will write in the present paper." ], [ "We will now consider the quantum toroidal algebra of type ${\\mathfrak {gl}}_1$ (also known as the Ding-Iohara-Miki algebra).", "Consider the rational function: $\\zeta (x) = \\frac{(1-xq_1)(1-xq_2)}{(1-x)(1-xq)}$ and the formal delta series $\\delta (z) = \\sum _{k \\in {\\mathbb {Z}}} z^k$ .", "Definition 2 ([14], [35]) Let: ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}= {\\mathbb {Q}}(q_1,q_2) \\Big \\langle e_k, f_k, h_{m}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{k \\in {\\mathbb {Z}}, m \\in {\\mathbb {Z}}\\backslash 0} \\Big /^{c_1, c_2 \\text{ central}}_{\\text{relations (\\ref {eqn:rel tor 1})-(\\ref {eqn:rel tor 6})}}$ where we construct the power series $e(z) = \\sum _{k \\in {\\mathbb {Z}}} \\frac{e_k}{z^k}$ , $f(z) = \\sum _{k \\in {\\mathbb {Z}}} \\frac{f_k}{z^k}$ , and let: $[h_m, h_{m^{\\prime }}] = \\frac{\\delta _{m+m^{\\prime }}^0 m \\left(c_2^m - c_2^{-m} \\right)}{\\left(q_1^{\\frac{m}{2}} - q_1^{-\\frac{m}{2}} \\right) \\left(q_2^{\\frac{m}{2}} - q_2^{-\\frac{m}{2}} \\right)\\left(q^{-\\frac{m}{2}} - q^{\\frac{m}{2}} \\right)}$ [hm, ek] = ek+m {ll 1 if m>0 -c2m if m<0 .", "[hm, fk] = fk+m {ll 1 if m<0 -c2m if m>0 .", "$e(z)e(w) \\zeta \\left(\\frac{z}{w} \\right) = e(w) e(z) \\zeta \\left(\\frac{w}{z} \\right)$ $f(z)f(w) \\zeta \\left(\\frac{w}{z} \\right) = f(w) f(z) \\zeta \\left(\\frac{z}{w} \\right)$ $[e_k, f_{k^{\\prime }}] = \\frac{\\left(q_1^{\\frac{1}{2}} - q_1^{-\\frac{1}{2}} \\right) \\left(q_2^{\\frac{1}{2}} - q_2^{-\\frac{1}{2}} \\right)}{\\left(q^{- \\frac{1}{2}} - q^{ \\frac{1}{2}} \\right)} \\left(\\underbrace{\\psi _{k+k^{\\prime }} c_1 c_2^{-k^{\\prime }}}_{\\text{if }k+k^{\\prime } \\ge 0} - \\underbrace{\\psi _{k+k^{\\prime }} c_1^{-1} c_2^{-k}}_{\\text{if }k+k^{\\prime } \\le 0} \\right)$ where the elements $\\psi _m$ are defined by the generating series: $\\sum _{m=0}^\\infty \\psi _{\\pm m} \\cdot x^m = \\exp \\left[ \\sum _{m=1}^\\infty \\frac{h_{\\pm m}}{m} \\cdot x^m \\left(q_1^{\\frac{m}{2}} - q_1^{-\\frac{m}{2}}\\right)\\left(q_2^{\\frac{m}{2}} - q_2^{-\\frac{m}{2}}\\right) \\left(q^{\\frac{m}{2}} - q^{- \\frac{m}{2}} \\right) \\right]$ To make sense of relations (REF ) and (REF ), one clears denominators in the rational functions $\\zeta $ and identifies the coefficients of $z^k w^l$ in the left and right-hand sides." ], [ "We note that ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ is a bialgebra, with coproduct: $\\Delta (c_1) = c_1 \\otimes c_1 \\qquad \\Delta (c_2) = c_2 \\otimes c_2$ $\\Delta (h_m) = {\\left\\lbrace \\begin{array}{ll} h_m \\otimes 1 + c_2^m \\otimes h_m &\\text{if } m > 0 \\\\ h_m \\otimes c_2^m + 1 \\otimes h_m &\\text{if } m < 0 \\end{array}\\right.", "}$ (ek) = ek 1 + m=0c1 c2k-m m ek-m (fk) = 1 fk + m=0fk+m c1-1 c2k+m -m and counit determined by $\\varepsilon (c_1) = \\varepsilon (c_2) = 1$ , $\\varepsilon (e_k) = \\varepsilon (f_k) = \\varepsilon (h_m) = 0$ .", "Note that the coproduct is defined in a topological sense, as it takes values in the completion: $\\Delta : {U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\rightarrow {U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\widehat{\\otimes }{U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ It is easy to see that the coproduct preserves the subalgebras: Uq1,q2(gl1)= Q(q1,q2) ek, hm, c11, c21 k Z, m N Uq1,q2(gl1)= Q(q1,q2) fk, h-m, c11, c21 k Z, m N of ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ , and it is well-known that we have a triangular decomposition: ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}= {U^\\ge _{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\otimes {U^\\le _{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\Big /(c_i \\otimes 1 - 1 \\otimes c_i)_{i \\in \\lbrace 1,2\\rbrace }$ Moreover, there exists a bialgebra pairing: $\\langle \\cdot , \\cdot \\rangle : {U^\\ge _{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\otimes {U^\\le _{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\rightarrow {\\mathbb {Q}}(q_1,q_2)$ determined by the assignments: $\\langle c_i, - \\rangle = \\langle -, c_i \\rangle = \\varepsilon (-)$ $\\Big \\langle e_k, f_{-k} \\Big \\rangle = \\frac{\\left(q_1^{\\frac{1}{2}} - q_1^{-\\frac{1}{2}} \\right) \\left(q_2^{\\frac{1}{2}} - q_2^{-\\frac{1}{2}} \\right)}{\\left( q^{\\frac{1}{2}} - q^{-\\frac{1}{2}} \\right)}$ $\\Big \\langle h_m, h_{-m} \\Big \\rangle = \\frac{m}{\\left(q_1^{\\frac{m}{2}} - q_1^{-\\frac{m}{2}} \\right) \\left(q_2^{\\frac{m}{2}} - q_2^{-\\frac{m}{2}} \\right) \\left( q^{\\frac{m}{2}} - q^{-\\frac{m}{2}} \\right)}$ (all other pairings between the generators $e_k, f_k, h_m$ are 0).", "Note that (REF ) is the Drinfeld double with respect to the datum above.", "Therefore, to construct and study the universal $R$ -matrix of the quantum toroidal algebra, we must find dual bases with respect to the pairing (REF ).", "To achieve this, we now turn to another incarnation of the quantum toroidal algebra, namely the elliptic Hall algebra." ], [ "Let us consider the following half planes: ${\\mathbb {Z}}_+^2 = \\lbrace (n,m) \\in {\\mathbb {Z}}^2 \\text{ s.t. }", "n>0 \\text{ or } n=0,m>0\\rbrace $ ${\\mathbb {Z}}_-^2 = \\lbrace (n,m) \\in {\\mathbb {Z}}^2 \\text{ s.t. }", "n<0 \\text{ or } n=0,m<0\\rbrace $ Definition 5 ([13]) The elliptic Hall algebra is: ${\\mathcal {A}}= {\\mathbb {Q}}(q_1,q_2) \\Big \\langle P_{n,m}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{(n,m) \\in {\\mathbb {Z}}^2 \\backslash (0,0)} \\Big /^{c_1, c_2 \\text{ central}}_{\\text{relations (\\ref {eqn:relation 1}), (\\ref {eqn:relation 2})}}$ where we impose the following relations: $[P_{n,m}, P_{n^{\\prime },m^{\\prime }}] = \\delta _{n+n^{\\prime }}^0 \\frac{d \\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right)}{\\left(q^{-\\frac{d}{2}} - q^{\\frac{d}{2}} \\right)} \\left(c_1^n c_2^m - c_1^{-n} c_2^{-m} \\right)$ if $nm^{\\prime }=n^{\\prime }m$ and $n>0$ , with $d = \\gcd (m,n)$ .", "The second relation states that whenever $nm^{\\prime }>n^{\\prime }m$ and the triangle with vertices $(0,0), (n,m), (n+n^{\\prime },m+m^{\\prime })$ contains no lattice points inside nor on one of the edges, then we have the relation: $[P_{n,m}, P_{n^{\\prime },m^{\\prime }}] = \\frac{\\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right)}{\\left(q^{- \\frac{1}{2}} - q^{ \\frac{1}{2}} \\right)} Q_{n+n^{\\prime },m+m^{\\prime }}$ $\\cdot \\ {\\left\\lbrace \\begin{array}{ll}c_1^{-n^{\\prime }} c_2^{-m^{\\prime }} & \\text{if } (n,m) \\in {\\mathbb {Z}}_\\pm ^2, (n^{\\prime },m^{\\prime }) \\in {\\mathbb {Z}}_\\mp ^2, (n+n^{\\prime },m+m^{\\prime }) \\in {\\mathbb {Z}}_\\pm ^2 \\\\c_1^n c_2^m & \\text{if } (n,m) \\in {\\mathbb {Z}}_\\pm ^2, (n^{\\prime },m^{\\prime }) \\in {\\mathbb {Z}}_\\mp ^2, (n+n^{\\prime },m+m^{\\prime }) \\in {\\mathbb {Z}}_\\mp ^2 \\\\1 & \\text{otherwise}\\end{array}\\right.", "}$ where $d = \\gcd (n,m)\\gcd (n^{\\prime },m^{\\prime })$ (by the assumption on the triangle, we note that at most one of the pairs $(n,m), (n^{\\prime },m^{\\prime }), (n+n^{\\prime },m+m^{\\prime })$ can fail to be coprime), and: $\\sum _{k=0}^{\\infty } Q_{ka,kb} \\cdot x^k = \\exp \\left[ \\sum _{k=1}^\\infty \\frac{P_{ka,kb}}{k} \\cdot x^k \\left(q^{\\frac{k}{2}} - q^{- \\frac{k}{2}} \\right) \\right]$ for all coprime integers $a,b$ .", "Note that $Q_{0,0} = 1$ .", "Remark 6 In the notation of [13], we have: $P_{n,m} = \\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right) u_{n,m}$ where $d = \\gcd (n,m)$ , as well as $c_1^n c_2^m = \\kappa _{n,m}$ .", "As shown in loc.", "cit., when we specialize the parameters $q_1$ and $q_2$ to the Frobenius eigenvalues of an elliptic curve ${\\mathcal {E}}$ over the finite field ${\\mathbb {F}}_q$ , and set $c_1,c_2 \\mapsto 1$ , the algebra ${\\mathcal {A}}$ matches a certain subalgebra of the Drinfeld double of the Hall algebra of the category of coherent sheaves over ${\\mathcal {E}}$ .", "The fact that the group $SL_2({\\mathbb {Z}})$ acts on the derived category of coherent sheaves on ${\\mathcal {E}}$ translates into the fact that the universal cover of this group acts on the algebra ${\\mathcal {A}}$ by automorphisms (the reason one needs the universal cover is the presence of the central elements): Pn,m = Pan+cm,bn+dm (c1an+cmc2bn+dm )# c1 = c1a c2b,        c2 = c1c c2d (see (6.16) of [13] for how to define the integer $\\#$ ) where: $\\widetilde{\\gamma } \\in \\widetilde{SL_2({\\mathbb {Z}})} \\quad \\text{is a lift of} \\quad \\gamma = \\begin{pmatrix}a & c \\\\ b & d \\end{pmatrix} \\in SL_2({\\mathbb {Z}})$" ], [ "It was shown in [42] that there exists an algebra isomorphism: ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\stackrel{\\sim }{\\rightarrow }{\\mathcal {A}}$ generated by: $e_k \\mapsto P_{1,k}, \\quad f_k \\mapsto P_{-1,k}, \\quad h_m \\mapsto \\frac{P_{0,m}}{\\left(q_1^{\\frac{m}{2}} - q_1^{-\\frac{m}{2}}\\right)\\left(q_2^{\\frac{m}{2}} - q_2^{-\\frac{m}{2}}\\right)}$ This isomorphism allows us to transport the bialgebra structure from ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ to ${\\mathcal {A}}$ , as well as the decomposition (REF ) and the pairing (REF ): ${\\mathcal {A}}= {\\mathcal {A}}^\\ge \\otimes {\\mathcal {A}}^\\le \\Big /(c_i \\otimes 1 - 1 \\otimes c_i)_{i \\in \\lbrace 1,2\\rbrace }$ $\\langle \\cdot , \\cdot \\rangle : {\\mathcal {A}}^\\ge \\otimes {\\mathcal {A}}^\\le \\rightarrow {\\mathbb {Q}}(q_1,q_2)$ where we consider the subalgebras: A= Q(q1,q2) Pn,m, c11, c21 (n,m) Z+2 A= Q(q1,q2) Pn,m, c11, c21 (n,m) Z-2 of ${\\mathcal {A}}$ .", "Moreover, in terms of the $P_{n,m}$ generators, the pairing takes the form: $\\Big \\langle P_{n,m}, P_{n^{\\prime },m^{\\prime }} \\Big \\rangle = \\delta _{n+n^{\\prime }}^0 \\delta _{m+m^{\\prime }}^0 \\frac{d \\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right)}{\\left( q^{\\frac{d}{2}} - q^{-\\frac{d}{2}} \\right)}$ where $d = \\gcd (n,m)$ , for all $m,m^{\\prime },n,n^{\\prime }$ .", "Just like in the case of ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ , the decomposition (REF ) realizes ${\\mathcal {A}}$ as the Drinfeld double with respect to the data above." ], [ "The relevance of the generators $P_{n,m}$ is that ordered products of these elements give rise to an orthogonal basis of ${\\mathcal {A}}$ .", "Explicitly, it was shown in [13] that: $P_{\\pm v} = P_{\\pm n_1, \\pm m_1} \\dots P_{\\pm n_t, \\pm m_t}$ for any convex lattice path: $v = \\Big \\lbrace (n_1,m_1), \\dots ,(n_t,m_t)\\Big \\rbrace , \\quad \\frac{m_1}{n_1} \\le \\dots \\le \\frac{m_t}{n_t}, \\quad (n_i,m_i) \\in {\\mathbb {Z}}^2_+$ (we identify convex paths up to permuting those edges $(n_i,m_i)$ with the same slope, which does not change the product (REF ) due to (REF )) give rise to a basis: ${\\mathcal {A}}^{\\ge } = \\left( \\bigoplus _{v \\text{ convex}} {\\mathbb {Q}}(q_1,q_2) \\cdot P_{v} \\right) \\otimes _{\\mathbb {Q}}{\\mathbb {Q}}[c_1^{\\pm 1}, c_2^{\\pm 1}]$ as well as the analogous statement involving ${\\mathcal {A}}^\\le $ and $P_{-v}$ .", "Moreover, these bases are orthogonal with respect to the pairing (REF ) (see Proposition 5.7 of [36] for a proof, although it is already implicit from [13]): $\\Big \\langle P_{v}, P_{-v^{\\prime }} \\Big \\rangle = \\delta _{v^{\\prime }}^v z_v$ In the formula above, for a convex lattice path (REF ) we define: $z_v = v!", "\\prod _{i=1}^t \\frac{d_i \\left(q_1^{\\frac{d_i}{2}} - q_1^{-\\frac{d_i}{2}} \\right) \\left(q_2^{\\frac{d_i}{2}} - q_2^{-\\frac{d_i}{2}} \\right)}{\\left( q^{\\frac{d_i}{2}} - q^{-\\frac{d_i}{2}} \\right)}$ where we denote $d_i = \\gcd (n_i, m_i)$ , and $v!$ is the product of factorials of the number of times each vector $(n,m)$ appears in the path $v$ .", "Therefore, by analogy with (REF ), we call the following tensor the universal$^*$ $R$ -matrix: ${\\ddot{R}}:= \\sum _{v \\text{ convex}} \\frac{P_{v} \\otimes P_{-v}}{z_v} =$ $= \\prod _{\\text{coprime } (a,b) \\in {\\mathbb {N}}\\times {\\mathbb {Z}}\\sqcup (0,1)} \\exp \\left[ \\sum _{d=1}^\\infty \\frac{P_{da,db} \\otimes P_{-da,-db}}{d} \\cdot \\frac{\\left( q^{\\frac{d}{2}} - q^{-\\frac{d}{2}} \\right)}{\\left(q_1^{\\frac{d}{2}} - q_1^{-\\frac{d}{2}} \\right) \\left(q_2^{\\frac{d}{2}} - q_2^{-\\frac{d}{2}} \\right)} \\right]$ where the product on the second line is taken in increasing order of $\\frac{b}{a}$ .", "Because of the isomorphism (REF ), we will refer to (REF ) as lying in either algebra: ${\\ddot{R}}\\in {\\mathcal {A}}\\widehat{\\otimes }{\\mathcal {A}}\\cong {U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}\\widehat{\\otimes }{U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ As explained in Subsection REF , the actual universal $R$ -matrix of ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ is: $R_{{U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}} = {\\ddot{R}}\\cdot q^{\\sum _{i=1}^2 \\log _q c_i \\otimes \\log _q d_i + \\log _q d_i \\otimes \\log _q c_i}$ where $d_1, d_2$ are elements that one must add to the algebra ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ (and work over ${\\mathbb {Q}}((\\hbar _1, \\hbar _2))$ instead of over ${\\mathbb {Q}}(q_1,q_2)$ , where $\\hbar _i = \\log q_i$ ).", "We refer the reader to Section 2.2 of [18] for details as to the correct setup, and henceforth focus on (REF )." ], [ "As is clear from (REF ), understanding the generators $P_{n,m} \\in {\\mathcal {A}}$ is key.", "To make them explicit, we turn to the shuffle algebra incarnation of ${\\mathcal {A}}$ .", "Specifically, the following is a trigonometric degeneration of the $\\ddot{{\\mathfrak {gl}}}_1$ version of [20].", "Definition 10 ([16]) Consider the ${\\mathbb {Q}}(q_1,q_2)$ -vector space: $V = \\bigoplus _{k \\ge 0} {\\mathbb {Q}}(q_1,q_2)(z_{1}, \\dots ,z_{k})^{\\emph {Sym}}$ of rational functions which are symmetric in the variables $z_1, \\dots ,z_k$ , for any $k$ .", "We endow $V$ with an algebra structure by the shuffle product: $R(z_{1}, \\dots ,z_{k}) * R^{\\prime }(z_{1}, \\dots ,z_{k^{\\prime }}) =$ $= \\frac{1}{k!", "k^{\\prime }!}", "\\cdot \\emph {Sym}\\left[R(z_{1}, \\dots ,z_{k})R^{\\prime }(z_{k+1}, \\dots ,z_{k+k^{\\prime }}) \\prod _{i=1}^k \\prod _{j = k+1}^{k+k^{\\prime }} \\zeta \\left( \\frac{z_i}{z_j} \\right) \\right]$ where Sym denotes the symmetrization operator: $\\emph {Sym}\\left( R(z_1, \\dots ,z_k) \\right) = \\sum _{\\sigma \\in S(k)} R(z_{\\sigma (1)}, \\dots ,z_{\\sigma (k)})$ The shuffle algebra ${\\mathcal {S}}\\subset V$ is defined as the set of rational functions of the form: $R(z_{1}, \\dots ,z_{k}) = \\frac{r(z_{1}, \\dots ,z_{k})}{\\prod _{1 \\le i \\ne j \\le k} (z_{i} - z_{j}q)}$ where $r$ is a symmetric Laurent polynomial that satisfies the wheel conditions: $r(z_1, \\dots ,z_k) \\Big |_{\\left\\lbrace \\frac{z_1}{z_2}, \\frac{z_2}{z_3}, \\frac{z_3}{z_1} \\right\\rbrace = \\left\\lbrace q_1,q_2, \\frac{1}{q} \\right\\rbrace } = 0$ Condition (REF ) is a trigonometric version of Condition 3 in $§$ 1.3 of [20]." ], [ "It was observed in [21], [43] that there are algebra homomorphisms: A- -S,          P-1,k z1k A+ +Sop,        P1,k z1k where the subalgebras ${\\mathcal {A}}^\\pm \\subset {\\mathcal {A}}$ are defined by: ${\\mathcal {A}}^\\pm = {\\mathbb {Q}}(q_1,q_2) \\Big \\langle P_{n,m} \\Big \\rangle _{\\pm n > 0, m \\in {\\mathbb {Z}}} \\\\$ The maps (REF ) and (REF ) were shown in [36] to be isomorphisms, and the images of the generators $P_{n,m}$ under these maps were also computed: $\\Upsilon ^\\pm ( P_{\\pm n,m} ) = q^{\\frac{n-d}{2}} R_{n,m}(z_1, \\dots ,z_n)$ where for all $n \\in {\\mathbb {N}}$ and $m\\in {\\mathbb {Z}}$ , we let $d = \\gcd (n,m)$ , $a = \\frac{n}{d}$ and: $R_{n,m} = \\textrm {Sym}\\left[ \\frac{\\prod _{i=1}^n z_i^{\\left\\lfloor \\frac{im}{n} \\right\\rfloor - \\left\\lfloor \\frac{(i-1)m}{n} \\right\\rfloor }}{\\prod _{i=1}^{n-1} \\left(1 - \\frac{qz_{i+1}}{z_i} \\right)} \\sum _{s=0}^{d-1} q^{s} \\frac{z_{a(d-1)+1} \\dots z_{a(d-s)+1}}{{z_{a(d-1)} \\dots z_{a(d-s)}}} \\prod _{1 \\le i < j \\le n} \\zeta \\left( \\frac{z_i}{z_j} \\right) \\right]$ We have a triangular decomposition: ${\\mathcal {A}}= {\\mathcal {A}}^+ \\otimes {\\mathcal {A}}^0 \\otimes {\\mathcal {A}}^-, \\quad {\\mathcal {A}}^0 = {\\mathbb {Q}}(q_1,q_2)\\Big \\langle P_{0,m}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{m \\in {\\mathbb {Z}}\\backslash 0}$ which matches the well-known triangular decomposition of ${U_{q_1,q_2}(\\ddot{{\\mathfrak {gl}}}_1)}$ under (REF )." ], [ "Let us recall the bijection between partitions $\\lambda = (\\lambda _1 \\ge \\lambda _2 \\ge \\dots )$ and Young diagrams.", "The latter are sets of $1 \\times 1$ boxes placed in the first quadrant of the plane, with $\\lambda _1$ boxes placed on the first row, $\\lambda _2$ boxes on the second row etc.", "For example, the following is the Young diagram associated to the partition $(4,3,1)$ : $(4,0)$$(4,1)$$(3,1)$$(3,2)$$(1,2)$$(1,3)$$(0,3)$The hollow circles in the figure above will be called the inner corners (abbreviated by “i.c.\")", "and the full circles will be called the outer corners (abbreviated “o.c.\")", "of the partition.", "The weight of a box is defined as the quantity: $\\chi _{\\square }= u q_1^x q_2^y$ where $(x,y)$ are the coordinates of the bottom left corner of the box, and $u$ is a parameter.", "Given two Young diagrams, we will write $\\mu \\subset \\lambda $ if $\\mu $ is contained in $\\lambda $ .", "If this happens, and $R$ is a symmetric rational function in $|{\\lambda \\backslash \\mu }|$ variables, write: $R({\\lambda \\backslash \\mu }) = R(\\dots ,\\chi _{\\square },\\dots )_{{\\square }\\in \\lambda \\backslash \\mu }$ We set $R({\\lambda \\backslash \\mu }) = 0$ if $\\mu \\lnot \\subset \\lambda $ or if $|{\\lambda \\backslash \\mu }|$ is not equal to the number of variables of $R$ .", "Definition 2 ([21], [43], see also [37]) Let $F_u^\\uparrow $ be a vector space with a basis $|\\lambda \\rangle $ indexed by partitions.", "Then the following formulas determine an action ${\\mathcal {A}}\\curvearrowright F_u^\\uparrow $ : $c_1 \\mapsto q^{\\frac{1}{2}}, \\qquad c_2 \\mapsto 1,$ $\\langle \\mu | P_{0,\\pm m} | \\lambda \\rangle = \\pm \\delta _\\lambda ^\\mu q^{\\mp \\frac{m}{2}} \\left( \\sum _{{\\square }\\text{ i.c.", "of }\\lambda } \\chi _{\\square }^{\\pm m} - \\sum _{{\\square }\\text{ o.c.", "of }\\lambda } \\chi _{\\square }^{\\pm m} \\right)$ Given an endomorphism $S$ of $F_u^\\uparrow $ , we write $\\langle \\mu | S |\\lambda \\rangle $ for the coefficient of $|\\mu \\rangle $ in $S \\left(|\\lambda \\rangle \\right)$ .", "and for all $X \\in {\\mathcal {A}}^-$ with $\\Upsilon ^-(X) = R$ (resp.", "$Y \\in {\\mathcal {A}}^+$ with $\\Upsilon ^+(Y) = R$ ), we have: | X | = R() || o.c.", "of (1 - ) i.c.", "of (1 - ) | Y | = R() || i.c.", "of (1 - q ) o.c.", "of (1 - q ) where $\\sigma = \\frac{(1-q_1)(1-q_2)}{1-q}$ and $\\bar{\\sigma } = \\sigma q^{\\frac{1}{2}}$ .", "The reader might ask how to interpret the evaluation $R({\\lambda \\backslash \\mu })$ , given that elements $R$ of the shuffle algebra take the form (REF ), and thus have poles at $z_i - z_j q$ .", "The answer lies in the wheel conditions.", "One first defines the specialization: $\\rho (y_1,y_2,\\dots ) = R \\left(y_1 q_1^{\\lambda _1 - \\mu _1}, \\dots ,y_1 q_1^{\\lambda _1-1}, y_2 q_1^{\\lambda _2 - \\mu _2}, \\dots ,y_2 q_1^{\\lambda _2-1}, \\dots \\right)$ (which is allowed, because $R$ has no poles at $z_i - z_j q_1$ ) and then invoke the wheel conditions (REF ) to conclude that $\\rho $ has no poles at $y_i - y_j q_2$ .", "Then we define: $R({\\lambda \\backslash \\mu }) = \\rho (1,q_2,q_2^2,\\dots )$ Remark 3 If we replace individual partitions $\\lambda $ by $r$ -tuples of partitions ${{{\\lambda }}}= (\\lambda ^1,\\dots ,\\lambda ^r)$ , then straightforward analogues of formulas (REF )-(REF ) yield an action: ${\\mathcal {A}}\\curvearrowright F_{u_1}^\\uparrow \\otimes \\dots \\otimes F_{u_r}^\\uparrow $ To this end, one must replace the weight (REF ) of a box in an individual partition by the weight $\\chi _{\\square }= u_i q_1^x q_2^y$ of a box ${\\square }$ located at coordinates $(x,y)$ in the $i$ -th constituent partition of an $r$ -tuple of partitions ${{{\\lambda }}}$ .", "We refer the reader to [37] for details, and for the connection to moduli spaces of rank $r$ sheaves on the affine plane." ], [ "Let us use the notation $F_u^\\rightarrow = {\\mathbb {Q}}(q_1,q_2)[p_1,p_2,\\dots ]$ for the Fock space (REF ).", "Definition 5 ([16], [18]) The following formulas determine an action ${\\mathcal {A}}\\curvearrowright F_u^\\rightarrow $ : $c_1 \\mapsto 1, \\qquad c_2 \\mapsto q^{\\frac{1}{2}},$ P0,-m multiplication by pm P0,m - m (q1m2 - q1-m2 ) (q2m2 - q2-m2 ) pm and $\\forall X \\in {\\mathcal {A}}^-$ with $\\Upsilon ^-(X) = R(z_1,\\dots ,z_n)$ (respectively $Y \\in {\\mathcal {A}}^+$ with $\\Upsilon ^+(Y) = R$ ): $X \\mapsto \\frac{(u q^{-\\frac{1}{2}})^n}{n!}", "\\int ^{|q_1|,|q_2| < 1}_{|z_1| = \\dots = |z_n| = 1} \\frac{R(z_1,\\dots ,z_n)}{\\prod _{1\\le i \\ne j \\le n} \\zeta \\left( \\frac{z_i}{z_j} \\right)} \\prod _{a=1}^n \\frac{dz_a}{2\\pi i z_a}$ $\\exp \\left[\\sum _{k=1}^\\infty \\frac{z_1^k+\\dots +z_n^k}{k} \\cdot q^{\\frac{k}{2}} p_k \\right] \\exp \\left[-\\sum _{k=1}^\\infty (z_1^{-k}+\\dots +z_n^{-k}) \\cdot q^{-\\frac{k}{2}} (1-q_1^k)(1-q_2^k) \\frac{\\partial }{\\partial p_k} \\right]$ $Y \\mapsto \\frac{(-u^{-1}q^{\\frac{1}{2}})^{n}}{n!}", "\\int ^{|q_1|,|q_2| > 1}_{|z_1| = \\dots = |z_n| = 1} \\frac{R(z_1,\\dots ,z_n)}{\\prod _{1\\le i \\ne j \\le n} \\zeta \\left( \\frac{z_i}{z_j} \\right)} \\prod _{a=1}^n \\frac{dz_a}{2\\pi i z_a}$ $\\exp \\left[-\\sum _{k=1}^\\infty \\frac{z_1^k+\\dots +z_n^k}{k} \\cdot p_k \\right] \\exp \\left[\\sum _{k=1}^\\infty (z_1^{-k}+\\dots +z_n^{-k}) (1-q_1^{-k})(1-q_2^{-k}) \\cdot \\frac{\\partial }{\\partial p_k} \\right]$ The integrals in (REF ) and (REF ) are contour integrals, and the parameters $q_1$ and $q_2$ are interpreted as complex numbers satisfying the conditions displayed in the superscripts of the integral signs.", "We have an isomorphism of vector spaces: $\\Psi : F_u^\\uparrow \\stackrel{\\sim }{\\longrightarrow }F_u^\\rightarrow $ obtained by sending $|\\lambda \\rangle $ to the modified Macdonald polynomial associated to the partition $\\lambda $ (see [25], [28]) for parameters $(q,t) \\leftrightarrow (q_1^{-1},q_2^{-1})$ .", "However, the isomorphism (REF ) also respects the ${\\mathcal {A}}$ actions, up to rotation by 90 degrees: $\\Psi (P_{n,m} \\cdot x) = q^{\\frac{m \\varepsilon _{n,m}}{2}}P_{-m,n} \\cdot \\Psi (x)$ for any $x \\in F_u^\\uparrow $ and $(n,m) \\in {\\mathbb {Z}}^2 \\backslash (0,0)$ , where $\\varepsilon _{n,m}$ is defined to be $-1$ if $(n,m)$ lies in the second or fourth quadrant (including the horizontal axis, but excluding the vertical axis) and 0 otherwise.", "To prove (REF ), note that the algebra ${\\mathcal {A}}$ is generated by $P_{n,m}$ with $(n,m) \\in (\\pm 1, 0), (0,\\pm 1)$ .", "Moreover: $P_{n,m} \\mapsto (c_1^n c_2^m)^{\\varepsilon _{n,m}} P_{-m,n}$ is an algebra automorphism, namely the particular case of (REF ) for: $\\gamma = \\begin{pmatrix}0 & -1 \\\\ 1 & 0 \\end{pmatrix} \\in SL_2({\\mathbb {Z}})$ Therefore, in order to prove (REF ) for a general lattice point $(n,m)$ , it suffices to prove it for the four special lattice points $(\\pm 1, 0), (0,\\pm 1)$ .", "All four of these statements are well-known facts in Macdonald polynomial theory.", "Remark 6 The gist of (REF ) and (REF ) is that $F_u^\\uparrow $ and $F_u^\\rightarrow $ can be perceived as the same module, up to the automorphism provided by the matrix (REF ).", "One could turn the problem on its head, and define for any $\\gamma \\in SL_2({\\mathbb {Z}})$ the module: $F_u^\\gamma = F_u^\\rightarrow $ but any element $a \\in {\\mathcal {A}}$ acts on the left-hand side just like $\\gamma (a) \\in {\\mathcal {A}}$ acts on the right-hand side, where $\\gamma (a)$ denotes the automorphism (REF ) for some lift of $\\gamma $ .", "Up to a simple isomorphism (namely conjugation by powers of the famous $\\nabla $ operator, see [9]), this module structure only depends on: $\\gamma \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} a \\\\ b \\end{pmatrix} \\in {\\mathbb {Z}}^2$ i.e.", "the choice of $b/a \\in {\\mathbb {Q}}\\sqcup \\infty $ .", "It is an interesting open problem to present the module structure on $F_u^\\gamma $ in such a way that any $X \\in {\\mathcal {A}}^\\pm $ acts by an explicit formula that essentially involves only the rational function $R = \\Upsilon ^\\pm (X)$ .", "This was provided in (REF )-(REF ) for the vertical slope and in (REF )-(REF ) for the horizontal slope.", "Such rotated modules for the quantum toroidal algebra play an important role in the study of refined link invariants, [26], [27]." ], [ "We will now compute the matrix coefficients of the universal$^*$ $R$ -matrix (REF ) in the two types of Fock spaces considered in the present paper.", "Recall that convex paths go over ${\\mathbb {Z}}_+^2$ , and we will use the term convex$^*$ path to refer to those whose edges do not point directly up (i.e.", "we restrict to $n_i > 0$ in (REF )).", "The size of such a path is the $x$ -coordinate of the lattice point where it terminates, i.e.", "the number $n_1+\\dots +n_t$ in (REF ).", "Any convex path $v$ can be obtained by concatenating a convex$^*$ path $v^*$ with a vertical path, hence we can uniquely write: $P_v = P_{v^*} P_{0,\\bar{n}}$ where $P_{0,\\bar{n}} = P_{0,n_1} \\dots P_{0,n_t}$ for any partition $\\bar{n} = (n_1 \\ge \\dots \\ge n_t)$ .", "Therefore: $\\boxed{{\\ddot{R}}= R^{\\prime } R^{\\prime \\prime }}$ where: R' = v convex* Pv P-vzv A+ A- Sop S R” = n partition P0,n P0,-nzn A0 A0 We will now compute the matrix coefficients of $R^{\\prime }$ and $R^{\\prime \\prime }$ in the module $F^\\uparrow _u$ .", "Take any two partitions $\\mu $ and $\\lambda $ , and regard them as a vector and covector: $\\langle \\lambda | \\in (F_u^\\uparrow )^\\vee \\qquad |\\mu \\rangle \\in F_u^\\uparrow $ Then we have (recall (REF ) for notation pertaining to matrix coefficients of $R^{\\prime }, R^{\\prime \\prime }$ in one of the two tensor factors): |R'| = v convex* Pvzv | P-v | |R”| = n partition P0,nzn | P0,-n | By formulas (REF ) and (REF ), the right-hand sides above are non-zero only if $\\mu \\subset \\lambda $ .", "Claim 8 The right-hand side of (REF ) is equal to: $\\delta _\\lambda ^\\mu \\cdot \\exp \\left[ \\sum _{k=1}^\\infty \\frac{P_{0,k}}{k} \\cdot q^{\\frac{k}{2}} \\left( \\sum _{{\\square }\\text{ o.c.", "of }\\mu } \\chi _{\\square }^{-k} - \\sum _{{\\square }\\text{ i.c.", "of }\\mu } \\chi _{\\square }^{-k} \\right) \\right]$ The claim above is a simple exercise, which we leave to the interested reader.", "Meanwhile, (REF ) implies that the right-hand side of (REF ) is equal to: $\\left[ \\sum _{v \\text{ convex}^* \\text{ of size }|{\\lambda \\backslash \\mu }|} \\frac{P_v}{z_v} \\cdot \\Upsilon ^-(P_{-v})({\\lambda \\backslash \\mu }) \\right] \\sigma ^{|{\\lambda \\backslash \\mu }|} \\prod _{{\\blacksquare }\\in {\\lambda \\backslash \\mu }} \\frac{\\prod _{{\\square }\\text{ o.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)}{\\prod _{{\\square }\\text{ i.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)} \\qquad \\quad $ We note that expression (REF ) is an infinite sum of $P_v$ 's, over convex$^*$ paths starting at 0 and ending at a point on the line $x = |{\\lambda \\backslash \\mu }|$ .", "Therefore, it can only act on graded modules where such $P_v$ 's act locally nilpotently, which will henceforth be called good modules (for example, $F_{u^{\\prime }}^\\rightarrow $ is good).", "Then let us consider the expression: $W(x) = \\sum _{k \\in {\\mathbb {Z}}} P_{1,k}x^k$ In any good module, the expression $W(x_1) \\dots W(x_n)$ makes sense when expanded in $|x_1| \\gg \\dots \\gg |x_n|$ .", "Moreover, it is clear from the shuffle algebra incarnation that: $W(x_1,\\dots , x_n) = W(x_1) \\dots W(x_n) \\prod _{1 \\le i < j \\le n} \\zeta \\left(\\frac{x_j}{x_i} \\right)$ is a symmetric expression in $x_1,\\dots ,x_n$ , at least formally.", "In good representations, this means that the expression (REF ) has the property that all its matrix coefficients are symmetric rational functions in $x_1,\\dots ,x_n$ .", "Remark 9 In the good representation $F_{u^{\\prime }}^\\rightarrow $ , the expression $W(x_1,\\dots ,x_n)$ acts by: $\\exp \\left[-\\sum _{k=1}^\\infty \\frac{x_1^{-k}+\\dots +x_n^{-k}}{k} p_k \\right] \\exp \\left[\\sum _{k=1}^\\infty (x_1^{k}+\\dots +x_n^{k}) (1-q_1^{-k})(1-q_2^{-k})\\frac{\\partial }{\\partial p_k} \\right]$ times $(-{u^{\\prime }}^{-1} q^{\\frac{1}{2}})^n$ .", "As a consequence of formula (7.7) of [40], we have the following formula for the pairing: $\\Big \\langle W(x_1,\\dots ,x_n), a \\Big \\rangle = \\Upsilon ^-(a)(x_1,\\dots ,x_n) \\qquad \\forall a \\in {\\mathcal {A}}^-$ Letting $a = P_{-v}$ for any convex$^*$ path $v$ , and recalling that such convex paths give rise to orthogonal bases (REF ), we obtain the following formula: $W(x_1, \\dots , x_n) = \\sum _{v \\text{ convex}^* \\text{ of size }n} \\frac{P_v}{z_v} \\cdot \\Upsilon ^-(P_{-v})(x_1,\\dots ,x_n)$ If we plug this formula in (REF ), then we conclude: Theorem 10 For any partitions $\\lambda , \\mu $ , we have the following identity in ${\\mathcal {A}}^+$ : $_{\\langle \\lambda |}R^{\\prime }_{|\\mu \\rangle } = \\underline{W(\\dots ,\\chi _{\\blacksquare },\\dots )_{{\\blacksquare }\\in {\\lambda \\backslash \\mu }}} \\cdot \\sigma ^{|{\\lambda \\backslash \\mu }|} \\prod _{{\\blacksquare }\\in {\\lambda \\backslash \\mu }} \\frac{\\prod _{{\\square }\\text{ o.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)}{\\prod _{{\\square }\\text{ i.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)}$ Meanwhile, $_{\\langle \\lambda |}R^{\\prime \\prime }_{|\\mu \\rangle }$ is given by (REF ).", "In the module $F_{u^{\\prime }}^\\rightarrow $ , the underlined term in (REF ) is precisely the normal ordered product (3.12) of [3], which plays a key role in the construction of the intertwiner: $F^\\uparrow _u \\otimes F^\\rightarrow _{u^{\\prime }} \\longrightarrow F^{\\nearrow }_{-uu^{\\prime }}$ where $F^{\\nearrow }_{-uu^{\\prime }}$ is the module (REF ) for: $\\gamma = \\begin{pmatrix} 1 & 0 \\\\ 1 & 1 \\end{pmatrix}$ This module was denoted by $F^{(1,1)}_{-uu^{\\prime }}$ in loc.", "cit.", "Remark 11 A similar formula to (REF ) holds if we replace the module $F_u^\\uparrow $ by the MacMahon module of [19], in which case the variables of the underlined term should go over the set of weights of boxes of a skew 3-dimensional partition ${{{\\lambda \\backslash \\mu }}}$ .", "The generalization is straightforward, but the product of factors in (REF ) is more involved in the MacMahon case, and so we leave the details as an exercise." ], [ "We will now consider the Fock space of Subsection REF , and identify it with: $F_u^\\rightarrow = {\\mathbb {Q}}(q_1,q_2)[p_1,p_2,\\dots ] \\stackrel{\\sim }{\\longrightarrow }{\\mathbb {Q}}(q_1,q_2)[x_1,x_2,\\dots ]^{\\textrm {Sym}} = \\Lambda $ via $p_n = x_1^n+x_2^n+\\dots $ .", "We note that a linear basis of $F_u^\\rightarrow \\cong \\Lambda $ is given by: $p_{\\bar{n}} = p_{n_1} \\dots p_{n_t}$ as $\\bar{n}= (n_1 \\ge \\dots \\ge n_t)$ goes over partitions.", "We regard elements of $F_u^\\rightarrow \\cong \\Lambda $ as functions $f[X]$ , where $X$ is shorthand for the variable set $x_1,x_2,\\dots $ .", "We adopt “plethystic notation\", according to which one defines, for any symbol $z$ : $f[X \\pm z] \\in \\Lambda [[z^{\\pm 1}]]$ to be the image of $f[X]$ under the ring homomorphism $\\Lambda \\rightarrow \\Lambda [[z^{\\pm 1}]]$ that sends: $p_n \\mapsto p_n \\pm z^n$ In other words, the way one computes (REF ) is to expand $f[X]$ in the basis (REF ), and then replace each $p_n$ therein according to (REF ).", "Morally, the plethysm (REF ) means “add/remove $z$ from the list of variables\", hence the notation.", "Claim 13 For any $f[X] \\in \\Lambda $ and any variables $z_1,\\dots ,z_n$ , we have: $\\exp \\left[\\sum _{k=1}^\\infty (z_1^{-k}+\\dots +z_n^{-k}) (1-q_1^{-k})(1-q_2^{-k})\\frac{\\partial }{\\partial p_k} \\right] \\cdot f[X] = \\\\ = f \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i^{-1} \\right] $ It is enough to prove the claim above for: $f[X] = \\prod _{i=1}^{\\infty } \\prod _{j=1}^t (1-x_i a_j) = \\exp \\left[- \\sum _{k=1}^\\infty \\frac{a_1^k + \\dots + a_t^k}{k} \\cdot p_k \\right]$ because the coefficients of such expressions in the variables $a_i$ provide a linear basis of $\\Lambda $ .", "The computation of (REF ) for $f$ as in the formula above is a straightforward exercise, which we leave to the interested reader.", "The following is also obvious.", "Claim 14 The operators $-\\frac{p_n}{n}$ and $(1-q_1^{-n})(1-q_2^{-n}) \\frac{\\partial }{\\partial _{p_n}}$ are adjoint with respect to the (modified) Macdonald inner product: $\\langle \\cdot , \\cdot \\rangle : \\Lambda \\otimes \\Lambda \\rightarrow {\\mathbb {Q}}(q_1,q_2)$ $\\Big \\langle p_{\\bar{n}}, p_{\\bar{n}^{\\prime }} \\Big \\rangle = \\delta _{\\bar{n}^{\\prime }}^{\\bar{n}} \\bar{n}!", "\\prod _{i=1}^t \\Big [ - n_i (1-q_1^{-n_i})(1-q_2^{-n_i}) \\Big ]$ We will now use the language above to compute the universal$^*$ $R$ -matrix in the module $F_u^\\rightarrow $ .", "Let us recall the decomposition (REF ).", "Then we have: $^{\\langle f|}{R^{\\prime \\prime }}^{|g \\rangle } = \\sum _{\\bar{n}\\text{ partition}} \\frac{P_{0,-\\bar{n}}}{z_{\\bar{n}}} \\cdot \\langle f , P_{0,\\bar{n}} g \\rangle $ According to Claim REF and formula (REF ), we obtain (recall (REF ) for notation pertaining to matrix coefficients of $R^{\\prime \\prime }$ in one of the two tensor factors): $^{\\langle f|}{R^{\\prime \\prime }}^{|g \\rangle } = \\sum _{\\bar{n}\\text{ partition}} \\frac{P_{0,-\\bar{n}}}{z_{\\bar{n}}} \\cdot q^{\\frac{|\\bar{n}|}{2}} \\langle p_{\\bar{n}} f , g \\rangle = \\left\\langle \\exp \\left[ \\sum _{n=1}^\\infty \\frac{P_{0,-n}q^{\\frac{n}{2}}}{n} \\cdot p_n \\right] \\cdot f, g \\right\\rangle \\qquad $ where $p_n$ acts on symmetric functions as multiplication by $p_n$ , while the symbols $P_{0,-n} \\in {\\mathcal {A}}^0$ are unaffected by their interaction with $f$ and $g$ , or the pairing.", "To compute the matrix coefficients of $R^{\\prime }$ , we need to understand the action: ${\\mathcal {A}}^+ \\cong {\\mathcal {S}}^{\\text{op}} \\curvearrowright F_u^\\rightarrow \\cong \\Lambda $ of (REF ) in the language of the present Subsection.", "Claims REF and REF imply: $\\langle f | Y | g \\rangle = \\frac{(-u^{-1}q^{\\frac{1}{2}})^{n}}{n!}", "\\int ^{|q_1|,|q_2| > 1}_{|z_1| = \\dots = |z_n| = 1} \\frac{\\Upsilon ^+(Y)(z_1,\\dots ,z_n)}{\\prod _{1\\le i \\ne j \\le n} \\zeta \\left( \\frac{z_i}{z_j} \\right)} \\prod _{a=1}^n \\frac{dz_a}{2\\pi i z_a}$ $\\left\\langle f \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i \\right], g \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i^{-1} \\right] \\right\\rangle $ for any $Y \\in {\\mathcal {A}}^+$ , $f = f[X], g = g[X] \\in \\Lambda $ , where the inner product is given by (REF ).", "Let us explain the gist of (REF ): for any two fixed symmetric polynomials $f$ and $g$ , the second line of (REF ) is a Laurent polynomial in the variables $z_1,\\dots ,z_n$ , which one then integrates against the rational function on the first line of (REF ).", "With this in mind, we may use the definition of $R^{\\prime }$ in (REF ) to compute: $^{\\langle f|}{R^{\\prime }}^{|g\\rangle } = \\sum _{n=0}^\\infty \\frac{(-u^{-1}q^{\\frac{1}{2}})^{n}}{n!}", "\\sum _{v \\text{ convex}^* \\text{ of size }n} \\frac{P_{-v}}{z_v} \\int ^{|q_1|,|q_2| > 1}_{|z_1| = \\dots = |z_n| = 1} \\frac{\\Upsilon ^+(P_v)(z_1,\\dots ,z_n)}{\\prod _{1\\le i \\ne j \\le n} \\zeta \\left( \\frac{z_i}{z_j} \\right)} \\prod _{a=1}^n \\frac{dz_a}{2\\pi i z_a}$ $\\left\\langle f \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i \\right], g \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i^{-1} \\right] \\right\\rangle $ To evaluate the sum above, we will use the following formula, which one can prove using the machinery of [36] (similarly with the way one proves (REF )).", "Proposition 15 Consider the following element of ${\\mathcal {A}}^- \\cong {\\mathcal {S}}$ , $\\forall d_1,\\dots ,d_n \\in {\\mathbb {Z}}$ : $S_{d_1,\\dots ,d_n} = \\emph {Sym}\\left[ z_1^{d_1} \\dots z_n^{d_n} \\right]$ This element admits the following decomposition in the basis (REF ): $S_{d_1,\\dots ,d_n} = \\sum _{v \\text{ convex}^*\\text{ of size }n} \\frac{P_{-v}}{z_v} \\int _{|y_1| = \\dots = |y_n| = 1}^{|q_1|,|q_2|>1} \\frac{\\Upsilon ^+(P_v)(y_1,\\dots ,y_n) y_1^{-d_1}\\dots y_n^{-d_n}}{\\prod _{1\\le i \\ne j \\le n} \\zeta \\left( \\frac{y_i}{y_j} \\right)} \\prod _{a=1}^n \\frac{dy_a}{2\\pi i y_a}$ Therefore, if we write: $S(w_1,\\dots ,w_n) = \\sum _{d_1,\\dots ,d_n \\in {\\mathbb {Z}}} S_{d_1,\\dots ,d_n} w_1^{d_1}\\dots w_n^{d_n}$ as a formal series of elements of ${\\mathcal {A}}^- \\cong {\\mathcal {S}}$ , then we conclude the following.", "Theorem 16 For any $f,g \\in \\Lambda $ , we have the following identity in ${\\mathcal {A}}^- \\cong {\\mathcal {S}}$ : $^{\\langle f|}{R^{\\prime }}^{|g\\rangle } = \\sum _{n=0}^\\infty \\frac{(-u^{-1}q^{\\frac{1}{2}})^{n}}{n!}", "\\int ^{|q_1|,|q_2| > 1}_{|z_1| = \\dots = |z_n| = 1} S(z_1,\\dots ,z_n) \\prod _{a=1}^n \\frac{dz_a}{2\\pi i z_a}$ $\\left\\langle f \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i \\right], g \\left[ X + \\left(1-q_1^{-1} \\right)\\left(1- q_2^{-1} \\right) \\sum _{i=1}^n z_i^{-1} \\right] \\right\\rangle $ Meanwhile, $^{\\langle f|}{R^{\\prime \\prime }}^{|g\\rangle }$ is given by (REF ).", "We emphasize the fact that for any fixed $f$ and $g$ , the $n$ -th summand of (REF ) is a finite linear combination of elements of ${\\mathcal {A}}^- \\cong {\\mathcal {S}}$ .", "To see this, we note that the second line of (REF ) is a Laurent polynomial in $z_1,\\dots ,z_n$ , hence only finitely many terms of the formal series $S(z_1,\\dots ,z_n)$ survive the contour integral.", "Remark 17 Formula (REF ) is reminiscent of [17], [22], where the weighted trace of ${\\ddot{R}}^{F^\\rightarrow _u}$ in the first tensor factor was computed (e.g.", "Proposition 3.3 of [17])." ], [ "In [18], the authors studied the object $^{\\langle 1|}{{\\ddot{R}}}^{|g\\rangle }$ for an arbitrary $g \\in \\Lambda $ , but instead of regarding it as an element of ${\\mathcal {A}}^- \\cong {\\mathcal {S}}$ , they regard it as an element of: ${\\mathcal {A}}= {\\mathcal {A}}^\\uparrow \\otimes {\\mathbb {Q}}(q_1,q_2)\\Big \\langle P_{n,0}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{n \\ne 0} \\otimes {\\mathcal {A}}^\\downarrow $ where ${\\mathcal {A}}^\\uparrow $ (resp.", "${\\mathcal {A}}^\\downarrow $ ) is the subalgebra generated by $P_{n,m}$ for all $n \\in {\\mathbb {Z}}$ and $m > 0$ (resp.", "$m<0$ ).", "Because of the automorphisms (REF ), these subalgebras are also isomorphic to the shuffle algebra and its opposite, so we have an isomorphism: ${\\mathcal {A}}\\cong {\\mathcal {S}}\\otimes {\\mathbb {Q}}(q_1,q_2)\\Big \\langle P_{n,0}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{n \\ne 0} \\otimes {\\mathcal {S}}^{\\text{op}}$ If we compose the inclusion ${\\mathcal {A}}^- \\subset {\\mathcal {A}}$ with (REF ), then we obtain: ${\\mathcal {A}}^- \\stackrel{\\iota }{\\hookrightarrow }{\\mathcal {S}}\\otimes {\\mathbb {Q}}(q_1,q_2)\\Big \\langle P_{n,0}, c_1^{\\pm 1}, c_2^{\\pm 1} \\Big \\rangle _{n \\ne 0} \\otimes {\\mathcal {S}}^{\\text{op}}$ As a consequence of (REF ), it is not hard to see that: $\\iota \\left( ^{\\langle 1|}{{\\ddot{R}}}^{|g\\rangle } \\right) \\in {\\mathcal {S}}\\otimes {\\mathbb {Q}}(q_1,q_2)\\left[ P_{n,0} \\right]_{n < 0}$ The element (REF ) is related to off-shell Bethe vectors in loc.", "cit.", "We do not have a closed formula for it, but we will now explain how formula (REF ) allows one to obtain an explicit sum over convex paths.", "Explicitly, for any $f$ and $g$ , we have: $^{\\langle f |}{\\ddot{R}}^{|g \\rangle } = \\sum _{v = \\left\\lbrace \\frac{m_1}{n_1} \\le \\dots \\le \\frac{m_t}{n_t} \\right\\rbrace , (n_i,m_i) \\in {\\mathbb {Z}}_+^2} \\frac{P_{-v}}{z_v} \\cdot \\langle f | P_{v} | g \\rangle $ Let $f = \\bar{\\lambda } := \\Psi ( |\\lambda \\rangle )$ and $g = \\bar{\\mu } := \\Psi ( |\\mu \\rangle )$ be the modified Macdonald polynomials associated to partitions $\\lambda $ and $\\mu $ .", "Then we can invoke (REF ) to obtain the following: $^{\\langle \\bar{\\lambda } |}{\\ddot{R}}^{|\\bar{\\mu } \\rangle } = \\sum _{v = \\left\\lbrace \\frac{m_1}{n_1} \\le \\dots \\le \\frac{m_t}{n_t} \\right\\rbrace , (n_i,m_i) \\in {\\mathbb {Z}}_+^2} \\frac{P_{-v}}{z_v} \\cdot \\langle \\lambda | P_{m_1,-n_1}\\dots P_{m_t,-n_t} |\\mu \\rangle q^{\\sum _i \\frac{n_i \\varepsilon _{m_i,-n_i}}{2}}$ where now the matrix coefficients are calculated in the representation $F_u^\\uparrow $ , i.e.", "according to formulas (REF )-(REF ).", "Note that all but finitely many convex paths have trivial matrix coefficient for any given $\\lambda $ and $\\mu $ , as can be seen from the fact that $P_{m_t,-n_t}|\\mu \\rangle $ is a linear combination over Young diagrams with $m_t$ boxes fewer than $\\mu $ .", "Akin to (REF ), we have the following factorization of the operator (REF ): Proposition 19 We have the following formula for the matrix coefficients of ${\\ddot{R}}$ in terms of the decomposition (REF ): $^{\\langle \\bar{\\lambda } |}{\\ddot{R}}^{|\\bar{\\mu } \\rangle } = \\sum _{\\text{Young diagrams }\\nu \\subset \\lambda , \\mu } A_{\\lambda ,\\nu } \\cdot D_\\nu \\cdot B_{\\nu , \\mu }$ where: A, = v = {m1n1 ...mtnt }, mi < 0, ni > 0 P-vzv | Pm1,-n1...Pmt,-nt | B, = v = {m1n1 ...mtnt },mi > 0, ni 0 P-vzv | Pm1,-n1...Pmt,-nt |q-i ni2 and $D_\\nu = \\sum _{\\bar{n} = (n_1 \\ge \\dots \\ge n_t)} \\frac{P_{-\\bar{n},0}}{z_{\\bar{n}}} \\langle \\nu | P_{0,-\\bar{n}} | \\nu \\rangle $ .", "By analogy with Claim REF , it is easy to see that: $D_\\nu = \\exp \\left[ \\sum _{k=1}^\\infty \\frac{P_{-k,0}}{k} \\cdot q^{\\frac{k}{2}} \\left( \\sum _{{\\square }\\text{ o.c.", "of }\\nu } \\chi _{\\square }^{-k} - \\sum _{{\\square }\\text{ i.c.", "of }\\nu } \\chi _{\\square }^{-k} \\right) \\right]$ which matches formula (4.8) of [18] for the operator $L_{\\emptyset , \\emptyset }$ in the terminology of loc.", "cit.", "As for the operators (REF ) and (REF ), formulas (REF ) and (REF ) imply that: $A_{\\lambda , \\nu } = \\sum _{\\left\\lbrace \\frac{m_1}{n_1} \\le \\dots \\le \\frac{m_t}{n_t} \\right\\rbrace , m_i < 0, n_i > 0} \\frac{P_{-n_1,-m_1}\\dots P_{-n_t,-m_t}}{z_v} \\cdot $ $\\Upsilon ^-(P_{m_1,-n_1} \\dots P_{m_t,-n_t}) (\\lambda \\backslash \\nu ) \\sigma ^{|{\\lambda \\backslash \\nu }|} \\prod _{{\\blacksquare }\\in {\\lambda \\backslash \\nu }} \\frac{\\prod _{{\\square }\\text{ o.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)}{\\prod _{{\\square }\\text{ i.c.", "of }\\lambda } \\left(1 - \\frac{\\chi _{\\square }}{\\chi _{\\blacksquare }} \\right)}$ $B_{\\nu , \\mu } = \\sum _{\\left\\lbrace \\frac{m_1}{n_1} \\le \\dots \\le \\frac{m_t}{n_t} \\right\\rbrace ,m_i > 0, n_i \\ge 0} \\frac{P_{-n_1,-m_1}\\dots P_{-n_t,-m_t}}{z_v} \\cdot $ $\\Upsilon ^+(P_{m_1,-n_1} \\dots P_{m_t,-n_t}) (\\mu \\backslash \\nu ) \\bar{\\sigma }^{|{\\mu \\backslash \\nu }|} \\prod _{{\\blacksquare }\\in {\\mu \\backslash \\nu }} \\frac{\\prod _{{\\square }\\text{ i.c.", "of }\\nu } \\left(1 - \\frac{\\chi _{\\square }}{q\\chi _{\\blacksquare }} \\right)}{\\prod _{{\\square }\\text{ o.c.", "of }\\nu } \\left(1 - \\frac{\\chi _{\\square }}{q\\chi _{\\blacksquare }} \\right)} q^{-\\sum _i \\frac{n_i}{2}}$ where $z_v$ denotes (REF ) for the path $v=\\lbrace (n_1,m_1),\\dots ,(n_t,m_t)\\rbrace $ .", "Then one can plug in formula (REF ) in order to explicitly compute the second lines of (REF ) and (REF )." ] ]

[ [ "Drift-preserving numerical integrators for stochastic Poisson systems" ], [ "Abstract We perform a numerical analysis of a class of randomly perturbed {H}amiltonian systems and {P}oisson systems.", "For the considered additive noise perturbation of such systems, we show the long time behavior of the energy and quadratic Casimirs for the exact solution.", "We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two.", "These properties are illustrated with numerical experiments." ], [ "Introduction", "Hamiltonian systems are widely used models in science and engineering.", "In the deterministic case, one main feature of such models is that the solution conserves exactly the Hamiltonian energy for all times.", "The design and study of energy-preserving numerical methods for such problems as attracted much attention in the recent years, see for instance [7], [8], [12], [17], [22], [23], [29], [30], [37], [38], [34], [39], [49] and references therein.", "For an additive white noise perturbation of such Hamiltonian systems, the energy is no longer constant along time, but grows in average linearly for the exact solution, which reveals non trivial to reproduce by numerical methods, see [9], [14], [21], [19], [28], [44], [43], [13], and extensions to the case of stochastic partial differential equations in [3], [4], [15], [18], [41].", "In this paper, we propose and analyze a drift-preserving scheme for stochastic Poisson systems subject to an additive noise.", "Such problems are a direct generalization of the stochastic differential equations (SDEs) studied recently in [13], as well as in all the above references, but the proposed numerical integrator is not a trivial generalization of the one given in [13].", "In Section  we propose a new numerical method that exactly satisfies a trace formula for linear growth for for all times of the expected value of the Hamiltonian energy and of the Casimir of the solution.", "Such long time behavior corresponds to the one of the exact solution of stochastic Poisson systems and can also be seen as a long time weak convergence estimate.", "For the sake of completeness, in Section , we prove mean-square and weak orders of convergence of the proposed numerical method under classical assumptions on the coefficients of the problem.", "Finally, Section  is devoted to numerical experiments illustrating the main properties of the new numerical method for stochastic Hamiltonian systems and Poisson systems." ], [ "Drift-preserving scheme for stochastic Poisson problem", "This section presents the problem, introduces the drift-preserving integrator and shows some of its main geometric properties." ], [ "Setting", "For a fixed dimension $d$ , let $W(t) \\in {\\mathbb {R}}^d$ denote a standard $d$ -dimensional Wiener process defined for $t>0$ on a probability space equipped with a filtration and fulfilling the usual assumptions.", "For a fixed dimension $m$ and a smooth potential $V\\colon {\\mathbb {R}}^m\\rightarrow {\\mathbb {R}}$ , we consider the separable Hamiltonian function of the form $H(p,q)=\\frac{1}{2}\\sum _{j=1}^mp_j^2+V(q).$ We next set $X(t)=(p(t),q(t))\\in {\\mathbb {R}}^m\\times {\\mathbb {R}}^m$ and consider the following stochastic Poisson system with additive noise $\\text{d}X(t)=B(X(t))\\nabla H(X(t))\\,\\text{d}t+\\begin{pmatrix}\\Sigma \\\\ 0\\end{pmatrix}\\,\\text{d}W(t).$ Here, $B(X)\\in {\\mathbb {R}}^{2m\\times 2m}$ is a smooth skew-symmetric matrix and $\\Sigma \\in {\\mathbb {R}}^{m\\times d}$ is a constant matrix.", "The initial value $X_0=(p_0,q_0)$ of the problem (REF ) is assumed to be either non-random or a random variable with bounded moments up to any order (and adapted to the filtration).", "For simplicity, we assume in the analysis of this paper that $(x,y) \\mapsto B(x)\\nabla H(y)$ is globally Lipschitz continuous on ${\\mathbb {R}}^{2m}\\times {\\mathbb {R}}^{2m}$ and that $H$ and $B$ are $C^7$ , resp.", "$C^6$ -functions with all partial derivatives with at most polynomial growth.", "This is to ensure existence and uniqueness of solutions to (REF ) for all times $t>0$ as well as bounded moments at any orders of such solutions.", "These regularity assumptions on the coefficients $B$ and $H$ will also be used to show strong and weak convergence of the proposed numerical scheme for (REF ).", "We observe that one could weaken these assumptions, but this is not the aim of the present work.", "The present setting covers, for instance, the following examples: simplified versions of the stochastic rigid bodies studied in [45], [47], the stochastic Hamiltonian systems considered in [13] by taking $B(X)=J=\\begin{pmatrix} 0& -Id_m\\\\ Id_m& 0\\end{pmatrix}$ the constant canonical symplectic matrix, for which the SDE (REF ) yields $\\text{d}p(t) = - \\nabla V(q(t))\\,\\text{d}t + \\Sigma \\,\\text{d}W(t), \\qquad \\text{d}q(t) = p(t)\\,\\text{d}t,$ the Hamiltonian considered in [9] (where the matrix $\\Sigma $ is diagonal), the linear stochastic oscillator from [44], and various stochastic Hamiltonian systems studied in [36], see also [35], or [42], [50], [27], [26].", "Remark 1 We emphasize that our analysis is not restricted to the above form of the Hamiltonian.", "Indeed, the results below as well as the proposed numerical scheme can be applied to the more general problem (no needed of partitioning the vector $X$ neither to have the separable Hamiltonian (REF )) $\\text{d}X(t)=B(X(t))\\nabla H(X(t))\\,\\text{d}t+\\begin{pmatrix}\\Sigma \\\\ 0\\end{pmatrix}\\,\\text{d}W(t),$ as long as the Hessian of the Hamiltonian has a nice structure.", "One could for instance consider a (linear in $p$ ) term of the form $\\tilde{V}(q)p$ or most importantly the case when the Hamiltonian is quadratic as in the example of a stochastic rigid body problem.", "See below for further details.", "Applying Itô's lemma to the function $H(X)$ on the solution process $X(t)$ of (REF ), one obtains $\\text{d}H(X(t))&=\\left(\\nabla H(X(t))^\\top B(X(t))\\nabla H(X(t))+\\frac{1}{2}\\operatorname{Tr}\\left( \\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}^\\top \\nabla ^2H\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\right) \\right)\\,\\text{d}t \\nonumber \\\\&\\quad +\\nabla H(X(t))^\\top \\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\,\\text{d}W(t).$ Using the skew-symmetry of the matrix $B(X)$ , we have $\\nabla H(X)^TB(X)\\nabla H(X)=0$ .", "Furthermore, using that the partial Hessian $\\nabla _{pp}^2H(X)=Id_m$ is a constant matrix, thanks to the separable form of the Hamiltonian (REF ), and rewriting the above equation in integral form and taking the expectation, one finally obtains the so-called trace formula for the energy of the stochastic Poisson SDE (REF ): ${\\mathbb {E}}\\left[H(X(t))\\right]={\\mathbb {E}}\\left[H(X_0)\\right]+\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)t.$ This shows that the expected energy of the exact solution of (REF ) grows linearly with time for all $t>0$ .", "Remark 2 Observe that the form of the noise term in equation (REF ) makes the term $\\operatorname{Tr}\\left(\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}^\\top \\nabla ^2H\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\right)=\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)$ in (REF ) independent of the stochastic process $X(t)$ .", "Hence one obtains the linear growth along time of the expected energy in (REF ).", "In general, this is not the case if one would consider a non-zero additive noise in all the component or a multiplicative noise in (REF ).", "Note however that the linear growth property of the expected energy is still valid if one considers a more general Hamiltonian function (REF ) with kinetic energy given by $\\frac{1}{2}p^\\top M^{-1}p$ , with a given invertible mass matrix $M$ .", "Our objective is to derive and analyze a new numerical scheme for (REF ) that possesses the same trace formula for the energy for all times." ], [ "Definition of the numerical scheme", "The numerical integrator studied in [13] cannot directly be applied to the stochastic Poisson system (REF ).", "Our idea is to combine a splitting scheme with one of the (deterministic) energy-preserving schemes from [17].", "Observe that a similar strategy was independently presented in [20] in the particular context of the Langevin equation with other aims than here.", "We thus propose the following time integrator for problem (REF ), which is shown in Theorem REF below to be a drift-preserving integrator for all times: $\\begin{split}Y_1&:=X_n+\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\left(W(t_n+\\frac{h}{2})-W(t_n)\\right),\\\\Y_2&:=Y_1+hB\\left(\\frac{Y_1+Y_2}{2}\\right)\\int _0^1\\nabla H(Y_1+\\theta (Y_2-Y_1))\\,\\text{d}\\theta ,\\\\X_{n+1}&=Y_2+\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\left(W(t_{n+1})-W(t_n+\\frac{h}{2})\\right).\\end{split}$ In the above formulas, we denote the step size of the drift-preserving scheme with $h>0$ and discrete times with $t_n=nh$ .", "Remark 3 (Numerical implementation) The middle step in equation (REF ) requires, in general, the solution to a nonlinear system of equations.", "Even in higher dimension, if the problem is not stiff, this can be solved by fixed point iterations rather than Newton iterations, which makes the computational complexity similar to that of an implicit Runge–Kutta scheme with two stages, see [17] or [25] for instance.", "Remark 4 (Further extensions) Let us observe that the (deterministic) energy-preserving scheme from [17] present in the term in the middle of (REF ) could be replaced by another (deterministic) energy-preserving scheme for (deterministic) Poisson systems, see for example: [8], [6], [48], [10] or a straightforward adaptation of the energy-preserving Runge–Kutta schemes for polynomial Hamiltonians in [11].", "Let us further remark that it is also possible to interchange the ordering in the splitting scheme by considering first half a step of the (deterministic) energy-preserving scheme, then a full step of the stochastic part, and finally again half a step of the (deterministic) energy-preserving scheme.", "Finally, let us add that one could add a damping term in the SDE (REF ) to compensate for the drift in the energy thus getting conservation of energy for such problems (either in average or a.s.).", "In this case, one would add the damping term in the first and last equations of the numerical scheme (REF ) in order to get a (stochastic) energy-preserving splitting scheme.", "An example of application is Langevin's equation, a widely studied model in the context of molecular dynamics.", "We do not pursue further this question.", "We now show the boundedness along time of all moments of the numerical solution given by (REF ).", "Lemma 5 Let $T>0$ .", "Apply the drift-preserving numerical scheme (REF ) to the Poisson system with additive noise (REF ) on the compact time interval $[0,T]$ .", "One then has the following bounds for the numerical moments: for all step sizes $h$ assumed small enough and all $m\\in {\\mathbb {N}}$ , ${\\mathbb {E}}[|X_n|^{2m}] \\le C_m,$ for all $t_n=nh \\le T$ , where $C_m$ is independent of $n$ and $h$ .", "To show boundedness of the moments of the numerical solution given by (REF ), we use [36], which states that it is sufficient to show the following estimates: $\\left|{\\mathbb {E}}\\left[X_{n+1}-X_n|X_n\\right]\\right|\\le C\\left(1 + |X_n|\\right)h\\quad \\text{and}\\quad \\left|X_{n+1}-X_n\\right|\\le M_n (1 + |X_n|)\\sqrt{h},$ where $C$ is independent of $h$ and $M_n$ is a random variable with moments of all orders bounded uniformly with respect to all $h$ small enough.", "Since the numerical scheme (REF ) is a splitting method, it is more convenient to apply [36] to the Markov chain $\\lbrace X_0,Y_1,Y_2,X_1,\\ldots \\rbrace $ instead of the Markov chain $\\lbrace X_0,X_1,\\ldots \\rbrace $ .", "This makes the verification of the above estimates immediate using the linear growth property of the coefficients of the SDE (REF ), a consequence of their Lipschitzness." ], [ "Exact drift preservation of energy", "We are now in position to prove the main feature of the proposed numerical method (REF ) which benefits from the very same trace formula for the energy as the one for the exact solution to the stochastic Poisson problem (REF ), hence the name drift-preserving integrator for this numerical scheme.", "Theorem 6 Consider the numerical scheme (REF ) applied to the Poisson system with additive noise (REF ).", "Then, for all time steps $h$ assumed small enough, the expected energy of the numerical solution satisfies the following trace formula ${\\mathbb {E}}\\left[H(X_n)\\right]={\\mathbb {E}}\\left[H(X_0)\\right]+\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)t_n$ for all discrete times $t_n=nh$ , where $n\\in \\mathbb {N}$ .", "The first step of the drift-preserving scheme can be rewritten as $Y_1=X_n+\\int _{t_n}^{t_n+\\frac{h}{2}}\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\,\\text{d}W(s)$ and an application of Itô's formula gives ${\\mathbb {E}}\\left[H(Y_1)\\right]={\\mathbb {E}}\\left[H(X_n)\\right]+\\frac{h}{4}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right).$ Since the second step of the drift-preserving scheme (REF ) is the deterministic energy-preserving scheme from [17], one then obtains ${\\mathbb {E}}\\left[H(Y_2)\\right]={\\mathbb {E}}\\left[H(Y_1)\\right].$ Finally, as in the beginning of the proof, the last step of the numerical integrator provides ${\\mathbb {E}}\\left[H(X_{n+1})\\right]&={\\mathbb {E}}\\left[H(Y_2)\\right]+\\frac{h}{4}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)={\\mathbb {E}}\\left[H(Y_1)\\right]+\\frac{h}{4}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)\\\\&={\\mathbb {E}}\\left[H(X_n)\\right]+\\frac{h}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right).$ The identity (REF ) then follows by induction on $n$ .", "A recursion now completes the proof." ], [ "Splitting methods with deterministic symplectic integrators and backward error analysis: linear case", "As symplectic integrators for deterministic Hamiltonian systems or Poisson integrators for deterministic Poisson systems have proven to be very successful [24], it may be tempting to use them in a splitting scheme for the SDE (REF ).", "One could for instance replace the (deterministic) energy-preserving scheme in the middle step of equation (REF ) by a symplectic or Poisson integrator, such as for instance the second order Störmer–Verlet method [25] which turns out to be explicit in the context of a separable Hamiltonian (REF ).", "Using a backward error analysis, see [40], [24], [32], or [5], one arrives at the following result in the case of a linear Hamiltonian system with additive noise (REF ) (i. e. for a quadratic potential $V$ ), where the proposed splitting scheme is drift-preserving for a modified Hamiltonian.", "Proposition 7 For a quadratic potential $V$ in (REF ), consider the numerical discretization of the Hamiltonian system with additive noise (REF ) (where $B(x)=J$ for ease of presentation) by the drift-preserving numerical scheme (REF ), where the energy-preserving scheme in the middle $Y_1 \\mapsto Y_2$ is replaced by a deterministic symplectic partitioned Runge–Kutta method of order $p$ .", "Then, there exists a modified Hamiltonian $\\widetilde{H}_h$ which is a quadratic perturbation of size $\\mathcal {O}(h^p)$ of the original Hamiltonian $H$ , such that the expected energy satisfies the following trace formula for all time steps $h$ assumed small enough, ${\\mathbb {E}}\\left[\\widetilde{H}_h(X_n)\\right]={\\mathbb {E}}\\left[\\widetilde{H}_h(X_0)\\right]+\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\widetilde{\\sigma }_h\\Sigma \\right)t_n,$ for all discrete times $t_n=nh$ , where $n\\in \\mathbb {N}$ , and $\\widetilde{\\sigma }_h=\\nabla ^2_{pp} \\widetilde{H}_h(x)$ is a constant matrix (independent of $x$ ).", "By backward error analysis and the theory of modified equations, see for instance [24], the symplectic Runge–Kutta method $Y_1\\mapsto Y_2$ solves exactly a modified Hamiltonian system with initial condition $Y_1$ and modified Hamiltonian $\\widetilde{H}_h(x)=H(x) + \\mathcal {O}(h^p)$ given by a formal series which turns out to be convergent in the linear case for all $h$ small enough (and with a quadratic modified Hamiltonian).", "Following the lines of the proof of Theorem REF applied with the modified Hamiltonian $\\widetilde{H}_h$ , and observing that the partial Hessian $\\nabla ^2_{pp} \\widetilde{H}_h(x)$ is a constant matrix independent of $x$ (as $\\widetilde{H}_h$ is quadratic), we deduce the estimate (REF ) for the averaged modified energy.", "Observe in (REF ) that the constant scalar $\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\widetilde{\\sigma }_h \\Sigma \\right)=\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)+\\mathcal {O}(h^p)$ is independent of $x$ and a perturbation of size $\\mathcal {O}(h^p)$ of the drift rate for the exact solution of the SDE in (REF ).", "Finally, note that an analogous result in the nonlinear setting (with nonquadratic potential $V$ in (REF )) does not seem straightforward due in particular to the non-boundedness of the moments of the numerical solution over long times and the fact that the modified Hamiltonian $\\widetilde{H}_h(p,q)$ is nonquadratic with respect to $p$ in general for a nonquadratic potential $V$ ." ], [ "Exact drift preservation of quadratic Casimir's", "We now consider the case where the ordinary differential equation (ODE) coming from (REF ), i. e. equation (REF ) with $\\Sigma =0$ , has a quadratic Casimir of the form $C(X)=\\frac{1}{2}X^\\top AX,$ with a symmetric constant matrix $A=\\begin{pmatrix}a & b\\\\b^\\top & c\\end{pmatrix}$ with $a,b,c\\in \\mathbb {R}^{m\\times m}$ .", "Recall that a function $C(X)$ is called a Casimir if $\\nabla C(X)^\\top B(X)=0$ for all $X$ .", "Along solutions to the ODE, we thus have $C(X(t))=\\text{Const}$ .", "This property is independent of the Hamiltonian $H(X)$ .", "In this situation, one can show a trace formula for the Casimir as well as a drift-preservation of this Casimir for the numerical integrator (REF ).", "Proposition 8 Consider the numerical discretization of the Poisson system with additive noise (REF ) with the Casimir $C(X)$ by the drift-preserving numerical scheme (REF ).", "Then, the exact solution to the SDE (REF ) has the following trace formula for the Casimir ${\\mathbb {E}}\\left[C(X(t))\\right]={\\mathbb {E}}\\left[C(X_0)\\right]+\\frac{a}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)t,$ for all times $t>0$ .", "the numerical solution (REF ) has the same trace formula for the Casimir, for all time steps $h$ assumed small enough, ${\\mathbb {E}}\\left[C(X_n)\\right]={\\mathbb {E}}\\left[C(X_0)\\right]+\\frac{a}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)t_n,$ for all discrete times $t_n=nh$ , where $n\\in \\mathbb {N}$ .", "The above results can be obtain directly by applying Itô's formula and using the property of the Casimir function $C(X)$ .", "Stochastic models with such a quadratic Casimir naturally appear for a simplified version of a stochastic rigid body motion of a spacecraft from [45] which has the quadratic Casimir $C(X)=\\left\\Vert X\\right\\Vert ^2_2$ or a reduced model for the rigid body in a solvent from [47].", "See also the numerical experiments in Section REF ." ], [ "Convergence analysis", "In this section, we study the mean-square and weak convergence of the drift-preserving scheme (REF ) on compact time intervals under the classical setting of globally Lipschitz continuous coefficients." ], [ "Mean-square convergence analysis", "In this subsection, we show the mean-square convergence of the drift-preserving scheme (REF ) on compact time intervals under the classical setting of Milstein's fundamental theorem [36].", "Theorem 9 Let $T>0$ .", "Consider the Poisson problem with additive noise (REF ) and the drift-preserving integrator (REF ).", "Then, for all time steps $h$ assumed small enough, the numerical scheme (REF ) converges with order 1 in the mean-square sense: $\\left({\\mathbb {E}}[\\left\\Vert X(t_n)-X_n\\right\\Vert ^2]\\right)^{1/2}\\le Ch,$ for all $t_n=nh\\le T$ , where the constant $C$ is independent of $h$ and $n$ .", "Denoting $f(x)=B(x)\\nabla H(x)$ , a Taylor expansion of $f$ in the exact solution of (REF ) gives $X(h)&=X_0+hf(X_0)+\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}W(h)+hf^{\\prime }(X_0)\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}\\int _0^h W(t)\\,{\\rm d}t+REST_1,$ where the term (denoting $f^{\\prime \\prime }$ the bilinear form for the second order derivative of $f$ ) $REST_1=f^{\\prime }(X_0)\\int _0^h\\int _0^tf(X(s))\\,\\text{d}s+ \\int _0^h\\int _0^1 (1-\\theta )f^{\\prime \\prime }(X_0+\\theta (X(t)-X_0) )\\left(X(t)-X_0,X(t)-X_0\\right){\\rm d}\\theta \\,{\\rm d}t$ is bounded in the mean and mean-square sense as follows: ${\\mathbb {E}}[REST_1]\\le Ch^2 \\quad \\text{and}\\quad {\\mathbb {E}}[\\left\\Vert REST_1\\right\\Vert ^2]^{1/2}\\le Ch^{2},$ where $C$ is a constant independent of $h$ , but that depends on $X_0=x$ with at most a polynomial growth.", "Performing a Taylor expansion of $f$ in the numerical solution (REF ) gives, after some straightforward computations, $X_1&=X_0+hf(X_0)+\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}W(h)+hf^{\\prime }(X_0)\\begin{pmatrix}\\Sigma \\\\0\\end{pmatrix}W\\left(\\frac{h}{2}\\right)+REST_2,$ where the term $REST_2$ analogously satisfies the bounds (REF ).", "The above computations result in the following local error estimates, ${\\mathbb {E}}[X(h)-X_1]=\\mathcal {O}(h^2),\\qquad {\\mathbb {E}}[\\left\\Vert X(h)-X_1\\right\\Vert ^2]^{1/2}=\\mathcal {O}(h^{3/2}),$ where the constants in $\\mathcal {O}$ depend on $X_0=x$ with at most a polynomial growth.", "An application of Milstein's fundamental theorem, see [36], finally shows that the scheme (REF ) converges with global order of convergence 1 in the mean-square sense, as consequence of the local error estimates (REF ) and Lemma REF .", "This concludes the proof." ], [ "Weak convergence analysis", "The proof of weak convergence of the drift-preserving scheme (REF ) on compact time intervals easily follows from [46], where convergence of the Strang splitting scheme for SDEs is shown.", "See also [31], [2] for related results.", "Theorem 10 Let $T>0$ .", "Consider the Poisson problem with additive noise (REF ) and the drift-preserving integrator (REF ).", "Then, there exists $h^*>0$ such that for all $0<h\\le h^*$ , the numerical scheme converges with order 2 in the weak sense: for all $\\Phi \\in C_P^6({\\mathbb {R}}^{2m},{\\mathbb {R}})$ , the space of $C^6$ functions with all derivatives up to order 6 with at most polynomial growth, one has $\\left|{\\mathbb {E}}[\\Phi (X(t_n))]-{\\mathbb {E}}[\\Phi (X_n)]\\right|\\le Ch^2,$ for all $t_n=nh\\le T$ , where the constant $C$ is independent of $h$ and $n$ ." ], [ "Numerical experiments", "In this section, we illustrate numerically the above analysis of the proposed drift-preserving scheme (REF ), denoted by DP below.", "Furthermore, we compare it with the well known integrators, in particular the Euler–Maruyama scheme (denoted by EM) and the backward Euler–Maruyama scheme (denoted by BEM).", "The first and second Hamiltonian test problems considered (linear stochastic oscillator and stochastic mathematical pendulum) use parameter values similar to those in [13].", "The third test problem is a stochastic rigid body model which is a Poisson system perturbed by white noise, but not a Hamiltonian system.", "For nonlinear problems, we use fixed-point iterations for the implementation of the schemes, but one could use Newton iterations as well, see Remark REF ." ], [ "The linear stochastic oscillator", "The linear stochastic oscillator has extensively been used as a test model since the seminal work [44].", "We thus first consider the SDE (REF ) with $B(X)=J$ the constant $2\\times 2$ Poisson matrix and the following Hamiltonian $H(p,q)=\\frac{1}{2}p^2+\\frac{1}{2}q^2.$ Furthermore, the initial values are $(p_0,q_0)=(0,1)$ and we consider a one dimensional noise with parameter $\\Sigma =1$ .", "For this problem, the integral present in the drift-preserving scheme (REF ) can be computed exactly, resulting in an explicit time integrator: $\\begin{split}Y_1&:=X_n+\\begin{pmatrix}\\left(W(t_n+\\frac{h}{2})-W(t_n)\\right)\\\\0\\end{pmatrix},\\\\Y_2&:=\\frac{1}{1+\\frac{h^2}{4}}\\begin{pmatrix}1-\\frac{h^2}{4} & -h\\\\ h & 1-\\frac{h^2}{4} \\end{pmatrix}Y_1,\\\\X_{n+1}&=Y_2+\\begin{pmatrix}\\left(W(t_{n+1})-W(t_n+\\frac{h}{2})\\right)\\\\0\\end{pmatrix}.\\end{split}$ This numerical scheme is different from the one proposed in [13].", "In Figure REF , we compute the expected values of the energy $H(p,q)$ for various numerical integrators.", "This is done using the step sizes $h=5/2^4$ , resp.", "$h=100/2^8$ , and the time intervals $[0,5]$ , resp.", "$[0,100]$ .", "For the numerical discretization of the linear stochastic oscillator, we choose the (backward) Euler–Maruyama schemes (EM and BEM), the drift-preserving scheme (DP), and also the stochastic trigonometric method from [14] (STM).", "For the considered problem, the stochastic trigonometric method (STM) also has an exact trace formula for the energy, see [14] for details.", "We approximate the values of the expected energies using averages over $M=10^6$ samples.", "Similarly to the stochastic trigonometric method (STM) from [14], one can observe the perfect long time behavior of the drift-preserving scheme with exact averaged energy drift along time, as stated in Theorem REF .", "In contrast, the left picture of Figure REF illustrates that the expected energy of the classical Euler–Maruyama scheme drifts exponentially with time, while the backward Euler–Maruyama scheme exhibits an inaccurately slow growth rate, as emphasized in [44].", "Figure: Linear stochastic oscillator: numerical trace formulas for𝔼[H(p(t),q(t))]{\\mathbb {E}}[H(p(t),q(t))] on the interval [0,5][0,5] (left) and [0,100][0,100] (right).Comparison of the Euler–Maruyama scheme (EM), the stochastic trigonometric method (STM), the drift-preserving scheme (DP), the backward Euler–Maruyama scheme (BEM), and the exact solution.In Figure REF , we illustrate numerically the strong rate of convergence of the drift-preserving scheme (REF ) and compare with the other schemes.", "To this aim, we discretize the linear stochastic oscillator on the time interval $[0,1]$ using step sizes ranging from $h=2^{-6}$ to $h=2^{-10}$ and we use as a reference solution the stochastic trigonometric method with small time step $h_{\\text{ref}}=2^{-12}$ .", "The expected values are approximated by computing averages over $M=10^6$ samples.", "One can observe the mean-square order 1 of convergence of the drift-preserving scheme (REF ) with lines of slope 1 in Figure REF , which corroborates Theorem REF .", "Figure: Linear stochastic oscillator: mean-square convergence rates for the backward Euler–Maruyama scheme (BEM), the Euler–Maruyama scheme (EM), the drift-preserving scheme (DP),and the stochastic trigonometric method (STM).", "Reference lines of slopes 1, resp.", "1/2.Next, Figure REF illustrates the weak convergence rate of the drift-preserving scheme (REF ).", "For simplicity, we only display the errors in the first and second moments since explicit formulas are available for these quantities.", "We take the noise scaling parameter $\\Sigma =0.1$ and step sizes ranging from $h=2^{-4}$ to $h=2^{-16}$ .", "The remaining parameters are the same as in the previous numerical experiment.", "The lines of slope 2 in Figure REF illustrates that the drift-preserving scheme has a weak order of convergence 2 in the first and second moments, as stated in Theorem REF .", "Figure: Linear stochastic oscillator: weak convergence ratesfor the backward Euler–Maruyama scheme (BEM), the Euler–Maruyama scheme (EM), the drift-preserving scheme (DP),and the stochastic trigonometric method (STM).", "Reference lines of slopes 1, resp.", "2.As symplectic integrators for deterministic Hamiltonian systems have proven to be very successful [24], it may be tempting to use them in a splitting scheme for the SDE (REF ).", "To study this, in Figure REF , we compare the behavior, with respect to the trace formula, of the drift-preserving scheme and of the symplectic splitting strategies discussed in Section REF .", "We use the classical Euler symplectic and Störmer–Verlet schemes for the deterministic Hamiltonian and integrate the noisy part exactly.", "These numerical integrators are denoted by SYMP, resp.", "ST below.", "As a comparison with non-geometric numerical integrators, we also use the classical Euler and Heun's schemes in place of a symplectic scheme.", "These numerical integrators are denoted by splitEULER and splitHEUN.", "We discretize the linear stochastic oscillator on the time interval $[0,100]$ with $2^7$ step sizes.", "It can be observed that the splitting method using the non-symplectic schemes splitEULER or splitHEUN behaves as poorly as standard explicit schemes for SDEs: we hence display in Figure REF only part of their numerical values due to their exponential growth.", "Although not having the exact growth rates, the two symplectic splitting schemes behave much better than the classical Euler–Maruyama scheme with a linear drift in the averaged energy with a perturbed rate, as predicted by Proposition REF .", "Figure: Linear stochastic oscillator: numerical trace formulas for 𝔼[H(p(t),q(t))]{\\mathbb {E}}[H(p(t),q(t))] on the interval [0,100][0,100].Comparison of the drift-preserving scheme (DP), the splitting methods with, respectively, the symplectic Euler method (SYMP), theStörmer–Verlet method(ST), the explicit Euler method (splitEULER), the Heun method (splitHEUN), and the exact solution." ], [ "The stochastic mathematical pendulum", "Let us next consider the nonlinear SDE (REF ) (with $B(X)=J$ the constant canonical Poisson matrix) with the Hamiltonian $H(p,q)=\\frac{1}{2}p^2-\\cos (q)$ and a noise in dimension one with parameter $\\Sigma =1$ .", "We take the initial values $(p_0,q_0)=(1,\\sqrt{2})$ .", "We again compare the behavior, with respect to the trace formula, of the DP, SYMP, ST and splitEULER schemes.", "To do this, we integrate numerically the stochastic mathematical pendulum on the time interval $[0,100]$ with $2^7$ step sizes.", "The results are presented in Figure REF .", "Again, we recover the fact that the drift-preserving scheme exhibits the exact averaged energy drift, as predicted in Theorem REF .", "Furthermore, one can still observe a good behavior of the symplectic strategies from Section REF analogously to the linear case in Section REF , although the analysis in Proposition REF is only valid for the linear case.", "Figure: Stochastic mathematical pendulum: numerical trace formulas for 𝔼[H(p(t),q(t))]{\\mathbb {E}}[H(p(t),q(t))] on the interval [0,100][0,100].Comparison of the drift-preserving scheme (DP), the splitting methods with, respectively, the symplectic Euler method (SYMP), theStörmer–Verlet method(ST), the explicit Euler method (splitEULER), and the exact solution." ], [ "Stochastic rigid body problem", "We now consider an Itô version of the stochastic rigid body problem studied in [33], [1], [16] for instance.", "This system has the following Hamiltonian $H(X)=\\frac{1}{2}\\left(X_1^2/I_1+X_2^2/I_2+X_3^2/I_3\\right),$ the quadratic Casimir $C(X)=\\frac{1}{2}\\left(X_1^2+X_2^2+X_3^2\\right),$ and the skew-symmetric matrix $B(X)=\\begin{pmatrix} 0 & -X_3 & X_2 \\\\ X_3 & 0 & -X_1 \\\\ -X_2 & X_1 &0\\end{pmatrix}.$ Here, we denote the angular momentum by $X=(X_1,X_2,X_3)^\\top $ and take the moments of inertia to be $I=(I_1,I_2,I_3)=(0.345, 0.653, 1)$ .", "The initial value for the SDE (REF ) is given by $X(0)=(0.8, 0.6, 0)$ and we consider a scalar noise $W(t)$ with $\\Sigma =0.25$ (acting on the first component $X_1$ only).", "Observe that, even if the Hamiltonian has not the desired structure (REF ), one still has a linear drift in the energy since the Hamiltonian is quadratic and thus the Hessian matrix present in the derivation of the trace formula has the correct structure as noted in Remark REF .", "In Figure REF , we compute the expected values of the energy $H(X)$ and the Casimir $C(X)$ using $N=2^5$ step sizes on the time interval $[0,4]$ (in order to see also the behavior of the Euler–Maruyama scheme) along various numerical solutions.", "The expected values are approximated by computing averages over $M=10^6$ samples.", "The exact long time behavior with respect to the energy and the Casimir averaged growths of the drift-preserving scheme can be observed in Figure REF , which corroborates Theorem REF and Proposition REF .", "As in the previous numerical experiment, one can also see that the growth rates of the Euler–Maruyama schemes are in contrast qualitatively wrong.", "Figure: Stochastic rigid body problem: numerical trace formulasfor the energy 𝔼[H(X(t))]{\\mathbb {E}}[H(X(t))] (left) and for the Casimir 𝔼[C(X(t))]{\\mathbb {E}}[C(X(t))] (right) for the drift-preserving scheme (DP),the Euler–Maruyama scheme (EM), the backward Euler–Maruyama scheme (BEM), and the exact solution.Similarly to the previous example, we numerically illustrate in Figure REF the strong convergence rate of the drift-preservation scheme (REF ) for the stochastic rigid body problem.", "To this aim, we discretize the problem on the time interval $[0,0.75]$ using step sizes ranging from $h=2^{-6}$ to $h=2^{-10}$ and compare with a reference solution obtained with scheme (REF ) with $h_{\\text{ref}}=2^{-12}$ .", "We compute averages over $M=10^5$ samples to approximate the expected values present in the mean-square errors.", "One observes again mean-square convergence of order 1 for the drift-preserving scheme.", "Figure: Stochastic rigid body problem: mean-square convergence ratesfor the backward Euler–Maruyama scheme (BEM),the drift-preserving scheme (DP),and the Euler–Maruyama scheme (EM).Reference lines of slopes 1, resp.", "1/2.Next, Figure REF illustrates the weak convergence rate of the drift-preserving scheme (REF ).", "We plot the weak errors in the first and second moments of the first component of the solutions using the parameters: $\\Sigma =0.1$ , $T=1$ , $M=2500$ samples, and step sizes ranging from $h=2^{-10}$ to $h=2^{-20}$ .", "The rest of the parameters are as in the previous numerical experiment.", "One can observe weak order 2 in the first and second moments for the drift-preserving scheme (up to Monte-Carlo errors), which confirms again the statement of Theorem REF .", "Figure: Stochastic rigid body problem: weak convergence rates in the first moment 𝔼[X 1 (t n )]{\\mathbb {E}}[X_1(t_n)] (left) and second moment 𝔼[X 1 (t n ) 2 ]{\\mathbb {E}}[X_1(t_n)^2] (right) for the drift-preserving scheme (DP),the Euler–Maruyama scheme (EM), and the backward Euler–Maruyama scheme (BEM).", "Reference lines of slopes 1, resp.", "2.Finally, in Figure REF , we take the same parameters as in the first experiment of this subsection but we consider a noise in dimension two with the matrix $\\Sigma =\\begin{pmatrix}0.25 & 0\\\\ 0 &0.25\\end{pmatrix}.$ We then compute the expected values of the energy $H(X)$ and the Casimir $C(X)$ using $N=2^6$ step sizes along various numerical solutions and display the trace formula for the energy ${\\mathbb {E}}\\left[H(X(t))\\right]={\\mathbb {E}}\\left[H(X_0)\\right]+\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\begin{pmatrix}1/I_1 & 0\\\\ 0 & 1/I_2\\end{pmatrix}\\Sigma \\right)t$ and the trace formula for the Casimir ${\\mathbb {E}}\\left[C(X(t))\\right]={\\mathbb {E}}\\left[C(X_0)\\right]+\\frac{1}{2}\\operatorname{Tr}\\left(\\Sigma ^\\top \\Sigma \\right)t.$ Figure: Stochastic rigid body problem with two dimensional noise: numerical trace formulas for the energy 𝔼[H(X(t))]{\\mathbb {E}}[H(X(t))] (left) and for the Casimir 𝔼[C(X(t))]{\\mathbb {E}}[C(X(t))] (right) for the Casimir 𝔼[C(X(t))]{\\mathbb {E}}[C(X(t))] (right) for the drift-preserving scheme (DP),the Euler–Maruyama scheme (EM), the backward Euler–Maruyama scheme (BEM), and the exact solution.Again, one can observe in Figure REF the excellent behavior of the drift-preserving scheme as stated in Theorem REF and Proposition REF ." ], [ "Acknowledgements", "We appreciate the referees' comments on an earlier version of the paper.", "The work of DC was supported by the Swedish Research Council (VR) (projects nr.", "$2018-04443$ ).", "The work of GV was partially supported by the Swiss National Science Foundation, grants No.", "200020_184614, No.", "200021_162404 and No.", "200020_178752.", "The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N, Umeå University and UPPMAX, Uppsala University." ] ]

[ [ "The role of fragment shapes in the simulations of asteroids as\n gravitational aggregates" ], [ "Abstract Remote measurements and in-situ observations confirm the idea that asteroids up to few hundreds of meters in size might be rubble piles.", "The dynamics of these objects can be studied using N-body simulations of gravitational aggregation.", "We investigate the role of particle shape in N-body simulations of gravitational aggregation.", "We study contact interaction mechanisms and the effects of parameters such as surface friction, particle size distribution and number of particles in the aggregate.", "We discuss the case of rubble pile reshaping under its own self-gravity, with no spin and no external force imposed.", "We implement the N-body gravitational aggregation problem with contact and collisions between particles of irregular, non-spherical shape.", "Contact interactions are modeled using a soft-contact method, considering the visco-elastic behavior of particles' surface.", "We perform numerical simulations to compare the behavior of spherical bodies with that of irregular angular bodies.", "The simulations are performed starting from aggregates in non-equilibrium state.", "We allow particles to settle through reshaping until they reach an equilibrium state.", "Preliminary tests are studied to investigate the quantitative and qualitative behavior of the granular media.", "The shape of particles plays a relevant role in the settling process of the rubble pile aggregate, affecting both transient dynamics and global properties of the aggregate at equilibrium.", "In the long term, particle shape dominates over simulation parameters such as surface friction, particle size distribution and number of particles in the aggregate.", "Spherical particles are not suitable to model accurately the physics of contact interactions between particles of N-body aggregation simulations.", "Irregular particles are required for a more realistic and accurate representation of the contact interaction mechanisms." ], [ "Introduction", "In the last few decades, remote measurements and in-situ observations have contributed to build evidence [57], [31] upon the idea that asteroids up to few hundreds of meters in size might be aggregates of loosely consolidated material, or `rubble piles' [15].", "Due to their properties, the dynamical evolution of these objects can be studied using numerical N-body simulations.", "This technique has been used successfully in the past to simulate a wide range of phenomena, including planetary ring dynamics [54], [66], [7], [42] and planetesimal dynamics [58].", "In the context of rubble-pile dynamics, N-body simulations have been crucial to provide insights on phenomena involving natural reshaping processes towards equilibrium shapes [56], [71], , spin evolution and rotational breakup [60], [62], [17], [5], , , towards the formation of binary systems .", "These give insights on the internal structure of such bodies, which to date remains largely unknown [13], [63], [65].", "The intrinsic granularity of N-body models makes them ideal candidates to investigate scenarios that involve the disruption of rubble-pile objects, and subsequent reaccumulation after catastrophic [47], [48], [46], [49] or non-catastrophic events [27], to investigate the formation of satellites [20], exposed internal structure [11], [10] and interactions with external objects, including tidal disruption events [3], [4], and low-speed collisions [40], [6].", "All these studies consider contact and collision interactions between particles of spherical shape.", "The use of spheres in N-body simulations is very beneficial from the computational point of view, especially when dealing with a high number of bodies, since it reduces dramatically the complexity of the collision detection and contact solver algorithm.", "However, the use of spheres might affect significantly the realism of the contact interactions [46] which are over-simplified by disregarding the geometrical effects due to the irregularity and angularity of real bodies [70], [55].", "This is confirmed by experiments in terrestrial applications of granular dynamics [53], [19].", "Few studies have been performed in the past involving non-spherical particles.", "[38] studied low-speed collisions between gravitational aggregates made of polyhedral particles and showed a substantial difference between the contact interaction mechanism between polyhedra, as compared to spheres.", "In particular, they highlighted how off-center collisions, which are only possible between non-spherical objects, produce significantly lower restitution at contact and consequently a higher aggregation probability after collision.", "In a follow-up study, [39] managed to obtain a substantial speed-up of their code, by employing the ODE (Open Dynamics Engine) physics engine.", "[52] use similar aggregates to investigate tidal disruption events, showing how the use of non-spherical particles produces a better estimate of the bulk density of the aggregate object compared to using spheres.", "Other studies make use of correction parameters applied to spherical objects to model rolling friction [67], of spheres.", "Compared to perfect spheres, these provide a more realistic representation of the rolling motion, and may be used to model dissipation phenomena occurring between particles.", "However, using a coefficient of rolling friction to model non-spherical shape is not realistic and cover only a limited range of phenomena due to the angularity of particle shape.", "In particular, rolling friction is dissipative only and always acts against rolling motion, whereas shape could also enhance rolling .", "For the same reason, it cannot be used to reproduce accurately interlocking between particles.", "Also, it does not provide any means to model the dissipation of energy due to off-center collisions [38], which are always towards-center when using spheres.", "As discussed, this would result in a substantially lower coefficient of restitution for the case of angular bodies.", "In this paper, we investigate the role of particle shape in the numerical N-body problem with gravity and contact/collision dynamics.", "We model particles as rigid bodies with six degrees of freedom, including translational and rotational motion.", "We compare and discuss the behavior of spheres against particles with randomly-generated irregular shapes.", "Additionally, we study the effects of physical parameters of granular media, such as particle size distribution (mono- vs poly-disperse) [70], [71], [9] and resolution in terms of number of bodies involved in the simulation.", "The goal is to provide insights, identify general rules and key parameters that play a relevant role in the setup of a numerical simulation of gravitational aggregation.", "We first introduce the case study and useful definitions in Section .", "The setup of numerical simulations performed is described in Section , including a brief introduction on the code used (Section REF ) and preliminary tests run (Section REF ).", "Section REF introduces the parameter space investigated, while Sections REF and REF discuss the numerical tools and routines used, respectively, to setup the simulations and retrieve the final results.", "Section  discusses the outcome of simulations and presents aggregated results.", "Conclusions are finally presented in Section ." ], [ "Statement of the problem", "The primary objective is to identify the role of particle shape in numerical simulations of gravitational aggregation.", "To this goal, we perform several numerical simulations, to explore a wide range of parameter values and dynamical scenarios.", "After running some preliminary tests, we perform a simulation campaign to reproduce the self-reshaping dynamics of a rubble pile object within a range of parameter values, as detailed in Section .", "At the beginning of each simulation, the particles are arranged to form an elongated ellipsoidal aggregate, which is initially in a non-equilibrium state.", "No external force or angular momentum are applied to the particles, which are subject to their mutual gravity only.", "The dynamics are propagated forward until the aggregate reach a stable equilibrium condition.", "More detail on the initial rubble pile model and its creation can be found in Section ." ], [ "Definitions", "For the sake of clarity and to have a unique reference, we provide here definitions of quantities and symbols in use throughout the paper.", "Inertial elongation $\\lambda $ : it is used as a measure of the elongation of the full aggregate.", "It is defined as the ratio between minimum and maximum moments of inertia of the aggregate ($\\lambda \\le 1$ ).", "The inertia tensor of the aggregate is computed taking into account for the actual inertia tensors of all particles, whether they be spheres or angular bodies.", "Porosity $\\phi $ : it is used as a measure of the void fraction inside the aggregate.", "It can be written in terms of the material density $\\rho $ and bulk density $\\rho _b$ of the aggregate as $\\phi =1-\\rho _b/\\rho $ .", "Its value depends on the definition of the surface to envelope the aggregate.", "We compute ranges of porosity, based on a minimum and a maximum volume surface, as discussed in Section REF .", "Packing index $\\psi $ : it is defined for each particle as the mean distance with its closest twelve neighbors (twelve is the number of bodies in contact in a dense spherical packing).", "Its value is nondimensional and normalized to the characteristic radius of the body.", "Distances are evaluated from center of mass to center of mass.", "Contact force $F_c$ : it is the resulting contact force acting on a single particle." ], [ "Numerical simulations", "This section highlights the main features of the N-body code used and introduces the rationale of the study, providing its assumptions and limiting boundaries." ], [ "N-body code", "The numerical model is based on the GRAINS N-body code [25], [24], a rework of the multi-physics engine Project Chrono .", "GRAINS handles the dynamics of particles with six degrees of freedom each (translation and rotation), as they interact through mutual gravity and contacts/collisions.", "Gravitational interactions are computed by means of either direct N$^2$ integration, or using a GPU-parallel implementation of the Barnes-Hut octree method [8], [12], [24].", "Contact and collision interactions are computed based on the actual shape model of the body, whether it be a sphere or an angular body.", "The shape of angular bodies is created numerically as the convex envelope of a cloud of randomly-generated points.", "All clouds of points share the same statistical properties of size and shape of their bounding domain: bodies have different shape but similar characteristic size and axis ratios.", "For an angular body, these are defined as the size and axis ratios of its bounding domain: in this case, the maximum size of the angular body is always smaller then (or equal to) its characteristic size.", "Collision detection is performed into two hierarchical steps: a broad phase, where close pairs are identified, and a narrow phase, where contact points between irregular shapes are found precisely using a GJK algorithm , [25].", "Collision detection takes advantage of a thread-parallel implementation based on the subdivision of the domain to speed up computations.", "The contact dynamics are resolved by means of a force-based soft-body DEM method [26].", "This approach is coherent with the features of the dynamical problem we study.", "In the past, studies of gravitational aggregation have been performed using hard-sphere methods [47], , [59] or non-smooth contact dynamics [25].", "Both methods consider impulsive collisions between rigid bodies [1], [36], [19] and are suitable to reproduce non-smooth dynamics.", "However, they are not accurate to reproduce smooth dynamical processes [28].", "For these problems, soft-contact methods [18] are best suited [61], [67], [69].", "Our problem studies the motion of particles within an aggregate, as it slowly settles under its own gravitational force, before reaching equilibrium.", "Due to the long-lasting contact interactions between particles, addressing this problem would require a soft-contact method to consider non-impulsive smooth collisions between bodies.", "Accordingly, we use a force-based DEM method, which takes into account for the visco-elastic behavior of the surface material at contact.", "Normal and tangential visco-elastic actions are exchanged between bodies at each contact point: the dynamics are modeled based on a two-way normal-tangent spring-dashpot Hertzian system.", "Friction is also modeled at each contact point, based on a Coulomb model.", "Overall, the contact interaction is set by selecting the normal/tangential stiffness and damping coefficients of the spring-dashpot systems ($K_n$ ,$K_t$ ,$G_n$ ,$G_t$ ), and the coefficients of static/dynamic friction ($\\eta _s$ ,$\\eta _d$ ).", "A more detailed description of the contact/collision methods used can be found in [26] and in the documentation of Project Chrono .", "The code has been extensively validated in the past, and proved its capability to address accurately problems of gravitational [25], [24] and granular dynamics [45], [44], including successful comparative studies with laboratory experiments [53].", "GRAINS provides double precision accuracy and ensure satisfactorily the conservation of energy and angular momentum through collisions [24].", "In the context of N-body simulations of non-spherical objects interacting through mutual gravity and contacts/collisions, attempts have been made in the past using customized integration schemes [38] or physics/video game engines [39], [52] to simulate hard-contact interactions between polyhedral bodies.", "However, in these papers approximations were made to the dynamics, so that angular momentum was not sufficiently well conserved to adequately reproduce gravitational dynamics.", "In this context, our work is the first to use a N-body code to study the dynamics of non-spherical bodies interacting through mutual gravity and contacts/collisions, providing a sufficient level of accuracy to reproduce accurately problems of gravitational dynamics." ], [ "Preliminary tests", "Before studying the full gravity/contact problem, we focus here on granular dynamics only, without considering mutual gravity between particles.", "As mentioned, the code is fully validated against typical granular dynamics benchmarks [45], [44] and laboratory experiments [53].", "We perform preliminary tests to complement such validation studies, with the goal to investigating the behavior of the granular media in term of angle of repose and angle of slide.", "In particular, we simulate the dynamics of granular terrain, as it settles dynamically (angle of repose test) and quasi-statically (angle of slide test) under a uniform gravity field.", "These will contribute to the discussion of the results of numerical simulations reported in Section .", "Figure: Angle of slide test: finding slope distribution of the surface.", "(a) Enveloping surface that encloses all terrain particles; (b) grid of surface points; (c) slopes computed between points in the grid.Figure: Angle of slide test: slope distribution examples.", "The higher the resolution of the model (more points in the histograms), the higher slopes can be attained.First, we simulate the dynamics of granular material as it piles up under uniform vertical gravity, as shown in Figure REF .", "This test gives information on the angle of repose of the granular media, i.e.", "the slope at which a landslide stops and the pile comes to rest.", "In our test, 3,000 particles fall from a height of 4 m on a planar surface.", "We use a polydisperse size distribution of particles, with average characteristic size (or diameter for the case of spheres) of 30 cm (see Section REF for details on the distribution function used).", "The coefficient of friction is set to $\\eta =0.6$ .", "Figure REF shows that angular particles reach a non-zero angle of repose, at a value of about 30 deg.", "This result is consistent with the coefficient of friction selected, since $\\arctan (\\eta )\\simeq 30$ deg.", "On the other hand, Figure REF shows that spheres are not able to pile up dynamically, showing a zero angle of repose.", "As shown in , spheres exhibit a non-zero angle of repose when initially arranged in a compact crystallized configuration, but are not able to attain such configuration dynamically.", "Finally, Figure REF shows the comparison between angle of repose attained by angular particles with no friction ($\\eta =0$ ) and the same case with $\\eta =0.6$ .", "This shows that, even with no surface friction, angular particles are able to attain a non-zero angle of repose, thanks to purely geometrical effects related to their shape (e.g.", "interlocking).", "In this simple example, the effects of particle shape are quantified to an angle of repose of about 20 deg, not far from the 30 deg obtained with $\\eta =0.6$ .", "The second test we performed is used to estimate the maximum angle of slide attainable by our granular media with angular particles.", "Unlike the angle of repose, which is attained dynamically, the angle of slide is attained statically, as the maximum slope that a granular material can withstand before landsliding occurs.", "To estimate this quantity, we simulate the slow dynamical modification of the morphology of a granular terrain.", "As shown in Figure REF , we simulate a granular terrain made of angular bodies, which lies initially on a planar floor.", "The terrain is settled under uniform vertical gravity until reaching equilibrium.", "The floor is made of several cubic blocks, which can be moved independently and whose interstitial distance is much lower than the characteristic size $l_p$ of the terrain particle.", "The side length $l_c$ of the cubic blocks is chosen such that $l_c>10 \\ l_p$ .", "After settling, each block (subscript $j$ ) moves downwards with a velocity $v_j=h_j/t_{\\text{sim}}$ , where $t_{\\text{sim}}$ is the simulation time (equal for all blocks) and $h_j$ is the final vertical displacement of block $j$ .", "The value for $h_j$ is chosen randomly, with a uniform distribution between 0 and $l_c$ , which is the maximum vertical displacement allowed.", "The downwards velocity of the blocks is very slow, to guarantee a quasi-static settling of the terrain during the whole simulation.", "In particular, the parameters of the simulation are chosen such that the value of $v_j$ is much lower than the free fall velocity of particles.", "In our simulations, we use 25,000 to 100,000 equally sized particles, and floors made of either 4-by-4 or 5-by-5 blocks.", "Inter-particle coefficient of friction is set to $\\eta =0.6$ for all simulations.", "Maximum computation times (for the case of 100,000 particle simulation, with a 24 thread-parallel architecture), are in the order of one week.", "Once settled to its final configuration (Figure REF shows the envelope of all particles at the end of the simulation), the morphology of the terrain is acquired by means of the following procedure.", "The upper surface is sampled, in order to acquire surface points within a user-defined grid (Figure REF ).", "The sampling grid is consistent with the size of the particle and typically each grid patch contains about 4 to 16 particles (2-by-2 up to 4-by-4).", "The highest particle point within each patch is the sampling point.", "The three-dimensional slopes on the surface are computed as the gradient between sampling points (Figure REF ).", "The procedure of sampling patches instead of taking directly vertices or positions of fragments, is implemented to filter out slopes related to the shape of the single particle, and to acquire the granular morphology of the terrain only.", "Figure REF shows examples of slope distributions obtained.", "The graph shows that higher slopes are attained in higher resolution models (more sampling points/particles in the simulation).", "For a lower number of particles and higher slopes, the measure points in the distribution histogram become too few and the uncertainty increases.", "The curves are mostly accurate in the range 0-30 degrees, where they match nicely the qualitative and quantitative behavior of measured slope distributions of small bodies [23].", "In a future investigation, we plan to increase the number of bodies in the simulation, to investigate the slope distribution curve beyond 30 deg, which is currently out of the scope of this work.", "Although not providing accurate information on the higher slope distribution, our simulations do provide meaningful information on the maximum slopes attainable by the granular medium, which are observed in a wide range between 35 and 65 deg.", "As expected, these are higher than the angle of repose , which is obtained dynamically." ], [ "Parameter space investigated", "The goal is to study the dynamical behavior of a rubble pile aggregate under its own gravity, as it evolves from a non-equilibrium initial state to a self-achieved equilibrium condition.", "To investigate the role of particle shape in the dynamical process of adjustment towards equilibrium, we performed an extensive simulation campaign, including several sets of simulations.", "In order to compare results of such simulation sets and to have comparable dynamics in terms of characteristic times, we use initial aggregates with the same bulk properties.", "For all cases, we use the same total mass, bulk density and overall shape.", "In particular, the initial aggregate is shaped as a prolate ellipsoid (a$>$ b=c), with b/a=c/a=0.4.", "More details on the initial aggregates used are provided in Section REF .", "To perform comparative analyses on simulation parameters, we investigate the effects of shape, size distribution, number of particles and coefficient of static/dynamic friction.", "More details are provided here.", "Shape: spheres vs angular particles.", "Angular particles are created as the convex envelope of 15-20 points created randomly using a uniform distribution within a cubic box.", "Their aspect ratio is therefore close to one.", "Such angular bodies have on average 10 vertices.", "Particle size distribution: mono vs polydisperse.", "When mentioned, monodisperse distribution refer to particles with same radius (spheres) or characteristic size (angular particles).", "In all polydisperse cases, the particle size distribution is set using the Zhang probability distribution function : $P(s)=e^{-\\frac{s-s_{\\text{m}}}{\\bar{s}-s_{\\text{m}}}} \\quad \\text{with } s\\ge s_{\\text{m}}$ In this case, the distribution function can be set by selecting the ratio between the average size $\\bar{s}$ and the minimum size $s_{\\text{m}}$ .", "In our simulations, we use a value of $\\bar{s}/s_{\\text{m}}=2.70$ , which corresponds to the so-called modified Zhang distribution [22].", "The Zhang distribution comes with the important assumption that a minimum particle size exists in the fragment population.", "This is very convenient when performing numerical simulations.", "Also, this assumption has proven consistent with the mechanics of fragmentation , and show good agreement with laboratory experiments , [29].", "These have proven the accuracy of the Zhang distribution to match size distribution after fragmentation processes , [22].", "Number of particles: ranging between 1,000 and 2,000.", "Coefficient of friction: ranging from 0 to 1.", "In our simulations, to reduce the amount of free parameters, the dynamic coefficient of friction $\\eta _d$ is always set equal to the static coefficient of friction $\\eta _s$ : for this reason, from now on, we use the term $\\eta $ to indicate both.", "Additional parameters include coefficients related to the surface/contact interactions.", "In our simulations, both normal and tangential stiffness coefficients are set to $K_n=K_t=2 \\times 10^5$  N/m, while the damping coefficients are set to $G_n=20$  Ns/m and $G_t=40$  Ns/m.", "Values chosen are commensurate to the case study, and are within the range of values used in previous works: as reported by [61], the $K_t/K_n$ ratio is typically between 2/3 and 1 for most materials [50].", "Moreover, [68] reports that “the contact dynamics are not very sensitive to the precise value of this ratio” and thus the choice of these values is not critical for our simulations.", "From the numerical point of view, the dynamics of the system are propagated forward in time using a symplectic semi-implicit Euler integrator [2], [43].", "The time step is selected consistently with characteristic times of the dynamics involved: gravitational dynamics, collision detection and contact dynamics.", "Gravitational dynamics is typically very slow and in our case its characteristic time can be estimated to be no lower than: $T_g\\simeq \\frac{1}{\\sqrt{G\\rho }}>10^3 \\text{ s}$ where $\\rho $ is the material density of particles and $G$ is the universal gravitational constant.", "Contact/collision dynamics is typically much faster.", "The time stepping must be fast enough to avoid missing any collision and to properly reproduce the visco-elastic behavior of the material at contact.", "Equation (REF ) provides an estimate of the characteristic time between two consecutive collisions $T_d$ considering the limiting case of grazing collisions between spherical particles of radius $R$  [61], with relative velocity $v$ and maximum overlapping allowed $\\delta $ .", "In our simulations the settling dynamics is very slow and the relative velocity $v$ between particles is always below 1 m/s.", "We choose this value ($v=1$ m/s) as a limiting value to estimate the shortest characteristic time $T_d$ in the system.", "The maximum overlapping allowed is chosen as $\\delta =R/100$ , which proved to be consistent with parameters in our simulation, as demonstrated by previous similar tests [24].", "Given the range of parameters in use, we can estimate characteristic times for collision detection and contact dynamics and derive a lower-bound constraint, respectively as [24]: $ T_d\\simeq \\frac{2\\sqrt{4R\\delta -\\delta ^2}}{v}> 10 \\text{ s} \\\\ T_k\\simeq \\frac{2\\pi }{\\omega _k}=2\\pi \\sqrt{\\frac{m_r}{k}}>10^2 \\text{ s}$ where $m_r$ is the reduced mass between the two least massive particles of the system (in our case the tightest constraint applies on simulations with polydisperse size distribution) and $k$ is the highest stiffness in the system (in our case is $K_n=K_t$ ).", "With the parameter values chosen for our case study, the tightest constraint on the time step is derived from collision detection characteristic time $T_d$ .", "Accordingly, time steps are chosen in the order of few seconds.", "The maximum simulation time is set to 250 h (real world time) for all simulations, in order to leave enough time for all bodies to settle after reaching an equilibrium state." ], [ "Creation of initial aggregate", "The initial aggregate is obtained as the result of a process of gravitational aggregation.", "The purpose of this is to have realistic rubble pile models with internal voids randomly distributed between irregular fragments.", "The gravitational aggregation process is simulated for a high number of bodies, created at random positions in a cubic physical domain, with zero velocity and spin rate, and therefore with no orbital angular momentum.", "After settling under self-gravity, the bodies reach a stable and nearly-spherical aggregate.", "Figure REF shows snapshots from the gravitational aggregation simulation (in this example to form a 5,000-body parent aggregate).", "The rubble pile used in our simulations is extracted from the parent aggregate using a ray-tracing algorithm, which helps identifying particles that are inside a given 3d mesh.", "We validated this shape extracting algorithm by testing it on complex and non-convex three-dimensional shapes.", "Figure REF shows a test performed using the shape of comet 67P/Churyumov-Gerasimenko (also referenced as 67P/C-G in the followings): in this case a 10,000-body rubble pile model of the comet is extracted from a 64,000-body parent aggregate of monodisperse angular bodies.", "Figure REF shows two ellipsoidal aggregates used in our simulations as initial rubble pile models, and their parent aggregates.", "In this example, a 1,000-body monodisperse rubble pile model (Figure REF ) is extracted from a 5,000-body parent aggregate (Figure REF ), and a 2,000-body polydisperse model (Figure REF ) is extracted from a 10,000-body parent aggregate (Figure REF ).", "The process of creating the initial aggregate is carried both for the case of spheres and angular particles.", "As already mentioned, the extracted aggregate is a prolate ellipsoid (a$>$ b=c), with b/a=c/a=0.4, for all cases." ], [ "Identification of final aggregate", "The identification of the final aggregate in terms of number of bodies and mass is straightforward.", "Conversely, the identification of its overall shape is arbitrary and relies on the definition of its external surface.", "This definition is non-unique, given the information available (shape and position of each particle in the aggregate).", "This arbitrary process affects the computation of global properties such as volume and porosity.", "To provide a proper estimate of global properties, we provide ranges of values for these quantities, referring to two limiting surfaces: a maximum and a minimum volume surface.", "The maximum volume surface is the convex envelope of the aggregate, which is unique, when considering all vertices of angular bodies or the external surface of spheres.", "The minimum volume surface is computed using an alpha-shape algorithm [21]: intuitively, the alpha-shape envelope is the surface created by a sphere of radius similar to the characteristic radius of particles, as it rolls over the cloud of points, whether they be the vertices of angular bodies or the external surface of spheres.", "To test the ability of the alpha-shape algorithm to identifying concave enveloping surfaces, we use it to find the minimum volume surface of comet 67P/C-G, starting from its rubble-pile model shown in Figure REF .", "Using this method, the surface of the comet is obtained very accurately, within the resolution given by the characteristic size of particles.", "Figure REF shows the minimum volume surface (black surface), together with the convex envelope of the comet (light gray surface).", "In this case, since 67P/C-G is non-convex, its convex envelope does not reproduce accurately its shape.", "However, in our simulations, all aggregates are ellipsoid (convex) and none of them ever reach a non-convex shape during (or at the end) of the simulation.", "Therefore, in our case, it makes sense to use the convex envelope to set an upper limit, as a reliable measure of the maximum volume surface the aggregate.", "Figures REF and REF show examples of, respectively, the minimum and maximum volume surfaces of an ellipsoidal aggregate.", "In this case the differences are limited to the roughness of the envelope's surface and appear to be minimal.", "Still, the change in volume (and therefore in bulk density or porosity, since the total mass is constant) can be significant.", "Figure: (a) Testing the alpha-shape algorithm using the shape of comet 67P/Churyumov-Gerasimenko: minimum volume shape vs convex envelope.", "Examples of (b) minimum volume shape and (c) convex envelope for the case of ellipsoidal aggregate." ], [ "Simulations of self-gravitational settling", "The results of the simulation campaign are discussed here.", "In particular, we highlight how the shape of the particles affects the properties of the aggregate during its dynamic evolution and after the transient phase, when reaching its final equilibrium state.", "As described in Section REF , the aggregate used in these simulations is extracted from a larger parent, which is the result of a gravitational aggregation process.", "This child aggregate (also referenced as initial aggregate in the followings) is elongated and thus not in equilibrium: its self-gravitational settling is studied here." ], [ "Time evolution of the aggregate", "First, we analyze the evolution of the aggregate as it reshapes towards an equilibrium condition under its own gravity.", "The overall shape of the aggregate can be monitored using its inertial elongation $\\lambda $ , as defined in Section .", "Figure REF shows the short- (2 hours) and long-term (250 hours, on a semi-logarithmic scale) time evolution of $\\lambda $ , for different levels of friction and using both angular bodies (Fig.", "REF and REF ) and spheres (Fig.", "REF and REF ).", "Results shown in Figure REF refer to aggregates with 1,000 monodisperse bodies only.", "Some considerations can be made based on long-term evolution graphs in Figure REF and REF : angular bodies contribute to reaching lower values of $\\lambda $ compared to spheres at any friction level; angular bodies contribute to reaching a steady equilibrium state earlier compared to spheres at any friction level; both simulation sets show a substantially different dynamical behavior between cases with friction versus case with no friction; both simulation sets show that, when friction is present, the value of $\\eta $ does not play a relevant role to assessing the final shape of the aggregate.", "Counter-intuitively, in the long-term there is no monotonic trend between $\\eta $ and $\\lambda $ of the aggregate at equilibrium.", "Most importantly, results in Figure REF indicate clearly that substantial differences exist in the evolution of the aggregate due to the angular/spherical shape of particles.", "The reason for this is purely geometrical and mainly due to two effects: the number of points of contact between bodies and the interlocking mechanism.", "In particular, each pair of spheres has only one point of contact, while two angular bodies can have multiple points of contacts, including edges and surfaces.", "Dissipation forces related to friction are exchanged at points of contact and having more points means more forces acting on the bodies: friction is acting on a different scale.", "The second effect is due to the geometrical interlocking between angular bodies, which hinder their motion and does not occur between spherical bodies.", "These are indeed the reason behind (REF ) and (REF ): an overall larger dissipation effect acts on angular bodies, which as a result, reach equilibrium before spheres and form more elongated aggregates.", "Considerations (REF ) and (REF ) are closely related to the role of friction in the simulation.", "To better investigate it, we focus on the initial phases of the reshaping process.", "Fig.", "REF and Fig.", "REF show the first few hours of simulation.", "These graphs show that, at the very beginning of the simulation, the trend between $\\eta $ and $\\lambda $ is monotonic both for spheres and for angular shapes, as intuition would suggest.", "However, on a longer time scale, the chaotic nature of contact interactions between bodies smooths these differences until the monotonic trend is lost.", "According to our results, and in agreement with results by [62], friction shall never be omitted when simulating a real-world scenario, since even a small dissipation produces a relevant difference in the dynamical outcome.", "On the other hand, the value of $\\eta $ does not appear to be relevant on a long-time scale, especially for the case of angular bodies.", "Our results show that the relevance of $\\eta $ depends on the time scale considered in the simulation.", "For a 2-hour simulation (short-term plots) there is a difference between having $\\eta =0$ , $\\eta =0.2$ and $\\eta \\ge 0.4$ (above this value we have very similar behaviors), while for a 250-hour simulation significant differences occur only between $\\eta =0$ and $\\eta \\ne 0$ .", "In principle, given sufficient simulation time, this simplifies greatly the setup of the numerical problem, since it allows to avoid expensive parametric studies to select the value of $\\eta $ .", "We performed additional simulations to investigate the effect of a non-uniform size distribution and number of particles.", "We use a modified Zhang distribution (see Section REF ) and we simulated scenarios with 1,000, 1,500 and 2,000 particles.", "As discussed, all simulations are performed under the same values of total mass $M$ and bulk density $\\rho _b$ of the aggregate.", "For the case of polydisperse and higher number of particle simulations, material density and particle size are adjusted to match $M$ and $\\rho _b$ .", "All additional simulations are performed using angular bodies and a coefficient of friction $\\eta =0.6$ .", "Figure REF shows the time evolution of the inertial elongation $\\lambda $ of the aggregate.", "Two effects are worth discussing.", "Our first considerations concern the settling time and time scales involved in the simulations.", "A meaningful metric we can use is the “knee” between the steep and mild slope of the curve.", "This is better visible in the short-term plots (Figures REF and REF ) and identifies the transition point from an initial phase of fast settling to a phase of slower settling.", "In the case of monodisperse distribution (Fig.", "REF ) the knee occurs earlier for higher number of bodies, suggesting that a higher number of bodies contribute to shorten the fast settling phase.", "This is due to the increased surface interactions, which are the main source of motion dissipation.", "As expected, a higher number of contact interaction reduces the settling time.", "On the other hand, the same does not apply (or applies to a much lower extent) in the polydisperse case (Fig.", "REF ), where the knee occurs nearly at the same time for all cases.", "However, in these simulations the knee occurs earlier compared to monodisperse cases, for any N. This is consistent with the increased number of contact points in polydisperse simulations compared to monodisperse ones: smaller particles fill within the interstices of larger bodies, thus increasing the overall particle surface area at contact within the aggregate.", "The second aspect to be discussed concern the elongation $\\lambda $ of the aggregate.", "In this case, the curves appear to be translated vertically.", "Also, they appear to converge to a single solution for higher N. At a careful analysis of the numbers, this appear to be motivated by the resolution of the aggregate.", "The term “resolution” here refers to the relative size between a single particle and the whole aggregate: the higher N, the better the resolution.", "In fact, when computing the shape of the aggregate, the effect of resolution is not negligible: for a 1000-body aggregate, the error due to resolution could be as high as 10-20% (size of a single particle vs size of the aggregate).", "These errors are consistent with the vertical shift in the curves.", "Consistently, polydisperse fragments converges to a solution for higher N compared to monodisperse.", "In fact, the polydisperse mixture contains particles of bigger size (and thus have lower resolution) compared to the monodisperse ones (whose particle size is the mean of polydisperse distribution).", "As observed for the case of inter-particle friction, these effects are clearly visible, but limited to the first few hours of the numerical simulation.", "After that, the problem becomes highly chaotic and the effects of single parameters are not identifiable unequivocally any longer.", "In fact, at equilibrium, none of the aforementioned relations exist, except for the case of polydisperse simulations, where an increase in the number of bodies can be still correlated monotonically to a higher motion dissipation, and thus a more elongated final shape.", "Monodisperse simulations show a behavior similar to what already observed when comparing spheres with angular particles: aggregates that settle faster at the beginning of the simulation, reach equilibrium earlier compared to slower settling aggregates, which in the long term form more rounded shapes." ], [ "Final aggregate at equilibrium", "After the transient phase, the aggregate reaches an equilibrium condition.", "We discuss here the properties and internal rubble-pile structure of the final aggregate.", "To provide quantitative means of comparison between simulations, we compute quantities defined in Section  and discuss the results both from a global perspective, i.e.", "by considering the aggregate as a whole, and from a local perspective, i.e.", "by tracking each body in the aggregate.", "From a global point of view, the results of simulations are used to constraint relevant quantities into ranges of values.", "This approach is motivated by the arbitrariness of the aggregate's surface definition, as discussed in Section REF .", "For each simulation and for each final aggregate, we compute the minimum and maximum volume aggregates.", "A global information of interest is the porosity $\\phi $ or bulk density of the aggregate.", "Since the mass of the aggregate does not change between minimum and maximum volume aggregates, $\\phi $ is minimum for minimum volume aggregates and maximum for maximum volume aggregates.", "Figure REF shows the estimate of porosity range for final aggregates made of spheres or angular bodies, for different levels of friction.", "In agreement with results on time evolution of the aggregate, friction plays a very minor role, while the shape of the particle affects significantly the result.", "As expected, the packing of angular bodies is more compact and efficient with respect to spheres and they occupy a higher fraction of total volume, thus leading to lower values of porosity ($\\phi _{\\text{ang}}\\simeq 0.29\\pm 0.02$ vs $\\phi _{\\text{sph}}\\simeq 0.33\\pm 0.02$ ).", "As local metrics, we use the packing index $\\psi $ associated to each particle and the resulting contact force $F_c$ acting on each particle.", "For both quantities, we look into their values and distribution within the aggregate.", "As defined in Section , $\\psi $ is computed for each body in the aggregate as the mean distance (from center to center of the bodies) between the body and its 12 closest neighbors.", "All distances are normalized to $R=D/2$ , where $D$ is the characteristic size of the angular body or the diameter of the sphere.", "Figure REF and REF show the packing distribution of the particles as function of their distance from the center of the aggregate for the case of, respectively, angular bodies and spheres.", "In both cases the distribution cloud has a plateau at lower distance from center and a steep ascent at higher distance.", "The plateau indicates a diffusely regular packing in the inner layers of the aggregates up to its external layers and surface, identified by the ascending part of the graph.", "Although sharing a similar behavior with spheres, angular bodies confirm to be more densely packed.", "Also, spheres appear to have a more regular packing in the inner layers, where all bodies are nearly at the condition of optimal packing ($\\sim 2$ ), while a more chaotic packing is observed on angular bodies due to their uneven and unequal shape.", "The same is confirmed in the histograms in Fig.", "REF and REF , where spheres show a larger discontinuity between the cluster around optimal packing conditions and the external layers.", "As in previous discussions, all graphs in Figure REF confirm that the value of friction coefficient $\\eta $ is not relevant to the local packing distribution of the aggregate.", "Similarly to the packing index, the contact force distribution among the particles can give useful insights on the internal structure of the aggregate.", "Contact forces are produced between bodies due to gravitational pulls acting on them.", "Ideally, for an aggregate at equilibrium, the net contact force on each internal particle equals the gravity force at the field point where the particle is.", "The aggregate is a distributed source of gravity and its internal gravity field can be approximated by the gravity field inside a solid shape.", "The gravity field inside a solid sphere follows the well-known relation $a_s(r)=\\frac{GM_s}{R_s^3}r$ where $G$ is the universal gravitational constant, $M_s$ is the total mass of the solid sphere, $R_s$ is its radius and $r$ is the distance of between the field point and the center of the sphere.", "The gravitational force acting on a body of mass $m$ located inside the solid sphere at distance $r$ is linearly dependent on its distance from the center of the sphere $F_s(r)=qr \\qquad \\text{with} \\qquad q=\\frac{GM_s m}{R_s^3}$ In our case, the final aggregates are not perfectly spherical but their elongation is small and Eq.", "(REF ) can be used as a comparison mean against force distribution inside the aggregate.", "In particular, the linear coefficient $q$ is computed for two sample spheres, representative of aggregates with angular and spherical particles.", "These consider the total mass of the aggregate $M=M_s$ , the mass of each body $m$ (we consider here cases with 1,000 monodisperse particles, where both $M$ and $m$ are equal for all simulations) and the mean radius of the aggregate $R_{\\text{agg}}=R_s$ .", "The mean radius of the aggregate $R_{\\text{agg}}$ is computed from its mean volume (between maximum and minimum volume aggregate).", "Since aggregates with angular particles are smaller in size, $q_{\\text{ang}}>q_{\\text{sph}}$ and forces inside them are expected to be higher.", "This is confirmed by results of simulations in Figure REF .", "Figure REF and REF show the distribution of forces inside the aggregate, as function of the distance between the body and the center of the aggregate.", "As expected, the distributions behave linearly with $r$ and are coherent with the gravity field inside the sample solid sphere (black line).", "Based on Fig.", "REF , the choice of particle shape appears to be very relevant to assessing the distribution of contact forces within the aggregate.", "In the case of spheres, the distribution of forces is extremely regular, suggesting regular internal packing, while angular bodies show a more chaotic distribution.", "The chaotic nature of interactions between angular bodies, due to a higher number of contact points and to the interlocking mechanism, causes very high forces on a number of particles in the inner layers of the aggregate.", "This does not happen in the regularly packed spheres, where highest forces are found on the external surface of the aggregate.", "High forces appear to be caused by the angularity of contacts involving vertices and edges and are not observed on the smoother spherical surfaces.", "A more detailed analysis would be beneficial to fully investigate this behavior: we highlight it here as an interesting point for a follow-up analysis, which is however out of the scope of the current work.", "The regularity/irregularity of force distributions are in agreement with results on packing index shown in Figure REF .", "Again, the value of $\\eta $ does not play a relevant role.", "No relevant differences are observed in the internal packing distribution when dealing with a higher number of particles.", "Instead, as expected, aggregates with polydisperse material have smaller porosity (5-15% smaller with respect to monodisperse aggregates).", "In this case, smaller particles fill the voids between bigger particles and help avoiding the formation of crystallized packing." ], [ "Considerations on the angle of friction", "When dealing with self-gravitating objects, the classical approach is to compare their shape to hydrostatic sequences of equilibrium.", "The theory of continuum shows how self-gravitating fluids follow minimum energy configurations (as Jacobi and MacLaurin sequences), which correlate their angular momentum or spin with their shape [14].", "Our simulations show equilibrium configurations of non-rotating aggregates.", "In the context of the continuum model, the equilibrium figure of a non-rotating fluid is a perfect sphere.", "However, in our simulations, the kinetic energy of the system is such that the granular material does not behave as a fluid, but rather as a granular solid.", "In fact, shapes of rubble piles can be very different from hydrostatic equilibrium ones [51], [30], [64] and no direct comparison is possible.", "In general, self-gravitating granular aggregates relax towards a more spherical shape but, even those formed by frictionless particles, never reach precisely the hydrostatic spherical shape , [60].", "This is due to mechanisms that hinder the motion of particles within the granular solid.", "To provide a better theoretical model, capable of dealing with this phenomenon, Holsapple has applied the continuum theory to non-fluid (solid) bodies.", "In this effort, he extended the range of configurations attainable by a self-gravitating object by introducing the effects of a non-zero angle of friction [32], , [34], .", "The angle of friction is a quantity not directly related to surface friction, but it is rather a measure of the motion-hindering effects occurring within the interior of the object.", "In a direct comparison with a granular system, it would include all such mechanisms related to contact interactions.", "As shown by [35], the range of admissible shapes of a self-gravitating ellipsoid is greatly enhanced by a non-zero angle of friction.", "For example, for a friction angle of 30 deg (which is rather common in soil-like material), a non-rotating ellipsoid can attain almost any shape, except for extremely elongated ones, when c/a is very close to zero [35].", "However, although this model is certainly more accurate than using simpler fluid equilibrium sequences, it still relies on major simplifications of the granular problem.", "In particular, the model implies that the equilibrium shape attained by an object depends only on its angular momentum/spin and friction angle.", "In principle, this is true for granular systems as well.", "However, the friction angle of granular media is strongly dependent on the kinetic energy of the particles: according to its energy level, the granular media can behave as a gas, fluid or solid, each having a very different friction angle.", "To demonstrate it, we provide a simple example, considering two different dynamical scenarios.", "The first scenario considers an initially dispersed cloud of particles, collapsing under self-gravity.", "In this case, spherical (or nearly spherical) shapes can be attained for the aggregate at equilibrium .", "The greater contribution to the aggregate's final shape is provided by the initial phase, where particles follow free gravitational dynamics, while contact interactions contribute marginally to it.", "In this first phase, the dispersed particles act as a granular gas and then transition to a granular fluid as soon as the contacts becomes more numerous.", "In these initial phases, the angle of friction is extremely low, and this allows to minimize the motion-hindering mechanisms that act against reaching hydrostatic equilibrium.", "This is what we obtain in our simulations of parent aggregate formation (Figure REF ), as shown in Section REF .", "Consistently, we form nearly spherical aggregates (with $\\lambda \\simeq 1$ ), as shown in Figures REF and REF .", "The second scenario considers the same particles, but arranged in a different initial configuration, such that simulations starts from an already formed aggregate.", "After settling, the aggregate reaches the form of a granular solid, with a much higher angle of friction compared to the gaseous and fluid phases.", "This makes a substantial difference, since particles are experiencing contact interactions and interlocking from the beginning.", "They are not free to move in the entire space but only to roll/slide over each other.", "In this case, the initial shape of the aggregate strongly biases the final shape achieved after the reshaping process, which is no longer close to spherical.", "So, unlike the fluid case and the Holsapple case, the granular problem does not admit a unique equilibrium solution.", "The shape at equilibrium still depends on its angular momentum and angle of friction of the granular media, but also on the complex contact history of the aggregate.", "In our case study, this refers mainly to the way the initial aggregate is created and how the numerical simulation is initiated.", "Since we are not studying the failure conditions of our aggregates, the approach of Holsapple cannot be used directly to infer the theoretical angle of friction of our ellipsoidal aggregates based on their axes ratio.", "However, this approach can provide useful information to constraint the angle of friction of our aggregate.", "Holsapple investigates the limiting failure conditions of objects, using either the Drucker-Prager or the Mohr-Coulomb failure criterion.", "These are based on the idea that failure is due to an exceeding shear stress.", "This limiting condition provides a lower limiting value for the angle of friction: regardless of the contact history of the aggregate, we can state that a certain shape (value of semiaxes of the ellipsoid) implies that the angle of friction can never be less than the value found with the continuum model by Holsapple.", "Should it be lower than that, the aggregate would have been further relaxed towards hydrostatic equilibrium.", "In particular, we refer here to the Drucker-Prager (DP) criterion, applied to ellipsoidal shapes, and we look for its limiting condition (we do not consider any cohesive term): $\\sqrt{J_2}+3sp=0$ where $J_2$ is the second invariant of the deviator stress tensor, $p$ is the hydrostatic pressure (or mean normal stress) and $s$ is a constant that can be related to the angle of friction $\\theta _f$ of the aggregate.", "Respectively, they can be expressed as: $&J_2=\\frac{1}{6} \\left[ (\\sigma _1-\\sigma _2)^2 + (\\sigma _1-\\sigma _3)^2 + (\\sigma _2-\\sigma _3)^2 \\right] \\\\&p=\\frac{1}{3}(\\sigma _1+\\sigma _2+\\sigma _3) \\\\ &s=\\frac{2 \\sin (\\theta _f)}{\\sqrt{3}(3-\\sin (\\theta _f))}$ where $(\\sigma _1,\\sigma _2,\\sigma _3)$ are the principal stresses.", "We remark that the expression for $s$ is not unique [16]: we use Eq.", "(REF ) to be consistent with results obtained in [33], [35].", "To have a meaningful means of comparison with the continuum model, we apply the DP criterion to an equivalent ellipsoid, built upon our final rubble-pile aggregate.", "The equivalent ellipsoid is defined as the one having the same principal moments of inertia of the rubble-pile aggregate.", "In the literature this is often referred to as the Dynamically Equivalent Equal Volume Ellipsoid (DEEVE).", "The use of an equivalent ellipsoid for our case study is justified by the fact that we use ellipsoidal aggregates: all of their equilibrium shapes can be well approximated using ellipsoids.", "To find the semi-axes of the equivalent ellipsoid, we first compute the inertia tensor and principal moments of inertia of the rubble pile aggregate.", "When performing this operation, we consider the full inertia tensor of each single particle.", "The semi-axes of the equivalent ellipsoid are then computed from principal inertia moments, as if they belonged to a homogeneous ellipsoid.", "The principal stresses of the ellipsoid can be expressed in terms of its gravity potential terms.", "The gravity potential $U$ of an ellipsoid can be written in a co-rotating frame (rigidly rotating with the body) as [14]: $U=\\pi \\rho G (- A_0 + A_x x^2 + A_y y^2 + A_z z^2)$ where $\\rho $ is the homogeneous density of the ellipsoid, $G$ is the universal gravitational constant and terms $(A_0,A_x,A_y,A_z)$ depend only on the semi-axes of the ellipsoid.", "We evaluate the stress tensor at the center of a non-rotating ellipsoid, which can be written as [35]: $\\mathbf {\\sigma }_c=-\\pi \\rho ^2 G\\begin{bmatrix}A_x & 0 & 0 \\\\ 0 & A_y & 0 \\\\ 0 & 0 & A_z\\end{bmatrix}\\begin{bmatrix}a^2 & 0 & 0 \\\\ 0 & b^2 & 0 \\\\ 0 & 0 & c^2\\end{bmatrix}$ All terms in Eq.", "(REF ) can be expressed in terms of the semi-axes of the ellipsoid, which we denote with $a>b>c$ .", "Finally, we can find the angle of friction $\\theta _f$ that satisfies Eq.", "(REF ), as a function of semi-axes only.", "Figure: Shape of equilibrium aggregates in terms of semi-axes of equivalent ellipsoid, with a>b>ca>b>c.", "Particle shape and cases with no surface friction η=0\\eta =0 are highlighted.Figure REF shows the shape of final aggregates at equilibrium.", "All of them result close to the b/a=c/a line, i.e.", "nearly prolate ellipsoids.", "This is clearly biased by the choice of the initial shape, which is a prolate ellipsoid (with b/a=c/a=0.4) for all simulations.", "As discussed this prevents us from drawing general conclusion based on their final shape only, but it allows to derive a lower limiting value for their angle of friction.", "If excluding cases with no friction (marked with $\\eta =0$ in the figure), the semi-axes are in the range [0.83-0.92] for aggregates with spherical particles and [0.77-0.84] for angular bodies.", "These correspond to minimum angles of friction of about 4 deg for the spheres.", "As expected, this value is consistent with the findings of [71], when one considers in their article the evolution of the ellipsoid having similar flattening and the lowest angular momentum.", "The interesting difference that we find is the friction angle for angular bodies, around 6 deg, which denote a modest but relevant (+50%) increase provided by the angularity of particles.", "An accurate evaluation of the angle of friction of the aggregates is out of the scope of this paper, which is rather focused on highlighting the role of surface parameters and particle shape in the dynamical reshaping process.", "Also, we recall here the results of preliminary tests discussed in Section REF , which are performed using the same surface parameters used in ellipsoid simulations.", "In particular, the angle of slope test provides an upper limit in terms of slope attainable by the granular medium, which are observed up to values of about 65 deg.", "In this context, we highlight that a more comprehensive treatment of the problem would consider many other degrees of freedom when modeling the rubble-pile aggregate.", "A very relevant one concerns the internal structure of these bodies.", "Elongated shapes might be sustained by inhomogeneous internal structures (or density distributions).", "To provide few examples, [41] speculated about the interior of the highly elongated asteroid Itokawa based on its global shape and spinning dynamics: according to them a strong inhomogeneity within Itokawa (two separate bodies with different density) could possibly explain their measures of spin dynamics.", "[37] also proposed an inhomogeneous model for Itokawa, to cope with its gravitational potential and surface properties.", "In this respect, our approach is clearly simplified and do not address the problem in its whole complexity (we make the bulk hypothesis that such bodies have a full rubble-pile structure with homogeneous density).", "However, our ultimate goal is not to reproduce the high fidelity internal structures, but rather to exploit the rubble-pile structure to reproduce the self-gravitational dynamics of these objects." ], [ "Conclusions", "In this paper we study the role of particle shape and surface/contact parameters in N-body simulations, considering the full gravity-contact problem.", "We discuss the results of test scenarios of granular dynamics to quantify and qualitatively characterize the angle of slide and angle of repose of a realistic granular media, made of angular particles.", "As a case study, we reproduce the natural reshaping process of an elongated ellipsoidal rubble-pile object by numerical simulations.", "We investigate the effects of simulation parameters, including particle shape, surface friction, particle size distribution and number of particles, and discuss their contribution to the overall angle of friction within the granular medium.", "We summarize here the main outcomes of the paper: Friction.", "The presence/absence of friction is very relevant, but not its value.", "This is supported by results in all simulations and test scenarios.", "Friction is observed at a different scale for the case of angular bodies compared to spheres.", "This is due to the increased number of contact points (many, compared to only two per sphere couple) and enhanced by the chaotic nature of interactions between angular bodies.", "The overall effect is that angular bodies contribute to reaching equilibrium earlier than spheres and, in the long time, the value of friction does not affect the final shape of the aggregate.", "Porosity.", "As expected, aggregates with spheres have higher porosity, while packing of angular bodies is more efficient in terms of volume occupied.", "This appears to be a purely geometrical feature of the granular medium and has direct consequences on the dynamics and evolution of rubble pile objects.", "However, based on this consideration only, it is hard to speculate about global properties of small celestial bodies and compare with their high porosity values, due to the lack of data about their internal structure.", "This is the missing ingredient we would need to formulate hypotheses of correlation between local (internal) geometrical structure and global properties.", "Final shape of aggregate.", "Aggregates with spheres are more rounded and closer to hydrostatic equilibrium shape.", "Also, their reach equilibrium later in time.", "This is because they move freely on the surface, due to a lower friction (less contact points) and to the absence of mechanisms to hinder their motion such as geometrical interlocking, which instead play a relevant role for the case of angular bodies.", "Aggregate internal structure.", "Spheres have a more regular packing as they form ordered crystal structures.", "Angular bodies have a more chaotic packing structure.", "This has consequences on the internal properties of the aggregate: a clear example is shown on its internal force distribution, which is very different in the two cases.", "While assessing the role of particles in N-body simulations, we remark that these results are valid for the specific scenario studied, where all particles have equal density within a homogeneous aggregate that slowly settles under its own gravity, and with no other external actions.", "As mentioned for the case of porosity, any direct generalization of these results to the case of small celestial bodies would be possible provided that we have a better understanding and data available on the internal structure of such bodies.", "This work is part of a basic research effort, aimed at investigating the dynamical behavior of gravitational aggregates.", "In this context, general results and considerations clearly emerge.", "Angular bodies are certainly more realistic than spheres since they are capable to simulate to a more complex extent the contact interaction mechanism.", "The results give a clear indication towards the use of angular bodies for a better understanding of the dynamics involving the internal structure and global properties of rubble piles asteroids.", "A further argument of support is provided by the analysis of short- versus long- term dynamics.", "We observe that, although clear trends may exists within the first few hours of simulation, most of them disappear in the long term.", "For instance, considering reshaping process, we observe that after two hours of simulation clear trends exist between the shape of the aggregate and simulation parameters: in this case a more elongated aggregate is produced by (i) higher surface friction between particles, (ii) higher number of fragments (higher resolution of the aggregate) or (iii) non-uniform size distribution of particles.", "As expected, these three phenomena contribute to hindering the motion of particles within the aggregate, either by a direct increase of surface friction (i), or by its indirect increase after enlarging the surface area at contact (ii,iii).", "However, in all cases, when looking at results at equilibrium (after 250 hours of simulations), these trends are no more observed.", "This is interpreted as an effect of the extremely chaotic dynamical environment, which on the long-term dominates over the effects of simulation parameters.", "The only effect that survives in the long term is that of particle shape, which is observed in all simulations.", "Our simulations indicate clearly that relevant differences exist when using angular bodies compared to using spheres.", "Despite the importance of spherical particles in disclosing the physics of granular materials, both in laboratory and numerical simulations, our results highlight that relevant differences exist when irregular fragments are adopted.", "The additional level of complexity that non-spherical shapes bring in numerical simulations, appears to be an unavoidable step to better reproduce real-world scenarios." ], [ "Acknowledgment", "This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 800060.", "Part of the research was carried out at Laboratoire Lagrange, Observatoire de la Côte d'Azur, acting as secondment institution of the Marie Skłodowska-Curie project GRAINS.", "Part of the research work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.", "F.F.", "would like to thank Anton Ermakov for the fruitful discussions that inspired the setup of the angle of slide test scenario.", "The authors would like to thank the anonymous reviewers for their comments and suggestions that helped to increase the quality of the paper." ] ]

[ [ "A case study on stochastic games on large graphs in mean field and\n sparse regimes" ], [ "Abstract We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph.", "We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph's normalized Laplacian matrix.", "This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail.", "In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense.", "Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices.", "Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence." ], [ "Introduction", "Mean field game (MFG) theory has enjoyed rapid development and widespread application since its introduction over a decade and a half ago in [23], [29].", "It provides a systematic framework for studying a broad class of stochastic dynamic games with many interacting players, in terms of limiting models featuring a continuum of players which are often more tractable.", "There are by now various rigorous results justifying the MFG approximation.", "On the one hand, the equilibria of $n$ -player games can be shown to converge to the MFG limit under suitable assumptions.", "On the other hand, a solution of the continuum model may be used to construct approximate equilibria for the $n$ -player model with the particularly desirable properties of being decentralized and symmetric, in the sense that each player applies an identical feedback control which ignores the states of all other players.", "We refer to the recent book [9] for a thorough account of MFG theory and its many applications.", "A key structural assumption of the MFG paradigm is that the players interact symmetrically, i.e., through an empirical measure which weights each player equally.", "In many natural situations, however, players do not view each other as exchangeable and instead interact directly only with certain subsets of players to which they are connected, e.g., via some form of a graph or network.", "This is the purview of the broad field of network games, and we refer to [24] for a representative overview of mostly static models.", "Our paper contributes to a very recent line of work bridging MFG theory and network games by studying $n$ -player stochastic dynamic games in which interactions are governed by a graph $G_n$ on $n$ vertices.", "(When $G_n$ is the complete graph we recover the traditional MFG setting.)", "Roughly speaking, the goal is to understand the robustness of the mean field approximation and, when it fails, a substitute.", "Somewhat more precisely, two central questions are: For what kinds of graph sequences $\\lbrace G_n\\rbrace $ is the usual MFG approximation still valid?", "What is the right limit model for a general sequence $\\lbrace G_n\\rbrace $ , and how well does it approximate the corresponding $n$ -player game?", "Little progress has been made so far toward a systematic understanding of these questions.", "The recent paper [13] addresses (1) when $G_n=G(n,p)$ is the Erdős-Rényi graph on $n$ vertices with fixed edge probability $p \\in (0,1)$ , showing that the usual MFG limit is still valid.", "More recently, [17] studies a linear-quadratic model very similar to ours, but only considering directed path or cycle graphs.", "In another direction, recent efforts on (2) have proposed continuum models based on graphons, which describe limit objects for general dense graph sequences [30].", "See recent work on graphon games for the static case [8], [36] or graphon MFGs in the dynamic case [5], [19], [40].", "The combination of network, dynamic, and game-theoretic effects is essential for many recent models of large economic and financial systems, and several recent studies have attacked specific models combining some of these features; see [6], [10], [16], [33] and references therein.", "Even without game-theoretic (strategic) features, incorporating network effects into large-scale dynamic models already presents many mathematical challenges, which very recent work has begun to address; see [2], [12], [32] and [14], [28], [34] for studies of dense and sparse graph regimes, respectively.", "Notably, prior work studied dense graph regimes, and most questions in the sparse regime (roughly defined as finite limiting neighborhood sizes) remain open, as was highlighted in particular in the recent paper [5].", "The purpose of our article is to give comprehensive answers to (1) and (2), in both dense and sparse regimes, in the setting of a specific yet rich linear-quadratic model, inspired by the systemic risk (flocking) model of [10].", "For a suitably dense graph sequence $\\lbrace G_n\\rbrace $ (meaning roughly that the degrees diverge as $n\\rightarrow \\infty $ ), we show (in Theorem REF ) that the classical construction of MFG theory is still valid: The MFG equilibrium gives rise to a sequence of decentralized and symmetric approximate Nash equilibria for the $n$ -player games.", "The dense regime includes the complete graph, the Erdős-Rényi graph $G_n=G(n,p_n)$ with $np_n \\rightarrow \\infty $ , and many others.", "Our findings in the dense case conform to an increasingly well-understood principle of statistical physics, that (static) interacting particle systems (e.g., the Ising model) on sufficiently dense graphs tend to behave like their mean field counterparts (e.g., the Curie-Weiss model); see [1] and references therein.", "The case of sparse graphs is more delicate, and the MFG approximation is no longer valid.", "Here we restrict our attention to (vertex) transitive graphs, which intuitively “look the same\" from the perspective of any vertex (see Definition REF ); transitive graphs have rich enough symmetry groups to make up for the lack of exchangeability.", "We compute the Markovian Nash equilibrium explicitly (in Theorem REF ), up to the solution of a one-dimensional ordinary differential equation (ODE) governed by the empirical eigenvalue distribution of the Laplacian of the graph (i.e., the rate matrix of the simple random walk).", "As a consequence, we show (in Theorem REF ) that for a given graph sequence $\\lbrace G_n\\rbrace $ , the limiting law can be computed for a typical player's state process, under the assumption that the empirical eigenvalue distributions of the Laplacian matrices of the graphs converge weakly.", "We also discuss (in Section REF ) similar and much simpler results for the corresponding cooperative problem, which we can explicitly solve for any graph (not necessarily transitive).", "The eigenvalue distribution of a graph Laplacian is reasonably tractable in many interesting cases.", "The dense graph case is precisely the case where the eigenvalue distribution converges weakly to a point mass.", "In the sparse case, the eigenvalue distribution converges to a non-degenerate limit, and we characterize the much different $n\\rightarrow \\infty $ behavior in terms of this limit.", "We do not have a complete answer to (2) in the sparse case, as it remains unclear how to identify the limiting dynamics intrinsically, without relying on $n\\rightarrow \\infty $ limits of $n$ -player models.", "In contrast, the MFG framework identifies an intrinsic continuum model, the solution of which agrees with the $n\\rightarrow \\infty $ limit of the $n$ -player equilibria.", "A crucial challenge in the sparse setting is that equilibrium controls are not local, even at the limit.", "Even though each player's cost function depends only on the player's neighbors, the equilibrium (feedback) control depends on the entire network, requiring each player to look beyond its nearest neighbors or even its neighbors' neighbors.", "That said, we prove a correlation decay estimate (Proposition REF ), which shows that the covariance of two players' equilibrium state processes decays to zero with graph distance between these players.", "Correlation decay is interesting in its own right, as it illustrates that asymptotic independence of players can arise both in a dense graph (because degrees are large) and a sparse graph (because typical vertices are very far apart).", "In addition, we use correlation decay crucially in proving the convergence of the empirical distribution of state processes in equilibrium to a non-random limit (propagation of chaos), for large graph sequences $\\lbrace G_n\\rbrace $ .", "The key mathematical difficulty in the paper lies in the semi-explicit solution of the $n$ -player game.", "As is standard for linear-quadratic $n$ -player games, we reduce the problem to solving a coupled system of $n$ matrix differential equations of Riccati type.", "Riccati equations of this form do not often admit explicit solutions, but assuming the graph is transitive gives us enough symmetry to work with to derive a solution." ], [ "Main results", "In this section we present all the main results of the paper, and we defer proofs to later sections.", "We first give the precise setup of the $n$ -player game (Section REF ).", "After describing the semi-explicit solution of the equilibrium for transitive graphs (Section REF ), we then consider the large-$n$ behavior (Sections REF and REF ), paying particular attention to the distinction between the sparse and dense regimes.", "Finally, we discuss the analogous cooperative game (Section REF )." ], [ "The model setup", "In this section we define a stochastic differential game associated to any finite graph $G = (V,E)$ .", "All graphs will be simple and undirected.", "We abuse notation at times by identifying $G$ with its vertex set, e.g., by writing $v \\in G$ instead of $v \\in V$ .", "Similarly, we write $|G|=|V|$ for the cardinality of the vertex set, and ${\\mathbb {R}}^G={\\mathbb {R}}^V$ for the space of vectors indexed by the vertices.", "Each vertex $v \\in V$ is identified with a player, and we associate to this player a state process on time horizon $T > 0$ with dynamics $dX^G_v(t) = \\alpha _v(t, \\mathbf {X}^G(t)) dt + \\sigma dW_v(t), \\quad t \\in [0,T], $ where $\\sigma > 0$ is given, $(W_v)_{v \\in V}$ are independent one-dimensional standard Brownian motions defined on a given filtered probability space $(\\Omega , {\\mathcal {F}},{\\mathbb {F}}, {\\mathbb {P}})$ , and $\\mathbf {X}^G = (X^G_v)_{v \\in V}$ is the vector of state processes.", "Players choose controls $\\alpha _v$ from the set of (full-information) Markovian controls ${\\mathcal {A}}_G$ , defined as the set of Borel-measurable functions $\\alpha : [0,T] \\times {\\mathbb {R}}^V \\rightarrow {\\mathbb {R}}$ such that $\\sup _{(t,\\mathbf {x}) \\in [0,T] \\times {\\mathbb {R}}^V} \\frac{|\\alpha (t,\\mathbf {x})|}{1 + |\\mathbf {x}|} < \\infty .$ For any $\\alpha _1,\\ldots ,\\alpha _n \\in {\\mathcal {A}}_G$ , the SDE system (REF ) has a unique strong solution by a result of Veretennikov [41] (see also [27]).", "The given initial states $\\mathbf {X}^G(0)=(X^G_v(0))_{v \\in V}$ are assumed non-random, and in many cases we will set them to zero for simplicity.", "Each player $v \\in V$ faces a quadratic cost function $J_v^G : {\\mathcal {A}}_G^V \\rightarrow {\\mathbb {R}}$ that depends on the state processes of her nearest neighbors.", "For a non-isolated vertex $v$ , we set $J_v^G((\\alpha _u)_{u \\in V}) := \\frac{1}{2}{\\mathbb {E}}\\left[ \\int _0^T |\\alpha _v(t, \\mathbf {X}^G(t))|^2dt + c\\left|X^G_v(T)-\\frac{1}{\\mathrm {deg}_G(v)} \\sum _{u \\sim v} X^G_u(T) \\right|^2 \\right], $ where $c > 0$ is a fixed constant, $\\mathrm {deg}_G(v)$ denotes the degree of vertex $v$ , and $u \\sim v$ means that $(u,v)$ is an edge in $G$ .", "For an isolated vertex $v$ (i.e., if $\\mathrm {deg}_G(v)=0$ ), we set $J_v^G((\\alpha _u)_{u \\in V}) := \\frac{1}{2}{\\mathbb {E}}\\left[ \\int _0^T |\\alpha _v(t, \\mathbf {X}^G(t))|^2dt + c \\left|X^G_v(T) \\right|^2 \\right].", "$ Remark 2.1 We gain little generality by allowing $G$ to be disconnected.", "Indeed, restricting attention to the connected components of $G$ yields decoupled games of the same form, which we can study separately.", "But when we discuss Erdős-Rényi and other random graphs, it is useful to fix a convention for how to handle isolated vertices.", "In comparison to the usual settings of mean field games (MFGs), the key feature here is that the players do not interact with each other equally, but rather each player interacts (directly) only with her nearest neighbors in the graph.", "The form of the cost function implies indeed that each player, in addition to minimizing a standard quadratic energy term, will try to be as close as possible to the average of her nearest neighbors at the final time.", "For this reason, we can think of this as a flocking model.", "The benchmark case to keep in mind is where $G$ is the complete graph on $n$ vertices, which corresponds to the usual MFG setup.", "The first goal is to find a Markovian Nash equilibrium for this game, formally defined as follows, along with some generalizations.", "We write ${\\mathbb {R}}_+=[0,\\infty )$ throughout the paper.", "Definition 2.2 For a graph $G$ on vertex set $V=\\lbrace 1,\\ldots ,n\\rbrace $ and a vector $\\mathbf {\\epsilon }=(\\epsilon _i)_{i=1}^n \\in {\\mathbb {R}}_+^n$ , we say that a vector $\\mathbf {\\alpha }^* = (\\alpha ^{*}_i)_{i=1}^n \\in {\\mathcal {A}}_G^n$ of admissible strategies is a (Markovian) $\\mathbf {\\epsilon }$ -Nash equilibrium on $G$ if $J_i^G(\\mathbf {\\alpha }^*) \\le \\inf _{\\alpha \\in {\\mathcal {A}}_G}J_i^G(\\alpha ^{*}_1, ..., \\alpha ^{*}_{i-1}, \\alpha , \\alpha ^{*}_{i+1}, ..., \\alpha ^{*}_n) + \\epsilon _i, \\quad \\forall \\, i=1,\\ldots ,n.$ The corresponding equilibrium state process $\\mathbf {X}^G=(X^G_i)_{i =1}^n$ is the solution of the SDE $dX^G_i(t) = \\alpha ^*_i(t,\\mathbf {X}^G(t))dt + \\sigma dW_i(t).$ When the graph is understood from context, we may omit the qualifier “on $G$ .\"", "When $\\epsilon _1=\\cdots =\\epsilon _n=\\epsilon $ for some $\\epsilon \\ge 0$ , we refer to $\\mathbf {\\alpha }^*$ as a $\\epsilon $ -Nash equilibrium instead of a $(\\epsilon ,\\ldots ,\\epsilon )$ -Nash equilibrium.", "Naturally, a 0-Nash equilibrium is simply called a Nash equilibrium.", "The notion of $\\epsilon $ -Nash equilibrium for $\\epsilon \\ge 0$ is standard and means that no player can reduce her cost by more than $\\epsilon $ by a unilateral change in control.", "The more general notion of $\\mathbf {\\epsilon }=(\\epsilon _i)_{i=1}^n$ -Nash equilibrium stated here is less standard, and it simply means that different players may stand to improve their costs by different amounts.", "Of course, a $\\mathbf {\\epsilon }=(\\epsilon _i)_{i=1}^n$ -Nash equilibrium is also a $\\delta $ -Nash equilibrium for $\\delta =\\max _{i=1}^n\\epsilon _i$ .", "But this distinction will be useful in asymptotic statements (as in the discussion after Theorem REF ), because the statement $\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{i=1}^n\\epsilon ^n_i =0$ is of course much weaker than $\\lim _{n\\rightarrow \\infty }\\max _{i=1}^n\\epsilon ^n_i =0$ for triangular arrays $\\lbrace \\epsilon ^n_i : 1 \\le i \\le n\\rbrace \\subset {\\mathbb {R}}_+$ ." ], [ "The equilibrium", "To solve the game described in Section REF , we impose a symmetry assumption on the underlying graph.", "Let ${\\mathrm {Aut}}(G)$ denote the set of automorphisms of the graph $G=(V,E)$ , i.e., bijections $\\varphi : V \\rightarrow V$ such that $(u,v) \\in E$ if and only if $(\\varphi (u),\\varphi (v)) \\in E$ .", "One should think of an automorphism as simply a relabeling of the graph.", "Definition 2.3 We say $G$ is (vertex) transitive if for every $u,v \\in V$ there exists $\\varphi \\in {\\mathrm {Aut}}(G)$ such that $\\varphi (u)=v$ .", "Essentially, a transitive graph “looks the same\" from the perspective of each vertex.", "Importantly, the game we are studying is clearly invariant under actions of ${\\mathrm {Aut}}(G)$ , in the sense that the equilibrium state process (if unique) should satisfy $(X^G_v)_{v \\in V} \\stackrel{d}{=} (X^G_{\\varphi (v)})_{v \\in V}$ for each $\\varphi \\in {\\mathrm {Aut}}(G)$ .", "In the MFG setting, i.e., when $G$ is the complete graph, ${\\mathrm {Aut}}(G)$ is the set of all permutations of the vertex set, and the ${\\mathrm {Aut}}(G)$ -invariance of the random vector $\\mathbf {X}^G$ is better known as exchangeability.", "For a general graph, ${\\mathrm {Aut}}(G)$ is merely a subgroup of the full permutation group, and we lose exchangeability.", "While the transitivity of $G$ is a strong assumption, it is not surprising that a sufficiently rich group of symmetries would help us maintain some semblance of the tractability of MFG theory which stems from exchangeability.", "Transitivity, in particular, ensures that we still have $X^G_v \\stackrel{d}{=} X^G_u$ in equilibrium, for each $v,u \\in V$ .", "We need the following notation.", "Let $A_G$ denote the adjacency matrix of a graph $G$ on $n$ verties, and let $D_G = \\mbox{diag}(\\mathrm {deg}_G(1), ..., \\mathrm {deg}_G(n))$ be the diagonal matrix of the degrees.", "If $G$ has no isolated vertices (i.e., all degrees are nonzero), we define the LaplacianIn the literature, there are several different matrices derived from a graph which go by the name Laplacian.", "Our matrix $L_G$ is sometimes called the random walk normalized Laplacian (or the negative thereof).", "by $L_G := D_G^{-1}A_G - I,$ where $I$ is the identity matrix.", "It is easy to see that a transitive graph $G$ is always regular, meaning each vertex has the same degree, which we denote $\\delta (G)$ .", "The Laplacian matrix then becomes $L_G = \\tfrac{1}{\\delta (G)}A_G-I$ , which is notably a symmetric matrix.", "Remark 2.4 Throughout the paper, we will make frequent use of the fact that $L_G$ has real eigenvalues, all of which are between $-2$ and 0.", "Indeed, note that $L_G = D_G^{-1/2}\\widetilde{L}_G D_G^{1/2}$ where $\\widetilde{L}_G=D_G^{-1/2}A_GD_G^{-1/2} - I$ is the symmetric normalized Laplacian, and thus the eigenvalues of $L_G$ and $\\widetilde{L}_G$ are the same; the properties of $\\widetilde{L}_G$ are summarized in [11].", "Note that the all-ones vector is an eigenvector of $L_G$ with eigenvalue 0.", "Our first main result is the following: Theorem 2.5 (Characterization of equilibrium on transitive graphs) Suppose $G$ is a finite transitive graph on $n$ vertices without isolated vertices.", "Define $Q_G : {\\mathbb {R}}_+ \\rightarrow {\\mathbb {R}}_+$ by $Q_G(x) := (\\det (I - x L_G))^{1/n}, \\ \\ \\text{ for } x \\in {\\mathbb {R}}_+.", "$ Then $Q_G : {\\mathbb {R}}_+ \\rightarrow {\\mathbb {R}}_+$ is well defined and continuously differentiable, and there exists a unique solution $f_G : [0,T] \\rightarrow {\\mathbb {R}}_+$ to the ODE $f_G^{\\prime }(t) = c Q_G^{\\prime }(f_G(t)), \\hspace{14.22636pt} f_G(0) = 0.$ Define $P_G : [0,T] \\rightarrow {\\mathbb {R}}^{n \\times n}$ by $P_G(t) := - f_G^{\\prime }(T-t) L_G \\big (I - f_G(T-t) L_G \\big )^{-1}, $ and finally define $\\alpha ^{G}_i \\in {\\mathcal {A}}_G$ for $i \\in G$ by $\\alpha ^{G}_i(t, \\mathbf {x}) = - e_i^T P_G(t) \\mathbf {x},$ where $(e_v)_{v \\in G}$ is the standard Euclidean basis in ${\\mathbb {R}}^G$ .", "Then $(\\alpha ^{G}_i)_{i \\in G}$ is a Nash equilibrium.", "For each $t \\in (0,T]$ , the equilibrium state process $\\mathbf {X}^G(t)$ is normally distributed with mean vector $(I - f_G(T-t)L_G) (I - f_G(T)L_G)^{-1}\\mathbf {X}^G(0)$ and covariance matrix $\\sigma ^2 (I - f_G(T-t)L_G)^2 \\int _0^t (I - f_G(T-s)L_G)^{-2}ds.$ Finally, writing $|\\cdot |$ for the Euclidean norm, the time-zero average value is $\\mathrm {Val}(G) := \\frac{1}{n}\\sum _{v \\in G} J^G_v((\\alpha ^{G}_i)_{i \\in G}) = \\frac{|P_G(0)\\mathbf {X}^G(0)|^2}{2{\\mathrm {Tr}}(P_G(0))} - \\frac{\\sigma ^2}{2} \\log \\frac{{\\mathrm {Tr}}(P_G(0))}{nf^{\\prime }_G(T)}.", "$ The proof is given in Section .", "As usual, we first reduce our (linear-quadratic) game to a system of matrix differential equations of Riccati type in Section REF .", "In our setting we can explicitly solve these Riccati equations using symmetry arguments based on the transitivity assumption.", "It is interesting to note that the equilibrium controls $\\alpha ^G_i$ obtained in Theorem REF are nonlocal, in the sense that the control of player $i$ depends on the states of all of the players, not just the neighbors.", "Naive intuition would suggest that players should only look at the states of their neighbors, because the objective of each player is to align at time $T$ with those neighbors.", "On the contrary, a rational player anticipates that her neighbors will in turn try to align with their own neighbors, which leads the player to follow the states of the neighbors' neighbors, and similarly the neighbors' neighbors' neighbors, and so on.", "It is worth noting that in the setting of Theorem REF we have ${\\mathbb {E}}\\left[\\frac{1}{n}\\sum _{v \\in G}X^G_v(t)\\right] = \\frac{1}{n}\\sum _{v \\in G}X^G_v(0).$ That is, the average location of the players stays constant over time, in equilibrium.", "Indeed, this follows easily from the formula for the mean ${\\mathbb {E}}[\\mathbf {X}^G(t)]$ and from the fact that the vector of all ones is an eigenvector with eigenvalue 0 for the symmetric matrix $L_G$ .", "We suspect that the Markovian Nash equilibrium identified in Theorem REF is the unique one.", "This could likely be proven along the lines of [9], but for the sake of brevity we do not attempt to do so." ], [ "Asymptotic regimes", "The form of the equilibrium computed in Theorem REF lends itself well to large-$n$ asymptotics after a couple of observations.", "First, for simplicity, we focus on the case $\\mathbf {X}^G(0)=\\mathbf {0}$ .", "Transitivity of the graph $G$ (or Lemma REF ) ensures that $X^G_i(t) \\stackrel{d}{=} X^G_j(t)$ for all $i,j \\in G$ and $t > 0$ , and we deduce that each $X^G_i(t)$ is a centered Gaussian with variance ${\\mathrm {Var}}( X^G_i(t)) &= \\frac{1}{n}\\sum _{k=1}^n {\\mathrm {Var}}( X^G_k(t) ) \\nonumber \\\\&= \\frac{\\sigma ^2}{n}{\\mathrm {Tr}}\\left[ (I-f_G(T-t)L_G)^2 \\int _0^t (I-f_G(T-s)L_G)^{-2}ds\\right] \\nonumber \\\\&= \\frac{\\sigma ^2}{n} \\sum _{k=1}^n \\int _0^t \\left(\\frac{1 - f_G(T-t)\\lambda ^G_k}{1 - f_G(T-s)\\lambda ^G_k}\\right)^2 ds, $ where $\\lambda _1^G,\\ldots ,\\lambda _n^G$ are the eigenvalues of $L_G$ , repeated by multiplicity.", "The average over $k=1,\\ldots ,n$ can be written as an integral with respect to the empirical eigenvalue distribution, $\\mu _G := \\frac{1}{n}\\sum _{i=1}^n\\delta _{\\lambda ^G_i}, $ which we recall is supported on $[-2,0]$ , as in Remark REF .", "The other quantities in Theorem REF can also be expressed in terms of $\\mu _G$ .", "Indeed, the value $\\mathrm {Val}(G)$ becomes $\\mbox{Val}(G) = - \\frac{\\sigma ^2}{2} \\log \\frac{{\\mathrm {Tr}}{(P_{G}(0))}}{n f^{\\prime }_{G}(T)} = - \\frac{\\sigma ^2}{2} \\log \\int _{[-2,0]} \\frac{- \\lambda }{1 - f_{G}(T)\\lambda } \\mu _{G}(d\\lambda ), $ and the function $Q_G$ defined in (REF ) becomes $Q_G(x) &= \\left(\\prod _{i=1}^n(1-x\\lambda ^G_i)\\right)^{1/n} = \\exp \\int _{[-2,0]} \\log (1-x\\lambda ) \\,\\mu _G(d\\lambda ).$ Thus, if we are given a sequence of graphs $G_n$ such that $\\mu _{G_n}$ converges weakly to some probability measure, it is natural to expect the equilibrium computed in Theorem REF to converge in some sense.", "This is the content of our next main result, which we prove in Section : Theorem 2.6 (Large-scale asymptotics on transitive graphs) Let $\\lbrace G_n\\rbrace $ be a sequence of finite transitive graphs without isolated vertices, with $\\lim _{n\\rightarrow \\infty }|G_n|=\\infty $ .", "Let $\\mathbf {X}^{G_n}$ denote the equilibrium state process identified in Theorem REF , started from initial position $\\mathbf {X}^{G_n}(0)= \\mathbf {0}$ .", "Suppose $\\mu _{G_n}$ converges weakly to a probability measure $\\mu $ , and define $Q_\\mu : {\\mathbb {R}}_+ \\rightarrow {\\mathbb {R}}_+$ by $Q_\\mu (x) := \\exp \\int _{[-2,0]} \\log (1-x\\lambda ) \\,\\mu (d\\lambda ).$ Then the following holds: There exists a unique solution $f_\\mu : [0,T] \\rightarrow {\\mathbb {R}}_+$ of the ODE $f^{\\prime }_\\mu (t) = cQ^{\\prime }_\\mu (f_\\mu (t)), \\quad f_\\mu (0)=0.", "$ For any vertex sequence $k_n \\in G_n$ and any $t \\in [0,T]$ , the law of $X^{G_n}_{k_n}(t)$ converges weakly as $n\\rightarrow \\infty $ to the Gaussian distribution with mean zero and variance ${V}_\\mu (t) = \\sigma ^2\\int _0^t\\int _{[-2,0]} \\left(\\frac{1 - \\lambda f_\\mu (T-t)}{1 - \\lambda f_\\mu (T-s)}\\right)^2\\mu (d\\lambda ) ds.", "$ For any $t \\in [0,T]$ , the (random) empirical measure $\\frac{1}{|G_n|}\\sum _{i \\in G_n} \\delta _{X^{G_n}_i(t)}$ converges weakly in probability as $n\\rightarrow \\infty $ to the (non-random) Gaussian distribution $\\mathcal {N}(0,{V}_\\mu (t))$ .", "The time-zero values given in (REF ) with $\\mathbf {X}^{G_n}(0)= 0$ converge: $\\lim _{n\\rightarrow \\infty }\\mathrm {Val}(G_n) = - \\frac{\\sigma ^2}{2} \\log \\int _{[-2,0]} \\frac{-\\lambda }{1-\\lambda f_\\mu (T)}\\mu (d\\lambda ).", "$ There are many concrete graph sequences $\\lbrace G_n\\rbrace $ for which $\\mu _{G_n}$ can be shown to converge to a tractable (typically continuous) limiting measure, and we document several notable cases in Section REF .", "The Laplacian spectrum is in fact quite tractable and well-studied.", "There is a substantial literature on the eigenvalues of Laplacian (and other) matrices of graphs [11], [22], which are well known to encode significant structural information about the graph.", "It is arguably more natural a priori to state instead that the graph sequence $G_n$ itself has some asymptotic structure in some sense, which might be formalized using one of the many several notions of graph limits in the literature.", "In particular, the notion of local weak convergence has proven to be quite powerful and natural in the sparse setting; see [39] for a thorough treatment, as well as [28], [34] for some recent work applying this framework to analyze large-scale interacting diffusion models on sparse graphs.", "We will not define local weak convergence here, but we highlight that the local weak convergence of $G_n \\rightarrow G$ is known to imply the weak convergence of $\\mu _{G_n}$ to a certain spectral measure $\\mu $ , under mild technical conditions [3].", "This spectral measure is defined intrinsically on a general finite or countable (locally finite) graph $G=(V,E)$ by applying the spectral theorem to an appropriately defined Laplacian operator on the complex Hilbert space $\\ell ^2(V)$ , and in the finite (transitive) graph case this coincides exactly with the empirical eigenvalue distribution defined in (REF ).", "Remark 2.7 We develop in Section some noteworthy qualitative and quantitative properties of the equilibrium variance ${V}_\\mu (t)$ given in (REF ).", "We show in Proposition REF that ${V}_\\mu (0)=0$ , ${V}^{\\prime }_\\mu (0)=\\sigma ^2$ , and ${V}^{\\prime \\prime }_\\mu (0)=-2\\sigma ^2 (f^{\\prime }_\\mu (T))^2/cQ_\\mu (f_\\mu (T))$ .", "In particular, for short times, the leading-order behavior ${V}_\\mu (t) = \\sigma ^2 t + o(t)$ does not depend on the underlying graph.", "It is only at the second order or at longer time horizons that the influence of the graph is felt.", "Remark 2.8 The restriction to $\\mathbf {X}^{G_n}(0)=\\mathbf {0}$ in Theorem REF is merely to simplify the resulting formulas.", "One could easily accommodate the more general setting in which the empirical measure of initial states converges to some limiting distribution.", "In addition, a functional version of Theorem REF can likely be derived under no further assumptions, in which the Gaussian process $(X^{G_n}_{k_n}(t))_{t \\in [0,T]}$ converges weakly in $C([0,T])$ to a limiting Gaussian process.", "We omit these generalizations, as the more complicated statements do not shed any light on the role of the network structure, which is the main focus of our work." ], [ "Dense graphs", "If $G_n$ is the complete graph, then it turns out that $\\mu _{G_n} = \\frac{1}{n}\\delta _0 + \\frac{n-1}{n}\\delta _{-n/(n-1)} \\rightarrow \\delta _{-1}$ , which leads to a simpler form for the limiting law in (REF ).", "More generally, the case $\\mu _{G_n} \\rightarrow \\delta _{-1}$ represents a “dense\" regime, as described in the following result.", "Recall that all transitive graphs are regular, meaning each vertex has the same degree.", "The following is proven in Section REF : Corollary 2.9 (Large-scale asymptotics on dense transitive graphs) Suppose $\\lbrace G_n\\rbrace $ is a sequence of transitive graphs, and suppose each vertex of $G_n$ has common degree $\\delta (G_n) \\ge 1$ .", "Then $\\mu _{G_n} \\rightarrow \\delta _{-1}$ if and only if $\\delta (G_n)\\rightarrow \\infty $ .", "In this case, the limiting variance (REF ) and value (REF ) simplify to ${V}_{\\delta _{-1}}(t) = \\sigma ^2 t \\frac{ 1 + c(T-t) }{1+cT}, \\qquad \\lim _{n\\rightarrow \\infty }\\mathrm {Val}(G_n) = \\frac{\\sigma ^2}{2} \\log (1+ cT).", "$ Moreover, there is a constant $C < \\infty $ , depending only on $c$ and $T$ , such that $|{V}_{G_n}(t) - {V}_{\\delta _{-1}}(t)| + \\left|\\mathrm {Val}(G_n) - \\tfrac{\\sigma ^2}{2} \\log (1+ cT)\\right| \\le C/\\delta (G_n), \\quad \\forall n \\in {\\mathbb {N}}, \\ t \\in [0,T].", "$ Finally, the Gaussian law $\\mathcal {N}(0,{V}_{\\delta _{-1}}(t))$ is precisely the time-$t$ law of the unique solution of the SDE $dX(t) = -\\frac{cX(t)}{1+c(T-t)}dt + \\sigma dW(t), \\quad X(0)=0.", "$ Corollary REF shows that the dense regime is particularly tractable.", "In particular, the mean field case (where $G_n$ is the complete graph) is universal in the sense that the same limit arises for any other transitive graph sequence with diverging degree.", "Moreover, the rate $C/\\delta (G_n)$ in (REF ) becomes $C/n$ in the mean field case, which is the best-known convergence rate for the value functions of well-behaved MFGs [7].", "Remark 2.10 We show in Proposition REF that the dense graph regime uniquely achieves the lowest possible variance; precisely, we have ${V}_\\mu (t) \\ge {V}_{\\delta _{-1}}(t)$ , recalling the definitions (REF ) and (REF ), with equality only when $\\mu =\\delta _{-1}$ .", "The example of the torus graphs in Section REF below illustrates what appears to be a general principle, that a highly connected graph has smaller variance in equilibrium.", "This makes intuitive sense, as a higher degree means each player has a larger set of neighbors to be attracted toward.", "Our next result shows that in the dense regime we may use the limiting object to construct approximate equilibria for $n$ -player games on general large dense graphs (not necessarily transitive), in the same way that the equilibrium of a MFG can be used to build approximate equilibria for finite games.", "Theorem 2.11 (Approximate equilibria on general dense graphs) Suppose $G$ is a finite graph.", "For each vertex $v$ of $G$ , define a control $\\alpha _v^{\\mathrm {MF}}(t,\\mathbf {x}) := \\frac{- cx_v}{1+c(T-t)}, \\quad t \\in [0,T], \\ \\ \\mathbf {x} = (x_u)_{u \\in G} \\in {\\mathbb {R}}^G.$ Finally, define $\\mathbf {\\epsilon }^G=(\\epsilon ^G_v)_{v \\in G} \\in {\\mathbb {R}}_+^G$ by $\\epsilon ^G_v := \\left\\lbrace \\begin{array}{ll}\\sigma ^2\\frac{cT}{1+cT}\\sqrt{\\frac{cT(2+cT)}{ \\mathrm {deg}_G(v)}} & \\mbox{ if } \\mathrm {deg}_G(v) \\ge 1 \\\\0 & \\mbox{ if } \\mathrm {deg}_G(v) = 0.\\end{array}\\right.$ Then, for each $n$ , $(\\alpha _v^{\\mathrm {MF}})_{v \\in G}$ is an $\\mathbf {\\epsilon }^G$ -Nash equilibrium on $G$ .", "In particular, ifAs usual, we write $a \\vee b := \\max \\lbrace a,b\\rbrace $ .", "$\\epsilon _G := \\sigma ^2\\frac{cT}{1+cT}\\sqrt{\\frac{cT(2+cT)}{1 \\vee \\delta (G)}}, \\qquad \\text{where } \\ \\ \\delta (G):= \\min _{v \\in G}\\mathrm {deg}_G(v),$ then $(\\alpha _v^{\\mathrm {MF}})_{v \\in G}$ is a $\\epsilon _G$ -Nash equilibrium on $G$ .", "We use the notation $\\alpha _v^{\\mathrm {MF}}$ because this is precisely the control one obtains from the corresponding MFG (see Lemma REF ).", "Hence, Theorem REF says that on a graph sequence with “diverging degree\" in some sense, the MFG provides a (decentralized, symmetric) approximate Nash equilibrium.", "More precisely, if $\\lbrace G_n\\rbrace $ is a sequence of graphs with diverging minimal degree $\\delta (G_n) \\rightarrow \\infty $ , then the controls $\\mathbf {\\alpha }^n:=(\\alpha _v^{\\mathrm {MF}})_{v \\in G_n}$ form an approximate equilibrium, in the sense that $\\mathbf {\\alpha }^n$ is an $\\epsilon _{G_n}$ -Nash equilibrium for each $n$ with $\\lim _n\\epsilon _{G_n} = 0$ .", "Of course, the now-classical theory of MFGs tells us the same thing when $G_n$ is the complete graph (see, e.g., [9], or [23] for the standard rate of $\\epsilon _{G_n} = O(1/\\sqrt{n})$ ), but Theorem REF gives a threshold of how dense the graph needs to be in order for the mean field approximation to remain valid.", "The constant $\\epsilon _{G_n}$ shows quantitatively how the accuracy of the mean field approximation depends on the “denseness\" of the graph, as measured by the minimal degree.", "Some examples beyond the complete graph will be discussed in Section REF below.", "In fact, we may relax the denseness threshold if we are happy to assert that $(\\alpha _v^{\\mathrm {MF}})_{v \\in G_n}$ form an approximate equilibrium in a weaker sense, suggested by the most general form of Definition REF .", "A small fraction of players (namely, those with small degree) might have a lot to gain by deviating, but this potential gain from deviation is small when averaged over all players.", "Precisely, suppose that instead of the minimum degree diverging, we suppose merely that degrees diverge in the following averaged sense: $\\lim _{n\\rightarrow \\infty }\\frac{1}{|G_n|}\\sum _{v \\in G_n}(\\mathrm {deg}_{G_n}(v))^{-1/2} = 0.", "$ Then $(\\alpha _v^{\\mathrm {MF}})_{v \\in G_n}$ is an $\\mathbf {\\epsilon }^{G_n}$ -Nash equilibrium, and $\\lim _n\\frac{1}{n}\\sum _{i=1}^n\\epsilon ^{G_n}_i = 0$ .", "In summary, different manners of quantifying the concept of approximate equilibrium lead to different sparsity/denseness thresholds for the validity of the mean field approximation.", "The Erdős-Rényi case in Section REF gives a concrete example." ], [ "Correlation decay and asymptotic independence", "Before discussing examples, we lastly present an estimate of correlation decay, which is crucial in the proof of convergence of the empirical measure in Theorem REF , and which also reveals what form of asymptotic independence between the players can be expected.", "Proposition 2.12 (Correlation decay on transitive graphs) Let $G$ be a finite transitive graph without isolated vertices, and let $\\mathbf {X}^G$ denote the equilibrium state process identified in Theorem REF .", "Suppose each vertex of $G$ has degree $\\delta (G) \\in {\\mathbb {N}}$ .", "For vertices $u,v \\in G$ , let $d_G(u,v)$ denote the graph distance, defined as the length of the shortest path from $u$ to $v$ (and $\\infty $ if no such path exists).", "Let $\\gamma = cT/(1+cT) \\in (0,1)$ .", "Then $|{\\mathrm {Cov}}(X^G_u(t), X^G_v(t))| \\le 2\\sigma ^2 t\\frac{\\gamma ^{d_G(u,v)} \\big (1 + d_G(u,v)(1 - \\gamma )\\big )}{\\delta (G)(1 - \\gamma )^2}1_{\\lbrace d_G(u,v) < \\infty \\rbrace }.", "$ Note that the right-hand side of (REF ) is a bounded function of $\\delta (G)$ and $d_G(u,v)$ .", "If $G_n$ is the complete graph on $n$ vertices (i.e., the mean field case), then $\\delta (G_n)=n-1\\rightarrow \\infty $ , and each pair of players (in equilibrium) becomes asymptotically independent as $n\\rightarrow \\infty $ .", "This is an instance of the phenomenon of propagation of chaos for mean field systems.", "More generally, this remains true for any dense graph sequence, i.e., whenever $\\delta (G_n) \\rightarrow \\infty $ .", "On the other hand, the picture is rather different for a sparse graphs sequence, i.e., when $\\sup _n\\delta (G_n) < \\infty $ .", "An arbitrary pair of players can no longer be expected to become asymptotically independent as $n\\rightarrow \\infty $ , but only distant players.", "More precisely, two players $u_n,v_n \\in G_n$ become asymptotically independent only if $d_{G_n}(u_n,v_n) \\rightarrow \\infty $ (since $\\gamma < 1$ ).", "For a transitive graph sequence with $|G_n| \\rightarrow \\infty $ and $\\sup _n\\delta (G_n) < \\infty $ , it is always the case that the distance between two uniformly random vertices converges to infinity in probability, and it follows that two (uniformly) randomly chosen players are asymptotically independent, which in turn implies the convergence of the empirical measure to a non-random limit in Theorem REF .", "See Section for details.", "In summary, for a sequence of (transitive) graphs $G_n$ with $|G_n|\\rightarrow \\infty $ , asymptotic independence of a typical pair of players arises for one of two quite distinct reasons.", "Either: The degree diverges, and each player interacts with many other players, with pairwise interaction strengths of order $1/\\delta (G_n) \\rightarrow 0$ .", "The degrees stay bounded, but typical players are very far apart in the graph and thus very weakly correlated." ], [ "Examples", "In this section we specialize the results of Section REF to a short (and by no means exhaustive) list of somewhat tractable natural large-graph models.", "We focus on cases where the minimum degree and/or the empirical eigenvalue distribution of the graph are tractable, as these quantities are particularly relevant to the main results of Section REF ." ], [ "The complete graph", "Let us summarize what we have mentioned regarding the simplest (mean field) case, where $G_n$ is the complete graph on $n$ vertices.", "In this case, the Laplacian matrix takes the form $L_{G_n}=\\frac{1}{n-1}(J-I) - I,$ where $J$ is the matrix of all ones.", "From this we easily deduce that the eigenvalues of $L_{G_n}$ are 0 and $-\\tfrac{n}{n-1}$ , with respective multiplicities 1 and $n-1$ .", "Hence, $\\mu _{G_n} \\rightarrow \\delta _{-1}$ , and the degree $\\delta (G_n)=n-1 \\rightarrow \\infty $ .", "The complete graph is of course transitive, and all of our main theorems apply, in particular Corollary REF .", "In the complete graph setting, our model essentially becomes the $\\epsilon =q=0$ case of [10].", "The only difference is that in [10] each player is included in the empirical average; that is, the terminal cost of player $k$ is $|\\frac{1}{n}\\sum x_i - x_k|^2$ instead of $|\\frac{1}{n-1}\\sum _{i\\ne k}x_i - x_k|^2$ .", "This can easily be fit into our framework, as the following remark explains.", "Remark 2.13 For a finite transitive graph $G$ without isolated vertices, every vertex has a common degree $\\delta (G)$ .", "Letting $N_v(G)$ denote the union of $\\lbrace v\\rbrace $ and the set of neighbors of a vertex $v$ in $G$ , we can write the terminal cost function for player $v$ as $\\left|\\frac{1}{\\delta (G)}\\sum _{u \\sim v}x_u - x_v \\right|^2 = \\left(\\frac{\\delta (G)+1}{\\delta (G)} \\right)^2\\left|\\frac{1}{\\delta (G)+1}\\sum _{u \\in N_v(G)}x_u - x_v \\right|^2.$ Hence, we can modify our setup so that each each player is included in the average in the terminal cost, simply by modifying the constant $c$ by a factor of $(1+1/\\delta (G))^2$ ." ], [ "The cycle graph", "Suppose now that $G_n=C_n$ is the cycle on $n$ vertices.", "This is a transitive graph in which every vertex has common degree $\\delta (C_n)=2$ .", "The adjacency matrix $A_{C_n}$ is a circulant matrix, which makes it easy to calculate the eigenvalues as $2 \\cos ( 2\\pi k/n)$ for $k=1,\\ldots ,n$ .", "The eigenvalues of the Laplacian $L_{C_n} = \\tfrac{1}{2} A_{C_n}-I$ are thus $\\lambda ^{C_n}_k=\\cos (2\\pi k/n) - 1$ for $k=1,\\ldots ,n$ .", "In this case, for a bounded continuous function $f$ we compute $\\int f\\,d\\mu _{C_n} = \\frac{1}{n}\\sum _{k=1}^n f(\\cos (2\\pi k/n) - 1) \\rightarrow \\int _0^1 f(\\cos (2\\pi u) - 1) \\,du, \\ \\text{ as } n \\rightarrow \\infty ,$ which shows that $\\mu _{C_n}$ converges weakly to the probability measure $\\mu $ given by the law of $\\cos (2\\pi U) - 1$ , where $U$ is uniform in $[0,1]$ , i.e., $\\mu (dx)=1_{[-2,0]}(x)\\frac{dx}{\\pi \\sqrt{-x(2+ x)}}$ .", "The function $Q_\\mu $ in Theorem REF is then $Q_\\mu (x) = \\exp \\int _0^1 \\log \\big (1 + x - x\\cos (2\\pi u)\\big ) du, \\qquad x \\ge 0.", "$ In Section REF we derive a semi-explicit solution of the ODE (REF ) in this setting: Proposition 2.14 Define $Q_\\mu $ as in (REF ).", "Then $Q_\\mu (x)=\\tfrac{1}{2}(\\sqrt{1+2x}+x+1)$ , and the unique solution of the ODE $f^{\\prime }_\\mu (t)=cQ^{\\prime }_\\mu (f_\\mu (t))$ with $f_\\mu (0)=0$ is given by $f_\\mu (t) = \\Phi ^{-1}\\left(\\log 2 +\\frac{ct-1}{2} \\right),$ where $\\Phi ^{-1}$ is the inverse of the strictly increasing function $\\Phi : {\\mathbb {R}}_+\\rightarrow {\\mathbb {R}}_+$ defined by $\\Phi (x) := \\log (1+\\sqrt{1+2x}) - \\sqrt{1+2x} + x + \\tfrac{1}{2}.$ The variance from (REF ) then becomes ${V}_\\mu (t) = \\sigma ^2\\int _0^t\\int _0^1 \\left(\\frac{1 - (\\cos (2\\pi u) -1) \\Phi ^{-1}\\left(\\log 2 + \\frac{c(T-t)-1}{2} \\right)}{1 - (\\cos (2\\pi u) -1) \\Phi ^{-1}\\left(\\log 2 + \\frac{c(T-s)-1}{2} \\right)}\\right)^2 du \\, ds.", "$ This does not appear to simplify further, but Figure REF gives plots for various $c$ .", "Note that the variance in the dense case is always lower than in the cycle case, as we show in Proposition REF .", "In both cases, the variance at any fixed time $t$ decreases with $c$ .", "As $c\\rightarrow \\infty $ , the variance ${V}_\\mu (t)$ in both the dense and cycle graph cases can be shown to converge to $\\sigma ^2 t(T-t)/T$ , which is the same as that of a Brownian bridge.", "Figure: Variance of a typical player over time in the dense graph (dashed lines, equation ()) and in the cycle graph (solid lines, equation ()) for different values of cc.", "Here T=σ=1T=\\sigma =1." ], [ "The torus", "For $d \\in {\\mathbb {N}}$ , consider the torus $G_n={\\mathbb {Z}}^d_n:= {\\mathbb {Z}}^d/ n {\\mathbb {Z}}^d$ .", "That is, this graph is the subgraph of the integer lattice ${\\mathbb {Z}}^d$ with vertex set $\\lbrace 1,\\ldots ,n^d\\rbrace $ and with “wrapping around\" at the boundary.", "The eigenvalues of $L_{{\\mathbb {Z}}^d_n}$ are easily computed from those of $L_{C_n}$ , the cycle graph from the previous section, after noting that ${\\mathbb {Z}}^d_n$ is the $d$ -fold Cartesian product of the cycle $C_n$ with itself.", "In particular, if $G$ and $H$ are two graphs, and $A_G$ and $A_H$ have eigenvalues $(\\eta ^G_v)_{v \\in G}$ and $(\\eta ^H_v)_{v \\in H}$ respectively, then the eigenvalues of the adjacency matrix of the Cartesian product of $G$ and $H$ are given by $(\\eta ^G_v+\\eta ^H_u)_{u \\in G, v \\in H}$ .See Chapter 7.14 of [22] for definition of the Cartesian product of graphs and Chapter 9.7 for a derivation of the eigenvalues of the adjacency matrix of a Cartesian product.", "In particular, the eigenvalues of $A_{{\\mathbb {Z}}^d_n}$ are $\\sum _{i=1}^d 2\\cos (2\\pi k_i/n), \\qquad \\mathbf {k}=(k_1,\\ldots ,k_d) \\in {\\mathbb {Z}}^d_n.$ Noting that each vertex in ${\\mathbb {Z}}^d_n$ has degree $2d$ , we find that the eigenvalues of $L_{{\\mathbb {Z}}^d_n}=\\tfrac{1}{2d}A_{{\\mathbb {Z}}^d_n} - I$ are $\\lambda ^{{\\mathbb {Z}}^d_n}_{\\mathbf {k}} = \\frac{1}{d}\\sum _{i=1}^d \\cos (2\\pi k_i/n) - 1, \\qquad \\mathbf {k}=(k_1,\\ldots ,k_d) \\in {\\mathbb {Z}}^d_n.$ Hence, for a bounded continuous function $f$ we compute $\\int f\\,d\\mu _{{\\mathbb {Z}}^d_n} &= \\frac{1}{|{\\mathbb {Z}}^d_n|}\\sum _{\\mathbf {k} \\in {\\mathbb {Z}}^d_n} f_\\mu (\\lambda ^{{\\mathbb {Z}}^d_n}_{\\mathbf {k}}) = \\frac{1}{n^d}\\sum _{k_1,\\ldots ,k_d=1}^n f\\left(\\frac{1}{d}\\sum _{i=1}^d \\cos (2\\pi k_i/n) - 1\\right) \\\\&\\rightarrow \\int _{[0,1]^d}f\\left(\\frac{1}{d}\\sum _{i=1}^d \\cos (2\\pi u_i) - 1\\right)du, \\qquad \\text{ as } \\ \\ n\\rightarrow \\infty ,$ which shows that $\\mu _{{\\mathbb {Z}}^d_n}$ converges weakly to the probability measure $\\mu $ given by the law of $\\frac{1}{d}\\sum _{i=1}^d \\cos (2\\pi U_i) - 1$ , where $U_1,\\ldots ,U_d$ are independent uniform random variables in $[0,1]$ .", "The function $Q_\\mu $ in Theorem REF is then $Q_\\mu (x) = \\exp \\int _{[0,1]^d} \\log \\left(1 + x - \\frac{x}{d}\\sum _{i=1}^d \\cos (2\\pi u_i)\\right) du.", "$ We cannot evaluate (REF ) or the solution $f$ of the ODE (REF ) explicitly, for the torus of dimension $d > 1$ (the case $d=1$ is the cycle graph).", "But we can easily do so numerically.", "Figure REF shows the variance ${V}_\\mu (t)$ of (REF ) for the torus of various dimensions, compared with the dense graph case.", "Notably, the variance decreases with the dimension $d$ , supporting the intuition that a more highly connected graph leads to a behavior closer to the mean field regime.", "Figure: Variance of a typical player over time in the torus of dimensions d=1,2,3,4d=1,2,3,4 and the dense case.", "Here T=c=σ=1T=c=\\sigma =1." ], [ "Erdős-Rényi graphs", "Most of our main results require a transitive graph and thus have little to say about classical random graph models, such as Erdős-Rényi, random regular graphs, or the configuration model, which generate graphs which are non-transitive with high probability.", "In particular, in applications of Theorems REF and REF we cannot take any advantage of the vast body of literature on the behavior of the eigenvalue distribution of the adjacency and Laplacian matrices of these (non-transitive) random graph models.", "That said, we mention here some noteworthy dense random graph models, to which Theorem REF applies and shows that the MFG approximation is valid.", "For the Erdős-Rényi graph $G(n,p_n)$ , it is known that as $n\\rightarrow \\infty $ with $\\liminf _n np_n/\\log n > 1$ , the minimal degree converges to infinity in probability [15].", "Hence, Theorem REF applies in this regime to give a random sequence $\\epsilon _n \\ge 0$ converging to zero in probability such that $(\\alpha ^{\\mathrm {MF}}_v)_{v \\in G_n}$ is an $\\epsilon _n$ -Nash equilibrium for each $n$ .", "This is sharp in a sense, because $p_n > \\log n /n$ is precisely the threshold for connectedness: If $\\limsup _n np_n/\\log n < 1$ , then $G(n,p_n)$ contains isolated vertices with high probability (in particular, the minimal degree is 1), and we cannot expect $\\alpha ^{\\mathrm {MF}}_v$ to be near-optimal for $v$ in a small connected component.", "This might be compared to the main result of [13], which keeps $p_n=p$ constant as $n\\rightarrow \\infty $ and finds likewise that the usual MFG approximation is valid for a class of games on the Erdős-Rényi graph.", "If we relax our concept of approximate equilibrium, as in the discussion after Theorem REF , then we may push the denseness threshold all the way to $np_n\\rightarrow \\infty $ .", "That is, if $np_n\\rightarrow \\infty $ , then a straightforward calculation using the fact that $\\mathrm {deg}_{G_n}(v) \\sim \\mathrm {Binomial}(n-1,p_n)$ shows that ${\\mathbb {E}}\\left[\\frac{1}{n}\\sum _{v =1}^n (1 \\vee \\mathrm {deg}_{G_n}(v))^{-1/2}\\right] = \\sum _{k=0}^{n-1}{n-1 \\atopwithdelims ()k}p_n^k(1-p_n)^{n-k-1}(1 \\vee k)^{-1/2} \\rightarrow 0,$ which ensures by Theorem REF that there exist random (graph-dependent) variables $\\mathbf {\\epsilon }^n=(\\epsilon ^n_v)_{v=1}^n$ such that $\\frac{1}{n}\\sum _{v=1}^n\\epsilon ^n_v \\rightarrow 0$ in probability and $(\\alpha ^{\\mathrm {MF}}_v)_{v \\in G_n}$ forms a $\\mathbf {\\epsilon }^n$ -Nash equilibrium.", "Note that this threshold $np_n\\rightarrow \\infty $ means that the expected degree (of a randomly chosen vertex) diverges.", "In the extremely sparse (diluted) regime $\\lim _n np_n = \\theta \\in (0,\\infty )$ , the degree of a typical vertex converges in law to Poisson($\\theta $ ), and Theorem REF yields no information.", "These thresholds are in line with recent work on interacting particle systems (without game or control).", "For example, interacting particle systems on either the complete graph or the Erdős-Rényi graph $G(n,p_n)$ converge to the same mean field (McKean-Vlasov) limit as soon as $np_n\\rightarrow \\infty $ [2], [35].", "This is clearly the minimal sparsity threshold for which we can expect a mean field behavior, as evidenced by recent work in the extremely sparse (diluted) regime where $np_n$ converges to a finite non-zero constant [28], [34]." ], [ "Random regular graphs", "Random regular graphs are well known to admit tractable large-scale behavior, in both the dense and sparse regimes.", "Let $d_n \\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace $ , and let $R(n,d_n)$ denote a uniformly random choice of $d_n$ -regular graph on $n$ vertices.", "In the dense regime, where $d_n \\rightarrow \\infty $ , Theorem REF lets us construct approximate equilibria.", "In the sparse regime, when $d_n =d$ is constant, the empirical measure $\\mu _{R(n,d)}$ is known to converge weakly to an explicit continuous probability measure $\\mu (d\\lambda )$ known as (an affine transformation of) the Kesten-McKay law [25], [31], with density given by $\\lambda \\mapsto \\frac{\\sqrt{4(d-1)-d^2(\\lambda +1)^2}}{2\\pi (1 - (\\lambda +1)^2)} 1_{\\lbrace |1+\\lambda | \\le 2\\sqrt{d-1}/d\\rbrace }.$ The same limit $\\mu $ arises for any sequence $G_n$ of $d$ -regular graphs satisfying $\\lim _{n\\rightarrow \\infty }\\tfrac{C_k(G_n)}{|G_n|} = 0$ for each $k \\in {\\mathbb {N}}$ , where $C_k(G_n)$ is the number of cycles of length $k$ in $G_n$ , by [31].", "Note that we cannot apply our main results to the random regular graph $G_n=R(n,d)$ for $d$ fixed, because $G_n$ is then transitive with vanishing probability as $n\\rightarrow \\infty $ ; in fact, $G_n$ has trivial automorphism group with high probability [26]." ], [ "The cooperative game", "For comparison, we discuss the corresponding cooperative game, which can be solved easily even without assuming transitivity of the underlying finite graph $G=(V,E)$ .", "In the setup of Section REF , consider the optimal control problem $\\inf _{\\mathbf {\\alpha }\\in {\\mathcal {A}}_G^V} \\sum _{v \\in V}J_v^G(\\mathbf {\\alpha }).$ Let us abbreviate $L=L_G$ for the Laplacian.", "The corresponding Hamilton-Jacobi-Bellman (HJB) equation is $\\left\\lbrace \\begin{array}{ll}&\\partial _t v (t,\\mathbf {x}) - \\frac{1}{2} |\\nabla v(t,\\mathbf {x})|^2 + \\frac{1}{2} \\sigma ^2 \\Delta v(t,\\mathbf {x}) = 0, \\quad (t,\\mathbf {x}) \\in (0,T) \\times {\\mathbb {R}}^V, \\\\& v(T,\\mathbf {x}) = \\frac{c}{2}|L\\mathbf {x}|^2 = \\frac{c}{2}\\mathbf {x}^\\top L^\\top L\\mathbf {x},\\end{array}\\right.$ and the optimal control is $\\mathbf {\\alpha }^* = - \\nabla v$ .", "Using the ansatz $v(t,\\mathbf {x}) = \\tfrac{1}{2} \\mathbf {x}^\\top F(t) \\mathbf {x} + h(t)$ , for some symmetric matrix $F(t)$ , the HJB becomes $\\frac{1}{2} \\mathbf {x}^T F^{\\prime }(t)\\mathbf {x} + h^{\\prime }(t) - \\frac{1}{2} \\mathbf {x}^\\top F^2(t) \\mathbf {x} + \\frac{1}{2}\\sigma ^2 {\\mathrm {Tr}}(F(t)) = 0, \\quad (t,\\mathbf {x}) \\in (0,T) \\times {\\mathbb {R}}^V,$ with terminal conditions $F(T) = cL^\\top L$ and $h(T) = 0$ .", "Matching coefficients, we deduce that $F$ and $h$ must solve $\\left\\lbrace \\begin{array}{ll}& F^{\\prime }(t) - F^2(t) = 0, \\hspace{14.22636pt} F(T) = cL^\\top L, \\\\& h^{\\prime }(t) + \\frac{1}{2} \\sigma ^2 {\\mathrm {Tr}}(F(t)) = 0, \\hspace{14.22636pt} h(T) = 0.\\end{array}\\right.$ We find that the solution to this system is given by $F(t) &= c L^\\top L(I + c(T-t) L^\\top L)^{-1} = - \\frac{d}{dt} \\log (I+c(T-t)L^\\top L),$ where the log of the positive definite matrix is defined via power series, and $h(t) &= \\frac{\\sigma ^2}{2} {\\mathrm {Tr}}\\log (I + c(T-t)L^\\top L) = \\frac{\\sigma ^2}{2} \\log \\det (I + c(T-t)L^\\top L).$ The optimal control is $\\mathbf {\\alpha }(t,\\mathbf {x})=-F(t)\\mathbf {x}$ , and the optimal state process follows $d\\mathbf {X}(t) = - F(t) \\mathbf {X}(t) + \\sigma d\\mathbf {W}(t).$ This SDE can be explicitly solved, and the law of $\\mathbf {X}(t)$ is Gaussian with mean $(I+cTL^\\top L)(I+c(T-t)L^\\top L)^{-1}\\mathbf {X}(0)$ and covariance matrix $\\sigma ^2 \\int _0^t (I+c(T-t)L^\\top L)^2(I+c(T-s)L^\\top L)^{-2} ds.$ In particular, if the graph $G$ is transitive, we compute for each $i \\in V$ as in the beginning of Section REF that ${\\mathrm {Var}}(X_i(t)) = \\sigma ^2 \\int _0^t\\int _{[-2,0]}\\left(\\frac{1+c(T-t)\\lambda ^2}{1+c(T-s)\\lambda ^2}\\right)^2\\mu _G(d\\lambda )ds.$ And if $\\mathbf {X}(0)=0$ , then the per-player value is $\\frac{1}{|V|}\\inf _{\\mathbf {\\alpha }\\in {\\mathcal {A}}_G^V} \\sum _{v \\in V}J_v^G(\\mathbf {\\alpha }) = \\frac{1}{|V|}v(0,0) = \\frac{\\sigma ^2}{2} \\int _{[-2,0]} \\log (1 + cT\\lambda ^2) \\,\\mu _G(d\\lambda ).$ It is interesting to compare these outcomes to the competitive equilibrium computed in Theorem REF .", "In the competitive case, the function $f(t)=f_\\mu (t)$ may be seen as determining the rate of flocking; as $f(t)$ increases over time, players expend more effort to move toward the average.", "In the competitive case, this function depends crucially on the graph, and it simplifies to $f(t)=ct$ in the dense graph case.", "In the cooperative case, we always have $f(t)=ct$ , but the solution is governed by the squared Laplacian instead of the Laplacian itself.", "Figure REF shows the variance over time for the competitive and cooperative solutions on the (limiting) cycle graph.", "Figure: Variance of a typical player over time in the cycle graph, in the competitive versus cooperative regimes.", "Here T=σ=c=1T=\\sigma =c=1.In addition to being an interesting point of comparison for the competitive case, we highlight the cooperative model also in connection with mean field control theory.", "We are not the first to study stochastic optimal control problems on large graphs, but we do not know of much other work other than the recent papers [20], [21], which focus on the graphon setting and do not seem to offer any explicit examples.", "Organization of the rest of the paper.", "The rest of the paper is devoted to the proofs of the results stated in this section.", "We begin in Section with an analysis of the functions $Q_G$ and $Q_\\mu $ and the ODEs for $f_G$ and $f_\\mu $ which appear in Theorems REF and REF .", "Section then gives the proof of Theorem REF .", "This proof is essentially a verification argument and does not explain how to derive the announced solution, so in Section we give a sketch of a direct derivation.", "Section proves the covariance bound of Proposition REF .", "Finally, Section proves Theorem REF , Theorem REF , and some claims of Section REF ." ], [ "Analysis of the ODE", "In this section, we derive several useful results about the ODEs encountered in Theorems REF and REF .", "For a probability measure $\\mu $ on $[-2,0]$ , define the function $Q_\\mu (x) = \\exp \\int _{[-2,0]} \\log (1-x\\lambda )\\,\\mu (d\\lambda ), \\quad x \\ge 0.", "$ Note that the support of $\\mu $ ensures that $Q_\\mu (x)$ is well-defined and infinitely differentiable for $x \\ge 0$ .", "Note that if $\\mu =\\mu _G$ for a finite graph $G$ , recalling the notation of Section REF , then $Q_\\mu =Q_G$ takes the form of a normalized determinant as in (REF ).", "We restrict to probability measures on $[-2,0]$ in this section precisely because $\\mu _G$ is supported on $[-2,0]$ for every finite graph $G$ , as discussed in Remark REF .", "In addition, because the adjacency matrix of a (simple) graph has zero trace, the Laplacian matrix of a graph on $n$ vertices therefore has trace $-n$ .", "In particular, $\\int _{[-2,0]} x\\,\\mu _G(dx) = \\frac{1}{n}\\sum _{i=1}^n\\lambda ^G_i = \\frac{1}{n}{\\mathrm {Tr}}(L_G)=-1,$ for a finite graph $G$ on $n$ vertices with Laplacian eigenvalues $(\\lambda ^G_1,\\ldots ,\\lambda ^G_n)$ .", "We thus restrict our attention to the set ${\\mathcal {P}}_{\\mathrm {Lap}}$ of probability measures on $[-2,0]$ with mean $-1$ , equipped with the topology of weak convergence (i.e., $\\mu _n \\rightarrow \\mu $ in ${\\mathcal {P}}_{\\mathrm {Lap}}$ if $\\int f\\,d\\mu _n \\rightarrow \\int f\\,d\\mu $ for each continuous function $f : [-2,0] \\rightarrow {\\mathbb {R}}$ ).", "We will make repeated use of the following formulas for the first two derivatives of $Q_\\mu $ , computed via straightforward calculus: $Q^{\\prime }_\\mu (x) &= Q_\\mu (x)\\int _{[-2,0]} \\frac{-\\lambda }{1-x\\lambda }\\,\\mu (d\\lambda ), \\\\Q^{\\prime \\prime }_\\mu (x) &= Q_\\mu (x)\\left[\\left(\\int _{[-2,0]} \\frac{-\\lambda }{1-x\\lambda }\\,\\mu (d\\lambda )\\right)^2 - \\int _{[-2,0]} \\frac{\\lambda ^2}{(1-x\\lambda )^2}\\,\\mu (d\\lambda ) \\right].", "\\nonumber $ Proposition 3.1 For each $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ , the function $Q_\\mu : {\\mathbb {R}}_+ \\rightarrow {\\mathbb {R}}_+$ defined in (REF ) satisfies $1 &\\le Q_\\mu (x) \\le 1+x, \\qquad 0 < Q^{\\prime }_\\mu (x) \\le 1, \\qquad -4 \\le Q^{\\prime \\prime }_\\mu (x) \\le 0,$ for all $x \\in {\\mathbb {R}}_+$ , as well as $Q^{\\prime }_\\mu (x) &\\ge 1 - (x + \\tfrac{1}{2} x^2){\\mathrm {Var}}(\\mu ), \\quad \\text{where}\\quad {\\mathrm {Var}}(\\mu ) := \\int _{[-2,0]} (\\lambda +1)^2\\mu (d\\lambda ).", "$ First note that the support of $\\mu $ ensures that $Q_\\mu (x) \\ge 0$ for all $x \\ge 0$ .", "Jensen's inequality yields $Q_\\mu (x) \\le \\int _{[-2,0]} (1-x\\lambda )\\,\\mu (d\\lambda ) = 1 + x.$ and $Q^{\\prime \\prime }_\\mu (x) \\le 0$ .", "Since $Q^{\\prime }_\\mu (0)=1$ , we deduce that $Q^{\\prime }_\\mu (x) \\le 1$ for all $x \\ge 0$ .", "The next claims follow from the fact that, for each $x \\ge 0$ , the function $[-2,0] \\ni \\lambda \\mapsto -\\lambda /(1-x\\lambda ) \\quad \\text{is nonnegative and strictly decreasing.}", "$ Indeed, this first implies that $Q^{\\prime }_\\mu (x) \\ge 0$ , and in fact the inequality must be strict because $\\mu $ has mean $-1$ and is thus not equal to $\\delta _0$ .", "In addition, (REF ) implies that $Q_\\mu ^{\\prime \\prime }(x) &\\ge -Q_\\mu (x)\\int _{[-2,0]} \\frac{\\lambda ^2}{(1-x\\lambda )^2}\\,\\mu (d\\lambda ) \\ge -Q_\\mu (x) \\frac{4}{(1+2x)^2} \\ge -4,$ where the last step uses $Q_\\mu (x) \\le 1+x\\le (1+2x)^2$ .", "To prove the final claim, define $\\theta (x):=\\int _{[-2,0]}\\frac{-\\lambda }{1-x \\lambda }\\,\\mu (d\\lambda ), \\quad x \\ge 0.$ Using the Harris inequality (known as Chebyshev's sum inequality in the discrete case), we get $-2\\theta (x)\\theta ^{\\prime }(x) &= \\int _{[-2,0]}\\frac{-\\lambda }{1-x \\lambda }\\,\\mu (d\\lambda )\\int _{[-2,0]}\\frac{2\\lambda ^2}{(1-x \\lambda )^2}\\,\\mu (d\\lambda ) \\\\&\\le \\int _{[-2,0]}\\frac{-2\\lambda ^3}{(1-x \\lambda )^3}\\,\\mu (d\\lambda ) = \\theta ^{\\prime \\prime }(x),$ since both integrands are decreasing functions of $\\lambda $ .", "Note that $\\theta (0)=1$ and $\\theta ^{\\prime }(0)={\\mathrm {Var}}(\\mu )+1$ , and integrate the above inequality to find $\\theta ^2(x) + \\theta ^{\\prime }(x) \\ge -{\\mathrm {Var}}(\\mu )$ for $x \\ge 0$ .", "The identity $Q^{\\prime \\prime }_\\mu (x)=Q_\\mu (x)(\\theta ^2(x)+\\theta ^{\\prime }(x))$ thus yields $Q^{\\prime \\prime }_\\mu (x) \\ge -(1+x){\\mathrm {Var}}(\\mu )$ .", "Integrate, using $Q^{\\prime }_\\mu (0)=1$ , to complete the proof.", "Using these properties, we next justify the existence, uniqueness, and stability for the ODEs appearing in Theorems REF and REF : Proposition 3.2 Let $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ .", "There is a unique continuous function $f_\\mu : {\\mathbb {R}}_+ \\rightarrow {\\mathbb {R}}_+$ , continuously differentiable on $(0,\\infty )$ , satisfying $f^{\\prime }_\\mu (t) = cQ^{\\prime }_\\mu (f_\\mu (t)), \\ \\ \\ t > 0, \\qquad f_\\mu (0)=0.$ Moreover, we have the bounds $0 \\le f_\\mu (t) \\le ct, \\qquad f_\\mu (t) \\ge ct - (\\tfrac{1}{2} c^2t^2 + \\tfrac{1}{6} c^3t^3){\\mathrm {Var}}(\\mu ).$ Finally, if $\\mu _n$ is a sequence in ${\\mathcal {P}}_{\\mathrm {Lap}}$ converging to $\\mu $ , then $Q^{\\prime }_{\\mu _n}$ and $f_{\\mu _n}$ converge uniformly to $Q^{\\prime }_{\\mu }$ and $f_{\\mu }$ , respectively, on compact subsets of ${\\mathbb {R}}_+$ .", "By Proposition REF , $Q^{\\prime }_\\mu $ is nonnegative and Lipschitz (with constant 4) on ${\\mathbb {R}}_+$ .", "A standard Picard iteration yields existence and uniqueness of the ODE in question.", "Next we prove the estimates for $f_\\mu $ .", "Note first that the bound $Q^{\\prime }_\\mu \\le 1$ from Proposition REF ensures that $0 \\le f_\\mu (t) \\le ct$ for all $t \\ge 0$ .", "Use the lower bound on $Q^{\\prime }_\\mu $ from (REF ) to get $f^{\\prime }_\\mu (t) &= cQ^{\\prime }_\\mu (f_\\mu (t)) \\ge c - c\\left(f_\\mu (t) + \\frac{1}{2} f_\\mu (t)^2\\right) {\\mathrm {Var}}(\\mu ) \\ge c - c(ct + \\tfrac{1}{2} c^2t^2){\\mathrm {Var}}(\\mu ).$ Integrate both sides to get the desired lower bound on $f_\\mu (t)$ .", "We next prove that $Q^{\\prime }_{\\mu _n}$ converges to $Q^{\\prime }_{\\mu }$ uniformly on compacts.", "Because $Q^{\\prime }_{\\nu }$ is 4-Lipschitz and $Q^{\\prime }_{\\nu }(0)=1$ for each $\\nu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ , the equicontinuous family $\\lbrace Q^{\\prime }_{\\nu } : \\nu \\in {\\mathcal {P}}_{\\mathrm {Lap}}\\rbrace \\subset C({\\mathbb {R}}_+;{\\mathbb {R}})$ is precompact in the topology of uniform convergence on compacts, by the Arzelà-Ascoli theorem.", "Hence, we need only prove the pointwise convergence of $Q^{\\prime }_{\\mu _n}$ to $Q^{\\prime }_{\\mu }$ .", "But this follows easily from the assumed convergence $\\mu _n \\rightarrow \\mu $ and the form of $Q^{\\prime }$ in (REF ).", "Finally, since $Q^{\\prime }_{\\mu _n}$ is 4-Lipschitz, for $t \\ge 0$ we have $|f_{\\mu _n}(t)-f_\\mu (t)| &\\le c\\int _0^t|Q^{\\prime }_{\\mu _n}(f_{\\mu _n}(s)) - Q^{\\prime }_{\\mu _n}(f_{\\mu }(s))|ds + c\\int _0^t|Q^{\\prime }_{\\mu _n}(f_{\\mu }(s)) - Q^{\\prime }_{\\mu }(f_\\mu (s))|ds \\\\&\\le 4c\\int _0^t| f_{\\mu _n}(s) - f_\\mu (s) |ds + c\\sup _{u \\in [0,ct]}|Q^{\\prime }_{\\mu _n}(u) - Q^{\\prime }_{\\mu }(u)|.$ Use Gronwall's inequality and the uniform convergence of $Q^{\\prime }_{\\mu _n}$ to $Q^{\\prime }_{\\mu }$ on compacts to deduce that $f_{\\mu _n} \\rightarrow f_\\mu $ uniformly on compacts.", "This concludes the basic analysis of $f_\\mu $ and $Q_\\mu $ needed for the proofs of the main results of Sections REF and REF .", "The rest of the section is devoted to some properties of the variance computed in Theorem REF , which will not be needed in the subsequent sections but which justify the claims in Remarks REF and REF .", "Extending the formula (REF ) for the variance, we define for $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ and $t \\in [0,T]$ the quantity ${V}_\\mu (t) := \\sigma ^2\\int _0^t\\int _{[-2,0]} \\left(\\frac{1 - \\lambda f_\\mu (T-t)}{1 - \\lambda f_\\mu (T-s)}\\right)^2\\mu (d\\lambda ) ds.", "$ Proposition 3.3 For each $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ , $f_\\mu $ is strictly increasing and concave, with $0 < f^{\\prime }_\\mu (t) \\le c$ for all $t \\ge 0$ .", "Moreover, ${V}_\\mu $ satisfies ${V}_\\mu (0)=0, \\quad {V}^{\\prime }_\\mu (0)=\\sigma ^2, \\quad \\text{and } \\ {V}^{\\prime \\prime }_\\mu (0) = -2\\sigma ^2\\frac{(f^{\\prime }_\\mu (T))^2}{cQ_\\mu (f_\\mu (T))}.$ Fix $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ .", "First, we note that $f^{\\prime }_\\mu (t)=cQ^{\\prime }_\\mu (f_\\mu (t)), \\qquad \\text{and} \\qquad f^{\\prime \\prime }_\\mu (t) &= cQ^{\\prime \\prime }_\\mu (f_\\mu (t))f^{\\prime }_\\mu (t).$ Using Proposition REF we deduce that $ 0 <f^{\\prime }_\\mu (t) \\le c$ and $f^{\\prime \\prime }_\\mu (t) \\le 0$ for all $t \\ge 0$ , which proves the first claim.", "Differentiating in (REF ) using Liebniz's rule yields ${V}^{\\prime }_\\mu (t) &= \\sigma ^2 - 2\\sigma ^2f^{\\prime }_\\mu (T-t) \\int _0^t\\int _{[-2,0]} \\frac{-\\lambda (1 - \\lambda f_\\mu (T-t))}{(1 - \\lambda f_\\mu (T-s))^2}\\mu (d\\lambda ) ds.$ The claim ${V}^{\\prime }_\\mu (0)=\\sigma ^2$ follows.", "Differentiate again at $t=0$ to get ${V}^{\\prime \\prime }_\\mu (0) &= - 2\\sigma ^2f^{\\prime }_\\mu (T) \\int _{[-2,0]} \\frac{-\\lambda }{ 1 - \\lambda f_\\mu (T) }\\mu (d\\lambda ) \\\\&= - 2\\sigma ^2\\frac{d}{dt}\\Big |_{t=T} \\int _{[-2,0]} \\log ( 1 - \\lambda f_\\mu (t))\\mu (d\\lambda ) \\\\&= - 2\\sigma ^2\\frac{d}{dt}\\Big |_{t=T} \\log Q_\\mu (f_\\mu (t)) \\\\&= - 2\\sigma ^2 \\frac{Q^{\\prime }_\\mu (f_\\mu (T))f^{\\prime }_\\mu (T)}{Q_\\mu (f_\\mu (T))} \\\\&= - 2\\sigma ^2 \\frac{(f^{\\prime }_\\mu (T))^2}{cQ_\\mu (f_\\mu (T))}.$ We next show that the dense graph regime uniquely minimizes the variance, as announced in Remark REF .", "Proposition 3.4 Let $\\mu \\in {\\mathcal {P}}_{\\mathrm {Lap}}$ .", "For each $t \\in (0,T]$ we have ${V}_\\mu (t) \\ge \\sigma ^2t \\frac{1+c(T-t) }{1+cT} = {V}_{\\delta _{-1}}(t),$ with equality if and only if $\\mu =\\delta _{-1}$ .", "Note from Proposition REF that $Q_\\mu (x) \\le 1+x = Q_{\\delta _{-1}}(x)$ and $Q^{\\prime }_\\mu (x) \\le 1 = Q^{\\prime }_{\\delta _{-1}}(x)$ for $x \\ge 0$ .", "By a standard comparison argument, it follows that $f_\\mu (t) \\le f_{\\delta _{-1}}(t)=ct$ for all $t \\ge 0$ .", "Next, note that ${\\mathbb {R}}_+ \\ni t \\mapsto \\frac{1-\\lambda f_\\mu (t)}{1-\\lambda f_{\\delta _{-1}}(t)} = \\frac{1-\\lambda f_\\mu (t)}{1-\\lambda ct} \\ \\ \\text{ is non-increasing for each } \\lambda \\in [-2,0].", "$ Indeed, the derivative is $\\frac{ -\\lambda f^{\\prime }_\\mu (t)(1-\\lambda ct) + \\lambda c(1-\\lambda f_\\mu (t))}{(1-\\lambda ct)^2} = \\frac{-\\lambda (f^{\\prime }_\\mu (t) - c) + c\\lambda ^2(tf^{\\prime }_\\mu (t) - f_\\mu (t)) }{(1-\\lambda ct)^2}.$ This is at most zero, because we know $f^{\\prime }_\\mu (t) \\le c$ by Proposition REF , and the concavity of $f_\\mu $ together with $f_\\mu (0) = 0$ imply $tf^{\\prime }_\\mu (t) \\le f_\\mu (t)$ .", "This proves (REF ), which is equivalent to the fact that $\\frac{1-\\lambda f_\\mu (T-t)}{1-\\lambda f_\\mu (T-s)} \\ge \\frac{1-\\lambda f_{\\delta _{-1}}(T-t)}{1-\\lambda f_{\\delta _{-1}}(T-s)} = \\frac{1-\\lambda c(T-t)}{1-\\lambda c(T-s)}, \\quad \\text{for } t > s > 0.$ As both sides are non-negative, this implies $\\int _{[-2,0]} \\left(\\frac{1 - \\lambda f_\\mu (T-t)}{1 - \\lambda f_\\mu (T-s)}\\right)^2\\mu (d\\lambda ) \\ge \\int _{[-2,0]} \\left(\\frac{1-\\lambda c(T-t)}{1-\\lambda c(T-s)} \\right)^2\\mu (d\\lambda ).", "$ Finally, the function $[-2,0]\\ni \\lambda \\mapsto \\left(\\frac{ 1- \\lambda c(T-t)}{1- \\lambda c(T-s)}\\right)^2$ is strictly convex for $t > s > 0$ , which by Jensen's inequality implies $\\int _{[-2,0]} \\left(\\frac{1-\\lambda c(T-t)}{1-\\lambda c(T-s)} \\right)^2\\mu (d\\lambda ) \\ge \\left(\\frac{1 +c(T-t)}{1 +c(T-s)}\\right)^2 $ since $\\mu $ has mean $-1$ .", "Combine (REF ) and (REF ) with the definition of ${V}_\\mu $ to get ${V}_\\mu (t) \\ge {V}_{\\delta _{-1}}(t)$ , as desired.", "The inequality (REF ) is strict unless $\\mu =\\delta _{-1}$ ." ], [ "The equilibrium on finite graphs: Proof of Theorem ", "This section gives the proof of Theorem REF .", "We begin with some general symmetry considerations in Section REF .", "We then derive the HJB system in Section REF and reduce it to a system of Riccati equations in Section REF ; these two steps are standard for linear-quadratic games.", "The explicit resolution of the system of Riccati equations is where the difficulty lies.", "In Section REF we show that the proposed solution Theorem REF does indeed work, and the remaining sections REF and REF provide the remaining computations of the equilibrium state process dynamics and the average value of the game.", "The proof given in this section, while complete and rigorous, is opaque in the sense that it does not give any idea of how one might arrive at the solution of the system of Riccati equations.", "For this reason, we give in Section a sketch a direct derivation of the solution.", "We fix throughout the section a finite transitive graph $G = (V,E)$ , and write $V=\\lbrace 1,\\ldots ,n\\rbrace $ for some $n \\in {\\mathbb {N}}$ .", "We may assume without loss of generality that $G$ is connected (see Remark REF ).", "Throughout this entire section, we omit $G$ from the notation by writing, e.g., $L=L_G$ and $\\mathbf {X}=\\mathbf {X}^G$ ." ], [ "Symmetries", "We first discuss some basic symmetry properties.", "Since the graph $G$ is transitive and thus regular, the Laplacian matrix is symmetric, i.e., $L=L^\\top $ .", "We will make some use of the so-called regular representation of the automorphism group.", "Recall from the beginning of Section REF that ${\\mathrm {Aut}}(G)$ denotes the set of automorphisms of $G$ .", "To each $\\varphi \\in {\\mathrm {Aut}}(G)$ we associate an invertible $n \\times n$ matrix $R_\\varphi $ , defined by requiring $R_\\varphi e_i = e_{\\varphi (i)}$ for each $i \\in V$ , where we recall that $(e_1, ..., e_n)$ denotes the standard Euclidean basis in ${\\mathbb {R}}^n$ .", "It is clear that $R_\\varphi R_\\psi = R_{\\varphi \\circ \\psi }$ , and in particular $R_\\varphi ^{-1} = R_{\\varphi ^{-1}}$ .", "We also have $R_\\varphi ^\\top = R_{\\varphi ^{-1}}$ , because $e_i^\\top R_\\varphi e_j &= e_i^\\top e_{\\varphi (j)} = 1_{\\lbrace i = \\varphi (j)\\rbrace } = 1_{\\lbrace j = \\varphi ^{-1}(i)\\rbrace } = e_j^\\top R_{\\varphi ^{-1}}e_i.$ The following elementary lemma summarizes some uses of transitivity (Definition REF ).", "The third property will be used only in the alternative proof of Theorem REF given in Section .", "Lemma 4.1 Assume that $G=(V,E)$ is transitive, with $V=\\lbrace 1,\\ldots ,n\\rbrace $ .", "$L$ commutes with $R_\\varphi $ for each $\\varphi \\in {\\mathrm {Aut}}(G)$ .", "If $Y \\in {\\mathbb {R}}^{n \\times n}$ commutes with $R_{\\varphi }$ for every $\\varphi \\in {\\mathrm {Aut}}(G)$ , then $Y_{ii} = \\frac{1}{n}{\\mathrm {Tr}}(Y), \\qquad \\forall \\, i\\in V.$ If $Y^1, \\ldots , Y^n \\in {\\mathbb {R}}^{n \\times n}$ satisfy $R_{\\varphi } Y^i = Y^{\\varphi (i)}R_{\\varphi }$ for every $\\varphi \\in {\\mathrm {Aut}}(G)$ and $i \\in V$ , then $Y^{i}_{ii} = Y^{j}_{jj}, \\qquad \\forall \\, i,j \\in V.$ For $\\varphi \\in {\\mathrm {Aut}}(G)$ we compute $e_i^\\top R_\\varphi ^\\top LR_\\varphi e_j = e_{\\varphi (i)}^\\top Le_{\\varphi (j)} = e_i^\\top Le_j, \\qquad \\forall i,j \\in V,$ with the last equality using the fact that $\\varphi $ is an automorphism.", "This shows that $R_\\varphi ^\\top LR_\\varphi =L$ .", "Since $R_\\varphi ^\\top =R_\\varphi ^{-1}$ , this shows $LR_{\\varphi } = R_{\\varphi }L$ .", "It suffices to apply (iii) with $Y^1=\\cdots =Y^n=Y$ to get $Y_{ii}=Y_{jj}$ for all $i,j \\in V$ .", "Let $i,j \\in V$ .", "By transitivity, there exists $\\varphi \\in {\\mathrm {Aut}}(G)$ such that $\\varphi (i) = j$ .", "Then $e_i^\\top Y^i e_i = e_i^\\top R_{\\varphi }^T Y^{\\varphi (i)} R_{\\varphi } e_i = e_{\\varphi (i)}^\\top Y^{\\varphi (i)} e_{\\varphi (i)} = e_j^\\top Y^j e_j.$" ], [ "The corresponding system of HJB equations", "We can write the cost function of (REF ) for player $i \\in V$ as $J_i(\\alpha _1,\\ldots ,\\alpha _n)=\\frac{1}{2}{\\mathbb {E}}\\left[ \\int _0^T |\\alpha _i(t, \\mathbf {X}(t))|^2dt + c\\left|e_i^\\top L\\mathbf {X}(T)\\right|^2 \\right].$ A standard argument associates this $n$ -player game to the following system of $n$ coupled PDEs: $\\begin{split}&0 = \\partial _t v_i(t,\\mathbf {x}) - \\frac{1}{2}(\\partial _i v_i(t,\\mathbf {x}))^2 - \\sum _{k \\ne i} \\partial _k v_k(t,\\mathbf {x}) \\partial _k v_i(t,\\mathbf {x}) + \\frac{\\sigma ^2}{2} \\sum _{k=1}^n \\partial _{kk} v_i(t,\\mathbf {x}), \\\\&v_i(T,\\mathbf {x}) = \\frac{1}{2} c(e_i^\\top L\\mathbf {x})^2, \\qquad i=1,\\ldots ,n.\\end{split} $ Here, we write $\\mathbf {x}=(x_1,\\ldots ,x_n)$ for a typical vector in ${\\mathbb {R}}^n$ , and for the functions $v_i : [0,T] \\times {\\mathbb {R}}^n \\rightarrow {\\mathbb {R}}$ we write $\\partial _t$ and $\\partial _k$ for the derivative with respect to $t$ and $x_k$ , respectively.", "If $(v_1,\\ldots ,v_n)$ is a classical solution of (REF ), then the controls $\\alpha _i(t,\\mathbf {x}) = -\\partial _i v_i(t,\\mathbf {x}), \\quad i =1,\\ldots ,n, $ form a Markovian Nash equilibrium.", "For a thorough derivation of PDEs of this form and a verification theorem, we refer to [9].", "But let us briefly explain how to check if some $(\\alpha _1,\\ldots ,\\alpha _n) \\in {\\mathcal {A}}_G^n$ forms a Nash equilibrium.", "Considering player $i$ 's optimization problem, $\\inf _{\\alpha _i \\in {\\mathcal {A}}_G}J_i(\\alpha _1,\\ldots ,\\alpha _{i-1},\\alpha _i,\\alpha _{i+1},\\ldots ,\\alpha _n).", "$ Standard stochastic control theory (see, e.g., [18], [37]) leads to the HJB equation $\\begin{split}&0 = \\partial _t v_i(t,\\mathbf {x}) - \\frac{1}{2}(\\partial _i v_i(t,\\mathbf {x}))^2 + \\sum _{k \\ne i} \\alpha _k(t,\\mathbf {x}) \\partial _k v_i(t,\\mathbf {x}) + \\frac{\\sigma ^2}{2} \\sum _{k=1}^n \\partial _{kk} v_i(t,\\mathbf {x}), \\\\&v_i(T,\\mathbf {x}) = \\frac{1}{2} c(e_i^\\top L\\mathbf {x})^2.\\end{split} $ Indeed, the (reduced) Hamiltonian for player $i$ is $H_i(\\mathbf {x},\\mathbf {p}) = \\inf _{a \\in {\\mathbb {R}}}\\left(\\frac{1}{2} a^2 + a p_i + \\sum _{k \\ne i}\\alpha _k(t,\\mathbf {x})p_k\\right) = - \\frac{1}{2} p_i^2 + \\sum _{k \\ne i}\\alpha _k(t,\\mathbf {x})p_k, \\qquad \\mathbf {x}, \\mathbf {p} \\in {\\mathbb {R}}^n,$ with the infimum attained at $a=-p_i$ .", "Hence, after solving the PDE (REF ), the optimal control in (REF ) is given by $\\alpha _i(t, \\mathbf {x}) = -\\partial _iv_i(t,\\mathbf {x})$ .", "Applying this optimality criterion for each player $i=1,\\ldots ,n$ couples the PDEs (REF ), leading to the system (REF )." ], [ "Reduction to Riccati equations", "Linear-quadratic control problems and games always relate to Riccati-type equations after a quadratic ansatz.", "For our PDE system (REF ), we make the ansatz $v_i(t,\\mathbf {x}) = \\frac{1}{2} \\mathbf {x}^\\top F^i(t) \\mathbf {x} + h_i(t), \\qquad (t,\\mathbf {x}) \\in [0,T] \\times {\\mathbb {R}}^n, $ where $F^i : [0,T] \\rightarrow {\\mathbb {R}}^{n \\times n}$ and $h_i : [0,T] \\rightarrow {\\mathbb {R}}$ are functions to be determined.", "We assume without loss of generality that $F^i(t)$ is symmetric for each $t$ .", "Recall that $L=L^\\top $ , and note that the boundary condition $v_i(T,\\mathbf {x})=\\frac{1}{2} c(e_i^\\top L\\mathbf {x})^2 = \\frac{1}{2} c \\mathbf {x}^\\top Le_i e_i^\\top L\\mathbf {x}$ leads to the boundary conditions $F^i(T) = cLe_i e_i^\\top L, \\qquad h_i(T)=0.", "$ We write $\\dot{F^i}$ and $\\dot{h}_i$ for the derivatives of these functions.", "Once we check that this ansatz is correct, the equilibrium controls are given by $\\alpha _i(t,\\mathbf {x}) = - \\partial _i v_i(t,\\mathbf {x})= - e_i^\\top F^i(t)\\mathbf {x}, \\quad i=1,\\ldots ,n. $ Applying the ansatz (REF ) to the PDE system (REF ), noting that $\\partial _kv_i(t,\\mathbf {x}) = e_k^\\top F^i(t)\\mathbf {x}$ and $\\partial _{kk}v_i(t,\\mathbf {x}) = e_k^\\top F^i(t) e_k$ , leads to $\\frac{1}{2}\\mathbf {x}^\\top \\dot{F}^i(t) \\mathbf {x} + \\dot{h}_i(t) + \\frac{1}{2} (e_i^\\top F^i(t) \\mathbf {x})^2 - \\sum _{k = 1}^n \\left(e_k^\\top F^k(t) \\mathbf {x} \\right)\\left(e_k^\\top F^i(t) \\mathbf {x} \\right) + \\frac{\\sigma ^2}{2} {\\mathrm {Tr}}(F^i(t)) = 0.$ Collecting like terms, we find $\\mathbf {x}^\\top \\Big ( \\frac{1}{2} \\dot{F}^i(t) - \\sum _{k = 1}^n F^k(t)e_k e_k^\\top F^i(t) +\\frac{1}{2} F^i(t)e_i e_i^\\top F^i(t) \\Big ) \\mathbf {x} + \\dot{h}_i(t) + \\frac{\\sigma ^2}{2}{\\mathrm {Tr}}(F^i(t)) =0.$ This must hold for each $\\mathbf {x} \\in {\\mathbb {R}}^n$ , and we note that a square matrix $A$ satisfies $\\mathbf {x}^\\top A \\mathbf {x}=0$ if any only if $A+A^\\top =0$ .", "Recalling that $F^i(t)$ is symmetric, we arrive at the two equations: $0 &= \\dot{F}^i(t) - \\sum _{j = 1}^n F^j(t)e_j e_j^\\top F^i(t) - F^i(t)\\sum _{j = 1}^n e_j e_j^\\top F^j(t) + F^i(t)e_i e_i^\\top F^i(t), \\\\0 &= \\dot{h}_i(t) + \\frac{\\sigma ^2}{2} {\\mathrm {Tr}}(F^i(t)), \\qquad i=1,\\ldots ,n. \\nonumber $ Once we solve for $F^i$ using the first equation, the second equation and the boundary condition $h_i(T)=0$ yield $h_i(t) &= \\frac{\\sigma ^2}{2}\\int _t^T {\\mathrm {Tr}}(F^i(s))ds.", "$ Hence, the main task ahead is to solve the system (REF ).", "A key role will be played by the matrix $P(t) = \\sum _{j = 1}^n F^j(t)e_j e_j^\\top ,$ which in equilibrium will agree with $P_G(t)$ defined in (REF )." ], [ "Checking the solution", "It follows from the results of Section that the ODE $f^{\\prime }(t) = c Q^{\\prime }(f(t)), \\qquad f(0) = 0,$ is well-posed, where $Q(x)=Q_G(x)=(\\det (I-xL))^{1/n}$ , and the solution $f$ is nonnegative and strictly increasing since $Q^{\\prime } > 0$ .", "As a result, the matrix-valued function defined in (REF ) by $P(t) =P_G(t):= - f^{\\prime }(T-t) L\\big (I - f(T-t) L\\big )^{-1} $ is well-defined; the symmetric matrix $I - f(T-t) L$ is invertible because $L$ is negative semidefinite and $f \\ge 0$ .", "Moreover, $P(t)$ satisfies the following useful properties for each $t \\in [0,T]$ : $e_i^T P(t) e_i = \\frac{1}{n} {\\mathrm {Tr}}(P(t)) > 0$ for all $i = 1,\\ldots , n$ .", "$P(t)$ is symmetric and positive semidefinite.", "Both $L$ and $P(t)$ have 0 as an eigenvalue, with the same multiplicity.", "$P(t)$ and $L$ commute.", "Indeed, the third property is trivial.", "The second follows from the facts that $L$ is negative semidefinite, $f \\ge 0$ , and $f^{\\prime } > 0$ ; note that the vector of all ones is an eigenvector of $L$ with eigenvalue zero, and thus also an eigenvector of $P(t)$ with eigenvalue zero.", "To derive the first property, note that $L$ commutes with $R_{\\varphi }$ for all $\\varphi \\in {\\mathrm {Aut}}(G)$ by Lemma REF (i), and thus so does $P(t)$ , which means Lemma REF (ii) applies.", "The strict positivity of ${\\mathrm {Tr}}(P(t))$ follows from the fact that $P(t)$ is positive semidefinite and is not the zero matrix.", "Since ${\\mathrm {Tr}}(P(t)) > 0$ by property (i), we may define $F^i$ for $i=1,\\ldots ,n$ by $F^i(t) := \\frac{1}{{\\mathrm {Tr}}(P(t))/n} P(t)e_ie_i^\\top P(t).", "$ Using property (i), we compute $e_i^\\top F^i(t) &= \\frac{1}{{\\mathrm {Tr}}{(P(t))}/n}\\left(e_i^\\top P(t) e_i\\right)e_i^\\top P(t) = e_i^\\top P(t).$ In other words, the $i^\\text{th}$ column of $F^i(t)$ is the same as that of $P(t)$ .", "In particular, the control $\\alpha ^*_i(t,\\mathbf {x}) = - e_i^\\top P(t) \\mathbf {x}$ defined in Theorem REF satisfies $\\alpha ^*_i(t,\\mathbf {x}) = - e_i^\\top F^i(t) \\mathbf {x}$ .", "Recall from (REF ) that this was indeed the form dictated by the PDE.", "We next check that $(F^1,\\ldots ,F^n)$ solve the equations (REF ) and boundary conditions (REF ).", "Beginning with the boundary condition (REF ), note first that $f(0)=0$ and thus $P(T) = - f^{\\prime }(0)L= - c Q^{\\prime }(0) L= - c L.$ This implies $\\frac{{\\mathrm {Tr}}{(P(T)})}{n} = c$ , and from which we immediately compute $F^i(T)=cLe_ie_i^\\top L$ , as desired.", "We next turn to the equations (REF ).", "Using property (i) once again, we compute $\\sum _{i=1}^n F^i(t)e_ie_i^\\top = \\frac{1}{{\\mathrm {Tr}}{(P(t))}/n} \\sum _{i=1}^n P(t)e_i\\left(e_i^\\top P(t) e_i\\right)e_i^\\top = P(t) \\sum _{i=1}^n e_ie_i^\\top = P(t).$ Similarly, $F^i(t)e_ie_i^\\top F^i(t) = P(t) e_i e_i^\\top P(t).$ With these identifications, to show that $(F^1, \\ldots , F^n)$ solves (REF ), it suffices to check that $\\dot{F^i}(t) - P(t) F^i(t) - F^i(t) P(t) + P(t) e_i e_i^T P(t) = 0, \\qquad t \\in (0,T).$ For this purpose, let us define $\\tau (t) := {\\mathrm {Tr}}(P(t))/n > 0$ .", "Omitting the time-dependence, the equation (REF ) then becomes $- \\frac{\\tau ^{\\prime }}{\\tau ^2} P e_ie_i^\\top P + \\frac{1}{\\tau }\\dot{P}e_ie_i^\\top P + \\frac{1}{\\tau }Pe_ie_i^\\top \\dot{P} - \\frac{1}{\\tau }P^2 e_ie_i^\\top P - \\frac{1}{\\tau }P e_ie_i^\\top P^2 + P e_ie_i^\\top P =0,$ or equivalently, multiplying by $\\tau $ , $- \\frac{\\tau ^{\\prime }}{\\tau } P e_ie_i^\\top P + \\dot{P}e_ie_i^\\top P + P e_ie_i^\\top \\dot{P} - P^2e_ie_i^\\top P - Pe_ie_i^\\top P^2 + \\tau P e_ie_i^\\top P =0.$ Hence $(F^1, ..., F^n)$ are solutions of the ODEs (REF ) if and only if $P$ solves the ODEs (REF ) for $i =1, \\ldots , n$ , with $\\tau (t)={\\mathrm {Tr}}(P(t))/n$ .", "Now let $v_1,\\ldots ,v_n$ denote an orthonormal basis of eigenvectors of the symmetric matrix $L$ , with associated eigenvalues $\\lambda _1,\\ldots ,\\lambda _n$ .", "From the definition (REF ) of $P(t)$ it follows that $v_1,\\ldots ,v_n$ are eigenvectors of $P(t)$ , and the eigenvalue $\\rho _j(t)$ of $P(t)$ associated with the eigenvector $v_j$ is given by $\\rho _j(t) &= \\frac{- f^{\\prime }(T-t) \\lambda _j}{1 - f(T-t) \\lambda _j} = - \\partial _t \\log (1 - f(T-t) \\lambda _j).$ Note that because $P(t)$ is symmetric, we have not only $P(t)v_k=\\rho _k(t)v_k$ but also $v_k^\\top P(t) = \\rho _k(t) v_k^\\top $ for all $k$ .", "Now, $P(t)$ satisfies (REF ) for all $i=1,...,n$ if and only if for all $i,j,k$ we have $v_j^\\top \\left(-\\frac{\\tau ^{\\prime }}{\\tau } P e_ie_i^\\top P + \\dot{P}e_ie_i^\\top P + P e_ie_i^\\top \\dot{P} - P^2e_ie_i^\\top P - Pe_ie_i^\\top P^2 + \\tau P e_ie_i^\\top P\\right) v_k =0,$ which is equivalent to $(v_j^\\top e_i) (e_i^\\top v_k) \\left(- \\frac{\\tau ^{\\prime }}{\\tau } \\rho _j \\rho _k + \\rho _j^{\\prime }\\rho _k + \\rho _j\\rho _k^{\\prime } - \\rho _j^2 \\rho _k - \\rho _j \\rho _k^2 + \\tau \\rho _j \\rho _k \\right) = 0,$ where we omit the time-dependence from $\\rho _k=\\rho _k(t)$ .", "Notice that the term in the parenthesis can be written as $\\left(\\rho _k^{\\prime } - \\rho _k^2 - \\frac{\\rho _k}{2}\\left( \\frac{\\tau ^{\\prime }}{\\tau } - \\tau \\right) \\right) \\rho _j + \\left(\\rho _j^{\\prime } - \\rho _j^2 - \\frac{\\rho _j}{2}\\left( \\frac{\\tau ^{\\prime }}{\\tau } - \\tau \\right) \\right) \\rho _k.$ To complete the proof, it thus suffices to show that $\\rho _k^{\\prime } - \\rho _k^2 - \\frac{\\rho _k}{2}\\left( \\frac{\\tau ^{\\prime }}{\\tau } - \\tau \\right) =0.$ For $k$ such that $Lv_k=0$ , this holds trivially because $\\rho _k \\equiv 0$ ; for all other $k$ we have $\\rho _k(t) > 0$ for all $t \\in [0,T]$ , and by dividing by $\\rho _k$ , we may show instead that $\\frac{\\rho _k^{\\prime }}{\\rho _k} - \\rho _k = \\frac{1}{2}\\left(\\frac{\\tau ^{\\prime }}{\\tau } - \\tau \\right), \\ \\ \\text{ for } k \\text{ such that } \\rho _k(t) > 0 \\ \\forall t \\in [0,T],$ where $\\tau := \\frac{1}{n}{\\mathrm {Tr}}(P) = \\frac{1}{n}\\sum _{k=1}^n\\rho _k$ .", "Now, using the form of $\\rho _k$ from (REF ), a straightforward computation yields $\\frac{\\rho _k^{\\prime }(t)}{\\rho _k(t)} - \\rho _k(t) = - \\frac{f^{\\prime \\prime }(T-t)}{f^{\\prime }(T-t)}.", "$ To simplify the right-hand side of (REF ), we recall $Q(x):=(\\det (I-xL))^{1/n}=\\prod _{k=1}^n(1-x\\lambda _k)^{1/n}$ and compute $\\tau (t) &= \\frac{1}{n}\\sum _{k=1}^n\\rho _k(t) = - \\frac{1}{n}\\sum _{k = 1}^n \\partial _t \\log (1 - f(T-t)\\lambda _k) \\\\&= - \\partial _t \\log \\prod _{k=1}^n (1 - f(T-t)\\lambda _k)^{1/n}, \\\\&= - \\frac{\\partial }{\\partial t} \\log Q(f(T-t)) = \\frac{f^{\\prime }(T-t)Q^{\\prime }(f(T-t))}{Q(f(T-t))}.$ Therefore after computing the derivative of $\\tau $ and rearranging we obtain $\\frac{\\tau ^{\\prime }(t)}{\\tau (t)} - \\tau (t) = - \\frac{f^{\\prime \\prime }(T-t)}{f^{\\prime }(T-t)} - \\frac{f^{\\prime }(T-t)Q^{\\prime \\prime }(f(T-t))}{Q^{\\prime }(f(T-t))}.", "$ Now, since $f$ solves the ODE $f^{\\prime }(t)=cQ^{\\prime }(f(t))$ , we have also $f^{\\prime \\prime }(t) = c Q^{\\prime \\prime }(f(t))f^{\\prime }(t)$ , and we compute $\\frac{f^{\\prime }(T-t)Q^{\\prime \\prime }(f(T-t))}{Q^{\\prime }(f(T-t))} &= cQ^{\\prime \\prime }(f(T-t)) = \\frac{f^{\\prime \\prime }(T-t)}{f^{\\prime }(T-t)}.$ Returning to (REF ), we deduce $\\frac{\\tau ^{\\prime }(t)}{\\tau (t)} - \\tau (t) = - 2 \\frac{f^{\\prime \\prime }(T-t)}{ f^{\\prime }(T-t)}.$ Recalling (REF ), this proves (REF ), which completes the proof that $(F^1,\\ldots ,F^n)$ defined in (REF ) solves the desired equations.", "Thus, the controls $\\alpha _i^*(t,\\mathbf {x}) = -e_i^\\top P(t)\\mathbf {x}$ form a Nash equilibrium as discussed above." ], [ "State process dynamics in equilibrium", "The state process in equilibrium is $dX_i(t) = - e_i^TP(t) \\mathbf {X}(t) dt + \\sigma dW_i(t), \\qquad i = 1,\\ldots ,n,$ which can be written in vector form as $d\\mathbf {X}(t) = - P(t) \\mathbf {X}(t) dt + \\sigma d\\mathbf {W}(t).$ We can explicitly solve this SDE in terms of $P$ , noting that $P(t)$ and $P(s)$ commute for all $t,s \\in [0,T]$ , to obtain $\\mathbf {X}(t) = e^{- \\int _0^t P(s)ds}\\mathbf {X}(0) + \\sigma \\int _0^t e^{ -\\int _s^t P(u)du}d\\mathbf {W}(s).$ Thus we deduce that in equilibrium the law of $\\mathbf {X}(t)$ is $\\mathbf {X}(t) \\sim \\mathcal {N} \\left(e^{- \\int _0^t P(s)ds} \\mathbf {X}(0), \\sigma ^2\\int _0^t e^{-2 \\int _s^t P(u)du}ds\\right).$ Using the expression $P(t) = - \\partial _t \\log (I - f(T-t)L)$ , noting that the log of the positive definite matrix is well-defined via power series, we have $\\exp \\left(-\\int _s^tP(u)du\\right) &= (I - f(T-t)L)(I - f(T-s)L)^{-1}.$ Hence, $\\mathbf {X}(t)$ has mean vector ${\\mathbb {E}}[\\mathbf {X}(t)]=(I - f(T-t)L)(I - f(T)L)^{-1}\\mathbf {X}(0)$ and covariance matrix ${\\mathrm {Var}}(\\mathbf {X}(t))=\\sigma ^2(I - f(T-t)L)^2 \\int _0^t (I - f(T-s)L)^{-2} ds.$" ], [ "Computing the value of the game", "We next justify (REF ).", "Returning to the ansatz (REF ) and (REF ), and recalling the form of $F^i$ from (REF ), we find $\\frac{1}{n}\\sum _{i=1}^nv_k(t,\\mathbf {x}) &= \\frac{1}{n}\\sum _{i=1}^n\\left(\\frac{1}{2} \\mathbf {x}^\\top F^i(t)\\mathbf {x} + h_i(t) \\right) \\\\&= \\frac{1}{2{\\mathrm {Tr}}(P(t))}\\sum _{i=1}^n ( e_i^\\top P(t)\\mathbf {x})^2 + \\frac{\\sigma ^2}{2n}\\sum _{i=1}^n \\int _t^T {\\mathrm {Tr}}(F^i(s))ds.", "$ To compute the second term, we define $R(x) = \\log \\frac{1}{n}\\sum _{k=1}^n \\frac{-\\lambda _k }{1 - x\\lambda _k}$ for $x \\ge 0$ , so that $\\frac{1}{n}\\sum _{i=1}^n {\\mathrm {Tr}}(F^i(t)) &= \\frac{1}{n}\\sum _{i,k=1}^n e_k^\\top F^i(t)e_k = \\frac{1}{{\\mathrm {Tr}}(P(t))}\\sum _{i,k=1}^n e_k^\\top P(t) e_i e_i^\\top P(t)e_k \\\\&= \\frac{1}{{\\mathrm {Tr}}(P(s))}\\sum _{k=1}^n e_k^\\top P(t)^2e_k \\\\&= \\frac{{\\mathrm {Tr}}(P(t)^2)}{{\\mathrm {Tr}}(P(t))} \\\\&= -\\sum _{k=1}^n \\frac{(f^{\\prime }(T-t))^2\\lambda _k^2}{(1-f(T-t)\\lambda _k)^2} \\Bigg / \\sum _{k=1}^n \\frac{f^{\\prime }(T-t)\\lambda _k}{1-f(T-t)\\lambda _k} \\\\&= -R^{\\prime }(f(T-t))f^{\\prime }(T-t) \\\\&= \\partial _t R(f(T-t)) .$ Thus, using $f(0)=0$ and $R(0)=0$ , we get $\\frac{\\sigma ^2}{2n}\\sum _{i=1}^n \\int _t^T {\\mathrm {Tr}}(F^i(s))ds &= \\frac{\\sigma ^2}{2}[R(f(0)) - R(f(T-t))] = - \\frac{\\sigma ^2}{2}R(f(T-t)).$ Finally, note that $R(f(T-t)) = \\log \\frac{1}{n}\\sum _{k=1}^n \\frac{-\\lambda _k }{1 - f(T-t)\\lambda _k} = \\log \\frac{{\\mathrm {Tr}}(P(t))}{nf^{\\prime }(T-t)}.$ Returning to (REF ), we get $\\frac{1}{n}\\sum _{i=1}^nv_k(t,\\mathbf {x}) &= \\frac{|P(t)\\mathbf {x}|^2}{2{\\mathrm {Tr}}(P(t))} - \\frac{\\sigma ^2}{2}\\log \\frac{{\\mathrm {Tr}}(P(t))}{nf^{\\prime }(T-t)}.$ Plugging in $t=0$ and $\\mathbf {x}=\\mathbf {X}^G(0)$ , the left-hand side equals $\\mathrm {Val}(G)$ , and the proof of (REF ) is complete." ], [ "A direct but heuristic proof of Theorem ", "In this section we aim to give a more enlightening derivation of the solution given in Theorem REF , compared to the more concise “guess and check\" style of proof presented in Section .", "To keep this brief, we will avoid giving full details and instead treat this section as a heuristic argument.", "Suppose throughout this section that $G$ is a given finite transitive graph with vertex set $V=\\lbrace 1,\\ldots ,n\\rbrace $ , and omit $G$ from the notations as in $L=L_G$ ." ], [ "More on symmetries", "As a first step, we elaborate on the symmetry discussion of Section REF .", "Recall the notation ${\\mathrm {Aut}}(G)$ and $R_\\varphi $ introduced therein, and note that ${\\mathrm {Aut}}(G)$ acts naturally on ${\\mathbb {R}}^V$ via $(\\varphi ,\\mathbf {x}) \\mapsto R_{\\varphi } \\mathbf {x}$ .", "Suppose we have a solution $(v_1,\\ldots ,v_n)$ of the HJB system (REF ).", "The structure of the game described in Section REF is invariant under automorphisms of $G$ .", "More precisely, this should translate into the following symmetry property for the value functions: $v_i(t,\\mathbf {x}) = v_{\\varphi (i)}(t, R_{\\varphi } \\mathbf {x}), \\qquad i \\in V, \\ \\ \\varphi \\in {\\mathrm {Aut}}(G), \\ \\ \\mathbf {x} \\in {\\mathbb {R}}^V.", "$ In particular, if $(v_i(t,\\mathbf {x}))_{i \\in V}$ solves the HJB system (REF ), then a straightforward calculation shows that so does $(v_{\\varphi (i)}(t, R_{\\varphi } \\mathbf {x}))_{i \\in V}$ .", "Hence, if uniqueness holds for (REF ), then we would deduce (REF ).", "Plugging the quadratic ansatz (REF ) into both sides of (REF ) yields $\\mathbf {x}^\\top F^i(t) \\mathbf {x} + h_i(t) = \\mathbf {x}^\\top R_{\\varphi }^\\top F^{\\varphi (i)}(t) R_{\\varphi }\\mathbf {x} + h_{\\varphi (i)}(t), \\qquad i \\in V, \\ \\ \\varphi \\in {\\mathrm {Aut}}(G), \\ \\ \\mathbf {x} \\in {\\mathbb {R}}^V.$ Matching coefficients yields $F^i(t) = R_{\\varphi }^\\top F^{\\varphi (i)}(t)R_{\\varphi }, \\qquad h_i(t) = h_{\\varphi (i)}(t), \\qquad i \\in V, \\ \\ \\varphi \\in {\\mathrm {Aut}}(G).", "$ This immediately shows that the map $i \\mapsto h_i(t)$ is constant on orbits of the action of ${\\mathrm {Aut}}(G)$ on $V$ .", "That is, if $i \\in V$ and $\\mathrm {Orbit}(i) := \\lbrace \\varphi (i) : \\varphi \\in {\\mathrm {Aut}}(G)\\rbrace ,$ then $h_k(t) = h_j(t)$ for all $k,j \\in \\mathrm {Orbit}(i)$ .", "Note that the orbits $\\lbrace \\mathrm {Orbit}(i) : i \\in V\\rbrace $ form a partition $V$ , and the assumption that $G$ is transitive (Definition REF ) means precisely that $V$ itself is the only orbit.", "Similarly, elaborating on the first identity in (REF ), for any $i,j,k \\in V$ we find $e_k^\\top F^i(t) e_j = e_k^\\top R_{\\varphi }^\\top F^{\\varphi (i)}(t)R_{\\varphi } e_j = e_{\\varphi (k)}^\\top F^{\\varphi (i)}(t) e_{\\varphi (j)}.$ In the Riccati equation (REF ), a key role was played by the matrix $P(t) := \\sum _{j = 1}^n F^j(t)e_j e_j^\\top $ .", "Under a stronger transitivity assumption on the graph, the symmetry property (REF ) is enough to ensure that $P(t)$ is symmetric and commutes with $L$ for each $t$ : Definition 5.1 We say that a graph $G$ is generously transitive if for each $i,j \\in V$ there exists $\\varphi \\in {\\mathrm {Aut}}(G)$ such that $\\varphi (i)=j$ and $\\varphi (j)=i$ .", "Proposition 5.2 If $G$ is generously transitive and (REF ) holds, then for each $t \\in [0,T]$ the matrix $P(t) := \\sum _{j = 1}^n F^j(t)e_j e_j^\\top $ satisfies: $P(t) = P(t)^\\top $ .", "$P(t)$ and $L$ commute.", "For brevity, we write $F^k=F^k(t)$ and $P=P(t)$ .", "Note that $Pe_k=F^ke_k$ for each $k=1,\\ldots ,n$ .", "Let $j,k \\in V$ , and let $\\varphi \\in {\\mathrm {Aut}}(G)$ be such that $\\varphi (j)=k$ and $\\varphi (k)=j$ .", "We find from (REF ) that $e_j^\\top P e_k = e_j^\\top F^k e_k = e_{\\varphi (j)}^\\top F^{\\varphi (k)} e_{\\varphi (k)} = e_{k}^\\top F^{j} e_j= e_{k}^\\top P e_j.$ Using the identities $e_j = R_\\varphi e_k$ and $e_k = R_\\varphi e_j$ , then the fact that $L$ commutes with $R_{\\varphi }^\\top $ , and finally that $R_{\\varphi }^{\\top }F^{k}R_{\\varphi } = R_{\\varphi }^{\\top }F^{\\varphi (j)}R_{\\varphi } = F^j$ we get $e_j^\\top LP e_k = e_j^\\top LF^k e_k = e_k^\\top R_{\\varphi }^\\top LF^k R_\\varphi e_j = e_k^\\top LR_{\\varphi }^\\top F^k R_\\varphi e_j = e_k^\\top LF^j e_j = e_k^\\top LP e_j.$ In our “guess and check\" proof of Theorem REF given in Section , the two properties of Proposition REF followed automatically from the asserted formula (REF ) for $P(t)$ .", "Here we see, on the other hand, that these properties follow from purely algebraic arguments." ], [ "Heuristic solution of the HJB system", "In this section we will explain how to find the expressions of $P$ and $F^i$ .", "We start here from the Riccati equation (REF ), and assume from now on that the graph $G$ is generously transitive.", "The objective is to find $F^1,\\ldots , F^n$ such that each $F^i$ is a solution of the matrix differential equation $\\dot{F^i} - P F^i - F^i P^\\top + F^i e_i e_i^\\top F^i = 0,$ with terminal condition $F^i(T) = c Le_i e_i^\\top L$ , and with $P = \\sum _{i=1}^n F^ie_ie_i^\\top $ .", "We use a fixed point approach: Treating $P$ as given, this is a system of $n$ decoupled Ricatti differential equations which we can solve.", "The solutions $F^1, ..., F^n$ give rise to a new $P$ , which we then match to the original $P$ .", "Now fix $P$ , which by Proposition REF we can assume to be symmetric and commuting with $L$ , and let us solve for $F^1,\\ldots ,F^n$ .", "From [38], we know that if $(Y^i , {\\Lambda }^i)$ is solution of the equation $\\begin{bmatrix} \\dot{Y^i} \\\\ \\dot{{\\Lambda }^i} \\end{bmatrix} = \\begin{bmatrix} P & 0 \\\\ e_ie_i^\\top & -P\\end{bmatrix} \\begin{bmatrix} Y^i \\\\ {\\Lambda }^i\\end{bmatrix}.$ on $[0,T]$ with ${\\Lambda }^i$ nonsingular on $[0,T]$ then $F^i = Y^i ({\\Lambda }^i)^{-1}$ is a solution of (REF ).", "Since $P(t)$ is symmetric and commutes with $L$ , the two matrices are simultaneously diagonalizable.", "If we further assume that $P(t)$ and $P(s)$ commute for all $s$ and $t$ , then the matrices $L$ and $\\lbrace P(t) : t \\in [0,T]\\rbrace $ are all simultaneously diagonalizable.", "We can then choose an orthonormal basis $V=(v_1, \\ldots , v_n)$ of eigenvectors of $P(t)$ and $L$ , such that $L$ and $P(t)$ are diagonalizable in this basis for all $t \\in [0,T]$ .", "For each $t$ , let $\\rho _1(t), \\ldots , \\rho _n(t)$ denote the eigenvalues of $P(t)$ with corresponding eigenvectors $v_1, \\ldots , v_n$ .", "We can now easily solve the equation for $Y^i$ to get $Y^i(t) = \\tilde{P}(t) Y^i(T), \\qquad \\text{ where } \\ \\ \\ \\tilde{P}(t) := \\exp \\left(-\\int _t^T P(s)ds\\right).$ We deduce from the expression of $Y^i$ that ${\\Lambda }^i$ solves the equation $\\dot{{\\Lambda }}^i(t) = e_ie_i^\\top \\tilde{P}(t)Y^i(T) - P(t){\\Lambda }^i(t)$ , and it follows that ${\\Lambda }^i(t) = \\tilde{P}^{-1}(t) \\left[{\\Lambda }^i(T) - \\int _t^T \\tilde{P}(s)e_ie_i^\\top \\tilde{P}(s)Y^i(T)ds \\right].$ If we now choose the boundary values ${\\Lambda }^i(T) = I, \\hspace{28.45274pt} Y^i(T) = c Le_i e_i^\\top L,$ then $F^i=Y^i({\\Lambda }^i)^{-1}$ is symmetric and the terminal condition is satisfied.", "Plugging these terminal conditions into ${\\Lambda }^i$ and $Y^i$ yields ${\\Lambda }^i(t) = \\tilde{P}^{-1}(t) \\left[I - c \\int _t^T \\frac{\\mbox{Tr}(\\tilde{P}(s)L)}{n} \\tilde{P}(s)e_ie_i^\\top Lds \\right], \\quad Y^i(t) = c\\tilde{P}(t)Le_ie_i^\\top L.$ Here we used the identity $e_i^\\top \\tilde{P}(s)Le_i = \\frac{{\\mathrm {Tr}}(\\tilde{P}L)}{n}$ which holds by Lemma REF (ii), since $\\tilde{P}L$ commutes with $R_{\\varphi }$ for all $\\varphi \\in {\\mathrm {Aut}}(G)$ .", "Now that we have our explicit solutions $(Y^i, {\\Lambda }^i)$ , assuming that ${\\Lambda }^i(t)$ is invertible, we deduce that the Ricatti equation (REF ) admits the following solution $F^i(t) = c\\tilde{P}(t)Le_ie_i^\\top L\\left[I - c {\\Xi }^i(t) \\right]^{-1} \\tilde{P}(t),$ where we define ${\\Xi }^i(t) := \\int _t^T \\frac{{\\mathrm {Tr}}(\\tilde{P}(s)L)}{n}\\tilde{P}(s)e_ie_i^\\top Lds.$ The objective is now to solve for $P$ .", "To that end, we first note that $F^i(t)e_i = c \\eta _i(t) \\tilde{P}(t)Le_i, \\quad \\text{where} \\quad \\eta _i(t) := e_i^\\top L[I - c {\\Xi }^i(t) ]^{-1} \\tilde{P}(t)e_i .$ Recalling that both $L$ and $\\tilde{P}$ commute with $R_{\\varphi }$ , a simple computation shows that $R_\\varphi {\\Xi }^i(t) = {\\Xi }^{\\varphi (i)}(t)R_\\varphi $ for $\\varphi \\in {\\mathrm {Aut}}(G)$ .", "It follows that $R_\\varphi L[I - c {\\Xi }^i(t) ]^{-1} \\tilde{P}(t) = L[I - c {\\Xi }^{\\varphi (i)}(t) ]^{-1} \\tilde{P}(t)R_\\varphi $ , and using Lemma REF (iii) we deduce that $\\eta _1(t)=\\eta _2(t)=\\cdots =\\eta _n(t)$ , and we let $\\eta (t)$ denote the common value.", "We then compute $P(t) = \\sum _{i = 1}^n F^i(t)e_i e_i^\\top = c \\sum _{i = 1}^n \\eta (t) \\tilde{P}(t)Le_i e_i^\\top = c \\eta (t) \\tilde{P}(t) L.$ Because $P$ , $\\tilde{P}$ , and $L$ are simultaneously diagonalizable, we deduce the following relationship between the eigenvalues: $\\rho _k(t) e^{\\int _t^T \\rho _k(s)ds} = c \\eta (t) \\lambda _k, \\qquad k=1,\\ldots ,n,$ where $\\lambda _1,\\ldots ,\\lambda _n$ are the eigenvalues of $L$ , corresponding to eigenvectors $v_1,\\ldots ,v_n$ .", "Integrating from $t$ to $T$ , taking the logarithm, and finally differentiating leads to $\\rho _k(t) = \\frac{c \\lambda _k \\eta (t)}{1 + c \\lambda _k \\int _t^T \\eta (s)ds},$ which we can rewrite in matrix form as $P(t) = c \\eta (t) L\\left(I + c \\int _t^T \\eta (s)ds \\,L\\right)^{-1}.", "$ This completes our fixed point argument, provided we identify $\\eta $ .", "Using once more that $e_i^\\top L\\tilde{P}(s)e_i = \\frac{{\\mathrm {Tr}}(\\tilde{P}(s)L)}{n}$ for all $s$ , a quick computation shows $({\\Xi }^i(t))^2 = \\Big (\\int _t^T \\big (\\frac{{\\mathrm {Tr}}(\\tilde{P}(s)L)}{n}\\big )^2 ds \\Big ) {\\Xi }^i(t)$ .", "Thus, assuming the validity of the power series $(I-c{\\Xi }^i(t))^{-1} = \\sum _{k=0}^\\infty (c{\\Xi }^i(t))^k$ , we have $[I - c {\\Xi }^i(t) ]^{-1} &= \\sum _{k=0}^{\\infty } (c {\\Xi }^i(t))^k = I + c\\left[1 - c \\int _t^T \\Big (\\frac{{\\mathrm {Tr}}(\\tilde{P}(s)L)}{n}\\Big )^2 ds\\right]^{-1} {\\Xi }^i(t).", "$ Plugging this back into the definition (REF ) of $\\eta $ , we get (for any $i$ ) $\\eta (t) = e_i^\\top L[I-c{\\Xi }^i(t)]^{-1}\\tilde{P}(t) e_i = \\frac{{\\mathrm {Tr}}(\\tilde{P}(t)L)}{n} \\, \\frac{1}{1 - c \\int _t^T \\left(\\frac{{\\mathrm {Tr}}(\\tilde{P}(s)L)}{n}\\right)^2 ds }.$ Using (REF ) and (REF ) we get ${\\mathrm {Tr}}(\\tilde{P}(t)L) = \\frac{1}{c \\eta (t)} {\\mathrm {Tr}}(P(t)) = \\sum _{k=1}^n \\frac{\\lambda _k}{1 + c \\lambda _k \\int _t^T \\eta (s)ds}$ , and from (REF ) we then obtain $\\eta (t) = \\frac{\\frac{1}{n} \\sum _{k=1}^n \\frac{\\lambda _k}{1 + c \\lambda _k \\int _t^T \\eta (s)ds}}{1 - c \\int _t^T \\Big (\\frac{1}{n} \\sum _{k=1}^n \\frac{\\lambda _k}{1 + c \\lambda _k \\int _s^T \\eta (s)ds} \\Big )^2 ds}.$ Multiplying both sides by $\\frac{1}{n} \\sum _{k=1}^n \\frac{\\lambda _k}{1 + c \\lambda _k \\int _t^T \\eta (s)ds}$ , integrating from $t$ to $T$ , and exponentiating yield $\\prod _{k=1}^n \\left(1 + c \\lambda _k \\int _t^T \\eta (s)ds \\right)^{1/n} = \\frac{1}{1 - c \\int _t^T \\Big (\\frac{1}{n} \\sum _{k=1}^n \\frac{\\lambda _k}{1 + c \\lambda _k \\int _s^T \\eta (s)ds} \\Big )^2 ds}.$ Defining $Q(x)=(\\det (I - xL))^{1/n}$ and $f(t) = -c\\int _{T-t}^T\\mu (s)ds$ , we find from (REF ) and (REF ) that $f$ must satisfy $f(0) = 0$ and $f^{\\prime }(t)=cQ^{\\prime }(f(t))$ .", "Returning to (REF ), and noting that $f^{\\prime }(T-t)=-\\eta (t)$ , we have thus proved that $P$ can be written as $P(t) = - f^{\\prime }(T-t) L\\left(I - f(T-t) L\\right)^{-1},$ justifying the expression of $P$ in Theorem REF .", "We can also simplify the expression of $F^i$ in (REF ) to recover the expression we introduced in (REF ) our first the proof of Theorem REF .", "Use (REF ) to get $F^i(t) = \\frac{c}{1 - c \\int _t^T\\left(\\frac{{\\mathrm {Tr}}{(\\tilde{P}(s)L)}}{n}\\right)^2 ds } \\tilde{P}(t)Le_ie_i^\\top L\\tilde{P}(t).$ From (REF ) and (REF ), we deduce $1 - c \\int _t^T \\left(\\frac{{\\mathrm {Tr}}{(\\tilde{P}(s)L)}}{n}\\right)^2 ds = \\frac{1}{c\\mu (t)^2}\\frac{{\\mathrm {Tr}}{(P(t))}}{n}$ and $F^i(t) = \\frac{1}{{\\mathrm {Tr}}{(P(t))}/n} P(t) e_ie_i^\\top P(t).$" ], [ "Correlation decay: Proof of Proposition ", "The purpose of this section is to prove Proposition REF , which is essential in the proof of the convergence of the empirical measure given in the next section.", "Throughout this section, we fix a finite transitive graph $G = (V,E)$ without isolated vertices and with vertex set $V = \\lbrace 1, ..., n\\rbrace $ for some $n \\in {\\mathbb {N}}$ .", "We may without loss of generality assume $G$ to be connected, as otherwise $X^G_v(t)$ and $X_u^G(t)$ are independent for $u$ and $v$ in distinct connected components and the right-hand side of (REF ) is zero.", "Since $G$ is fixed throughout the section we omit it in the notations, e.g., $L= L_G$ , $\\mathbf {X}(t) = \\mathbf {X}^G(t)$ , $\\delta = \\delta (G)$ , and $f=f_G$ .", "As before, $(e_1, ..., e_n)$ denotes the standard Euclidean basis in ${\\mathbb {R}}^n$ .", "We make some use of the adjacency matrix $A=A_G$ in this section, and we repeatedly use the well known fact that $e_u^\\top A^\\ell e_v$ counts the number of paths of length $\\ell $ from $v$ to $u$ , for each $\\ell \\in {\\mathbb {N}}$ and vertices $v,u \\in V$ .", "Recall from (REF ) that $X_i(t)$ is Gaussian with variance $\\frac{\\sigma ^2}{n}\\sum _{k=1}^n \\int _0^t \\left(\\frac{1 - f(T-t)\\lambda _k}{1 - f(T-s)\\lambda _k}\\right)^2 ds \\le \\sigma ^2 T.$ Indeed, the inequality follows from the fact that $-2 \\le \\lambda _k \\le 0$ and $f$ is increasing.", "Since $0 < \\gamma < 1$ , this proves the claim (REF ) in the case $u=v$ (noting also that $\\delta (G) \\ge 1$ ).", "We thus focus henceforth on the case of distinct vertices.", "From Theorem REF , we know that in equilibrium the state process $\\mathbf {X}$ is normally distributed with covariance matrix $\\sigma ^2(I - f(T-t)L)^2 \\int _0^t (I - f(T-s)L)^{-2} ds.$ Our objective is to find a bound for the off-diagonal elements of this matrix.", "Fix two distinct vertices $u,v\\in V$ .", "We have ${\\mathrm {Cov}}(X_u(t), X_v(t)) = \\sigma ^2 \\int _0^t \\sum _{k=1}^n e_u^\\top (I - f(T-t)L)^2 e_k e_k^\\top (I - f(T-s)L)^{-2} e_v ds.$ Let us first develop $e_u^\\top (I - f(T-t)L)^2 e_k$ .", "Using $L= \\frac{1}{\\delta }A - I$ , where $A$ is the adjacency matrix of the graph, we find $\\begin{split}e_u^\\top (I - f(T-t)L)^2 e_k = ( 1 + f(T-t))^2 \\Bigg (e_u^\\top e_k &- 2 \\frac{f(T-t)}{\\delta ( 1 + f(T-t))} e_u^\\top Ae_k \\\\& + \\left(\\frac{f(T-t)}{\\delta ( 1 + f(T-t))} \\right)^2 e_u^\\top A^2 e_k\\Bigg ).\\end{split}$ Recall that $d(i,j) = d_G(i,j)$ denotes the distance between two vertices $i$ and $j$ in the graph, i.e., the length of the shortest path from $i$ to $j$ in $G$ .", "Let us write $P(i,m)$ the set of vertices which can be reached in exactly $m$ steps from $i$ .", "By definition of $A$ , $e_u^\\top A e_k = 1$ if $d(u,k) = 1$ and zero otherwise.", "Similarly, $e_u^\\top A^2 e_k$ is the number of paths of length 2 from $u$ to $k$ , so, in particular, $e_u^\\top A^2 e_k=0$ unless $k \\in P(u,2)$ .", "Plugging this into (REF ), we get ${\\mathrm {Cov}}(X_u(t), X_v(t)) &= \\sigma ^2 ( 1 + f(T-t))^2 \\int _0^t \\Bigg \\lbrace e_u^\\top (I - f(T-s)L)^{-2} e_v \\nonumber \\\\&\\quad - \\sum _{k \\in P(u,1)} 2 \\frac{f(T-t)}{\\delta ( 1 +f(T-t))} e_k^T (I - f(T-s)L)^{-2} e_v \\\\&\\quad + \\sum _{k \\in P(u,2)} \\left(\\frac{f(T-t)}{\\delta ( 1 +f(T-t))} \\right)^2 e_u^\\top A^2 e_k e_k^T (I - f(T-s)L)^{-2} e_v \\Bigg \\rbrace ds.", "\\nonumber $ Next, we estimate the term $e_k^T (I - f(T-s)L)^{-2} e_v = ( 1 +f(T-s))^{-2} e_k^T \\left(I - \\frac{f(T-s)}{\\delta ( 1 +f(T-s))}A\\right)^{-2} e_v.$ To simplify the notations, we define the function $\\gamma (s) := \\frac{f(T-s)}{ 1 +f(T-s)}.$ From Proposition REF , we know that $0 \\le f(t)\\le ct$ for all $t \\in [0,T]$ , and thus $0 \\le \\gamma (s) < 1$ for all $s \\in [0,T]$ .", "Moreover, the spectral radius of the adjacency matrix $A$ is always bounded by the degree $\\delta $ .", "We can thus use the series $\\frac{1}{(1-x)^2} = \\sum _{\\ell =0}^{\\infty } (\\ell +1)x^\\ell $ for $|x| < 1$ to get $\\begin{split}e_k^\\top (I - f(T-s)L)^{-2} e_v &= ( 1 +f(T-s))^{-2} \\sum _{\\ell = 0}^{\\infty } (\\ell +1) \\left( \\frac{\\gamma (s)}{\\delta } \\right)^\\ell e_k^\\top A^\\ell e_v \\\\&= ( 1 +f(T-s))^{-2} \\sum _{\\ell = d(k,v)}^{\\infty } (\\ell +1) \\left( \\frac{\\gamma (s)}{\\delta } \\right)^\\ell e_k^\\top A^\\ell e_v,\\end{split} $ where in the last line we used the fact that $e_k^T A^\\ell e_v = 0$ if $\\ell < d(k,v)$ , since the latter implies there are no paths of length $\\ell $ from $k$ to $v$ .", "Next, note the elementary bound $0 \\le e_k^T A^\\ell e_v \\le \\delta ^{\\ell - 1}$ , since each vertex has exactly $\\delta $ neighbors.", "Indeed, for each of the first $\\ell -1$ steps of a path of length $\\ell $ from $k$ to $v$ , there are at most $\\delta $ choices of vertex, and for the last step there is at most one choice which terminates the path at $v$ .", "Hence, using the identity $\\sum _{\\ell =k}^{\\infty } (\\ell +1)x^\\ell = \\frac{d}{d x} \\left(\\sum _{\\ell =k}^{\\infty } x^{\\ell +1} \\right) = \\frac{d}{d x} \\left(\\frac{x^{k+1}}{1 - x} \\right) = \\frac{x^k}{(1-x)^2} (1 + k(1-x))$ for $|x| < 1$ , we deduce from (REF ) that $0 \\le e_k^\\top (I - f(T-s)L)^{-2} e_v \\le \\frac{1}{\\delta ( 1 +f(T-s))^{2}} \\frac{\\gamma (s)^{d(k,v)} (1 + d(k,v)(1 - \\gamma (s))}{(1 - \\gamma (s))^2}.$ Now notice that in (REF ) the first term and all the terms in the last sum on the right hand side are positive, whereas all the terms in the second line sum are negative.", "Therefore we have the following upper bound ${\\mathrm {Cov}}(X_u(t), X_v(t)) \\le \\, &\\sigma ^2 \\int _0^t \\frac{(1 +f(T-t))^2}{\\delta ( 1 +f(T-s))^{2}} \\Bigg \\lbrace \\frac{\\gamma (s)^{d(u,v)} (1 + d(u,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2} \\\\&+ \\sum _{k \\in P(u,2)} e_u^\\top A^2 e_k \\left(\\frac{\\gamma (t)}{\\delta } \\right)^2 \\frac{\\gamma (s)^{d(k,v)} (1 + d(k,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2} \\Bigg \\rbrace ds.", "\\nonumber $ The function $d \\mapsto \\gamma (t)^d(1 + d(1 - \\gamma (t)))$ is non-increasing on $[0, \\infty )$ since $\\gamma (t) \\in [0,1)$ .", "Moreover, for all $k$ such that $d(k,u) \\le 2$ , we have $|d(k,v) - d(u,v)| \\le 2$ , and in particular $d(k,v) \\ge d(u,v) - 2$ .", "We consider two cases separately: First, suppose $d(u,v) \\ge 2$ .", "Since $d(k,v) \\ge d(u,v)-2$ by the above argument, monotonicity of $d \\mapsto \\gamma (t)^d(1 + d(1 - \\gamma (t)))$ lets us estimate $\\sum _{k \\in P(u,2)} & e_u^\\top A^2 e_k \\left(\\frac{\\gamma (t)}{\\delta } \\right)^2 \\frac{\\gamma (s)^{d(k,v)} (1 + d(k,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2} \\\\&\\le \\left(\\frac{\\gamma (t)}{\\delta } \\right)^2 \\frac{\\gamma (s)^{d(u,v)-2} (1 + (d(u,v)-2)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2}\\sum _{k \\in P(u,2)} e_u^\\top A^2 e_k \\\\&\\le \\gamma (t)^2 \\frac{\\gamma (s)^{d(u,v)-2} (1 + (d(u,v)-2)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2}.$ Indeed, the last step uses the fact that $\\sum _{k \\in P(u,2)} e_u^\\top A^2 e_k$ is precisely the number of paths of length two originating from vertex $u$ , which is clearly bounded from above by $\\delta ^2$ .", "Moreover, since $f$ is increasing by Proposition REF , it is straightforward to check that $\\gamma $ is decreasing.", "Therefore $0 \\le \\gamma (t) \\le \\gamma (s) \\le 1$ for all $t \\ge s \\ge 0$ , and the above quantity is further bounded from above by $\\frac{\\gamma (s)^{d(u,v)} (1 + d(u,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2}.$ Plugging this back in (REF ), and using the inequality $\\frac{1 +f(T-t)}{1 +f(T-s)} \\le 1$ which again follows from the fact that $f$ is increasing and nonnegative, we get ${\\mathrm {Cov}}(X_u(t), X_v(t)) \\le \\frac{2\\sigma ^2}{\\delta } \\int _0^t \\frac{\\gamma (s)^{d(u,v)}(1 + d(u,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2} ds.$ Suppose next that $d(u,v)=1$ .", "We then use the bounds $\\gamma (t)^d(1 + d(1 - \\gamma (t))) \\le 1$ and $\\sum _{k \\in P(u,2)} e_u^\\top A^2 e_k \\le \\delta ^2$ to estimate $\\sum _{k \\in P(u,2)} & e_u^\\top A^2 e_k \\left(\\frac{\\gamma (t)}{\\delta } \\right)^2 \\frac{\\gamma (s)^{d(k,v)} (1 + d(k,v)(1 - \\gamma (s)))}{(1 - \\gamma (s))^2} \\\\&\\le \\frac{\\gamma (t)^2}{(1 - \\gamma (s))^2} \\le \\frac{\\gamma (s)^2}{(1 - \\gamma (s))^2} \\le \\frac{\\gamma (s)^{d(u,v)+1}(1+d(u,v)(1-\\gamma (s))}{(1 - \\gamma (s))^2}.$ Plugging this into (REF ) shows that the same bound (REF ) is valid for $d(u,v)=1$ .", "Now, recall that $\\gamma (\\cdot )$ is decreasing.", "Since $f(T)\\le cT$ by Proposition REF , we have $0 < \\gamma (0) = \\frac{cf(T)}{1+cf(T)} \\le \\frac{cT}{1 + cT} < 1$ .", "Set $\\gamma := \\frac{cT}{1 + cT}$ .", "The function $y \\mapsto \\frac{y^d(1+d(1-y))}{(1 - y)^2}$ is easily seen to be increasing on $[0,1)$ for any $d \\ge 0$ .", "Thus, from (REF ) we finally deduce the desired upper bound ${\\mathrm {Cov}}(X_u(t), X_v(t)) \\le 2\\sigma ^2 t \\frac{ \\gamma ^{d(u,v)}(1 + d(u,v)(1-\\gamma ))}{\\delta (1 - \\gamma )^2}.$ Using the same arguments for the second term in (REF ), which is the only negative term, we obtain a similar lower bound for ${\\mathrm {Cov}}(X_u(t), X_v(t))$ , which concludes the proof." ], [ "Asymptotic regimes", "In this section, we provide the derivations of the large-$n$ asymptotics of the in-equilibrium processes.", "We will first prove Theorem REF , then Theorem REF , and lastly we will focus on the examples we discussed in Section REF , and in particular prove Corollary REF and Proposition REF ." ], [ "Large-scale asymptotics on transitive graphs: Proof of Theorem ", "Part (1) of Theorem REF is a consequence of Proposition REF , so we focus on parts (2–4).", "Let $\\lbrace G_n\\rbrace $ be a sequence of finite transitive graphs, and let $\\lbrace \\mu _{G_n}\\rbrace $ be the corresponding sequence of empirical eigenvalue distributions defined by (REF ).", "We assume that $\\lbrace \\mu _{G_n}\\rbrace $ converges weakly to a probability measure $\\mu $ .", "Recall that the initial states are $\\mathbf {X}^{G_n}(0) = \\mathbf {0}$ , and recall from (REF ) that each $X^{G_n}_i(t)$ is Gaussian with mean zero and variance ${V}_{G_n}(t) &:= \\sigma ^2 \\int _0^t \\int _{[-2,0]} \\left(\\frac{1 - f_{G_n}(T-t)\\lambda }{1 - f_{G_n}(T-s)\\lambda }\\right)^2 \\mu _{G_n}(d\\lambda )ds.$" ], [ "Convergence of $X^{G_n}_{k_n}(t)$ : Proof of (2)", "By Proposition REF , we know $f_{G_n}=f_{\\mu _{G_n}}$ converges uniformly to the function $f_\\mu $ given by (REF ).", "Defining ${V}_\\mu (t)$ as in (REF ), it follows from this uniform convergence and the weak convergence of $\\mu _{G_n}$ to $\\mu $ that ${V}_{G_n}(t) \\rightarrow {V}_\\mu (t)$ .", "Therefore $X^{G_n}_{k_n}(t) \\sim {V}_{G_n}(t)$ converges weakly to $\\mathcal {N}(0,{V}_\\mu (t))$ as $n\\rightarrow \\infty $ ." ], [ "Convergence of the empirical measure: Proof of (3)", "We next show that the (random) empirical measure $m^{G_n}(t) := \\frac{1}{|G_n|}\\sum _{v \\in G_n} \\delta _{X^{G_n}_v(t)}$ converges to the Gaussian measure $\\mathcal {N}(0,{V}_\\mu (t))$ , for each $t \\in [0,T]$ .", "In fact, it suffices to show that $m^{G_n}(t)$ concentrates around its mean, in the following sense: For any bounded 1-Lipschitz function $h$ , it holds that $\\lim _{n\\rightarrow \\infty }{\\mathbb {E}}\\left[\\left|\\int h \\, dm^{G_n}(t) - {\\mathbb {E}}\\int h \\, dm^{G_n}(t)\\right|^2\\right] = 0.", "$ Indeed, once (REF ) is established, it follows from the transitivity of $G_n$ that ${\\mathbb {E}}\\int h \\, dm^{G_n}(t) = {\\mathbb {E}}[h(X^{G_n}_{k_n}(t))]$ , where $k_n \\in G_n$ is arbitrary.", "Since the law of $X^{G_n}_{k_n}(t)$ converges weakly to $m(t):=\\mathcal {N}(0,{V}_\\mu (t))$ , we deduce that $\\int h \\, dm^{G_n}(t) \\rightarrow \\int h \\, dm(t)$ in probability, and the claim follows.", "Before proving (REF ), we digress to state a lemma pertaining to the degrees.", "Recall that each vertex in the transitive graph $G_n$ has the same degree, denoted $\\delta (G_n)$ .", "Lemma 7.1 We have $\\mu _{G_n} \\rightarrow \\delta _{-1}$ if and only if $\\delta (G_n) \\rightarrow \\infty $ .", "If $\\mu _{G_n} \\rightarrow \\mu \\ne \\delta _{-1}$ , then $\\sup _n \\delta (G_n) < \\infty $ .", "Recall that $L_{G_n}=\\frac{1}{\\delta (G_n)}A_{G_n}- I$ , where $A_{G_n}$ is the adjacency matrix of the graph $G_n$ .", "Then ${\\mathrm {Var}}(\\mu _{G_n}) &= \\int _{[-2,0]} (1+\\lambda )^2 \\,\\mu _{G_n}(dx) = \\frac{1}{n}{\\mathrm {Tr}}\\Big (\\frac{1}{\\delta (G_n)^2}A_{G_n}^2\\Big ) = \\frac{1}{n\\delta (G_n)^2}\\sum _{i=1}^n (A_{G_n}^2)_{ii}.$ Since $(A_{G_n}^2)_{ii}=\\delta (G_n)$ is exactly the number of paths of length 2 starting and ending at vertex $i$ , we get ${\\mathrm {Var}}(\\mu _{G_n}) = 1/\\delta (G_n)$ .", "Thus, if $\\mu _{G_n} \\rightarrow \\mu $ weakly for some probability measure $\\mu $ on $[-2,0]$ , we have ${\\mathrm {Var}}(\\mu )=\\int _{[-2,0]} (1+\\lambda )^2 \\,\\mu (dx) &= \\lim _{n\\rightarrow \\infty }\\frac{1}{\\delta (G_n)}.", "$ The second claim follows immediately.", "It is straightforward to check that $\\mu _{G_n} \\rightarrow \\delta _{-1}$ if and only if ${\\mathrm {Var}}(\\mu _{G_n}) \\rightarrow 0$ , and the first claim follows.", "We now turn toward the proof of (REF ), for a fixed bounded 1-Lipschitz function $h$ .", "We achieve this by applying the Gaussian Poincaré inequality and then using the covariance estimate of REF .", "To this end, fix $t \\in [0,T]$ and $n$ , and suppose for simplicity that $G_n$ has $n$ vertices.", "Let $\\Sigma ^n$ denote the $n \\times n$ covariance matrix of $\\mathbf {X}^{G_n}(t)$ , and let $M$ denote its symmetric square root.", "Then $\\mathbf {X}^{G_n}(t) \\stackrel{d}{=} M \\mathbf {Z}$ for a standard Gaussian $\\mathbf {Z}$ in ${\\mathbb {R}}^n$ .", "Define $F : {\\mathbb {R}}^n \\rightarrow {\\mathbb {R}}$ by $F(\\mathbf {x}) := \\frac{1}{n}\\sum _{i=1}^n h(e_i^\\top M \\mathbf {x}).$ Then $F(\\mathbf {Z}) \\stackrel{d}{=} \\int h\\,dm^{G_n}(t)$ .", "Noting that $\\partial _i F(x) = \\frac{1}{n} \\sum _{j=1}^n h^{\\prime }(e_j^\\top M \\mathbf {x}) M_{ji}$ , we get $\\begin{split}|\\nabla F(\\mathbf {x})|^2 &= \\sum _{i=1}^n \\frac{1}{n^2} \\Big (\\sum _{j=1}^n h^{\\prime }(e_j^\\top M \\mathbf {x}) M_{ji}\\Big )^2 \\le \\frac{1}{n^2} \\sum _{j,k = 1}^n h^{\\prime }(e_j^\\top M \\mathbf {x}) h^{\\prime }(e_k^\\top M \\mathbf {x}) \\Sigma _{jk} \\\\& \\le \\frac{1}{n^2} \\sum _{j,k = 1}^n |\\Sigma _{jk}^n|,\\end{split}$ where in the last inequality we used the fact that $h$ is 1-Lipschitz.", "Now, applying the Gaussian Poincaré inequality [4], we find ${\\mathbb {E}}\\left[\\left|\\int h \\, dm^{G_n}(t) - {\\mathbb {E}}\\int h \\, dm^{G_n}(t)\\right|^2\\right] &= {\\mathrm {Var}}(F(\\mathbf {Z})) \\le {\\mathbb {E}}[|\\nabla F(\\mathbf {Z})|^2] \\le \\frac{1}{n^2} \\sum _{j,k = 1}^n |\\Sigma _{jk}^n|.", "$ It remains to show that this converges to zero as $n\\rightarrow \\infty $ .", "Let $\\epsilon > 0$ .", "By Proposition REF , $|\\Sigma _{jk}^n|$ converges to 0 as $d_{G_n}(j,k) \\rightarrow \\infty $ , where $d_{G_n}$ denotes the graph distance in $G_n$ .", "Choose $m \\in {\\mathbb {N}}$ large enough so that $|\\Sigma _{jk}^n| \\le \\epsilon $ for all $n \\in {\\mathbb {N}}$ and $j,k \\in G_n$ with $d_{G_n}(j,k) > m$ .", "For $k \\in G_n$ let $B_n(j,m)$ denote the set of vertices in $G_n$ of distance at most $m$ from $j$ .", "Because $G_n$ is transitive, the cardinality $|B_n(j,m)|$ does not depend on $j \\in G_n$ , and we denote by $|B_n(m)|$ this common value.", "Then, we use the bound on $|\\Sigma ^n_{jk}|$ from Proposition REF to get $\\begin{split}\\frac{1}{n^2} \\sum _{j,k = 1}^n |\\Sigma ^n_{jk}| &\\le \\epsilon + \\frac{1}{n^2} \\sum _{j=1}^n \\sum _{k \\in B_n(j,m)} |\\Sigma ^n_{jk}| \\le \\epsilon + \\frac{|B_n(m)|}{n\\delta (G_n)} \\frac{2\\sigma ^2T}{(1-\\gamma )^2},\\end{split}$ where we used the fact that $\\gamma ^d(1+d(1-\\gamma ))$ is decreasing in $d \\ge 0$ since $0 \\le \\gamma <1$ and is thus bounded by 1.", "We now distinguish two cases.", "If $\\delta (G_n) \\rightarrow \\infty $ , then we use $|B_n(m)|/n \\le 1$ to send $n\\rightarrow \\infty $ and then $\\epsilon \\rightarrow 0$ to get that (REF ) converges to zero.", "On the other hand, suppose $\\delta (G_n)$ does not converge to infinity.", "Then necessarily $\\sup _n\\delta (G_n) < \\infty $ by Lemma REF , since $\\mu _{G_n}$ converges weakly by assumption.", "Using the obvious bound $|B_n(m)| \\le \\delta (G_n)^m$ , we can again send $n\\rightarrow \\infty $ and then $\\epsilon \\rightarrow 0$ to get that (REF ) converges to zero.", "This completes the proof of part (3) of Theorem REF ." ], [ "Convergence of the value: Proof of (4)", "Recall the identity for the value of the game from (REF ).", "Since $f_{G_n}(T) \\rightarrow f_\\mu (T)$ by Proposition REF and $\\mu _{G_n} \\rightarrow \\mu $ weakly, it follows that $\\mbox{Val}(G_n) = - \\frac{\\sigma ^2}{2} \\log \\int _{[-2,0]} \\frac{- \\lambda }{1 - f_{G_n}(T)\\lambda } \\mu _{G_n}(d\\lambda ) \\rightarrow - \\frac{\\sigma ^2}{2} \\log \\int _{[-2,0]} \\frac{- \\lambda }{1 - f_\\mu (T)\\lambda } \\mu (d\\lambda ).$ This gives (4) and completes the proof of Theorem REF ." ], [ "Approximate equilibria: Proof of Theorem ", "We begin toward proving Theorem REF by first studying the control $\\alpha ^{\\mathrm {MF}}$ introduced therein.", "The following lemma shows that it arises essentially as the equilibrium of a mean field game, or equivalently as the optimal control for a certain control problem: Lemma 7.2 Define $\\alpha ^{\\mathrm {MF}} : [0,T] \\times {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ $\\alpha ^{\\mathrm {MF}}(t,x) := - \\frac{cx}{1+c(T-t)}.", "$ Let $(\\Omega ^{\\prime },{\\mathcal {F}}^{\\prime },{\\mathbb {F}}^{\\prime },{\\mathbb {P}}^{\\prime })$ be any filtered probability space supporting an ${\\mathbb {F}}^{\\prime }$ -Brownian motion $W$ and a ${\\mathbb {F}}^{\\prime }$ -progressively measurable real-valued process $(\\beta (t))_{t \\in [0,T]}$ satisfying ${\\mathbb {E}}\\int _0^T\\beta (t)^2dt < \\infty $ .", "Let $X$ be the unique strong solution of the SDE $dX(t) &= \\alpha ^{\\mathrm {MF}}(t,X(t))dt + \\sigma dW(t), \\quad X(0)=0,$ and define $(Y(t))_{t \\in [0,T]}$ by $dY(t) &= \\beta (t)dt + \\sigma dW(t), \\quad Y(0)=0.$ Then $\\frac{1}{2}{\\mathbb {E}}\\left[\\int _0^T|\\alpha ^{\\mathrm {MF}}(t,X(t))|^2dt + c|X(T)|^2\\right] \\le \\frac{1}{2}{\\mathbb {E}}\\left[\\int _0^T|\\beta (t)|^2dt + c|Y(T)|^2\\right]$ We study the HJB equation corresponding to this control problem, which is $\\partial _t v(t,x) + \\inf _{a \\in {\\mathbb {R}}} \\left(a \\partial _x v(t,x) + \\frac{1}{2} a^2 \\right) + \\frac{1}{2} \\sigma ^2 \\partial _{xx} v(t,x) = 0,$ or equivalently $\\partial _t v(t,x) - \\frac{1}{2} |\\partial _x v(t,x)|^2 + \\frac{1}{2} \\sigma ^2 \\partial _{xx} v(t,x) = 0,$ with terminal condition $v(T,x) = cx^2/2$ .", "The ansatz $v(t,x)=a(t)x^2 + b(t)$ yields a classical solution, where $a$ and $b$ are functions satisfying $a^{\\prime }(t) - 2 a(t)^2 = 0, \\qquad b^{\\prime }(t) + \\sigma ^2 a(t) = 0,$ with terminal conditions $a(T) = c/2$ and $b(T)=0$ .", "We deduce $a(t) = \\frac{1}{2}\\frac{c}{1+c(T-t)}, \\quad b(t) = \\frac{\\sigma ^2}{2}\\log (1+c(T-t)).$ Therefore the optimal Markovian control is $- \\partial _x v(t,x) = - 2 a(t) x = \\alpha ^{\\mathrm {MF}}(t,x)$ .", "This completes the proof, by a standard verification argument [37].", "Now let $G$ be a fixed finite graph with vertex set $V=\\lbrace 1,\\ldots ,n\\rbrace $ .", "Let us again omit the $G$ superscripts from the notation, with ${\\mathcal {A}}={\\mathcal {A}}_G$ and $J=J^G$ denoting the control set and value function from Section REF .", "Let $\\mathbf {\\alpha }= (\\alpha ^{\\mathrm {MF}}_i)_{i=1}^n$ , and for $\\beta \\in {\\mathcal {A}}$ and $i \\in V$ let $(\\beta ,\\mathbf {\\alpha }^{-i}) := (\\alpha ^{\\mathrm {MF}}_1,\\ldots ,\\alpha ^{\\mathrm {MF}}_{i-1},\\beta ,\\alpha ^{\\mathrm {MF}}_{i+1},\\ldots ,\\alpha ^{\\mathrm {MF}}_n)$ .", "To be clear about the notation, we write $\\alpha ^{\\mathrm {MF}}$ without a subscript to denote the control in (REF ), whereas $\\alpha ^{\\mathrm {MF}}_i(t,\\mathbf {x})=\\alpha ^{\\mathrm {MF}}(t,x_i)$ for $i \\in V$ and $\\mathbf {x} \\in {\\mathbb {R}}^n$ .", "If $v \\in V$ has $\\mathrm {deg}_G(v) = 0$ , then (recalling the definition (REF ) of the cost function for isolated vertices) Lemma REF ensures that $J_v(\\mathbf {\\alpha }) = \\inf _{\\beta \\in {\\mathcal {A}}}J_v(\\beta ,\\mathbf {\\alpha }^{-v}),$ so we may take $\\epsilon _v=0$ as claimed in Theorem REF .", "Thus, we assume henceforth that $v \\in G$ is a fixed non-isolated vertex, so that $\\mathrm {deg}_G(v) \\ge 1$ .", "Now define $\\beta := \\arg \\!\\min _{\\beta ^{\\prime } \\in {\\mathcal {A}}} J_v(\\beta ^{\\prime }, \\mathbf {\\alpha }^{-v}).$ (Or take a $\\delta $ -optimizer in case no optimizer exists, and send $\\delta \\rightarrow 0$ at the end of the proof.)", "We aim to prove that $J_v(\\beta ,\\mathbf {\\alpha }^{-v}) &\\ge J_v(\\mathbf {\\alpha }) - \\frac{\\sigma ^2 cT}{1+cT} \\sqrt{ \\frac{cT(2+cT)}{\\mathrm {deg}_G(v)}}.", "$ Define the state processes $\\mathbf {X}=(X_1,\\ldots ,X_n)$ and $\\mathbf {Y}=(Y_1,\\ldots ,Y_n)$ as the unique strong solutions of the SDEs $dX_i(t) &= \\alpha ^{\\mathrm {MF}}(t, X_i(t))dt + \\sigma dW_i(t), \\quad X_i(0)=0, \\ \\ i \\in V, \\\\dY_i(t) &= \\alpha ^{\\mathrm {MF}}(t, Y_i(t))dt + \\sigma dW_i(t), \\quad Y_i(0)=0, \\ \\ i \\in V \\setminus \\lbrace v\\rbrace , \\nonumber \\\\dY_v(t) &= \\beta (t, \\mathbf {Y}(t))dt + \\sigma dW_v(t), \\quad Y_v(0)=0.", "\\nonumber $ Note that $Y_i\\equiv X_i$ for $i \\ne v$ .", "The values for the player $v$ , under $\\alpha ^{\\mathrm {MF}}$ and the deviation $\\beta $ , are then, respectively, $J_v(\\mathbf {\\alpha }) &= \\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\alpha ^{\\mathrm {MF}}(t,X_v(t))|^2dt + c \\left|\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - X_v(T)\\right|^2\\right], \\\\J_v(\\beta , \\mathbf {\\alpha }^{-v}) &= \\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2dt + c \\left|\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - Y_v(T)\\right|^2\\right],$ where we recall that $u \\sim v$ means that $u$ is adjacent to $v$ .", "We prove (REF ) in three steps: We show that $J_v(\\beta , \\mathbf {\\alpha }^{-v}) \\ge \\ &\\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\beta (t, \\mathbf {Y}(t))|^2dt + c |Y_v(T)|^2\\right] + \\frac{c\\sigma ^2T}{2\\,\\mathrm {deg}_G(v)(1+cT)} \\\\&- c \\sqrt{\\frac{\\sigma ^2 T^2}{\\mathrm {deg}_G(v)(1+ cT)} {\\mathbb {E}}\\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2 dt }.$ We then estimate ${\\mathbb {E}}\\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2 dt \\le c\\sigma ^2T\\frac{2+cT}{1+cT}.$ Finally, we show that $\\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2dt + c |Y_v(T)|^2\\right] \\ge J_v(\\mathbf {\\alpha }^{MF}) - \\frac{c\\sigma ^2T}{2\\,\\mathrm {deg}_G(v)(1+cT)}.$ which will conclude the proof.", "Step 1.", "We start with $& J_v(\\beta , \\mathbf {\\alpha }^{-v}) - \\frac{1}{2} {\\mathbb {E}}\\left[\\int _0^T |\\beta (t, \\mathbf {Y}(t))|^2dt + c |Y_v(T)|^2\\right] \\\\&\\quad = \\frac{c}{2} {\\mathbb {E}}\\left[ \\left|\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - Y_v(T)\\right|^2 - |Y_v(T)|^2\\right] \\\\&\\quad = \\frac{c}{2} {\\mathbb {E}}\\left[ \\left(\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T)\\right)^2\\right] - c {\\mathbb {E}}\\left[\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) Y_v(T) \\right].$ From the form of the SDE (REF ) and the definition of $\\alpha ^{\\mathrm {MF}}$ , we find that $(X_u(T))_{u \\in V}$ are i.i.d.", "Gaussians with mean zero and variance $\\sigma ^2 T/(1+cT)$ .", "We deduce ${\\mathbb {E}}\\left[ \\left(\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T)\\right)^2\\right] = \\frac{\\sigma ^2T}{\\mathrm {deg}_G(v)(1+cT)}.", "$ For the second term, we note ${\\mathbb {E}}\\left[\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) Y_v(T)\\right] = {\\mathbb {E}}\\left[\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) \\int _0^T \\beta (t,\\mathbf {Y}(t)) dt\\right],$ and we use Cauchy-Schwarz to bound this term in absolute value by $\\sqrt{\\frac{\\sigma ^2T^2}{\\mathrm {deg}_G(v)(1+cT)} {\\mathbb {E}}\\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2 dt }.$ Plugging these two terms back in our first inequality we obtain claim (1) above.", "Step 2.", "To find a bound for ${\\mathbb {E}}\\int _0^T|\\beta (t,\\mathbf {Y}(t))|^2dt$ , we use the definition of $\\beta $ as the minimizer of $J_v(\\beta ,\\mathbf {\\alpha }^{-v})$ .", "In particular, $J_v(\\beta ,\\mathbf {\\alpha }^{-v}) \\le J_v(0,\\mathbf {\\alpha }^{-v})$ .", "Expand this inequality, discarding the non-negative terminal cost on the left-hand side, and noting that the state process of player $v$ when adopting the zero control is precisely $(\\sigma W_v(t))_{t \\in [0,T]}$ , to get ${\\mathbb {E}}\\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2dt &\\le c {\\mathbb {E}}\\left[ \\Big |\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - \\sigma W_v(T)\\Big |^2\\right].$ Using (REF ) and the independence of the $X_u(T)$ and $W_v(T)$ , the right-hand side equals $\\frac{c\\sigma ^2T}{\\mathrm {deg}_G(v)(1+cT)} + c\\sigma ^2 T \\le c\\sigma ^2T\\frac{2+cT}{1+cT}.$ Step 3.", "We use Lemma REF to deduce that $\\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\beta (t,\\mathbf {Y}(t))|^2dt + c |Y_v(T)|^2\\right] &\\ge \\frac{1}{2} {\\mathbb {E}}\\left[ \\int _0^T |\\alpha ^{\\mathrm {MF}}(t,X_v(t))|^2dt + c |X_v(T)|^2\\right].$ The right-hand side can be written as $J_v(\\mathbf {\\alpha }) + \\frac{c}{2}{\\mathbb {E}}\\left[ |X_v(T)|^2 - \\Big |\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - X_v(T) \\Big |^2\\right].$ Since $(X_u(T))_{u \\in V}$ are i.i.d.", "with mean zero and variance $\\sigma ^2 T/(1+cT)$ , as in (REF ) we get $\\frac{c}{2}{\\mathbb {E}}\\left[ |X_v(T)|^2 - \\Big |\\frac{1}{\\mathrm {deg}_G(v)}\\sum _{u \\sim v} X_u(T) - X_v(T) \\Big |^2\\right] = - \\frac{c \\sigma ^2 T}{2 \\mathrm {deg}_G(v) (1+cT)}.$" ], [ "Examples", "We next specialize the results to the examples discussed in Section REF .", "In particular, we prove Corollary REF and Proposition REF ." ], [ "Dense case: Proof of Corollary ", "As in Corollary REF , let $\\lbrace G_n\\rbrace $ be a sequence of transitive graphs such that each vertex of $G_n$ has common degree $\\delta (G_n) \\ge 1$ .", "The claim that $\\mu _{G_n} \\rightarrow \\delta _{-1}$ if and only if $\\delta (G_n) \\rightarrow \\infty $ holds as a consequence of Lemma REF .", "In this case, we apply Theorem REF with $\\mu =\\delta _{-1}$ .", "The function $Q_\\mu $ therein is then $Q_{\\delta _{-1}}(x) = 1+x$ , and the function $f_{\\delta _{-1}}$ satisfies $f^{\\prime }_{\\delta _{-1}}(t)=c$ with $f_{\\delta _{-1}}(0)=0$ .", "Hence, $f_{\\delta _{-1}}(t)=ct$ , and the variance in (REF ) simplifies to ${V}_{\\delta _{-1}}(t) &= \\sigma ^2 \\int _0^t \\left(\\frac{1 + c(T-t)}{1 + c(T-s)}\\right)^2 ds = \\sigma ^2\\frac{t(1 + c(T-t))}{1+cT}.$ Recall now from the proof of Lemma REF that ${\\mathrm {Var}}(\\mu _{G_n})=1/\\delta (G_n)$ .", "Letting $C_0=\\tfrac{1}{2} c^2t^2+\\tfrac{1}{6} c^3T^3$ , the bounds of Proposition REF show that $|f_{G_n}(t)-ct|\\le C_0/\\delta (G_n)$ .", "The function $(1-\\lambda x)/(1-\\lambda y)$ is Lipschitz in $(x,y) \\in [0,cT]^2$ , uniformly in $\\lambda \\in [-2,0]$ .", "From this it is straightforward to argue that $|{V}_{G_n}(t)-{V}_{\\delta _{-1}}(t)|\\le C_1/\\delta (G_n)$ for some constant $C_1$ , and similarly for the convergence of the value using the identity (REF ).", "Lastly, the SDE (REF ) admits the solution $X(t)=\\int _0^t\\frac{1+c(T-t)}{1+c(T-s)}\\sigma dW(s)$ , and it is then straightforward to identify the Gaussian law $X(t)\\sim \\mathcal {N}(0,{V}_{\\delta _{-1}}(t))$ ." ], [ "Cycle graph case: Proof of Proposition ", "We begin by simplifying the expression of $Q^{\\prime }$ , with $Q=Q_\\mu $ defined as in REF .", "Differentiate under the integral sign to get $Q^{\\prime }(x) = Q(x) \\int _0^1 \\frac{1 - \\cos {\\left(2\\pi u\\right)}}{1 + x - x \\cos {\\left(2\\pi u\\right)}}du = \\frac{Q(x)}{2\\pi } \\int _{-\\pi }^{\\pi } \\frac{1 - \\cos {\\left(u\\right)}}{1 + x - x \\cos {\\left(u\\right)}}du.$ We will first assume that $x > 0$ and then show that the formula is still valid for $x =0$ .", "We perform the change of variable $t=\\tan (u/2)$ , using $\\cos (u) = \\frac{1-t^2}{1+t^2}$ , to get $\\int _{-\\pi }^{\\pi } \\frac{1 - \\cos (u)}{1 + x - x \\cos (u)}du &= \\int _{-\\infty }^{\\infty } \\frac{1 - \\frac{1-t^2}{1+t^2}}{1 + x - x \\frac{1-t^2}{1+t^2}} \\frac{2}{1+ t^2} dt = \\int _{-\\infty }^{\\infty } \\frac{4t^2}{(1 + t^2(1+2x))(1+t^2)}dt \\\\&= \\frac{2}{x}\\int _{-\\infty }^{\\infty } \\left(\\frac{1}{1+t^2} - \\frac{1}{1+ t^2(1+ 2x)} \\right)dt\\\\&= \\frac{2}{x} \\left[\\arctan (t) - \\frac{1}{\\sqrt{1+2x}}\\arctan (t\\sqrt{1+2x}) \\right]_{t=-\\infty }^{\\infty } \\\\&= 2\\pi \\frac{1 + 2x - \\sqrt{1+2x}}{x(1+2x)} =: 2\\pi h(x).$ We thus find $Q^{\\prime }(x)=Q(x)h(x)$ for $x > 0$ , and if we define $h(0):=1$ then the formula extends by continuity to $x=0$ .", "Using $Q(0)=1$ , we find $Q(x)=\\exp \\int _0^x h(u)du$ , and we compute this integral using the change of variables $v=\\sqrt{1+2u}$ : $\\int _0^x h(u) du &= \\int _0^x \\frac{1 + 2u - \\sqrt{1+2u}}{u(1+2u)}du = \\int _1^{\\sqrt{1+2x}} \\frac{2}{v+1} dv = \\log \\left(\\frac{1}{2}(\\sqrt{1+2x} + 1 + x) \\right).$ Therefore $Q(x) = \\frac{1}{2}(\\sqrt{1+2x} + 1 + x), \\quad \\text{and} \\quad Q^{\\prime }(x) = \\frac{1}{2}\\left(\\frac{1}{\\sqrt{1+2x}} + 1\\right).$ Recall from Proposition REF that we defined $\\Phi (x)=\\log (1+\\sqrt{1+2x}) -\\sqrt{1+2x} + x + \\tfrac{1}{2}$ , and $f(t):=f_\\mu (t):=\\Phi ^{-1}\\left(\\log 2 +\\frac{ct-1}{2} \\right)$ .", "It remains to show that $f$ satisfies the desired ODE $f^{\\prime }(t)=cQ^{\\prime }(f(t))$ with $f(0)=0$ , where we recall that this ODE is well-posed by Proposition REF .", "Note that $\\Phi $ is continuous and increasing on ${\\mathbb {R}}_+$ and maps ${\\mathbb {R}}_+$ to $[\\log 2 - 1/2, \\infty )$ .", "Hence, the inverse $\\Phi ^{-1}$ is well defined on $[\\log 2 - 1/2, \\infty )$ with $f(0)=\\Phi ^{-1}(\\log 2 - 1/2)=0$ .", "Straightforward calculus yields $\\Phi ^{\\prime }(x)=\\frac{\\sqrt{1+2x}}{1+\\sqrt{1+2x}}$ and thus $Q^{\\prime }(x)=\\frac{1}{2\\Phi ^{\\prime }(x)}$ , and we find that indeed $f^{\\prime }(t) = \\frac{c}{2\\Phi ^{\\prime }\\left(f(t)\\right)} = cQ^{\\prime }(f(t))$ , which concludes the proof of Proposition REF ." ] ]

[ [ "Neural Network Flame Closure for a Turbulent Combustor with Unsteady\n Pressure" ], [ "Abstract In this paper, neural network (NN)-based models are generated to replace flamelet tables for sub-grid modeling in large-eddy simulations of a single-injector liquid-propellant rocket engine.", "In the most accurate case, separate NNs for each of the flame variables are designed and tested by comparing the NN output values with the corresponding values in the table.", "The gas constant, internal flame energy, and flame heat capacity ratio are estimated with 0.0506%, 0.0852%, and 0.0778% error, respectively.", "Flame temperature, thermal conductivity, and the coefficient of heat capacity ratio are estimated with 0.63%, 0.68%, and 0.86% error, respectively.", "The progress variable reaction rate is also estimated with 3.59% error.", "The errors are calculated based on mean square error over all points in the table.", "The developed NNs are successfully implemented within the CFD simulation, replacing the flamelet table entirely.", "The NN-based CFD is validated through comparison of its results with the table-based CFD." ], [ "Nomenclature", "@l @   =    l@ $a_\\gamma $ ratio of specific heat capacity coefficient, $C$ progress variable $e$ relative distance between two signals $e_m$ relative distance between two signals mean value $e_f$ flame internal energy, $P$ pressure, $R$ gas constant, $\\eta $ the ratio of one signal rms to the other one's $T_f$ flame temperature, $Z$ mixture fraction $Z^{\\prime \\prime 2}$ mixture fraction variance from the mean $\\gamma $ ratio of specific heat capacity $\\kappa $ correlation between two signals $\\lambda $ thermal conductivity, $\\Omega $ vorticity, $\\dot{\\omega }_C$ progress variable reaction rate, $\\Phi $ flamelet output set $\\Psi $ flamelet input set @l @   =    l@ $\\widetilde{{white}{-}}$ density-weighted Favre average $\\overline{{white}{-}}$ Reynolds average $\\widehat{{white}{-}}$ empirical mean of a set" ], [ "Introduction", "Computationally efficient and accurate flamelet models for a turbulent combustor are needed for useful large eddy simulations (LES).", "Promise is offered through the use of deep neural networks (NN).", "Here, a well-studied configuration through prior LES and experiment is used but now with NN providing the sub-grid model for the flamelets.", "It is of special interest to extend the use of NN modeling for unsteady behavior of mean pressure and velocity fields.", "Both transient and dynamic equilibrium oscillatory conditions are considered.", "Combustion instability is an acoustical phenomenon caused by the high rate of energy release that increases pressure oscillation amplitude inside a combustion chamber; i.e., combustion excites and sustains an unstable high-amplitude pressure oscillation, which can be destructive.", "While most rockets and jet engines can oscillate, their unstable behaviors can differ.", "The configuration here is a single-injector liquid-propellant rocket engine (LPRE), see [1], [2].", "Models for LPRE combustion instabilities has been a research subject for decades [3], [4], [5].", "Computational fluid dynamics (CFD) models have been proposed for detecting combustion instabilities in LPREs.", "The Continuously Variable Resonance Combustor (CVRC) experiment of Purdue University provides a fundamental test case with available data [6], [7].", "CVRC is a combustion chamber with a single-element injector/oxidizer post, in which the oxidizer post length can be varied, resulting in configurations with different stability characteristics.", "Although it uses gaseous injection, the CVRC is an experiment that is widely accepted as a valid benchmark for computational methods that address combustion instability in liquid-propellant rocket engines.", "Current computational capabilities for reacting flows in realistic combustion chambers do not allow for resolution of the smallest scales of importance.", "Therefore, LES rather than direct numerical simulations (DNS) are common [8], [9].", "This requires models for sub-grid phenomena, especially for the combustion process.", "For gaseous reactants, the flamelet model ([10], [11]) enjoys popularity.", "The sub-grid model has been employed through a look-up table approach ([12], [13], [14]) that avoids the need for a computational time step determined by the chemical kinetics time scales, which typically are shorter than the ones from the numerically resolved physics.", "Nguyen et al.", "developed a hybrid LES/Reynolds-Averaged Navier-Stokes code, capturing combustion instabilities in CVRC at a much lower computational cost compared to prior works [12].", "The results from [12] were compared favorably with experimental data and numerical simulations of CVRC experiment developed in Purdue University [15], [6].", "Implementing the flamelet model was a key step in reducing the computational cost.", "[10] introduced the flamelet concept for turbulent combustion modeling.", "The flamelet model for turbulent combustion, based on the non-premixed flame physics, has been developed further by [11] for the integration of a sub-grid model with LES.", "The flamelet model is a sub-grid model, and its raison d'etre is that one cannot afford the grid resolution needed to incorporate combustion details in the LES.", "With the required resolution, the sub-grid model would not be needed.", "The flamelet model is based on the assumption that time scales for heat and mass diffusion, advection, and strain are larger than the chemical times; thereby, quasi-steadiness for chemistry is used.", "However, the chemistry is localized in narrow regions, and the stronger assumption of chemical equilibrium throughout the flow is not employed; i.e., there are narrow flame regions.", "While using flamelet models results in computational efficiency, they are the product of simplifying assumptions and they have certain limitations.", "There are uncertainties associated with presumed sub-grid PDF distributions as well as the progress variable definition.", "Flamelet theory assumes an axisymmetric strain field while three-dimensional behavior is commonly found in practice.", "A single diffusion flamelet occurs with fuel only on one side of the flame and oxygen only on the other side.", "Experience indicates that combustible mixtures can exist on one or both sides; premixed flames, diffusion flames, and multi-branched flames (e.g., triple flames) can occur.", "New flamelet models addressing these issues are in development ([16], [17], [18], [19]).", "Neural networks have been used in many different applications for the purpose of classification, clustering, and regression analysis.", "Previous studies have also used NNs as a tool to develop closure models in fluid mechanics and even combustion problems.", "Recursive NNs were utilized in an early work to predict the unsteady boundary-layer development [20].", "NN closure models have been incorporated for turbulence modeling in [21], [22], [23].", "Menon and co-workers ([24], [25]) used NN for the coupling of the linear eddy mixing (LEM) sub-grid model with the LES.", "[26] had developed optimal artificial NNs to compute flame variables and compare their performances with flamelet tables with different resolutions for a stably burning flame.", "Among the findings in [26] are that: (1) using NNs requires much less memory than a look-up flamelet table, (2) NNs can obtain a smoother flow field solution, and (3) NNs require higher computational cost than the flamelet table; yet the computational cost of using NNs is still significantly lower than solving the flame equations.", "Given the positive findings of [24], [25], [26], we envision that future versions of the flamelet model might best be introduced through the use of NNs, allowing nonlinear interpolation that is not easily obtained with the table, which now uses linear interpolation.", "NN can also allow both experimental and theoretical data to be used.", "Here, we use the data from the look-up table for training the NNs.", "In attempting to study the capability of the NNs not only to reproduce combustion behavior but also its integration into CFD calculation, flamelet tables provide a good stepping stone, as evidenced by the encouraging results shown in this work.", "In the future with new flamelet models, we propose moving directly to the flamelet calculations (possibly augmented by experimental data).", "The objective of our work is to develop NN-based closure models suitable for studying combustion instability.", "The key factor in such a model is the coupled relation between pressure and flame variables, specifically the progress variable reaction rate (PVRR) and heat release rate (HRR).", "Combustion instability leads to a considerably huge change in pressure, which affects the flame behavior.", "Thus, it requires the flamelet table to cover a large range of pressure; in fact, [12] needed to generate a flamelet table that takes pressure as an input.", "As an example, the maximum amount of HRR at 30 atm is 650.5% of the maximum HRR in 8 atm; 8 atm and 30 atm determine the pressure range for the CFD simulation of CVRC model with a 14-cm oxidizer post after reaching dynamic equilibrium state.", "A goal is to develop NNs for calculating flame variables based on a pressure-dependent flamelet model.", "In the earlier stages of the work, the approach was to develop a model that calculates flame related variables from the CFD simulation, replacing the function described within the dashed line in the schematic in  REF , [27], [28].", "In this paper, however, the approach is to study the data inside the flamelet table directly and to develop NNs that focus only on the flamelet model outputs; therefore the NNs are trained on the flamelet table data, that was used before in [12].", "Several different NN structures are developed.", "In the most accurate case, separate NNs for each of the flame variables are designed and tested by comparing their output with the output of the similar variable in the table.", "In Sec.", ", our numerical simulation and combustion modeling method is discussed.", "In Sec.", ", the design and evaluation of the proposed NNs are discussed.", "In Sec.", ", the NNs are tested on the flamelet table and are also implemented into different CFD simulations.", "The paper concludes with Sec. .", "The CVRC experiment is a single-injector co-axial dump combustor [29].", "Methane is injected through the outer concentric tube at 300 .", "The oxidizer, which is injected through the inner tube, is composed of 58% $H_2O$ and 42% $O_2$ at 1030 .", "Both reactants are injected at constant mass flow rates of $0.32$   and $0.027$   for the oxidizer and fuel, respectively.", "The 0.8 equivalence ratio makes the flow globally fuel-lean.", "Here, an unstable configuration with a 14-cm oxidizer post length and a 38-cm combustion chamber is the test case; see  REF .", "Also, a stable configuration is discussed with a 9-cm oxidizer post length and a similar 38-cm combustion chamber.", "Figure: Overview of the computational domain for the CVRC experiment and its CFD code architectureA constant mass flow rate inlet boundary condition is implemented using the Navier-Stokes Characteristic Boundary Conditions [30] at both reactant inlets.", "To save computational resources, a short-choked-nozzle [31] outlet boundary condition is used instead of an actual convergent-divergent nozzle computational domain; this promotes a high amplitude pressure oscillation.", "The mesh consists of 137,494 grid points.", "Its structure is based on the mesh used in the 3D calculations of Srinivasan et al.", "[32].", "The smallest radial grid size is 0.05 , located around the mixing shear layer or any walls.", "The smallest axial grid size is 0.2 , located both upstream and downstream of the back step.", "The maximum grid stretching factor along any direction is 1.05, thus ensuring high-quality mesh.", "Our code is a multi-block, structured finite-difference solver with axisymmetric cylindrical coordinates.", "(See  REF for a high-level view.)", "The overall accuracy is second-order in space and fourth-order in time.", "An important feature is the shock-capturing capability of the code longitudinal-mode.", "The entrance-to-throat area ratio is 5.", "On the wall, the no-slip boundary condition is applied.", "The walls are also assumed to be adiabatic and impermeable.", "Data is acquired from the CFD simulation at a 200 sampling rate from all points." ], [ "Combustion Modeling and Challenges", "The flamelet approach models the turbulent flame as a collection of laminar flamelets, where the chemical time scales are shorter than the turbulent time scales.", "Accordingly, the chemistry-related calculation occurs prior to the main flow simulation, through a quasi-steady flamelet table for a diffusion flame under normal compressive strain [10], [11], [12].", "Decoupling flamelet solution and the resolved LES flow simulation allows the prediction of the complex mechanism at a much less computational cost, yet with much less computational cost, which is the advantage of flamelet modeling ([13]).", "As shown in  REF , the inputs to the flamelet model are the flame state variables ($\\widetilde{C}$ , $\\widetilde{Z}$ , and $\\widetilde{Z\"^2}$ ) and $\\overline{P}$ .", "$\\widetilde{Z}$ indicates the mixture character at each point.", "$\\widetilde{Z\"^2}$ defines the deviation from the mean of $\\widetilde{Z}$ and is important for turbulent combustion.", "Essentially, in turbulent non-premixed modeling, laminar flamelet solutions are convoluted by a subgrid beta probability density function (pdf) of Z to model turbulence-chemistry interaction.", "The beta pdf parameters are functions of $\\widetilde{Z}$ and $\\widetilde{Z\"^2}$ .", "$\\widetilde{C}$ determines how much of the combustion process has been conducted at each time and point in space.", "A Dirac delta pdf is assumed to relate $C$ to $\\widetilde{C}$ according to the discussions in [11].", "The inputs of the flamelet model are collected in $\\Psi =[\\widetilde{Z},B,\\widetilde{C},\\overline{P}]$ , where $B=\\frac{\\widetilde{Z\"^2}}{\\widetilde{Z}-\\widetilde{Z}^2}$ is a surrogate variable that replaces $\\widetilde{Z\"^2}$ .", "In the input set, $\\overline{P}$ varies between 1 atm and 30 atm, $B$ and $\\widetilde{Z}$ range between 0 and 1, and $\\widetilde{C}$ varies from 0 to 0.261.", "The table includes 30 grid points for $\\overline{P}$ , 85 grid points for $\\widetilde{Z}$ , 26 grid points for $\\widetilde{Z\"^2}$ , and 156 grid points for $\\widetilde{C}$ .", "The output set for this flamelet model is defined as $\\Phi =[\\widetilde{\\dot{\\omega }}_C,\\widetilde{T}_f,\\widetilde{e}_f, \\widetilde{R},\\widetilde{\\lambda },\\widetilde{\\gamma },\\widetilde{a}_\\gamma ]$ .", "The proposed NN-based models in this work take exactly the same inputs and outputs as the inputs and outputs of the flamelet model stored in $\\Psi $ and $\\Phi $ , respectively.", "In the CFD structure, flow internal energy along with the above quantities are used to calculate flow temperature and enthalpy.", "Pressure is calculated based on the ideal gas law after extracting data from the flamelet table in the CFD.", "The main difficulty with the flamelet model is that the HRR becomes a one-dimensional quantity, although embedded in a three-dimensional flow field.", "Furthermore, in the case of combustion instability, the HRR interactions with turbulence and acoustical phenomena result in extremely nonlinear behaviors.", "Implementing the flamelet model helps to preserve the multi-scale and highly nonlinear behavior of HRR in the CFD simulation.", "In the flamelet-based simulation, HRR is not playing a direct role in the governing equation; instead, the PVRR is the variable that affects the governing equations as the source term in the $\\widetilde{C}$ transport equation.", "The flamelet solutions use a complex chemical mechanism (72 reactions with 27 species).", "Heat release rate relates to the products of the reaction rates of all the species and their corresponding enthalpies; in contrast, the progress variable, defined as the summation of CO2 and H2, only represents the major global chemical reactions.", "HRR accounts for more detailed (including less significant) reactions than the PVRR, which indicates the progress of the global methane reaction, using only simple global oxidation chemistry.", "However, it still has similar multi-dimensional, multi-scale, and highly nonlinear behavior, yet less severe and less costly to model than HRR." ], [ "Rayleigh Index as an Instability Criterion", "The most important cause of the high-frequency instability is the coupling between the HRR and the acoustic pressure wave.", "The Rayleigh Index (RI) measures this coupling based on the HRR and pressure fluctuation correlation determining if the HRR drives or damps the pressure wave.", "The time-averaged local Rayleigh Index is defined over a time period ($\\tau $ ), typically few cycles, starting from an initial time $t_o$ as: $RI=\\frac{1}{\\tau }\\int _{t_o}^{t_o+\\tau }\\frac{\\widetilde{\\gamma }+1}{\\widetilde{\\gamma }}\\times \\frac{p^{\\prime }}{\\widehat{P}}\\times \\frac{HRR^{\\prime }}{\\widehat{HRR}}dt$ where $p^{\\prime }$ , and $HRR^{\\prime }$ are the local fluctuations in $\\overline{P}$ and HRR, respectively [12].", "Also, $\\widehat{P}$ and $\\widehat{HRR}$ are the global time averages of $\\overline{P}$ and HRR.", "A positive (negative) value of RI conveys that the pressure oscillation is driven (damped) by HRR.", "The importance of RI led us to discuss a similar measure for PVRR oscillations by replacing HRR with $\\widetilde{\\dot{\\omega }}_C$ in Eq.", "(REF ) to get a modified Rayleigh Index (mRI).", "The objective here is to design a NN-based closure model to replace the flamelet table in  REF .", "Therefore, exactly the same sets of inputs and outputs of the flamelet model, i.e., $\\Psi $ and $\\Phi $ , are selected as the NN-based model input and output sets.", "A NN is a computational unit that replaces an input/output block in a system implementing its original task.", "After training, the NN learns to perform a task based on input-output examples without knowing the algorithm that led to those examples.", "A deep NN contains a series of layers, each of which has several nodes (neurons).", "Each neuron gets a linear combination of the outputs from neurons in the previous layer as its input, and provides an output, i.e., a specific nonlinear function (activation) of its input used for the next-layer calculations.", "The objective of the training process is to find the coefficients of those linear combinations, i.e., the weights, through solving an optimization problem to minimize the square of absolute error between NN outputs and the originally provided examples, i.e., the training data.", "The square of absolute error is selected as the cost function of the optimization problem to assure better estimation performance for the data with higher magnitude.", "The training is performed with the back-propagation method using the RMSProp update rule [33].", "We used the Leaky Rectified Linear ($LReLU(x)=max(x,0.001x)$ ) Activation Function in the hidden layers, and linear activation function at the output layer [34].", "The NN weights are given the random Xavier initialization [35].", "If the training process is overextended, typically, the NN performance is only good for the training set; this phenomenon is called overfitting.", "To avoid this, a validation error is computed on an independent set of samples and used to monitor overfitting [33].", "Providing the appropriate training and validation data sets are key steps in designing a NN.", "The optimal NN weights are obtained as those minimizing the error on the validation set.", "The inputs and outputs are standardized before introduction to the NN, by being centered around their mean and normalized by their standard deviation, to improve training performance [33]." ], [ "Sample Selection", "Enough samples must be provided to capture the problem complexity, while limiting the computational cost of the training.", "The flame-variable modeling is challenging because it is multi-scale; e.g., PVRR varies from $-284$   to $1.28e5$  .", "Combustion instability occurs from the coupling of high HRR and pressure; thus, the NN must accurately predict points with high HRR.", "However, the population of these points is relatively smaller than other points, requiring a nonuniform sample selection approach.", "The samples should be selected to provide a rich data set from high HRR points, with enough samples that should be selected from low HRR points to avoid biasing the NN towards uniformly high HRR values.", "Based on the designed table resolution, there exist around 10 million cells in the table, which are used for selecting training and validation data sets.", "Initially, a data set is selected uniformly from the available data in the table, then enriched with more points from higher energy release zones.", "At this stage, around 500,000 data points are selected for training, and around 200,000 data points are selected for validation representing 5% and 2% of the available points, respectively.", "After several iterations, when the learning progress starts to slow down, the partially trained NN is tested on additional data from the table.", "Then, points with errors higher than a specified bound (e.g., 5%) are added to the training set and points with errors higher than a lower bound (e.g., 2%) are added to the validation set.", "Here, different NNs are trained for each output of the table; so, after this step, the training and validation sets might be different among outputs.", "A maximum limit of 1.5 million points per set was established after the modification steps." ], [ "Network Structure", "A general feed-forward NN comprises of layers of different numbers of neurons, which communicate through the weighted links.", "The focus of this work is to explore the effectiveness of NNs as closure models in turbulent combustion; optimizing the NN structure was beyond the scope of the work.", "We arrived at the proposed structures in trial-and-error fashion by increasing the number of neurons in each layer and the number of layers to a point that good accuracy on the training and validation sets was achieved.", "Our observation is that increasing the number of NN layers leads to better accuracy than increasing the number of neurons in a single layer.", "Hence, NNs with more layers and fewer neurons on each layer are selected.", "In particular, two NN structures are introduced here.", "In the first NN, based on the physical behavior of flamelet data, the outputs are grouped as $G_1$ : $\\widetilde{\\dot{\\omega }}_C$ , $G_2$ : $\\widetilde{e}_f$ , $\\widetilde{R}$ , $G_3$ : $\\widetilde{T_f}$ , $\\widetilde{\\lambda }$ , $G_4$ : $\\widetilde{\\gamma }$ and $G_5$ : $\\widetilde{a}_\\gamma $ .", "($G_i$ denotes a group), and five different NNs (6-layer) are designed, one for each of these sets of outputs, each having 5 hidden layers with 15, 20, 25, 20, and 15 neurons, respectively.", "The input layer has 4, and the output layer has 1 or 2 neurons depending on the output group.", "A single output NN with the above structure requires 1771 floating point operations (flop) for data retrieval, while a double-output NN requires 1787 flop.", "This NN model, referred as $NN_a$ , requires 8887 flop for overall flame modeling.", "Another set of NNs, referred as $NN_b$ , improves the accuracy by increasing the number of layers and using 7 single-output NNs (8-layer) each with 7 hidden layers with 15, 20, 25, 30, 25, 20, and 15 neurons, respectively.", "Data retrieval in this NN requires 3326 flop for a single variable, and 23,282 flop for all variables.", "While the structure proposed in $NN_b$ increases the computational cost of retrieving flame data by 162% over $NN_a$ , it gives an approximately 40% reduction in the reconstruction Mean Square Error (MSE)." ], [ "Results ", "The closure model for flame variables must be implemented inside the CFD.", "Both the single-output and the multi-output NNs discussed in Sec.", "are implemented to replace the flamelet table.", "The designed NNs are tested in a stand-alone phase and an in-situ phase.", "In the stand-alone phase (offline), the NN is tested on all the available table data regardless of whether they are used or not for the training and validation, giving error bounds for each NN output compared to the corresponding table output.", "The second (online) phase tests whether the NN model can satisfactorily model combustion instability in the single injector LPRE case.", "Particularly, the highly coupled dynamics governing the LPREs makes combustion modeling very sensitive and complicated.", "Through the online test, NNs are successfully implemented within the CFD, and the are compared with the table-based simulations.", "In the following, the offline and online test results for $NN_a$ and $NN_b$ are provided." ], [ "Offline test", "The NNs are tested on all table data, while at most 15% of the data is used for the training process.", "REF presents the relative MSE for each variable calculated from Eq.", "(REF ); where $x_i$ is an element from the $n$ elements in the flamelet table, and $y_i$ is the NN-estimated value for that element.", "$e(\\%)=100\\times \\frac{\\sqrt{\\sum _i^n{(y_i-x_i)^2}}}{\\sqrt{\\sum _i^n{x_i^2}}}$ Table: Comparison of relative MSE (%) results for NN a NN_a and NN b NN_b on all data in the flamelet table-based CFD REF shows better performance for $NN_b$ than $NN_a$ by 36% to 44%.", "Having more layers and neurons, besides separate NNs for each variables in $NN_b$ provides more degrees of freedom for the model while, on the downside, it increases the computational cost relative to $NN_a$ by around 2.4 times.", "As $NN_b$ provides better estimation, the data from the flamelet table is compared with its counterpart calculated from $NN_b$ in  REF for $\\widetilde{e}_f$ , $\\widetilde{T_f}$ , $\\widetilde{\\lambda }$ , $\\widetilde{\\gamma }$ , $\\widetilde{a}_\\gamma $ , and PVRR.", "In the graphs, the x-axis shows the value of an output variable in the table, and the y-axis shows the value of the same variable estimated from NNs.", "Perfect estimation implies that all points lie on the $y=x$ line in each graph.", "On the graphs the lines $y=(1\\pm 1\\%) x$ , $y=(1\\pm 2\\%) x$ , $y=(1\\pm 5\\%) x$ , and $y=(1\\pm 10\\%) x$ are provided as guidelines for evaluating the NN performance in estimating the value of each point.", "Variable $\\widetilde{R}$ was estimated with the lowest error (less than 1%) and not plotted here.", "REF also shows that the performance of similar NN structures in estimating distinct variables is different.", "Some variables are estimated much better than others.", "One of the factors in determining NN performance is the variable range relative to its average value.", "$\\widetilde{R}$ and $\\widetilde{\\gamma }$ have relatively short ranges, and they are estimated with the lowest errors.", "PVRR, on the other hand, varies by five orders of magnitudes, thus it is estimated with the worst error.", "Figure REF shows that the worst relative accuracy in PVRR estimation occurs for low values, since higher values are dominating the training.", "Although each variable is estimated with different errors, our studies showed that required levels of accuracy are different for each variable for successfully implementing the model in CFD.", "The studies involved a sensitivity analysis of CFD to perturbations in the outputs of the flamelet table.", "Different simulations are conducted, and in each of them, one of the outputs of the flamelet table is perturbed with random Gaussian noise with bounded amplitudes.", "Then, we measured how that affects the pressure signal.", "For instance, our modeling error sensitivity analysis in CFD showed that estimating $\\widetilde{e}_f$ accurately is more important than others.", "Figure: Offline result comparison for internal energy,flame temperature,thermal conductivity, ratio of heat capacity,ratio of heat capacity coefficient,and PVRR" ], [ "Online Test", "Next, $NN_a$ and $NN_b$ are introduced in the CFD simulations for replacing the flamelet table.", "According to [12], the 14-cm oxidizer post is a CVRC configuration with a very high level of instability.", "In the 14-cm oxidizer post, for each of the NN sets, two different simulations have been conducted starting from two distinct initial conditions.", "The two initial conditions are selected at a time before the emerging instability and at a time after the instability has been fully developed.", "The first is called the \"transient case,\" and the second is called the \"dynamic equilibrium case.\"", "The pressure signals represent the time waveform of pressure at each of the spatial grid points over the combustor geometry.", "To validate the NN-based flame models, the results from each simulation are evaluated by investigating the similarity between pressure signals from NN-based and table-based CFD simulations.", "To this end, first, we looked at the overall relative error between pressure signals calculated from each CFD.", "Assume $x$ is the pressure signal from table-based CFD, and $y$ is the pressure signal from the NN-based CFDs.", "The relative error of signal $y$ with respect to $x$ is calculated based on Eq.", "(REF ), where the index i runs over the time signals.", "$n$ is the number of time snapshots, and $x_i$ and $y_i$ are the values of the signals at the i-th time sample.", "This gives an overall measure of the distance between the two signals relative to the magnitude of the target signal.", "As shown later in  REF , the third column, the pressure signals in the dynamic equilibrium case are estimated on average with 7.18% and 6.23% error by the $NN_a$ -based and the $NN_b$ -based simulations, respectively.", "For the transient case, the average error resulted from both simulations is 5.02%.", "This close results from both NN-based models raise the question of whether the relative error is a comprehensive measure for the purpose of modeling combustion instability.", "Equation (REF ) normalizes the distance of two signals by the magnitude of the reference.", "For oscillating signals with high mean values, such a measure ignores the fluctuations errors.", "In the combustion instability study, it is of high significance for a model to capture the phase and amplitude of the pressure fluctuation.", "In that regard, the root mean square (rms) of the pressure signals and their correlations are compared.", "The similarity between the phases of two signals ($x$ and $y$ ) can be measured through their correlation (Pearson's linear correlation coefficient), calculated as: $\\kappa (x,y)=\\frac{\\sum _{i=1}^n(x_i-\\bar{x}_i)(y_i-\\bar{y}_i)}{\\sqrt{\\sum _{i=1}^n(x_i-\\bar{x}_i)^2\\sum _{i=1}^n(y_i-\\bar{y}_i)^2}}$ where $\\bar{x}$ and $\\bar{y}$ are their empirical mean values.", "The similarity between the fluctuation amplitudes of the two signal is measured by comparing their rms.", "The rms ratio of the fluctuations of $y$ with respect to $x$ , is defined as: $\\eta (x,y)=\\frac{rms(y)}{rms(x)}=\\frac{\\sqrt{\\frac{1}{n}\\sum _{i=1}^n(y_i-\\bar{y}_i)^2}}{\\sqrt{\\frac{1}{n}\\sum _{i=1}^n(x_i-\\bar{x}_i)^2}}$ In Eq.", "(REF ) and Eq.", "(REF ), the time averages of the signals are subtracted from the main signal to calculate the fluctuation signals.", "Hence, the similarity between their mean values is also measured.", "The relative errors of the mean values ($e_m$ ) of the signals are calculated through Eq.", "(REF ) by replacing $x$ and $y$ by $\\bar{x}$ and $\\bar{y}$ .", "When signal amplitude is not constant over time (transient signal), the $\\bar{x}_i$ and $\\bar{y}_i$ are smoothed versions of the actual signals.", "Otherwise, the actual means $\\bar{x}_i$ and $\\bar{y}_i$ are equal for all indexes (dynamic equilibrium).", "REF summarizes the values of the above criteria for each of the simulations for dynamic equilibrium and transient cases.", "Each criterion is measured at each grid points on the CVRC geometry.", "The mean and standard deviation of all the measurements for each criterion is presented in  REF .", "The first and second columns identify the case and NN model used in each case.", "The third main column presents the overall error statistics by their mean and standard deviation in the sub-columns.", "Similarly, the fourth, fifth, and sixth columns present the correlation, rms ratio, and error in mean values.", "Note that the desired value for correlation and the rms ratio is 100%.", "Table: Error analysis summary for the 14-cm oxidizer post configuration test baseIn the dynamic equilibrium case, $NN_b$ has a slightly better performance in all criteria.", "However, $NN_a$ is selected as the representing flamelet model considering its computational cost, which is around 40% of the $NN_b$ -based simulation computational cost.", "The results from the $NN_a$ -based simulations and the table-based simulations are discussed in more detail in Sec.", "REF .", "In the transient case, both NN-based models perform similarly with respect to the overall error, mean, and correlation criteria.", "But for the pressure fluctuation, the rms ratio captured by the $NN_b$ -based and the $NN_a$ -based simulations are 103.63% and 75.84%, respectively.", "Therefore, $NN_b$ is selected for representing the flamelet model.", "The results from the $NN_b$ -based and the table-based simulations are discussed in more detail in Sec.", "REF .", "$NN_b$ , which shows higher capability in capturing flame dynamics, is also implemented in the 9-cm oxidizer post configuration, which, according to [12] is recognized as a stable configuration.", "This is discussed later in Sec.", "REF ." ], [ " Dynamic Equilibrium Case", "In combustion instability studies, the accuracy in simulating pressure is perhaps the most significant for validating the model.", "The relative error between pressure signals from the $NN_a$ -based and the table-based simulations, calculated by Eq.", "(REF ), is shown in  REF .", "The highest error happens near the centerline.", "In the axisymmetric configuration, the centerline ($r=0$ ) can be considered as a singularity and is prone to numerical errors.", "Accordingly, near the centerline, the signals are noisier and can be modeled with less quality.", "The correlation of the fluctuation of the pressure signal at each grid point with the pressure signal at the same point from the original table-based simulation is calculated and shown in the contour plot in  REF .", "The points with lower than 80% correlation are located in the pressure node, where the amplitudes of pressure harmonics are almost equal and smaller relative to other locations; therefore, the fluctuation signals there can be considered as noise.", "Moreover,  REF , compares the ratio of pressure fluctuation rms calculated from the $NN_a$ -based simulation to the rms calculated from the table-based simulation; the rms of fluctuations of the two simulations are shown in  REF and  REF , respectively.", "The parts of the combustor in which the rms of pressure fluctuations is lower is consistent with the regions where correlation is lower.", "These errors that are calculated from Eq.", "(REF ) to Eq.", "(REF ) are demonstrated by comparing pressure signals calculated from the table-based and the NN-based simulations at few points in the following.", "Figure: 14-cm, dynamic equilibrium: The distribution of κ\\kappa calculated from NN a NN_a-based () to the one calculated from table-based () simulationsFigure REF and  REF compare the pressure signals at the antinode (10 ), and near the nozzle at the pressure tab (37 ) on the top wall.", "At the antinode, the correlation is 87.73%, and near the nozzle, the correlation is 91.21%.", "At these points, the overall relative errors are 4.91% and 4.98%, respectively.", "The rms ratio is 95%, and the mean value is estimated with a 1.4% error at $x=10$ .", "At $x=37$ , the rms ratio is 94.74%, and the mean value is estimated with a 1.14% error.", "The first and second modes of oscillations are captured with high accuracy in these signals.", "The antinode and pressure tab are the common points that are investigated for each test because of their geometrical importance and to provide reference points among different cases.", "Figure REF and  REF compare pressure signals at points with lower correlations and higher error values.", "One of the points with a lower correlation is located on the centerline at 21.8 , i.e., the pressure node vicinity.", "Figure REF compares the pressure signal at this point between the NN-based and the table-based simulations.", "The correlation at this point is 40.07%.", "The overall error at this point is 9%, the rms ratio is 96.72%, and the mean value error is 0.9%.", "At 21.8 , looking at the pressure signal in the frequency domain, the power of the signal is distributed almost evenly among different modes, and the first mode is weaker relative to other locations.", "Essentially the higher frequencies are more significant at this point; however, the rms of the signal is captured to a good extent.", "The other point is located on the centerline at 1.14 , which is right after the dump plane near the mixing layer.", "Figure REF compares the pressure signal at this point with 53.64% correlation.", "The overall error at this point is 17.72%, the rms ratio is 89.03%, and the mean value error is 2.15%.", "This point is located near the interface of the oxidizer post and combustor, at $r=0$ ; a singular point with a very high density mesh.", "The spikes are associated with quantitative excursions of pressure at and beyond the limits used for the construction of the table (30 atm).", "Spikes do occur in close-neighboring physical locations with the NN as well.", "The spikes fortunately occur only in regions where little reaction occurs—near the centerline and near the entrance for the propellant flow into the main chamber.", "Thereby, with no reaction locally, these spikes have no consequence on the Rayleigh Index and the stability of the combustion chamber.", "Note that the physical behavior is chaotic, and therefore each realization can differ in detail.", "Spikes cannot be expected to occur always at the same location when the small quantitative differences (i.e., numerical errors) in constraints can appear between NN and the table.", "Figure: 14-cm, dynamic equilibrium: comparison of pressure signalsat different points on top wall and centerline between the NN a NN_a-based and the table-based simulationsAlso, to compare the frequency content of pressure signals, the mode shapes of the signals over the centerline are compared.", "Pressure mode shapes are demonstrated by plotting the modulus of the Fourier spectrum peaks at each grid point along the centerline.", "First, the time-average of the signals, which are plotted in  REF , are subtracted from it.", "Then, the Fourier spectrum is calculated for each of the centered signals.", "The first and the second modes are plotted in  REF and  REF .", "The $NN_a$ -based simulation gives a good estimate of the mean ($e_m$ ranges between 0.66% to 2.46%), the first, and the second modes; it starts to deviate as for the third mode; however, the effect of the third mode in the overall results is considered negligible.", "Figure: 14-cm, dynamic equilibrium: comparison of pressure mean, the first, and the second mode shapes between NN a NN_a-based and table-based simulationsAs discussed before, it is important for a model to provide a good estimation of RI to be useful in combustion stability research.", "The local mRI and RI are compared for the NN-based and the table-based simulations in  REF .", "Comparing  REF with  REF and comparing  REF with  REF show that the $NN_a$ -based simulation captures the location of the flame that drives instability, yet underestimates the flame near the corner.", "Particularly, the $NN_a$ -based simulation underestimates mRI from the table-based simulation near the corner and dump plane.", "HRR is calculated after the CFD simulation was conducted, only for the purpose of calculating RI.", "For RI, there exists a good correlation in the flame zone, but discrepancies downstream.", "The slight underestimation of RI and mRI are consistent with the slight underestimation of the rms value of the limit cycle with the NN-based simulation.", "Figure: 14-cm, dynamic equilibrium: comparison of RI and mRI from the NN a NN_a- and the table-based simulationsIn the following, other significant variables are briefly compared between the $NN_a$ -based and table-based simulations by demonstrating their time-averaged behavior.", "Figure REF compares the contour plots of time-averaged PVRR ( REF and  REF ), time-averaged vorticity ( REF and  REF ), time-averaged progress variable ( REF and  REF ), and time-averaged mixture fraction ( REF and  REF ).", "In all these figures, although there are differences in the detailed behavior, the overall shapes are very similar between the two simulations.", "Figure: 14-cm, dynamic equilibrium: time-averaged values of: PVRR, progress variable, mixture fraction, vorticity from the NN a NN_a-based and the table-based simulations" ], [ " Transient Case ", "As discussed before, the $NN_a$ -based simulation underestimates the rms value of pressure fluctuation in the transient case, while the $NN_b$ -based simulation has a similar rms with the table-based one.", "Accordingly, $NN_b$ is selected as the flame model for the transient simulation.", "The relative error between the $NN_b$ -based and the table-based simulations is shown in  REF .", "Similar to the dynamic equilibrium case, in the axisymmetric configuration, the centerline ($r=0$ ) can be considered as a singularity and prone to numerical error and noisier signals.", "Correlation measures how two signals are similar after their mean values have been subtracted.", "In the transient case, using a global mean value will distort the ability of correlation in measuring the performance of a model.", "To calculate the local mean value, the signals are smoothened using local quadratic regression [36].", "The mean values are subtracted from each signal at each spatial point to get the fluctuation signals.", "The correlation of the fluctuation of the pressure signal at each grid point from the $NN_b$ -based and the original table-based simulations are calculated and shown in the contour plot in  REF .", "In addition, the pressure fluctuation rms values calculated from the $NN_b$ -based simulation ( REF ) are compared to those calculated from the table-based simulation ( REF ) through their ratio at each grid point.", "The rms ratio is shown in  REF .", "Figure: 14-cm, transient: The distribution of κ\\kappa calculated from the NN b NN_b-based () to the one from the table-based () simulationsFigure REF and  REF compare the pressure signals at the antinode (10 ) and near the nozzle (37 ) on the top wall for the transient case.", "At the antinode, the correlation is 87.65%, and near the nozzle, the correlation is 91.46%.", "At these points, the overall relative errors are 3.65% and 4.02%, respectively.", "The rms ratio is 104.41%, and the mean value is estimated with 0.95% error at $x=10$ .", "At $x=37$ , the rms ratio is 100.42%, and the mean value is estimated with a 1.43% error.", "Figure REF compares the pressure signal located on the centerline near the pressure node (17.1 ) between the NN-based simulation and the original one, where the correlation is 12.48%.", "The overall error at this point is 7.33%, the rms ratio is 112.02%, and the mean value error is 1.08%.", "Another point with low correlation is located above the centerline at 2.58 , which is right after the dump plane near the centerline.", "Figure REF compares the pressure signal at this point with a correlation of 71.36%.", "The overall error at this point is 7.42%, the rms ratio is 105.79%, and the mean value error is 1.15%.", "These points are selected to demonstrate the need for different criteria to measure error.", "In  REF , the overall errors are similar, yet the correlations are drastically different.", "Figure: 14-cm, transient: the NN b NN_b-based and the table-based simulations comparison of pressure signalat different pointsThe frequency content of pressure signals are compared through the mode shapes over the centerline.", "In the analysis here, only the steady-state parts of the simulations are considered.", "The time-average of each of the signals is plotted in  REF .", "The value of the Fourier spectrum for the first and the second modes are plotted in  REF and  REF .", "The NN-based simulation presents a good estimate of the mean ($e_m$ ranges between 1.15% to 2.01%), first and second modes.", "The discrepancies between the two NN-based and table-based simulations start to grow from the third mode.", "The local RI and mRI are compared for the NN-based simulation in  REF .", "Comparing  REF with  REF , and comparing  REF with  REF , show a great similarity in flame location and magnitude between the two simulations.", "Figure: 14-cm, transient: comparison of pressure mean, the first, and the second mode shapes between NN b NN_b-based and table-based simulations (last 4 msms)Figure: 14-cm, transient: comparison of RI and mRI between the NN b NN_b-based and the table-based simulationsAt three time points, from the transient simulations the snapshots of pressure and vorticity ( REF ), progress variable, and PVRR ( REF ) are compared between the $NN_b$ -based and the table-based simulations.", "The time snapshots are selected at $t_1=0.5$ , $t_2=3.5$ , and $t_3=6.5$ , where the last time point is associated with the time that signals have reached to the limit cycle, while the other two time points are associated with the growing parts of the pressure signal.", "In the progress variable graphs, the flame front shape is predicted with a great accordance.", "The flame front is the curve (thin flame) ( REF ) that separates the low and high temperature zones.", "These quantities are governed by both turbulence and acoustic behavior.", "The turbulent combustion causes the system to become chaotic, while the acoustic phenomena causes the system to resemble wave forms and modal behavior.", "Accordingly, for variables that are dominantly affected by turbulence, the performance of the NN-based model can be assessed by the statistical characteristics and consequences of those variables; while for variables where the acoustic phenomena is the significant deriver, the NN-based model can be compared in a point-wise manner.", "One major statistical consequence for a variable such as PVRR is the modified RI that was shown in REF and REF for the NN-based and table-based simulations, respectively.", "The $mRI$ calculated from NN-based simulation shows highly consistent behavior to the $mRI$ calculated from table-based Simulation, despite the differences that can be spotted in the detailed behavior of PVRR in the time snapshots (e.g., REF vs. REF ).", "Figure: 14-cm, transient: pressure and vorticity snapshots from the NN b NN_b-based and the table-based simulationsFigure: 14-cm, transient: progress variable and PVRR snapshots from the NN b NN_b-based and the table-based simulations" ], [ " 9-cm case", "NN training was designed mostly for the purpose of reproducing the CFD simulation results of the CVRC experiment under unstable pressure oscillation conditions.", "This happens when the oxidizer post length is 14 .", "If the length is 9 , the pressure oscillations grow to a limit cycle, which is much smaller than in the 14-cm case.", "So the 9-cm is considered to be stable.", "Next, to validate its generality, the $NN_b$ model, which was successfully implemented in the 14-cm transient case, is also implemented in the 9-cm case.", "The relative error between the NN-based and the table-based simulations is shown in  REF .", "The highest error occurs in the oxidizer post, as the NN is not forced to have certain values there.", "On the other hand, the table values are enforced at the non-reacting zones.", "In our future work, we will include such constraints in the NN development.", "The correlation of the fluctuations of the pressure signals from their mean value from the $NN_b$ -based and table-based simulations is shown in  REF .", "The oscillations are not that well correlated.", "The reason is that the oscillation amplitude (roughly about 50 ) is very small relative to the mean value (roughly about 1600 ).", "Essentially, the maximum oscillation amplitude is around 3% of the mean value, and it can be considered as noise.", "Figure: Case with 9 cm oxidizer post: distribution of relative error (%) and fluctuation correlation between pressure signals calculated from the NN b NN_b-based and the table-based simulationsThe distribution ratio of pressure fluctuation rms in the $NN_b$ -based simulation ( REF ) to that in the table-based simulation ( REF ) is shown in  REF .", "Most of the points have a similar rms value.", "Here again, the $NN_b$ -based simulation overestimated the rms values.", "Both the rms values and their spatial gradients in the 9-cm case are also lower than the ones in the 14-cm unstable case.", "The pressure rms values are higher at the inlet, and they become smaller throughout the combustor.", "Figure: Case with 9 cm oxidizer post: The distribution of κ\\kappa calculated from NN b NN_b-based () to the one calculated from table-based () simulationsFigure REF compares the pressure signal at the antinode (10 ), and near the nozzle (37 ) on the top wall, calculated from the $NN_b$ -based simulation with that from the table-based simulation.", "At the antinode, the correlation is -3.68%, and near the nozzle, the correlation is 0.32%.", "At these points, overall relative errors are 2.71% and 3.25%, respectively.", "The rms ratio is 108.32%, and the mean value is estimated with 0.8% error at $x=10$ .", "At $x=37$ , the rms ratio is 125.11%, and the mean value is estimated with 0.8% error.", "The low correlations between the fluctuations of signals are because of the noisy nature of the pressure waveform in this configuration.", "Here, we are not interested in capturing noise and we did not pursue refining the NN for a better performance.", "However, although it is harder to generate a fully correlated signal, the modal frequencies and signal trends are similar between the NN-based and table-based simulations.", "Figure: Case with 9 cm oxidizer post: comparison of pressure signalson the wall, at pressure antinode and near the nozzle, between the NN b NN_b-based and the table-based simulationsThe pressure mode shape shows similar behavior in  REF for the mean value and the first and the second mode shapes, where the modes are not predicted as well as the mean.", "The first and the second modes in the 9-cm case have relatively small magnitudes; also, the magnitude of the modes are approximately equally distributed.", "In the combustion zone, the first mode is approximately 0.31% of the mean in the NN-based simulation, and 0.22% in the table-based simulation.", "Similarly, the second mode is approximately 0.25% of the mean in the NN-based simulation, and 0.19% in the table-based simulation in the combustion zone.", "Figure: 9-cm case: comparison of pressure mean and the first and second mode shape between NN b NN_b-based and table-based simulationsSince the 9-cm oxidizer post configuration leads to a stable configuration, the observed pressure fluctuations resembles relatively small amplitude.", "Essentially, the dynamic of the system is governed significantly by the turbulent combustion rather than acoustic behavior.", "The turbulence leads to a chaotic behavior in fluctuations for different quantities.", "Therefore, it is expected from a model to regenerate similar statistical behavior, such as Rayleigh Index, rather than regenerating exactly the same solution as the reference model.", "The local mRI and RI are compared for the two simulations in  REF .", "Comparing  REF with  REF and comparing  REF with  REF show that the $NN_b$ -based simulation is very similar to the table-based one in capturing the behavior near the corner and the RI shape.", "Since the 9-cm case is a stable one, the magnitude of RI and mRI are lower than those in the 14-cm case.", "Low RI shows that there is no instability occurring.", "Figure: Case with 9 cm oxidizer post: comparison of RI and mRI from the NN b NN_b-based and the table-based simulationsThe computational cost for the NN-based and table-based simulations are compared in REF .", "One millisecond (200 snapshots) of data is generated from the dynamic equilibrium test cases for each of the simulations.", "The required CPU time to generate each of the snapshots is calculated for each of the three simulations; the minimum, average, standard deviation, and the maximum of the required CPU time for generating each simulation time step are reported in REF .", "The table-based simulation is considered as the reference, and the relative values of these quantities to the reference are also reported for the $NN_a$ and $NN_b$ .", "The average CPU times per time step for the $NN_a$ -based and the $NN_b$ -based simulations are around 39.2 and 100.7 times the average CPU time per time step of the table-based simulation.", "The table-based simulation is essentially the same code that was developed and discussed in [12].", "[12], reported a 0.28 core-hour per millisecond for the computational cost of an axisymmetric simulation with $6.26e4$ grid points based on a flamelet model for combustion of 27 species, while a similar work, discussed in [15], reported 53 core-hour per millisecond for the computational cost of an axisymmetric simulation with $5.5e4$ grid points based on a general equation and mesh solver (GEMS) for combustion of 4 species.", "The current work uses $1.375e5$ grid points.", "Also, one should note that the computational costs for the NN-based simulations reported here are based on a non-optimized code; there is a potential to reduce the NN-based simulation computational cost considerably by applying methods to parallelize and optimize the code.", "Yet, optimizing the computational cost was not in the scope of this work.", "Table: CPU time per time step between two consecutive time samples statistics comparison among table-based, NN a NN_a-based, and NN b NN_b-based simulations for 14-cm dynamic equilibrium caseAlthough NN-based closure models increase the computational cost over a flamelet table model, they reduce the required memory significantly.", "The flamelet model requires around 1.2 GB of the hard disk memory, while $NN_a$ and $NN_b$ require around 144.2 kB and 376 kB, respectively, to be stored on our computer machine.", "We estimated that loading NN-based models into CFD causes the required RAM to increase less than 1 MB, while loading the table-based model into CFD increases the required RAM around 276 MB.", "The difference between the volume of data when it is stored on the hard disk vs. when it is loaded in the RAM is because of the data on hard disk is stored as text file, while it is loaded as numbers in computer RAM.", "The very low required memory by NN-based models paves the way for utilization of Graphic Processing Units (GPUs) and consequently high levels of parallelization, which can significantly reduce the computational cost.", "Particularly, in 2010, [37] provided examples of 16 to 137 times speed up in fluid mechanics related problems when they are implemented on GPU." ], [ "Conclusion", "In this study, two deep learning neural network sets with different levels of complexity were developed to represent a flamelet model in turbulent combustion with unsteady pressure.", "The design of the networks was explained, including input-output and training set selection.", "The goal is to explore the capability of neural networks as a tool for combustion modeling.", "The two developed models ($NN_a$ and $NN_b$ ) are first validated by testing on the flamelet table data, as an offline test, and then validated by being implemented in CFD simulations of different cases and compared with the table-based simulations.", "These simulations include dynamic equilibrium and transient simulations on an unstable rocket configuration (14-cm oxidizer CVRC), and transient simulation in a stable configuration (9-cm oxidizer CVRC).", "$NN_b$ , which contains more layers than $NN_a$ , has shown a better performance both in offline and online validation.", "In the offline test, the difference between their error is very small (up to 2%).", "$NN_b$ wins over the $NN_a$ model in the accuracy competition.", "However, its required computational cost is 2.5 times of that for $NN_a$ for retrieving one set of outputs.", "In the dynamic equilibrium case, the two NN-based models have similar performance according to measures such as overall relative error and fluctuation correlation.", "The $NN_a$ -based simulation results were in agreement with the table-based simulation results.", "However, mRI was underestimated near the dump plane through the $NN_a$ -based simulation.", "In the transient case, the main shortcoming of the $NN_a$ -based simulation is in predicting the pressure limit cycle or rms of pressure fluctuation.", "On average, only 75% of the fluctuation energy is captured by the $NN_a$ simulation; whereas, the $NN_b$ simulation can capture the amplitude with great consistency, close to 100%.", "In the transient case, $NN_b$ is the proper model as it is capable of faithfully reproducing the unstable acoustic behavior.", "The $NN_b$ -based model was also implemented in a case with 9-cm oxidizer post, which was characterized as a stable configuration of CVRC.", "Although, the mean pressure were predicted in this test, the pressure fluctuation of the NN-based simulation did not follow those from table-based simulation exactly.", "In the stable case, the dynamics are significantly governed by the turbulent combustion rather than acoustic behavior.", "The significance of turbulence combustion causes the simulation to be resemble more chaotic behavior in different quantities.", "Particularly, our analysis in all three simulations show that NN-based simulation provides more correlated results with the table-based simulation, when the acoustic phenomenon dominates the system over the turbulence.", "Flamelet models provide a good stepping stone, as evidenced by the encouraging results shown in this work.", "The authors also hold the view that the framework presented in this paper can be applied on high-quality data from sources other than the flamelet table such as high fidelity LES or even DNS, where the cost saving of using machine learning models can be highly advantageous.", "NN models require much lower memory than the flamelet table.", "Although the data retrieval from a NN model is more time costly than reading the data from a look-up table, the computational cost is still lower than using the chemical kinetics solver.", "Our data retrieval code for NN-based CFD is not optimized.", "In our future work, the computational cost of NN modeling will be revisited after parallelizing and optimizing the codes for a GPU implementation." ], [ "Acknowledgments", "This research was supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-18-1-0392, with Mitat Birkan as the scientific officer." ] ]

[ [ "Top Hadrons in Lorentz-Violating Field Theory" ], [ "Abstract If there is Lorentz symmetry violation in the $t$ quark sector of the standard model, changes to particles' dispersion relations might allow for the existence of stable top-flavored hadrons.", "Observations of the survival of high-energy $\\gamma$-rays over astrophysical distances can be used to place one-sided constraints on certain linear combinations of Lorentz violation coefficients in the $t$ sector at the $\\sim 10^{-4}$ level of precision." ], [ "Introduction", "Since the introduction of the special theory of relativity, there has always been interest in understanding whether the Lorentz symmetry of special relativity is exact, or whether it is merely an excellent approximation.", "Approximate symmetries have turned out to be an extremely important topic in particle physics.", "Thanks to advances in effective field theory over the last quarter century, there has been an explosion of interest in the quantitative problem of evaluating how well Lorentz invariance has been confirmed.", "The fundamental physics that we currently understand is based on two very different theories, the standard model, describing particle physics, and general relativity, which describes gravitation.", "Understanding how these two theories can be reconciled into a theory of quantum gravity is probably the greatest remaining challenge in fundamental physics.", "However, despite their puzzling differences, the two basic theories actually have a number of important features in common.", "These include a number of spacetime symmetries; both the standard model and general relativity are invariant under rotations, Lorentz boosts, and CPT.", "Experimental searches have not turned up any compelling evidence of Lorentz violation; however, it remains a topic of theoretical and experimental interest.", "If any deviations from Lorentz invariance were conclusively demonstrated, that would be a discovery of tremendous significance and would open a new window into our understanding of the laws of physics at the most elemental level.", "Using the machinery of effective theory, it is possible to describe deviations from these symmetries, whether involving standard model quanta or gravitational effects.", "The general effective field theory that can be used to describe such effects is called the standard model extension (SME).", "The full SME contains an infinite hierarchy of Lorentz-violating operators.", "However, the action for the minimal SME—which contains only those operators constructed out of known standard model fields that are gauge symmetric, translation invariant, and power-counting renormalizable—contains a finite number of coupling parameters.", "The SME operators resemble the operators that appear in the usual standard model action, except that they may break Lorentz symmetry when they have residual tensor indices.", "The Lagrange density may be written in a form in which the residual indices on the operators are contracted with tensor-valued coefficients, so that the coefficients indicate the presence of preferred vector and tensor backgrounds [1], [2].", "If the origin of the Lorentz violation is a form of spontaneous symmetry breaking, then the preferred background tensors are set by the vacuum expectation values of tensor-valued dynamical fields.", "The minimal SME is well suited for comparing the results of experimental Lorentz tests.", "Experiments in many different areas of physics have been used to place bounds on the minimal SME coefficients.", "Information about the best current bounds are collected and summarized in [3].", "It happens that many of the strongest bounds on various SME coefficients come from astronomy.", "These bounds typically take advantage of either the exceeding long travel distances or very high energies that are available to certain extraterrestrial quanta.", "Observations of photons from astrophysical sources have thus been used to place many of the best constraints on Lorentz violation.", "By examining the photons emitted by very distant sources, we may learn a great deal about the dispersion relation of the photons themselves, to see whether it has the expected Lorentz-invariant form $E_{\\gamma }=|\\vec{p}\\,|$ .", "However, observing that a photon was emitted and that it made its way to our detectors can also tell us quite a bit about the behavior of the other particles that might interact with such a photon.", "The observation of TeV-scale $\\gamma $ -rays reveals information about the processes that might have produced them—typically either inverse Compton scattering, $e^{-}+\\gamma \\rightarrow e^{-}+\\gamma $ , involving the upscattering of a low-energy photon by an exceedingly energetic electron, or neutral pion decay, $\\pi ^{0}\\rightarrow \\gamma +\\gamma $ .", "The fact that such photons are produced at all tells us things about energy-momentum relations of all the particles that are involved in the reaction.", "The experimental observation of the spectra of astrophysical $\\gamma $ -ray sources can therefore be used to place bounds on Lorentz violation in the lepton and hadron sectors of the SME [4], [5], [6], [7].", "However, even when the production methods of these $\\gamma $ -rays are unclear, they can still tell us interesting things about Lorentz violation in other sectors, merely based on the fact that the photons live long enough to reach us.", "For the massless photon, any kind of decay process that occurs in vacuum and involves one or more massive daughter particles is forbidden by energy-momentum conservation, if all the quanta have their usual Lorentz-invariant dispersion relations.", "A decay such as $\\gamma \\rightarrow e^{+}+e^{-}$ or $\\gamma \\rightarrow t+\\bar{t}$ (with a top-flavored quark-antiquark pair), or an emission process such as $\\gamma \\rightarrow \\gamma +\\pi ^{0}$ , cannot occur without another constituent (such as a nearby heavy nucleus) to take up the photon's excess momentum.", "However, if the energy-momentum relations are Lorentz violating—so that, at large momenta, the energies of the daughter particles increase more slowly as a function of $p$ than the energy of the parent photon $E_{\\gamma }(\\vec{p}\\,)$ —then the process may be permitted above some threshold.", "(Decay of a photon into other strictly massless quanta—that is, photon splitting, $\\gamma \\rightarrow N\\gamma $ —is separately forbidden, not by energy-momentum conservation, but by vanishing of the relevant on-shell matrix element [8].", "Yet for that kind of process also, Lorentz-violating radiative corrections may again make the process possible, even when then photon sector itself is not modified [9].", "However, since the process is always precisely at threshold, there are phase space issues, and the process would manifest itself not as an irreversible decay, but via oscillation phenomena, such as $\\gamma \\rightarrow 2\\gamma \\rightarrow \\gamma \\rightarrow \\cdots $ .)", "In this paper, we shall expand the use of observations of the survival of ultra-high-energy propagating photons to place constraints on Lorentz violation in the $t$ sector.", "Section  describes the structure and one-particle kinematics of the minimal SME operators of interest.", "The kinematic details of the single-photon $t$ -$\\bar{t}$ production process, and the bounds on SME parameters that result from the observed absence of this process are discussed in section .", "The $t$ sector is more complicated than others, because $t$ quarks do not normally exist as constituents of asymptotic hadron states; a $t$ produced in a reaction will ordinarily decay weakly before hadronization occurs.", "This will make the analysis for this sector somewhat more intricate than that required in other sectors of the SME.", "Our conclusions, and the outlook for further improvements, are discussed in secton ." ], [ "Lorentz Violation in the Minimal SME", "The Lagrange density for the minimal SME contains operators of mass dimensions 3 and 4, with their outstanding indices contracted with those of tensor-valued coefficients.", "For the QED sector of the minimal SME, with a single fermion species, the Lagrange density is ${\\cal L}=-\\frac{1}{4}F^{\\mu \\nu }F_{\\mu \\nu }-\\frac{1}{4}k_{F}^{\\mu \\nu \\rho \\sigma }F_{\\mu \\nu }F_{\\rho \\sigma }+\\frac{1}{2}k_{AF}^{\\mu }\\epsilon _{\\mu \\nu \\rho \\sigma }F^{\\nu \\rho }A^{\\sigma }+\\bar{\\psi }(i\\Gamma ^{\\mu }D_{\\mu }-M)\\psi ,$ where the SME coefficients for the fermion are collected in $M & = & m+a^{\\mu }\\gamma _{\\mu }+b^{\\mu }\\gamma _{5}\\gamma _{\\mu }+\\frac{1}{2}H^{\\mu \\nu }\\sigma _{\\mu \\nu }+im_{5}\\gamma _{5} \\\\\\Gamma ^{\\mu } &= & \\gamma ^{\\mu }+c^{\\nu \\mu }\\gamma _{\\nu }+d^{\\nu \\mu }\\gamma _{5}\\gamma _{\\mu }+e^{\\mu }+if^{\\mu }\\gamma _{5}+\\frac{1}{2}g^{\\lambda \\nu \\mu }\\sigma _{\\lambda \\nu }.$ The gauge-covariant derivate $D_{\\mu }=\\partial _{\\mu }+iqA_{\\mu }$ contains the charge of the fermion species, with $q=-\\frac{2}{3}e=\\frac{2}{3}|e|$ for the $t$ quark.", "At ultrarelativistic energies, the dimension-4 coefficients ($k_{F}$ for photons and those comprising $\\Gamma $ for the fermion field) dominate the dispersion relations for the quanta, so we may neglect all the dimension-3 operators except the usual $M\\approx m$ .", "Moreover, while the $e^{\\mu }$ , $f^{\\mu }$ , and $g^{\\lambda \\nu \\mu }$ coefficients are acceptable in a pure QED theory, they are not consistent with the $SU(2)_{L}$ gauge invariance of the standard model.", "They can only arise as vacuum expectation values of higher-dimensional operators involving the Higgs field and thus actually excluded from the truly minimal SME.", "This leaves the fermion-sector Lorentz violation at ultrarelativistic energies predominantly controlled by $c^{\\nu \\mu }\\gamma _{\\nu }+d^{\\nu \\mu }\\gamma _{5}\\gamma _{\\mu }$ .", "For the electromagnetic sector, the dimension-4 $k_{F}^{\\mu \\nu \\rho \\sigma }$ tensor (which, in the most general case, may be taken to have the same symmetries as a Riemann curvature tensor and a vanishing double trace) may be split into two-parts (with Weyl-like and Ricci-like structures, respectively).", "The ten-components of the Weyl-like part (as well as the four possible dimension-3 $k_{AF}^{\\mu }$ terms) give rise to photon birefringence in vacuum, which has not been seen, even for polarized sources at cosmological distances [10], [11].", "The bounds on certain linear combinations of these dimensionless coefficients are down at the $10^{-38}$ level.", "We shall therefore neglect all these birefringent terms.", "The remaining Ricci-like terms have the structure $k_{F}^{\\mu \\nu \\rho \\sigma }=\\frac{1}{2}\\left(g^{\\mu \\rho }\\tilde{k}_{F}^{\\nu \\sigma }-g^{\\mu \\sigma }\\tilde{k}_{F}^{\\nu \\rho }-g^{\\nu \\rho }\\tilde{k}_{F}^{\\mu \\sigma }+g^{\\nu \\sigma }\\tilde{k}_{F}^{\\mu \\rho }\\right),$ where $\\tilde{k}^{\\mu \\nu }=k_{F\\alpha }\\,^{\\mu \\alpha \\nu }$ .", "The bounds on these $\\tilde{k}_{F}^{\\mu \\nu }$ coefficients are weaker, although for some terms, the bounds are still quoted at the at the quite respectable $10^{-22}$ level [12], based on a Michelson-Morely analysis of Laser Interferometer Gravitational-Wave Observatory (LIGO) data (although it should be noted these bounds are actually necessarily on differences between the electromagnetic $\\tilde{k}_{F}^{\\mu \\nu }$ coefficients and some set of aggregate coefficients for everyday matter).", "With flavor-changing SME coefficients neglected, there are separate sets of background tensors for each fermion species.", "However, because of the $SU(2)_{L}$ gauge invariance, which connects fermions with different charges, not all the coefficients for different flavors are truly independent.", "Between the $t$ and $b$ quarks, there are really only three independent $c^{\\mu \\nu }$ and $d^{\\mu \\nu }$ tensors.", "The gauge invariance ensures that the chiral SME coefficint $c_{L}^{\\mu \\nu }=c^{\\mu \\mu }+d^{\\mu \\nu }$ is the same for the left-handed $t$ and $b$ quarks that form an $SU(2)_{L}$ doublet.", "However, the $c_{R}^{\\mu \\nu }=c^{\\mu \\mu }-d^{\\mu \\nu }$ for the two species are, generally speaking, unrelated.", "The tensor $\\frac{1}{2}\\tilde{k}_{F}^{\\mu \\nu }$ is the photonic analogue of the $c^{\\mu \\nu }$ for fermions.", "Each of them represents a spin-independent modification of the ultrarelativistic dispersion relation.", "For the fermions, the modified dispersion relation is most straightforwardly expressed in terms of the maximum achievable velocity (MAV) of a particle (which can depend on the direction of its motion and its spin).", "The MAV along the direction $\\hat{v}$ is $1+\\delta (\\hat{v})$ , where [13] $\\delta (\\hat{v})=-c_{00}-c_{(0j)}\\hat{v}_{j}-c_{jk}\\hat{v}_{j}\\hat{v}_{k}+sd_{00}+sd_{(0j)}\\hat{v}_{j}+sd_{jk}\\hat{v}_{j}\\hat{v}_{k},$ and where $s$ is the product of the particle helicity and fermion number (e.g.", "$+1$ for a quark, $-1$ for an antiquark).", "The parentheses, as in $c_{(0j)}$ , indicate a symmetrized sum $c_{0j}+c_{j0}$ .", "Note that $c^{\\mu \\nu }$ and $d^{\\mu \\nu }$ appear in $\\delta (\\hat{v})$ only via contractions $c^{\\mu \\nu }v_{\\mu }v_{\\nu }$ and $d^{\\mu \\nu }v_{\\mu }v_{\\nu }$ , where $v^{\\mu }=(1,\\hat{v})$ .", "The dispersion relation for a relativistic quantum is $E=\\sqrt{m^{2}+p^{2}[1+2\\delta (\\hat{p})]},$ up to corrections of ${\\cal O}\\left(m^{4}/E^{4}\\right)$ .", "For one-dimensional motion restricted to the $\\hat{p}$ -direction, this just looks like ordinary special relativity, but with a modified top speed $1+\\delta (\\hat{v})$ .", "The actual velocity at these ultrarelativistic energies is $\\vec{v}=\\left[1+\\delta (\\hat{p})-\\frac{m^{2}}{2p^{2}}\\right]\\hat{p},$ which is simply the usual ultrarelativistic velocity plus the Lorentz-violating correction to the MAV, $\\delta (\\hat{p}\\,)\\hat{p}$ .", "An electromagnetic decay such as $\\gamma \\rightarrow t+\\bar{t}$ would occur extremely rapidly if it were allowed.", "The survival of GeV-scale laboratory photons over everyday distances has been used to place a bound on certain SME coefficients [14].", "So if a photon, with a certain energy and traveling in a certain direction, manages to reach us from an astronomical source, that is essentially certain evidence that there are no photon decay products that are kinematically permitted for that particular combination of energy and direction.", "It is well established that photons with TeV energies can survive over astrophysical distances, and this has been used to place bounds on the differences between certain Lorentz violation coefficients for photons and the kinematically equivalent coefficients for leptons or hadrons [15], [16].", "The absence of the process $\\gamma \\rightarrow X^{+}+X^{-}$ in a direction $\\hat{v}$ places restrictions on the linear combination $\\left(c_{X}^{\\mu \\nu }-\\frac{1}{2}\\tilde{k}_{F}^{\\mu \\nu }\\right)\\!v_{\\mu }v_{\\nu }$ of SME parameters.", "The strength of the bound depends on the mass $m_{X}$ of the charged daughter particles, with a characteristic ${\\cal O}(m_{X}^{2}/E_{\\gamma }^{2})$ dependence.", "The strongest individual bounds come from the photons with the highest observed energies, but $\\gamma $ -rays coming from a variety of different directions are required to get the best overall constraints on all the components of the traceless, symmetric Lorentz tensor $c_{X}^{\\mu \\nu }$ .", "Lorentz violation for photons, and for the first-generation fermions that are the primary fermionic constituents of everyday matter, is generally fairly well constrained.", "Bounds for second- and, especially, third-generation fermions are much less comprehensive.", "It can be quite challenging to do high-precision experiments with short-lived particles.", "Lorentz violation for the top sector is currently rather poorly constrained, with all the existing bounds in the literature being derived in one of two fashions.", "There are direct measurements of the $t$ -$\\bar{t}$ production cross section [17], with sensitivities to the coefficients of dimension-4 SME operators at the $10^{-1}$ –$10^{-2}$ level.", "The fact that $t$ quarks do not exist as external states is not an issue for these measurements, because the SME coefficients can affect the physical cross sections not primarily through the kinematics but also directly via the dynamical matrix element for the $t$ -$\\bar{t}$ production process [18].", "The other bound arises from radiative corrections.", "The $t$ quark field, like all other charged fermion fields, can appear via a virtual particle-antiparticle pair in the vacuum polarization diagram of the photon self-energy.", "This will transfer any Lorentz violation in the charged matter sector to the photon sector, including the radiative generation of dimension-6 operators in the electromagnetic Lagrange density.", "An analysis of these higher-dimensional radiative corrections—generalizing earlier perturbative demonstrations of the one-loop renormalizability of the quantum electrodynamics (QED) sector of the minimal SME [19]—placed a bound on a dimension-4 SME operator in the $t$ sector at the $10^{-7}$ level, based on the relative arrival times of TeV $\\gamma $ -rays from an active galactic nucleus [20].", "However, the calculations leading to this bound relied an an assumption of spatial isotropy.", "Moreover, since radiative corrections from multiple species are involved, there could potentially be cancelations between the effects of different types of virtual particles.", "The bound is thus best characterized as a constraint on a particular SME coefficient $\\left|c_{t}^{00}\\right|$ , under the rather restrictive assumption that this is the only nonzero SME parameter." ], [ "Photon Decay into Top-Flavored Hadrons", "Normally, when a $t$ -$\\bar{t}$ pair is produced at an accelerator, the large mass $m_{t}$ ensures that the quark and antiquark decay via the weak interaction before they can hadronize to produce mesons or baryons.", "However, the Lorentz-violation scenario in which $\\gamma \\rightarrow t+\\bar{t}$ is possible in vacuum corresponds to one in which the $t$ and $\\bar{t}$ energies grow more slowly with energy than the energy of the parent photon—that is, $\\delta _{t},\\delta _{\\bar{t}}<\\delta _{\\gamma }$ .", "A significantly negative $\\delta _{t}$ also corresponds, in fact, to precisely the kind of scenario in which the weak decay of a $t$ might be energetically forbidden.", "Due to the slower than normal growth of $E_{t}(\\vec{p}\\,)$ , a sufficiently-fast-moving $t$ might not possess enough energy to be able to produce a virtual $W^{+}$ and a light quark with the same total momentum!", "Thanks to ultrarelativistic beaming, all the constituents in a putative photon decay process like $\\gamma \\rightarrow t+\\bar{t}$ are moving essentially collinearly; angular deviations near threshold are typically at a ${\\cal O}(m_{t}/E_{\\gamma })$ level.", "This has important implications for the kinematics.", "Obviously, only the $\\delta (\\hat{p})$ parameters for the unique direction of motion will enter into the kinematics.", "However, there can also be other, more subtle, effects.", "In the decay of a photon into leptons, such as $\\gamma \\rightarrow e^{+}+e^{-}$ , the collinearity limits which SME coefficients may have observable effects.", "In this process, without strong final-state interactions, the exiting quanta must essentially both have the same helicity.", "Since the initial photon is in a spin-1 state with allowed helicities $\\pm 1$ , the two spin-$\\frac{1}{2}$ daughter particles must have identical helicities at threshold, where all the motion is along a single direction.", "(Angular-momentum nonconservation is, in principle, possible in Lorentz-violating processes; however, that would entail an invariant matrix element squared that would be of second order in the Lorentz violation, and thus negligibly small.)", "This means that the $d^{\\mu \\nu }$ coefficients for the leptons have no effect on the threshold, since any $d^{\\mu \\nu }$ term will shift the dispersion relations for the particle and antiparticle in opposite directions [21].", "Thus the observation of the absence of a $\\gamma \\rightarrow e^{+}+e^{-}$ threshold provides a relatively clean constraint on a linear combination of just $c_{e}^{\\mu \\nu }-\\frac{1}{2}\\tilde{k}_{F}^{\\mu \\nu }$ components.", "The situation is potentially rather different for a putative photon decay into quarks, because the quarks do not themselves exist as external states.", "The ultimate daughter state following $\\gamma \\rightarrow t+\\bar{t}$ will consist of hadronized particles, most generally forming jets.", "However, at the threshold, the minimal final-state configuration will include two top-flavored mesons.", "The identities of the other quarks present in these mesons are relatively unimportant, since the mesons' masses (and thus their kinematics) will be totally dominated by the current mass of the top field, $m_{t}\\approx 173$ GeV.", "However, the mere presence of these additional quarks, making the daughter particles at threshold composite states, with more than just the $t$ and $\\bar{t}$ carrying angular momentum, changes the analysis so that the $d_{t}^{\\mu \\nu }$ components could conceivably also be involved.", "Unlike a freely propagating lepton, a quark in a bound state does not need to have a single, well-defined angular momentum state, because the other partons can also carry angular momentum.", "It thus may not be instantly obvious that we could not produce $t$ -$\\bar{t}$ pairs that will settle down with opposite, rather than aligned, helicities.", "In fact, after hadronization, the $t$ and $t$ -bar constituents of two different hadrons could have instantaneously different helicities.", "However, the exchange of virtual gluons, which are components of a spin-1 field, between the quark constituents of a hadron ensures that a given quark cannot be maintained in a single consistent spin state.", "Inside a spin-0 meson, for instance, the spin of each of the valance quarks must average to zero, and because of this, the spin-dependent effects of any quark $d^{\\mu \\nu }$ coefficients must also vanish.", "We therefore introduce the spin-averaged quantity $\\overline{\\delta }(\\hat{v})=-c_{00}-c_{(0j)}\\hat{v}_{j}-c_{jk}\\hat{v}_{j}\\hat{v}_{k},$ as it turns out that for quarks, just like leptons, this is the particular combination that photon survival observations are sensitive to.", "We now must look in detail at the kinematics that might allow for the existence of top-flavored meson daughter particles in a photon decay.", "We consider a configuration with two mesons—one of them $T$ , which may have the quark content $t\\bar{q}$ for any other flavor $q$ , and the other the antiparticle $\\overline{T}$ .", "To the order of magnitude precision we need, $m_{T}\\approx m_{t}$ .", "For the $T$ -$\\overline{T}$ pair to be produced, the energy of a photon moving in the $\\hat{p}$ -direction must exceed $E_{{\\rm th}}=\\frac{2m_{T}}{\\sqrt{2\\delta _{\\gamma }(\\hat{p})-\\delta _{T}(\\hat{p})-\\delta _{\\overline{T}}(\\hat{p})}}.$ As expected, a real value for the threshold energy $E_{{\\rm th}}$ only exists if the MAV parameters satisfy $2\\delta _{\\gamma }(\\hat{p})-\\delta _{T}(\\hat{p})-\\delta _{\\overline{T}}(\\hat{p})>0$ , indicating a steeper dispersion relation $E(\\vec{p}\\,)$ for the photon than for the mesons.", "In order that the $c_{t}^{\\mu \\nu }$ coefficients should determine the kinematics, it must also be the case that the usual weak decay of the $t$ be forbidden.", "The most accessible decay modes of the bare, unhadronized $t$ are of the form $t\\rightarrow q+l+\\nu $ , with a quark, a lepton, and a neutrino.", "The lightest decay products occur via a radiative decay, in which $q=u$ and the lepton is actually a second neutrino.", "This is rather unlike the usual $t$ decay modes, which proceed through $t\\rightarrow W^{+}+b$ , with subsequent $W^{+}\\rightarrow l^{+}+\\nu $ or $W^{+}\\rightarrow {\\rm jets}$ .", "However, under the assumption that the Lorentz violation in the entire process is dominated by the $c_{t}^{\\mu \\nu }$ coefficients, the specific identities of the $t$ -decay daughter particles are relatively unimportant, provided they are all much lighter than $m_{t}$ .", "For the specific $t\\rightarrow u+\\nu _{1}+\\nu _{2}$ decay process to occur, we must have $(1+\\delta _{t})p_{t}+\\frac{m_{t}^{2}}{2p_{t}}>(1+\\delta _{u})p_{u}+\\frac{m_{u}^{2}}{2p_{u}}+(1+\\delta _{\\nu _{1}})p_{\\nu _{1}}+(1+\\delta _{\\nu _{2}})p_{\\nu _{2}},$ with $p_{u}+p_{\\nu _{1}}+p_{\\nu _{2}}=p_{t}$ and taking the neutrinos to be effectively massless.", "Presuming a negative $\\delta _{t}$ and negligible Lorentz violation for the neutrinos (flavor-diagonal Lorentz violation in the neutrino sector being well constrained, typically at the $10^{-17}$ level, by cosmic ray observations [22], [23]), the threshold configuration at which equality holds puts virtually all the momentum in the $u$ .", "That means that the $t$ decay is allowed if $\\delta _{t}-\\delta _{u}>\\frac{m_{u}^{2}-m_{t}^{2}}{2p_{t}^{2}}\\approx -\\frac{m_{t}^{2}}{2p_{t}^{2}}.$ Since (REF ) is the condition for the $t$ to decay weakly (via one particular radiative channel) prior to hadronization, it can be generalized and turned around to give a stability condition, for a $t$ -containing meson $T$ to have time to exist.", "Direct decay of a $t$ with momentum $p_{t}$ will be forbidden (and thus it will hadronize) if $\\overline{\\delta }_{t}-\\overline{\\delta }_{q}<-\\frac{m_{t}^{2}}{2p_{t}^{2}},$ where $\\delta _{q}$ is the modification to the MAV for any lighter quark that the $t$ might decay into.", "In fact, the $c^{\\mu \\nu }$ coefficients for all the lighter quarks have been previously constrained in various ways.", "For the $u$ and $d$ , many strong constraints are available, and even for the $s$ , $c$ , and $b$ , there are the observed absences of the photon decays $\\gamma \\rightarrow K^{+}+K^{-}$ , $\\gamma \\rightarrow D^{+}+D^{-}$ , and $\\gamma \\rightarrow B^{+}+B^{-}$ .", "Since all these mesons are much lighter than $m_{t}$ , the nonoccurrence of these decays leads to bounds on the $c_{q}^{\\mu \\nu }-\\frac{1}{2}\\tilde{k}^{\\mu \\mu }$ parameters that are—according to the characteristic ${\\cal O}(m_{X}^{2}/E_{\\gamma }^{2})$ strength of the bounds—orders of magnitude better than the $m_{t}$ bounds we are primarily interested in here.", "Moreover, the observation of a given source, with source-to-Earth direction $\\hat{p}$ , will constrain the same linear combinations of coefficients $\\overline{\\delta }$ for all the quarks.", "By the same token, the gluon configuration in a light-heavy meson such as $D$ or $B$ has been found not to contribute significant Lorentz violation, and thus, to level of precision relevant here, the stability of the $t$ against immediate decay is indicative of a bounds on just the combination linear combination $\\overline{\\delta }_{t}-\\delta _{\\gamma }$ of $t$ and photon SME parameters.", "We are now in position to assemble all our results into final bounds.", "Neglecting the electromagnetic Lorentz violation (because of the strong bounds on the $\\tilde{k}_{F}$ coefficients), if the $T$ is a viable asymptotic state, but photons with energy $E_{\\gamma }$ moving in the direction $\\hat{p}$ do not decay via $\\gamma \\rightarrow T+\\overline{T}$ , then according to (REF ), $\\delta _{T}(\\hat{p})+\\delta _{\\overline{T}}(\\hat{p})>-\\frac{4m_{T}^{2}}{E_{\\gamma }^{2}}.$ In an ordinary meson, the valance quarks each carry about one quarter of the momentum, with the other half belonging to the gluon field.", "It is straightforward to see from (REF ) that with the two quarks carrying essentially equal momentum fractions, $x_{t}=p_{t}/p_{T}\\approx x_{\\bar{q}}=p_{\\bar{q}}/p_{T}$ , and with Lorentz violation existing at the threshold level, the two valance quark constituents are moving at the same velocity—which is the overall velocity of the meson.", "So the internal momentum distribution in the $T$ meson is not greatly affected by the Lorentz violation, at the $T$ -$\\overline{T}$ decay threshold.", "This then simplifies (REF ) to $\\overline{\\delta }_{t}(\\hat{p})\\gtrsim -\\frac{8m_{t}^{2}}{E_{\\gamma }^{2}}.$ However, in order for the top-flavored mesons to exists as asymptotic states, the bare $t$ that is initially produced (with momentum $\\frac{1}{2}E_{\\gamma }$ at threshold) must not be susceptible to weak decay.", "According to (REF ), the condition for the bare $t$ to be stabilized against weak decay, so that $\\gamma \\rightarrow T+\\overline{T}$ may occur, is actually less stringent than (REF ) by a factor of 4.", "Thus (REF ) represents our final analytical result." ], [ "Conclusions", "The form of the bound (REF ) is similar to that for other $\\gamma \\rightarrow X^{+}+X^{-}$ reactions given in [16].", "The scale of the bounds is comparable, although the more complicated details associated with the hadronization around the $t$ appear to lead to about a factor of 4 loss in sensitivity.", "However, that factor really just corresponds to the difference between the coefficients for the $T$ mesons and the underlying $t$ quarks (representing the inverse of the momentum fraction carried by the $t$ ).", "Essentially the same factor would exist in comparing the sensitivities of the $\\delta _{B}$ for $B^{\\pm }$ mesons, versus the the $\\overline{\\delta }_{b}$ for their heavy quark.", "Moreover, this analysis still provides bounds that are better than any others available for most of the $c_{t}^{\\mu \\nu }$ coefficients.", "A sample of astrophysical sources of high-energy photons, coming from a wide range of directions, was listed in [16]; the energies involved extend up to the tens of TeV.", "At this scale, the sensitivity scale is given by $\\sim 8m_{t}^{2}/(20\\,{\\rm TeV})\\approx 6\\times 10^{-4}$ .", "The one unfortunate feature of these bounds (which a common issue with any constraints based solely on the absence of the photon decay process), is that all the bounds on the $\\bar{\\delta }(\\hat{p})$ are one sided.", "By making use of a wide variety of sources, it is possible to constrain the allowed parameter space significantly more strongly, but it is evident, for example from (REF ) and (REF ), that it will never be possible to rule out a negative value of $c_{t}^{00}$ .", "A Lorentz-violating theory will just a negative $c_{t}^{00}$ indicates an isotropic speedup of the $t$ excitations, with a steeper dispersion relation for any top-flavored quanta than for photons.", "It is obvious in this case that the anomalous photon decay process will never occur; it is even more strongly forbidden than in the Lorentz-invariant theory.", "So the absence of photon decay provides no information about this corner of the parameter space.", "The bound from [20] based on radiative corrections can be seen as complementary to the collection of photon survival bounds derived from observations of different sources.", "In particular, [20] provides a two-sided bound on $c_{t}^{00}$ , which is the one specific $c_{t}^{\\mu \\nu }$ coefficient that cannot, in principle, be constrained on both sides by the method discussed in this paper.", "However, those results and these cannot be directly combined, since the analysis in [20] assumed rotational isotropy from the start.", "The precision of the photon survival bounds will naturally be improved by the observation of ever higher-energy cosmic ray photons, coming from a wide range of approach directions.", "However, improvements of this type are ultimately limited by the availability of such photons, and there is thus an eventual physical cutoff beyond which the photon survival bounds cannot be further strengthened.", "For this reason, the bounds coming from radiative corrections and affecting photon arrival times may, in the long run, provide a systematically better approach.", "However, the calculations in of [20] would need to be extended to apply to situations without spherical isotropy in order for this improvement to be effected, and there would still be the necessity of observing photon arrival times along multiple different directions, not merely from a single source.", "In this paper, we have generalized the previous analysis of how the observed absence of the photon decay process $\\gamma \\rightarrow X^{+}+X^{-}$ can be used to place constraints on Lorentz violation.", "The generalized analysis has allowed us to treat possible decays into $t$ quarks, which were not previously included because the $t$ ordinarily decays too quickly to hadronize into a suitable external color-confined state.", "However, the same Lorentz-violating kinematics that could make $\\gamma \\rightarrow t+\\bar{t}$ possible would also tend to make the ordinary weak decay of the bare top energetically impossible.", "This means that much of the analysis applied to ordinarily stable daughter particles can be carried over, and the results are that there are bounds on $t$ sector Lorentz violation coefficients $c_{t}^{\\mu \\nu }$ at a characteristically $\\sim 10^{-4}$ level of precision." ] ]

[ [ "Some $q$-supercongruences modulo the fourth power of a cyclotomic\n polynomial" ], [ "Abstract In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]\\Phi_n(q)^3$.", "Here $[n]=(1-q^n)/(1-q)$ and $\\Phi_n(q)$ is the $n$-th cyclotomic polynomial in $q$.", "In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence.", "The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser." ], [ "Introduction", "For a complex variable $x$ , define the shifted-factorial to be $(x)_{0}=1\\quad \\text{and}\\quad (x)_{n}=x(x+1)\\cdots (x+n-1)\\quad \\text{when}\\quad n\\in \\mathbb {N}.$ In his second letter to Hardy on February 27, 1913, Ramanujan mentioned the identity $\\sum _{k=0}^{\\infty }(-1)^k(4k+1)\\frac{(\\frac{1}{2})_k^5}{k!^5}=\\frac{2}{\\Gamma (\\frac{3}{4})^4},$ where $\\Gamma (x)$ is the Gamma function.", "In 1997, Van Hamme [21] conjectured that (REF ) possesses the nice $p$ -adic analogue: $\\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)\\frac{(\\frac{1}{2})_k^5}{k!^5}\\equiv {\\left\\lbrace \\begin{array}{ll} \\displaystyle -\\frac{p}{\\Gamma _p(\\frac{3}{4})^4} \\pmod {p^3}, &\\text{if $p\\equiv 1\\pmod {4}$,}\\\\[10pt]0\\pmod {p^3}, &\\text{if $p\\equiv 3\\pmod {4}$.}\\end{array}\\right.", "}$ Here and throughout the paper, $p$ always denotes an odd prime and $\\Gamma _p(x)$ is the $p$ -adic Gamma function.", "The supercongruence (REF ) was later proved by McCarthy and Osburn [17].", "In 2015, Swisher [19] showed that (REF ) also holds modulo $p^5$ for $p\\equiv 1\\pmod {4}$ and $p>5$ .", "Recently, Liu [14] found another partial generalization of it: for $p\\equiv 3\\pmod {4}$ and $p>3$ , $\\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)\\frac{(\\frac{1}{2})_k^5}{k!^5}\\equiv -\\frac{p^3}{16}\\Gamma _p\\bigg (\\frac{1}{4}\\bigg )^4\\pmod {p^4}.$ It is known that some of the truncated hypergeometric series are related to the number of rational points on certain algebraic varieties over finite fields and further to coefficients of modular forms.", "For example, on the basis of the result of Ahlgren and Ono in [1], Kilbourn [2] proved Van Hamme¡¯s (M.2) supercongruence: $\\sum _{k=0}^{(p-1)/2}\\frac{(\\frac{1}{2})_k^4}{k!^4}\\equiv a_p\\pmod {p^{3}},$ where $a_p$ is the $p$ -th coefficient of a weight 4 modular form $\\eta (2z)^4\\eta (4z)^4:=q\\prod _{n=1}^{\\infty }(1-q^{2n})^4(1-q^{4n})^4,\\quad q=e^{2i\\pi z}.$ In 2011, Long [16] obtained the following two supercongruences: $\\sum _{k=0}^{(p-1)/2}(4k+1)\\frac{(\\frac{1}{2})_k^4}{k!^4} &\\equiv p\\pmod {p^{4}}, \\\\[5pt]\\sum _{k=0}^{(p-1)/2}(4k+1)\\frac{(\\frac{1}{2})_k^6}{k!^6} &\\equiv p\\sum _{k=0}^{(p-1)/2}\\frac{(\\frac{1}{2})_k^4}{k!^4}\\pmod {p^4}, $ where $p>3$ .", "According to (REF ), the supercongruence () can be written as $\\sum _{k=0}^{(p-1)/2}(4k+1)\\frac{(\\frac{1}{2})_k^6}{k!^6} \\equiv pa_p\\pmod {p^4}\\quad \\text{for}\\ p>3.$ For two complex numbers $x$ and $q$ , define the $q$ -shifted factorial to be $(x;q)_{0}=1\\quad \\text{and}\\quad (x;q)_n=(1-x)(1-xq)\\cdots (1-xq^{n-1})\\quad \\text{when}\\quad n\\in \\mathbb {N}.$ For shortening many of the formulas in this paper, we adopt the notation $(x_1,x_2,\\dots ,x_r;q)_{n}=(x_1;q)_{n}(x_2;q)_{n}\\cdots (x_r;q)_{n}.$ Following Gasper and Rahman [3], define the basic hypergeometric series by $_{r}\\phi _{s}\\left[\\begin{array}{c}a_1,a_2,\\ldots ,a_{r}\\\\b_1,b_2,\\ldots ,b_{s}\\end{array};q,\\, z\\right] =\\sum _{k=0}^{\\infty }\\frac{(a_1,a_2,\\ldots , a_{r};q)_k}{(q,b_1,b_2,\\ldots ,b_{s};q)_k}\\bigg \\lbrace (-1)^kq^{\\binom{k}{2}}\\bigg \\rbrace ^{1+s-r}z^k.$ Then the $q$ -Chu–Vandermonde sum (cf.", "[3]), a terminating $q$ -analogue of Whipple's $_3F_2$ sum (cf.", "[3]) and Watson's $_8\\phi _7$ transformation (cf.", "[3]) can be expressed as follows: $&\\qquad \\qquad \\qquad \\qquad _{2}\\phi _{1}\\!\\left[\\begin{array}{cccccccc}q^{-n}, &b \\\\ &c\\end{array};q,\\, \\frac{cq^n}{b} \\right]=\\frac{(c/b;q)_{n}}{(c;q)_{n}}, \\\\[10pt]&\\quad _{4}\\phi _{3}\\!\\left[\\begin{array}{cccccccc}q^{-n}, &q^{1+n}, &b, & -b \\\\&c, &b^2q/c, &-q\\end{array};q,\\, q \\right]=q^{\\binom{1+n}{2}}\\frac{(b^2q^{1-n}/c, cq^{-n};q^2)_{n}}{(b^2q/c, c;q)_{n}}, \\\\[10pt]& _{8}\\phi _{7}\\!\\left[\\begin{array}{cccccccc}a,& qa^{\\frac{1}{2}},& -qa^{\\frac{1}{2}}, & b, & c, & d, & e, & q^{-n} \\\\& a^{\\frac{1}{2}}, & -a^{\\frac{1}{2}}, & aq/b, & aq/c, & aq/d, & aq/e, & aq^{n+1}\\end{array};q,\\, \\frac{a^2q^{n+2}}{bcde}\\right] \\\\[5pt]&\\quad =\\frac{(aq, aq/de;q)_{n}}{(aq/d, aq/e;q)_{n}}\\,{}_{4}\\phi _{3}\\!\\left[\\begin{array}{c}aq/bc,\\ d,\\ e,\\ q^{-n} \\\\aq/b,\\, aq/c,\\, deq^{-n}/a\\end{array};q,\\, q\\right].", "$ Recently, Guo [8] proved that, for any positive odd integer $n\\equiv 3\\pmod {4}$ , $\\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\\equiv [n]\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\pmod {\\Phi _n(q)^3}$ and proposed the following conjecture: for any positive odd integer $n\\equiv 3\\pmod {4}$ , $\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^4)_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^k\\equiv [n]^2q^{(1+n)/2}\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\pmod {[n]\\Phi _n(q)^3},$ which is a $q$ -analogue of (REF ).", "Here and throughout the paper, $M$ is always equal to $(n-1)/2$ or $(n-1)$ .", "Some different works can be stated as follows.", "Guo and Zudilin [12] found the formula with two parameters: for a positive odd integer $n$ , $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/c,q;q^2)_k}{(q^2/a,aq^2,cq^2,q^2;q^2)_k}c^k\\\\[5pt]&\\:\\equiv [n]\\frac{(c/q)^{(n-1)/2}(q^2/c;q^2)_{(n-1)/2}}{(cq^2;q^2)_{(n-1)/2}}\\pmod {[n](1-aq^n)(a-q^n)}.$ Guo and Wang [11] achieved a $q$ -analogue of (REF ): for any positive odd integer $n$ , $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}\\equiv [n]q^{(1-n)/2}+[n]^3q^{(1-n)/2}\\frac{(n^2-1)(1-q)^2}{24}\\pmod {[n]\\Phi _n(q)^3}.$ Guo and Schlosser [10]) proved that, for a positive odd integer $n$ , $&\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^6}{(q^2;q^2)_k^6}q^k\\equiv [n]q^{(1-n)/2}\\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k}\\hspace*{-4.2679pt}\\pmod {[n]\\Phi _n(q)^2},\\\\[5pt]&\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^5}{(q^2;q^2)_k^5}q^{k^2+k}\\equiv [n]q^{(1-n)/2}\\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}q^{2k}\\hspace*{-4.2679pt}\\pmod {[n]\\Phi _n(q)^2}.$ The $q$ -supercongruence (REF ) is a $q$ -analogue of (), where the modulo $p^4$ condition is replaced by the weaker condition modulo $p^3$ .", "An indeed $q$ -analogue of (REF ) (cf.", "[7] and [22]) can be expressed as follows: for any positive odd integer $n$ , $&\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^4)_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^k\\\\[5pt]&\\equiv {\\left\\lbrace \\begin{array}{ll} \\displaystyle [n]\\frac{(q^2;q^4)_{(n-1)/4}^2}{(q^4;q^4)_{(n-1)/4}^2} \\pmod {[n]\\Phi _n(q)^2}, &\\text{if $n\\equiv 1\\pmod {4}$,}\\\\[15pt]0 \\pmod {[n]\\Phi _n(q)^2}, &\\text{if $n\\equiv 3\\pmod {4}$.}\\end{array}\\right.", "}$ We point out that more $q$ -analogues of supercongruences can be found in [5], [6], [9], [15], [18], [20], [23], [24] with various techniques.", "Inspired by these work just mentioned, we shall establish the following theorem.", "Theorem 1.1 Let $n$ be a positive odd integer.", "Then $&\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^4(cq,dq;q^2)_k}{(q^2;q^2)_k^4(q^2/c,q^2/d;q^2)_k}\\bigg (\\frac{q}{cd}\\bigg )^k\\\\[5pt]\\:&\\:\\equiv \\bigg \\lbrace [n]q^{(1-n)/2}+[n]^3q^{(1-n)/2}\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3(q/cd;q^2)_k}{(q^2;q^2)_k^2(q^2/c,q^2/d;q^2)_k}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.", "$ When $cd=q$ , the $q$ -supercognruence (REF ) reduces to (REF ).", "When the parameters $c$ and $d$ are further specified, we can confirm Guo's conjecture (REF ), a $q$ -analogue of (), five new $q$ -analogues of (REF ), and some other conclusions from this theorem.", "The rest of the paper is arranged as follows.", "We shall display several concrete $q$ -supercongruences from Theorem REF in Section 2.", "Via the creative microscoping method, a $q$ -supercongruence with four parameters modulo $\\Phi _n(q)(1-aq^n)(a-q^n)$ , which includes (REF ) and (REF )–(REF ) as special cases, will be derived in Section 3.", "Then we utilize it and the Chinese remainder theorem for coprime polynomials to deduce a $q$ -supercongruence with three parameters modulo $\\Phi _n(q)^2(1-aq^n)(a-q^n)$ and prove Theorem REF in Section 4." ], [ "Concrete $q$ -supergruences from Theorem ", "Nine $q$ -supercongruences modulo $[n]\\Phi _n(q)^3$ from Theorem REF will be laid out.", "Above all, we give the following lemma.", "Lemma 2.1 Let $n$ be a positive odd integer.", "Then $[n]^2\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\equiv 0\\pmod {[n]}.$ For two nonnegative integer $s,t$ with $s\\le t$ , it is well known that the $q$ -binomial coefficient $\\genfrac[]{0.0pt}1{t}{s}$ is a polynomial in $q$ and $\\frac{(q;q^2)_t}{(q^2;q^2)_t}=\\frac{1}{(-q;q)_t^2}\\genfrac[]{0.0pt}0{2t}{t}.$ By specifying the parameters in (), Guo [8] discovered the identity $& _{4}\\phi _{3}\\!\\left[\\begin{array}{cccccccc}q^{1-n}, &q^{1+n}, &q, &-q \\\\&q^{2+n}, &q^{2-n}, &-q^2\\end{array};q^2,\\, q^2 \\right]=[n]\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}.$ For proving Lemma REF , it is sufficient to show that $[n]\\frac{(q^{1-n},q^{1+n};q^2)_{k}}{(q^{2-n},q^{2+n};q^2)_{k}}\\equiv 0\\pmod {[n]},$ where $0\\le k\\le (n-1)/2$ .", "Through (REF ), we have $[n]\\frac{(q^{1-n},q^{1+n};q^2)_{k}}{(q^{2-n},q^{2+n};q^2)_{k}}&=q^{-k}[n]^2\\frac{(q;q^2)_{(n-1)/2-k}}{(q^2;q^2)_{(n-1)/2-k}}\\frac{(q^2;q^2)_{(n-1)/2+k}}{(q^3;q^2)_{(n-1)/2+k}}\\\\[5pt]&=\\frac{(q;q^2)_{(n-1)/2-k}}{(q^2;q^2)_{(n-1)/2-k}}\\\\[5pt]&\\quad \\times \\sum _{j=0}^{(n-1)/2+k}(-1)^jq^{j^2+j-k}\\genfrac[]{0.0pt}0{(n-1)/2+k}{j}_{q^2}\\frac{[n]^2}{[1+2j]}.$ Thus we verify the correctness of (REF ).", "This finishes the proof of Lemma REF .", "It is easy to understand that the factor $(q^2/c,q^2/d;q^2)_M$ in the denominator of the left-hand side of (REF ) is relatively prime to $\\Phi _n(q)$ as $c\\rightarrow 1,d\\rightarrow -1$ (some similar discussion will be omitted in the rest of the paper).", "Choosing $c=1,d=-1$ in Theorem REF , we obtain $\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^4)_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^k&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace $ $&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ By means of Lemma REF , (REF ) and the last relation, we get the formula: for a positive odd integer $n\\equiv 3\\pmod {4}$ , $\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^4)_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^k&\\equiv [n]^2q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\pmod {[n]\\Phi _n(q)^3}.$ It is routine to verify that $&[n]^2q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\\\[5pt]&\\quad \\equiv [n]^2q^{(1+n)/2}\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}\\pmod {[n]\\Phi _n(q)^3}.$ The combination of (REF ) and (REF ) confirms Guo's conjecture (REF ).", "Fixing $c=d=1$ in Theorem REF , we achieve the $q$ -analogue of ().", "Corollary 2.2 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^6}{(q^2;q^2)_k^6}q^k&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Setting $c=d=-1$ in Theorem REF , we gain the first new $q$ -analogue of (REF ).", "Corollary 2.3 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^2(q^2;q^4)_k^2}{(q^2;q^2)_k^2(q^4;q^4)_k^2}q^k&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^4}{(q^4;q^4)_k^2}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Letting $c=-1,d\\rightarrow \\infty $ in Theorem REF , we obtain the second new $q$ -analogue of (REF ).", "Corollary 2.4 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^3(q^2;q^4)_k}{(q^2;q^2)_k^3(q^4;q^4)_k}q^{k^2+k}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace $ $&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3}{(q^2;q^2)_k(q^4;q^4)_k}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Letting $c=-1,d\\rightarrow 0$ in Theorem REF , we get the third new $q$ -analogue of (REF ).", "Corollary 2.5 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^3(q^2;q^4)_k}{(q^2;q^2)_k^3(q^4;q^4)_k}q^{-k^2}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3}{(q^2;q^2)_k(q^4;q^4)_k}(-q)^{k}\\pmod {[n]\\Phi _n(q)^3}.$ Letting $c\\rightarrow \\infty ,d\\rightarrow \\infty $ in Theorem REF , we are led to the fourth new $q$ -analogue of (REF ).", "Corollary 2.6 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k^2+k}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3}{(q^2;q^2)_k^2}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Letting $c\\rightarrow 0,d\\rightarrow 0$ in Theorem REF , we arrive at the fifth new $q$ -analogue of (REF ).", "Corollary 2.7 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{-2k^2-k}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3}{(q^2;q^2)_k^2}(-q)^{-k^2}\\pmod {[n]\\Phi _n(q)^3}.$ The case $c=d=q^{-2}$ of Theorem REF yields the following result.", "Corollary 2.8 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^4(q^{-1};q^2)_k^2}{(q^2;q^2)_k^4(q^4;q^2)_k^2}q^{5k}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3(q^5;q^2)_k}{(q^2;q^2)_k^2(q^4;q^2)_k^2}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Choosing $M=(p^r-1)/2$ and letting $q\\rightarrow 1$ in the above $q$ -supercongruence, we obtain $\\sum _{k=0}^{(p^r-1)/2}(4k+1)\\frac{(\\frac{1}{2})_k^4(-\\frac{1}{2})_k^2}{k!^4(k+1)!^2}\\equiv p^r\\sum _{k=0}^{(p^r-1)/2}\\frac{(\\frac{1}{2})_k^3(\\frac{5}{2})_k}{k!^2(k+1)!^2}\\pmod {p^{r+3}}\\quad \\text{for}\\ p>3.$ The case $c=q^{-2},d=1$ of Theorem REF is the following result.", "Corollary 2.9 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(q;q^2)_k^5(q^{-1};q^2)_k}{(q^2;q^2)_k^5(q^4;q^2)_k}q^{3k}&\\equiv [n]q^{(1-n)/2}\\bigg \\lbrace 1+[n]^2\\frac{(n^2-1)(1-q)^2}{24}\\bigg \\rbrace \\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^3(q^3;q^2)_k}{(q^2;q^2)_k^3(q^4;q^2)_k}q^{2k}\\pmod {[n]\\Phi _n(q)^3}.$ Fixing $M=(p^r-1)/2$ and letting $q\\rightarrow 1$ in the above $q$ -supercognruence, we get $\\sum _{k=0}^{(p^r-1)/2}(4k+1)\\frac{(\\frac{1}{2})_k^5(-\\frac{1}{2})_k}{k!^5(k+1)!", "}\\equiv p^r\\sum _{k=0}^{(p^r-1)/2}\\frac{(\\frac{1}{2})_k^3(\\frac{3}{2})_k}{k!^3(k+1)!", "}\\pmod {p^{r+3}}\\quad \\text{for}\\ p>3.$" ], [ "A $q$ -supergruence with four parematers modulo {{formula:154cc54f-7e00-4814-83b5-0f5b4bbca5aa}}", "In this section, we shall deduce a $q$ -supercongruence with four parameters modulo $\\Phi _n(q)(1-aq^n)(a-q^n)$ in terms of the creative telescoping method.", "When the parameters are specified, it can produce (REF ) and (REF )–(REF ).", "We first require the following lemma (see [10]).", "Lemma 3.1 Let $n$ be a positive odd integer and $k$ an integer with $0\\le k\\le (n-1)/2$ .", "Then $\\frac{(xq;q^2)_{(n-1)/2-k}}{(q^2/x;q^2)_{(n-1)/2-k}}\\equiv (-x)^{(n-1)/2-2k}\\frac{(xq;q^2)_{k}}{(q^2/x;q^2)_{k}}q^{(n-1)^2/4+k}\\pmod {\\Phi _n(q)}.$ We also need two more lemmas.", "Lemma 3.2 Let $n$ be a positive odd integer.", "Then $ \\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\equiv 0\\pmod {\\Phi _n(q)}.$ Let $\\alpha _q(k)$ denote the $k$ -th term on the left-hand side of (REF ), i.e., $\\alpha _q(k)=[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k.$ According to Lemma REF , we obtain $\\alpha _{q}((n-1)/2-k)\\equiv -\\alpha _{q}(k)\\pmod {\\Phi _n(q)}.$ When $(n-1)/2$ is odd, it is not difficult to verify that $\\sum _{k=0}^{(n-1)/2}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\equiv 0\\pmod {\\Phi _n(q)}.$ When $(n-1)/2$ is even, the central term $\\alpha _{q}((n-1)/4)$ will remain.", "Since it has the factor $[n]$ , (REF ) is also true in this instance.", "If $n>(n-1)/2$ , the factor $(1-q^n)$ appears in the numerator of $\\alpha _q(k)$ .", "This indicates $\\sum _{k=0}^{n-1}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\equiv 0\\pmod {\\Phi _n(q)},$ as desired.", "Lemma 3.3 Let $n$ be a positive odd integer.", "Then $\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\equiv 0\\pmod {[n]}.$ For $n>1$ , let $\\zeta \\ne 1$ be an $n$ -th root of unity, which is not necessarily primitive.", "This means that $\\zeta $ is a primitive root of unity of odd degree $m|n$ .", "Lemma REF with $n=m$ implies that $\\sum _{k=0}^{m-1}\\alpha _{\\zeta }(k)=\\sum _{k=0}^{(m-1)/2}\\alpha _{\\zeta }(k)=0.$ By means of the relation: $\\frac{\\alpha _{\\zeta }(jm+k)}{\\alpha _{\\zeta }(jm)}=\\lim _{q\\rightarrow \\zeta }\\frac{\\alpha _{q}(jm+k)}{\\alpha _{q}(jm)}=\\alpha _{\\zeta }(k),$ we achieve $\\sum _{k=0}^{n-1}\\alpha _{\\zeta }(k)=\\sum _{j=0}^{n/m-1}\\sum _{k=0}^{m-1}\\alpha _{\\zeta }(jm+k)=\\sum _{j=0}^{n/m-1}\\alpha _{\\zeta }(jm)\\sum _{k=0}^{m-1}\\alpha _{\\zeta }(k)=0,$ $\\sum _{k=0}^{(n-1)/2}\\alpha _{\\zeta }(k)=\\sum _{j=0}^{(n/m-3)/2}\\alpha _{\\zeta }(jm)\\sum _{k=0}^{m-1}\\alpha _{\\zeta }(k)+\\sum _{k=0}^{(m-1)/2}\\alpha _{\\zeta }((n-m)/2+k)=0.$ The last two equations tell us that $\\sum _{k=0}^{n-1}\\alpha _{q}(k)$ and $\\sum _{k=0}^{(n-1)/2}\\alpha _{q}(k)$ are both divisible by the cyclotomic polynomials $\\Phi _m(q)$ .", "Because this is correct for any divisor $m>1$ of $n$ , we conclude that they are divisible by $\\prod _{m|n,m>1}\\Phi _m(q)=[n].$ This completes the proof.", "We now state our main result in this section.", "Theorem 3.4 Let $n$ be a positive odd integer.", "Then $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\\\[5pt]\\:&\\:\\equiv [n](b/q)^{(n-1)/2}\\frac{(q^2/b;q^2)_{(n-1)/2}}{(bq^2;q^2)_{(n-1)/2}}\\\\[5pt]&\\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(aq,q/a,q/b,q/cd;q^2)_k}{(q^2,q^2/b,q^2/c,q^2/d;q^2)_k}q^{2k}\\pmod {\\Phi _n(q)(1-aq^n)(a-q^n)}.", "$ When $a=q^{-n}$ or $a=q^n$ , the left-hand side of (REF ) is equal to $&\\sum _{k=0}^{M}[4k+1]\\frac{(q^{1-n},q^{1+n},q/b,cq,dq,q;q^2)_k}{(q^{2+n},q^{2-n},bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\\\[5pt]\\:&= {_8\\phi _7}\\!\\left[\\begin{array}{cccccccc} q,& q^{\\frac{5}{2}},&-q^{\\frac{5}{2}}, & cq, & dq, & q/b, & q^{1+n}, & q^{1-n}\\\\[5pt]& q^{\\frac{1}{2}}, & -q^{\\frac{1}{2}}, & q^2/c, & q^2/d, & bq^2, & q^{2-n}, & q^{2+n}\\end{array};q^2,\\, \\frac{bq}{cd}\\right].", "$ Via (), the right-hand side of (REF ) can be rewritten as $[n](b/q)^{(n-1)/2}\\frac{(q^2/b;q^2)_{(n-1)/2}}{(bq^2;q^2)_{(n-1)/2}}{_4\\phi _3}\\!\\left[\\begin{array}{cccccccc}q/cd, &q/b, &q^{1+n}, &q^{1-n}\\\\[5pt]&q^2/c,&q^2/d, &q^2/b\\end{array};q^2,\\, q^2\\right].$ This proves that the $q$ -supercongruence (REF ) holds modulo $(1-aq^n)$ or $(a-q^n)$ .", "Lemma REF implies that (REF ) is also true modulo $\\Phi _n(q)$ .", "Since $\\Phi _n(q)$ , $(1-aq^n)$ , and $(a-q^n)$ are pairwise relatively prime polynomials, we obtain (REF ).", "In the light of Lemma REF , the $q$ -supercongruence (REF ) becomes (REF ) when $cd=q$ .", "Letting $a\\rightarrow 1, b=c=d=1$ in (REF ) and using (REF ) and Lemma REF , we get (REF ).", "If we take $a\\rightarrow 1,b=c=1$ , $d\\rightarrow \\infty $ , then the $q$ -supercongruence (REF ) reduces to () under (REF ) and Lemma REF .", "Moreover, letting $a\\rightarrow 1$ and putting $b=c=1$ , $d=-1$ in (REF ) and employing (REF ) and Lemma REF , we have $&\\sum _{k=0}^{M}(-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^2)_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^{k}\\\\[5pt]&\\quad \\equiv [n]q^{(1-n)/2}\\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\\hspace*{-4.2679pt}\\pmod {[n]\\Phi _n(q)^2}.$ In view of the following $q$ -supercongruence (see [13]) $\\sum _{k=0}^{(n-1)/2}\\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\\equiv {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\frac{(q^2;q^4)_{(n-1)/4}^2}{(q^4;q^4)_{(n-1)/4}^2}q^{(n-1)/2} \\hspace{-8.53581pt} \\pmod {\\Phi _n(q)^2}, &\\text{if $n\\equiv 1\\pmod {4}$,}\\\\[15pt]0 \\pmod {\\Phi _n(q)^2}, &\\text{if $n\\equiv 3\\pmod {4}$,}\\end{array}\\right.", "}$ we immediately get (REF )." ], [ "Proof of Theorem ", "In this section, we shall establish a $q$ -supercongruence with three parameters modulo $\\Phi _n(q)^2(1-aq^n)(a-q^n)$ by using Theorem REF and the Chinese remainder theorem for coprime polynomials.", "Then we utilize it to give the proof of TheoremREF .", "In order to reach the goals, we need the following lemma.", "Lemma 4.1 Let $n$ be a positive odd integer.", "Then $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\\\[5pt]&\\quad \\equiv [n]\\frac{(q;q^2)_{(n-1)/2}^2}{(aq^2,q^2/a;q^2)_{(n-1)/2}} \\\\[5pt]&\\quad \\quad \\times \\sum _{k=0}^{(n-1)/2}\\frac{(aq,q/a,q/b,q/cd;q^2)_k}{(q^2,q^2/b,q^2/c,q^2/d;q^2)_k}q^{2k}\\pmod {(b-q^n)}.$ When $b=q^{n}$ , the left-hand side of (REF ) is equal to $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q^{1-n},cq,dq,q;q^2)_k}{(q^{2}/a,aq^{2},q^{2+n},q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{q^{n+1}}{cd}\\bigg )^k\\\\[5pt]\\:&= {_8\\phi _7}\\!\\left[\\begin{array}{cccccccc} q,& q^{\\frac{5}{2}},&-q^{\\frac{5}{2}}, & cq, & dq, & aq, & q/a, & q^{1-n}\\\\[5pt]& q^{\\frac{1}{2}}, & -q^{\\frac{1}{2}}, & q^2/c, & q^2/d, & q^2/a, & aq^{2}, & q^{2+n}\\end{array};q^2,\\, \\frac{q^{n+1}}{cd}\\right].", "$ Via (), the right-hand side of (REF ) can be rewritten as $[n]\\frac{(q;q^2)_{(n-1)/2}^2}{(aq^2,q^2/a;q^2)_{(n-1)/2}}{_4\\phi _3}\\!\\left[\\begin{array}{cccccccc}q/cd, &aq, &q/a, &q^{1-n}\\\\[5pt]&q^2/c,&q^2/d, &q^{2-n}\\end{array};q^2,\\, q^2\\right].$ This proves that Lemma REF holds modulo $(b-q^n)$ .", "We now give a parametric generaliztation of Theorem REF .", "Theorem 4.2 Let $n$ be a positive odd integer.", "Then, modulo $\\Phi _n(q)^2(1-aq^n)(a-q^n)$ , $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,cq,dq;q^2)_k(q;q^2)_k^2}{(q^2/a,aq^2,q^2/c,q^2/d;q^2)_k(q^2;q^2)_k^2}\\bigg (\\frac{q}{cd}\\bigg )^k\\\\[5pt]\\:&\\:\\equiv [n]\\Omega _q(a,n)\\sum _{k=0}^{(n-1)/2}\\frac{(aq,q/a,q/cd,q;q^2)_k}{(q^2/c,q^2/d;q^2)_k(q^2;q^2)_k^2}q^{2k},$ where the notation on the right-hand side denotes $\\Omega _q(a,n)=q^{(1-n)/2}+q^{(1-n)/2}\\frac{(1-aq^n)(a-q^n)}{(1-a)^2}\\bigg \\lbrace 1-\\frac{n(1-a)a^{(n-1)/2}}{1-a^n}\\bigg \\rbrace .$ It is clear that the polynomials $(1-aq^n)(a-q^n)$ and $(b-q^n)$ are relatively prime.", "Noting the $q$ -congruences $&\\frac{(b-q^n)(ab-1-a^2+aq^n)}{(a-b)(1-ab)}\\equiv 1\\pmod {(1-aq^n)(a-q^n)},\\\\[5pt]&\\qquad \\qquad \\frac{(1-aq^n)(a-q^n)}{(a-b)(1-ab)}\\equiv 1\\pmod {(b-q^n)}$ and employing the Chinese remainder theorem for coprime polynomials, we can derive, from Lemmas REF and REF and Theorem REF , the following $q$ -congruence: modulo $\\Phi _n(q)(1-aq^n)(a-q^n)(b-q^n)$ , $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,q/b,cq,dq,q;q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\\bigg (\\frac{bq}{cd}\\bigg )^k\\\\[5pt]\\:&\\quad \\equiv [n]R_q(a,b,n)\\sum _{k=0}^{(n-1)/2}\\frac{(aq,q/a,q/b,q/cd;q^2)_k}{(q^2,q^2/b,q^2/c,q^2/d;q^2)_k}q^{2k},$ where $R_q(a,b,n)&=\\frac{(b-q^n)(ab-1-a^2+aq^n)}{(a-b)(1-ab)}\\frac{(b/q)^{(n-1)/2}(q^2/b;q^2)_{(n-1)/2}}{(bq^2;q^2)_{(n-1)/2}}\\\\[5pt]&\\quad +\\frac{(1-aq^n)(a-q^n)}{(a-b)(1-ab)}\\frac{(q;q^2)_{(n-1)/2}^2}{(aq^2,q^2/a;q^2)_{(n-1)/2}}.$ Letting $b\\rightarrow 1$ in (REF ), we conclude that, modulo $\\Phi _n(q)^2(1-aq^n)(a-q^n)$ , $&\\sum _{k=0}^{M}[4k+1]\\frac{(aq,q/a,cq,dq;q^2)_k(q;q^2)_k^2}{(q^2/a,aq^2,q^2/c,q^2/d;q^2)_k(q^2;q^2)_k^2}\\bigg (\\frac{q}{cd}\\bigg )^k\\\\[5pt]\\:&\\:\\equiv [n]S_q(a,n)\\sum _{k=0}^{(n-1)/2}\\frac{(aq,q/a,q/cd,q;q^2)_k}{(q^2/c,q^2/d;q^2)_k(q^2;q^2)_k^2}q^{2k},$ where $S_q(a,n)=\\frac{(1-q^n)(1+a^2-a-aq^n)}{(1-a)^2}q^{(1-n)/2}-\\frac{(1-aq^n)(a-q^n)}{(1-a)^2}\\frac{(q;q^2)_{(n-1)/2}^2}{(aq^2,q^2/a;q^2)_{(n-1)/2}}.$ Two formulas due to Guo [4] can be stated as $&(aq^2,q^2/a;q^2)_{(n-1)/2}\\equiv (-1)^{(n-1)/2}\\frac{(1-a^n)q^{-(n-1)^2/4}}{(1-a)a^{(n-1)/2}}\\pmod {\\Phi _n(q)},\\\\[5pt]&(aq,q/a;q^2)_{(n-1)/2}\\equiv (-1)^{(n-1)/2}\\frac{(1-a^n)q^{(1-n^2)/4}}{(1-a)a^{(n-1)/2}}\\pmod {\\Phi _n(q)},$ from which we deduce that $\\frac{(q;q^2)_{(n-1)/2}^2}{(aq^2,q^2/a;q^2)_{(n-1)/2}}\\equiv \\frac{n(1-a)a^{(n-1)/2}}{(1-a^n)q^{(n-1)/2}}\\pmod {\\Phi _n(q)},$ and so $[n]S_q(a,n)\\equiv [n]\\Omega _q(a,n)\\pmod {\\Phi _n(q)^2(1-aq^n)(a-q^n)}.$ Using (REF ) and (REF ), we gain the desired result.", "[Proof of Theorem REF ] By L'Hôspital's rule, we have $&\\lim _{a\\rightarrow 1}\\frac{(1-aq^n)(a-q^n)}{(1-a)^2}\\bigg \\lbrace 1-\\frac{n(1-a)a^{(n-1)/2}}{1-a^n}\\bigg \\rbrace \\\\[5pt]&=\\lim _{a\\rightarrow 1}\\frac{(1-aq^n)(a-q^n)\\lbrace 1-a^n-n(1-a)a^{(n-1)/2}\\rbrace }{(1-a)^2(1-a^n)}\\\\[5pt]&=[n]^2\\frac{(n^2-1)(1-q)^2}{24}.$ Letting $a\\rightarrow 1$ in Theorem REF and utilizing the above limit, we arrive at (REF ) through (REF ) and Lemma REF .", "On the basis of numerical calculations, we would like to put forward the following conjecture.", "Conjecture 4.3 Let $n$ be a positive integer with $n\\equiv 3\\pmod {4}$ .", "Then $\\sum _{k=0}^{M} (-1)^k[4k+1]\\frac{(q;q^2)_k^4(q^2;q^{4})_k}{(q^2;q^2)_k^4(q^4;q^4)_k}q^k \\equiv [n]^2\\frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}q^{(1-n)/2}\\pmod {[n]\\Phi _n(q)^4},$ where $M=(n-1)/2$ or $n-1$ .", "In particular, for $p\\equiv 3\\pmod {4}$ , $\\sum _{k=0}^{(p^r-1)/2}(-1)^k(4k+1)\\frac{(\\frac{1}{2})_k^5}{k!^5}\\equiv p^{2r}\\frac{(\\frac{3}{4})_{(p^r-1)/2}}{(\\frac{5}{4})_{(p^r-1)/2}}\\pmod {p^{r+4}}.$" ] ]

[ [ "Topological valley plasmon transport in bilayer graphene metasurfaces\n for sensing applications" ], [ "Abstract Topologically protected plasmonic modes located inside topological bandgaps are attracting increasing attention, chiefly due to their robustness against disorder-induced backscattering.", "Here, we introduce a bilayer graphene metasurface that possesses plasmonic topological valley interface modes when the mirror symmetry of the metasurface is broken by horizontally shifting the lattice of holes of the top layer of the two freestanding graphene layers in opposite directions.", "In this configuration, light propagation along the domain-wall interface of the bilayer graphene metasurface shows unidirectional features.", "Moreover, we have designed a molecular sensor based on the topological properties of this metasurface using the fact that the Fermi energy of graphene varies upon chemical doping.", "This effect induces strong variation of the transmission of the topological guided modes, which can be employed as the underlying working principle of gas sensing devices.", "Our work opens up new ways of developing robust integrated plasmonic devices for molecular sensing." ], [ "ol Topological valley plasmon transport in bilayer graphene metasurfaces for sensing applications [1]Yupei Wang [1]Jian Wei You [1]Zhihao Lan [1,*]Nicolae C. Panoiu [1]Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom [*]Corresponding author: [email protected] Topologically protected plasmonic modes located inside topological bandgaps are attracting increasing attention, chiefly due to their robustness against disorder-induced backscattering.", "Here, we introduce a bilayer graphene metasurface that possesses plasmonic topological valley interface modes when the mirror symmetry of the metasurface is broken by horizontally shifting in opposite directions the lattice of holes of the top layer of the two freestanding graphene layers.", "In this configuration, light propagation along the domain-wall interface of the bilayer graphene metasurface shows unidirectional features.", "Moreover, we have designed a molecular sensor based on the topological properties of this metasurface using the fact that the Fermi energy of graphene varies upon chemical doping, namely molecular adsorption in our case.", "This effect induces strong variation of the transmission of the topological guided modes, which can be employed as the underlying working principle of gas sensing devices.", "Our work opens up new ways of developing robust integrated plasmonic devices for molecular sensing.", "Research in topological photonics, inspired by the theory of quantum Hall effect in solid-state physics, has led to the discovery of novel and unique phenomena, such as unidirectional, defect-immune, and scattering-free propagation of light [1], [2], [3], [4], [5], [6], which have the potential to contribute to the development of robust on-chip ultracompact nanophotonic devices.", "Topological photonic modes could be achieved by gapping out symmetry-protected Dirac cones, for example, through time-reversal symmetry breaking induced by magneto-optical effects under an external static magnetic field, or spatial-inversion symmetry breaking induced by spatially asymmetric perturbations [1], [2], [3].", "Currently, a variety of experimental platforms are available for topological photonics, including metamaterials, photonic crystals, and evanescently coupled waveguides and optical-ring resonators [3].", "Valley degree of freedom, which is associated to the conduction-band minima (or valence-band maxima) in graphene-like two-dimensional (2D) materials [4], has recently been introduced to photonics [5], too.", "These materials exhibit nontrivial Berry curvature distribution in the momentum space around each valley, which gives rise to a valley-dependent topological index associated to the integral of Berry curvature around a valley [4].", "Furthermore, a domain-wall interface separating two topologically distinct valley photonic crystals can support valley-momentum locked modes localized at the interface, similar to the quantum-valley Hall effect [5].", "Until now, valley-Hall photonic modes have been mostly studied in bulk materials, such as photonic crystals [1], [2], [3], being less explored in 2D photonic platforms, including graphene [7].", "This 2D material is becoming a promising platform to achieve passive and active topologically protected plasmonic modes [8], [9], due to its high carrier mobility and long relaxation time [10], [11].", "Equally important, recent advances in nanofabrication techniques make it possible to achieve graphene based plasmonic nanostructures with particularly complex geometrical configurations[12], [13].", "In this paper, we design a bilayer graphene metasurface to realize, to the best of our knowledge for the first time, valley topological plasmonic modes by utilizing a novel mechanism of mirror symmetry breaking between the top and bottom freestanding layers of a graphene metasurface, by horizontally shifting in opposite directions the lattice of holes of the top layer.", "As such, the symmetry-protected Dirac cones are gapped out and, consequently, a topological nontrivial frequency gap emerges.", "Furthermore, topologically guided valley modes are observed along a domain-wall interface with respect to which the composite metasurface is mirror symmetric.", "Our full-wave numerical simulations, based on solving the full set of 3D Maxwell equations via the finite element method, verify that the light propagation along the domain-wall interface shows indeed unidirectional feature.", "Employing this unique feature of unidirectional propagation and the tunable optical response of graphene, a molecular sensor based on this newly proposed topological metasurface is designed and its sensitivity and functionality are quantitatively characterized.", "Figure: Schematic of the bilayer graphene metasurface.", "(a) The metasurfacecontains a domain-wall interface oriented along the xx-axis, which is constructed by shifting thehole lattices of the two halves of the top graphene layer (purple) w.r.t.", "the bottom layer (green)along the positive and negative directions of the yy-axis.", "(b) Top view of the unit cell withlattice constant, aa, and horizontal shift, ss.", "Hole centers OO and O ' O^{\\prime } correspond tothe unit cells of the bottom and top layer, respectively.", "(c) Bird's eye view of the unit cell witha separation distance, hh, between the two layers.The schematic of the proposed topological bilayer graphene metasurface is shown in Fig.", "REF .", "It consists of two freestanding, optically coupled graphene plasmonic crystals with the same unit cell.", "Note that the conclusions of this study remain qualitatively valid if one assumes that the two graphene layers are separated by a certain dielectric material instead of air, the only changes being of quantitative nature.", "Moreover, the hole lattice of the left- and right-hand side domains of the top graphene layer are horizontally shifted, in opposite directions and normally onto an interface lying along the $x$ -axis, by a certain distance, $s$ .", "Each domain consists of a hexagonal graphene plasmonic crystal with a hole in the unit cell.", "The top and bird's-eye views of the unit cell are given in Figs.", "REF (b) and REF (c), respectively.", "In this work, we fix the lattice constant $a={400}{}$ , the radius of holes $r={100}{}$ , and the distance between the top and bottom graphene layers $h={90}{}$ .", "The optical properties of graphene are described by its electric permittivity, $\\epsilon _g$ , which is given by Kubo's formula [11]: g()=1-e240hg( -i+i ) +ie2kBT02hg[ ckBT+2(e-ckBT+1) ] where $\\omega $ is the frequency, $T$ is the temperature, $\\mu _c$ is the chemical potential, $h_g={0.5}{}$ is the graphene thickness, $\\bar{\\omega }=1-i\\omega \\tau $ , and $\\xi =2\\tau |\\mu _c|/\\hbar $ , with $\\tau $ being the relaxation time.", "Note that dispersive and dissipative effects are incorporated in our simulations via the frequency-dependent complex surface conductivity of graphene, defined as $\\sigma _s=-i\\epsilon _0\\omega h_g(\\epsilon _g-1)$ .", "In our analysis, $T={300}{}$ and $\\tau ={50}{}$ , and we set $\\mu _c={0.2}{}$ unless otherwise stated.", "Unlike the case of the mirror-symmetric bilayer graphene metasurface ($s=0$ ), in the case of a metasurface with $s\\ne 0$ , the frequency maxima and minima are not necessarily located at the high-symmetry points of the first Brillouin zone (FBZ) [14].", "Thus, in order to properly identify the frequency band gap, the plasmonic bands of this bilayer graphene metasurface have been evaluated in the entire FBZ and the results are given in Fig.", "REF (we have used the Wave Optics Module of COMSOL Multiphysics 5.4).", "When the distance $h$ between the top and bottom graphene layers in Fig.", "REF (c) is large, the optical near-field coupling between the two layers can be neglected.", "As a consequence, each graphene layer, which is a plasmonic crystal, possesses decoupled Dirac cones protected by $D_{6h}$ point symmetry group [15].", "This is indeed verified by the bands presented in Fig.", "REF (a), where the Dirac cones located at 14 of each graphene plasmonic crystal perfectly overlap.", "In order to enhance the optical coupling between the top and bottom graphene layers, the distance $h$ is reduced to $h={90}{}$ and, to break the mirror symmetry between the top and bottom graphene layers, a shift of $s={100}{}$ is introduced as explained above.", "As a result, the $D_{6h}$ -symmetry-protected Dirac cones are gapped out, and a frequency band gap emerges.", "Specifically, the band diagram of the bilayer graphene metasurface exhibits a 0.21 topological gap from 13.9614.17, as depicted in Fig.", "REF (b).", "Note that all the results reported here remain qualitatively the same for smaller $s$ but the frequency bandgap would be narrower.", "Thus, if $s$ is reduced to ${70}{}$ , the bandgap would decrease by $\\sim $ 10.", "Since the bilayer graphene metasurface has hexagonal symmetry, it possesses six Dirac cones (see Fig.", "REF ) with two non-equivalent valleys at $K$ and $K^{\\prime }$ symmetry points.", "The integral of the Berry curvature around each valley defines the valley Chern number of $C_{K,K^{\\prime }}=\\pm 1/2$ [5].", "Moreover, the two valleys at $K$ and $K^{\\prime }$ are related to each other via rotations of the metasurface by $\\pi /3$ , $\\pi $ , and $5\\pi /3$ .", "Therefore, in order to construct a domain-wall interface that can possess topological interface modes, one can place together two bilayer graphene metasurfaces with $s\\ne 0$ in a mirror-symmetric manner, i.e.", "rotated by $\\pi $ w.r.t.", "each other, as per Fig.", "REF (a).", "Consequently, the difference of the valley Chern number across the domain-wall interface at each valley is $+1$ or $-1$ .", "In this way, we can obtain a pair of valley-momentum locked interface states, where the interface state at one valley has a positive velocity whereas the other has a negative one.", "Figure: (a) Band diagram of a bilayer graphene metasurface with s=0s=0, in whichthe coupling between the top and bottom graphene nanohole crystals is very weak.", "(b) Band diagramof a composite bilayer graphene metasurface in which the coupling between the top and bottomgraphene plasmonic crystals is relatively strong, namely h=90h={90}{} ands=100s={100}{}, as depicted in Fig. (c).", "Since the mirror symmetry of thecomposite graphene metasurface is broken in this case, a nontrivial bandgap corresponding to thebeige region emerges.The projected band diagram of a finite bilayer graphene metasurface consisting of 20 unit cells along the $y$ -axis and periodic along the $x$ -axis is computed, and the results are presented in Fig.", "REF (a).", "In this figure, the green regions represent the bulk states and the topological interface modes are marked by red lines.", "Note that since the bilayer graphene metasurface has a finite number of unit cells along the $y$ -axis, there are additional edge modes in Fig.", "REF (a).", "More specifically, the blue lines in this figure indicate edge modes confined at the metasurface boundaries rather than the domain-wall interface.", "In particular, we find that the edge modes generally appear in pairs, at the top and bottom boundaries, and their influence on the domain-wall interface topological modes can be neglected when the number of unit cells of each domain along $y$ -axis is larger than about 7.", "However, additional functionality can be achieved in a photonic system in which these modes become optically coupled [16], [17].", "In order to gain deeper physical insights into the properties of these interface and edge modes, their corresponding field distribution are further investigated.", "To be more specific, the field distribution of the interface mode 1 in Fig.", "REF (a), given in Fig.", "REF (b), is highly confined at the domain-wall interface, whereas the field distribution of the edge mode 2 in Fig.", "REF (a), presented in Fig.", "REF (c), is confined at the boundary of the finite metasurface.", "Figure: (a) Projected band diagram (green region), topological interface modes(red lines), and non-topological edge modes (blue lines), determined for a finite bilayer graphenemetasurface with width of 20 unit cells and s=100s={100}{}.", "(b), (c) Field distributionsof a topological interface mode and edge mode, marked by 1 and 2 inFig.", "3(a), respectively.Importantly, we also demonstrated the unidirectional character of light propagation along the domain-wall interface.", "To illustrate this, we simulated the composite bilayer graphene metasurface with absorbing boundary conditions, but left a certain air space along the $z$ -axis to allow for the radiation losses.", "A monochromatic light source with frequency of 14.16 was used to excite the proposed composite bilayer graphene metasurface.", "In order to study the chirality-momentum locking property, which arises from the valley-Hall effects induced by the intrinsic chirality associated to each valley [4], the excitation source was constructed by placing at the corners of a small hexagon six electric dipoles, marked by circles in Fig.", "REF .", "In our simulations, the phase difference between neighboring dipoles is set to $\\pm \\pi /3$ , so as to implement right-circularly polarized (RCP) and left-circularly polarized (LCP) sources, respectively.", "More specifically, as illustrated in Fig.", "REF (a), a RCP light source is placed at the center of the composite bilayer graphene metasurface, and unidirectional propagation of light along the negative direction of the $x$ -axis of the domain-wall interface is observed.", "Similarly, as shown in Fig.", "REF (b), a LCP light source located at the center of the metasurface, created by reversing the phase difference between adjacent dipoles, excites at the interface a topological mode that propagates along the positive direction of the $x$ -axis.", "The unidirectional propagation feature of the topological interface mode of the bilayer graphene metasurface investigated in this work can find applications to efficient photonic nanodevices.", "To illustrate this, in what follows we demonstrate how the interfacial topological mode can be used as the key component of a molecular sensor.", "Thus, graphene is a particularly promising 2D material for sensing applications, chiefly due to its tunable chemical potential and high optical damage threshold [18].", "Generally, the chemical potential $\\mu _c$ of graphene is proportional to the Fermi velocity and the carrier density $n_0$ , which can be tuned via molecular doping, which is a particular type of chemical doping [19].", "As such, graphene based sensors can be used to detect the concentration of specific gases in the environment, by measuring the concentration of the corresponding molecules adsorbed onto a graphene sheet.", "To be more specific, as shown in Fig.", "REF (a), the proposed bilayer graphene metasurface is used to design a molecular sensor based on the large variations of its optical properties induced by small changes of its chemical characteristics.", "Figure: (a) Unidirectional propagation along the negative direction of thexx-axis, when the finite metasurface is excited by a right-circularly polarized source.", "(b) Thesame as in (a), but for a left-circularly polarized source.", "In this case, the topologicalinterfacial mode propagates along the positive direction of the xx-axis.The sensor consists of three bilayer graphene metasurfaces, marked as regions I, II, and III, the lengths of these regions being $l_1$ , $l_2$ , and $l_3$ , respectively, and the corresponding chemical potentials $\\mu _{c1} = \\mu _{c2} = \\mu _{c3} ={0.2}{}$ .", "The gas molecules can be adsorbed only in the region II, and upon their adsorption $\\mu _{c2}$ varies.", "In practice, this can be achieved by covering the regions I and III with some material, e.g., polymethyl methacrylate – PMMA.", "To add specificity to our analysis, we assume that the gas is NO$_2$ .", "The relation between the variation of the chemical potential induced by NO$_2$ gas with concentration, $C_{\\mathrm {NO_2}}$ , is $\\Delta \\mu _c=\\alpha C_{\\mathrm {NO_2}}$ , where the experimentally determined value of $\\alpha $ is $\\alpha \\approx {5.4e-3}{}$ /p.p.m [20], [21], [22].", "The variation of the chemical potential in the region II, in turn, leads to a variation of the graphene permittivity, and consequently to a shift of the frequency of the topological band gap associated to region II.", "This means that, if the frequency of the input light in region I is in the band gap of this region, the corresponding topological interfacial mode can be switched to a leaky bulk mode in the region II.", "Since the leaky bulk modes are particularly lossy, the output power $P_{out}$ collected in the region III will sharply decrease.", "Figure: (a) Schematic of the proposed molecular sensor.", "The topologicalinterfacial mode carries an input power and output power in the regionsI and III, respectively.", "Anadditional bilayer graphene metasurface in the region II issandwiched in-between the regions I andIII, and is used to detect the concentration of adsorbedmolecules of a certain gas (NO 2 _2 in our case).", "(b) Light transmission, defined as the ratiobetween the output and input power, vs. the concentration C NO 2 C_{\\mathrm {NO_{2}}} of NO 2 _2 gas,determined for l 2 =8al_2=8a and l 2 =14al_2=14a.", "(c) Dependence of the sensitivity of the molecular sensor onthe concentration C NO 2 C_{\\mathrm {NO_{2}}} of NO 2 _2.In order to validate these ideas, we have studied the light transmission in the proposed graphene metasurface based molecular sensor.", "To this end, a monochromatic light source with frequency of 14.16 is placed in the center of the region I.", "The lengths of the region I ($l_1$ ) and region III ($l_3$ ) are $18a$ and $12a$ , respectively.", "Moreover, we computed the transmission of the optical power, $\\eta $ , defined as the ratio between the output power, $P_{out}$ , collected in the region III and the input power $P_{in}$ in the region I, namely $\\eta =P_{out}/P_{in}$ .", "These calculations were performed for two different values of the length of region II, namely for $l_2=8a$ and $l_2=14a$ , and the corresponding results are summarized in Fig.", "REF .", "It can be seen in Fig.", "REF (b), where we plot the dependence of the transmission on the concentration of molecules adsorbed in region II, that the transmission $\\eta $ decreases steeply when the concentration $C_{\\mathrm {NO_2}}$ of the NO$_2$ gas adsorbed in this region increases.", "In Fig.", "REF (b), the dots represent the numerically computed data, whereas the solid lines indicate the fitting of the results via a third-order polynomial.", "These results prove that, as expected, the longer the length of the region II is, the larger the slope of the transmission curve is, which means that the radiation loss of the input power in the region II is larger.", "Note also that when $C>{5}$ p.p.m., the transmission in the case when $l_{2}=14a$ is larger than when $l_{2}=8a$ , which is attributable to the constructive interference of the mode propagating in region II, and which undergoes multiple reflections at the interfaces between this region and regions I and III.", "When the concentration $C_{\\mathrm {NO_2}}$ is larger than about 9 p.p.m, most of the input power is scattered out into radiation modes, so that the transmitted power is almost zero in this case.", "Moreover, we have also studied the sensitivity of the metasurface sensor, $\\sigma $ , which is defined as the absolute value of the first-order derivative of the transmission with respect to the concentration of NO$_2$ molecules adsorbed in region II, that is, $\\sigma =\\vert d\\eta /dC_{NO_2}\\vert $ .", "As shown in Fig.", "REF (c), the proposed molecular sensor can be used to detect the gas variations in a broad range of molecular concentrations, its sensitivity being particularly large for small concentrations of adsorbed molecules.", "In conclusion, we have proposed a novel mechanism to realize valley-Hall topological plasmon transport in a bilayer graphene metasurface.", "In order to create a topological nontrivial valley bandgap, the lattice of holes of the top layer of the two freestanding graphene layers is horizontally shifted by a certain distance with respect to the bottom layer, such that the mirror symmetry between the top and bottom layers is broken.", "Moreover, to produce a valley-Hall topological plasmon mode within the nontrivial bandgap, a domain-wall interface is constructed by placing together two bilayer graphene metasurfaces in a way in which the composite metasurface is mirror-symmetric with respect to the interface.", "The results of our numerical computations show that the proposed domain-wall interfacial waveguide supports topological modes that exhibit unidirectional propagation feature.", "This property is further used to design a molecular sensor based on the fact that the chemical potential of graphene can by tuned via gas molecule adsorption.", "Our work could have an important impact on the development of integrated plasmonic devices and key applications pertaining to molecular sensing.", "Funding.", "European Research Council (ERC) (ERC-2014-CoG-648328); China Scholarship Council (CSC); University College London (UCL).", "Disclosures.", "The authors declare no conflicts of interest." ] ]

[ [ "Band power modulation through intracranial EEG stimulation and its\n cross-session consistency" ], [ "Abstract Background: Direct electrical stimulation of the brain through intracranial electrodes is currently used to probe the epileptic brain as part of pre-surgical evaluation, and it is also being considered for therapeutic treatments through neuromodulation.", "It is still unknown, however, how consistent intracranial direct electrical stimulation responses are across sessions, to allow effective neuromodulation design.", "Objective: To investigate the cross-session consistency of the electrophysiological effect of electrical stimulation delivered through intracranial EEG.", "Methods: We analysed data from 79 epilepsy patients implanted with intracranial EEG who underwent brain stimulation as part of a memory experiment.", "We quantified the effect of stimulation in terms of band power modulation and compared this effect from session to session.", "As a reference, we applied the same measures during baseline periods.", "Results: In most sessions, the effect of stimulation on band power could not be distinguished from baseline fluctuations of band power.", "Stimulation effect was also not consistent across sessions; only a third of the session pairs had a higher consistency than the baseline standards.", "Cross-session consistency is mainly associated with the strength of positive stimulation effects, and it also tends to be higher when the baseline conditions are more similar between sessions.", "Conclusion: These findings can inform our practices for designing neuromodulation with greater efficacy when using direct electrical brain stimulation as a therapeutic treatment." ], [ "Introduction", "About 35% of patients with epilepsy are drug-resistant and require additional treatment [1], [2].", "In this context, direct electrical stimulation through intracranial electroencephalography (iEEG) has become an invaluable tool for clinicians.", "Direct electrical stimulation is currently used in three ways.", "First, functional mapping of the cortex so that eloquent cortical areas are preserved in resective epilepsy surgery [3], [4].", "Second, measuring the “epileptogenicity” of the stimulated and surrounding areas [5].", "Third, exploring the neuromodulatory potential of direct electrical stimulation which can be the basis for therapeutic interventions [6], [7].", "In this work we will focus on the neuromodulatory potential of intracranial electric stimulation.", "Arguably, to achieve any therapeutic goals, the effect of stimulation should be consistent across multiple sessions [8].", "To our knowledge, the consistency of iEEG stimulation effect has not been studied systematically.", "Neuromodulation has been explored as an alternative treatment for patients with non-conclusive pre-surgical evaluation of the epileptogenic zone [9].", "In such cases, without any candidate resection area, the goal is to modulate the epileptic network in a way that enhances physiological neural activity, and prevents pathological, or seizure activity.", "It is currently unknown how a targeted modulatory effect can be achieved a priori, but several studies have begun to map out how stimulation affects the brain both electrophysiologically as well as behaviourally.", "For instance, Keller et al.", "showed that repeated stimulation modulated the excitability of neighbouring areas around the stimulation site [6].", "Memory enhancement has been reported after using a closed-loop electrical stimulation of the lateral temporal cortex [10].", "Furthermore, stimulation applied to the posterior cingulate cortex induced an increase of low gamma power in hippocampus which correlated with the magnitude of memory impairment [11].", "Muller and colleagues have reported a correlation between the modulation of high gamma frequencies and somatosensory perception, both induced by direct current stimulation [12].", "Khambhati and colleagues demonstrated functional reconfiguration of brain networks after stimulation as indicated by alterations in band-specific functional connectivity [7], while Huang and colleagues further demonstrated the close relationship of functional connectivity and stimulation-induced band power modulation [13].", "Similarly, another study showed that temporal cortex stimulation increased theta band power in remote areas predicted by functional connectivity, especially when the stimulation was delivered close to white matter [14].", "These studies show the potential of using direct electrical stimulation in therapeutic neuromodulation, and intracranial stimulation through iEEG can be a useful tool to rapidly explore possible stimulation locations and parameters for the design of effective neuromodulation.", "Consistent stimulation effects -electrophysiologically or behaviourally- across sessions are crucial for developing therapeutic neuromodulation treatments.", "For example, understanding the underlying electrophysiological effect of transcranial stimulation and its consistency is an important step towards taking advantage of its already demonstrated benefits on motor rehabilitation [15], [16].", "Relevant investigations on cross-session consistency have been reported in non-invasive stimulation modalities (for a review see [8]).", "For instance, while the electrophysiological effect of transcranial magnetic stimulation has been reported to be highly consistent across sessions [17], while transcranial direct current stimulation (tDCS) effect was found to be inconsistent [18], [19] (but see also [20]).", "The sources of such variability have been discussed extensively in the context of inter-individual studies but some of them apply on an intra-individual basis as well (e.g., baseline physiological state, cognitive task at hand; for a review see [21]).", "However, to our knowledge, the cross-session consistency of the electrophysiological effects of iEEG stimulation has not yet been systematically investigated.", "Here we investigate the consistency of the iEEG stimulation effect in terms of band power modulations between stimulation sessions from the same subject.", "We measure how stimulation modulates band power in five different frequency bands and investigate whether these modulations vary from one stimulation session to the next for the same subject and stimulation location.", "We introduce a measure of consistency that accounts for the distributed stimulation effects recorded across multiple iEEG channels.", "We finally investigate which features of the stimulation protocol, the measured stimulation effect, and the baseline conditions most influence between-session consistency." ], [ "Electrophysiological and cortical surface data", "We used data that are publicly available as part of the Restoring Active Memory (RAM) project (managed by the University of Pennsylvania; http://memory.psych.upenn.edu/RAM).", "As stated in the project's website \"Informed consent has been obtained from each subject to share their data, and personally identifiable information has been removed to protect subject confidentiality\".", "The original research protocol for data acquisition was approved by the relevant bodies at the participating institutions.", "Furthermore, the University Ethics Committee at Newcastle University approved the current project involving the data analysis reported here (Ref: 12721/2018).", "We extracted data from all patients (n=87) that underwent at least one stimulation session while performing memory tasks.", "We excluded 8 patients that either had substantial stimulation artefacts in almost all channels or their data were limited (single session with <18 stimulation trials).", "Thus, we analysed data from 79 subjects from which 36 had at least 2 stimulation sessions with the same stimulation location (totalling 101 pairs of sessions with same stimulation location)." ], [ "Stimulation Paradigm", "Stimulation was delivered using charge-balanced biphasic rectangular pulses (300 $\\mu $ s pulse width) at 10, 25, 50, 100, or 200 Hz frequency 0.25–3.5 mA amplitude.", "The duration of the stimulation was 500 ms or 4.6 s, depending on the experiment." ], [ "Preprocessing", "To measure stimulation effect, 1-second segments were extracted from the iEEG signals around every stimulation trial; that is, we extracted one segment before (pre) and one after (post) the stimulation event, with a 50ms buffer between each segment and the event.", "To assess baseline fluctuations, ‘pre’ and ‘post’ segments were also extracted from the baseline activity during baseline epochs, with a pre-post interval equal to the one around the stimulation trials of the same session.", "A baseline epoch was considered to be any inter-stimulus interval which was at least 20 sec long and 5 sec away from the stimulation itself.", "Figure REF shows a schematic of the session timeline and the process of segment extraction.", "Since the stimulation trials were temporally organised in groups of three in a typical session (i.e., less than 10s interval between trials in the same group), we extracted baseline pre/post segments from each baseline epoch in groups of three as well (see Fig.", "S1 in Supplementary Material), such that the number of segments taken around stimulation and the number of baseline segments were approximately equal in each session.", "Figure: Measuring band power changes in response to stimulation and band power baseline fluctuations.Top panel: Timeline of a typical stimulation session.", "The schematic also shows how pre- and post-stimulation segments are extracted from each stimulation trial and analysed in terms of their band power.", "While only three trials are shown here, a typical stimulation session had 60 trials (median value with 13.9 SD).", "Lower panels: Band power in five different frequency bands was calculated and log-transformed for each extracted segment.", "The effect of stimulation on band power, and equivalently, band power's fluctuations during baseline, are expressed by the effect U, which is derived from a non-parametric test applied to the paired differences between pre and post segments.The time series of each segment were centred around zero and de-trended.", "De-trending was achieved by applying linear regression and then removing the least-squares fit from the signal.", "Any channels with repeated artefacts were excluded (see below).", "A common average re-referencing was applied to the remaining set of channels.", "The stimulation channels were excluded from the common average calculation, but the calculated common average was applied to them.", "The band power of each segment was calculated in 5 different bands [delta (2-4 Hz), theta (4-8 Hz), alpha (8-12 Hz), beta (12-25 Hz), and gamma (25-55 Hz)] after estimating the power spectral density of the segment using Welch’s method (with window length equal to half of the segment length and overlap length equal to a quarter of the segment length).", "Finally, the band powers were log-transformed.", "Figure REF shows a schematic of the preprocessing.", "Channels with repeated stimulation artefacts (i.e., voltage deflection) were excluded.", "A repeated stimulation artefact was detected based on two criteria.", "Either one of these two criteria was sufficient to indicate a channel with repeated artefacts.", "First, a strong effect of stimulation on the average (across time) voltage of the first half of the post segment compared to the average (across time) voltage of the second half of the pre segment.", "The effect was quantified using the t-statistic of a paired t-test.", "Second, the second half of the average (across trials) post signal had a slow return to the average (across time) voltage value of the pre segments.", "This was detected by linear regression." ], [ "Box Plots", "Box plots were used to summarise various distributions in the Results.", "Central lines indicate median values, while the boxes extend from the 25th to 75th percentile (interquartile range) of the distribution.", "Whiskers extend to the upper and lower adjacent values, that is, the most extreme values that are not outliers.", "Outliers are considered to be values that lie more than 1.5$\\times $ [interquartile range] away from the 25th or 75th percentile." ], [ "Effect measures", "The effect of stimulation on band power from pre to post was considered as the z-statistic (indicated by U throughout) produced by the Wilcoxon sign rank test (paired non-parametric test; signrank function in MATLAB).", "A positive U indicates an increase in band power from pre to post, whereas a negative U indicates a decrease from pre to post.", "To also quantify the baseline fluctuations of band power, the same measure was used on the ‘pre/post’ pairs of the baseline activity (Fig.", "REF ).", "The overall difference in stimulation effect between two sessions (across all channel/band combinations) in Fig.", "REF B was quantified by using the absolute t-statistic of a paired t-test on the absolute effect U of the two sessions.", "We used absolute effects as we wanted to generally assess changes in effect magnitude." ], [ "Consistency coefficient", "The consistency of stimulation effect was measured for each pair of sessions with the same stimulation location in the same subject.", "All possible combinations of 2 sessions were considered, totalling 101 pairs.", "The consistency was computed by first pairing the effect values of corresponding channel/band combinations between the two sessions.", "Note that only the intersection of valid channels between the two sessions was considered (a channel can be excluded due to artefacts in one session but not the other).", "The consistency coefficient was given by the Fisher-transformed zero-centred Pearson’s correlation.", "Considering the effect values of the two sessions as random variables $S_1$ and $S_2$ , then the consistency coefficient is given by: $r_0 = \\mathbf {E}[S_1S_2]/(\\hat{\\sigma }_1\\hat{\\sigma }_2)$ , where $\\mathbf {E}$ denotes expected value and $\\hat{\\sigma }$ refers to the average deviation from 0 ($\\hat{\\sigma } = \\sqrt{(\\sum ^n_{i=1}{s^2_i})/n}$ ).", "We use the zero-centred Pearson’s correlation to only detect a zero-translated agreement between the random variables, that is, in the form of $S_1 = kS_2$ , with 0 intercept and $k$ a non-zero constant." ], [ "Consistency curve", "The consistency curve was used to express the consistency between two sessions by gradually considering fewer pairs of effect values at low effect sizes.", "Considering a scatter plot of all the effect value pairs, it was computed by gradually increasing the radius of an exclusion circle emanating from (0,0).", "The consistency curve at radius = 0 gives the consistency when all points are included in the consistency calculation, whereas the consistency curve at radius = x expresses the consistency as computed after excluding every pair of effect values that lie inside a circle with centre (0,0) and radius x.", "The circle was gradually enlarged with a step of 0.2 and the enlargement stopped just before covering 98% of the scattered values.", "We used this procedure to ensure that we can detect consistency even if only a few channels exhibited consistency, without the consistency being masked by low effect channels.", "Each consistency curve is represented by its maximum consistency coefficient.", "The maximum consistency coefficient is the value on the curve that deviates the most from 0, being positive or negative.", "Thus, it expresses the strongest correlation or anti-correlation found between the effect values of the two sessions." ], [ "Multiple Linear Regression Analysis", "To explore which factors determine consistency across all 101 session pairs, we modelled the maximal value of consistency as a linear combination of the following variables: session time difference: absolute time difference between the sessions' starting timestamps.", "difference in baseline (band power) means: mean absolute paired difference between the sessions' mean values of band power during baseline (both ‘pre’ and ‘post’).", "difference in baseline (band power) standard deviations: mean absolute paired difference between the sessions' standard deviations of band power during baseline (both ‘pre’ and ‘post’).", "average max effect: average (between sessions) maximum effect (across all channel/band combinations).", "average min effect: average (between sessions) minimum effect (across all channel/band combinations).", "average stimulation amplitude: average stimulation amplitude between sessions.", "stimulation amplitude difference: difference in stimulation amplitude between sessions.", "stimulation frequency: frequency of stimulation pulse train (always common between examined session pairs).", "depth of the stimulation location: distance of stimulation location (midpoint between anode and cathode) from brain surface.", "task difference: difference in memory tasks (categorical variable) carried out by the subject during recording; that is, 0 for same and 1 for different tasks between sessions.", "The stimulation depth was computed as the Euclidean distance of the anode-cathode midpoint from the subject's brain surface.", "If that midpoint was found to be outside the provided surface, its depth was set to negative (minus the Euclidean distance)." ], [ "ANOVA test", "In order to quantify the explanatory power of all the different independent variables on the consistency we used ANOVA test on the model built by the Multiple Linear Regression Analysis.", "We built the model and assessed the ANOVA effects 200 times through bootstrapping.", "We used this bootstrapping approach to check for the robustness of the model.", "The ANOVA effect, the $R^2$ , and the Adjusted $R^2$ are reported." ], [ "Stimulation elicits a weak effect in most sessions and across frequency bands", "Figure REF shows the measured stimulation effects across channels and frequency bands for one example session in each of two example subjects 1022 and 1069.", "These example sessions represent sessions with weak (Fig.", "REF , left) and strong (Fig.", "REF , right) stimulation effects.", "As a reference, the upper panels show the “effect” during baseline, that is, the background fluctuations of band power.", "The lower panels show the stimulation effect in terms of band power changes, based on multiple pre- and post-stimulation pairs (see example inset panels on the right and Fig.", "REF ).", "Notice that, even in the example subject 1069, where some strong stimulation effects are seen, these are restricted to a handful of channels and specific frequency bands.", "This observation is typical for all the sessions that exhibited a strong effect.", "Similarly, the example session on the left is a typical example of all the sessions that have a stimulation effect that is indistinguishable from the baseline fluctuations.", "Figure: Examples of sessions with low and high stimulation effect.The heat maps show the stimulation effect in two example sessions: one from subject 1022 with low effect and one from subject 1069 with high effect.", "The effect was measured for all combinations of channels and frequency bands.", "Notice that the effect can be positive or negative, indicating increase or decrease of band power from pre to post stimulation (see example distributions of the differences (post-pre) in the rightmost panels).", "The fluctuations of band power during baseline are also shown for comparison.", "The channels are sorted based on their Euclidean distance from the stimulation site.", "The lower panels show the spatial distribution of the stimulation effect on theta band across the cortex.The lower panels in Fig.", "REF show the spatial layout of the iEEG stimulation and recording channels in the brain, with electrodes colour-coded by their corresponding stimulation effect sizes.", "Note that a strong stimulation effect, in this case on theta band, is not limited to contacts close to the stimulation site but also affected remote contacts (lower right panel).", "In order to assess if the effect of stimulation exceeded baseline fluctuations in general across all 165 sessions and 79 patients, we compared the extrema of the stimulation effect to the extrema of the baseline fluctuations for each frequency band.", "Figure REF A shows the distributions of minima and maxima effect on theta band for baseline and stimulation.", "These extrema were taken across channels to capture the strongest effect during a session.", "Generally, it is evident that, even the channel with the strongest stimulation effect does not have a substantially larger effect size compared to the baseline fluctuations.", "In theta band, only 10.2% of the sessions exhibit a minimum (negative) stimulation effect that exceeds the adjacent value of the baseline minima.", "Similarly, only 18.1% of the sessions exhibit a maximum (positive) stimulation effect that exceeds the adjacent value of the baseline maxima (see Fig.", "REF A).", "The limited stimulation effect across all sessions was also evident when we computed the paired differences in effect between stimulation and baseline conditions.", "The histograms for the effect minima and maxima in Figure REF B indicate that, in most sessions, even the most extreme effect sizes do not exceed the band power fluctuations during baseline.", "However, these distributions are not zero-centred (paired t-test for minima: $p = 2.4 \\cdot 10^{-5}$ , effect size for minima: -0.430; paired t-test for maxima : $p = 4.9 \\cdot 10^{-5}$ , effect size for maxima: 0.437), indicating that across patients and sessions there is a small but significant difference between baseline and stimulation conditions in our dataset.", "Similar results were found for all frequency bands (see Fig.", "S2 in Supplementary Material).", "Figure: Low stimulation effect on theta band found in most sessions.", "AAcross all sessions, the distributions of their extrema effect values on theta band are compared between baseline and stimulation.", "Each point corresponds to the minimum (left) or maximum (right) effect value U of all recording channels in a given session.", "A minority of sessions have stimulation extrema (10.2% for min and 18.1% for max) that are more extreme than the adjacent values seen in baseline distributions (adjacent values being the most extreme values that are not outliers).", "B The histograms present the paired (per session) differences in extreme values of effect on theta band (session stimulation effect – session baseline effect)." ], [ "Limited effect of stimulation is not due to low stimulation amplitude", "Next, we investigated whether the low stimulation effect size in most sessions can be attributed to the stimulation amplitude of the session.", "Figure REF A shows that there is no correlation between the effect size achieved in the session and the session stimulation amplitude.", "The distributions of effect sizes for each session, across all channels and frequency bands, are represented by their minima and maxima.", "Neither of these two measures tend to increase or decrease with the stimulation amplitude (range: 0.25 - 3.5 mA; see also Fig.", "S3 in Supplementary Material for band specific results).", "Furthermore, we considered all the pairs of stimulation sessions with the same stimulation location in the same subject (101 pairs).", "We tested whether their difference in effect size is correlated with the difference of stimulation amplitude between the sessions.", "Figure REF B shows that the absolute difference in effect size is not correlated with the absolute difference in stimulation amplitude.", "Thus, even for the same subject and the same stimulation location, an increase in stimulation amplitude does not necessarily produce a stronger effect.", "Figure: No correlation found between stimulation amplitude and effect.A The effect minima and maxima (across all channels and frequency bands) from each session is scattered against the stimulation amplitude of the session.", "BFor all the pairs of sessions that come from the same subject and have the same stimulation location, the difference in their stimulation effects is scattered versus the difference in their stimulation amplitudes." ], [ "Weak stimulation effects are inconsistent across sessions", "To investigate the consistency of stimulation effect across sessions, we focused on pairs of sessions in the same subject and stimulation location.", "Figure REF shows two examples of these session pairs.", "The example on the left (subject 1022) does not show positive correlation between the two sessions in terms of stimulation effect in different channels and frequency bands.", "The example on the right (subject 1069) shows that the patient’s two sessions are positively correlated.", "Notice that this correlation is mainly driven by channels that exhibit a strong positive stimulation effect in the first place.", "Figure: Examples of session pairs with low and high effect consistency.The stimulation effect in two pairs of sessions, from twodifferent subjects, is shown in the top panels as examples.", "The pair of sessions on the left (subject 1022) has low consistency whereas the pair on the right (subject 1069) has high consistency.", "This disparity is more clearly shown in the lower panels, where the corresponding effect pairs for each channel and frequency band are scattered.", "The low consistency on the left is shown by a circular cloud of points, whereas the high consistency on the right is shown by an elongated cloud of points.In order to assess the level of consistency in stimulation effect across all 101 session pairs, we computed the consistency curve for each pair.", "Figure REF A shows the consistency curves of the two session pair examples in Fig.", "REF alongside some illustrations on how the curve is computed: a circle of exclusion emanating from (0,0) is gradually enlarged and the consistency coefficient is calculated for varying values of the circle’s radius (see Methods).", "The consistency curve (as a function of the radius) captures the consistency coefficient of the session pair when all effect values are considered (at radius 0) but also while increasingly excluding channel and frequency band combinations with weaker stimulation effect (at higher radii).", "The exclusion of channel/band combinations with weak effect serves to minimise the influence of the inherently inconsistent band power fluctuations on the consistency calculation.", "In addition, considering the selective connectivity of brain areas, it is expected that only a subset of channels will respond to a localised stimulation.", "The two curves shown in Figure REF A capture the difference between high and low consistency as shown in the scatter plots of Fig.", "REF , not only when all values are considered, but also when only the strong effect values are considered.", "This approach of gradually excluding the weaker stimulation effects (around the level of baseline fluctuations) essentially allows us to capture consistency in the few channels that display a discernible stimulation effect in the first place.", "Figure REF B shows the consistency curves for all 101 session pairs and the confidence interval of consistency coefficients of the baseline periods (blue background).", "The overall consistency in this dataset is not high: 32.7% of the session pairs have higher consistency than the $97.5^{th}$ percentile of the baseline `effect’ at radius = 0; 12.9% of the session pairs have higher consistency than the $97.5^{th}$ percentile of the baseline `effect’ at radius = 3; and only 34.6% of the session pairs have a maximum consistency that is higher than the maximum value of the baseline `effect’ confidence interval.", "Four examples of maximum consistency coefficients on four of these curves are indicated with brown markers in Fig.", "REF B.", "We will consider these maximum consistency coefficients as a representative value of the session pair consistency in the following (i.e., highest consistency achieved after exclusion of some not stimulation-related channels).", "In Fig.", "REF C, we demonstrate a strong and significant correlation between the average maximum effect and the maximum consistency coefficients (Pearson's $r=0.536, p=7.4\\times 10^{-9}$ ).", "Theta and alpha bands contribute more to this correlation (see Fig.", "S4 in Supplementary Material).", "As a comparison, we applied the same procedure to simulated data (normal distribution with mean 0 and standard deviation matching the the sessions’ baseline), and the correlation is not present (Pearson's $r= -0.003, p=0.438$ ).", "Essentially, the stronger stimulation effects also tend to be more consistent across sessions.", "Finally, we built a multiple linear regression model to explain the maximum consistency coefficients as a linear combination of multiple independent variables including the average maximum effect ($R^2 = 0.406$ , Adjusted $R^2 =0.340$ ).", "The high explanatory power of the average maximum effect is also evident after running ANOVA on the multiple linear regression model, with the results shown in Fig.", "REF D (distributions produced after 200 bootstrap samples).", "Other than the strong effect of the average maximum effect on consistency, Fig.", "REF D shows a fair effect of both the task difference and the difference of baseline mean on consistency ($p=0.008$ and $p=0.017$ , respectively), which are both anti-correlated with the maximum consistency coefficient.", "Figure: Cross-session consistency is found in a minority of subjects while it relies heavily on strong (positive) effect.A Consistency curves were computed by gradually enlarging the circle of exclusion and calculating the consistency coefficient on the remaining scatter points.", "Three example radii for the circle of exclusion are shown.", "The plotted consistency curves represent the two example session pairs in Fig.", ".B All 101 consistency curves, one for each session pair, are shown with five examples of maximal consistency coefficients achieved (brown diamonds).", "The shaded blue region indicates the 95% two-sided confidence interval of the consistency coefficients of baseline activity.C Maximum consistency coefficient scattered versus average maximum effect reveals a strong correlation between them (Pearson's r=0.536,p=7.4×10 -9 r=0.536, p=7.4\\times 10^{-9}).D Distributions of ANOVA effect values (produced through bootstrapping - see Methods) for the independent variables used in the multiple linear regression model which was used to explain the maximum consistency coefficients.", "The average maximum effect between paired sessions has the strongest explanatory power over consistency.", "Both task difference and difference in baseline mean have a fair explanatory power over consistency.", "Outliers are omitted for clarity." ], [ "Discussion", "We showed that the cross-session consistency of stimulation effect (in terms of band power modulations) is relatively low in a group of 36 subjects who had multiple stimulation sessions through iEEG.", "A third of session pairs indicate a consistency that is above the baseline consistency (Fig.", "REF B).", "High consistency of stimulation effect was found to rely heavily on a strong positive effect of stimulation, that is, high increase of band power (Fig.", "REF D).", "Thus, given these findings, the low consistency levels would be expected in this dataset since the stimulation effect was limited (Fig.", "REF ).", "Other datasets with more pronounced stimulation effect in terms of band power changes may exhibit a higher level of consistency between sessions.", "Variability in the baseline brain state may have impacted the consistency of the stimulation responses in our data set.", "Even the stimulation response within a session has been repeatedly found to depend on the underlying brain state [22], [23], [24].", "This finding is corroborated here since consistency was found to be anti-correlated with both the difference in baseline mean band power and difference in memory task which can be understood as a difference in brain state (Fig.", "REF D).", "In other words, the more similar the brain states (as measured by task, or band power configurations) were in this dataset, the more consistent the stimulation effects tended to be.", "Therefore, a practical advice is to use the same task across stimulation sessions if consistency across sessions is desired.", "Our multiple linear regression model included the stimulation depth as one of the independent variables, and it did not exhibit a strong predictive power over consistency.", "This is not surprising since we did not find any strong relation between stimulation depth and the effect U in the first place (see Fig.", "S5 in Supplementary Material).", "However, it is worth noting that there was no distinction between stimulation through surface and depth electrodes in our analysis.", "The difference between these two types of electrodes cannot be fully captured by the stimulation depth variable.", "Other confounding characteristics, like the physical dimensions of the contacts and the average distance from other recording electrodes, were not accounted for.", "Future work can investigate further whether consistency depends on such factors.", "Considering the data across all subjects, the most represented stimulation site is the right medial temporal lobe, but several other areas were stimulated.", "In addition, the spatial extent of the recording electrodes across the dataset covers the whole cortex.", "A visual inspection of the stimulation sites and the highly responsive sites did not reveal any specific area that was associated with high effect or consistency (see Fig.", "S6 in Supplementary Material).", "Surprisingly, the effect on band power was not correlated with the amplitude of stimulation in this dataset.", "This finding agrees with the reported insensitivity of motor-cortical excitability to tDCS intensity increases [25].", "However, another iEEG study has found stimulation intensity to correlate with high frequency activity (30-100Hz), a frequency range which extends beyond those we investigated [26].", "Furthermore, multiple studies have reported correlations between stimulation intensity and motor improvements when deep brain stimulation of subthalamic nuclei is used for the treatment of Parkinson’s disease (e.g., [27]).", "This discrepancy might indicate a non-trivial or non-linear relationship between the electrophysiological and behavioural effects of an increasing stimulation intensity.", "The potentially `all-or-nothing' response may further depend on the stimulated area.", "In our study, the stimulation effect was measured based on the immediate responses within a session only.", "Arguably, the effect of stimulation can manifest at longer timescales or in other features and those effects may be more consistent across sessions [28], [29].", "This also relates to our definition of baseline in this study.", "Segments of baseline are taken from interstimulus intervals that may carry some post-stimulus modulations of band power.", "Any consistency in long-term changes due to stimulation should be investigated in future studies.", "Cross-session consistency of stimulation effect is critical for developing therapeutic neuromodulation treatments, both in terms of electrophysiological, as well as behavioural stimulation effect.", "This is supported by recent studies which established relationships between stimulation-induced modulation of specific frequency bands and behavioural outcomes [12], [11].", "Despite the fact that some anatomical factors (e.g., thicknesses of the skull and the cerebrospinal fluid layer) do not influence intracranial stimulation, as opposed to tDCS [30], we found that stimulation through iEEG still has low consistency in terms of band power modulations across sessions in our dataset, similar to tDCS [18], [19], [20].", "Our results suggest that ensuring a strong positive modulation of band power through stimulation, by choosing the appropriate stimulation location and parameters, is prerequisite for a high consistency across sessions.", "In addition, our results suggest that the dynamical brain state needs to be taken into account and a state-depended framework of stimulation may be required.", "The present and previous studies all show that more sophisticated protocol designs are needed to maximise the benefit of neurostimulation interventions." ], [ "Acknowledgments", "We thank the CNNP team (www.cnnp-lab.com) for discussions on the manuscript and the presentation of the results.", "CAP, PNT, and YW gratefully acknowledge funding from Wellcome Trust (208940/Z/17/Z and 210109/Z/18/Z).", "Supplementary Material Set of supplementary figures" ] ]

[ [ "Algorithm Selection Framework for Cyber Attack Detection" ], [ "Abstract The number of cyber threats against both wired and wireless computer systems and other components of the Internet of Things continues to increase annually.", "In this work, an algorithm selection framework is employed on the NSL-KDD data set and a novel paradigm of machine learning taxonomy is presented.", "The framework uses a combination of user input and meta-features to select the best algorithm to detect cyber attacks on a network.", "Performance is compared between a rule-of-thumb strategy and a meta-learning strategy.", "The framework removes the conjecture of the common trial-and-error algorithm selection method.", "The framework recommends five algorithms from the taxonomy.", "Both strategies recommend a high-performing algorithm, though not the best performing.", "The work demonstrates the close connectedness between algorithm selection and the taxonomy for which it is premised." ], [ "Introduction", "People, organizations and communities rely on the Internet of Things (IoT) to aid in almost any conceivable task that was previously performed manually.", "As technology advances, components of IoT have progressed into the wireless domain [10].", "These emerging systems are susceptible to attack by malicious actors wishing to degrade the system or steal proprietary information [7].", "Intrusion detection systems (IDS) are central to maintaining the security of modern computer networks from malicious actors [12].", "IDS have been successfully demonstrated in both the wired and wireless domain of IoT [10].", "The task assigned to an IDS is to classify network traffic as malicious or normal.", "Numerous studies [10] have explored meta-models to detect malicious behavior in computer networks.", "Maxwell et al.", "[12] further focused on intelligent cybersecurity feature engineering for various meta-models.", "Learning algorithms may be used to formulate a meta-model.", "Selection of the best machine learning (ML) algorithm, including hyper-parameters, for a particular problem instance is a difficult and time-consuming task [20].", "Cui et al.", "[6] has confirmed conclusions of [18] and [25] that meta-models' performance varies among problem types and problem instances.", "Wolpert et al.", "[28] uses The Extended Bayesian Formalism to show that given a set of learning algorithms and problems, each algorithm will outperform the others for some (equally sized) subset of problems.", "This phenomena has driven researchers to a trial-and-error strategy of identifying the best meta-model for a given problem.", "The preferred meta-model is selected by comparison of model performance metrics such as accuracy [5].", "Unfortunately, the computational run time and human investment required to select a learning algorithm by trial-and-error is generally prohibitive of finding the optimal choice.", "This paper aims to advance the IDS body of knowledge by incorporating recent work in algorithm selection.", "Accordingly, an algorithm selection framework is introduced.", "The algorithm selection framework leverages a taxonomy of ML algorithms.", "The framework narrows down the list of applicable algorithms based on problem characterization.", "Two strategies are presented to select the most preferred algorithm: rules-of-thumb and meta-learner.", "If successful, the algorithm selection framework promotes high-performance results of the IDS and assuages the computational cost of performing multiple ML algorithms.", "This paper includes Related Works in Section 2.", "The Methodology is presented in Section 3.", "Section 4 contains the Results and Section 5 is the Conclusion.", "As in any other domain, criminals and adversaries seek to inflict harm by exploiting weaknesses in cybersecurity systems.", "The rate of system intrusion incidents is increasing annually [16].", "Landwehr et al.", "[11] provides a taxonomy of all known flaws in computer systems.", "Special attention is provided to the category of flaws that allow exploitation by malicious actors.", "Commonly, a malicious actor, or their code, appears benign to a computer security system for a long enough time to exploit information or degrade the attacked system.", "The Trojan horse is among the most prevalent categories of a malicious attack on computer systems.", "It is characterized as a code that appears to provide a useful service but instead steals information or degrades the targeted system.", "A Trojan horse containing self replicating code is known as a virus.", "A trapdoor is a malicious attack in which an actor covertly modifies a system in such a way that they are permitted undetected access.", "Finally, a time bomb is an attack that accesses a system covertly and lies dormant until a detonation time.", "Upon detonation, the time bomb will inflict damage to the system either by disrupting service or destroying information.", "Intrusion detection systems are a layer of network security that tracks activity patterns in a computer system to detect malicious actors before they can inflict harm.", "Debar et al.", "[7] describes efforts as early as 1981 and Sobirey [21] maintains a repository of prominent IDS projects.", "According to Debar et al.", "[7], the success of these systems has spawned a commercial market of IDS software including brands such as Sysco Systems, Haystack Labs, Secure Networks, among others.", "Typically, the IDS employs a detector module that monitors system status.", "The detector catalogues patterns of both normal and malicious activity in a database.", "The detector also monitors patterns in the current system configuration.", "Further, the detector provides an audit of events occurring within the system.", "The detector leverages these data channels to generate an alarm for suspicious activity and countermeasures if necessary.", "An IDS is evaluated by its accuracy of attack detection (false positive), completeness to detect all threats (false negatives) and performance to detect threats quickly.", "NSL-KDD is a publicly available benchmark data set of network activity.", "NSL-KDD improves on several flaws of the well-known KDD Cup `99 benchmark data set.", "Most notably, NSL-KDD has rectified the 78% and 75% duplicate records in training and test sets, respectively [16].", "Four classes of attacks are recorded in the data set.", "Denial of service attacks bombard a network with an overwhelming quantity of data such that the computing resources are exhausted.", "As a result, the system cannot fulfil any legitimate computing processes.", "User to Root attackers enter the network disguised as a legitimate user but seek security vulnerabilities which grant them elevated system privileges.", "A remote to user attack is performed by sending data to a private network and identifying insecure access points for exploitation.", "Probing is the attack technique by which the assailant studies an accessible system for vulnerabilities which will be exploited at a later time [13].", "Maxwell et al.", "[12] and Viegas [24] describe IDS tools that incorporate ML models.", "Unfortunately, raw network traffic data is not a suitable input for building accurate and efficient ML models.", "Instead, the data must be transformed as a set of vectors representing the raw data.", "The process of constructing such vectors is known as feature engineering, which is a non-trivial task that requires both domain knowledge and mastery of ML to capture all available information in the model.", "It is shown experimentally that varying the feature engineering strategy does affect classification accuracy of the IDS but no single feature is known to be superior to others.", "Kasongo [10] explores IDS procedures catered for the wireless domain.", "The UNSW-NB15 data set was selected to derive both the training and test data sets.", "A wrapper-based feature extractor generated many feature vectors for comparison from a full set of 42 features.", "The experiment was performed for both binary and multi-class classification in which the type of attack was predicted.", "Candidate algorithms included decision Trees, Random Forest, Naïve Bayes, K-Nearest Neighbor, Support Vector Machines, and Feed-forward Artificial Neural Networks (FFANN).", "The optimal feature set consisted of 26 columns.", "The FFANN reflected the best classification accuracy on the full data set with 87.10% binary and 77.16% multi-class.", "Random Forest, Decision Tree, and Support Vector Machine were close behind.", "When the feature set and neural network hyper-parameters were optimized, the classification accuracy of the FFANN improved to 99.66% and 99.77% for binary and multi-class classification, respectively." ], [ "Algorithm Selection Problem", "Rice's algorithm selection framework was presented in 1976 [17].", "The framework is performed by employing all algorithms under consideration on all problems in a problem set.", "One or more performance metrics are chosen, and the performance of each algorithm on each problem is reported.", "Upon completion of the process, the preferred algorithm for each problem is taken as the one with the best performance metrics [17].", "Woods [29] presents a modern depiction of Rice's framework as phase 1 in Figure REF .", "Figure: The meta-learner version of Rice's framework .The classic approach of learning algorithms is known as base learning.", "That is an ML algorithm which builds a data-driven model for a specific application [5].", "Meta-learning, however, is an approach introduced by [23] which algorithms learn on the learning process itself.", "A meta-learning algorithm extracts meta-features $ f(x) \\in $ space $F$ from a problem x $\\in $ problem space $P $ .", "The meta-model is trained to recommend the best-known base learning algorithm $ a \\in A $ to solve $x$ .", "Works such as [15] and [3] further contribute to the theory of meta-learning recommendation systems [5].", "In 2014, Smith [19] proposes the concept of applying meta-learning to Rice's model.", "It was not until 2016, however, that [5] implemented the concept.", "Figure REF demonstrates that Cui et al.", "[5] trained a meta-learning model to correlate problem features to algorithm performance and that the trained model could be used to recommend the algorithm for unobserved problems within Rice's framework.", "The meta-learner correctly recommended the best algorithm in 91% of test problems.", "Further, it demonstrated that time to perform algorithm selection could be reduced from minutes to seconds compared to trial-and-error techniques [5].", "Follow on studies by [29] and [26] expanded on this work by exploring various meta-features and meta-learner response metrics." ], [ "Methodology", "The assigned task for an IDS is to classify network traffic records as normal or malicious.", "This task is investigated from the broader perch of the algorithm selection problem.", "Figure REF shows that within the analysis process, three factors drive the analytical approach and analytical technique selection.", "They are the input to the algorithm selection framework.", "Figure: The factors identified are superimposed with the stages of algorithm selection which they impact." ], [ "Characterizing the Problem", "The framework is a mechanism to characterize an analysis problem and to determine the algorithms that best matches the problem characterization.", "The three factors each drive analytical approach selection and analytical technique selection.", "The factor assigned task pertains to the problem provided by a decision-maker.", "The analyst must decipher the intent of the assignment from the lexicon of the decision-maker into specific analytical terms, which are listed under the Task.", "This list of terms, called considerations is shown in Figure REF for each factor.", "The considerations for the factor data describe the different formats analysts commonly receive data for analysis problems.", "The data factor is important because it relates to the problem's compatibility with the mathematical mechanics of the analysis technique.", "Likewise, the considerations for the resources factor help the analyst identify which algorithms are compatible with the available resources.", "The analyst should refer to Figure REF to evaluate and record the considerations for each factor prior to beginning step 1.", "Figure: The considerations are shown for each factor which drives analytical approach and analytical technique selection." ], [ "Step 1: Map Problem to Category and Approach", "Step 1 leverages information from the problem characterization to identify the appropriate analytical approaches.", "Each consideration selected from the assigned task factor maps to one or more categories of analysis.", "The categories of analysis describe the general goal of the analysis problem [4].", "Each category of analysis can be implemented by certain analytical approaches.", "The analytical approach a technique class referring to the specific type of response the techniques produce.", "Therefore, the framework leverages a hierarchical taxonomy that groups techniques grouped by both categories of analysis and analytical approaches.", "Figure REF shows the mapping from assigned task to category of analysis, and the mapping of category of analysis to analytical approach.", "Since the task of an IDS is to classify network users, the prescriptive and predictive categories of analysis are selected.", "Figure: The assigned task for an IDS is classify.", "Classify is one of 11 common assigned tasks.", "It belongs to the predictive and prescriptive categories of analysis.An excerpt of the proposed taxonomy is presented in Figure REF .", "The taxonomy is built with an object-oriented structure to promote flexibility and expandability.", "As an example, techniques are shown within the regression and classification analytical approaches.", "The text predictive and descriptive appears at the bottom edge of the regression panel to indicate that regression techniques produce results suitable for either of these two categories of analysis.", "The requirements, or required considerations, for each factor are presented with the technique.", "Compatible training styles are listed to the right of the technique name.", "The object-oriented structure allows new techniques to be easily added and new attributes to be included.", "Figure: A portion of the proposed taxonomy is high-lighted to show its structure." ], [ "Step 2: Rank Techniques", "The framework identifies a subset of techniques that are compatible for the problem according to application.", "Next, the framework leverages the remaining three factors data, resources and experience to discern aspects of technique compatibility relating to the mechanics of the mathematical model.", "Step two predicts the utility scores of each algorithm from these factors according to two strategies: rules of thumb and meta-learning.", "They are presented in parallel below." ], [ "Rules-of-Thumb", "A logical decision tree is used to assign a preference rank among candidate algorithms.", "The decision tree is built according to rules-of-thumb regarding features of the data.", "The features pertaining to data also impact the compatibility of techniques in respect to resources.", "Thus, it is justified to use the same decision tree, Figure REF , to adjudicate the scores for both factors.", "Figure: The decision tree represents the logical tests used to rank order the preference of each algorithm via the rules of thumb strategy" ], [ "Meta-Learning", "A meta-learner is constructed in Python 3.7.", "All data sets are pre-processed according to best practices for data mining.", "Base learning is performed on 14 benchmark data sets with 20 repetitions.", "For each repetition, the data sets were split into training and test sets with an 80/20 ratio and with stratification.", "The KDDTest+ and KDDTrain+ sets were obtained having already been split into a master test set and a training set.", "12 meta-features of each data set were stored as predictor data for the meta-learner; the mean observed recall was stored as the target.", "The meta-learner was trained to model recall as a function of meta-features using a support vector regression algorithm.", "The radial basis function kernel was selected.", "The regularization parameter was set at 1.0, and the kernel coefficient was auto-scaled as a function of the number of features and predictor variance.", "All other parameters followed Scikit-learn defaults [14].", "Pseudo-code of the meta-learner is presented in Algorithm 1. predrecalldataPredicted Recall obsrecalldataObserved Recall dataRepositorysetDatasettestset Test Dataset setsDatasetsUpupcolcolumnUnionUnionFindCompressFindCompress InputinputOutputoutputfeaturefeatureExtract()trainbasetrainBaseLearner()testbaseTestBaseLearner() trainmetaTrainMetaLearner() predictrecallPredictRecall() pcaPCA() minimaxminiMax(0,1) onehotOneHot() algAlgorithm alglistCandidate Algorithms recordRecordRecall() rankRankAlgorithms() & of using Pre-processing all in all in is numerical is categorical Feature Extraction all in Base Learning all all in Meta Learning all Pseudo-code of Meta-learner The candidate algorithms are selected because they are members analytical approaches derived in step 1 of Section 3.2.", "Clearly, these are not the only algorithms that fall into the applicable analytical approach.", "Rather, they represent a demonstrative taxonomy.", "Note that the Scikit-learn default settings are selected on each algorithm to show generality of the algorithm selection framework.", "Training data sets are selected from easily accessible benchmark repositories.", "The first five are selected to provide diversity of meta-features to the meta-learner and improve the robustness of the model.", "The nine subsets of KDDTrain+ are selected to provide statistical information consistent with the KDDTest+ data set.", "KDDTrain+ is split into subset to promote diversity of meta-features but also reduce the number of records in the training set which is an order of magnitude greater than the number of records in the test set.", "A uniform random number generator is used to determine the number of rows allocated to each training set.", "Rows are not re-arranged during the subset process.", "Training Data Heart: Predict presence of heart disease from 13 predictor variables [9] Framingham: Predict presence of heart disease in the Framingham study from 15 predictor variables [1] Spam: Predict if an email is spam based on six predictor variables [2] Loan: Predict whether a consumer purchases a loan from Thera Bank based on 12 predictor variables [8] Cancer: Predict whether a patient has breast cancer from 30 predictor variables collected in a fine needle aspirate procedure [27] Nine subsets of KDDTrain+: Predict whether a network activity record is normal or malicious from four categorical and 39 numerical predictor variables [22] Test Data KDDTest+: Predict whether a network activity record is normal or malicious from four categorical and 39 numerical predictor variables [22] The choice of meta-features was adopted from [26].", "The following meta-features were used as predictor data by the meta-learner to model expected recall.", "Number of Rows Number of Columns Rows to Columns Ratio Number of Discrete Columns Maximum number of factors among discrete columns Minimum number of factors among discrete columns Average number of factors among discrete columns Number of continuous columns Gradient average Gradient minimum Gradient maximum Gradient standard deviation" ], [ "Results", "The algorithm selection framework is applied to the task of classifying network traffic as malicious or normal.", "Problem characterization is performed in step 1 to identify prescriptive and predictive as the categories of analysis.", "This leads to four analytical approaches, namely regression, classification, multivariate, and reinforcement.", "Five example algorithms which meet this criteria are taken from a notional taxonomy.", "In step 2, candidate algorithms are ranked in order of preference by each recommendation strategy, rules-of-thumb and meta-learner.", "Both strategies yielded support vector regression as the most highly recommended algorithm.", "According to the mean observed recall, random forest was the best performing algorithm to detect malicious activity from the KDDTest+ data set.", "The standard deviations of observed recall on each algorithm were very low.", "The 90% Bonferroni confidence interval for support vector machine (SVM) and support vector regression (SVR) overlapped, indicating statistically identical recall performance.", "All other mean values were statistically unique.", "Further, the Spearman's coefficient of rank correlation was not statistically significant, largely due to the small size of the rank scheme.", "Since neither recommendation strategy succeeded in predicting the best performing algorithm, recall efficiency is introduced.", "Recall efficiency, Equation REF , is the ratio of the recall observed by the top recommended algorithm to best observed recall.", "$E_{R}=\\frac{R_{best Rec}}{R_{best Obs}}$ The recall efficiency for SVR, the top recommendation of both strategies, is 0.98.", "Table REF presents a summary of the results including observed mean recall, meta-learner predicted recall, mean runtime, standard deviation of observed recall, observed ranks, rule-of-thumb predicted ranks, and meta-learner predicted ranks.", "Figure REF outlines the mean observed recall and the recall predicted by the meta-learner for each algorithm.", "Table: Results compare the recommendations of each strategy to observed algorithm performance.Figure: The assigned task for an IDS is classify.", "Classify is one of 11 common assigned tasks.", "Classify falls into the predictive and prescriptive categories of analysis.It is difficult to ascertain whether either of the recommendation strategies employed in this study were successful.", "While the recall efficiency is very high, the rank correlation was not conclusive.", "All base learners produced very high and very similar recalls.", "It would therefore be difficult for any model to discern the true rank preference.", "The meta-model employed the meta-features according to the precedent set by [26], however there were many more proposed by [5] that were not used.", "Furthermore, the meta-learner was trained by only 14 data sets, significantly less than used in [29].", "Providing more training sets, especially from the domain of network traffic, would likely improve the predictive capability of the meta-learner.", "As a whole, the framework is beneficial even when it does not recommend the true best performer.", "The framework consistently filters techniques that are incompatible with the problem characterization.", "Further, the framework identifies five viable options, each of which perform excellently." ], [ "Conclusion", "Cyber attack detection from an IDS using any of the recommended algorithms could reasonably be deemed successful.", "Neither of the two recommendation strategies demonstrated perfect results.", "They did, however, show enough promise to motivate further investigations.", "Fundamentally, the meta-data and user input collected by the framework does contain information capable of consistently predicting a good analysis technique for a problem.", "Notably, there were algorithms from distinct analytical approaches that performed well on the same task.", "The process of problem characterization fits well into the framework but does require further refining.", "The rule-of-thumb decision tree provided intelligible recommendation logic whereas the meta-learner is a black box model.", "Future work should use the Gini criterion to optimize the decision tree.", "Further, the meta-learner should be improved to include more meta-features and training sets.", "There is a close connectedness in having a useful taxonomy of algorithms and a successful algorithm selection.", "This relationship is only beginning to be understood.", "Wireless IDS already provide good classification performance, however, algorithm selection, hyper-parameter tuning and feature engineering suffer from the time-costly trial-and-error practice.", "The algorithm selection framework may be a step forward in reducing this cost." ] ]

[ [ "Ship-track-based assessments overestimate the cooling effect of\n anthropogenic aerosol" ], [ "Abstract The effect of anthropogenic aerosol on the reflectivity of stratocumulus cloud decks through changes in cloud amount is a major uncertainty in climate projections.", "The focus of this study is the frequently occurring non-precipitating stratocumulus.", "In this regime, cloud amount can decrease through aerosol-enhanced cloud-top mixing.", "The climatological relevance of this effect is debated because ship exhaust does not appear to generate significant change in the amount of these clouds.", "Through a novel analysis of detailed numerical simulations in comparison to satellite data, we show that results from ship-track studies cannot be generalized to estimate the climatological forcing of anthropogenic aerosol.", "We specifically find that the ship-track-derived sensitivity of the radiative effect of non-precipitating stratocumulus to aerosol overestimates their cooling effect by up to 200%.", "This offsetting warming effect needs to be taken into account if we are to constrain the aerosol-cloud radiative forcing of stratocumulus." ], [ "Dataset", "This study is based on the ensemble of large-eddy simulations (LESs) described in reference [32].", "In comparison to the original dataset, we have excluded outliers in terms of above-cloud humidity, which results in a dataset of 144 LES runs.", "External, or large-scale, conditions are the same across the LES ensemble and are summarized in Table REF .", "Variability of LWP within the ensemble is achieved by varying the initial profiles of temperature and moisture; individual simulations vary in $N$ because they have been initialized with varying aerosol backgrounds (Table REF ).", "Following reference [33], we prevent co-variability among the initial conditions by means of a 6D latin-hypercube sampling of the internal factors listed in Table REF .", "For the derivation of the flow field, additional simulations have been added to achieve a better coverage of the $N$ -LWP space.", "The dataset and its variants are illustrated in Figure REF .", "The flow field $\\vec{v} = \\left(\\textrm {d} \\ln N /\\textrm {d} \\textrm {t}, \\textrm {d}\\ln \\textrm {LWP}/\\textrm {d}t\\right)^\\textrm {T}= \\left(v_\\textrm {N}, v_\\textrm {LWP}\\right)^\\textrm {T}$ shown in Figure REF (b) is based on separately deriving the components in the $N$ -direction, $v_\\textrm {N}$ , and the LWP-direction, $v_\\textrm {LWP}$ (Figure REF ).", "To derive these component fields, we first extract tendencies $\\textrm {d}\\ln N/\\textrm {d}t$ and $\\textrm {d}\\ln \\textrm {LWP}/\\textrm {d}t$ from the data and then interpolate the extracted tendencies by means of Gaussian-process regression to obtain emulators for the tendency surfaces.", "To derive tendencies, we split each simulated time-series into six intervals of $100\\,$ min duration, each of which contains 10 consecutive data points at a $10\\,$ min output frequency.", "For each of these we determine tendencies by fitting trend lines.", "We only consider significant trends (p-value $<0.05$ ) and assign a value of zero otherwise.", "To account for oscillatory behavior that occurs because some of our simulations remain influenced by spin-up processes, we assume that the last $100\\,$ min-segment of each time-series provides the correct sign of the evolution.", "The previous segments are then only considered if they feature the same sign.", "With these restrictions, we obtain a dataset of 828 LWP-tendencies and 783 $N$ -tendencies.", "We process these datasets largely in the same way as described in detail in reference [32].", "Here we only mention adaptations to the technical parameters mentioned therein.", "Instead of a 50%-50% split of the dataset into training and validation data, we use a smaller fraction of training data (33%) to ensure good validation.", "To obtain an ensemble of 5 emulated surfaces for $v_\\textrm {LWP}$ and $v_\\textrm {N}$ , we do not restrict the fraction of the training data to be used for individual ensemble members.", "As a result of our interpolation technique, we obtain a mean emulator surface $v(N, \\textrm {LWP}$ ), to which individual emulator ensemble members contribute according to their root-mean-square error (RMSE) in predicting the validation data [32].", "The central value for the LWP adjustment is based on the zero-contour of this mean surface.", "For the uncertainty percentiles, we determine LWP adjustments from the zero-contours of a specific sampling of the emulator ensemble.", "For this sampling, we take 100 samples of each of the 5 ensemble members and then select a subset of these 500 samples, such that each ensemble member contributes proportionally to its RMSE-based weight, i.e.", "for the ensemble member with the lowest RMSE, all 100 samples are considered, and for the ensemble member with the highest RMSE, no samples are considered.", "Table REF summarizes the uncertainty ranges obtained in this way for the LWP adjustment value.", "To derive the adjustment equilibration timescale, we determine the time required by the entire system of LWPs to reach their respective steady states.", "We assume that the LWP for any $N$ approaches its steady state $\\textrm {LWP}_\\infty (N)$ with approximately the same velocity $[\\textrm {LWP}_\\textrm {ini}-\\textrm {LWP}_\\infty (N_0)]/\\tau $ controlled by the characteristic equilibration timescale $\\tau = 9.6\\,\\textrm {h}$ (Figure REF ), where $\\textrm {LWP}_\\textrm {ini}$ is an initial non-steady-state LWP.", "Hence, the linearized exponential change in LWP yields $\\textrm {LWP}(t,N) = \\textrm {LWP}_\\textrm {ini} - \\frac{t}{\\tau } \\left[\\textrm {LWP}_\\textrm {ini} - \\textrm {LWP}_\\infty (N_0) \\right],$ where $\\textrm {LWP}_\\infty (N_0)$ is the steady-state LWP for the smallest $N$ in the non-precipitating regime, $N_0$ .", "Note that we focus on the LWPs that approach the steady state from larger values since only those require a longer time to reach the steady state for larger $N$ .", "The steady-state LWP as a function of $N$ is obtained by integrating and linearizing the adjustment $\\textrm {d}\\ln \\textrm {LWP}/\\textrm {d} \\ln N$ , $\\textrm {LWP}_\\infty (N) = \\textrm {LWP}_\\infty (N_0) \\left[1 + \\frac{\\textrm {d}\\ln \\textrm {LWP}_\\infty }{\\textrm {d} \\ln N} \\ln \\left(\\frac{N}{N_0}\\right)\\right],$ using the same constants of integration as above.", "Combining Equations REF and REF , and solving for $t\\equiv \\tau _\\textrm {adj}$ , gives the adjustment equilibration timescale necessary to equilibrate the entire system: $\\tau _\\textrm {adj} = \\tau \\left[1 - \\frac{\\textrm {d}\\ln \\textrm {LWP}_\\infty }{\\textrm {d} \\ln N} \\ln \\left(\\frac{N_2}{N_0}\\right) \\frac{\\textrm {LWP}_\\infty (N_0)}{\\textrm {LWP}_\\textrm {ini} - \\textrm {LWP}_\\infty (N_0)} \\right],\\nonumber $ where $N=N_2$ is the largest droplet concentration in the considered non-precipitating regime, resulting in the longest time to equilibrate the LWP.", "With $N_0=107\\,\\textrm {cm}^{-3}$ , $N_2=390\\,\\textrm {cm}^{-3}$ and $\\textrm {LWP}_\\infty (N_0)=89\\,\\textrm {g}\\,\\textrm {m}^{-2}$ in the smallest (index 0) and largest (index 2) $N$ -bin, and $\\textrm {d} \\ln \\textrm {LWP}_\\infty / \\textrm {d} \\ln N=-0.64$ , we obtain best fitting results for $\\tau _\\textrm {adj}$ when assuming $\\textrm {LWP}_\\textrm {ini} = 159\\,\\textrm {g}\\,\\textrm {m}^{-2}$ , which amounts to the 78th percentile of LWPs in the $N_2$ -bin.", "These numerical values provide the adjustment equilibration timescale stated in Equation REF .", "The effective evolution time $\\Delta t_\\textrm {lane}$ of a ship track in a shipping lane can be estimated based on Equation REF , $\\textrm {adj}(\\Delta t_\\textrm {lane}) = -0.24 \\Rightarrow \\Delta t_\\textrm {lane}\\approx 9\\,\\textrm {h},$ where we have used the adjustment value $\\textrm {d} \\ln \\textrm {LWP}/\\textrm {d} \\ln N = -0.24$ from reference [17].", "The cloud-mediated aerosol forcing depends on the aerosol sensitivity of the relative cloud radiative effect, $\\textrm {rCRE}$ , which relates downwelling short-wave radiative fluxes at the surface, $F$ , under clear-sky (index $\\textrm {clr}$ ) and all-sky (index $\\textrm {all}$ ) conditions [51], $\\textrm {rCRE}=\\frac{F_\\textrm {clr}-F_\\textrm {all}}{F_\\textrm {clr}} \\approx \\textrm {CF}\\cdot A_\\textrm {c}\\approx A_\\textrm {c}$ and amounts to cloud albedo $A_\\textrm {c}$ in fully overcast Sc with cloud fraction $\\textrm {CF}\\approx 1$ .", "We assume that climatological cloud properties can be approximated by steady-state values.", "With a steady-state cloud albedo of $A_\\textrm {c}=0.5$ based on Figure REF , Equation REF for the sensitivity of $A_\\textrm {c}$ , or rCRE, respectively, results in $S = \\frac{1}{N} \\left(\\frac{1}{12} + \\frac{5}{24} \\frac{\\textrm {d}\\ln \\textrm {LWP}}{\\textrm {d}\\ln N}\\right).$ With $\\textrm {d}\\ln \\textrm {LWP}/\\textrm {d}\\ln N \\approx -0.1$ (Equation REF ), ship-track studies imply $S_\\textrm {ship} \\approx 0.06/N>0$ and thus a cooling effect of anthropogenic aerosol via increased cloud brightness at almost constant LWP.", "In contrast, the steady-state adjustment value of $\\textrm {d}\\ln \\textrm {LWP}_\\infty /\\textrm {d}\\ln N=-0.64$ derived here as a lower bound results in $S_\\textrm {clim}=-0.05/N<0$ , which indicates that aerosol-induced cloud thinning overcompensates the brightening effect at constant LWP.", "Ship-track studies thus overestimate the cooling effect of aerosol on Sc by up to $|(-0.05-0.06)/(-0.05)| =220\\,\\% \\approx 200\\,\\%$ .", "FG thanks Tom Goren and Anna Possner for helpful discussions about the interpretation of satellite literature.", "FG acknowledges support by The Branco Weiss Fellowship – Society in Science, administered by the ETH Zürich, and by a Veni grant of the Dutch Research Council (NWO).", "FH holds a visiting fellowship of the Cooperative Institute for Research in Environmental Sciences (CIRES) at the University of Colorado Boulder, and the NOAA/Earth System Research Laboratory.", "Jill S. Johnson and Ken S. Carslaw were supported by the Natural Environment Research Council (NERC) under grant NE/I020059/1 (ACID-PRUF) and the UK-China Research and Innovation Partnership Fund through the Met Office Climate Science for Service Partnership (CSSP) China as part of the Newton Fund.", "Ken S. Carslaw is currently a Royal Society Wolfson Research Merit Award holder.", "This research was partially supported by the Office of Biological and Environmental Research of the U.S. Department of Energy Atmospheric System Research Program Interagency Agreement DE-SC0016275 and by an Earth's Radiation Budget grant, NOAA CPO Climate & CI #03-01-07-001.", "Marat Khairoutdinov graciously provided the SAM model.", "The University of Wyoming, Department of Atmospheric Science, is acknowledged for archiving the radiosonde data." ] ]

[ [ "User Behavior Retrieval for Click-Through Rate Prediction" ], [ "Abstract Click-through rate (CTR) prediction plays a key role in modern online personalization services.", "In practice, it is necessary to capture user's drifting interests by modeling sequential user behaviors to build an accurate CTR prediction model.", "However, as the users accumulate more and more behavioral data on the platforms, it becomes non-trivial for the sequential models to make use of the whole behavior history of each user.", "First, directly feeding the long behavior sequence will make online inference time and system load infeasible.", "Second, there is much noise in such long histories to fail the sequential model learning.", "The current industrial solutions mainly truncate the sequences and just feed recent behaviors to the prediction model, which leads to a problem that sequential patterns such as periodicity or long-term dependency are not embedded in the recent several behaviors but in far back history.", "To tackle these issues, in this paper we consider it from the data perspective instead of just designing more sophisticated yet complicated models and propose User Behavior Retrieval for CTR prediction (UBR4CTR) framework.", "In UBR4CTR, the most relevant and appropriate user behaviors will be firstly retrieved from the entire user history sequence using a learnable search method.", "These retrieved behaviors are then fed into a deep model to make the final prediction instead of simply using the most recent ones.", "It is highly feasible to deploy UBR4CTR into industrial model pipeline with low cost.", "Experiments on three real-world large-scale datasets demonstrate the superiority and efficacy of our proposed framework and models." ], [ "Introduction", "Click-through rate (CTR) prediction plays a key role in today's online personalization platforms (e.g., e-commerce, online advertising, recommender systems), whose goal is to predict a user's clicking probability on a specific item under a particular context.", "With more than a decade of development for the online personalization platforms, the amount of user behaviors logged on the platforms grows rapidly.", "There are 23% of users have more than 1000 behaviors during six months on Taobao [23].", "As there exist resourceful temporal patterns embedded in user behaviors, it becomes an essential problem for both industry and academia to build an effective and efficient model which could utilize user sequential behaviors to obtain accurate predictions of CTR.", "In the deep learning era, there are many deep neural network (DNN) based CTR prediction models such as Wide&Deep [4], FNN [43], DeepCross [36], DeepFM [7], PNN [21], [22] and xDeepFM [17], most of which have been deployed on commercial personalization platforms.", "These models emphasize mining feature interactions, and are proposed to utilize the multi-categorical features of the data better.", "Nonetheless, such models ignore the sequential or temporal patterns of user behaviors.", "As shown in [11], [16], [8], [1], the temporal dynamics of user behavior play a key role in predicting the user's future interests.", "These sequential patterns include concept drifting [37], long-term behavior dependency [16], [23], periodic patterns [26], etc.", "Thus, there are models proposed to capture the user's sequential patterns in both CTR prediction and sequential recommendation tasks.", "For CTR prediction, there are attention-based models like DIN [45] and DIEN [44], memory network-based models like HPMN [23].", "More user behavior modeling methods are proposed for sequential recommendation, which is a quite similar task to CTR prediction.", "There are RNN-based models [12], CNN-based models [31], Transformer-based models [15] and memory network-based models [35], [5].", "Figure: Comparison between traditional framework and UBR4CTR.However, most of the sequential models above have a common problem in real-world practice.", "When the platforms have logged a large number of user behaviors, the common industrial solution is to truncate the whole behavior sequence and only uses the most recent $N$ behaviors as the input to the prediction model [45], [44], as illustrated in the upper part of Figure REF .", "The strict requirements of online serving time plus bottleneck of system load and computational capacity put limits on the length of user sequence that could be used.", "As a result, in most of industrial cases, no more than 50 recent behaviors are used [44].", "Traditional framework that uses the most recent $N$ behaviors could cause negative issues.", "It is obvious that the effective sequential patterns may be not just embedded in the recent sequence of behaviors.", "It may be traced back into further history such as periodicity and long-term dependencies [23].", "If we try to use a longer sequence, however, a lot of irrelevant behaviors and noises could be introduced.", "Let alone the time and space complexity the longer history brings.", "In this paper, to tackle the above practical issues, we try to solve the problem from the data perspective instead of designing more sophisticated yet complicated model.", "Specifically, we target to design a framework to retrieve a limited number of historic behaviors that are most useful for each CTR prediction target.", "As shown in Figure REF , a prediction target consists of three parts, i.e.,the target user, the target item and the corresponding context.", "There are features of the prediction target, such as the user's location, gender, occupation, and item's category, brand, merchant, and context features such as time and scenario.", "Then we use a model to select a subset of these features, which builds a query to retrieve the relevant historical behaviors.", "All the user's behaviors are stored as the information items in a search engine, and we use the generated query to search from the historical records.", "The retrieved behaviors are used in CTR prediction.", "For every different candidate target item for the same user, we will retrieve different behaviors for prediction because the generated queries are different.", "This is a significant change compared with traditional framework that uses exactly the same recent $N$ behaviors for prediction on different items with the same user.", "The resulted solution framework is called User Behavior Retrieval for CTR (UBR4CTR).", "In UBR4CTR, the task is divided into two modules.", "The first is a learnable retrieval module which consists of a self-attentive network to select the features and form the query, and a search engine in which the user behaviors are stored in an inverted index manner.", "The other module is the prediction module in which an attention-based deep neural network is built to make the final prediction based on the retrieved user behaviors as well as the features of the prediction target.", "The contributions of the paper can be summarized in three-fold: [leftmargin=15pt] We reveal an important fact that it is important to retrieve more relevant user behaviors than just use the most recent behaviors in user response prediction.", "Instead of designing more sophisticated and complex models, we put more attention on retrieving user's behavioral data.", "We propose a new framework called UBR4CTR which manages to retrieve different behaviors for the same user when predicting her CTR to different items under different contexts.", "All the previous sequential models only use the most recent behaviors of a user.", "We propose a search engine based method and an effective training algorithm to learn to retrieve appropriate behavioral data.", "We conduct extensive experiments and compare our framework with several strong baselines using traditional framework over three real-world large-scale e-commerce datasets.", "The results verify the efficacy of UBR4CTR framework.", "The rest of the paper is organized as follows.", "In Section , we will introduce the preliminaries and some notations used in this paper.", "Section  is about the detailed description of our proposed framework and models.", "The experimental settings and corresponding results are shown in Section .", "The deployment feasibility is discussed in Section .", "In Section  we discuss about some important related works.", "Finally, we conclude the paper and discuss the future work in Section ." ], [ "Preliminaries", "In this section, we formulate the problem and introduce the notations.", "For CTR prediction task, there are $M$ users in $\\mathcal {U} = \\lbrace u_1,..., u_M\\rbrace $ and $N$ items in $\\mathcal {V} = \\lbrace v_1, ..., v_N\\rbrace $ .", "The user-item interactions are denoted as $\\mathcal {Y}=\\lbrace y_{uv} | u \\in \\mathcal {U}, v \\in \\mathcal {V} \\rbrace $ and, $y_{uv} = \\left\\lbrace \\begin{array}{rcl}1, & & u~ \\text{has clicked} ~v; \\\\0, & & \\text{otherwise.}", "\\\\\\end{array}\\right.$ Furthermore, each user-item interaction has a timestamp and context in which the interaction happened, thus the data is formulated as quadruples which is $\\lbrace u, v, c, ts\\rbrace $ indicating $u$ clicked $v$ at time $ts$ in context $c$ .", "To model user's evolving interests, we organize the user behavioral history as $H_u = \\lbrace b^u_1, b^u_2, ..., b^u_T\\rbrace $ in which $b^u_i$ stands for $i$ -th behavior record of user $u$ sorting by timestamp.", "As click-through rate is essentially a matching probability between user, item and context, each behavior record $b^u_i$ is consist of these three parts that $b^u_i = [u, v_i, c_i]$ where $v_i$ is the $i$ -th clicked item and $c_i$ is the context at which the interaction happened.", "At the feature level, each user $u$ is represented by a bunch of features that $u = [f^u_1, ..., f^u_{K_u}]$ where $f^u_p$ denotes the $p$ -th feature of $u$ .", "It is a common practice that all the features are multiple categorical features [45], [24].", "Numerical features, if any, are discretized to categorical features.", "Similarly, $v = [f^v_1, ..., f^v_{K_v}]$ and $c = [f^c_1, ..., f^c_{K_c}]$ .", "The goal of CTR prediction is to predict probability of target user $u$ clicking target item $v$ given historical behaviors of $u$ under context $c$ .", "It is formulated as $\\hat{y}_{uv} = \\mathit {f}(u, v, c| H_u; \\theta ),$ where $f$ is the learned function with parameters $\\theta $ .", "We conclude the notations and the corresponding descriptions in Table REF .", "Table: Notations and corresponding descriptionsFigure: Overall framework of the proposed UBR4CTR framework." ], [ "Methodology", "In this section, we describe our proposed UBR4CTR (User Behavior Retrieval for CTR prediction) framework in detail.", "We firstly give a big picture of the overall framework, then we give detailed descriptions on the user behavior retrieval module and the prediction module.", "Moreover, the training methods and some analysis about time complexity are given in the following sections." ], [ "Overall Framework", "The overall framework of UBR4CTR is illustrated in Figure REF .", "The framework can be divided into two major modules: user behavior retrieval module and prediction module.", "The user behavior retrieval module consists of a feature selection model, a search engine client and a user history archive.", "All the user historical behaviors are stored in the archive and they are organized in a feature-based inverted index manner which will be explained in details in Section REF .", "As shown in Figure REF , when we need to predict the click-through rate between the target user and target item in a certain context, all of the three parts of information are combined to form a prediction target.", "The prediction target essentially consists of a bunch of features of the target user, target item and the context.", "So the prediction target is then fed to the feature selection model which will select the appropriate features to form a query.", "Detailed design of the feature selection model is in Section REF .", "Then we use the query to search in the user history archive through a search engine client.", "The search engine client retrieved a certain number of user behavior records and these records are then used by the prediction module.", "In the prediction module, we use an attention-based deep model to distinguish the influence of each behavior to the clicking probability and make the final prediction which will be discussed in Section REF .", "The feature selection model and the prediction model are trained in turn.", "The goal of feature selection model is to select the most useful subset of features.", "The features of this subset will be combined to generate a query which is used to retrieve the most relevant user behaviors for the final prediction." ], [ "User Behavior Retrieval Module", "In this section, we introduce the user behavior retrieval module which consists of a feature selection model and a behavior searching process." ], [ "Feature Selection Model", "As shown in Figure REF , we regard all the features of target user $u$ , target item $v$ and the corresponding context $c$ as the input of the feature selection model.", "Without loss of generality, we set $f^u_1$ as the user id feature.", "User id is a special feature which we must select because we want to retrieve behaviors of user $u$ herself.", "All the other features are concatenated as a whole.", "For simplicity, we denote all the features $[f^u_2, ..., f^u_{K_u}, f^v_1, ..., f^v_{K_v}, f^c_1, ..., f^c_{K_c}]$ as $[f_1, ..., f_{K_q}]$ correspondingly where $K_q=K_u + K_v + K_c - 1$ .", "To better model the relationships and interactive patterns between the features, we use self-attention mechanism [32].", "Specifically, we define that $Q = K = V = \\left(\\begin{array}{c}\\mathbf {f}_{1} \\\\\\vdots \\\\\\mathbf {f}_{K_q}\\end{array}\\right)$ where $\\mathbf {f}_i$ is the dense embedding representation of the $i$ -th feature.", "And $K=Q=V \\in R^{K_q \\times d}$ where $d$ is the dimension of the embeddings.", "The self-attention is defined as, $\\text{SA}(Q, K, V) = \\operatorname{softmax}\\left(\\frac{Q K^{T}}{\\sqrt{d_{}}}\\right) V$ and multihead self-attention is $E = \\text{Multihead}(Q, K, V) = \\text{Concat}(head_1, ..., head_h)W^O,$ where $head_i = \\text{SA}(QW_i^Q, KW_i^K, VW_i^V)$ .", "The parameters matrixes are $W^O \\in R^{hd \\times d}$ , $W_i^Q \\in R^{d \\times d}$ , $W_i^K \\in R^{d \\times d}$ , $W_i^V \\in R^{d \\times d}$ .", "The output of multihead self-attention $E$ is then fed to a multi-layer perceptron with $ReLU(x)=max(0,x)$ activation function as $\\hat{E} = \\text{MLP}(E),$ where $\\hat{E} \\in R^{K_q \\times 1}$ .", "And the probability of selecting each corresponding feature is obtained by taking a sigmoid function as $P = \\left(\\begin{array}{c}p_{1} \\\\\\vdots \\\\p_{K_q}\\end{array}\\right) = \\sigma (\\hat{E}),$ where $\\sigma (x)=\\frac{1}{1+\\exp ^{-x}}$ .", "Then we sample the features according to these probabilities and thus get the selected subset of features.", "Note that the user id feature is always in the subset because we have to retrieve from the target user's own behaviors.", "Figure: Illustration of feature selection model." ], [ "Behavior Searching", "We use the typical search engine approach to store and retrieve the user behaviors.", "We regard each user behavior as a document and each feature as a term.", "Inverted index is used to track all the behaviors, which is shown in Figure REF .", "Each feature value is associated with a posting list which consists of all the behaviors that has this specific feature value.", "For instance, the \"user_id_1\" has the posting list of all the behaviors of the user whose id is 1 and \"Nike\" has the posting list which consists of the behavior records that brand ids are \"Nike\".", "The logic of the query is essentially formulated as $f_1^u~AND~(f_1~OR~f_2~OR ...OR~f_n),$ where $f_1^u$ is user id.", "The selected query feature set is $q = \\lbrace f_1, f_2, ...f_n\\rbrace $ .", "The posting list of $f_1^u$ and the union set of $f_1$ to $f_n$ 's posting lists are intersected.", "The intersection is the candidate behavior set.", "Then we use BM25 [29] to score every behavior document in the candidate set and the top $S$ are retrieved.", "The similarity $s$ between the query $q$ and a behavior document $D$ is calculated as, $s = \\sum _{i=1}^{n} \\operatorname{IDF}\\left(f_{i}\\right) \\cdot \\frac{tf\\left(f_{i}, D\\right) \\cdot \\left(k_{1}+1\\right)}{tf\\left(f_{i}, D\\right)+k_{1} \\cdot \\left(1-b+b \\cdot \\frac{|D|}{\\text{ avgdl }}\\right)},$ where $tf\\left(f_{i}, D\\right)$ is feature $f_i$ 's term frequency in $D$ which is 1 or 0.", "The length of a behavior document is defined as the number of the features in that behavior, so all the behavior documents have the same length.", "Thus the average length is each documents' length and $\\frac{|D|}{\\text{avgdl}}=1$ .", "$k_1$ and $b$ are free parameters.", "We set $k_1=1.2$ and $b=0.75$ .", "IDF is defined as, $\\operatorname{IDF}\\left(f_{i}\\right)=\\log \\frac{\\mathcal {N}-\\mathcal {N}\\left(f_{i}\\right)+0.5}{\\mathcal {N}\\left(f_{i}\\right)+0.5},$ where $\\mathcal {N}$ is the total number of the behavior documents and $\\mathcal {N}\\left(f_{i}\\right)$ is the number of documents that contains the feature $f_i$ .", "The IDF term gives the common features less importance than rare features.", "It makes sense that rare features imply stronger signals on user's preference compared to the commonly seen features.", "By ranking the candidate behaviors using BM25, we obtain the top $S$ documents as the retrieved behaviors.", "Figure: Illustration of feature based inverted index." ], [ "Prediction Module", "For the prediction module, we use an attention-based deep neural network to model the importance of different user behaviors to the final prediction.", "As shown in Figure REF , the comprehensive user representation $\\mathbf {r^u}$ is calculated by weighted sum pooling as $ \\mathbf {r}^u = \\sum _{i=1}^S \\alpha _i \\cdot \\mathbf {b}^u_i, \\mathbf {b}^u_i \\in B^u$ where $\\mathbf {b}^u_i = [\\mathbf {u}, \\mathbf {v}_i, \\mathbf {c}_i]$ and $\\alpha _i$ represents the contribution of $b^u_i$ to the comprehensive user representation.", "The attention weight $\\alpha _i$ is calculated as $\\alpha _i = \\frac{\\exp (w_i)}{\\sum _{j=1}^S \\exp (w_j)},$ where $w_i$ is defined as, $w_i = \\text{Att}(\\mathbf {b}^u_i, \\mathbf {t}),$ and prediction target $\\mathbf {t} = [\\mathbf {u}, \\mathbf {v}, \\mathbf {c}]$ .", "$\\text{Att}$ is a multi-layer deep network with ReLU activation function.", "Then the final prediction is calculated as $\\hat{y}_{uv}= f_{\\phi }\\left(B^u, \\mathbf {u}, \\mathbf {v}, \\mathbf {c}\\right) = f_{\\phi }\\left(\\mathbf {r^u}, \\mathbf {u}, \\mathbf {v}, \\mathbf {c}\\right)$ where $f$ is implemented as a three-layer perceptron, whose widths are 200, 80 and 1 respectively.", "$\\phi $ is the parameters.", "The output layer uses sigmoid function and other layers use ReLU as activation.", "Figure: Illustration of attention-based prediction network of the prediction module." ], [ "Model Training", "As the objective is to estimate the click-through rate accurately, we utilize the log-likelihood as our objective function which is defined as, $J^{\\pi _{\\theta }, f_{\\phi }} = \\max _{\\theta } \\max _{\\phi } \\sum _u \\sum _v E_{B^u \\sim \\pi _{\\theta }(B^u|q)} [LL(y_{uv}, f_{\\phi }(B^u, u, v, c))],$ where $LL$ is the log-likelihood of predicted score on target user-item pair $u$ , $i$ given retrieved user behaviors $B^u$ under context $c$ .", "It is defined as, $ \\begin{aligned}LL(y_{uv}, f_{\\phi }(B^u, u, v, c)) & = y_{uv} \\cdot \\log (f_{\\phi }(B^u, u, v, c)) \\\\& + (1-y_{uv}) \\cdot \\log (1-f_{\\phi }(B^u, u, v, c)).\\end{aligned}$ In Eq.", "REF , $q$ is the query which consists of the selected features as described in Section REF .", "Sampling is used to select features and then they form the query string.", "After the query string is formed, searching results, a.k.a, retrieved behaviors are deterministic.", "Thus we could regard the behaviors are sampled from a probability distribution $\\pi _{\\theta }(B^u|q)$ ." ], [ "Optimize the Prediction Module", "To optimize the attention-based prediction network $f_{\\phi }$ , we could have that, $\\begin{aligned}\\phi ^* &= \\text{arg}\\min _{\\phi } (L_{ce} + \\lambda L_r) \\\\& = \\text{arg}\\min _{\\phi } \\sum _u \\sum _v E_{B^u \\sim \\pi _{\\theta }(B^u|q)} [-LL(y_{uv}, f_{\\phi }(B^u, u, v, c))] \\\\& + \\frac{1}{2} \\lambda \\left( \\Vert \\mathbf {\\Phi } \\Vert _2^2 \\right) ~,\\end{aligned}$ where $L_{ce}$ is cross entropy loss and $L_r$ is regularization term.", "When we are optimizing the prediction network, the retrieval module remains unchanged, so if the function $f$ is differential with respect to $\\phi $ , the above optimization can be solved by typical stochastic gradient descent algorithm." ], [ "Optimize the Retrieval Module", "For the retrieval module, only the feature selection model needs to be optimized.", "As a sampling process is evolved, we could not directly use SGD to optimize it.", "Instead, we use the REINFORCE [38], [42] algorithm to deal with the discrete optimization.", "Specifically, while keeping the prediction network $f_{\\phi }$ fixed, the feature selection model is optimized via performing its maximization: $\\theta ^* = \\text{arg}\\max _{\\theta } \\sum _u \\sum _v E_{B^u \\sim \\pi _{\\theta }(B^u|q)} [LL(y_{uv}, f_{\\phi }(B^u, u, v, c))],$ we denote $LL(y_i, f_{\\phi }(B^K_i, u_i, i_i, c_i))$ as $LL(\\cdot )$ because it doesn't have parameter $\\theta $ .", "For each query $q$ , we denote objective function as $J^{q}$ which is $J^{q} = E_{B^u \\sim \\pi _{\\theta }(B^u|q)} [LL(\\cdot )].$ We regard the retrieval module $\\pi _{\\theta }(B^u|q)$ as a policy and use the likelihood ratio to estimate its gradients as, $\\begin{aligned}\\nabla _{\\theta }(J^{q}) &= \\nabla _{\\theta } E_{B^u \\sim \\pi _{\\theta }(B^u|q)} [LL(\\cdot )] \\\\&= \\sum _{B^u_i \\in \\mathcal {B}} \\nabla _{\\theta } \\pi _{\\theta }(B^u_i|q) [LL(\\cdot )] \\\\&= \\sum _{B^u_i \\in \\mathcal {B}} \\pi _{\\theta }(B^u_i|q) \\nabla _{\\theta } \\log (\\pi _{\\theta }(B^u_i|q)) [LL(\\cdot )]\\\\&= E_{B^u \\sim \\pi _{\\theta }(B^u|q)} \\nabla _{\\theta } \\log (\\pi _{\\theta }(B^u|q)) [LL(\\cdot )]\\\\& \\simeq \\frac{1}{L} \\sum _{l=1}^L \\nabla _{\\theta } \\log (\\pi _{\\theta }(B^u_l|q)) [LL(\\cdot )],\\end{aligned}$ which is an unbiased estimation on the gradients of Eq.", "REF .", "As the uncertainty of the entire retrieval module actually comes from the feature selection model, we could derive that, $\\pi _{\\theta }(B^u|q) = \\prod _{j=1}^n p(f_j),$ where $f_j \\in \\lbrace f_1,..,f_n\\rbrace $ and $p(f_j)$ is the sampling probability obtained in Eq.", "REF .", "So in Eq.", "REF the gradient of $\\theta $ can be further derive as $\\nabla _{\\theta }(J^{q}) \\simeq \\frac{1}{L} \\sum _{l=1}^L \\sum _{j=1}^n \\nabla _{\\theta } \\log (p(f^l_j)) [LL(\\cdot )],$ where $f^l_j$ is $j$ -th feature of $l$ -th query.", "To train the model with a reward that has a better scale, we replace $LL(\\cdot )$ with Relative Information Gain (RIG) as the reward function here.", "RIG is defined as $RIG = 1 - NE$ where NE is the normalized entropy [10].", "NE is calculated as, $NE = \\frac{LL(\\cdot )}{p \\log (p)+(1-p) \\log (1-p)}$ where $p$ is the average experienced CTR." ], [ "Pseudo Code of Training Process.", "In this subsection, we give a detailed pseudo code of the training process.", "First we pre-train the prediction network with the initial feature selection model.", "After the pre-train, the two models are optimized in turn.", "Algorithm REF shows the training process.", "[!h] Training the UBR4CTR framework [1] Dataset $\\mathcal {D} = (\\mathcal {U}_{target}, \\mathcal {V}_{target}, \\mathcal {C}_{target})$ containing all the target user-item-context triples; User history archive $H_u$ .", "final CTR prediction $\\hat{Y}$ between all the target user $u$ and target item $v$ .", "Initialize all parameters.", "Select the features and form the queries $\\mathcal {Q} = \\lbrace q,...\\rbrace $ for each prediction target $[u, v, c] \\in \\mathcal {D}$ using the initialized feature selection model.", "Obtain the retrieved behaviors $\\mathcal {B}=\\lbrace B^u,...\\rbrace $ of the queries $\\mathcal {Q}$ using the search engine as described in Section REF .", "Train the attention-based prediction network using Eq.", "REF for one epoch.", "Train retrieval model using Eq.", "REF for one epoch.", "Select the features and form the queries $\\mathcal {Q} = \\lbrace q,...\\rbrace $ for each prediction target $[u, v, c] \\in \\mathcal {D}$ using the feature selection model.", "Obtain the retrieved behaviors $\\mathcal {B}=\\lbrace B^u,...\\rbrace $ of the queries $\\mathcal {Q}$ using the search engine as described in Section REF .", "Train attention-based prediction network using Eq.", "REF for one epoch.", "convergence" ], [ "Model Analysis", "In this section, we analyze the time complexity and feasibility of our method.", "We use $\\mathcal {N}$ to denote the total number of all users' logged behaviors and use $F$ to denote the total number of unique features (equivalent to term) that have ever appeared in the whole dataset.", "Then the average length of the posting lists in the user history archive is $\\frac{\\mathcal {N}}{F}$ .", "Recall the searching operation described in Section REF and Eq.", "REF , we first retrieve all the postings of features in $q$ which takes $O(1)$ time.", "Then the interaction operation takes $O(T + K_q \\cdot \\frac{\\mathcal {N}}{F})$ time where $T$ is the average length of a user sequence and $K_q = K_u + K_v + K_c - 1$ is the upper bound of the number of selected features.", "The next scoring operations does not increase the complexity because it is linear to $O(T + K_q \\cdot \\frac{\\mathcal {N}}{F})$ .", "The complexity of the self-attention in feature selection model is $O(K_q^2)$ .", "The attention-based prediction network takes $O(C)$ time where $C$ is the cost of computing one Att operation in Eq.", "REF because all the attention operations can be paralleled.", "These two constants can be ignored and the total time complexity of UBR4CTR is $O(T + K_q \\cdot \\frac{\\mathcal {N}}{F})$ ." ], [ "Experiments", "In this section, we present our experimental settings and corresponding results in detail.", "We compare our model with several strong baselines and achieve the state-of-the-art performance.", "Furthermore, we have published our code for reproductionhttps://github.com/qinjr/UBR4CTR.", "We start with three research questions (RQ) and use them to lead the following discussions.", "RQ1 Does UBR4CTR achieves the best performance compared to other baselines?", "RQ2 What is the convergence performance of Algorithm REF ?", "Is the training process effective and stable?", "RQ3 What is the influence of the retrieval module in UBR4CTR and how the retrieval size affect the performance?" ], [ "Datasets", "We use three real-world and large-scale datasets of users online behaviors from three different platforms of Alibaba Group.", "The statistics of the datasets can be found in Table REF .", "[leftmargin=15pt] Tmall https://tianchi.aliyun.com/dataset/dataDetail?dataId=42 is provided by Alibaba Group which contains user behavior history on Tmall e-commerce platform from May 2015 to November 2015.", "Taobao [46] is a dataset consisting of user behavior data retrieved from Taobaohttps://tianchi.aliyun.com/dataset/dataDetail?dataId=649, one of the biggest e-commerce platforms in China.", "It contains user behaviors from November 25 to December 3, 2017, with several behavior types including click, purchase, add to cart and item favoring.", "Alipay https://tianchi.aliyun.com/dataset/dataDetail?dataId=53 is collected by Alipay which is an online payment application.", "The users online shopping behaviors are from July 1, 2015 to November 30, 2015.", "Table: The dataset statistics.Dataset Preprocessing.", "For UBR4CTR, the datasets are processed into the format of comma separated features.", "A line containing user, item and context features is treated as a behavior document.", "For baselines, the user behaviors are simply sorted by timestamp.", "As the datasets don't contain specific context features, we manually design some features using behavior timestamp to make it possible to capture periodicity.", "We design features such as season id (spring, summer, etc), weekend or not, and which half of the month it is.", "Search Engine.", "After the datasets are preprocessed, they are inserted into a search engine using a comma separated tokenizer.", "We use Elastic Search https://www.elastic.co as the search engine client which is based on Apache Lucenehttp://lucene.apache.org.", "Train & Test Splitting.", "We split the datasets using the time step.", "The training dataset contains the 1st to $(T-2)$ th user behaviors, in which the 1st to $(T-3)$ th behaviors are used to predict the behavior at $(T-2)$ th step.", "Similarly, the validation set uses 1st to $(T-2)$ th behaviors to predict $(T-1)$ th behavior and the test set uses 1st to $(T-1)$ th behaviors to predict $T$ th behavior.", "Hyperparameters.", "The learning rate of feature selection model of UBR4CTR is searched from $\\lbrace 1 \\times 10^{-6}, 1 \\times 10^{-5}, 1 \\times 10^{-4}\\rbrace $ , learning rate for attention based prediction network is from $\\lbrace 1 \\times 10^{-4}, 5 \\times 10^{-4}, 1 \\times 10^{-3}\\rbrace $ and the regularization term is from $\\lbrace 1 \\times 10^{-4}, 5 \\times 10^{-4}, 1 \\times 10^{-3}\\rbrace $ .", "The search space of learning rate and regularization term for baseline models are the same with prediction network in UBR4CTR.", "Batch size is from $\\lbrace 100, 200\\rbrace $ for all models.", "The hyperparameters of each model are tuned and the best performances have been reported in Section REF ." ], [ "Evaluation Metrics", "We evaluate the CTR prediction performance with two widely used metrics.", "The first one is area under ROC curve (AUC) which reflects the pairwise ranking performance between click and non-click samples.", "The other metric is log loss.", "Log loss is to measure the overall likelihood of the test data and has been widely used for the classification tasks [25], [24]." ], [ "Compared Baselines", "We compare our framework and models with seven different strong baselines from both sequential CTR prediction and recommendation scenarios.", "[leftmargin=40pt] GRU4Rec [12] is based on GRU and it is the first work using the recurrent cells to model sequential user behaviors for session-based recommendation.", "Caser [31] is a CNNs-based model that regards the user sequence as an image thus uses horizontal and vertical convolutional layers to capture temporal patterns of user behaviors.", "SASRec [15] uses Transformer [32].", "It regards the user behaviors as a sequence of tokens and uses self-attention mechanism and position embedding to capture the dependencies and relations between behaviors.", "HPMN [23] is a hierarchical periodic memory network that is proposed to handle very long user historical sequence.", "Moreover, the user memory state can be updated incrementally.", "MIMN [19] is based on Neural Turing Machine [6] which models multiple channels of user interests drifting.", "The model is implemented as a part of user interest center [19] which could model very long user behavior sequences.", "DIN [45] is the first model that uses attention mechanism in CTR prediction of online advertising.", "DIEN [44] uses two-layer RNNs with attention mechanism to capture evolving user interests.", "It uses the calculated attention values to control the second recurrent layer, which is called AUGRU.", "UBR4CTR is our proposed framework and models described in Section ." ], [ "Performance Comparison: RQ1", "We conduct two groups of comparisons between our UBR4CTR and baseline models.", "In the first group of experiments, all of the models are using the same amount of user behaviors which are 20, 20, 12 for the three datasets respectively.", "The only difference is that baselines are using the most recent behaviors (about 20% of the total length) and UBR4CTR retrieve 20% of behaviors from the whole sequence.", "The experimental results are shown in Table REF .", "From the table, we could find the following facts.", "(i) The performance is improved significantly compared to the baselines.", "AUC are improved by 4.1%, 10.9% and 22.3% on three datasets respectively, and log-loss are improved by 9.0%, 12.0% and 32.3% respectively.", "(ii) The vast improvement is a demonstration that the most recent behaviors do not embed enough temporal patterns so the baselines can not capture them effectively.", "Although some of the baselines are pretty complex and fancy, they cannot perform well if the patterns that they try to capture are not contained in the recent behavior sequence in the first place.", "Table: The first group of results of CTR prediction in terms of AUC (the higher, the better) and log-loss (LL, the lower, the better).", "Bold values are the best in each column, while the second best values are underlined.", "Improvements are against the second best results.Table: The second group of results of CTR prediction in terms of AUC (the higher, the better) and log-loss (LL, the lower, the better).", "Improvements are against the second best results.In the second group of experiment, we use different settings for the baselines and exactly the same settings for UBR4CTR as the first group of experiment.", "We feed the full-length sequences to all the baseline models which are 100, 100 and 60 respectively on three datasets.", "They are the maximum lengths of user behaviors in these datasets.", "And the sizes of the retrieved behaviors remain the same for UBR4CTR which are 20, 20 and 12.", "The results are shown in Table REF .", "In Table REF , the performance of UBR4CTR is the same as that in Table REF because we don't change any settings.", "From the table, we could find the following facts.", "(i) UBR4CTR still has the best performance even though it uses $80\\%$ less behaviors than other baselines.", "This shows that longer sequence may has more noise and irrelevant information thus it is necessary to obtain only the most useful data out of the whole sequence.", "(ii) Most of the baselines achieve better performance than themselves compared to Table REF , especially DIN and DIEN.", "This shows that behaviors from further history do contain richer information and patterns.", "And these patterns are easier to be captured using attention mechanism.", "(iii) Although the improvement of AUC is much smaller due to the better performance of baselines, log-loss still improves significantly.", "The reason is that the optimization objective of the retrieval module is RIG (equivalent to log-loss) after all." ], [ "Learning Process: RQ2", "To illustrate the convergence of our framework, we plot the learning curves of UBR4CTR.", "In Figure REF , the upper three subplots are the AUC curves of the attentive prediction network when training on three datasets, respectively.", "Each step of the x-axis is corresponding to the iteration over $4\\%$ of the training set.", "Figure: Learning curves of UBR4CTR.The lower three sub-figures show the \"reward\" of the REINFORCE algorithm w.r.t the feature selection model of the retrieval module.", "The \"reward\" is essentially RIG which is a variant of log-likelihood.", "Each step of the x-axis means the iteration over $4\\%$ of the training set.", "The rewards increase during the training process implying that the feature selection model actually learns useful patterns.", "From the AUC figures, we could find that our models converge effectively.", "For the prediction network, we can observe that there are flat areas in the middle of training process followed by a rapid increase, especially in the second and third AUC plots.", "Recall our training procedure described in Algorithm REF , the retrieval module is trained after the prediction network.", "It means that when the prediction network is about to converge, the retrieval module begins training, and after that, there will be a performance break though for the prediction network." ], [ "Extensive Study: RQ3", "In this section, we conduct some extensive and ablation studies on our framework.", "Figure REF illustrates the influence of different retrieval sizes on the prediction performance.", "From the figure, we can find that the fluctuation of AUC and log-loss is not very severe in terms of the absolute values.", "However, there exists an optimal size for each dataset.", "This reveals that smaller sizes may not contain enough behaviors and information, while too much retrieved behaviors are not always suitable for performance either.", "Because it will introduce too much noise.", "Figure: Influence of different retrieval size.To illustrate the importance of the retrieval module of our framework, we plot the performance comparisons between the sum pooling model and attention network with&without user behaviors retrieval.", "Sum pooling model just uses a very simple sum pooling operation on user behaviors which means the $\\alpha _i=1$ in Eq.", "REF .", "The results are shown in Figure REF .", "From the figure, we find that the sum pooling model without retrieving (SP) performs very poorly.", "Once equipped with the retrieval module, the performance of it increases significantly (UBR_SP).", "The same phenomenon applies for attention network which performance is largely improved when equipped with behavior retrieval module (ATT vs. UBR4CTR).", "Figure: Ablation study on the influence of retrieval module.", "Note: AUC, the higher, the better; log-loss, the lower, the better" ], [ "Deployment Feasibility", "The deployment of UBR4CTR has been made on the engineering schedule of a daily item recommendation platform under a mainstream bank company.", "In this section, we mainly discuss the feasibility of industrial deployment of UBR4CTR framework.", "First, it is not difficult to switch the current model pipeline to UBR4CTR because the main change brought from UBR4CTR is the how the historical user behaviors is obtained.", "To update the model pipeline to UBR4CTR, a search engine of historical user behaviors needs to be built, while the whole CTR prediction model pipeline remain almost the same but adding an additional retrieval module.", "As Figure REF shows, the prediction module is not different from that of the traditional solutions.", "Efficiency is another essential concerns in industrial applications.", "We analyze the time complexity of UBR4CTR which is $O(T + K_q \\cdot \\frac{\\mathcal {N}}{F})$ in Section REF .", "For most of the sequential CTR models which are normally based on RNN, time complexity of them is $O(C \\cdot T)$ where $T$ is the average length of user sequences and $C$ is the cost of one operation (e.g.", "GRU rolling).", "The time complexity of UBR4CTR is not totally infeasible because the term $\\frac{\\mathcal {N}}{F}$ is very closed to a constant as $F$ is a big number and it will slow the increase of this term.", "From the perspective of system load, UBR4CTR is better because it does not require maintaining all the $T$ behaviors in memory which is a common practice for the traditional methods.", "Moreover, we compare the actual inference time between our UBR4CTR and other sequential CTR baselines in experiments.", "The average inference time of the models are illustrated in Figure REF .", "The time is calculated by dividing the overall time (only the time that contains the forward computations and behavior searching) on test dataset with the number of prediction targets.", "From the figure, we could find that the absolute value of UBR4CTR's inference time on three datasets is less than 1ms which is efficient enough for online serving [34].", "The inference time of UBR4CTR is the longest among all the sequential CTR models but the gap is not that large.", "Particularly, compared with DIEN which has been deployed in Alibaba online advertising platform [44], the average inference time of UBR4CTR is about 15% to 30% longer, which can be optimized via further infrastructure implementation.", "Figure: Inference time of different models." ], [ "User Response Prediction", "For user response prediction, there are two main streams of methods.", "One stream is about modeling the interactions of multiple categorical features.", "The key point of these models is to design structure that could capture the cross feature interactions.", "Factorization machine [27] is the pioneer model which uses matrix factorization in CTR prediction task and a lot of variants of FM [14], [41], [30] are proposed in recent years.", "Except for the early feature interaction models, deep neural networks are also used in CTR prediction.", "Wide&Deep [4] is the first deep learning model which effectively transform the high dimensional discrete features into dense representation and achieve good prediction performance.", "After Wide&Deep and FM, more and more models which combine the feature interaction structure and deep neural networks are proposed.", "DeepFM [7] uses both FM and DNN to improve the CTR prediction performance.", "DeepCross [36] and PNN [21] automatically models the cross feature interactions by outer product and inner product respectively.", "There are similar models such as xDeepFM [17], FNN [43] etc.", "The other stream of models focuses more on mining temporal patterns from sequential user behaviors.", "DIN [45] is an attention-based network which attributes different weights on different items that user has interacted with.", "DIEN [44] utilizes two layers of GRU and attention mechanism to capture evolving user interests.", "HPMN [23] is a memory network-based method which model very long sequences for user response prediction.", "MIMN [19] is based on neural Turing machine [6] that models the multiple channels of user evolving interests." ], [ "Sequential User Modeling", "Sequential user modeling is about capture user's drifting dynamics of behaviors.", "It is a research hotspot for recommender systems recently.", "Multiple types of model are proposed.", "The first is temporal collaborative filtering [16] which considers the drifting user preferences.", "The second type is based on Markov chains [8], [9], [28] which models the user state dynamics in an implicitly way and drive the future behaviors.", "The third category is based on deep learning.", "There are RNN-based models [12], [11], [39], [13], [18], [2], [33] that regard user behaviors as sequence of tokens, CNN-based models [31], [15] which regard the behaviors as an image and Transformer-based models [15].", "Furthermore, there are models [40] that not only utilize user-side sequence but also item-side sequence.", "[20] propose dual side neighbor based CF for sequential user modeling.", "Memory networks are also used for sequential user modeling [5], [35], [3], [19] which aim to memorize longer sequence of user behaviors." ], [ "Conclusion And Future Work", "In this paper, we propose the UBR4CTR framework of user response prediction.", "The retrieval module of our framework generates a query to search from the whole user behaviors archive to retrieve the most useful behavioral data for prediction.", "The retrieved data is then used by an attention-based deep network to make the final prediction.", "Our framework overcomes the practical problems of traditional framework which simply uses most recent behaviors and significantly improves the CTR prediction performance.", "The deployment of UBR4CTR has been made on the engineering schedule of a daily item recommender system of a mainstream bank company.", "For the future work of research, we will put more efforts on distributed training algorithm which will make the framework more efficient to train in a mini-batch manner.", "Furthermore, we will explore new effective indexing and retrieval methods for storing and searching the user behavioral archive.", "Acknowledgement.", "The corresponding author Weinan Zhang thanks the support of National Natural Science Foundation of China (61702327, 61772333, 61632017)." ] ]

[ [ "Neutral and charged dark excitons in monolayer WS$_2$" ], [ "Abstract Low temperature and polarization resolved magneto-photoluminescence experiments are used to investigate the properties of dark excitons and dark trions in a monolayer of WS$_2$ encapsulated in hexagonal BN (hBN).", "We find that this system is an $n$-type doped semiconductor and that dark trions dominate the emission spectrum.", "In line with previous studies on WSe$_2$, we identify the Coulomb exchange interaction coupled neutral dark and grey excitons through their polarization properties, while an analogous effect is not observed for dark trions.", "Applying the magnetic field in both perpendicular and parallel configurations with respect to the monolayer plane, we determine the g-factor of dark trions to be $g\\sim$-8.6.", "Their decay rate is close to 0.5 ns, more than 2 orders of magnitude longer than that of bright excitons." ], [ "Introduction ", "Monolayers (MLs) of semiconducting transition metal dichalcogenides (S-TMDs) MX$_2$ where M=Mo or W and X=S, Se or Te, are direct band gap semiconductors with the minima (maxima) of conduction (valence) band located at the inequivalent K$^+$ and K$^-$ points of their hexagonal Brillouin zone (BZ) [1], [2].", "The strong spin-orbit interaction in the crystal lifts the spin-degeneracy of the bands, which leads, in particular, to the splitting of the valence ($\\Delta _v$ ) and the conduction ($\\Delta _c$ ) bands.", "While the former splitting is of the order of few hundreds of meV, the latter equals few tens of meV only.", "Moreover, the $\\Delta _c$ splitting can be positive or negative.", "As a result two subgroups of MLs can be distinguished: $bright$ (the excitonic ground state is optically active or bright) and $darkish$ (the excitonic ground state is optically inactive or dark).", "It is well established that WS$_2$ and WSe$_2$ MLs belong to the group of darkish materials [3], [4], while MLs of MoSe$_2$ and MoTe$_2$ belong to the family of bright materials [3], [1], [5].", "The assignment of the MoS$_2$ ML is still under debate as theoretical predictions and experimental results are contradictory [6], [3], [5].", "Figure: Schematic illustration of possible configurations for the neutral (blue shade) and the negatively charged dark (green shade) excitons located at the K + ^+ and K - ^- valleys, respectively.", "The grey (orange) curves indicate the spin-up (spin-down) subbands.", "The electrons (holes) in the conduction (valence) band are represented by blue (white) circles.The complex electronic structure of S-TMD MLs results in a wide variety of excitonic complexes, which can be formed from carriers at the vicinity of the conduction (CB) and valence band (VB) extrema.", "Their studies have been largely facilitated by the efforts to improve their optical quality, in particular by encapsultaing MLs of S-TMDs in thin layers of hexagonal BN (hBN).", "In particular, this improvement allowed to investigate several dark excitonic complexes [7], [8], [9], [10], [11], [12], [13], [14], [15], [11], [16], [17], [18], [5].", "Among them there are spin- or momentum-forbidden dark excitons, which can not recombine optically due to spin or momentum conservation laws.", "Moreover, these complexes can be neutral and charged.", "Fig.", "REF illustrates schematically the neutral and negatively charged spin-forbidden dark excitons in darkish monolayer ($i.e.$ WS$_2$ or WSe$_2$ ).", "The neutral dark exciton at K$^+$ valley is composed of an electron from the lowest-lying level of CB and a hole from the highest-lying level of VB.", "The negative trion at the K$^-$ point is formed similarly by the energetically lowest electron-hole ($e−h$ ) pair and an extra electron located in the K$^+$ valley.", "It was proposed theoretically and demonstrated experimentally that the spin-forbidden neutral dark excitons exhibit a double (fine) structure comprising so-called grey (X$^\\textrm {G}$ ) and dark (X$^\\textrm {D}$ ) complexes.", "These complexes are characterized by the out-of-plane and zero excitonic dipole momenta [19], [7], [12], respectively.", "Although substantial efforts have been made to study dark excitons in WSe$_2$ MLs [20], [7], [21], [12], [13], [14], [15], [11], [16], [17], [18], their properties in WS$_2$ are still far from complete understanding [20], [11], [17].", "For example, the reported energy difference between the bright and dark excitonic emission is not well determined as it varies significantly (55 meV [20] vs. 46 meV [11]).", "This motivates our addressing properties of dark states in WS$_2$ ML.", "In this work, we use polarization resolved photoluminescence spectroscopy with an applied magnetic field to investigate dark excitonic complexes through the magnetic brightening effect in a high quality and naturally $n$ -doped WS$_2$ ML encapsulated in thin hBN layers.", "The magnetic field has been applied in different geometries: perpendicular, parallel or at 45$^\\circ $ with respect to the monolayer plane.", "The emissions from both the neutral and the charged exciton complexes are activated by the in-plane magnetic field component.", "The double structure of the neutral dark exciton (grey and dark excitons) is observed while the dark trions do not present any fine structure.", "Our study also shows that the magnetic brightening of the dark excitons and dark trions depends on the carrier concentration." ], [ "Results ", "Fig.", "REF illustrates the brightening of neutral and charged dark excitons in a monolayer of WS$_2$ encapsulated in hBN by an in-plane magnetic field.", "The zero-field PL spectrum is composed of several emission lines.", "Based on the previous reports, [22], [23], [24], [11], [25] three peaks can be ascribed unquestionably to a bright exciton (X$^\\textrm {B}$ ) and singlet (T$^\\textrm {S}$ ) and triplet (T$^\\textrm {T}$ ) states of negative trions.", "The application of an in-plane magnetic field $B_{||}$ results in the appearance of three additional lines, labelled X$^\\textrm {G}$ , X$^\\textrm {D}$ and T$^\\textrm {D}$ .", "This can be appreciated in Fig.", "REF , which displays the spectrum measured at $B_{||}$ =10 T. The X$^\\textrm {G}$ and X$^\\textrm {D}$ peaks correspond to the dark and grey states of the neutral exciton, while the T$^\\textrm {D}$ peak is related to a dark state of the negative trion.", "Moreover, the detailed analysis of the X$^\\textrm {D/G}$ line, shown in the inset to Fig.", "REF (a), exhibits its fine structure, in line with previous studies on WSe$_2$ MLs [26], [12].", "These results can be summarized as follows: (i) the X$^\\textrm {D/G}$ line is red-shifted by 40 meV from the X$^\\textrm {B}$ peak; (ii) the energy separation between the X$^\\textrm {G}$ and X$^\\textrm {D}$ emissions $\\delta $ =530 $\\mu $ eV; (iii) the T$^\\textrm {D}$ resonance is red-shifted by 57 meV from the X$^\\textrm {B}$ one.", "Surprisingly, the energy separation between the negatively charged/neutral dark and bright neutral exciton in the studied WS$_2$ ML, which amounts to 57 meV/40 meV, are very similar to the corresponding values observed in a WSe$_2$ ML encapsulated in the same dielectric environment ($\\sim $ 57 meV/$\\sim $ 40 meV) [20], [7], [21], [12], [13], [14], [15], [11], [16], [18].", "The obtained value for the fine structure splitting of the neutral dark-grey exciton of about 530 $\\mu $ eV in the studied WS$_2$ ML is also very similar to that observed in WSe$_2$ (660 $\\mu $ eV) [7], [12].", "Figure: The effect of the in-plane magnetic field B || B_{||} on the integrated relative intensities of the charged and neutral dark excitons measured in magnetic fields up to (a) 10 T and (b) 30 T. The spectra were detected in (a) two linear polarizations for X G ^\\textrm {G} and X D ^\\textrm {D} emissions (see Fig.", "(a) for details), and otherwise detection was unpolarized.", "Note that the X D /G^\\textrm {D/G} intensities in both panels were multiplied by factor 5 for clarity.", "The solid black curves represent quadratic fits.Having identified the dark excitons, we now investigate these complexes at higher magnetic fields (up to 30 T) but with unpolarized optical detection.", "In fact, as can be seen in Fig.", "REF (b), the application of the in-plane magnetic field $B_{||}$ up to 30 T leads to a strong brightening effect, which is significantly different for the neutral and charged dark excitons.", "With increasing $B_{||}$ , the intensity of the dark trion increases substantially, while the intensity of the bright exciton and trions stays practically unchanged.", "As can be seen in Fig.", "REF (b), the emission of the neutral grey-dark exciton doublet is much smaller as compared to dark trion even at highest magnetic fields ($B$ =30 T).", "The $B_{||}$ evolution of the intensity of dark excitons is expected to be quadratic $I=\\alpha B_{||}^2$   [3], [4], [12].", "Note that we neglect the zero-field intensity of the grey components of the neutral and charged complexes, as we were not able to resolve it at zero magnetic field.", "Fig.", "REF demonstrates the $B_{||}^2$ dependence of the relative intensities of the charged and neutral dark excitons in magnetic fields up to 10 T and 30 T, which are accompanied with the quadratic fits.", "The integrated relative intensities of the T$^\\textrm {D}$ , X$^\\textrm {G}$ and X$^\\textrm {D}$ lines were obtained by fitting the relative spectra defined as (PL$_{B\\ne \\textrm {0 T}}$ - PL$_{B\\textrm {=0 T}}$ )/$I_{\\textrm {X}^\\textrm {B}}(B)$ using Lorentzian functions.", "The PL$_{B\\ne \\textrm {0 T}}$ and PL$_{B\\textrm {=0 T}}$ are correspondingly photoluminescence spectra measured at non-zero and zero magnetic fields, while $I_{\\textrm {X}^\\textrm {B}}(B)$ represents the integrated intensity of the bright neutral exciton as a function of magnetic fields.", "Note that the division by the $I_{\\textrm {X}^\\textrm {B}}(B)$ parameter allows us to eliminate variation of the signal intensity during measurements, $e.g.$ the measured signal in 30 T setup is affected by the Faraday effect.", "The fitted $\\alpha $ parameters for three analysed excitons observed in both experimental setups are coherent within experimental error.", "However, as it is shown in Figs REF (a) and REF (b), there is a well pronounced difference between the fitted $\\alpha $ parameters for three analyzed excitons, beyond experimental error.", "The difference between the two neutral dark components: $\\alpha _{\\textrm {X}^\\textrm {D}}$ =0.23/T$^2$ versus $\\alpha _{\\textrm {X}^\\textrm {G}}$ =0.15/T$^2$ , can be explained by the thermal occupation of these two states at $T$ =10 K corresponding to our experimental conditions.", "The calculated population ratio is $e^{{-\\delta }/{k_B T}}$ =0.54, which is in reasonable agreement with the measured ratio $\\alpha _{\\textrm {X}^\\textrm {G}} / \\alpha _{\\textrm {X}^\\textrm {D}}$ =0.65.", "The brightening rate of the dark trion with respect to dark excitons is much faster with $\\alpha _{\\textrm {T}^\\textrm {D}}$$\\sim 24$$\\alpha _{\\textrm {X}^\\textrm {G}}$ .", "This large difference may be understood considering the finite free electron concentration of 10$^{11}$  cm$^{-2}$ in the WS$_2$ ML [27].", "The formation of trions and of dark trions is then favored compared to neutral excitons and this is reflected in the much more pronounced brightening effect for T$^\\textrm {D}$ than for X$^\\textrm {D/G}$ .", "Note that in our previous studies devoted to the dark excitonic complexes in the WSe$_2$ ML [12], we observed the opposite situation, $i.e.$ the brightening of the X$^\\textrm {D/G}$ line was significantly larger as compared to the one of the charged complex.", "This may be understood if the doping level of the WSe$_2$ ML was close to the neutrality point, which favours the creation of the neutral dark excitons.", "The obtained results highlight a role of doping in MLs on the brightening of the neutral and charged dark exciton emission in the in-plane magnetic field.", "Figure: (a) False-color map of the charged (left panel) and neutral (right panel) dark exciton emissions as a function of the detection linear polarization angle measured at B || B_{||}=10 T. (b) The integrated intensities of the luminescence peaks measured at B || B_{||}=10 T as a function of the polarization.", "Solid black curves represent sine fits.The polarization properties of magnetically brightened neutral and charged dark excitons were also analysed (see Fig.", "REF (a)).", "The integrated PL intensity for each peak measured at $B_{||}$ =10 T as a function of the linear polarization is shown in Fig.", "REF (b).", "The detection angle is measured between the polarizer axis and the direction of the $B_{||}$ field.", "The X$^\\textrm {G}$ and X$^\\textrm {D}$ intensities are expected to follow the $I(\\phi _d)\\sim I_0 \\sin ^2(\\phi _d + \\theta )$ dependence, where $\\phi _d$ represents the detection angle, while $I_0$ and $\\theta $ are fitting parameters.", "Our measurements show that the neutral dark (grey) exciton at $B_{||}$ =10 T is linearly polarized along (perpendicularly to) the direction of the in-plane magnetic field, in line with our previous study of magnetic brightening in a monolayer of WSe$_2$  [12].", "In contrast to dark excitons, the emission from dark trions do not show sizable linear polarization.", "Figure: a) False-color map of the PL response as a function of out-of-plane component (B ⊥ B_\\perp ) of the applied tilted magnetic field.", "Note that the positive and negative values of magnetic fields correspond to σ ± \\sigma ^\\pm polarizations of detection.", "The intensity scale is normalized to subtract the Faraday effect affecting the experimental data.", "White dashed superimposed on the investigated transitions are guides to the eyes.", "(b) Transition energies of the σ +/- \\sigma ^{+/-} (red/blue points) components of the T D ^\\textrm {D} line as a function of the out-of-plane magnetic field.", "The solid lines represent fits according to the equation described in the text.In order to describe the T$^\\textrm {D}$ polarization hallmarks, we analyse the observed sine-square intensity profiles of neutral dark excitons presented in Fig.", "REF (b).", "The evolution results from two effects: i) the coupling between dark and bright excitons of the same valley by an in-plane magnetic field; and ii) the coupling of dark excitons of opposite valleys by short-range exchange interaction.", "The first effect allows to observe dark excitons by transferring part of the optical activity from bright to dark excitons through the mixing of the the two spin states of the conduction band.", "Then, dark excitons from K$^\\pm $ valleys get the ability to decay radiatively by emitting $\\sigma ^\\pm $ polarized photons in the direction, perpendicular to the monolayer's plane.", "The exchange interaction mixes the \"valley\" dark exciton states into new ”grey” and ”dark” states [19].", "The new states are no more degenerated in energy, are split by the exchange interaction [20], and are orthogonal to each other.", "As a result, dark excitons recombine by emitting linearly polarized light with orthogonal polarization.", "The linear polarization results in the sine-square profile of the PL intensity versus the detection angle, while the orthogonality is responsible for the $\\pi /2$ phase shift of corresponding curves.", "The lowest energy dark exciton state is more populated than the grey exciton state and hence emits more photons [12].", "The difference between the corresponding intensities becomes significant at low temperatures due to Boltzmann factor $\\exp (-\\delta /k_BT)$ , as it is observed in our study, see Figs REF and REF .", "Note that in the absence of the exchange interaction both sine-square curves would be characterised by the same intensity profile and energy.", "In such case the valley dark excitonic doublet would remain doubly degenerate and the PL intensity of dark exciton line would become angle-independent, since $\\sin ^2(\\phi _d)+\\sin ^2(\\phi _d+\\pi /2)=1$ .", "In fact, such a situation is realized in the case of negative dark trions.", "They also form the valley doublet, which is brightened by in-plane magnetic field.", "However, since the dark trions from opposite valleys are not influenced by exchange interaction [26], their emission intensity is not polarized, as it was observed in the experiment (see green points in Fig.", "REF (b)).", "In addition, it is expected that dark trions are characterized by an out-of-plane exciton dipole momentum, which leads to the reported in-plane emission at 55 meV below the X$^\\textrm {B}$ in WS$_2$ monolayer [20].", "To determine the dark trion g-factor, it is necessary to apply both an in-plane magnetic field allowing for its direct observation in a photoluminescence experiment and, at the same time, to apply a magnetic field perpendicular to the layer plane to couple to its magnetic moment.", "This is achieved in our experiment by tilting the layer by 45$^\\circ $ with respect to the magnetic field direction.", "Fig.", "REF (a) shows the evolution of the photoluminescence response as a function of the out-of-plane component of the magnetic field ($B_\\perp $ ), in the form of colour-coded map.", "The T$^\\textrm {D}$ emission grows due to the in-plane field and additionally splits into components due to the valley Zeeman effect.", "The extracted emission energies of the negative dark trion are presented in the Fig.", "REF (b).", "The energy evolutions ($E(B)$ ) in external out-of-plane magnetic fields ($B_\\perp $ ) can be described as $E(B)=E_0\\pm \\frac{1}{2} g \\mu _\\mathrm {B} B_\\perp $ , where $E_0$ is the energy of the transition at zero field, $g$ denotes the g-factor of the considered excitonic complex and $\\mu _{B}$ is the Bohr magneton.", "The black solid lines in Fig.", "REF (b) are fits to our experimental data using this equation.", "We find that the T$^\\textrm {D}$ g-factor in the studied WS$_2$ monolayer is on the order of -8.9, which is very similar to the reported g-factors for dark trions in WSe$_2$ MLs [21], [13], [14], [18], about two times bigger than the g-factors of the X$^\\textrm {B}$ , T$^\\textrm {S}$ and T$^\\textrm {T}$ lines (for details see Supplementary Material (SM)).", "Figure: Low-temperature normalized time-resolved PL of the T T ^\\textrm {T}, T S ^\\textrm {S} and T D ^\\textrm {D} lines measured on a WS 2 _2 ML encapsulated in hBN flakes at in-plane magnetic field B || B_{||}=10 T.To get more information on the dark trion, we have performed time-resolved measurements at in-plane magnetic field of $B_{||}$ =10 T. Fig.", "REF displays the normalized time-resolved traces of the bright singlet (T$^\\textrm {S}$ ) and triplet (T$^\\textrm {T}$ ), together with the dark (T$^\\textrm {D}$ ) charged excitons.", "Monoexponential fits of these traces show that the decay time of the two bright trions is of 30 ps, in line with previous studies for monolayers of other S-TMD[13], [21].", "Dark trions are characterized by a much slower decay time, reaching values close to 0.5 ns.", "The obtained decay times of the bright and dark trions are in good agreement with previously reported values for other S-TMD monolayer [21], [13]." ], [ "Methods ", "The studied sample is composed of WS$_2$ ML encapsulated in hBN flakes and supported by a bare Si substrate.", "The structure was obtained by two-stage polydimethylsiloxane (PDMS)-based [28] mechanical exfoliation of WS$_2$ and hBN bulk crystals.", "A bottom layer of hBN in the hBN/WS$_2$ /hBN heterostructure was created in the course of a non-deterministic exfoliation.", "The assembly of the hBN/WS$_2$ /hBN heterostructure was realized via succesive dry transfers of WS$_2$ ML and capping hBN flake from PDMS stamps onto the bottom hBN layer.", "Low-temperature micro-magneto-PL experiments are performed in the Voigt, Faraday and tilted geometries, $i.e.$ magnetic field oriented parallel, perpendicular, and 45$^\\circ $ with respect to ML's plane.", "Measurements (spatial resolution $\\sim $ 2 $\\mu $ m) were carried out with the aid of two systems: a split-coil superconducting magnet and a resistive solenoid producing fields up to 10 T and 30 T using a free-beam-optics arrangement and an optical-fiber-based insert, respectively.", "The sample was placed on top of a $x$ -$y$ -$z$ piezo-stage kept at $T$ =10 K or $T$ =4.2 K and was excited using a laser diode with 532 nm or 515 nm wavelength (2.33 eV or 2.41 eV photon energy).", "The emitted light was dispersed with a 0.5 m long monochromator and detected with a charge coupled device (CCD) camera.", "The linear polarizations of the emissions in the Voigt geometry were analyzed using a set of polarizers and a half wave plate placed directly in front of the spectrometer.", "In the case of the Faraday and tilted-field configurations, the combination of a quarter wave plate and a linear polarizer placed in the insert were used to analyse the circular polarization of signals (the measurements were performed with a fixed circular polarization, whereas reversing the direction of magnetic field yields the information corresponding to the other polarization component due to time-reversal symmetry).", "For time-resolved measurements, a femtosecond pulsed laser with excitation at 570 nm (2.18 eV photon energy) from frequency-doubled Ti:Sapphire Coherent Mira-OPO laser system operated at a repetition rate of 76 MHz and a synchroscan Hamamatsu streak camera were used correspondingly for excitation and detection.", "Note that the excitation power for experiments performed in magnetic fields up to 30 T and with time resolution was adjusted based on the comparison of the measured PL spectrum and the one obtained under excitation of laser with 532 nm in measurement in fields up to 10 T." ], [ "Summary", "We have presented a photoluminescence investigation of dark exciton complexes in a monolayer of WS$_2$ encapsulated in hBN.", "Based on polarization resolved measurements, we have identified both the dark and the grey excitons which are brightened by the in-plane component of the magnetic field.", "The dark trion is also observed in this magneto-brightening experiment.", "In contrast to dark excitons, dark trions do not show any fine structure nor linear polarization due to the absence of coupling between dark trions in the two valleys.", "Time resolved measurements indicate that the decay time of dark trions is close to 0.5 ns, which is more than two orders of magnitude larger than that of bright trions in the same monolayer.", "This study also indicates that monolayer of WS$_2$ encapsulated in hBN is naturally $n$ -doped as revealed by the much higher brightening rate of the dark trion with respect to dark excitons, in contrast to previous results obtained in WSe$_2$ ." ], [ "Acknowledgements", "The work has been supported by the the National Science Centre, Poland (grants no.", "2017/27/B/ST3/00205, 2017/27/N/ST3/01612 and 2018/31/B/ST3/02111), EU Graphene Flagship project (no.", "785219), the ATOMOPTO project (TEAM programme of the Foundation for Polish Science, co-financed by the EU within the ERD-Fund), the Nano fab facility of the Institut Néel, CNRS UGA, and the LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL).", "P. K. kindly acknowledges the National Science Centre, Poland (grant no.", "2016/23/G/ST3/04114) for financial support for his PhD.", "M. B. acknowledges the financial support of the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 (Grant No.", "LQ1601).", "K.W.", "and T.T.", "acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, (grant no.", "JPMXP0112101001), JSPS KAKENHI (grant no.", "JP20H00354), and the CREST (JPMJCR15F3), JST.", "Supplemental Material: Neutral and charged dark excitons in monolayer WS$_2$ M. Zinkiewicz,$^{1}$ A. O. Slobodeniuk,$^{2}$ T. Kazimierczuk,$^{1}$ P. Kapuściński,$^{3,4}$ K. Oreszczuk,$^{1}$ M. Grzeszczyk,$^{1}$ M. Bartos,$^{3,5}$ K. Nogajewski,$^{1}$ K. Watanabe,$^{6}$ T. Taniguchi,$^{7}$ C. Faugeras,$^{3}$ P. Kossacki,$^{1}$ M. Potemski,$^{1,3}$ A. Babiński,$^{1}$ and M. R. Molas $^{1}$ $^{1}$ Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul.", "Pasteura 5, 02-093 Warsaw, Poland $^{2}$ Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 5, Praha 2 CZ-121 16, Czech Republic $^{3}$ Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA-EMFL, 25, avenue des Martyrs, 38042 Grenoble, France $^{4}$ Department of Experimental Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, ul.", "Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland $^{5}$ Central European Institute of Technology, Brno University of Technology, Purkyňova 656/123, 612 00 Brno, Czech Republic National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan $^{6}$ Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan $^{7}$ International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan Figure: False-color map of the PL response as a function of B ⊥ B_\\perp .", "Note that the positive and negative values of magnetic fields correspond to σ ± \\sigma ^\\pm polarizations of detection.", "The intensity scale is logarithmic.", "White dashed lines superimposed on the investigated transitions are guides to the eyes." ], [ "g-factors of bright excitonic complexes", "To get more information on the properties of the bright excitons emissions in the studied WS$_2$ monolayer, we performed the magneto-photoluminescence experiment in magnetic fields up to 10 T oriented perpendicular to ML's plane.", "Fig.", "REF illustrates the measured PL spectra as a function of magnetic fields in the form of colour-coded map.", "Upon application of an out-of-plane magnetic field, the excitonic emissions split into two circularly polarized components due to the excitonic Zeeman effect [29].", "Their energies evolutions ($E(B)$ ) in external out-of-plane magnetic fields ($B_\\perp $ ) can be described as: $E(B)=E_0\\pm \\frac{1}{2} g \\mu _\\mathrm {B} B_\\perp ,$ where $E_0$ is the energy of the transition at zero field, $g$ denotes the g-factor of the considered excitonic complex and $\\mu _{B}$ is the Bohr magneton.", "The results of fitting to the experimental results denoted by red and blue points, are presented in Fig.", "REF as solid black curves.", "We found that the g-factors for the bright exciton (X$^\\textrm {B}$ ) and singlet (T$^\\textrm {S}$ ) and triplet (T$^\\textrm {T}$ ) states of negative trions are of about -3.5, -4.0 and -3.9, respectively.", "The obtained values are in in very close agreement to previous measurements on WS$_2$ monolayer [30], [31], [32], [29], [33].", "Summarizing, the g-factors of the bright excitons, $i.e.$ complexes for which recombining an electron and a hole posses the same sign of the spin, are very close to the theoretically predicted value of 4 [29].", "Figure: Transition energies of the σ +/- \\sigma ^{+/-} (blue/red points) components of the X B ^\\textrm {B}, T S ^\\textrm {S} and T T ^\\textrm {T} lines as a function of the out-of-plane magnetic field.", "The solid lines represent fits according to Eq.", "." ] ]

[ [ "Deceptive Deletions for Protecting Withdrawn Posts on Social Platforms" ], [ "Abstract Over-sharing poorly-worded thoughts and personal information is prevalent on online social platforms.", "In many of these cases, users regret posting such content.", "To retrospectively rectify these errors in users' sharing decisions, most platforms offer (deletion) mechanisms to withdraw the content, and social media users often utilize them.", "Ironically and perhaps unfortunately, these deletions make users more susceptible to privacy violations by malicious actors who specifically hunt post deletions at large scale.", "The reason for such hunting is simple: deleting a post acts as a powerful signal that the post might be damaging to its owner.", "Today, multiple archival services are already scanning social media for these deleted posts.", "Moreover, as we demonstrate in this work, powerful machine learning models can detect damaging deletions at scale.", "Towards restraining such a global adversary against users' right to be forgotten, we introduce Deceptive Deletion, a decoy mechanism that minimizes the adversarial advantage.", "Our mechanism injects decoy deletions, hence creating a two-player minmax game between an adversary that seeks to classify damaging content among the deleted posts and a challenger that employs decoy deletions to masquerade real damaging deletions.", "We formalize the Deceptive Game between the two players, determine conditions under which either the adversary or the challenger provably wins the game, and discuss the scenarios in-between these two extremes.", "We apply the Deceptive Deletion mechanism to a real-world task on Twitter: hiding damaging tweet deletions.", "We show that a powerful global adversary can be beaten by a powerful challenger, raising the bar significantly and giving a glimmer of hope in the ability to be really forgotten on social platforms." ], [ "Introduction", "*Both authors contributed equally and are considered co-first authors.", "Every day, millions of users share billions of (often personal) posts on online social media platforms like Facebook and Twitter.", "This information is routinely archived and analyzed by multiple third parties ranging from individuals to state-level actors  [20], [43], [44], [59], [58], [27], [63], [55].", "Although the majority of these social media posts are benign, users also routinely post regrettable content on social media  [60], [72], [22] that they later wish to retract.", "Subsequently, most social platforms provide user-initiated deletion mechanisms that allow users to rectify their sharing decisions and delete past posts.", "Not surprisingly, users take advantage of these deletion mechanisms enthusiastically—Mondal et al.", "[49] showed that nearly one-third of six-year-old Twitter-posts were deleted.", "In another work, Tinati et al.", "[62] showed that this number is much higher in Instagram, where almost half of the pictures posted within a six month period had been removed.", "Ironically, current user-initiated deletion mechanisms may have an unintended effect: third-party archival services can identify deleted posts and infer that deleted posts might contain damaging content from the post creator's point of view (i.e., having an adverse effect on the personal/professional life of the content creator).", "In other words, deletion might inadvertently make it easier to identify damaging content.", "Indeed, today it is possible to detect deletions at scale: Twitter, for one, advertises user deletions in their streaming API Twitter provides a random sample of the publicly posted Twitter data in real time to the third parties via streaming API.", "via deletion notifications [5], [6] so that third-party developers can remove these posts from their database.", "Similarly, Pushshift [23], [15] is an archival system for all the contents on Reddit and Removeddit [16] uses this archive to publicize all the deleted posts and comments on Reddit.", "A malicious data-collector can simply leverage these notifications to flag deleted posts as possibly damaging and further use them against the users [4], [3], [69].", "Importantly, the hand-picked politicians and celebrities are not the only parties at the receiving end of these attacks.", "We find that the malicious data-collector can develop learning models to automate the process and perform an non-targeted (or global) attack at a large-scale; e.g., Fallait Pas Supprimer [12] (i.e., “Should Not Delete” in English) is a Twitter account that collects and publishes the deleted tweets of not only the French politicians and celebrities but also noncelebrity French users with less than a thousand followers.", "Asking the users not to post regrettable content on social platforms in the first place may seem like a good first step.", "However, users cannot accurately predict what content would be damaging to them in the future (e.g., after a breakup or before applying to a job).", "Zhou et al.", "[72] and Wang et al.", "[65] propose multiple types of classifiers (Naive Bayes, SVM, Decision Trees, and Neural Networks) to detect regrettable posts using users' history and to proactively advise users even before the publication of posts.", "However, this proactive approach cannot prevent users from publishing future-regrettable posts.", "It is inevitable to focus on reactive mechanisms to assist users with protecting their post deletions.", "Recently Minaei et al.", "[48] proposed an intermittent withdrawal mechanism to tackle this challenge of hiding user-initiated deletions.", "They offer a deniability guarantee for user-initiated deletions in the form of an availability-privacy trade-off and ensure that when a post is deleted, the adversary cannot be immediately certain if it was actually deleted or temporarily made unavailable by the platform.", "Their trade-off could be useful for future social and archival platforms; however, in current commercial social media platforms like Twitter, sacrificing even a small fraction of availability for all the posts is undesirable.", "To this end, our research question is straightforward, yet highly relevant—can we enhance the privacy of the deleted and possibly damaging posts at scale without excessively affecting the functionality of the platform?", "Contributions.", "We make the following contributions.", "First, we demonstrate the impact of deletion detection attacks by performing a proof-of-concept attack on real-world social media posts to identify damaging content.", "Specifically, we use a crowdsourced labeled corpus of (non)damaging deleted posts from Twitter (more than $4,000$ tweets) to train an adversary (a classifier).", "We demonstrate that our adversary is capable of detecting damaging posts with high probability.", "More precisely, our adversary can increase its F-score by 27 percentage points (56% increase) compared to a baseline adversary which uses random guessing to detect damaging posts.", "Thus, it is indeed feasible for the adversary to use automated methods for detecting damaging posts at a large scale (e.g., when focusing only on deleted posts).", "In fact, we expect systems such as Fallait Pas Supprimer [12] to employ analogous learning techniques soon to improve their detection.", "Second, we identify that there are already deletion services which enable users to delete their content in bulk (e.g., “twitWipe” [11] and “tweetDelete” [8] for Twitter, “Social Book Post Manager” [17] for Facebook, “Cleaner for IG” [13] for Instagram, “Nuke Reddit History” [14], and multiple bots on RequestABot subreddit for Reddit).", "However, these bulk deletions provide a clear signal to an adversary that the user is trying to hide damaging content via deletion.", "To that end, we introduce a novel deletion mechanism, Deceptive Deletions, that raises the bar for the adversary in identifying damaging content.", "Given a set of damaging posts (posts that adversary can leverage to blackmail the user) that users want to delete, the Deceptive Deletion system (also known as a challenger) selects $k$ additional posts for each damaging post and deletes them along with the damaging posts.", "The system-selected posts, henceforth called the decoy posts, are taken from a pool of non-damaging non-deleted posts provided by volunteers.", "Since a global adversary can only observe all of these deletions together, his goal is to distinguish deleted damaging posts from the deleted (non-damaging) decoy posts.", "Intuitively, Deceptive Deletion is more effective if the selected decoy posts are similar to the damaging posts.", "These two opposite goals create a minmax game between the adversary and the challenger that we further analyze.", "Third, we introduce the Deceptive Learning Game, which formally describes the minmax game between the adversary and the challenger.", "We start by considering a static adversary that tunes the parameters of its system (e.g., classifier for determining the damaging posts) up until a certain point in time.", "However, powerful adversaries are adaptive and continually tune their models as they obtain more deletions including the decoy deletions made by the challenger.", "Therefore, in the second phase, we consider an adaptive adversary and describe the optimization problem of the adaptive adversary and challenger as a minmax game.", "We identify conditions under which either only the adaptive adversary or only the challenger provably wins the minmax game and discuss the scenarios in-between these two extremes.", "To the best of our knowledge, this is the first attempt to develop a computational model for quantitative assessment of the damaging deletions in the presence of both static and adaptive adversaries.", "Finally, we empirically demonstrate that with access to a set of non-damaging volunteered posts, we can leverage Deceptive Deletions to hide damaging deletions against both static and adaptive adversary effectively.", "We use real-world Twitter data to demonstrate the effectiveness of the challenger.", "Specifically, we show that even when we consider only two decoy posts per damaging deletion, the adversarial performance (F-score) drops to 42% from 75% in the absence of any privacy-preserving deletion mechanism.", "Today, most archival and social media websites (e.g., Twitter, Facebook) enable users to delete their content.", "Recent studies  [49], [18] show that a significant number of users deleted content—35% of Twitter posts are deleted within six years of posting them.", "This user-initiated deletion is also related to the “Right to be Forgotten\"  [69], [66].", "However, this user-initiated content deletion suffered from the Streisand effect – attempting to hide some information has the unintended consequence of gaining more attention [69].", "Consequently, there is a need to provide deletion privacy to users.", "In addition to user-initiated deletions, there exist some premeditated withdrawal mechanisms where all historical content is eventually deleted automatically to provide deletion privacy.", "These mechanisms can be broadly classified into two categories.", "First, in age-based withdrawal, platforms like Snapchat [1] and Dust [2] and systems like Vanish [38], [37] and EphPub [26] automatically withdraw a piece of content after a preset time.", "Second, to make premeditated withdrawal more usable, Mondal et al.", "[49] proposed inactivity-based withdrawal, where posts will be withdrawn only if they become inactive, i.e., there is no interaction with the post for a specified time period (e.g., no more views by other users).", "However, even the premeditated withdrawals are not free from problems of their own.", "First, all the posts will eventually get deleted, removing all archival history from the platform.", "Second, if posts are deleted before the preset time or in-spite of high interaction, the adversary can be certain that the deletion was user-intended, violating deletion privacy.", "Minaei et al.", "[48] attempted to enable users to delete their content while neither attracting attention to deleted content nor deleting full historical archives.", "They presented a new intermittent withdrawal mechanism for all non-deleted posts, which provides a trade-off between availability and deletion privacy.", "In a nutshell, their system ensures that if an adversary found that a post if not available, then the adversary cannot be certain if the post is user-deleted or simply taken down by platform temporarily.", "Although this mechanism is useful for large internet archives, in platforms such as Facebook and Twitter, where content availability is crucial to the users and platform, a privacy-availability trade-off might not be feasible.", "Furthermore, the intermittent withdrawal mechanism does not consider the adversary's background knowledge about other deleted posts.", "Our work aims to bridge this gap and provide a novel learning-based mechanism which considers an adaptive adversary who aims to uncover tweet deletion.", "However, our mechanisms is not without precedent, and it is inspired by earlier work of obfuscation by noise injection.", "Tianti et al.", "[62] offer intuitions for predicting posts deletions on Instagram with the goal of managing the storage of posts on the servers: Once a post is archived, it becomes computationally expensive to erase it; thus, predicting deletions can help in reducing the overheads of being compliant with the “right to be forgotten” regulations.", "These predictions in the non-adversarial setting, however, does not apply to our minmax game between the adversary and the challenger.", "Recently Garg et al.", "[36] formalize the right to be forgotten using platforms as a cryptographic game.", "While being interesting, their definitions and suggested tools such as history-independent data structures are not applicable to our setting where the adversary has continuous and complete access to the collected data." ], [ "Obfuscation using Noise Injection", "There has been a line of work in the area of (non-cryptographic) private information retrieval [42], [50], [34], [53] that obfuscates the users' interest using dummy queries as noise to avoid user profiling.", "Howe et al.", "proposed TrackMeNot [42], [7], which issues randomized search queries to popular search engines to prevent the search engines in building any practical profile of the users based on their actual queries.", "GooPIR [34] is a similar work that uses a Thesaurus to obtain the keywords to constructs $k - 1$ other queries (dummy ones) and submit all $k$ queries at the same time.", "This way, timing attacks by the search engines are eliminated.", "However, it only addresses single keyword searches; these schemes do not address full-sentence searches.", "Murugesan et al.", "propose “Plausibly Deniable Search\" (PDS) [50] that analogous to GooPIR generates $k-1$ dummy queries using latent semantic indexing based approach.", "In their mechanisms, each real query is converted into a canonical query which protects against deanonymization attacks based on typos and grammar mistakes.", "We note that all of the systems mentioned so far consider hiding each query separately.", "However, a determined adversary may be able to find a user's interests by observing a sequence of such obfuscated queries.", "Multiple works have investigated such weaknesses [53], [54], [21].", "Some relatively new techniques further try to overcome these shortcomings by smartly generating the $k-1$ queries.", "For example, Petit et al.", "proposed PEAS [56], where they provide a combination of unlinkability and indistinguishability.", "However, apart from introducing an overhead for encrypting the user queries, their method also requires two proxy servers that are non-colluding, hence weakening the adversarial model.", "K-subscription [51] is yet another work that proposes an obfuscation based approach that enables the user to follow privacy-sensitive channels in Twitter by requiring the users to follow $k-1$ other channels to hide the user interests from the microblogging service.", "However, the K-subscription has a negative social impact for the user as the user's social connections will see the user following these dummy channels.", "These shortcomings, both social and technical, motivated our particular design decision for Deceptive Deletions." ], [ "System", "We consider a data-sharing platform (e.g., Twitter or Facebook) as the public bulletin board where individuals can upload and view content.", "Users are the post owners that are able to publish/delete their posts, and view posts from other users.", "In this work, we consider discrete time intervals in which the users upload and delete posts (fig:systemoverview [baseline=(char.base)] shape=circle,fill,inner sep=0.8pt] (char) 1;).", "A time interval could be as small as a minute or even a week, depending on the platform.", "We define two types of posts.", "[leftmargin = *] User-deleted posts A user could delete a post for two primary reasons [48], [18], [49]: Damaging posts: the post contained damaging content to the user's personal or professional life, or Non-damaging posts: the post was out-dated, contained spelling mistakes, etc.", "An adversary's goal is to find the damaging posts among all the deleted ones that could be used to blackmail the corresponding owners of the post.", "Volunteered posts We consider a subset of non-deleted posts that users willingly offer to be deleted to protect the privacy of other users whenever needed.", "These volunteered posts are non-damaging and cannot be used by the adversary to blackmail the user of the post.", "We discuss the challenges of obtaining volunteered posts in sec:discussion.", "A challenger's goal is to select a subset of volunteered posts (i.e., non-damaging) and delete them such that the aforementioned adversary is unable to distinguish between the damaging and the non-damaging post deletions.", "We denote the posts selected by the challenger as decoy posts.", "Notation.", "We use a subscript $t $ to denote the time interval and superscripts $\\delta , +,\\text{v}, *$ to denote the post type.", "In particular, $\\mathbb {D}_t $ is all the uploaded and deleted posts in time interval $t $ .", "Then we denote all the deleted posts (user- and challenger-deleted) in that interval as $\\mathbb {D}_t ^\\delta $ , the damaging posts as $\\mathbb {D}_t ^+$ , and volunteered posts by $\\mathbb {D}_t ^\\text{v}$ .", "The decoy posts that a challenger selects for deletion to fool the adversary is denoted by ${\\mathbb {G}}_t ^*$ .", "Note that ${\\mathbb {G}}_t ^*\\subseteq \\mathbb {D}_t ^\\text{v}\\subseteq \\mathbb {D}_t {\\setminus } \\mathbb {D}_t ^\\delta $ ." ], [ "Adversary's Actions and Assumptions", "Task.", "At a given time interval, the task of the adversary is to correctly label all the deleted posts as being damaging to the post-owner or not.", "We do not focus on local attackers (or stalkers) targeting individuals or small groups of users.Such stalkers can easily label their posts manually, and protecting against such an attack is extremely hard if not impossible.", "For example, consider the case that a stalker continuously takes snapshots of its targeted user profile with the goal of identifying the user's deletions.", "With its background/auxiliary information about the user (i.e., knowing what contents are considered sensitive to the target), the stalker can effectively identify the damaging deletions.", "We claim that, in the current full-information model, protection against such a local adversary is impossible.", "Our global adversary instead seeks damaging deletions on a large scale, rummaging through all the deleted posts to find as many damaging ones as possible.", "Fallait Pas Supprimer [12] (from sec:intro) is a real-world example of the global adversary.", "Data access.", "At any given time interval, we assume that the adversary is able to obtain all the deleted posts by comparing different archived snapshots of the platform.", "Although this strong data assumption benefits the adversary tremendously, we show in sec:results that Deceptive Deletions can protect the users' damaging deletions.", "Labels.", "Our global, non-stalker adversary is not able to obtain the true label (damaging or non-damaging) of the post from the user.", "Instead, the adversary uses a crowdsourcing service like Mechanical Turk (MTurk) [19] to obtain a proxy for these true labels.", "Although the labels obtained from the Mechanical Turkers (MTurkers) reflect societal values and not the user's intention, following previous work [65], we assume they closely match the true labels in our experiments.", "This is reasonable as the adversary can expend a significant amount of effort and money to obtain these true labels, at least for a small set of posts, that will ultimately be used to train a machine learning model.", "Budget.", "Since there is a cost associated with acquiring label for each deleted post from the MTurkers, the aim of the adversary is to learn to detect the damaging deletions under a budget constraint.", "We consider two types of budget constraints: [leftmargin=*] limited budget where the adversary can only obtain the labels for a fixed number of posts $B_\\text{static}$ , and fixed recurring budget where the adversary obtains the labels for a fixed number of posts $B_\\text{adapt}$ in each interval.", "The adversary with a limited budget is called the static adversary since it does not train after exhausting its budget.", "On the other hand, the adversary with a fixed recurring budget keeps adapting to the new deletions in each time interval, and hence is dubbed the adaptive adversary.", "Player actions.", "At every time interval $t $ , the adversary obtains a set of posts ${\\mathbb {A}}_t ^\\delta $ for training by sampling part of the deleted posts, say $p$ , from $\\mathbb {D}_t ^\\delta $ , an operation denoted by ${\\mathbb {A}}_t ^\\delta ~\\overset{p}{\\sim }~\\mathbb {D}_t ^\\delta $ .", "The adversary uses MTurk to label the sampled dataset ${\\mathbb {A}}_t ^\\delta $ .", "After training, the task of the adversary is to classify the rest of the deleted posts of that time interval.", "Additionally, as the adversary gets better over time, it also relabels all the posts deleted from the past intervals.", "The test set for the adversary is all the deleted posts from current and previous time intervals that were not used for training; i.e., $\\bigcup _{t^{\\prime } \\le t} (\\mathbb {D}_{t^{\\prime }}^\\delta \\setminus {\\mathbb {A}}_{t^{\\prime }}^\\delta )$ .", "fig:systemoverview [baseline=(char.base)] shape=circle,fill,inner sep=0.8pt] (char) 2; shows the adversary's actions.", "Note that although an adaptive adversary can sample $p = B_\\text{adapt}$ deleted posts at every time interval and use MTurkers to label them, a static adversary can only obtain the labels until it runs out of the limited budget (after $\\tau = B_\\text{static}/p$ time intervals).", "After this period, a static adversary does not train itself with new deleted posts.", "Performance metrics.", "The adversary wishes to increase precision and recall for the classification of deleted posts into damaging and non-damaging sets.", "At every time interval $t $ , we report adversary's F-scoreF-score = $2\\cdot precision\\cdot recall/(precision + recall)$ over the test set described above: deleted posts of all the past intervals, i.e., $\\bigcup _{t^{\\prime } \\le t} (\\mathbb {D}_{t^{\\prime }}^\\delta \\setminus {\\mathbb {A}}_{t^{\\prime }}^\\delta )$ ." ], [ "Challenger's Actions and Assumptions", "Task.", "In the presence of an adversary as described above, the task of a challenger is to obtain volunteered posts (i.e.", "non-damaging and non-deleted posts) from users, select a subset of these posts and delete them in order to fool the adversary into misclassifying these challenger-deleted posts as damaging.", "The challenger is honest, does not collude with the adversary, and works with the users (data owners) to protect their damaging deletions.", "Other than the platforms themselves, third party services such as “tweetDelete” [8] can take the role of the challenger as well.", "In sec:gan, we discuss the flaws in a possible alternate approach where the challenger is allowed to generate tweets rather than select from pool of volunteered posts.", "Data access.", "The challenger can be implemented by the platform or a third-party deletion service [10], [8], [9], that has access to the posts of the users.", "Additionally, we assume that there are users over the platform who volunteer a subset of their non-damaging posts to be deleted anytime (or within a time frame) by the challenger, possibly, in return for privacy benefits for their (and other users') damaging deletions.", "Labels.", "The challenger is implemented as part of the platform (or a third-party service permitted by the user).", "Thus, unlike the adversary that obtains proxy labels from crowdsourcing platforms, it has access to the true labels— damaging or non-damaging, from the owner of the post.", "This is easily implemented: before deleting a post, the user can specify whether the post is damaging (and needs protection).", "Access to the adversary.", "The challenger not only knows the presence of a global adversary trying to classify the deleted posts into damaging and non-damaging posts but also can observe its behaviour.Fallait Pas Supprimer [12] posts all its output on Twitter itself.", "As a result, we consider three types of accesses to the adversary: [leftmargin=*] no access where the challenger has no information about the adversary.", "monitored black-box access with a recurring query budget of $B_g$ where the challenger can obtain the adversary's classification probability for a limited number of posts $B_g$ every time interval, but the access is monitored, i.e., the adversary can take note of every post queried and treat them separately.", "black-box access where the challenger can obtain the adversary's classification probabilities for any post.", "Here, no access is the weakest assumption that defines the lower-bounds for our challenger's success.", "Nevertheless, we expect the challenger to have some access to the adversary's classification.", "An unrestricted black-box access serves as an upper bound for the challenger assuming that it can train a precise surrogate model of the adversary's classifier using its own training data.", "While employing such a surrogate model is common practice in the literature [45], [52], it can be hard to obtain in real world without knowing the adversary's exact architecture and training data.", "Our monitored black-box assumption with a recurring query budget (henceforth, interchangeably called the restricted black-box access) balances practicality of the access versus the feasibility of defending against an adversary with that access.", "In sec:model, we introduce three challengers (oracle, $\\text{D}^{\\text{2}}$ and random) corresponding to the three types of accesses.", "Player actions.", "At every time interval $t $ , the challenger receives new volunteer posts from the users and adds them to a set that stores the volunteered posts collected up until this point.", "Next, based on the type of access, it obtains the adversary's classification probabilities for some number of volunteer posts (the number is dependent on the access which we detail in sec:model).", "Finally, it selects decoy posts, a subset of the volunteered posts collected up until this point and deletes these posts in interval $t {+}1$ (hence the adversary sees these challenger-deleted posts in interval $t {+}1$ as part of the deleted set $\\mathbb {D}_{t {+}1}^\\delta $ ).", "fig:systemoverview [baseline=(char.base)] shape=circle,fill,inner sep=0.8pt] (char) 3; shows the challenger's actions.", "Performance metrics.", "The challenger, in direct contrast to the adversary, wishes to decrease adversary's precision and recall for the classification of deleted posts.", "Adversary's precision will decrease if it classifies the injected decoy posts as damaging (increased false-positives).", "On the other hand, adversary's recall will decrease if it learns to be conservative in order to ignore the decoy posts (increased false-negatives)." ], [ "The Deceptive Learning Game", "The deceptive learning game is a two-player zero-sum non-cooperative game over time intervals $t=1, 2, ... $ (units) between an adversary who wishes to find users' damaging deletions, and a challenger who wishes to hide the said damaging deletions.", "The challenger achieves this by deleting volunteers' non-damaging posts as decoys.", "While the adversary's goal is to maximize its precision/recall scores on the classification task, the challenger's goal is to minimize them.", "We denote each post by $(x, y)$ , where $x \\in {\\mathbb {X}}$ represents the features of the post (i.e., text, comments, etc.)", "and $y \\in \\lbrace 0, 1\\rbrace $ denotes its true label such that $y=1$ if the post is damaging and $y=0$ if it is non-damaging.", "In the following subsections, we describe the actions of each player in the time interval $t$ ." ], [ "Adversary", "We denote the adversary's classifier at the beginning of interval $t $ by $a(~\\cdot ~; {{\\theta }}_{t-1}) : {\\mathbb {X}}\\rightarrow [0, 1]$ parameterized by ${{\\theta }}_{t-1}$ such that $a(x; {{\\theta }}_{t-1}) P(\\hat{y}=1 ~|~ x; {{\\theta }}_{t-1})$ is the predicted probability of the post $x$ being damaging.", "The adversary collects all the deletions that happen in this interval (i.e., $\\mathbb {D}_t ^\\delta $ ) and samples $p$ posts, denoted by ${\\mathbb {A}}_t ^\\delta $ .", "The adversary then uses MTurk to obtain a proxy for the true labels of these $p$ posts.", "The adversary uses this labeled training data in the following optimization problem to update its parameters, $ {{\\theta }}_{t} = \\operatornamewithlimits{arg\\,min}_{{{\\theta }}} \\mathcal {L}_\\text{NLL}({{\\theta }}; {\\mathbb {A}}_t ^\\delta ) \\:,$ where $\\mathcal {L}_\\text{NLL}$ is the standard negative log-likelihood loss for the classification task, given by, $\\mathcal {L}_\\text{NLL}&({{\\theta }}; {\\mathbb {A}}_t ^\\delta ) = \\sum _{(x, y) \\in {\\mathbb {A}}_t ^\\delta } -y \\log \\left(a(x; {{\\theta }})\\right) - (1 - y) \\log \\left(1 - a(x; {{\\theta }})\\right) \\:.$ After training, the adversary uses the trained model $a(~\\cdot ~; {{\\theta }}_{t})$ to predict the labels of the rest of the deleted posts of time interval $t$ , i.e., $\\mathbb {D}_t ^\\delta \\setminus {\\mathbb {A}}_t ^\\delta $ along with all the deleted posts that it had already predicted in the past.", "This way the adversary hopes to capture damaging posts that were missed earlier.", "Hence, we report the adversary's performance on all the past deletions (not including the training data): $\\bigcup _{t^{\\prime } \\le t} (\\mathbb {D}_{t^{\\prime }}^\\delta \\setminus {\\mathbb {A}}_{t^{\\prime }}^\\delta )$ .", "Static vs Adaptive Adversary.", "Since the static adversary has a limited budget, first it chooses the number of time intervals for training, say $\\tau $ , and accordingly samples $p = B_\\text{static}/ \\tau $ posts for querying MTurk to obtain labels.", "The adaptive adversary has a fixed recurring budget of $B_\\text{adapt}$ and hence, can sample $p=B_\\text{adapt}$ posts every interval.", "This allows the adaptive adversary to train itself with new training data (of size $B_\\text{adapt}$ ) every interval indefinitely.", "alg:adversary depicts adversary's actions within a time interval (subscript $t$ removed for clarity).", "[t] InputinputOutputoutput Sample $p$ posts ${\\mathbb {A}}^\\delta ~\\overset{p}{\\sim }~\\mathbb {D}^\\delta $ Query MTurk and obtain labels for ${\\mathbb {A}}^\\delta $ Obtain optimal parameters ${{\\theta }}^* $ by solving eq:adversary $a(~\\cdot ~;{{\\theta }}^*)$ Adversary" ], [ "Challenger", "In the presence of such an adversary, the challenger's goal is to collect volunteered posts (non-damaging) from users and selectively delete these posts in order to confuse the adversary.", "As described before, $\\mathbb {D}_t^\\text{v}$ is the set of posts volunteered by users in the time interval $t$ .", "Let ${\\mathbb {G}}_{\\le t}^*$ be the set of decoy posts deleted by the challenger in the current and past intervals.", "At the end of interval $t$ , the challenger collects all the volunteered posts from the current and past intervals (except the posts that it has already used as decoys).", "The available set of volunteered posts is denoted by $\\mathbb {D}_{\\le t}^\\text{v}\\equiv (\\bigcup _{t^{\\prime }\\le t} \\mathbb {D}_{t^{\\prime }}^\\text{v}) \\setminus (\\bigcup _{t^{\\prime }\\le t} {\\mathbb {G}}_{t^{\\prime }}^*$ ).", "Note that $(x, y) \\in \\mathbb {D}_{\\le t}^\\text{v}{\\Rightarrow } y=0$ , i.e., the volunteered posts are non-damaging by definition.", "For ease of notation, let $N^\\text{v}:= |\\mathbb {D}_{\\le t}^\\text{v}|$ be the number of volunteered posts collected till interval $t$ .", "Then, the goal of the challenger is to construct the decoy set ${\\mathbb {G}}^*_{t+1} \\subseteq \\mathbb {D}_{\\le t}^\\text{v}$ and delete these posts during the next time interval $t{+}1$ in order to fool the adversary into misclassifying these challenger-deleted non-damaging posts as user-deleted damaging posts.", "Formally, we want to choose $K$ decoy posts (denoted by a $K$ -hot vector ${\\mathbf {w}}$ ) that maximizes the negative-log likelihood loss for the adversary's classifier, given by the following optimization problem, $ {\\mathbf {w}}^* &= \\operatornamewithlimits{arg\\,max}_{{\\mathbf {w}}} V({\\mathbf {w}}; \\mathbb {D}_{\\le t}^\\text{v}) \\nonumber \\\\\\text{s.t.", "~~~} & ||{\\mathbf {w}}||_1 = K, ~~~~~~ {\\mathbf {w}}\\in \\lbrace 0, 1\\rbrace ^{N^\\text{v}} \\:,$ where $ V({\\mathbf {w}}; \\mathbb {D}_{\\le t}^\\text{v}) &= \\sum _{i=1}^{N^\\text{v}} {-}{w}_i \\cdot \\log (1 - a(x_i; {{\\theta }}_t)) \\:,$ and $x_i$ is the $i$ -th volunteered post in $\\mathbb {D}_{\\le t}^\\text{v}$ .", "The cost function $V({\\mathbf {w}}; \\mathbb {D}_{\\le t}^\\text{v})$ in eq:V is simply the negative log-likelihood of the adversary over the set $\\mathbb {D}_{\\le t}^\\text{v}$ weighted by a $K$ -hot vector ${\\mathbf {w}}$ .", "eq:V uses the fact that the set only contains non-damaging posts (i.e., $y_i = 0$ ).", "Consequently, ${\\mathbf {w}}^*$ optimized in such a fashion selects $K$ posts from the set $\\mathbb {D}_{\\le t}^\\text{v}$ that maximizes the adversary's negative log-likelihood loss.", "The set of $K$ selected posts can be trivially constructed as ${\\mathbb {G}}_{t+1}^* = \\lbrace x_i : i \\in \\lbrace 1, \\ldots , N^\\text{v}\\rbrace \\wedge w_i = 1\\rbrace $ .", "The challenger deletes ${\\mathbb {G}}_{t+1}^*$ over the next time interval $t{+}1$ (hence the adversary sees these posts as part of the deleted set $\\mathbb {D}_{t +1}^\\delta $ ).", "Note that the challenger uses the adversary's classifier $a(~\\cdot ~;~\\theta _t)$ to create decoy posts for $t{+}1$ .", "However, as per subsec:adversary, in interval $t{+}1$ the adversary first trains over a sample of the deleted posts (including the decoy posts) and updates its classifier to $a(~\\cdot ~; \\theta _{t+1})$ before classifying the rest of the deleted posts of $t{+}1$ .", "Hence, the challenger is always at a disadvantage (one step behind).", "Next, we describe three challengers corresponding to the access types discussed in subsec:threatchallenger: no access, black-box access and monitored black-box access with a query budget.", "[t] InputinputOutputoutput ${\\mathbb {G}}^*\\leftarrow \\emptyset $ accessType = none Random challenger ${\\mathbb {G}}^*~\\overset{K}{\\sim }~\\mathbb {D}^\\text{v}$ accessType = black-box Oracle challenger ${\\mathbb {G}}^* \\leftarrow \\lbrace x_i : x_i \\in \\mathbb {D}^\\text{v}\\wedge a(x_i; {{\\theta }}) \\text{ is in the top } K \\rbrace $ accessType = monitored black-box (budget $B_g$ ) $\\text{D}^{\\text{2}}$ challenger Sample $B_g$ posts for training $\\mathbb {D}^{\\text{v},\\text{train}} ~\\overset{B_g}{\\sim }~\\mathbb {D}^\\text{v}$ $\\mathbb {D}^{\\text{v},\\text{test}} \\leftarrow \\mathbb {D}^\\text{v}\\setminus \\mathbb {D}^{\\text{v}, \\text{train}}$ Query $a(x_i; {{\\theta }})$ for all $(x_i, y_i = 0) \\in \\mathbb {D}^{\\text{v}, \\text{train}}$ Obtain optimal parameters ${{\\phi }}^*$ by solving eq:continuousrelaxation ${\\mathbb {G}}^* \\leftarrow \\lbrace x_i : x_i \\in \\mathbb {D}^{\\text{v}, \\text{test}}\\wedge g(x_i; {{\\phi }}^*) \\text{ is in the top } K \\rbrace $ ${\\mathbb {G}}^*$ Challenger Random challenger (no access).", "We begin with the case where the challenger has no access to the adversary's classifier and there is no side-information available to the challenger.", "With no access to the adversary's classification probabilities $a(~\\cdot ~; {{\\theta }}_t)$ , the optimization problem in eq:optimizationproblem cannot be solved.", "We introduce the naive random challenger that simply samples $K$ posts randomly from the available volunteered posts $\\mathbb {D}_{\\le t}^\\text{v}$ and deletes them, i.e., ${\\mathbb {G}}_{t+1}^*~\\overset{K}{\\sim }~\\mathbb {D}_{\\le t}^\\text{v}$ .", "This is the only viable approach if the challenger has no information about the adversary's classifier.", "Oracle challenger (black-box access).", "Next we consider the challenger that has a black-box access to the adversary's classifier with no query budget, i.e., at any time interval $t$ , the challenger can query the adversary with a post $x$ and expect the adversary's predicted probability $a(x; {{\\theta }}_{t})$ in response without the adversary's knowledge.", "Armed with the black-box access, oracle challenger can simply maximize eq:optimizationproblem by choosing the top $K$ posts with highest values for $a(x_i; {{\\theta }}_{t})$ .", "$\\text{D}^{\\text{2}}$ Challenger (monitored black-box access with query budget $B_g$ ).", "The oracle challenger assumes an unmonitored black-box access to the adversary with an infinite query budget which can be hard to obtain in practice.", "In what follows, we relax the access and assume a monitored black-box access with a recurring query budget of $B_g$.", "In other words, queries to the adversary, while being limited per interval, are also monitored and possibly flagged by the adversary.", "The adversary can simply take note of these queries as performed by a potential challenger, hence negating any privacy benefits from injecting decoy posts.", "Whenever the adversary sees a deleted post identical to one that it was previously queried about, it can ignore the post as it is likely non-damaging.", "Here we design a challenger, henceforth dubbed $\\text{D}^{\\text{2}}$ , that trains to select decoy posts from any given volunteered set.", "In other words, the $\\text{D}^{\\text{2}}$ challenger makes use of the monitored black-box access to the adversary only during training.", "Hence it can be used to find the decoy posts without querying the adversary; for example in a held-out volunteered set (separate from the training set).", "Additionally, the $\\text{D}^{\\text{2}}$ challenger queries the adversary for only $B_g$ posts every time interval.", "We denote the challenger's model at the beginning of interval $t $ by $g(~\\cdot ~; {{\\phi }}_{t{-}1}) : {\\mathbb {X}}\\rightarrow {\\mathbb {R}}$ parameterized by ${{\\phi }}_{t{-}1}$ .", "For a given volunteer post $x$ , $g(x; {{\\phi }}_{t{-}1})$ gives an unnormalized score for how likely the post will be mislabeled as damaging; higher the score, higher the misclassification probability.", "First, the $\\text{D}^{\\text{2}}$ challenger samples $B_g$ posts for training from the available volunteered set $\\mathbb {D}_{\\le t}^{\\text{v}}$ collected till interval $t$ .", "We denote the train and test sets of the $\\text{D}^{\\text{2}}$ challenger as $\\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}$ and $\\mathbb {D}_{\\le t}^{\\text{v}, \\text{test}}$ of sizes $B_g$ and $N^\\text{v}- B_g$ respectively.", "Then, the goal of the $\\text{D}^{\\text{2}}$ is to find optimal parameters ${{\\phi }}_{t}$ by solving a continuous relaxation of eq:optimizationproblem presented below, ${{\\phi }}_{t} = \\operatornamewithlimits{arg\\,max}_{{{\\phi }}} \\tilde{V}({{\\phi }}; \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}})$ where $\\tilde{V}({{\\phi }}; \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}) &= \\sum _{i=1}^{B_g} {-} \\alpha (x_i; {{\\phi }}, \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}) ~ \\log (1 - a(x_i; {{\\theta }}_t))\\:, \\nonumber $ and $ \\alpha (x_i; {{\\phi }}, \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}) &= \\frac{\\exp {(g(x_i; {{\\phi }}))}}{\\sum _{j=1}^{B_g} \\exp {(g(x_j; {{\\phi }}))}} \\:,$ is a softmax over the challenger outputs for all the examples in $\\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}$ .", "The softmax function makes sure that $0 \\le \\alpha (~\\cdot ~; {{\\phi }}, \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}) \\le 1$ and $\\sum _{j=1}^{B_g} \\alpha (x_j; {{\\phi }}, \\mathbb {D}_{\\le t}^{\\text{v}, \\text{train}}) = 1$ .", "The continuous relaxation in eq:continuousrelaxation allows the $\\text{D}^{\\text{2}}$ challenger to train a neural network model parameterized by $\\phi $ via backpropagation.", "We now show that optimizing the relaxed objective in eq:continuousrelaxation results in the best objective value for eq:optimizationproblem.", "Proposition 1 proposition]prop:confuser For any given volunteered set $\\mathbb {D}^\\text{v}$ with $N$ non-deleted posts, $\\max _{{{\\phi }}} \\tilde{V}({{\\phi }}; \\mathbb {D}^\\text{v}) = \\max _{w_1, \\ldots , w_{N}} V(w_1, \\ldots , w_{N}; \\mathbb {D}^\\text{v})$ We present proof of the proposition in app:proofs.", "Finally, the $\\text{D}^{\\text{2}}$ challenger with optimal parameters ${{\\phi }}_{t}$ computes $g(x; {{\\phi }}_{t})$ for all $(x, y=0) \\in \\mathbb {D}_{\\le t}^{\\text{v}, \\text{test}}$ , and constructs $G_{t+1}^{*}$ by choosing the examples with top $K$ values for $g(~\\cdot ~; {{\\phi }}_{t})$ .", "alg:challenger shows the actions of the challenger within a time interval (subscript $t$ removed for clarity)." ], [ "Deceptive Learning Game", "alg:game presents the game between the adversary and the challenger.", "In each time interval, users independently delete and volunteer posts (line 4).", "The platform/deletion-service additionally deletes the challenger-selected decoy posts (line 5).", "The adversary obtains all the deleted posts and queries the MTurk with a small subset of the posts for labels (if the adversary has not exhausted the budget).", "With this labeled set of deleted posts, the adversary trains its classifier (lines 6-7).", "The challenger collects new volunteered posts (line 8) and builds decoy posts to be injected in the next interval (line 9).", "This results in a real-life game between the adversary and the challenger, where each adapts to the other.", "The deceptive learning game is different from the adversarial learning approaches as we detail in sec:adversariallearning.", "[t] InputinputOutputoutput ${\\mathbb {G}}_1^*\\leftarrow \\emptyset $ $\\mathbb {D}_{\\le 0}^\\text{v}\\leftarrow \\emptyset $ $t \\leftarrow $ 1 n $\\mathbb {D}_t ^\\delta , ~\\mathbb {D}_t ^\\text{v}\\leftarrow $ Users($t$ ) *[r]deleted and volunteered posts of the users at interval $t$ $\\mathbb {D}_t ^\\delta \\leftarrow \\mathbb {D}_t ^\\delta \\cup {\\mathbb {G}}_t ^*$ *[r]user- and challenger-deleted posts at interval $t$ Adversary's budget has not exhausted $a(~\\cdot ~,{{\\theta }}_t) \\leftarrow \\text{Adversary}(\\mathbb {D}_t ^\\delta )$ $\\mathbb {D}_{\\le t}^\\text{v}\\leftarrow (\\mathbb {D}^\\text{v}_{\\le t-1} \\setminus {\\mathbb {G}}_t ^*) \\cup \\mathbb {D}_t ^\\text{v}$ *[r]available volunteered set ${\\mathbb {G}}_{t +1}^*\\leftarrow \\text{Challenger} (\\mathbb {D}_{\\le t}^\\text{v}, K, \\text{accessType})$ Deceptive Game" ], [ "Analysis: Who Wins the Game?", "In what follows, we analyze the scenarios where either the adversary or the challenger wins the deceptive learning game.", "We show that the volunteered set, $\\mathbb {D}^\\text{v}$ , plays a significant role in deciding the winner of the game.", "First, we need the definition of support of a distribution.", "Definition 1 (Support) Let $\\Omega = \\lbrace x : \\forall x, p(x) > 0\\rbrace $ be the support of distribution $p(x)$ , i.e., the set of all possible features $x$ with non-zero probability.", "Let $p^+(x)$ be the distribution of the features of damaging posts, with the corresponding support denoted by $\\Omega ^+$ .", "Then, a post $x$ is in $\\Omega ^+$ if there is a non-zero probability that it is a damaging post.", "Similarly, $\\Omega ^\\text{v}$ is the support of the distribution of volunteered posts $p^\\text{v}$ .", "Next, we analyze the two extreme scenarios of non-overlapping supports (i.e., $\\Omega ^\\text{v}\\cap \\Omega ^+= \\emptyset $ ) and fully-overlapping supports (i.e., $\\Omega ^\\text{v}= \\Omega ^+$ ).", "These extreme scenarios correspond to the following simple questions respectively: (a) “what if all the posts volunteered by users have completely different features than the damaging posts?” and (b) “what if the volunteered posts have very similar or same features as those of damaging posts?”." ], [ "Non-overlapping Support: Adversary Wins", "Proposition 2 (Non-overlapping support) proposition]p:nonoverlapping Assume $\\Omega ^\\text{v}\\cap \\Omega ^+= \\emptyset $ , i.e., the supports of volunteered and damaging posts do not overlap.", "Then, there is always a powerful-enough adversary to defeat the challenger.", "[Proof sketch] Assume the most powerful challenger who can select any post features $x$ from an infinite supply of volunteered posts.", "However, since $\\Omega ^\\text{v}\\cap \\Omega ^+= \\emptyset $ , there is no sampling from $p^\\text{v}$ to generate decoy examples that look like they are sampled from $p^+$ .", "Hence, given enough data, an adversary can find a perfect decision boundary between the damaging posts and the decoy posts.", "Because neural networks are universal function approximators [41], this powerful adversary always exists and, thus, the challenger can always be defeated in the deceptive learning game.", "An Illustrative Example: Consider the example provided in fig:halfmoonadvdecision.", "The two classes (denoted by red circles and green crosses respectively) have non-overlapping support.", "We show the decision boundary of the adaptive adversary in this setting dataset after 50 intervals of the deceptive learning game.", "We see that the adversary can perfectly label the points even in the presence of the oracle challenger.", "Real-world scenario: The non-overlapping case could happen in an online social platform if its users are very conservative in volunteering posts to the challenger.", "Consider for example, none of the volunteered posts contained any sensitive keyword, whereas all the damaging posts had at least one sensitive keyword, a clear case of non-overlapping supports.", "In such a scenario, the adversary will win the game as detailed above." ], [ "Fully-overlapping Support: Challenger Wins", "Proposition 3 (Fully overlapping support) proposition]p:fullyoverlapping Assume $\\Omega ^\\text{v}=\\Omega ^+$ , i.e., the supports of volunteered and damaging posts fully overlap.", "Then, given enough volunteered posts in $\\mathbb {D}^\\text{v}$ , the challenger always defeats the adversary (in both static and adaptive scenarios).", "More precisely, if the challenger selects $k$ decoys per damaging post in $\\mathbb {D}^\\delta $ , then the adversary's probability of identifying a damaging post in $\\mathbb {D}^\\delta $ is in average at most $1/(k+1)$ .", "[Proof sketch] The proof relies on a property of rejection sampling, which states that if the support of two distributions $p_1$ and $p_2$ fully overlap, then one can selectively filter samples from $p_1$ to make the filtered samples have distribution $p_2$ (a proof of this principle is given in the Appendix).", "Asymptotically, for each damaging example $x$ in adversary's test data, there are $k$ indistinguishable decoy examples (from the adversary's perspective).", "This is because, by Bayes theorem $&p^\\delta (y=1|x) =\\\\&\\frac{p^\\delta (x | y =1) p^\\delta (y=1)}{p^\\delta (x | y =1)p^\\delta (y=1) + p^\\delta (x | y =0)p^\\delta (y=0)}\\le \\frac{1}{1 + k},$ where the superscript $p^\\delta $ indicates the distribution of deleted posts $\\mathbb {D}^\\delta $ .", "The inequality holds by construction, as for all $x \\in \\mathbb {D}^\\delta $ with label one, there are at least $k \\ge 1$ samples from $p^\\text{v}(x)$ with label zero.", "An Illustrative Example: Consider the example provided in fig:gaussianadvdecision where the two classes (red circles and green crosses respectively) have fully overlapping supports (as they are drawn from a Gaussian distribution with different means).", "We show the decision boundary of the adaptive adversary in this setting after 50 intervals of the deceptive learning game.", "We see that for any decision boundary, there exist points in $\\Omega ^\\text{v}$ that a challenger can choose such that the adversary mislabels them as damaging.", "Real-world scenario: The fully-overlapping case could happen in an online social platform if the definition of what constitutes as damaging varies across the platform's users.", "For example, user $A$ could consider a post with a single sensitive word (e.g., a swear word) as damaging, whereas another user $B$ from a different background might consider the same post as completely innocuous and volunteer the post.", "In such a scenario, the challenger will use volunteered posts from user $B$ to protect the damaging posts of user $A$ .", "Hence, the challenger will win the game against even the most powerful adversary with infinite data.", "p:fullyoverlapping,p:nonoverlapping are important to understand the two extreme cases —where either the challenger clearly wins or the adversary clearly wins— as important insights, even though these clear-cut cases are unlikely to happen in practice.", "Most real-world applications will likely fall between these two extremes, where the supports only partially overlap.", "In such scenarios, the adversary wins outside the overlap (i.e., can classify everything correctly outside the overlap), and the challenger wins inside the overlap.", "In other words, extremely sensitive and damaging posts cannot be protected as they will have no overlap with any of the volunteered posts.", "However, as we show in the next section, with a reasonable volunteered set, the challenger can make it hard for the adversary to detect damaging deletions.", "Figure: Two examples illustrating the two possible scenarios relating to the supports of volunteered posts and damaging posts: non-overlapping (left) and fully overlapping (right).", "The black line denotes the decision boundary of the adaptive adversary after 50 intervals of the deceptive learning game." ], [ "System Evaluation on Twitter Deletions", "In this section we evaluate the efficiency of an adversary when Deceptive Deletions is applied to the real-world problem of concealing damaging deletions in Twitter.", "In this evaluation we first create and prepare sets of (non)damaging tweets.", "Then we use these sets to train the challenger and adversary classifiers and analyze their performance." ], [ "Data Collection", "In this work, we select Twitter as our experimental social media platform.", "We note that it was certainly plausible to perform the exact experiment on other social platforms.", "However we chose Twitter due to its popularity and feasibility of data collection.", "Specifically, in order to evaluate the challenger we needed a real-world dataset which includes (i) both deleted and non-deleted tweets (i.e., Twitter posts) and (ii) deleted tweets that contain both damaging and non-damaging tweets.", "To that end, we use two data sources to create such a dataset.", "We collected 1% of daily random tweet samples from the Twitter API from Oct 2015 - May 2018.", "Eliminating non-English tweets, we accumulated over one billion tweets.", "In the next step, we construct the damaging and volunteered sets.", "To construct the damaging set, we first needed to identify the deleted tweets.", "We sampled 300,000 tweets from the aforementioned collected data, and leveraging the Twitter API, we identified the tweets that were deleted at the time of our experiment (Jan 30th, 2020).", "In total, we identified 92,326 deleted tweets.", "The next step was to obtain ground truth labels for the deleted tweets—i.e., detect and assign “true” labels to damaging tweets and “false” labels to rest.", "We used the crowdsourcing service Amazon Mechanical Turk (MTurk) [19] to obtain a proxy for these true labels.", "However, there were two challenges– First, it was impractical to ask our annotators to label 92,326 tweets.", "Second, since the dataset was highly imbalanced, a simple random sample of tweets for labeling would have resulted in a majority of non-damaging tweets.", "Thus, we followed prior work to create a more balanced sample dataset [65], [72].", "Specifically, we filtered the deleted tweets using a simple sensitive keyword-based approach [72] (i.e., identify posts with sensitive keywords) to have a higher chance of collecting possibly damaging tweets.", "This approach resulted in 33,000 potentially damaging tweets, and we randomly sampled 3,500 tweets to be labeled by annotators on MTurk.", "The mean number of sensitive keywords in each tweet within our data set was $2.55$ .", "Note that, in addition to the cursing and sexual keywords, our sensitive keyword-based approach also considered keywords related to the topics of religion, race, job, relationship, health, violence, etc.", "Intuitively, if a post does not contain any such sensitive keywords then the likelihood of the post being damaging is very low.", "We confirmed this intuition by asking MTurk annotators to label 150 tweets which did not contain any sensitive keyword as damaging/non-damaging.", "We noted that more than 97% of these 150 tweets were labeled as non-damaging by annotators.", "We surmised that in practice, the adversary will also leverage a similar filtering approach to reduce its overhead and increase its chances of finding damaging posts.", "In total, out of our sampled 3,500 deleted tweets, we obtained labels for 3,177 tweets (excluding annotations from Turkers who failed our quality control checks as described later).", "Among the labeled tweets, 1,272 were identified as damaging, and $1,905$ were identified as non-damaging.", "Data labeling using MTurk.", "We acknowledge that ideally, the tweet labels should have been assigned by the posters themselves.", "However, since we collected random tweets at large-scale using the Twitter API, we could not track down and pursue original posters to label their deleted tweets.", "To that end, we note that there is a crowdsourcing based alternative which is already leveraged by earlier work to assign sensitivity labels [29], [25], [65].", "Specifically, these studies determined the sensitivity of social media posts by simply aggregating crowdsourced sensitivity labels provided by multiple MTurk workers (Turkers).", "Thus we took a similar approach as mentioned next.", "On MTurk, tasks (e.g., completing surveys) are called Human Intelligence Tasks or HITs.", "Turkers can participate in a survey by accepting the corresponding HIT only if they meet all the criteria associated with that HIT (set by the person(s) who created the HIT).", "We leverage this feature to ensure the reliability of our results.", "Specifically we asked that the Turkers taking our survey should: (i) have at least 50 approved HITs.", "(ii) have an assignment approval rate higher than 90%, and (iii) have their location set to United States.", "This last criterion ensured consistency of our Turkers' linguistic background.", "In our experiment each HIT consisted of annotating 20 tweets with true (damaging) or false (non-damaging) labels.", "We allowed the Turkers to skip some tweets in case they feel uncomfortable for any reason.", "We compensated 0.5 USD for each HIT and on average it took the Turkers 193 seconds to complete each HIT.", "To control the quality of annotation by Turkers, we included two hand-crafted control tweets with known labels in each HIT.", "These control tweets were randomly selected from two very small sets of clearly non-damaging or damaging tweets and were inserted at random locations within the selection of 20 tweets.", "For example a damaging control tweet was: “I think I have enough knowledge to make a suicide bomb now!", "Might need it New Year's Eve\" and non-damaging control tweet was: “Prayers with all the people in the hurricane irma\".", "If for a HIT, the responses to these control tweets did not match the expected label, we conservatively discarded all twenty annotations in that HIT.", "We countered possible bias resulting from the order of presentation of tweets via randomizing the order of tweets in every HIT.", "Even if two Turkers annotated the same set of tweets, the order of those tweets was different.", "Furthermore, to ease the subjectivity of the labels from each participant, for each tweet we collected the annotations of multiple Turkers and took the majority vote.", "In our experiment, we created the HITs such that each tweet was annotated by 3 distinct Turkers.", "After receiving the responses, for each tweet we assigned the final label (indicating damaging or non-damaging) based on the majority vote.", "We emphasize that in the real world, the burden of labeling the posts via crowdsourcing is on the adversary.", "The challenger, on the other hand, can be implemented as a service within the platform and can obtain the true labels directly from the post-owners.", "Therefore, existence of any mislabeled data will negatively impact only the adversary." ], [ "#Donttweet dataset", "Recently Wang et al.", "[65] proposed “#Donttweetthis”.", "“#Donttweetthis” is a quantitative model that identifies potentially sensitive content and notifies users so that they can rethink before posting those content on social platforms.", "Wang et al.", "created the training data for their model by (i) identifying possibly sensitive tweets by checking for the existence of sensitive keywords within the text and then (ii) using crowd-sourcing (i.e., using MTurk) to annotate the sensitivity of each tweet by three annotators.", "The data collection approach used by “#Donttweetthis” (section 3 of [65]) is very similar to ours.", "Therefore, to enrich our dataset and be able to evaluate the challenger over more intervals, we acquired their labeled tweets.", "Using the Twitter API, we queried the tweets using their corresponding IDs and identified the deleted ones (at the time of writing, Jan 30th, 2020).", "In total, we obtained 851 deleted tweets, where 418 were labeled as sensitive (damaging), and the remaining 433 were labeled as non-sensitive (non-damaging).", "The mean of sensitive keywords in each tweet within this set was $1.7$ .", "Summary of collected data.", "In summary, combining the two datasets explained above, we obtained labels for $4,028$ deleted tweets establishing the user deleted set.", "Among the deleted tweets $1,690$ were labeled as damaging constructing our damaging set ($\\mathbb {D}^+$ ).", "As we will demonstrate in the results section, in our evaluation the four thousand labeled tweets (larger than that of prior works [65], [72]) allows for 10 intervals for the game between the adversary and challenger.", "Furthermore, for our experiment, we consider $k = 1,2,5$ (i.e., number of decoy posts for each damaging post).", "We needed to accommodate these values of $k$ and also a pool that the challenger can make meaningful selections from.", "Thus, we sampled $100,000$ non-deleted tweets uniformly at random from the 1% of daily random tweet sample posted between Jan 1st, 2018 – May 31st, 2018 to build the volunteered set.", "The non-deleted tweets are assumed to be non-damaging.", "We consider this assumption to be reasonable as if a tweet contains some damaging content then its owner would not keep that post on its profile.", "In practice, we can forgo this assumption as the volunteer users themselves offer the volunteer posts.", "The average number of sensitive keywords in each tweet in this set was $0.41$ .", "Recall that in order to create our evaluation dataset we needed to show some deleted tweets to Turkers for the annotation task.", "Thus, we were significantly concerned about the ethics of our annotation task.", "Consequently, we discussed at length with the Institutional Review Board (IRB) of the lead author's institute and deployed the annotation task only after we obtained the necessary IRB approval.", "Next we will detail, how, in our final annotation task protocol we took quite involved precautionary steps for protecting the privacy of the users who deleted their tweets.", "We recognize that, in the context of our evaluation, the primary risk to the deleted-tweet-owners was the possibility of linking deleted tweets with deleted-tweet-owner profiles during annotation.", "This intuition is supported by prior research [47], [49] who suggested applying selective anonymization for research on deleted content.", "Thus, we anonymized all deleted tweets by replacing personally identifiable information or PII (e.g., usernames, mentions, user ids, and links) with placeholder text.", "For example, we replaced user accounts (i.e., words starting with @) and url-links with “UserAccount\" and “Link” respectively.", "Moreover, one of the authors manually went over each of these redacted posts to ensure anonymization of PII before showing them to Turkers." ], [ "Experiment Setup", "Partitioning the data for different time intervals.", "Recall from sec:systhreatmodel that we discretize time into intervals.", "In our experiments, we choose $T=10$ intervals in total (a choice made based on the number of collected tweets).", "Consequently, we partition our dataset into 10 intervals.", "Ideally, the partitions should be based on the creation and deletion timestamps of the tweets.", "Unfortunately however, the Twitter API does not provide deletion timestamps.", "Hence, we randomly shuffle the tweets and divide them into 10 equally sized partitions.", "BERT model.", "In line with our approach to model the most-powerful adversary as best as we possibly can, we use a state-of-the-art natural language processing model: the BERT (Bidirectional Encoder Representations from Transformers) language model [32], both for the adversary and for the challenger.", "Specifically, we use $\\text{BERT}_\\text{BASE}$ model that consists of 12 transformer blocks, a hidden layer size of 768 and 12 self-attention heads (110M parameters in total).", "BERT has been shown to perform exceedingly well in a number of downstream NLP tasks [32].", "We use HuggingFace's [67] implementation of the BERT model that was already pre-trained on masked language modeling and next sentence prediction tasks.", "BERT uses WordPiece embeddings [68] to convert each word in the input tweet to an embedding vector.", "The concatenated embedding vector is passed to the BERT neural network model.", "In our experiments, we only give the text of the tweet as input to both the adversary and the challenger to make it amenable to the pre-trained BERT models.", "Other tweet features such as deletion timestamps, number of likes, etc.", "could be used by both the adversary and the challenger to improve their performance.", "Note however that p:nonoverlapping,p:fullyoverlapping still apply as long as the adversary and the challenger have the same information.", "We fine-tune the BERT model on our datasets as prescribed by Devlin et al [32].", "In each interval, the adversary's classifier is fine-tuned for the classification of tweets into damaging and non-damaging using the negative log-likelihood loss in eq:adversary.", "We use a batch size of 32 and sample equal number of damaging and non-damaging tweets in each batch.", "This procedure results in better trained models as it avoids the scenario where a randomly sampled batch is too imbalanced (for example, no damaging tweet sampled in the batch).", "A separate BERT model is fine-tuned for the challenger using the loss function in eq:continuousrelaxation.", "Note that no balancing is required here since all the input tweets to the challenger model are non-damaging.", "We note that explaining the exact strategy employed by BERT models to classify text is an active research topic and complementary to our efforts.", "However, we highlight that our challenger does not use any information about either the adversary's exact model or its parameters.", "Budget constraints: We allow a limited budget of $B_\\text{static}=200$ deleted tweets for the static adversary and set $\\tau =1$ , i.e., the static adversary only trains during the first out of the ten intervals.", "Similarly for the adaptive adversary, we allow a fixed recurring budget of $B_\\text{adapt}=200$ deleted tweets every interval.", "There are no budget restraints for random and oracle challengers (having no access and black-box access respectively).", "However, we restrict the $\\text{D}^{\\text{2}}$ challenger to have the same (recurring) query budget as the adaptive adversary's recurring budget to keep the game fair, i.e., $B_g= B_\\text{adapt}= 200$ .", "We simulate the deceptive learning game described in alg:game with an adversary and a challenger, both implemented as BERT language models, with 10 different random seeds.", "We repeat the experiments for $k=1, 2, 5$ where $k$ denotes the number of decoy posts added per damaging deletion." ], [ "Results", "fig:twitterresultsadversary,fig:entireresultsfigure show the F-scores (with 95% confidence intervals), precision and recall for different adversaries over 10 time intervals.", "We make the following key observations from the results.", "Figure: F-score of different adversaries (random, static, adaptive) when no privacy preserving deletion mechanism is in place.", "Shaded areas represent 95% confidence intervals.Figure: F-score (with 95% confidence intervals), precision and recall for the three adversaries (random, static and adaptive) in the presence of different challengers corresponding to different accesses with k=1,2,5k=1,2,5.", "Key observation: D 2 \\text{D}^{\\text{2}} challenger fools the adversaries almost as well as the oracle challenger but with a restricted black-box access.Detection of damaging deletions in social media platforms is a serious concern.", "We start by considering the case where no privacy-preserving deletion mechanism is in place (i.e., no challenger to inject decoy deletions).", "In such a scenario, we compare the efficiency of different types of adversaries ten intervals shown in fig:twitterresultsadversary.", "The random adversary labels the posts based on the prior distribution of the deleted tweets (around $42\\%$ damaging and $58\\%$ non-damaging every interval).", "As expected, the adversary achieves a $42\\%$ precision and $58\\%$ recall resulting in an F-score of about $48\\%$ in each interval.", "As shown in fig:twitterresultsadversary, in the first interval, the static adversary achieves a 17 percentage points (i.e., a 35%) increase in its F-score compared to the random adversary, and remains almost constant over the rest of the intervals.", "On the other hand, the adaptive adversary receives new training data every interval and trains its classifier continually, and hence is able to increase its F-score even further by about 10 percentage points (56% increase compared to the random adversary) at the end of the 10th interval.", "This shows that even normal users of social media platforms, not only celebrities and politicians, are vulnerable to the detection of their damaging deletions.", "Furthermore, the adversaries can automate this attack on a large-scale with an insignificant amount of overhead (access to a small dataset of posts with the corresponding labels), highlighting the necessity for a much-needed privacy-preserving mechanism for the users' damaging deletions in today's social platforms.", "Injecting decoy deletions decreases the adversarial performance.", "As explained in sec:systhreatmodel,sec:model, we consider three challengers corresponding to the three types of accesses to the adversary's model – no access, black-box access, and restricted black-box access.", "In the following, we compare the performance of the adversaries in the presence of the respective challengers against the adversaries' performance in the absence of any challenger.", "No access: The top row of fig:entireresultsfigure shows the performance of the three adversaries (random, static, and adaptive) in the presence of the random challenger.", "We observe that although the F-score of both the static and the adaptive adversary decreases for all values of $k$ , the reduction is not significant (only 7 percentage points for $k{=}1$ compared to the no-challenger case) In fact, both the adversaries still perform much better than the random adversary.", "This shows that protection of damaging deletions in the no-access scenario is possible but severely limited.", "Black-box access: The middle row of fig:entireresultsfigure shows the performance of the adversaries in the presence of an oracle challenger.", "Not surprisingly, this approach is very effective at lowering the (static and adaptive) adversaries' F-scores (close to random for $k{=}1, 2$ ; i.e., 20 and 35 percentage point reduction in the case of $k{=}1$ for the static and adaptive adversary respectively compared to the no-challenger case).", "We also observe a major difference between the static and the adaptive adversaries in the presence of a competitive challenger.", "The static adversary retains the same recall performance (as in the no-challenger case) but loses drastically in precision, i.e., it classifies a large number of decoy posts as damaging.", "On the other hand, the adaptive adversary tries to adapt to the presence of decoy posts and becomes highly conservative – retains the same precision performance (as in the no-challenger case) but suffers heavily in the recall performance, i.e., it classifies a large number of damaging posts as non-damaging.", "Restricted black-box access: The bottom row of fig:entireresultsfigure shows the performance of the adversaries in the presence of the $\\text{D}^{\\text{2}}$ challenger.", "The performance of the $\\text{D}^{\\text{2}}$ challenger is comparable to the oracle challenger.", "The adversaries' F-scores in the presence of the $\\text{D}^{\\text{2}}$ challenger is close to $45\\%$ for the case of $k{=}1$ (20 and 30 percentage point reduction for the static and adaptive adversaries respectively compared to the no-challenger case).", "We also observe a precision-recall trade-off separating the static and the adaptive adversary (i.e., the static adversary loses in precision, whereas the adaptive adversary loses in recall) similar to the one described in the presence of an oracle challenger .", "Overall, we conclude that the $\\text{D}^{\\text{2}}$ challenger is able to successfully raise the bar for the adversaries in identifying damaging deletions without requiring an unmonitored black-box access with infinite query budget.", "The increase of decoy posts ($k$ ) results in lower adversarial performance with diminishing returns.", "While examining each row of fig:entireresultsfigure individually, we see that the performance of the adversaries always decreases as $k$ , the number of decoy deletions per damaging deletion, increases.", "However, we also observe that $k=1$ is enough to reduce the F-scores of the adversaries to 45% (close to the random adversary).", "Since the goal of most social platforms is to retain as many posts as possible, it would not be in the platform's best interests to use much larger values of $k$ or to delete the entire volunteered set.", "Observation of damaging and decoy posts.", "In tab:sampleposts in the Appendix, we show damaging tweets (as labeled by the AMT workers) and decoy tweets (chosen by the $\\text{D}^{\\text{2}}$ challenger from a set of non-deleted tweets).", "We observe that even though the decoy tweets typically seem to have sensitive words, they do not possess content damaging to the owner." ], [ "Adversarial Deception Tactics", "The adversary can use different techniques to sabotage the challenger.", "Here, we mention some prominent systems attacks and their effects on the challenger.", "Denial of Service attack.", "One of such attacks could be a simple Denial of Service (DoS), where the attacker submits requests for many damaging deletions to consume all the volunteer posts.", "First, we remind that the volunteered posts are a renewable resource, not a finite resource, as the users create, volunteer and delete posts in each time interval.", "Regardless, a DOS attack is possible wherein the adversary can use up all volunteered posts collected up until this point.", "A standard way to avoid such attacks is to limit the number of damaging deletions that can be protected for each user in one time interval (we assume that the adversary can have many adversarial users to help with the DoS attack but is not allowed to use bots [61], [28], [33], [64], [24], [35]).", "As is clear from sec:whowins, the challenger's defense is dependent on the distribution and number of volunteered posts.", "If there are more adversarial users than volunteers, then the adversary can win the game.", "We implemented the DoS attack as follows: in every interval, the adversary deletes as much as the standard deletions.", "We observed that the F-score did not change in this situation.", "Volunteer Identification attack.", "In a volunteer identification attack, the adversary deletes a bunch of posts and uses the process of doing so to identify individuals who volunteer posts to the challenger for deletion.", "First, we note that in each time interval there is a large number of posts being deleted ($>100$ million tweets daily [48]).", "Thus the posts deleted by the adversary (to try to identify volunteers) and the corresponding decoy deletions are mixed with other (damaging/non-damaging/decoy) deletions.", "In such a case, identifying the volunteers is equivalent to separating the decoy deletions from the damaging deletions; reducing to the original task.", "Additionally, the challenger does not delete the decoy posts at the same time as the original damaging deletion but does so in batches spread out within the time interval.", "Further, the volunteers can also have damaging deletions of their own.", "Even if an adversary is able to identify volunteers, the adversary still needs to figure out which of the volunteer's deletions are decoys.", "If the adversary ignores all posts from volunteers, then a simple protection for the users is to become a volunteer, which helps our cause.", "Adversary disguising as volunteer.", "In this attack, the adversary can take the role of a volunteer (or hire many volunteers) to offer posts to the challenger.", "Subsequently, the challenger may select the adversary's posts as decoys in the later intervals; however, these posts do not provide deletion privacy as the adversary will be able to discard these decoy posts easily.", "This effect can be mitigated with the help of more genuine volunteers and increasing the number of decoys per damaging deletion.", "This points to a more fundamental problem with any crowdsourcing approach: if the number of adversarial volunteers is more than the number of genuine volunteers, the approach fails." ], [ "Obtaining volunteered posts from users", "Volunteer posts are a significant component of our system.", "In sec:intro, we describe the possibility of obtaining these volunteered posts via bulk deletions (i.e., whenever a user bulk-deletes, consider the posts as “volunteered” with a guarantee that they will be deleted within a fixed time period).", "However, other strategies could be more effective, for instance, one based on costs and rewards.", "Under such a strategy, each user seeking privacy for his/her damaging deletions is required to pay a cost for the service, whereas the users that volunteer their non-damaging posts to be deleted by the challenger (at any future point in time) are rewarded.", "The costs and rewards can be monetary or can be in terms of the number of posts themselves (i.e., a user has to volunteer a certain number of her non-damaging posts to protect her damaging deletion).", "Nevertheless, in an ideal world, the volunteered set could also be obtained from altruistic users who offer their non-damaging posts for the protection of other users' deletions.", "We contacted the deletion services mentioned in sec:intro and shared our proposal Deceptive Deletions, for the privacy of users' damaging deletions.", "We got responses from some that provide services for the mass deletions on Twitter, Facebook, and Reddit.", "The response that we received has been positive.", "They attest that, with Deceptive Deletions, an attacker that observes the deletion of users in large numbers will have a harder time figuring out which of the deleted posts contain sensitive material." ], [ "Conclusion", "In this paper, we show the necessity for deletion privacy by presenting an attack where an adversary is able to increase its performance (F-score) in identifying damaging posts by 56% compared to random guessing.", "Such an attack enables the system like Fallait Pas Supprimer to perform large-scale automated damaging deletion detection, and leaves users with “damned if I do, damned if I don't” dilemma.", "To overcome the attack, we introduce Deceptive Deletions (which we also denote as challenger), a new deletion mechanism that selects a set of non-damaging posts (decoy posts) to be deleted along with the damaging ones to confuse the adversary in identifying the damaging posts.", "These conflicting goals create a minmax game between the adversary and the challenger where we formally describe the Deceptive Learning Game between the two parties.", "We further describe conditions for two extreme scenarios: one where the adversary always wins, and another where the challenger always wins.", "We also show practical effectiveness of challenger over a real task on Twitter, where the bar is significantly raised against a strong adaptive adversary in automatically detecting damaging posts.", "Specifically, we show that even when we consider only two decoy posts for each damaging deletion the adversarial performance (F-score) drops to $65\\%$ , $42\\%$ and $38\\%$ where the challenger has no-access, restricted black-box access and black-box access respectively.", "This performance indicates a significant improvement over the performance of the same adversary ($75\\%$ F-score) when no privacy preserving deletion mechanism is in effect.", "As a result, we significantly raise the bar for the adversary going after damaging deletions over the social platform.", "Our work paves a new research path for the privacy preserving deletions which aim to protect against a practical, resourceful adversary.", "In addition, our deceptive learning game can be adapted for current/future works in the domain of Private Information Retrieval [42], [50], [34], [53] that have similar setting for injecting decoy queries to protect the users' privacy.", "=0mu plus 1mu Deceptive Learning Game vs Generative Adversarial Networks Recall that in our setting, the task of the challenger is to select posts from a pre-defined volunteered set $\\mathbb {D}^\\text{v}$ .", "An alternative approach is to use generative models [39], [31], [46], [70], [57] to generate fake texts —see Zhang et al.", "[71] for a recent survey and Radford et al.", "[57] for the state-of-the-art— enabling the challenger to generate decoy posts instead of selecting them from a pre-defined set.", "However, we note that such generative models might not be favorable or even effective in practical systems.", "Let us consider the case of generating decoy posts on Twitter.", "Twitter posts are attached with a persistent non-anonymous user identities [29].", "Since, uploading fake posts from real user accounts raises serious ethical concerns, one should create multiple bot accounts that will upload machine-generated fake posts to be used as decoy posts (by deleting them later).", "However, unfortunately, detection of bot accounts is a well studied problem [61], [28], [33], [64], [24], [35].", "Moreover, when an adversary detects a bot, any decoy post from that bot account will be similarly unmasked.", "Therefore, in non-anonymous platforms like Twitter, selecting the decoy posts from the posts of actual users is arguably a more practical approach.", "Deceptive Learning Game vs Adversarial Learning In traditional adversarial learning [30] setting, there are two players: a classifier and an attacker.", "The classifier seeks to label the inputs $x$ (for instance, labeling emails as spam or not spam).", "Now, given a set of test inputs $\\lbrace x_i\\rbrace _{i=1}^N$ , the attacker's goal is to modify them such that the classifier will misclassify these examples (for example, in [30], the attacker modifies spam emails to fool the spam-detector in labeling them as benign).", "The attacker is free to modify any example $x$ as long as humans would agree on its label (i.e., the attacker's modified email should still be considered as spam by humans).", "Our setting, however, is different in that we are not allowed to modify the examples.", "Rather, the challenger wishes to attack the adversary's classifier by injecting hard-to-classify examples into the adversary's dataset (i.e., the deletion set).", "A key constraint for the challenger is that it has to select the examples from a preexisting set of volunteered posts (i.e., $\\mathbb {D}^\\text{v}$ ).", "This is because the challenger can only delete existing posts, and cannot generate fake posts (as we discuss in the next paragraph).", "Proofs Proposition (prop:confuser.)", "For any given volunteered set $\\mathbb {D}^\\text{v}$ with $N$ non-deleted posts, $\\max _{{{\\phi }}} \\tilde{V}({{\\phi }}; \\mathbb {D}^\\text{v}) = \\max _{w_1, \\ldots , w_{N}} V(w_1, \\ldots , w_{N}; \\mathbb {D}^\\text{v})$ [Proof of prop:confuser] Let $S^*_1 = \\max _{{{\\phi }}} \\tilde{V}({{\\phi }}; \\mathbb {D}^\\text{v})$ and $S^*_2 = \\max _{w_1, \\ldots , w_{N}} V(w_1, \\ldots , w_{N}; \\mathbb {D}^\\text{v})$ be the optimum values for the respective objective functions.", "First, note that $S^*_1 \\ge S^*_2$ because the optimal assignment for the discrete objective lies within the solution space of the continuous relaxation.", "Next, let $L_i = \\log (1 - a(x_i; {{\\theta }}_t))$ , where $x_i$ is the $i$ -th post in $\\mathbb {D}^\\text{v}$ and let $\\pi $ denote a sorting over them such that $L_{\\pi _1} \\ge \\ldots \\ge L_{\\pi _{N}}$ .", "Then, two cases arise – (1) when the top $K$ elements are strictly greater than the rest, $L_{\\pi _1} \\ge \\ldots \\ge L_{\\pi _{K}} > L_{\\pi _{K+1}} \\ge \\ldots L_{\\pi (N)}$ , and (2) when there is atleast one element in the bottom $N-K$ elements that has the same value as one of the top $K$ elements, $L_{\\pi _1} \\ge \\ldots \\ge L_{\\pi _{K}} = L_{\\pi _{K+1}} \\ge \\ldots L_{\\pi (N)}$ .", "In the former case, the optimal solution is clearly to assign a weight of one to the top $K$ elements and zero to the rest.", "Any other assignment (even in the continuous solution space) is clearly suboptimal.", "In the latter case, although there are infinitely many optimal solutions in the continuous domain that distribute the weights differently among the equal elements, the value of the objective function is the same.", "[Proof of Proposition  (cont)] First we show the kind of test distribution shift introduced by the challenger.", "The challenger-injected distribution is given by the following hypothetical acceptance-rejection sampling algorithm: sample $x\\sim p^\\text{v}(x)$ sample $u\\sim Uniform(0,1)$ independently of $x$ while $u>p^+(x)/(M p^\\text{v}(x))$ , reject $x$ and GOTO 1, for some constant $M$ .", "Accept (output) $x$ as a sample from $p^+(x)$ but with label $y=0$ , as the sample came from $p^\\text{v}(x)$ .", "While number of samples less than $k |\\mathbb {D}^+|$ , GOTO 1 Next we prove that the above rejection sampling algorithm produces samples with distribution $p^+(x)$ from examples from decoy examples that have distribution $p^\\text{v}(x)$ .", "Let $X^{\\prime }$ be a sample from the algorithm described above and $X \\sim p^+(x)$ , then $ p(X^{\\prime } = x) = p(X = x| \\text{Accept}) = \\frac{p(X = x, \\text{Accept})}{p(\\text{Accept})} = p^+(x)$ because $\\frac{P(X = x, \\text{Accept})}{P(\\text{Accept})} &= \\frac{P(\\text{Accept}|X = x) p(X = x)}{P(\\text{Accept})} \\\\ &= \\frac{\\frac{p^+(x)}{M p^\\text{v}(x)} p^\\text{v}(x)}{P(\\text{Accept})} \\\\ &= \\frac{\\frac{p^+(x)}{M}}{P(\\text{Accept})}= p^+(x)$ as $P(\\text{Accept}) &=\\int P(\\text{Accept}|X = x)p(X = x) dx\\\\& = \\int \\frac{p(x)}{M q(x)} q(x) dx \\\\&= \\frac{1}{M} \\int p(x) dx = \\frac{1}{M}$ The above ideal accept-reject sampling procedure can be reproduced via noise contrastive estimation [40], which is method that can generate data from a known distribution without the need to know $p^+(x)/(M p^\\text{v}(x))$ in advance.", "A variant of the same statistical principle is used today in generative models using Generative Adversarial Networks [39], which uses a minimax game similar to our procedure.", "Because we train the challenger to mimic the classifier of the adversary, it is easy to construct such rejection sampling method, such that there are in average $k$ decoy examples for every damaging example in the original data.", "Table: Sample tweet text extracts from the damaging, decoy, and non-damaging datasets.", "The real user accounts within the tweets have been replaced with @UserAccount.", "Some letters in the offensive keywords have been replaced by *.", "Examples of Damaging and Decoy Posts tab:sampleposts presents the damaging tweets (deleted tweets that were labeled as damaging by the AMT workers), the decoy tweets (chosen by the $\\text{D}^{\\text{2}}$ challenger) and the non-damaging tweets." ] ]

[ [ "Unsupervised learning of multimodal image registration using domain\n adaptation with projected Earth Move's discrepancies" ], [ "Abstract Multimodal image registration is a very challenging problem for deep learning approaches.", "Most current work focuses on either supervised learning that requires labelled training scans and may yield models that bias towards annotated structures or unsupervised approaches that are based on hand-crafted similarity metrics and may therefore not outperform their classical non-trained counterparts.", "We believe that unsupervised domain adaptation can be beneficial in overcoming the current limitations for multimodal registration, where good metrics are hard to define.", "Domain adaptation has so far been mainly limited to classification problems.", "We propose the first use of unsupervised domain adaptation for discrete multimodal registration.", "Based on a source domain for which quantised displacement labels are available as supervision, we transfer the output distribution of the network to better resemble the target domain (other modality) using classifier discrepancies.", "To improve upon the sliced Wasserstein metric for 2D histograms, we present a novel approximation that projects predictions into 1D and computes the L1 distance of their cumulative sums.", "Our proof-of-concept demonstrates the applicability of domain transfer from mono- to multimodal (multi-contrast) 2D registration of canine MRI scans and improves the registration accuracy from 33% (using sliced Wasserstein) to 44%." ], [ "Introduction", "Gathering labelled training data for learning-based multimodal registration is very time-consuming and expensive.", "To train supervise methods either a large number of corresponding landmarks (cf.", "[13]) or detailed anatomical multi-label segmentations are required (cf.", "[5]), which often cause bias or under-coverage.", "To circumvent the need for corresponding labels in multimodal / multi-domain images, unsupervised domain adaptation based on classifier discrepancies has been popularised in computer vision for classification and segmentation tasks e.g.", "in [7].", "Variants of discrepancy measures include e.g.", "the Earth Mover's distance (EMD) [11] for 1D cases and specialised solutions for 2D histograms in [8], but they are in general computationally expensive, approximative or based on sensitive hyperparameters.", "Contribution: We are the first to propose domain adaptation for medical registration and adapt the task to a discrete displacement labelling.", "Using the maximum classifier discrepancy approach [9] together with a novel 2D histogram Earth Movers distance, we substantially improve over the sliced Wasserstein metric [7].", "Related Work: Recent methods for supervised learning of multimodal registration include [10], who use a twin CNN architecture to learn the similarity of patches using aligned multi-modal training data.", "[5] and [4], both use anatomical segmentations to train a U-net like registration network, while the latter add a normalised gradient metric.", "The use of discrete displacements in deep learning based registration was proposed in [3] to capture large deformations.", "Unpaired unsupervised learning for multi-modal medical images has so far been restricted to modality synthesis using e.g.", "Cycle-GANs in [12].", "Very recent methods have shown promise for unsupervised domain adaptation and knowledge distillation for medical image classification and multimodal segmentation [2]." ], [ "Methods and Material:", "Unsupervised domain adaption has so far mainly shown success for classification tasks.", "We hence adapt the task of multimodal image registration to a discrete labelling problem, similar as done in [3].", "Here, we restrict ourselves to 2D patch based registration to demonstrate a proof-of-concept.", "For training, we extract large patches with a random offset within a grid of 5x5 discrete displacements to pose registration as a 25-class classification problem.", "We add 3D affine augmentations to avoid trivial overlap within two patches.", "During training we have access to a labelled source domain dataset (in this case MRI T1 patches) with known displacements and an unlabelled target domain dataset (MRI T2 patches).", "For feature extraction, we use a feed-forward net comprising four blocks of Conv2d, InstanceNorm and PReLU (13k weights) within a twin architecture that shares weights across both patches.", "This feature network produces a 18x18 map with 16 channels.", "Subsequently, we concatenate both patches and feed them into a three block classification network (70k weights) that predicts a 25D classification vector (encoding the displacements).", "Figure: Our method comprises a shared feature network and two classifiers for maximum discrepancy domain adaption.", "The results demonstrate the superiority of our new p-EMD metric (44% vs 33% accuracy) compared to sliced Wasserstein (SWD).We employ the maximum discrepancy of classifiers approach of [9], which uses two similar classifiers (with different random initialisation).", "The training alternates between the following three steps (see also Figure ): 1) optimise features and classifier on labelled source data, 2) maximise discrepancy measure of both classifiers on target domain while freezing the feature weights and minimising the classification loss on the labelled source data, 3) minimise the discrepancy measure of classifiers while updating only the feature weights.", "This process helps to identify target samples outside the classifier's support region in step 2 and brings the target feature distributions closer to the source ones during step 3.", "Step 3 is repeated twice as proposed in [9].", "We make two modifications that greatly improve stability: 1) we only update the first classifier in step 1 to avoid overfitting (too similar decision boundaries) on the source domain before the domain adaptation begins to improve, 2) we use the cross entropy loss for classifiers, but scale the predictions by 0.1 before computing the softmax output for the discrepancy measures to reduce overly confident predictions as motivated by [6].", "Fast projected Earth Mover's distance (p-EMD) for multidimensional histograms: A disadvantage of the sliced Wasserstein distance [7] for our application is its invariance to permutations of histogram bins / classes.", "This may be beneficial when no natural measure of class proximity exists.", "Yet, in our case the prediction can be converted into a 2D spatial probability map for x- and y-displacements.", "We therefore propose a new approximate metric for higher-order histograms (p-EMD) that takes these specificities into account.", "It is faster and easier to differentiate than conventional algorithms.", "Given that the distributions are close to monomodal Gaussians and based on the fact that exact algorithms for computing EMD for normalised 1D histograms in linear complexity exist [11], we approximate the optimal transport cost by projecting the (softmax) normalised 2D histograms onto a number of rotated lines (we use either 2 or 16 projections with angles between 0 and 90 degrees and use bilinear interpolation).", "We then employ the L1 distance of their cumulative sums to compute the p-EMD and average the values across projections (see Figure ), this correlates nearly perfectly with exhaustive EMD computations and is much more stable in our experiments than the 2D diffusion distance of [8]." ], [ "Results and Discussion", "We created a multimodal dataset for patch registration based on 9 3D T1 and T2 MRI scans of canine legs as provided by the 2013 MICCAI SATA challenge [1] with 5120 patch pairs in each modality.", "T1to T2 MRI is a simpler domain adaptation task, we thus increase the complexity by applying slightly different normalisations to the patches (global mean and variance for T1, and patch-wise for T2).", "The range of displacements was $\\lbrace -38,-19,0,+19,+38\\rbrace ^2$ pixels (posing a very challenging large motion problem) and each patch comprises a region of 77x77 voxels downsampled to half resolution.", "The supervised training was restricted to monomodal data (T1$\\rightarrow $ T1), while the multi-modal tests were performed on T2$\\rightarrow $ T1, T1$\\rightarrow $ T2 and T2$\\rightarrow $ T2.", "The average accuracy (prediction of 1 of the 25 classes) across 5 runs is shown in Fig.", "(right), yielding only a modest improvement from 31.9% (no adaptation) for sliced Wasserstein (SWD) to 33.2%.", "Both of our variants p-EMD (2 or 16 projections) reach accuracies over 40%, adding the losses of p-EMD (#16) and SWD is best with 44.1% (see also Table REF ).", "Future work will focus on more elaborate experiments and evaluation, e.g.", "integrating the patch-wise displacement estimation into global transformation models (e.g.", "using the instance optimisation proposed in [3]), extending it to 3D and comparison to classical multimodal metrics.", "This work was in part supported by the German ministry of Education and Research (BMBF) within the project Multi-Task Deep Learning for Large-Scale Multimodal Biomedical Image Analysis (MDLMA) FKZ 031L0202B.", "Table: Overview of label accuracy results for multi-modal MRI registrations" ] ]